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Witelonis Perspectivae 
liber secundus 
et liber tertius 


OSSOLINEUM
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Biblioteka 
Główna 
UMK Toruń 


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OSSOLINEUM 


lA' I TE.....ł-1NłKI
>>>
STUDIA COPERNICANA publi- 
shed by the Polish Academy of 
Sciences present COPERNICUS' 
work and related problems of his 
times as well as the scientific deve- 
lopments preceding his discovery 
and the impact of his ideas. 


II 


Jacket; figure from Witelo's Per- 
spectiva III, 44; binding: figure 
from Copernicus' De revolutionibus 
III, 20.
>>>
. 


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>>>
WITELONIS PERSPECTIV AE 
LIDER SECUNDUS ET LIDER TERTIUS 


ENGLlSH TRANSLATION AND LATIN EDITION 


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>>>
POLSKA AKADEMIA NAUK 
INSTYTUT HISTORII NAUKI, OSWIATY I TECHNIKI 
ZAKLAD BADAŃ KOPERNIKAŃSKICH 


STUDIA COPERNICANA 


XXVIII 


WROCLA W . WARSZAWA. KRAKÓW 
ZAKLAD NARODOWY IMIENIA OSSOLIŃSKICH 
WYDAWNICTWO POLSKIEJ AKADEMII NAUK 


........
>>>
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POLISH ACADEMY OF SCIENCES 
THE INSTITUTE FOR THE HISTORY OF SCIENCE, 
EDUCA TrON AND TECHNOLOG Y 
CENTRE FOR COPERNICAN STUDIES 


WITELONIS PERSPECTIV AE 
LIBER SECUNDUS 
ET LIBER TERTIUS 


BOOKS II AND III OF WITELO'S PERSPECTIV A 
A CRITICAL LA TIN EDITION 
AND ENGLISH TRANSLATlON WITH INTRODUCTION, 
NOTES AND COMMENTARIES 


by 
SABETAI UNGURU 


WROCŁAW. WARSZAWA. KRAKOW 
OSSOLINEUM 
THE POLISH ACADEMY OF SCIENCES PRESS 
1991 


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EDITORIAL COMMITIEE 


MARIAN BISKUP, PA WEL CZARTORYSKI (Editor-in-chief), 
BOGDAN SUCHODOLSKI 


EDlTORS OF THE VOLUME 


JERZY BURCHARDT, PA WEL CZARTORYSKI 


JACKET AND COVER DESIGNED BY 


ANNA SZCZURKIEWICZ-MUSZALSKA 



 
 O:S-l1'ł- 
c& 


I 


E ITOR 


TERESA BIERNACKA 


TECHNICAL SECRETARY 


ADAM PRZYLlBSKI 



 Copyright by ZakJad Narodowy Im. O.",lili.klch - Wydawnictwo. Wroclaw 1991 


Printed In Pola"" 


PL ISSN 0081-6701 
I BN l 
J7Q

03086-1 
UNIWERSYT9CKA 
w TORUNIU 


Zaklad Narodowy im. Ouolińlkich - Wydawnictwo. Wrocław 1991. 
Obj
lo9ć: ark. wyd. 30,40; ark. druk 24.0+ I wkl. + I wkl.; ark. A, - 32. 
Wrocławlka Drukarnia NaukowL Zam. 33S8/88. 


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2. O - Oxford, Bodłeian Library, MS Ashmolean 424, 101. 3r. Witelo olTering his work to William 
of Moerbeke.
>>>
BERNARDO ET HELENE FRANKLE 
GRA TIA AMICITIA AMOREQUE 


.
>>>
TABLE OF CONTENTS 


Foreword (by Paweł Czartoryski) . 
Preface. . . . . . . . . . . 
I. Introduction . . . . . . . 
l. Subject Matter and Makeup of Books II and III . 
2. Witelo on Light, Vision, and Visual Perception . 
3. The Sources of Boks II and III . 
4. Editorial Procedures . . . 
II. English Translation of Book II 
III. English Translation of Book III . 
IV. Notes and Commentaries . 
BookII........ . 
BookIlI. . . . . . . . . 
V. Latin Text and Variant Readings 
Book II. . 
Book III. . 
VI. Bibliography 
Index of Names , 
List of Plates. . 


7 
9 
11 
11 
14 
31 
32 
39 
102 
188 
188 
206 
234 
234 
291 
374 
379 
381
>>>
FOREWORD 


The present volume, containing Books 11 and 111 oj Witelo's Perspectiva" 
Jollows in cogent sequence Book I, which was published by Sabetai Unguru 
in 1977 ("Studia Copernicana" vol. XV). Since Book V oj the "Perspectiva" 
prepared by A. Mark Smith appeared in 1983 ("Studia Copernicana" vol. 
XX111), it is now only Book IV which remains to be published in the same 
way in order to obtain a modern, critical sequence oj the jirst jive books oj 
Witelo's work, actually consisting oj ten books. 
In recent years work on the "Perspectiva" was undertaken at Toruń Uni- 
versity, leading to a Polish translation oj Books 11 and 111. This translation, 
accompanied by a comprehensive introduction, notes and appendices, and pre- 
pared by Lech Bieganowski, opthalmologist, Andrzej Bielski, proJessor oj 
physics, Roman Dygdala, M.D., and Witold Wróblewski, proJessor oj classics, is 
being simultaneously published as vol. XXIX oj "Studia Copernicana". 
Both volumes, the English - Latin and the Polish one, with their respective 
introductions, translations, notes, commentaries and bibliographies were prepared 
independently oj each other, though materials and information were systematical- 
ly exchanged between the authors oJboth versions.. Moreover, arising questions 
could be directly discussed during ProJessor Unguru's visit in Toruń, in theJall oj 
1986, and the editor's visit in Tel Aviv, as guest oj the Van Leer Foundation, in 
the spring oj 1989. 
Thanks are due to the Herzog August Bibliothek in Wolfenbiittel, where 
a substantial part oj the present book was actually written by its Author. Dr. 
Jerzy Burchardt Jrom Wrocław should be thanked Jor his expert work oj putting 
in place the variant readings oj the Latin text, and Mrs. Teresa Biernacka oj the 
Ossolineum Press Jor patience and competence in editing the book. 
Acknowledgements are due to the Vatican Library Jor granting permission to 
reproduce the initial and two pages Jrom MS Urbinas 265, as well as two pages 
Jrom MS Borghese 64. Acknowledgements are also due to thl Master and the 
Fellows oj Emmanuel College Library at Cambridge, the WissenschaJtliche 


· E.g. the Figures prepared by the authors oC the Polish version were inserted, with 
appropriate adjustments stemming from the maDuscript tradition, in the English text, while the 
Latin text established here stood at the disposal of the Polish team.
>>>
8 


Bibliotek at Erfurt, the Bod/eian Library at Oxford, and the Bibliotheque 
Nationa/e at ParlS for permission to reproduce relevant pages from manuscripts 
of the "Perspectiva" extant in their co/lections. 
Conc/uding, may I ap%gize to Professor Sabetai Ungurufor the regrettab/e 
de/ays occuring in the process oj printing his book, and express my adm;ration 
for his tenacity, wh;ch enab/ed him sig/ehanded/y to comp/ete this next stage of 
making Wite/o's work accesib/e to the modern, scientific reader. 


Pawel Czartoryski 


...... 
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>>>
PREFACE 


f. 


The present work is a continuation and outgrowth of the work begun 
with my edition of Book I of Witelo's Perspectiva in more than the obvious 
sense. It pursues the effort to present to the interested scholar a modern, 
reliable critical edition of all the ten books of the Perspectiva. It relies 
on the conelusions reached in the earlier book which it further substantiates 
and strengthens. attempting to avoid repetitiousness whenever such an 
avoidance is consistent with the self-contained character and the substantial 
independence of this edition. It takes into account the results reached by 
. other scholars lahoring in the field of Witelo studies since the publication 
of my earlier book in 1977. Finally. it renews the hope that the desideratum 
expressed in the preface to that book for a complete critical edition of 
the Perspecti1'a wiJl eventually be fu lfilIed. 
I have incurred many debts during the long drawn-out writing of this 
book. Ił all began at the lnstitute for Advanced Study in Princeton in 
1978. during a one-year association with the School of Historical Studies, 
where I benefitted from the unique atmosphere of that great institution 
and from the advice and interest of Marshall Clagett. Ił is elear to me 
that I owe the Institute. the National Endowment for Humanities and Pro- 
fessor Clagett personally a great debt of gratitude. After some additional 
part-time labor spent on the project, I was in 1982- 1983 the beneficiary 
ofa National Science Foundation Scholar's Award (Grant No. SES- 8205508). 
During that year. which I spent in Israel in elose association with the 
newly-founded Institute for the History and Philosophy of Science and 
Ideas at Tel Aviv University, I completed the writing of important parts 
of the book and the first draft of the remaining parts and enjoyed 
the support of the two co-directors of the Institute. Yehuda Elkana and 
Amos Funkenstein. whom I thank cordially. 
Finally. the ()ther institutionally-centered sponsors of my writing are: 
(I) The Herzog August Bibliothek, Wolfenbiittel. this distinctively original 
treasury of old books and new learning, which, by inviting me during 
two consecutive summers (1985. 1986) as one of its Stipendiaten, enabled 
me to put the finishing touches to the book. To the Bibliothek, its di- 
rector. Professor Dr. Paul Raabe. and to Dr. Sabine Solf. the Leiterin 
des Forschungsbereichs und Kulturbereichs. I express my deepest appreciation. 


)
>>>
10 


(2) The "Deutscher Akademischer Austauschdienst" which ofTered me a two- 
-m(mths' stipend during the spring of 1986, making it possible for me to 
come to Germany and undertake my research and local travels. I thank 
DAAD for its assistance. (3) The Wissenschaftskolleg zu Berlin. which 
extended to me its hospitality and admirabie facilities during October 1985, 
thus clearing the road for ,the speedy writing of the Notes and Commentaries 
to Book II. I thank the Kolleg. its former and present Rectors, Professor 
Dr. Peter Wapnewski and Professor Dr. Wolf Lepenies, and its Secretary, 
Dr. J. Nettelneck. for their invaluable assistance. 
There are a num ber of individuals to whom I also owe thanks and 
appreciation. Paweł Czartoryski, the Editor-in-Chief of the series Studia 
Copernicana. has been a constant friendly and wise adviser in editorial 
matters and responded cheerfully and amicably to my requests and in- 
quiries. His interest in seeing the work tinished in order to pave the 
way for completing the edition "of this import ant source for the history 
of science" according to "modern standards of historical scholarship" I, was 
crucial in tinishing the undertaking. David C. Lindberg. my friend and 
mentor. has. through his prolitic and influential writings in the history of 
optics and through his personal concern with my work, provided the main 
(modern) source of my own preoccupations with the history of perspectił'a. 
Yochi Unguru, my wife, has, literally, stood at my side through thick 
and thin. supplying timely proddings and constant support, making the whole 
enterprise bot h attainable and worthwhile. Helen Watanabe-O'Kelly has 
been instrumental in mak ing an amorphous longhand manuscript into somethtng 
readable. She has given of her own preciously limited time and has spent 
a lot of energy on meeting my own deadlines with the publisher. doing 
it all with grace. selflessness and efficiency. These are the ponderahi/ia. 
Beyond that are the imponderables. Between them they represent an unusually 
great lot. They justify seeing her as the other dedicatee of this book. 
As to the dedicatees whose names appear on the frontispiece of the 
book. all that needs to be said. besides the dedication itself. is that 
they are "Menschen" - decent and kind human beings. 


Sabetai Unguru 


October J 986 
Tel Aviv UniverSlty 
Institute for the History and 
Phil(ls(lphy of Science and Ideas 


I F(lrew(lrd t(l my editi(ln (lf B(lok I. p. 7 of Witelonis Perspecti\lae Liber Primus.
>>>
I. INTRODUCTION 


The main purpose of this Introduction is to describe succinctly the 
contents, structure. problematics and sources of books II and III of Witelo's 
Perspectiva and to elucidate all matters pertaining to the editorial procedures 
adopted in preparing thls edition. It is not my intention to deal again 
with Witelo's intellectual biography or to repeat otherwise well-known and 
not directly related matters I. 


I. SUBJECT MATTER AND MAKEUP OF BOOKS Ił AND IłI 


In A. Mark Smith's words, books II, III and IV. "following the course 
of direct vision from its physical origins to its epistemological completion, 
provide the theoretical and conceptual background for all that follows"2. 
In generał term s this is an accurate description of the broad subject matter 
of these books and of their place in the organization of the Perspectiva. 
And yet, as we shall see, as well as Smith 's synthetic assessment of the 
two books quoted below. it reflects poorly the rich variety and anarchie 
organization 3 of Witelo's books. What. then, are the specific topics covered 


I The facts, well established, or conjectural. of Witelo's life and works have been treated 
most recently in the introduction to my edition of Book I of the Perspecti\la, Witełonis 
Perspeeti\lae Liber Primus An Eng/ish Trans/arion with lntroduction and Commentary and Latin 
Edition ol the Mathematicał Book ol Wite/o's Perspecti\la (Studia Copernieana XV) (Wrocław, 
etc., 1977), henceforth cited as Wit. Pers p.. anew in A. Mark Smith. Witełonis Perspecti\lae 
Liber Quintus Book V ol Witeło's Perspecti\la An English Trans/ation with lntroduction and 
Commentary and Latin Edition ol the First Catoptricał Book ol Wite/o's Perspecti\la (Studia 
Copernicana XXIII) (Wrocław, etc., 1983), pp. 13-18. and. finally, in Jerzy Burchardt's 
artieles appearing in the Bibliography to my aforementioned edition and in the since published 
additional items by him: "Czas śmierci Witelona i schyłek jego życia w świetle dokumentu 
z r. ł29S oraz innych ś\\iadectw XIII- XIV w.", Kwartalnik Historii Nauki i Techniki. vol. 
21, no. 2 (1976). pp. 281-287. List Witelona do Ludwika we Lwówku Śląskim. Proh/ematyka 
teoriopoznawcza, kosmologiczna i medyczna (Studia Copernicana XIX) (Wrocław, etc.. 1979). 
"Przedmowa Witelona do »Perspektywy((", and "Związki Witelona z Legnicą", both in 
Wite/o-matematyk. fizyk. filozof, ed. Jan Trzynadlowski (Wrocław, 1979) and lastly his 
recent Wite/o. fi/osoJo delia natura deł XIII sec%: una biografia (Accademia Polacca delie 
Scienze. Biblioteca e Centro di Studi a Roma, Confercnze 87), (Wrocław. 1984). 
2 Op. cit., p. 46. 
3 The oxymoron is premeditatedly accurate. I think Smith exaggerates somewhat the 

reat structural merits of the Perspecti\la (cf. ibid.. p. 48) as will become elear in our
>>>
12 


. I 


in books II and III? In its 5) propositions, book II. which begins with 
a short preamble folIowed by ten definitions and eight postulates. treats, 
in this order. the following issues: the rectilinearity of propagation of 
light and of the multiplication of forms (II.). the infinite speed of unimpeded 
light (11.2). the "sheaf-character" of luminous rays (which are continuous 
"pencils" possessing width rathel' than geometrical lines. by means of which 
they are. however. representable. II. 3). the ultimate unalterability of trans- 
parent media and their independent. separate penetration by the forms 
of lights and colors (II. 4. 5). the uniform distribution of light throughout 
the matter of homogeneous luminous and illuminated bodies and its uni- 
formity of action in the latter (II. 6. 7), the mutual relationships of 
luminous and illuminated. opaque and transparent, bodies with respect to 
the production of lighted and shaded regions of space and related questions 
(II. 8- 34). the indefensible claim that. as the length of Jight rays issuing 
from a point source increases. the rays approach more and more the 
"state of being perceptibly parallei" (II. 35). radiation through apertures 
of different shapes (II. 36- 41). the irrefrangibiJity of perpendicuJar rays 
as they pass from one medium into another (II. 42, 44). the mode of 
refrangibility of oblique rays (II. 43, 45, 47). the copJanarity of the incident 
and refracted rays (11.46). some additional properties of refraction (11.48- 
50), and. lastly. the determination of heitzht hy the Jength of its shado\\- 
(II. 5). It is elear then that. if one chooses for good reasons to overlook 
the great variety and the reJative Jack of structure of the book. it is 
quite acceptabJe to sum it up by saying "book II provides an introductory 
physicaJ analysis of light (and. to some extent. co lor). treating propagation 
through one Ol' two transparent media. the casting of shadows. and radiation 
through apertures" 4. 
Book III comprises 73 propositions preceded. again. by a brief preamble 


consideralions below. (He is righl. though. in saying Ihat Witelo's primary aim is analysis 
rather than Alhazen's synthesis.) "Far more than Alhazen. Witelo is concerned with melhodology. 
with baring logical strucIure. crilicaUy fragment ing that structure. and reworking il in pro- 
posilional steps along purely analytical Jines... II is Ihis profound grasp of sequence. not 
lccuracy or scope l?] Ihat makes the Perspecti\la so compelIing as a synthesis Iso the 
treatise turns out. after aU. to be a rather synthetical analysis!]. Cerlainly. in its st ruct ural 
refinement Witelo 's work represents Ihe very last stage in the long process of assimilation 
and fusion leading to Ihe science of perspeeti\la (ibid.) 
4 Ihid. This iso in effect what Witelo himself says in his preamble to Ihe book. He 
then goes on to define luminous, transparent. and opaque bodies, primary and secondary 
light. minimai /i1!III. ra.l'. radial and rerraCled /inC'. p.I'ramid (cone) or radiatiołl. and prramid 
ol illumination. Hi
 l'('Istulales dislm(luish belween slrOn(ler. wmpressed and expanded (diffusedl 
Jight. the former iUuminaling more stron(lly than the latter, and belween Ihe opposites 
"Iight" and "shadow". considering shadows as Jimited by a point. He continues by asserting 
the uniform diffusion of JigIJt and its assumption of Ihe color of Ihe objects Ihrough which 
it passes, and ends with a kind of principle of natural parsirnony ("nature does nothing 
in vain" and "it does not leave undone anything necessary").
>>>
13 


and by ten postulates. without the inclusion of any definitions. The preamble 
asserts tl'at the subject matter of the book is "the mode of simple vision 
and the proper arrangement of the organ of sight". The postulates c1aim 
that sight is not completed before the arrival of the visible form at the 
soul. that light and co lor are the only visibles per sef then go on to 
enumerate the other 20 visible intentions (which are visible only accidentally 
and require for their comprehension the cooperation of the other senses) - 
like magnitude. position. shape. shadow. beauty. etc. - . demand that strong 
light damage the eye. that the eye perceive bigger objects than itself more 
than one at a time. and that it perceive them accurately; likewise that 
the same object be simultaneously seen properly by both eyes. that the 
visibility of color always involve light. that vision. like all other natural 
actions. be by contact. and that visual power be finite. The propositions 
c('ver the following topics: the presence of light as a necessary condition 
for sight (III. l). the necessity that the object be suitably opposed to the 
eye for it to be seen (III. 2). the spherical shape of the eye and its anatomy 
and physiology (III. 3- 4). the nonexistence of visual rays (III. 5) and the 
occurrence of vision only as a result of the action of the visible form 
on the eye (III. 6). the geometrical structure of the eye (111.7-12. 22- 
24. 30. 33. 34). the need of a transparent intermediary medium between 
the eye and the object (III. 13). a medium. moreover. whose transparency 
must differ from that of the visible object (III. 14) as well as the need 
of an average distance between eye and object (III. 15) and of the eye's 
"suffering" on accoun1 of light. reqUlring thus a healthy eye (III. 16). all 
prerequisites for vision to occur; next. distinct vision as a result of the 
perpendicularity of light rays on the eye (III. 17) and the formation of 
a visual cone whose vertex is in the center of the eye and base in the 
surface of the ob.iect (III. 18). the required minimaI size of the object 
for visibility (III. ! 9). the completion of vision at the common optical nerve 
(III. 20). and the difference in transparency between the glacial and vitreous 
hurnors (III. 21); subsequently the geometry ol vision. i.e.. the hallmarks 
of the visual process as a function of the various geometrical structures 
and properties of the eye (III. 27. 29. 31. 32, 35- 38, 40- 42, 46. 50. 54), 
the qua!Jtative differences of comprehension by the eye of the various 
rays mak ing up the cone ofradiation (III. 43- 45, 47), additional physiological 
features of the eye having to do primarily with ocular motion (III. 25- 
26. 53). binocular perception (III. 28) and the role of time in vision (III. 
48- 49). and a mere problem of geometrical construction couched in ocular 
terminology (III. 39); finally, the two kinds of vision. by aspectus simplex 
and intuitio and the certification of visual images. in other words. the 
epistem(Jlo
-" ol \'ision (III. 51. 55-73). 
Having witnessed in this way the internal makeup of the book. we can, 
without danger of oversimplification. adopt A. M. Smith's characterization: 
"Book III. after furnishing a detailed description of the anatomy and physio-
>>>
14 


logy of the eye, undertakes a subjective analysis of the act of sight, 
tracing the process of vision from immediate apprehension of visu al effects 
to the ultimate evaluation of those effects by the brain" 5. 


2. WITELO ON LlGHT. VISlON. AND VISUAL PERCEPTION 


What are Witelo's views on the nature of light, vision, and visual 
perception as they come to the fore in these two books, the broad theme 
of which is direct vision? 
Even though light as such is never explicitly defined. being taken, for 
all practical purposes, as a. primitive term of perspectiva. it is obvious 
that light is that which luminous bodies possess intrinsically and iIIuminated 
ones derivaiively (Definitions, book II) and that it is a necessary (but 
not sufficient) condition for sight (III. I). lt (or its form) is propagated 
reclilinearly in sheafs of rays representable by straight lines (II. '.3) and 
its unimpeded speed is infinite (II. 2)6. Transparent media are not permanently 
altered by its passage. the various forms of Iight and color 7 penetrating 
such media independently and separately (II. 4,5). Furthermore light is 
distriouted uniform'y throughout the matter of 
omogeneous bodies and acts 
uniformly in t11em (II. 6.7), being prevented from passing through the matter 
(,f opaque bodies (defs., Book II). the shadows of which it then casts 
according to well-established geometrical relationships (Ił. 8- 34)11. Also its 



 /h id.. p. 46. Among the propositions of book III are scattered the nine necessary 
conditions for the act of vision: (I) light, (2) a certain distance between eye and visible 
object, (3) opposition between eye and object, (4) location of object with respect to the 
common axis of sight. (5) a certain minimaI magnitude of the object, (6) solidity of the 
object, (7) transparency of the medium, (8) a certain time required for visual comprehension 
or intuitio, and (9) a healthy eye. Witelo himself, in IV.1, miscounts his preconditions 
and sets their num ber at eight: "patet octo esse necessaria ad perfectam operationem visus, 
quae sunt: lux... Item distantia visibilis a visu... hem situs oppositionis ipsius visus.. . vel 
situs respectu axis communis... Item magnitudo corporis... Item soliditas corporis videndi... 
hem diaphanitas aeris... Item tempus conveniens intuitioni faciende... Item sanitas visus..... 
(Opt. Thes. Wit., i.e., Risner's 1572 edition of the Perspecti\la, p. 118). A. Mark Smith 
repeats the mistake but gets his num ber right by eliminating light from the preconditions 
of si
hl (cf. op. cit.. p. 158. n. 2 lo prop. 7). . 
6 This last view is Aristotelian in character; cf. commentary on 11.2. 
7 Color, though visible per se, requires light for its propagation (cf. book III. postulates); 
without light none of the remaining 21 visible intentions (for which see belo w) is perceivable. 
8 It should be said, however, that Witelo is entirely ignorant of the graduality of 
illumination and "umbrageousness", i.e., of the variation in intensity of iIIumination from 
extended luminous sources. His igt\orance is inherited, though. It comes from AI-Kindi's 
De aspectibus, which, in its fast-to-become-classic drawings, displays nescience of what is 
called "penumbral regions" when dealing with iUuminating and shadow.
asting bodies of 
various sizes by means of conical and cylindrical shadows. David Lindberg identities the 
root-cause of the difficulty faced by perspectivists in analysing shadows as Iying in their 
"inability to deal successfully with radiation from extended sources" ("Laying the Foundation 
of Geometrical Optics: Maurolico, KepIer, and the Medieval Tradition", in Lindberg and
>>>
15 


rays issuing from a point source have the property of tending towards 
sensible parallelism as their length increases (II. 35)9, a feature which con- 
tributes as well towards an explanation of radiation through apertures of 
different shapes (II. 36- 41). Light arriving aIong perpendicuIar Iines at the 
interface of two different media preserves its rectiIinearity (II. 42, 44). whiIe 
oblique rays are refracted in the denser medium towards the perpendicular 
at the point of refraction and away from that perpendicular in the rarer 
medium (II. 43, 45, 47). Lastly it is light that aets on the eye (ef. prologue 
to Book III). which suffers on account of it (prologue to Book III and IlU 6); 
correIative to this view is the deniaI of the existence of visuaI rays (III. 5). 
It seems manifest that aIthough it is possible to obtain a rather good 
mental grasp of the properties and behaviour of Iight. there is nothing 
in the precedmg paragraph about the nature of light. What kind of entity 
is it? What is its role in the universe? In his dedicatory epistIe to William 
of Moerbeke, WiteIo calls Iight "the first of all sensibIe forms" 10 and 
.'the diffusion of supreme corporeaI forms attending to the Iconstitutive] 
matters of inferior bodies through the nature of corporeaI form and imprinting 
itseIf, as wen as the descended forms of the divine and indivisibIe artifiees. 
on perishabIe bodies in divisible manner. producing aIways with lits] ineor- 
poration in them new speeific or individuaI forms, in whieh there eomes 
into being through Iight's aetion. the divine system of both the moving 
orbs and of their moving powers. Therefore, since Iight has the actuality 
of corporeal form. it makes itself equal to the corporeal dimensions of 
the bodies into which it flows and it extends itself to the size of capa- 
cious bodies. and yet, as it scrutinizes the source from which it flows 
according to the origin of its power. it always assumes accidentally the 


Geoffrey Cantor, The Diseourse ol Light Irom the Middłe Ages to the Enlightenment (Los 
Angeles: Clark Memorial Library, 1985). pp. 1- 65. at p. 5) which they inherited from 
antiquity; as it is already present in Euclid's Optica, "the earliest surviving optical treatise" 
(ibid., p. 4). In his De specułis comburentibus, "Bacon was attempting. albeit unsuccessfully, 
to deal with radiation in the penumbral region... Despite his lack of success, Bacon is 
the first scholar [to present] any discussion at aU of the penumbral region. His prede- 
cessors had universally lumped the penumbra with the fully iIIuminated region and considered 
this composite in opposition to the umbra, or region of fuli shade" (ibid., p. 23). 
The story has a Happy End, however, the steps toward its proper solution having 
been taken. chronologically, by Pecham, Maurolico, and Kepier. Maurolico's Photismi 
de lumine et umbra, completed in 1521 but published only in 1611, and Kepler's Ad 
Vite/lionem parałipomena (1604) "revealed how to deal with the projection of shadows from 
extended luminous sources. Between them, they [also] provided a completely satisfactory 
theory of radiation througb apertures built upon the principles of punctiform analysis and 
the rectilinear propagation of ligbt" (ibid., p. 48). But, as Lindberg convincingly shows, 
their efforts fit niceI y in the long medieval tradition of perspecti\la. the successful inheritors 
and continuators of which they happen to be. 
9 cr. commentary on 11.35. 
10 ......Iumen sit primum omnium formarum sensibilium..... (Opt. Thes. Wit., p. 2). 


--- 


....
>>>
16 


l 
, 
I 


dimeosioo of a distaoce (which IS a straight lioe) aod thus it takes to 
itself the oame » ray «" II. 
These passages cootaio. io a outshell, a whole "metaphysics of light". 
It is oot. however. my ioteotioo to uopack them aod to offer the reader 
a full-fledged precis of this metaphysics. Such offeriogs exist 12. Suffice it 
to say here that a properly oamed light metaphysics eotails the view that 
a created poiot of Iight. multiplyiog itself spootaoeously aod iostaotaoeously 
io all directioos. supplies to the simple substaoce malter the form of corporeity 
that eodows that simple aod dimeosiooless matter with its three-dimeosiooal 
exteosioo: io a very real seose, theo. light is the very stuff of the world. 
Such a view the letter to Moerbeke embodies. The rest of the Perspectiva, 
however. is rather ioooceot of such views developiog its aoalysis io a mostly 
cool. matter-of-fact maooer. the tOplCS covered beiog overwhelmiogly specifi- 
cally oPtical rather thao philosophical. .Stlll. it does 'ieem to he the case 
that bot h the explicit coofessioos of philosophical faith of the prehitory 
letter aod the implicit iotellectual attachmeots of the bulk of the treatise 
are ultimately coosistent with the basic iogredieots of a light metaphysics 13. 


II This is a translation of a fragment of the critical text of the letter to Moerbeke 
established by Clemens Baeumker on p. 125, lines 6-17 of his Witeło, ein Phiłosoph 
und Naturlorseher des XIII. Jahrhunderts (Beitrage zur Geschichte der Phiłosophie des Mittełałters, 
vol. 3, pt. 2 (1908). Risner's text, scarcely different, appears on p. I of Opt. Thes. Wit.: 
"supremarum formarum corporalium diffusio per naturam corporalis formae m terijs inferiorum 
corporum se applicans. et secum delatus formas divinorum et indivisibilium artificium per 
modum divisibilem caducis corporibus imprimens, suique cum illis incorporatione novas 
semper formas specificas aut individuas producens. in quibus resultal per actum luminis di- 
vinum artificium tam motoru m orbium quam moventium vlrtutem. (Juia itaquc lumen cor- 
poralis formae actum habet: corporalibus dimensionibus corporum "(quibus influit) se coaequat, 
et extensione capacium corporum se extendit: attamen quia fontem (a quo profluit) habet 
semper secundum suae virtutis exordium: prospicere dimensionem distantiae (quae est linea 
recta) per accidens assumit sicque sibi nomen radij coaptat". (The only differences between 
Risner's and Baeumker's texts are orthographical and of punctuation.) 
J2 Among the most recent and best are David C. Lindberg's Theories ol Vision Irom 
A/-Kindi to Kepłer (Chicago and London, 1976), pp. 95-99 and Roger Bacon's Phiłosophy 
o.f Nature. A Critic'ał Edition. H'it" Engli.
h Tran.
/ation, IntroduC'tion and Notes. o.f De multiplicatione 
specierum and De speculis conhurentibus (Oxford. 1983), pp. XXXV- LIII: also his "The 
Genesis of Kepler's Theory of Light: Light Metaphysics from Plotinus to KepIer", Osiris, 
2nd series, vol. 2 (1986). pp. 5- 42. Among the older. but stm very useful and enlightening. 
I mention only Baeumker's Witeło. pp. 357-467 and Ludwig Baur, Die Phi/osophie des 
Rohert Grosseteste, Bischo.fvon Lincołn (Beitrage zur Gesehichte der Philosophie des Mittela/ters, 
vol. 18, pts. 4- 6 (1917»; and. finally, among the newer works, published between Baeumker's 
and Lindberg's, valuable are. in chronological order of appearance. A. C. Crombie. Robert 
Grosseteste and the Origins ol Experimental Science 1100- J7()() (Oxford. 1953), pp. 104- 
116 and 128-131. Franz-Norbert Klein. Die Lichtterminologie hei Phi/on \lon Alexandrien 
und in den hermetischen Schriften (Leiden, 1962). pp. 155-193. G. F. Vescovini. Studi 
.
ulla pro.
pC'ftil'a medievale (Turin. 1965). pp. 7-32. and. lastly. Franz R(\senthal. Knoll'ledl(e 
Triumplrant: The Concept ol Knowledge in Medie\la/Isłam (Leiden. 1970), pp. 155-193. 
I
 Cf. my Wit. Persp.. pp. 23- 25.
>>>
17 


We should now treat Witelo's theory of vision. How do we see? What 
are the processes occurring between the visible object, the eye, and the 
brain? The most significant and consequential ingredient of Wite lo 's views 
is his adherence to Alhazen's intromission theory of vision. Luminous objects, 
i.e., objects characterized by inherent luminosity, as wen as illuminated 
ones, are qualified by lux and intrinsic color. But such potentially visible 
objects will eventually become actuany visible due to the spontaneous eapacity 
(virtue) of lux to propagate (multiply) itself through the surrounding trans- 
parent medium by informing it with self-generated lumen, the species (form) 
of luminosity. Lumen, then. is the intentional representation of lux, which 
is its generating eause. It is lumen that affects the medium and, as we 
shall see, the eye, initiating the process of vision. This is how A. Mark 
Smith puts the relationship between lux and lumen: "Lux... is both the 
efficient and formai cause of lumen, as inherent color is of color in the 
medium" 14. 
When the healthy eye is suitably opposed to the visible objects, their 
forms of light and co lor, spreading themselves out throughout the transparent 
medium, will reach and enter it through the pupil. It is by means of 
those forms acting on the eye that sight is achieved and not through 
putative visual rays that do not exist (III. 5, 6). How do the forms of 
light and co lor self-propagate through the medium, what is their course 
in the eye, and what are the optieal. physiologieal, and epistemological 
processes they initiate and, eventuany. help fulfil in the eye and brain? 
In order to answer these questions adequately. we must first eonsider briefly 
the anatomy of the eye, as presented. chiefly. in proposition III. 4. 
The eye is spherical and consists of four tunics and three humors. 
The tunics are, from the outside, the consolidat;va, which surrounds the 
uvea, except in the front where the cornea is situated, and the aranea 
enveloping two of the three humors, namely the glacial and the v;treous 
located within the uvea 15. The third humor is the aqueous or albug;neous 
humor situated in front of the glacial between the open ing of the uvea, 
covered by the cornea, and the glaciał. The uvea is perforated in the 
front by the pupil faeing the extremity of the optic nerve. This nerve 
itself consists of two tunics, an internal and an external tunie, emerging 
from the anterior part of the brain; the two hollow optie nerves meet 
in the common nerve, or the optic ehiasma. becoming there one nerve 


14 Op. cit., p. 39. Like Alha:ren, Witelo uses exdusively the term forma rather than 
the Baconian species; for the propagation of light, Witelo's standard term is diffundere 
rather than the Baconian multiplicare. The signiticance of these terminological differcnces 
is touched upon by Smith in ibid., pp. S9-60; cf. prop. III. S and my commentary for 
Witelo's arguments against visual rays. 
IS Another way of referring to the glaciał and \litreous humors is to consider them 
both parts of the glacial and to distinguish between the anterior part of the glacial, 
i.e.. the cr_vstalline humor. and its posterior part. i.e., the \litreous. 


2 


.. lU... Penpectivac..: 
IllIlIOTEk.\ 
UNIWERSYTBCKA. 
w TORUNIU 


-- 


-"
>>>
18 


which. then, again divides into two nerves coming to the eye-sockets where 
they form in a sense, the innermost tier of the eye and the substructure 
upon which the eye as a whole is mounted 16. 
Now the geometrical structure of the eye is meant to insure accurate 
and trustworthy sight. Thus all the spherical surfaces of the eye with the 
exception of the spheres of the uvea (III. 8), the vitreous (III. II). and the 
surface of common section of the glacial and the vitreous spheres (III. 23) are 
concentric with the sphere of the whole eye (III. 7). Since the uvea is 
apertured by the pupil its eccentricity does not count; as to the eccentricity 
of the posterior surface of the glacial and the vitreous, this is precisely, what 
is needed (as we shall sce) for truthful vision. Furthermore, to complete the 
geometrical picture of the eye's structure, all centers of the various tunics 
and humors constituting the eye are colinear (III. 9- 10, 12, 24, 30, 33, 34). 
Iying on the line joining the center of the pupil with the "gyration-center", 
i.e., the center of the extremity of the optic nerve. This. together with the 
remaining geometrical corollaries of the eye's structure and wit h what I called 
above the geometry of vision, will secure veracious vision 17. 
Let us no\\' examine concisely the physiology accompanying the above 
anatomy cum geometry. Luminous and illuminated bodies, as we sa w, 
propagate spontaneously and rectilinearly their light and color forms in all 
directions of space. The eyes, exposed to these forms, undergo. therefore, 
ubiquitous bombardment by light rays reaching indiscriminately all their 
points to which straight lines can be drawn from the lighted bodies (II. 3, 
20; 111.2, 17). The part of the eye sensitive to light is the glacial humor 
(III. 4), located protectively in the middle of the eye, smali, whitish, and 
humid (to retain the visible forms), and of moderate transparency, similar 
to that of ice, from which its name. Ił is the glacial that is "the proper 
organ of visual power for its transparency alone is retentive of the visible 
forms". This is obvious by the fact that injury to other tunics and humors, 
without the glacial being affected, is curable and leads to restoration of 
normaI sight, while corruption of the glacial means necessarily loss of 
si12 h t. Moreover its special virtue is due to its innervation by the optic 


16 See figs. 2, 2A, and 2B of book III, pp. 109, 110, 207. 
11 Lindberg, in describing Alhazen's accounl of ocular anatomy, an account which heavily 
inf1uenced Witelo, - as a matter of facto determined would be a better word, as my com- 
mcntaries to the citcd propositions will show - relies on Matthias Schramm to characterize 
Alhazen's views, as follows: "This is a highly idealized anatomical scheme, and it is 
elear that it was dictated not by the results of disseclion or any other experimental 
technique, but by the theorelical necessities of Alhazen's intromission doctrine" (Theories 
ol Vision. p. 69). This, manifestly, applies also to Witelo's views described above. It. patently, 
does not mean that the fundamental knowledge of ocular anatomy contained in the doctrine 
was not acquired, in part, through dissection. What is markedly innocent of any empirical 
component, in Alhazen and Witelo alike, is precisely the geometrieal structure superimposed 
on Ihe anatomy.
>>>
19 


nerve which supplies it with the visual spirit carried in its hollow. 
What is the relation between the form of light and color of the - visible 
object and its impression on the glacial '! How. at all. is elear and distinct 
vision possible when at every point of the glacial arrive Iight forms from 
every point of the object? Ił is the latter question that points to the 
answer adopted by Witelo from Alhazen who was himself influenced by 
AI-Kindi II!. The radiation of light forms is a punctiform proce ss. i.e.. each 
and every point of the visible object irradiates in space its form which 
will, therefore, reach alł points of the glacial opposed to it; and this is 
true of alł the points of the visible object. How, then, is utter confusion 
avoided? From each point of the object there is only one perpendicular 
to the eye and, by the same token, to the glaciał. Such a perpendicular, 
along which the form of the point arrives at the glaciał. reaches the latter 
u nrefracted, enabling, thus. the establishment of a one-to-one correspondence 
between the points of the object and the points of its glacial impression, 
securing in such wise elarity and distinctness of vision (III. 17). Furthermore 
alł those perpendiculars. when extended. would meet in the center of the 
eye by their very nature, engendering a cone of radiation, the base of 
which is in the surface of the object while its vertex is in the center 
of the eye. Indeed it is the very existence of this cone, the axis of which 
passes through the centers of alł its tunics and humors. łhat insures the 
crucial one-to-one correspondence required for truthful vision (III. 18) II). 
But what entitles Wite lo to neglect so cavalierly the impressions made 
by refracted (i.e., nonperpendicular) rays on the glacial? For Alhazen, his 
model, this is due, on the one hand, to ontological indistinguishability 
of the indefinitely many refracted rays reaching any point of the glacial, 
rays that would, moreover, had they been effective, have prevenłed the 
achievement of a coherent image. and, on the other hand. to the weakening 
undergone by alł refracted rays which legitimizes their neglect in his rays' 
accounting 20 . Witelo's main justification is, so to speak, "axiomatic", following 
from the postulate demanding that the observed object be seen "according 
to lits] position, shape, and the order of its parts"21. Since this is achievable 
only by perpendicular rays, because perpendicularity alone preserves shape, 
order, position and arrangement ofparts. Ieading. thus, to elarity ofperception. 
the neglect of oblique rays folłows necessarily (III. 17). To this main argument, 
Witelo adds. in the context of his discussion of refraction (lI. 47). that 


18 Ibid., p. 73. 
19 There is also another cone, as WiteJo points out. having the same vertex as the 
former and whose base is the glacial image of the ohject. 
20 Cf. Opt. Thes. A/h., i.e., Risner's 1572 edition of Alhazen's Perspectjt'a (De aspe- 
ctibus), props. 1.18, p. 10 and VII. 8, p. 241; see also Lindberg's enlightening discussion 
in Theories oj Vision. pp. 74-78, dealing with the difficulties stemming from Alhazx:n's 
position. 
21 See postulate 111.5. 


C' 


--- 


...
>>>
20 


"perpendicular lines are stronger. because they are intensified by the universal 
celestial virtue flowin
 into every body lying beneath according to the shortest 
line". This statement. appearing where it does, is not specifically brought 
to bear on the issue at hand and does not, consequently. carry, at all. 
the weight it has in Alhazen's argumentation. 
In any case. by punctiform analysis and by considering perpendicular 
rays alone. Wite lo obtains a elear. distinct. and truthful image on the glaciał. 
This is a very significant success of the intromission doctrine 22 . What 
happens nex1 to the image on the glacial? The perpendicular forms consti- 
tuting it travel through the quasi-transparent body of the glacial re ach ing, 
still in proper arrangement. the interface between the glacial and the vitreous 
humors. Now, had the transparencies of these two humors been the same, 
the forms would continue their rectilinear course intersecting in the center 
of the eye. Were they to come to an apex in the center and end there, 
they would. in a sense, be abolished as guardians of the position, shape, 
and order of the parts of the visible object, which is absurd. Consequently 
the forms cannot terminate in a single point of light at the eye's center. 
Furthermore. should the unrefracted forms continue their rectilinear journey 
beyond the center of the eye. they would become inverted and reversed 
with respect to the object and its faithful image on the glacial. causing 
unacceptable distortions in the perceptions of the material world and con- 
tradicting the postulate that assures reliable and unspecious perception. 
There follows, therefore. inescapably that the transparency of the vitreous 
must be different from that of the glaciaJ23, causing the Jight forms to 
be properly refracted at their interface (III. 21) which is, conveniently enough, 
located heJore the center of the eye (III. 22)24. 
Vision is not. however, completed in the eye. In a very real sense 
we do not "see" with the eyes but rather with the brai n, the connection 
between the two organ s tak ing place by means of the optic nerves. What 
does actually happen. then. after the necessary refraction at the interface 
glacial-vitreous? The vitreous humor and the visual spirit (the "sentient 
body") filling the hollow optic nerves have practically the same transparency 
(III. 22). Hence the properly refracted forms will not undergo any additional 
refraction. but. rather, preserving their proper arrangement, will be carried 
by the visual spirit to the common nerve where, as Wite lo puts it. "vision 
is... completed" since it is only "when the arrangement of the form re- 


22 See Lindberg, Theories oj Vision, p. 78. 
23 Lindberg thinks, without elaborating, that the vitreous must be denser than the glacial 
(ihid.. p. 244, n. 106). Witelo, in his 111.22. points out that the vitreous has the greater 
transparency, i.e., is subtler (han the glacial so that the forms are refracted away from 
(he perpendicular at the interface glacial-vitreous. As far as I can see, depending on the 
actual transparencies and distances involved, both alternatives are possible, as long as crossing 
over of the forms is thereby avoided. 
24 See my notes to and commentaries on 111.22 and 37.
>>>
21 


ceived in the surface of the glaciał will have reached to the common 
nerve" (III. 20) that the two identical forms from the two eyes will be 
united. engendering a single image that will be perceived by the ultimate 
sentient power (the u/timum sentiens) and thus terminate the process of 
sight. 
Like Alhazen. his model. Witelo hedges on a number of issues bearing 
on Ihe c()mpleti()n ()f the act ()f visi()n tha1 we have just described. Thus 
he is not entirely resolute about the location of the II/timllm sentiens. equivo- 
cal ing in the same proposition (III. 20) between the anterior part ()f the 
brain and the oplic chiasma as the płace where sight is compleled:!5. And 
he also tergiversates on the topie of transmission of the forms through 
the optic nerves. In proposition III. 22 he follows Alhazen in the belief 
that the transference of forms through the vitreous humor and the visual 
spirit is not simply the result of the basically identical diaphanous character 
of the two substances. but also of their inherent receptivity or sentient 
power. This. together with the fact that the two visu al forms coalesce 
in the optic chiasma to produce one truthful representation of the visu al 
object (which presupposes one-to-one binocular correspondence between points 
on the glacial humors and points in the common nerve, requiring sym- 
metrical arrangement of the visible object wit h respect to both eyes). 
entails nonrectilinearity of transmission of the light forms through the optic 
nerves. And yet. in proposition III. 37, Wite lo does not hesitate to "prove"26 
that the complete homology of the structures of the forms on both surfaces 
of the eyes and on the glacial humors with their counterpart in the common 
nerve follows from the recti/inearity of transmission of those very forms 
through Ihe optic nerves. 
What is remarkabłe about this Alhazenian scheme is whal Lindberg 
called ils "quasi-optical character". Indeed. "The radiation of forms according 
to the laws of oplics does not cease at the crystalline lens... to be fol- 
lowed by a nonoptical transmission of nervous impułses. Rather. forms proceed 
according lo Ihe laws of optics (though with some important modifications...) 
until Ihey encounter Ihe final sentient power in the optic chiasma" 27. 
We musI now examine succinctly Ihe epistemological aspects of Witelo's 
account of visual perception. And we should begin. perhaps. by pointing 


2
 The enunciation of 111.20 locates the completion of the act of sight in the common 
nerve, while the fjrst paragraph of the "proof' (or rather the argument) stales that "t he 
visual power (that] perceives and discerns aU that is visible is constituted when both oPlical 
nerves meet in the anterior part of the brain". Alhazen has supplied Witelo with Ihis 
hesilation: cf. Opt. Thes. Alh.. props. I. 25, 26, pp. 15- 16 and II. 16. pp. 34- 35. See 
also Lindberg. Theories oJ Vision. p. 81. The lack of precision concerning the localion 
of the ultimum sentiens can be accounted for by seeing in this designation a generic re- 
ference lo the brain (or to the mind), as a whole. 
16 That the so-called "proof' is wholly inadequate is certain; cf. my analysis. 
27 INd. 


--
>>>
22 


out that Witelo. "the appropriative spirit"28, works always with the mentality 
of the eelectic compiler. the guiding spirit of which is an-inelusiveness 
and reconciliation of counteracting views. We should add to this that it 
was the same kind of mentality that was at work in the Islamie tra- 
dition that. primarily through Alhazen, influenced Wite lo heavily. By the 
eleventh century. at the latest. the tightly-knit distinction between Aristo- 
telianism and Platonism (assuming there ever was such) had conapsed in 
Islamie thought. having been replaced by the powerful hybrid of Neoplatonized 
Aristotelianism. Witelo's Weltanschauung 29 shows elearly the weight of these 
syncretistic influences. Ił is, therefore, in light of these limiting parameters 
that ascription of Platonie or Aristotelian (or any other kind ot) ascendency 
upon specific Witelian views should be seen when encountered in this 
Introduction as wen as in my Notes and Commentaries. To this one must 
add that. in generał. Witelo 's strength (or main concern in the Perspectiva) 
- does not lie in the presentation of accurate philosophical conceptual dis- 
tinctions but rather in the attempt at systematic. encyclopedic treatment of 
his topies. This was both his strength and his weakness. 
Vision is accomplished by either aspectus simplex or intuitio (III. 51). 
The former, simple sight, takes place instantaneously (III. 55) and involves 
the "painful" reception of the visual form by the glacial humor, "from 
which it is elear that the eye ought to be of an adequate disposition in 
[its] health in order to prosecute completely the [proces s ot] vision"30. 
The role of the glacial is entirely passive; it suffers in being stamped 
wit h the form of light and color. What is it, then. that this form brings 
with it to the glacial? The twenty-two visual intentions, light and color 
being the only ones that are visible per se 31 , as wel1 as remoteness, magnitude, 
position, corporeity, shape, continuity, separation or division, number, motion, 
rest, asperity, smoothness, transparency, density, shadow, obscurity, beauty, 
deformity, similitude. and diversity, that are only visible mediately, per 
accidens, with the active assistance of the distinguishing power of the soul, 
the virtus distinctiva 32 . Just by the mere imprint of the form of light 


28 Wit. Persp., pp. 39-40. 
29 lhid.. pp. 23-25. 
30 Prop. 111.16, enunciation. 
31 This is an Alhazenian novelty which runs counter to the traditional Aristotelian 
view according to which only color is visible per se, light merely serving as a necessary 
condition for actualizing the potential transparency of the medium, making it amenable 
to the instantaneous "propagation" of color to the eye (De ałlima 418 b 9- 13. 418 a 32- 
418 b 3. 419 a 14-15). In Greek antiquity, then, and this includes both Euclid and 
Ptolemy. the proper object of vision is not light but color; they will both acquire equal 
status as \lisibilia per se only during the Middle Ages. Good recent analyses of Aristotle's 
theory of vision appear in Theories ol Vision, pp. 6-9 and Smith, op. cit., pp. 21- 
22. Smith's work also contains an iIIuminating account of Aristotle's theory of percep- 
tion (ihid.. pp. 31- 35). 
32 Cf. postulates to book III.
>>>
23 


and co lor on the glacial, the eye distinguishes, broadly and indecisively, 
the generał light and co lor outline of the object. It is only the first stage 
of visual cognition and it iso by its very nature, only partial and super- 
ficial: "The eye cannot comprehend the true form of the visible object 
by means of a first simple sight, but [only] after diligent intuition "33. 
However, even this primary, incomplete visual cognition cannot actually 
occur without the nine necessary preconditions for sight being fulfilled 34 . 
The visible intentions, then, represent the primary stutt: the elementary 
data. the building blocks of visual (and not onJy visual) cognition. They 
stem historically from and are an enlargement upon the Aristotelian eommon 
sensih/es. Thus, while the pro per sensibies (iOta ałaBT)T(l) are those entities 
suited tor unisensory perception without error (color for slght. sound for 
hearing. flavor for taste. odor for smell and the tangible qualities for 
touch). the eommon sensibies (K01va ałaBT)ta) are those entities suited for 
multi-sensory perception Iike movement. rest. number, shape and size. in 
which the soul has a signiflcant role to play and with respect to which there 
may arise errors 3S . Witelo.s vlsual pereeptihilia, then, are an essentially 
Aristotelian ingredient of his visual theory. Alternatively, his generał meta- 
physical outlook in the introductory letter to William of Moerbeke is 
Neoplatonic in tenor. And what I have called his epistemology of vision, 
based as it is on a process of abstraction, is, again, fundamentally Aris- 
tote1ian 3(). But this we shall see in detail helow. when we shall consider 
the workings of "diligens intuitio" in which the operative tool is the ab- 
stracting process. Before that. however, we must follow the course of the 
form imparted to the glacial on its way to becoming "the true form of 
the visible object" in the brain. 
It is well known that AristotIe's psychology is organically centered In 


33 Prop. 111.57. 
34 See n.S, supra. 
35 Aristotle, De anima, 428 I 17-19. In an interesting note on a passage from 
prop. V.37 (op. cit., pp. 176- 177, n. 1 S), A. M. Smith establishes, at least to my satisfaction, 
that there is a close relationship between Aristotle's categories and Witelo's visible intentions: 
"Clearly then, for Witelo the visible intentions correspond in intent, if not in num ber, 
with the Aristotelian categories. since they serve exactly the same purpose in permitting 
us to perceive visually. Thus, the \lirtus distincli\la... judges the attributes (e.g., specific 
color, size, shape, etc.) of simple perceptions and then applies them to their proper specific 
subject. And it is through this process of judgment that we are capable of visually re- 
cognizing specific differences between those subjects and, therefore, making fuli sense of 
them" (p. 177). Smith's conclusion, to which I subscribe, is that "the ontologicaI and 
logical underpinnings of the Perspectivist species-theory... are fundamentally Aristotelian 
and center on the categorical nature of » quiddity «..." (p. 176). 
36 G. F. Vescovini in her Studi sulla prospt'ctti\la medie\lale thinks differently. Her 
mistake should not detain us here as attention has already been drawn to it by A. M. 
S m i t h in op. cit., p. 65: more on the process of abstraction in the completion of the 
act of sight. below. 


-
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24 


the heart. Thus even the faculty of aicr3TJ(J1; KOlVi) is located there 37 . Bo w, 
then. can we brand Witelo's epistemology as Aristotelian? The answer 
lies in the historical metamorphosis of Aristotle's theory which underwent 
significant changes in the direction of its increasing physicalization at. the 
hands of Galen, who grafted onto it considerable chunks of his anatomy 
and physiology. As a result of this transmogrification, the brai n took the 
place of the heart as the seat of perception, reasoning, imagination and 
memory38. These faculties were located in various parts of the three-cham- 
bered brain. The sensus communis, in which all the five senses come together, 
being responsible for distinguishing between the various sensory qualities 
and for adding the ingredient of consciousness to sensation, was situated 
at the front of the anterior chamber (ventric1e) in the area of emergence 
of the optic nerves; at the back of the same ventric1e was ensconced 
the retc;ntiva, retentive imagination, in charge of retaining the impressions 
of the sensihilia received by the common sense. The middle chamber was 
the abode of the reasoning power, consisting of compositive imagination, 
which constructs "new unreal images out of real images (i.e., discursive 
reasoning after the fashion. more or less. of association of ideas)". and 
estimation. which "perceives the insensible forms connected with sensible 
objects (e.g., harmful intentions which are inferred by instinct...) and knows 
what is to be pursued and what is to be avoided'-
9. Lastly, the back 
chamber of the brain is the seat of memory which "retains the forms 
of estimation"40. It is possible, then. in light of the above, to distinguish 
two principal stages in the fulfillment of the act of vision: "The first 
of these. which was purely visual and » external «, occupied the space 
between the crystalline lens and the sensus communis; and the second, 
which was » internal « and common to all sense-perception, occupied the 
space between the sensus communis and the memoria/is"41. And these stages, 
in which the physiological and the epistemological were intimately inter- 
twined. ended. in good Aristotelian fashion. with the comprehension of 
the true forms. i.e.. the cognition of universals. How did it happen? 


.\1 De iUl'entute et seneclute 469 a 10- 12: "Moreover in all sanguineous anjmals the 
supremc organ of the sense-faculties lies in the heart; for in this part must lie the com mon 
sensorium of all the sense organs". This is W. S. Hett"s translation in vol. VIII. pp. 420- 
421 (Cambridge. Mass.. 1975) of the Loeb Classical Library edition of Aristotle's works 
in 23 vols. 
3ti In what fp110 ws, I am relying on Harry A. Wolfson. "The Internal Senses in 
Latin. Arabic and Hebrew Philosophic Texts", Har\lard The%gica/ Re\liew, vol. 28. no. 2 
(1935). pp. 69-133. See also Nicholas Steneck. "I"he Problem of the Internal Scnses in 
the Fourteenth Century". unpubli
hed Ph. D. dissertation. University of Wisconsin (Madison, 
1970). 
.Ił,) Wolfson. op. dl.. p. 96. 
-It, lhid. 
41 A. M. Smith. op. cit.. p. 38. 



 

 



 
, 
J 


, 
\
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25 


The whole process has to do with the doctrine of certification. The 
form of light and color imparted to the glacial humor by the visible 
object has enabled, by its very embodiment, the achievement of the primary 
level of visual cognition by aspectus simplex (III. 51, 52). This is imperfect 
and incomplete cognition, that is needed for gaining fuli optical knowledge. 
This form arrived at the glacial along perpendicular rays to the concentric 
surfaces of the eye and the glacial, unrefracted because of their perpen- 
dicularity and forming the cone of radiation with base on the object and 
vertex in the center of the eye. Only the rays constituting this cone 
are effective in producing clear and distinct vision due to the truthful 
form they, and only they, imprint on the glaciał. The form arrives at 
the interface of the glacial and vitreous humors which is eccentric to the 
eye, due to the eccentricity of the vitreous sphere (III, 11 and 4). There 
the light rays are necessarily refracted to avoid their intersection and crossing 
over in the center of the eye. Only one ray of the visual pyramid, its 
ax:is. enjoys special status, reaching unrefracted to the gyration aperture 
of the optic nerve: "It is necessary that amongst all the lines of the 
cone of radiation, only the ax:is passing through the centre of the opening 
of the uvea be perpendicular to the com mon surface of the glacial and 
the vitreous and to the posterior surface of the vitreous" (III. 24)42. As 
such it occupies a unique position in the rays' economy of the visual 
pyramid and boasts of a unique role in the process of certification of 
the visual forms. 
Aspectus simplex, simple sight, is arrived at instantly "along the entire 
extant visual cone, 'while] intuition takes place only according to the incidence 
of the axis of the visu al cone" (III. 52). In other words the incomplete 
and superficiaI. and yet crucial. first tier of visual cognition is reached 
by means of the generator-lines of the cone of radiation. while intuitio 
is the resuIt of the exclusive involvement of the ax:is of that cone in the 
completion of the act of sight 43 . What. then. is intuitio and how is it 
achieved? Whereas simple sight is "that act by means of which the form 
of the visible object is received for the first time directly in the surface 
of the eye", intuition is the act "by means of which the eye inquires 
diligently and thoroughly after the comprehension of the form of the ohject. 
not 'being] content with the mere reception but [striving for] a profound 
examination" (III. 51). And in the same proposition Witelo goes on to 
say that "the eye grasps by simple sight the manifest intentions which 
are in things. hut does not certify them: however through intuition it 



 


41 In Alhazen, as known, the weakening caused by refraction plays a fundament al 
role; it is this unweakened character that sets the axis apart. Of course the axis is also 
"special" due to its tixity during ocular motion (111.53). 
43 Aspectus simplex without intuitio is possible; however, "intuition cannot be without 
simple sight" (111.51). 


--
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26 


looks c10sely at all the visual intentions of the part of the form hidden 
from sight and certifies all the qualities of that visual form". So, having 
acquired, by sheer exposure to the visible objeet, a superficial knowledge 
of it, the eye, through its exceedingly fast and imperceptible motion (III. 53), 
covers with the axis of the visual pyramid, in successive rapid shifts 
of gaze, the various parts of the objeet, thus enabling the brain or mind, 
referred to sometimes generically as the u/timum sentiens 44 , to intuit, i.e.. 
fulły know visually the object. This fuli visual cognition, labelled intuition, 
is tantamount to an exhaustive awareness of all twenty-two visible intensions 
of the object and is achieved by perceiving each and all of the points 
of the object through the axis of the visual pyramid. The axis is the 
instrument of certain visual knowledge. 
As such those rays of the visual pyramid which are c10ser to the axis 
lead to better vision: "The comprehension of forms by the eye is accomplished 
with certainty along all the lines of the cone of radiation, with greater 
(certainty]. however, along Iines cIoser to the axis. and wit h greatest (certainty] 
along the axis passing through the centre of aperture of the uvea" 45. 
This is so because rays cIoser to the axis undergo lesser refraction than 
more remote rays and ""the quality of bent forms is not like the quality 
of forms extended (throughout] straight, because the bending alters the same 
necessarily with a certain alteration in the certainty of comprehension... n 
and ""the forms, the bending of which is smalIer, manifest themselves more 
than the forms the bending of which is greater" 46. 
We still have to sharpen the understanding we have gained so far 
and c1arify the exact sense in which Witelo's theory of visual perception 
is basically an Aristotelian theory. in spite of the obvious differences be- 
tween AristotIe's and Witelo's theory of vision 47 . To begin with.as we 
already saw, just as for Aristotle, for Wite lo too the completion of the 
act of sight involves heavy doses of physiological and epistemological 
ingredients so compounded as to ultimately lead to the knowledge of uni- 
versals. Moreover, as we are about to see, the road to the cognition 
of universals traverses the peaks of abstraction, a crucial elemel1t, if ever 
there was one. in Aristotle's account of intellection. The act of sight begins 


44 For instance in 111.69. 
4
 This is prop. 111.43. A whole series of other props. reinforces the message of this 
one. e.g.. III. 44. 45. 47. q.v. 
4/) lhid., 111.43. 
47 According to Aristotle light is the actualization of the potentially transparent medium: 
..... Aristotle's theory of vision involved a causal concatenation of actualizations, first 
of transparency and then of color, extending from the colored object through the transparent 
medium to the eye. Within this context light was the efficient cause, color the formaI 
cause, the medium and the eye the respective material causes, and the act of vision the 
final cause" (Smith, op. cit., p. 21). In some of the immediately following views, I am 
seriously indebted to Smith's iIIuminating discussion of the issues. 


........
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27 


with the world of external objects that propagate spontaneously, through 
the transparent medium 48 they inform, their forms (species) of light and 
color to the eye. The raw data of sight, then, are light and color which 
carry potentialIy with them the other twenty visual intentions. But it is 
light alone that is the sine qua non of the entire process, for even color 
needs light to be seen, light being the hypostasis of colors (II. l). How, 
then. can the process of visual cognition that begins externalIy in the 
object and ends internalIy in the acquisition of the universal form be 
accurately and succinctly described? In the folIowing words of A. Mark 
Smith that deserve quotation in fulI: 


[AJny physical object. insofar as it can be visually apprehended, must be viewed as 
a compendium of twenty-two defining and definitive properties. What any such object 
really is, both as individual and type. IS visually ascertained solely through these properties; 
but we do not have immedlate visual access to all of them, only to łux and intrinsic 
color in their intentionaJ forms as lumen and visible co lor. These two, intentionr IIi separated 
from their inherent causal agent s, impress a visible form upon the crystalline lens; and 
that form presents thc remaining visible intentions by implication to thc ultimum sentiens, 
where they are abstracted (or re-presented) and judged. Thc resulting perceptible representation 
in turn presents the intelligible form intentionally, which form, once abstracted and ascertained 
through judgement, yields the universal form. Thus, łux and inherent color imply or intend 
the physical form of any object immediately, its visible form secondarily, its perceptible 
form next, and its intelligible form last, just as the physical form implies or intends its 
visible form immediately. its perceptible form secondarily, and finally its intelligible form, 
and so on-W. 


Let us now illustrate selectively the pertinence of the above description 
for the Witelian theory of perception. I shalI let Wite lo speak without 
too much paraphrase, but with the exercise of some abbreviatory grace, so 
that the reader can grasp easily both the generaIly Aristotelian tenor of 
Witelo's views and the looseness of his speech about the brai n (mind) 
and its functiens: it is a discourse which. though consistent with the 
"internal-senses-model", does not refer to it expressly and is couched with 
slovenliness and remarkable lack of sharpness and philosophical sophisti- 
cation. In this. too. Witelo is the faithful disciple of Alhazen. 


48 While for Witelo, following in Alhazen 's footsteps. transparency is an inherent quality 
of certain media enabling the unimpeded passage of Iight through them (second def. of 
book II; see also props. 11.4, S), for Aristotle, transparency is a mere potentiality requiring 
light for its actualization (cf. pr
vious note). - 
49 Op. cit., p. 40. The description is fully consistent with Witelo's statcments in the 
Perspecti\la. though Wite lo is never quite so explicit and perspicuous. In generał it is the 
case that. though naturally wordy, Witelo gets mum when it comes to broad philosophical 
stands and underlying explanatory theories for the phenomena he treats. Also, it has already 
been pointed out that his Perspecti\la displays a "total absence of any explicit. coherent 
theoretical framework upon which the optical portion hangs" (ibid., p. 60).
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28 


hIt is necessary that any intuition be accomplished in time", the duration 
depending on "the diversity of the intentions of the intuited form". In 
the case of a many-Iegged animai, "the comprehension of its animality" 
requires "a smaller time and the comprehension of the multitude of the 
feet... a greater time...", while "the [grasping of the] number of feet... 
a [still] greater time..." (111.56). Animality is easily discernible from motion 
by a modicum of intuition, then follows the discernment of the sheer 
multitude of feet, and later, still, by more "intuitive etfort", the discrimination 
of the exact number of feet. As we saw, "the true form" is not attainable 
by simple sight alone; "diligent intuition" must intervene. "For the eye 
does not grasp the true form of the object except through grasping all the 
true particular intentions which are in that form". But "the form of the 
object in which there are subtIe intentions is not grasped by the eye, 
in conformity with the truth of the object's being. by a first glance... 
And since even in forms in which there are no subtIe intentions, the eye 
cannot assess correctly their absence at a glance, therefore even then there 
is also acting by intuition. for [the eye] cannot certify the truth of the 
form, except after the diligent intuition of any part of the form of the 
object" (III. 57). 
Now then. the true assessment and certification of the visual form 
stands in need of intuitio, exercised in the middle ventricłe of the brain 
by the compositiva and estimativa (Witelo will speak of the soul and of the 
virtus distinctiva), which produce conjointIy the intelligihle form. abstracted 
and ascertained to yield the universal form (Witelo's "true form"), subsequentIy 
conveyed to the memorialis, where it remains as a lasting imprint. subject 
abidingly to recall. Furthermore since the universal form is stamped on 
the soul in a process reminiscent of etching, its cłear-and-distinct everlasting 
availability to me mory is enhanced by the acuity of the first imprint and 
by its subsequent repetitions, restorations, and refinements. Indeed the manu- 
facture of the universal form itself seems to be the outcome of repeated 
abstractions undertaken somehow by the virtus dist inctiva. Besides. the universal 
form serves as the standard. the prototype against which the newly arriving 
forms of individual ohjects are properly assessed and recognized. 
Let us now see again. by means of selective examples, how all of 
the above is expressed in the Perspectiva. "Repeated looks [at objects] 
are imprinting and certifying better ltheir] sensible forms remaining in the 
soul. Indeed when the eye grasps a certain object and its true form will 
have been certified at the [ultimate] sentient [body] then the form of that 
object remains in the soul and is shaped in the imagination... And if 
the grasping of the object will be repeated, then its form will be better 
imprinted in the soul than the form of the object seen lonly] once, 
because... on account of the i1eration of vision. the form arrives always 
anew at the sNII and the previously seen form is refurhished in the soul;
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29 


and if any of the intentions of that form was surrendered to oblivion, 
it is lthus] restored, and if it was not previously seen, it is reclaimed... 
when the occurrence of the same intention is repeated many times in 
the soul, the soul will be remembering more that intention, and thus that 
form will be more [strongly] impressed in the soul. but also better cer- 
tified. .:' (III. 58). 
Essentially the same message is delivered i"n I II. 61, the enunciation 
of which reads: "Out of the frequently intuited intentions of individual 
forms, there remains in the soul the fixation and certification of the uni- 
versal form, [which is] for the extant eye the [fundamental] principle by 
means of which all the individuals of the same species are to be known". 
And the argument points out that "any of the ind iv idu al optical phenomena 
has a form... in which come together all the individuals of that species 
which differ only in the particular intentions understood by the sense of 
sight. Thus the soul will grasp by intuition the diversity of ir,dividuals 
of the same species, out of the diversity of the particular forms arriving 
at the eye with the universal forms, and, through the coming together 
of visible accidents in diverse individuals it will grasp that the form in 
which all individuals of that species come together is the universal form... 
Thus... the universal form remains in the soul and in its imaginative 
power. . ." 
Only light and color are understood by sight alone: "For first light 
is comprehended by the eye from the illumination of the sentient body, 
which is [fashioned] out of the substance of the eye, and color from 
the alteration of the form of the same sentient body and of its coloration 
with the admixture of light, which is the hypostasis of co lor" (III. 59). 
It follows, then, that "Every visible object is grasped either simply by sight, 
or with [accompanying] reason and discrimination... when the eye com- 
prehends two individuals of the same species and form at the same time, ... 
it will comprehend them as individuals and it will comprehend that they 
are slmilar. But the simitarity of the two forms is not .lltself] the same 
two forms, nor one of those. neither [is it] a third form proper to ltheir] 
similarity. but it consists of those two forms coming together in some 
thing. .. the comprehension of the similitude is... a consequence of the 
soul's power, which we cali reason.... an act of ratiocination comparing 
the various visual forms to one another... Istemming] from another power 
of the soul which we cali the discerning [power]. .. For sight bv itself 
has no discerning power, but [only] the discerning power of the soul distin- 
guishes all those [things] by means of sight" (III. 60). 
"[T]rue comprehension of the visible forms is [gained] either through 
intuition alone or... together with foreknowledge". Intuition alone is at 
work when objects are perceived for the first time; on other occasions, 
it is intuition w;th foreknowledge (recognition) that leads to comprehension 


........
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30 


..through the recollection of that objecf' and its certification (III. 62). Fur- 
thermore "Visual comprehension through recognition is always accomplished 
by a certain manner of participation of the reasoning power. 
Indeed recognition is the comprehension of the similarity of two forms, 
namely. .. the form which the eye now grasps... and of the previously 
understood form quiescent in the soul... recognition is always accomplished 
as a result of the comparison of the form quiescent in the soul to the form 
seen outside..." (III. 63)50. 
Quiddities of substances, not being accidents, are not visible by them- 
selves, but only through their ..sensible intentions". What the eye perceives 
directly are accidents, while quiddities are only comprehended ..through 
the intrinsic cognition of the soul... Consequently the comprehension of the 
quiddity of the seen substance... is... but... a resuJt of the... comparison 
of the form of the object to one of the universal forms quiescent in 
the soul and fixed in the imagination...", an operation performed by the 
virtus distinctiva (III. 66). ..The grasping of color as such is prior to the 
comprehension of the quiddity of color; from which it is elear that the 
comprehension of all visible phenomena as such, which are visible in [their] 
own genus. is prior to [the comprehension] of their special quiddities" 
(1Ił. 68). This is so because "t he first thing that the eye grasps from the 
form of color is the change of the sensing member and its coloring, 
because the eye is colored with the advent of the form in the eye, which, 
sensing itself colored, immediately senses the color, and after that under- 
stands the quiddity of color from the distinction and comparison of that 
[color] with the colors known to the eye" (ibid.) 5 I. This should be enough. 
Finally the Aristotelianism of the Perspectiva is further enhanced by 
the e1ear-cut grasp of the ontological differences between mathematics and 
optics (physics)52, by the analogical use of geometry in discussing optical 
phenomena 53 , and by the appeal to mechanical analogies when treating 


50 In the following prop., 111.64. Witelo argues that visu al comprehension by reco- 
gnition takes less time than by intuition alone. In the course of his argument he says: 
". .. when the discerning virtue will have grasped in the form coming to it a certain 
intention appropriate to that form, it will be reminded of the tirst form and it will re- 
cognize all the forms coming to it. because any inlention appropriate to some form is setting 
an imprint over those forms. .." (emphasis provided). 
51 Cf. also n.35 supra. 
52 In books II and III geometry is a representational and demonstrative tool. lines 
being only representative of real visual rays, etc. (See prop. 11.3 and my commentary 
on it.) For a discussion with examples of Witelo's perception of the ditferent natures 
of mathematics and physics, see Wit. Persp., pp. 29-30. 176 (comm. on 1.21), 185 (comm. 
on 1.72),195-196 (comm. on 1.113),196-197 (comm. on 1.114) and 207-212 (comm. 
on 1.135). 
53 Cf. the def. of minimałlighl (Book II). the postulates about "compressed" and "expanded" 
light, in the same book, as well as prop. 11.24: also, my "General Comment" following 
prop. 11.41.
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31 


- reflection and refraction and the behavior of łight with transparent and 
opaque bodies 54 . 
In łight of the above discussion of Witeło's epistemołogy of vision, 
A. Mark Smith's words on the generał character of the perspectivists' 
discourse seem to me entireły pertinent: 


II is important to understand that for the perspectlvlsts, as for all previous optical 
thinkers. the primary intent was not to establish a scientitic theory of light and color. Rather. 
it was to establish a coherent account of visu al cognition [...J on the basis of a scien- 
tific theory of light and color ss . 


3. THE SOURCES OF BOOKS II AND III 


A more appropriate titłe for this brief section woułd be "The Source 
of Books II and III". There is one paramount source of these two books: 
it is Ałhazen's Pen.pel,tiva (De aspectihus). Even a cursory look at my 
Notes and Commentaries shoułd suffice to convince the reader of the imma- 
cułate truth of this determination. Witeło follows Ałhazen wherever the 
Mosłim thinker łeads him with onły a few minor exceptions 56 . Occasionally 
Witeło will compress the Ałhazenian passage or proposition he copies 57 , 
more typically he will expatiate on it 58; sometimes he will endow the 
pureły rhetoricał exposition of his model with some welcome geometricał 
scaffołding 59 . but mostły, he will just eling faithfully to his master 60. 
In addition to Ałhazen, Witeło rełies heaviły (as he does throughout 
the Perspectiva) on Euelid's E/ements, which he cites expłicitły (a procedure 
not conformed wit h in the case of Ałhazen. his main source), and on 
his own results in the mathematicał book of the Perspectiva, Book I. 


S4 See pro ps. 11.4. S. Another Aristotelian facet is the corporeality fo)f light. On this 
issue Lindberg says: "Bacon's resolute defense of the corporeality of light or species, sup- 
ported by the authority of Aristotle. Grosseteste, Bonaventure, and others, determined the 
majority view of generations of European scholars. Two immediate folIowers were his younger 
contemporary Witelo. .. and John Pecham,. '.' though. in truth. neither man wasted a great 
deal of ink on what must have seemed a thorougly settled issue... The corporeal nature 
of light is reinforced in Witelo's theory of refraction, where the resistance of the medium 
to the passage of light is a central axiom" ("The Genesis of Kepler's Theory of Light", 
p. 22). 
ss Op. cit.. p. 31. 
56 See my comm. on 111.68; also comms. on 11.6 and 44 (especially n.l5) and 111.4, 
37 and 55. 
S7 A striking example is 111.14; other instances are 11.45. III.S2, 56 (slight compresssion) 
and' 58. See my commentaries. 
58 A few examples from book III are props. 4, 9, 33 and S4. 
59 Cr. my "General Comment" after 11.41; see also props. 111.32, 36 and 44 and the 
commentaries on them. 
60 Sometimes, of course, he also provides some original, quasi-geometrical propositions, 
based primarily on his mathematical book (Book I); 8 few cases in point are 11.2, 6- 
8, 9. 10, etc.
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32 


He also knows AI-Kindi's De aspectihus 61 , Pseudo-Euelid's De speculis 62 , 
Euelid's Optica 63 , and Theon's recension of Euelid's Optica 64 . 


4. EDlTORIAL PROCEDURES 


According to the last count. the number of complete and incomplete 
extant manuscripts of Witelo's Perspectiva is twenty-nine t1 :'. They are enume- 
rated and/or described in various places. the most recent lists being those 
of Lindherg. Ungurtl and Smith M . I ,hall not repeat here en Noc their 
enumeration and description. Suffice it to say that this critical edition 
is based on the fuli collation of six of those manuscripts: 
I. C - Cambridge, Emmanuel College Library, MS 20, fols. Ir-20Iv. 
Date: Late XIIIth century. 
Catalog description: M. R. James. The Western Manuscripts in the Lihrary 
oj Emmanuel College: A Descriptive Catalogue (Cambridge, 1904). p. '16. 
This is, presumably. the earliest extant manuscript. 
It is written on vellum quite legibly but with !1umerous insertions and 
iIIegible words running into the gutter. In most cases they can be reasonably 
guessed at from the remaining decipherable letters; when they cannot, the 
authoritative readings of the other manuscripts together with the generally 
correct grammatical character of C enable one to come up with appropriate 
readings 67 . 
2. p - Paris, Bibliotheque Nationale, MS Fonds Latin 14739, fols. 
lr- 266v. 
Date: XIVth centur"y- 
Description: Written in a elear hand. Contains many insertions. After fol. 
S7v there is a blank folio side where there should have been fol. S8r; 
moreover alater hand has numbered it as fol "57his". The continuation 


61 See comm. on prop. 11.21. 
62 See comm. on 11.35. 
63 See comms. on props. 11.51, IIU8 and 48. 
64 See my comm. on 11.51. This would seem to strengthen my conjecture about the 

vailability of Theon's recension of the Optica to Witelo; see my Wit. Persp., pp. 28, 
176. and 180. 
"" A. Mark Smith. op. cit., p. 73. gives the total number of manuscripts as 26. Three 
addilional manuscripls have been identitied by Dr. Jerzy Burchardt and Prof. Pawel C7arto- 
ryski, all located in the Bibliotheque Nationale: Paris, tonds Latin 14740; Paris Fonds Latm 
16213' Paris. Fonds Lalin 16214. This informalion was communicated lo me by Prof. Czar- 
toryski on March 17, 1989. 
"" David C. Lindberg, "Introduction" to the reprint edition (New York. 1972) of 
Risner's O/Jticae Thesaurus (Basel, 1572), containing the Perspecti\lae of Alhazen and Witelo. 
pp. XXVII; idem, A Catalogue oj Medie\lal and Rl'naissance Optieal Manuscripts (Subsidia 
Mediael'alia. vol. IV) (Toronto. 1975). pp. 77-79; Unguru, Wit. Persp., pp. 40-43; Smith, 
op. cit.. pp. 73-76. 
67 Other salient points of C are described in the appropriate places of the Critical 
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14. O - Oxford, Bodleian Library, MS Ashmolean 424, fol. 63r. Incipit of Book III.
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L 111 l! 1\ S£CVNDVS. J8 
!\1111'cmd,
du'OI(urmc don
cfccer c.ircum!ttrnt-'
 in pun
o c,&. 3 "'\ln,'(o c duea.. 
rurlinta ad dlamcrnłm m R,&' ufql ad Clrcumtcmmaqu:r fir Imea c n,feeam d'.\l1\:rni 
angin pundo I raIJttr,q. linea ł n lir zqualislina h dat".ł' per I J J .huius , &: d\lC3tUI h- 
ora g,& produearurdn 1in
a coneurrrnscum linea a gin punłto q.Cum iglrur an
 
krsd m cfir:rqu;a1is angulodncper 16.m-b),cadunrenim incundcm arcum quicltd c: 
f111m,ąuaa 
ir anK';'Iu.. q a I zqualis a
8U.lod;a '1.& angulus a 91 elł :rqualis angulod 
q a ('fi' I f. primi, cnt CfRO per J 1 . prlml angullls n q ł 
qUlan
ulus trian
\lIt) d (I 
a,iglrlll' pcr ....fat! mrproporrlo lina a 'lad q n,ficut linea- a d ad a l. Sed cum angu.. 
lus d ITI g fil :cąualis angulod n g
r 16.ttrrij,qui cadunt incundcm arcum d g.dł.autc 
per 29.primi angulusd m 
 zqnalis angulo b a R:p;ar,r,quia angulusq n g :rqualis ;an- 
gulob;ag.Sitiraą1 t pundus,mqllo linc;ad mcocurritcuma b,criltl;pcr I f .pt imi .an
 
gulusrlj a zqualis angu10 nq g,ergo per J 1.primi cric rrian&u1us t 'la 
t''1uiangulus tri 
anguJog q n,ait 
rgo per ".kxti proporrio1in
:r aq ad Ime;amqn fkut Imea: tq ad li. 
nwn qg,dt igirur per , , .quind proporrio lincz tlI ad lineam a g,Gem lincz a d ad li.. 
mamnlled linea n I dł Irquahs h alIUmpra: lin::;,: per J .hUlllS,&: .pporcio IinC".r' a d ad 
lirxam h.dl ficut lina c ad linea z,eR ergo proporno 1m.-;y t CI ad hncam K a .fiem linc:c 
c ad lincam %,"d dl .ppofitii.Ęt U (olingat 'ł' 
 p'ićło e ponim duci da
 Ima (Imiks li 
na' I n,nit polJibild pudo d duci duas bne:as fimilcs linc
 t q,ita Umihtcr ,ut utriuf. 
Ifad pam qua (cm ex bafe: a g Ur .p
rrio ficut lincz c ad linea :,8. nit cadi dcmólłra 
tIo.PlurcsaiilJ1uius1incasqidu.lsnoe:fipolJibilc:duci,utparuic R , J }.hui9, patet ergo 
.ppolitii,& bect hoc qd'hic ,pponii,nóuidai penttus unillcrfakquantii ad qu.zlibcc pii 

 d.ua,&. quaslibtt lineas datu,ad qu:a", ,pportiono: ficri de:bcat ipfiw bafis ,pponio. 
IW' tn hoc ,ppoUtO thcon:mate: nifi modo cóucn/cnri &. poou,ili in fcquctibus utemur. 


LIBER SECVNDVS 


PIUlSP£CTIVAB VITELLIONIS 


i Niuc:rfalibus buius fcimti
 axiomattoos mathcmaticis pnrmil1is,!n hoc 
I kamdo libro.ut pr.rmiUmus,uniucrfali aćłioni fcnfibilium formaru qu;!' 
- _ 'dam przambula narunlia przmin
tts,dnnodo proKdlols 1urninis pn; 
mrdiD111 unius diaphorU,ud p1urium fuper dillufas liR\lJ:ts corporum, &: 
de:prou:dionc:umbra,.,&dc IiRuration
 1uciseadmtis per fcncftras a
:. 
grtd1a11\Ll1" rraćł:łntm,ut de: ijs fincquibus f
rmoncm uilJbrlium formaR: .łggrct!1 cun- 
uenims non fuir,ptour in ptocdfu polłmodU patwlt,ąuor uefO pr:rmicrimus, ut DOla 
feoflli fllllr ilła. 
DIPPINITIONES. 
COłpw luminofum. dtciwr omoc corpu. qd' dłfut luminis dilfufiuum. Corptls 
tllalOnul1l1:1icinrr omlK corpus perquod lumini p:attt tranfiUl:s . 
rpus umbroli.tm 
diriNr corpw,pa quodlumini non paret tr.106tw. Lux pnma dlCltur III.. qu
 
tll. 
tit(rClll\lU,fieut luz; tntrans domum prr kndłri,& il1umm.lOs domum rcfidua
 l? lo 
(O aJ! fncidlt dicitur pritn3,in angulis urro domus drcitutlux f«unda. Lor. n:'1r.tm
 
dtcinrr,quz rł diutdl int
lliganrr,non habcbit amplius aćłum luci!. Radnls. dKIItI
 h 
Na kuninofa. Lora. radialis diałUt linea. per quam fit ddfuflO fOrmajt. Lnc;a rdr.ł 
& dic1Nr linl:a,cuiui partes angulumcontinmr.. Pyramis rad,ali"dici[ur pł raml
 
LIIUJ baCudł in fupctficiC corporis fll3l11 furmJ dlfłUndcntls.8. UCTtC1C 10 pum'hs alta, 
Ul cOI'pOtil cuiufcunqJ. Pyr:amil iIIuminaaorUs dicinu i!la,CUlus ucrtcX cfi In pun(ro 

rporirlluminofi.&. baliJ jn fupcrłicic rei ilIuminatz. 
PBTITIONU. 
PctJmuJ IUłCmIHK 
 pcrfcfaW(nQta luccm C'Omprdlim fOrriomnr!K 11,Cf:dif. 
· · k a &TCK:ml 


fi 


, 


15. I - Printed edition. Nuremberg 1535. ed. Georgius Tanstetter, Pel rus Apianus. fol. 38r. Incipit 
of Book 11.
>>>
LIBER TER TIVS 
..r.S..CTIVAB VlrSLLIOIU' 


c 


n N przmil1isltbńl macbmtaricalia & n
rura1fa pńn
'pfa pnnnffill\łit. pa 
,prOtIt nofira polIjb
ir.u &rr.noR:TI pr
fi
 eofc. Da inrmdjmw 
. d::łaran.Vollfnrts aUlt formaronaruralliJadlonea'tubUlPlld\lidmdl 
modo profcqul.f.fl1oqul lit pu I1mpl'eem ull1onr,& co qui ptr refltxlone 
. & i'lo qui pa rdTadioncm.ln hoc urtlo libro proftquimur modum fnn. 
placi, ui6.nis,& difpQlitioni .ppri2 orj;tani aifiui. Supponimus aur tRe quz fcqullllf 
ar, Joeis .bjs dcc1arata,ud ut per fc ipfa nora. Vifionem non eompkri mfi apud pn- 
umrumbrmzuilibilludanimam. Itemq. I'!"fc uifibdla funuantumduo,fcilicet 
lux& color,quoniam lux f t 'pfa uidcrur,&: ipfa elł hypofihafJs eoloru.aha UtrO pff ac- 
cidls aifibiha funr,ulpotlf rtmOtlo,magniludo,firus,corl)ritas,fi)tura, eorinuiru,fepa_ 
rario ud d'uifio,nurnaus .moru..,(\uilf' ,afpcritas,knira,.diaronitas,t.1i'fius,umbra. ob 
(curius,pulcritudo,ddi:Jrmiras,confimllirudo&: diuc:r
t
l. .!izc enim non 
olum Ulfu, 
fed alljs fcnfibus eOpr
nduO(ul. hem pttlmu5luce 'orte ledcrc ui(um dluUu. imu 
Ifntem. htm rem maioris quanlitatis, qu
m lir oculus,oculo u,dft" hnnrrmul. 
(am fccundii firum,figuram &: orda nem fU2
 piluium uider'. hem uifum fimul di. 
Utrfa uilibalia uidat. hi ab ambobw uihbus Iim\11 unam rem u,drd. hi q. colorn3 
dl moriuusu,fuf niflftUl1dUadulucldi. Irem fineeonradu uilion.: no fitłl,finn l1tC 
ali')uJ adlonc naturalc. bem \lirtutC uiliu2 finit:łmd1C.&. non exrcndi ininfil1iM1l. 
T HE O 1\1,14 A I. 
V ifibiJi lucern adu non participantc:,ipfum impoffibile efi uic3cri . 

zenirn.ur {uppoGnun en.per fe funt uifibilia,funt1ux &: color : lux autnn non 
cftuiflbilis przur rc,& ctiam lux cli lit hypofta. 
_ fis eolorum,non cfł poffibikeolorcs uidni fłndu. 
ce,forma cnim co10rls dł forma dcbillor
fit"r' 
matucis.cumcolor fit qwrdam lux incorpor.ltł 
corporibw mixris.V ifuI ago non recipil torm1 
coJoris rri uif:r .mli ex: lucc admixla cum forma co 
to D IOlis,&: proptcr hoc alternantur colorcs mulurii 
. ..lfrum a pud uifum pa altanatiUllc lucil arielIIi. 
(:.arc- Ipas:&: fi color,qu' ńł pcr Ituifibil..,non efi mot/uus ipfius uifuI,n'fi {rCtmdum 
a .lum lucldWar"r? om
j IIIGbd. adu !ucannon participanrc ipfum impofiibtlc cIł 
uldcrJ,patcr crgo propor.&um. 
II. 
Inn::
quod"bct punaum rupcr6cici rci uitlbilis,& aJiquod pum!iul1'lfu. 
pcrf!clel Ul fus produci pon fe Jim:as rcdas efi necdTe , ut res adu uidcatur, 
cx VO patet,fołum in oppofitionuei uif:r ad uifum neri uifionC'l11. 
160 mim fiU( fiat ex co Ił radlj cgrcdiunrur j uifi.r !Uprr pun{la rei ui{;r, RUf R 
D hoc.q. formor pundo".m rCI ulfor pa lInras radi:ll 
I ItR pt
uenjlnt ad fi.ptrnciem organi uiliuiJcmpn 
necelk c:fi inrn 'Iuodlibtt punlltl fupcrnciri mul 
(iMIS,&: lli,!uod pundum fi.ptrnciei uifi.as produd 
połk hntas rdiu,ut rcs uikantur adu:unde cum 
h:r 1i1a' fccundii qllodrun'}l propofitii modum "'- 
duci pollUnt,fi r ui60,nifi forre .ppta a'tcnus impc 
d.menri ,cfiftrntlam uifuJ fuerir impcd'nls. Cum 
A:cii
k :. . £: iracpui!UlfueritoppofinLtreiul&,uldebnil'fa
: 
nur clwoppoficlonc.nonfcntiCł iptam,& QUO rcucrtc:lur adoPlY.lficiolll", 
rcu£l 


! 


16. 1- Printed edition, Nuremberg 1535 ed Geor g ius Tanstetter Petr us A . r ł I .. 
, . ,planus,; .0.. nClplt of 
Book III.
>>>
I 


6. 


V I T E L L O N I S F I. 
L II T H V R I N G O R V M E T P O.. 


LOtlOIlVa O'.TrCAI 


L I . . ... I . C V II D " lo 


II 
....{;Ji'. ,,- jii""",uio-łi.", f1JIItbmUltitir,,-!/Iir. ;" ł- 
.{truk I;łro ( 1Il,,-f-) "';lUTjtłli M.'li",; [m{,J,iJ;- fi- 
".,," p,"'.."lii fMI",.II. ,,_umt,:, Je ",ou .J,oi,ai.,. łu.;,,;, 
" ",..._4;;' ",.;'" 
"'fb_,łUl pz"ri;,{"P" Ji",,/iu fi,l"" "'1""-' Ó" 
Jeprowuu,,,, -........ Ć1' Je fi!."'.n"" Iwil r.Jmtil l" Jmtlfr., "U"4;",,,,tr... 
8M-. Nr Je I' ,firN 'ł"'.'- {"".orli _jij"li_ fi"""';' lip rfrHU1łims ""Ił ftit. 
prorll"lł prortJlM ,.a-J- plłt'
r.,,,,, ....0 """,ilri_oJIl rłoU /;..r- .[MIII ifł.. 
D II . I Ił r T r O " II l. 
l. CorpulluminoCum,diciruromne corpu.,quod ei\ fui lumlnis dilf'uliull. I.Cor 
PUi diaphanul'A dlcitur omne corpu., per quod lumini pareuranlirul. ł. Corpus 
umbro(um dicirurcorplll,per quod lumini non parer rranliru..... Lllx prima dici_ 
rurilla,quz dłicir fecundS,licur IW! innan. domu per fene/trl, & illum;nS. domO 
ttliduS inloco,cui incidir, diarur pnma:in angIIIi. ucro domu. dicltur lu][ Ceeun- 
da.).LW! minima dicirur , 'luz li diuidi inrelligarur, n6 habebir ampliu. aaii IIIci.. 
o.Radiu. dicimr linea lumino(:\.,.LII1C:& radiali. dicirur linca,pcr quam fir dilf'ulio 
formaru.I.Linea renaul dlorur hnea,cuius partc. angIlIii conuni!r.9. Pyrami. ra_ 
diali. dlcirur pyramil,cuiu. balii clłin luper/kie eorporil (uS forIUJ dltfundenti., 
6: uena in pundo alreriu. corpor;1 euill(cunq;. 10. Pyumi. dluminari6il dicinu 
iUa,cuiuI ucm:x e/t m plic10 corpori.luminoJi,& balii in (uperficie rei i1Il1minat
: 
. II T I T r O " B l. 
Petimu. allte hzc, ut per fe (enll,i nota: I. LucE coprelTam forliori! cITe luce dir. 
,regara.a.bem lucem :;)rliorelU IIchClncnriu. dlumill.ue, & liigiu
 fcd,ITllndcrc. 
!.lr:m in ;1bli:ntia huuil1ll umbnm heri. ...!rem in "lIo1riol1: lumini. umbram dcfi 
cerc.,.lr
 aliquam umbr.m In rui rernuno acui,6: ad punfum Icrminari.6.1rem 
lud
 ad omnf polirionlS dijf
rcntiam rqualit:rdiffundi.7 .Irem luci! rCI coloraru 
!,elrlii:unrf iIIaru coloribus colorari, ur parct dc luce rrl(eW\te uirrc:&. fencnn!. 
'lur illorii uirrorii"coJorib. informai', (ccli formai ,lIoro :olorll {uI.er obicaa cor- 
pora dcfcrcndo.l.lrC' quodnatura nihil fruftra agir./kur nec del1:it in neceITarij.. 


T H E O R E M A T A
 


,. R6"'1f""..'.." Ut..,,,.... &....",,11(,,'_" fi""",r....,jir..J... rtl1.Ii",. 
I"""J.",,,,. I.Alh6J£'." .1. 
IIł O' quod hicproponiłur. non dł'monnll(ion
. (ed infłrome-,,'aliłł'pottA: dlclanric 
.hułr{illlllmca &nt;quorU.d hoc probandu plunbUI &: di\łł'rfi. uf. "ft inlł:rum.nri.. 
n
'lMrd urim\łr.f\o. quod hu: (ubfcrtbua".. 'łudd reluluiul huic JfPoł.i1ocr,.d,mul 
c.:ournir,.AfTu.m.lur il.q. Ul' .neum rotundii (ou.mł'n,er; laU""al! modem mIU.I 
.' aGrol,b'J.CU'UI fundi I.urudofil un_u, cubni,lUcJ majot, Ic: .fcitudo 0'2 f'IUI fitaoqUia. 
II, łat
łgdłn.i dua,u d!I!;1tOrii p"'pidlcul.na (uf'l' bólfi
 \llfi5: ac: in m.dio dorfi hUll
J u.6 I fir pn... 
pend.culanł.rcrr-8u ahqułI.1 CQrp\lf rJ':'"I!U rotun.!u column.re, c:uru..lonlltutlofit _qu.ri. J. 
llrud.n. rrium dil'corU.I.lllłl.U!.... t.at.rc. .tu. tir mmol'unochllłO:1c pOfl.tl1r ho, .a.lrcUdii (Ul pon.. 
tb....dIIInIOrnaaorlo.. rornctnr qtJourq. i'IU'I,hf'na enn li. In.rlnrceu. ac: e.ł.rnn{ccul Uff.ro.. 
łVinditari..& .lJcqu.ncurrlln.'upft'ficl
" i.l1th. &: corJ'u" coiuInnar. . quod en In'ł1'lcJlo cłom 

łtrOIQ.ndu. S'gnenł

ru:'q. In inłtrlori (U)'tł'I".tłUG rtln..łi kt.:IUł \1.16.. duz diamełli urrhO!Onłln.; 
It, ftuntcf qUllrfif11 . b Ic c d: r11jm, Quoru.m lU, ,!łamnn uan(łunł pe, Cmłf\lm (ircula łun
 
c1.1.quod lite:dł.nde fig. f'nw in ł-i..fi Dra: lillUl ulI1li'.quz C'łłcirculu. a, b d, in d.O.n'ta ełCtrr",it.. 
'1' ałltrl\lt dJ.l.lncuuł:-łlm produtłanun, ur d,.m.ln. b, fłC:lałnJum IUII.ud.nem uruu. dl.iłl pUG... 
a.: dum, 


17. R - Printed edition, Baseł 1572, ed. Friedrich Risner, p. 61. Incipit of Book II.
>>>
'4 
,..
,,;.ion
",. Simili19r"""'I'd"...qald r.'''I''Iu.fd,'rmal.r'''rlofo.. ".id....."",.. 
.tłllnalor '1\.,110 p al.Fllr 
jm fupt'r punacł. ..arminu lift'" x. ....Bul....1iu.\tl.lII&w.y: '''pet 
'. P I,qui r.canJulul. h lJa ,m"IAlwrq, rad.ut h. In ,sm8, .: COftCVrrlłql £WD 1a.n..J.in,...,nao 
k'."'i! par F"-" p.ne. hll,u...nll""" p' II _"...!,..n""" p d II' .lh".I.nll"'...... ....._ 
anr10 p l': non tiUlIł cD .,1II&1I1.qIlOft.l.. I\IIU: e. ,.....lła.. (tqu.ł'I'Cr"r .n"'ul in-cia-aa... 
equ.ln''Iuod .I\,on... hypOłhc6m,Cu"ł .nim r.pp06acłr. inf\".I..,C.d n,'I' ..nor! 'I""n.l&' 
li.... r.lr.ft,.. i....I"I.""'I,,..d .1\ có". 41. 4fl..i...,.a.'I" ...",:er",. "'r lu . p d I .ft.. 
iD. P 'IIJdć 'Iu0'ł' pot,ltd.móftnri fa....... ,,"Ii ."!UI". fu ii.. ."......""""'0 f.. per I' ,,- 
fOle r. .r(lK. c &: c c .ffam,,,cłilr aqullrr.cClc ."im '"pi p I.ac p c I.ru-nt pc, p"roulratClu.ł...; 
a.nlulu. Uł'rO p d I 1'nlftOl'ci't iilngulo P c I: quod patel cni.fi anpli ""'&:'011 pon.nl
rctlc Ir,ua. 
Jel. Dr hAC !tUICln matct'ia h,c; (&ł,n.tnarl.llJlqullnur. 4juQnił .,r... ID 10 hwu kb.... ubi l... pro- 
prium h.bc',pcń.a.u. ,..r.'Iu........I'...' ...'" 'ropo"..... . 
". DtI,,,,. ."i'._","It-łr_,...'''''i'"S''tłft"'
 
II ."..",'.E.'''''..,'bn.¥'II',..... 
Sil dal. ałlll"do. b.qua"I proponimv.l. q...nu Gr rOlnOrccrc ro.. 
I, app.rilc: ac fi ma .llil.do eft rtca.. tupet fv. r. Irł\ci
m-hori1OoEi'" 
cI"urur in ilJaruf'cr6c1chnc& bd pcrpłndicłI ani (upct,crminum 
altitu.dini. a b.qu, li.. b:& Inc.dill ndiu. {olań. pcr tlenlc.m Ił b (,w 
'I a)'p6..ruao d:1c lir. d:c'IO pUllhulu..erillmu b d umbn alEI,"" 
tlini. .pflu. a b
 criB.r\lr
. nora I-n... e E inufumbri b III. red
 e ci 
equicliaanrcf alulllldlnl . b I ut fi z e tił bl,,,J,,. notz ,u.nliladl. 
Erilrrlorr'aonUI dz, pCrl? P' aqulanrluarr1gono abd:erl;D 
pt... p 6.u.r pcr 9 hlliu...r.. proponio d ud ze .licul d b.J b.: rod 
d oz ad I C' proponia eR: nOłl:'!uon..l.m cum z c li. atTumpra norl. po- 
IrA: &I.ac. umb,.. fu... qUIr ctł J. d. Inodiu mcn(unrionc ficń nat.: 
.'ło d b ad b. proport:lo en nora:lcd d b pot..n m_hiurando fi"n no 
tl.-Err". b erllnora. Q.uodc9: propolilum.ułłihne. ablitalri- 
łudo abC"IUllU"11 ueJ PUiClil. fł'l' gaJcll.Jlrl ad mODNrand.i rp... , 
li. uU1
raruln.. 


v T T E L L O N IS. O P T Je AE 


\. 


--. 


VIT ELLO NIS FILII 
T H V R I N G O R V M E T P O.. 
1.0NOĄV1I1 OPTICAE LIIER TEJIoTIVI. 
lfj 'Npr_rjllllibril m4rh"".licJiIl & rull.,..Ii.pri"trpi/l prttmifr-/'" 
/fliS, prOll1 rt(]ffrlll poP. . ł,lttal frrl,wjłTi p"
pofir' co"frI/NrrII,. mlmdm'Nf 
auLtr..,.r . VDI,IłI,III-ł,.", firm4",'" 1I4lIł,..Lt_ .Efiorw f,,),rrrpl...,.u- 
. r t d",d/ ",odo pro!eq,.;,/r.Lcrr ,110.9N,fl prT pmplrc"" Nlfio"",,, & 10.1/'" 
prT rrjłexion,m,(9' ,/Jo'9"1 frr Ttftp,U.oni: in hoc rm,o Irh-o prof;"",mNr ",oa- P'" 
plrlll UJponu, (9' drfP0jitllmms pro F'.'" org"n' '"POI', SlIpptmi_ ..Iem h.c'l/a. 
ft'l".."I.r,iIJllIclI P,lrjl decł..ru...,llll prr /, 'i/A lUlI". 
P II T I T lON I S. 
I Vdion
m non compl
ri,nili apud p
ru
nfUm formz uilibilis ad animam. i. 
Jr
m quod per fc utlib.halilnł ran"im duo , fcihccr lux &color :quoniam lux 
cs fc ipf:t uidcrur: 6l"1pla cn hypolhlis colorum : ah. u
ro per ae.:id
ns u.I'bilia 
funr. lIrpore remorio, magmludo ,IiIUS, eorporC:llas . figura, ccnlłOu.I.'. fepua- 
tlO ud diuilio,numc:rus.motul,qules,alpcrllu,leniras, diaphaniru.denliros,um. 
bra,obfcurilu,pulchrirudo,dcformiru,c6limtlicudo 110: diu
rfius.H..,e 
rllm non 
lol"m uilu,rc:d ali;s frn/ibu5 eomprehcndunrur. ł. Jtrm pC:limullucrm fort
m 
Ia:dcrc Ullum diuriuI intuC'nrem. 4. Jtc:m rrm maiori. quańt1tatis, quam IiI 0- 
CUJlU,QCUJO uidc:ri. j. lr
m rem uifam Iccundum Iirum,figur3m& ordłOI! fua. 
rul11 r3rtium uldc:rL CI. Jrem ullum limu! diucrC.. uilibilia !liderr. 7. Irem ab 
ambobu5 nilibus limul unam rrm uideri. I. itI! qui,'" color n6 c:1I: motiuu. ui- 
li",l1Ili Ii:cundii a("lii lueidi. 9. ItI! (inr conrafiu "ino nI! n6 fic:ri, licur nre a!lqul 
acłlonl! natur
i!. JO. hem uirtutl! uiliuam finiti effr.& non nrend.1 in infinitCi. 
'f ił E 0- 


-:1 
'" 


18. R - Printcd edition. Basel 1572, ed. Fricdrich Risncr, p. 84. Incipit oC Book III.
>>>
33 


\ 
of the text of proposition III. 52. appears on a duly numbered folio ("58", 
Le., 58r), which is preceded, however, by the text of proposition III. 54 
and part of III. 55 on what is fol. 57 bis v; the continuation of III. 55 
appears, then, normallyon 58v and the copying becomes regular. Starting 
with fol. 62r the hand seems to have changed. 
3. v" - Vatican, Biblioteca Apostolica Vaticana, MS Urbinas Lat. 265, 
fols. lr-182v. 
Date: XIVth century. 
Catalog description: Cosimus Storna jolo, Codices Urbinates Lat in i, vol. I 
(Rome. 1902). p. 247. 
Description: Divided into columns and sometimes quite difficult to read. 
Between fols. 28 and 29 there is a hlank unnumhered folio. 
4. V h - Vatican, Biblioteca Apostolica Vaticana, MS Borghese 64, fols. 
Ir- 289r. 
Date: XIVth century. 
Description: Divided into columns, like v" , written in a careless hand 
with numerous interlinear and marginal insertions; contains a great many 
deletions. 
5. E - Erfurt, Wissenschaftliche Bibliothek, MS Amploniana F. 374, 
fols. lr-l7lv. 
Date: Second half of XIVth century. 
Catalog description: Wilhelm Schum, Beschreibendes Verzeichnis der Amplo- 
nianischen Handschriften-Sammlung zu Erfurt (Berlin, 1887), p. 259. . 
Descript ion: Written in a elear and appealing hand on parchment. Lacks 
books VI - X. Begins by dra win g the required diagrams but ceases to do so 
after proposition l. 12. 
6. O - Oxford, Bodleian Library, MS Ashmolean 424, fols. 3r- 355r. 
Date: End of XIVth century. 
Catalog description: W. H. Black, A Descriptive, Analytical and Critical 
Catalogue of the Manuscripts Bequeathed unto the University of Oxford 
by Elias Ashmole (Oxford. 1845), col. 340. . 
Description: Written on vellum in a quite legible hand; diagrams are occasion- 
ally in error. 
In addition to the integraI collation of the six manuscripts described 
above, I have also fully collated two of the three printed editions of 
the Perspectiva: 
7. J - First printed edition, entitled Vite/lionis mathematici doctissimi 1tEp! 
01t't1.lCijC;, id est de natura, ratione, et proiectione radiorum visus, luminum, 
c%rum atque formarum, quam vulgo Perspectivam vocant, libri X, ed. Georgius 
Tanstetter and Petrus Apianus (Norimbergae, 1535)68. 


68 The second printed edition, also published at Nuremberg in 1551 by the same publisher 
(Joann. Petreium) is, but for the first and last pages, a reprint of the first (cf. Wit. Persp., 
p. 12). 


- 3 - WiteloniB Penpectivae...
>>>
34 


8. R - Friedrich Risner's edition of Witelo's Perspect;va, V;tellon;s Thur;n- 
gop%n; Opt;CCJR L;br; Decem. published together with Alhazen's Perspec- 
tb'a (De a.
peclihus) and Ibn Mu'adh's De crespusculis et nuhium ascens;onihus 
under the collective title Opt;cae Thesaurus (Basileae. 1572). 
This is in itself a quasi-critical edition on which the editor bestowed great 
care and thoughtfulness 69 . 
In making my selections of manuscripts and printed editions for fuli 
collation. I examined lIlI the e.\tant manuscrip1s and printed editions known 
to me. I was strud hy the overall textual agreement of these varied sources, 
which does 1101 mean 1hat 1here are no signiticant differences between them. 
as my edition dearl) shows. I was also guided in my deliherations hy the 
conCIUSlons I haye reached in my edition of Book po and by A. Mark 
Smith's convincing considerations in the introduction to his edition of Book 
V of Witelo's Perspect;\'a 71 . Finally, I also took into account Lindberg's 
views arrived at on the basis of the comparison he made between some 
manuscripts and the Risner edition 72, as well as those of Baeumker 73 . 
On the basis of a careful evaluation of al1 those factors, I have come 
to believe that the attempt to establish a trustworthy slemma cod;cum 
is futile until and unless all the ten books of the Perspect;va are cri- 
tically weighed in the process. Since. moreover, as A. Mark Smith rightIy 
points out, such amonumental work as the Perspect;va (that was also 
used as a text in university teaching 74) must have been copied by pecia. 
making it, therefore. likely that a complete manuscript of the work might 
be the outcome of an assemblage of pieces of varied and tattered origins, 
any sten-tn1a elicited from such materials stands a good chance of being 
of doubtful value. Indeed. "every book might yield an autonomous stemma"75. 
What, then. served as specific criteria for selecting the manuscripts and 
printed editions used in this critical edition? Having decided to accept 
Smith's statistical analysis of the variations appearing in a group of 15 
manuscripts. as modified by the complete transcription of book V from 


6
 For a short and accurate description see A. Mark Smith. op. cit:, p. 76; for a fuller 
assessment and comparison with some manuscripts and wit h I. see Lindberg, "Introduction" 
(n. 66. sI/pru). pp. XXVIII-XXXIV; see also Baeumker, Wite/o (fuli reference in n.ll. 
supra). pp. XXI. 224- 225. 
711 Wit. Persp., pp. 41-43. 
71 Op. dt., pp. 77-82. 
72 "Introduction", pp. XXXII. XXXIV. 
73 Baeumker, Wite/o, pp. 127-179. 
74 Wit. Persp., p. 37. 
n Smith. op. cit.. p. 77. "Worse, if even a few copies of Book V itself were rendered 
in the same piecemeal fashion. the resulting stemma would doubtless represent a hodgepodge 
of nesting. sometimes conflicting, traditions rather than a true succession of trunks and 
branches" (ihid.); of course. the same applies. in principle. to any of the ten books con- 
stitutin" the Penl'C'("/il'{l.
>>>
35 


five selected manuscripts belonging to the group 76, I. also had to assent 
to the conc1usion of that analysis, i.e., the division of the five manuscripts 
into three families: C- Q, P, F- V" 77. Next I chose from each of these 
families one manuscript, namely C, P, and V". I then added to this representa- 
tive choice VI, a manuscript Smith was unable to consult, as well as two 
additional manuscripts, E and 0 78 , representing potentially different branches 
of the manuscript tradition of the Perspectiva. To complete the picture, 
I resolved. besides, to transcribe fully Risner's text (R) of books II and III, 
because of the importance, generał reliability. ..critical" character and in- 
fluence of this edition, as well as the first printed edition (I). precisely 
because it is the first. 
In establishing my critical text, I took C, like Smith, as ..the standard 
of critical authority" 79 and always weighed judiciously R. taking agreement 
between C, R and V" that made good mathematical and optical sense 
as very weighty indeed. When P and either E or O lent their support 
to the aforesaid concurrence of C. R. and V,.. this made the critical decision 
cut and dried. In this tentative hierarchy Sil, v" occupied. typically, the lowest 
rank, competing sometimes with I which had no firm relative place of its 
own in this rather moveable sequence that was. moreover. steadily invested 
wit h the spotlight of subjective retlections, stemming from a thorough fami- 
Iiarity with the text. that lead when most needed to an ineffable feeling for 
..the right word" 81. It is my conviction that the critical text I prepared as 
a result of all these deliberations is faithful to the manuscript tradition and 
reasonably c10se to the autograph. 
My aim has been throughout, as in the case of my edition of Book 
I of the Perspectiva. ..to pro vide an intelligible edition accompanied by 
a faithful translation which is not only trustworthy hut which also tries. 
within reas onable limits. to catch something of the spirit of the original"82. 
7" Op. cit., pp. 81- 82. 
77 Q is the siglum of a Paris manuscript. namely MS Fonds lalin 7248. while F is 
Ihat l,f a Florence manuscript. namely MS Gaddiana Reliqui III: 111} sigla P and VII appear 
in Smilh"s edition of Book V as S and U respectively. 
78 Smith's siglum for O is A. 
79 Op. cir., p. 82. The reasons for the choice are enumerated there. 
80 .'Tentative" because not mechanical. automatic, always depending on considerations 
of meaning. context. and "form" (especially when tied to diagrams and geometrical no- 
tation) and. therefore, almost always pervious to subjective pronouncements. 
81 I am fully aware of the potentially distortive dangers involved in tell ing the reader 
explicitly and repeatedly that the process of establishing the critical lext is far from mechanical 
and that subjectivity has its place of honor in the operation. Ali I can do is to try to 
prevent such distortions by making it elear that (I) the procedure is not arbitrary. not 
everything goes, and there are quite a few mechanical components sharing in the decision- 
-making procedure and yet (2) this procedure is not only not immune to subjectivity but 
actively requires its intercession to produce an authoritative text. (In light of Ihe above, 
I refrained from drawing a linear scheme representing, as it were, the hierarchy "Ieading" 
to the critical text.) 
82 Wit. Persp.. pp. 43-44. 


....
>>>
36 


Accordingly, I supplied an adequate punctuation and capitalization of the 
Latin text without including the numerous departures of the manuscripts 
and printed editions from my standards in the critical apparatus. On the 
other hand, I preserved the medieval orthography in every respect and 
included extant orthographical variations among the variant readings. Square 
brackets have been employed aU over exclusively for c1arificatory, or other 
kinds of, editorial additions, while parentheses belong inherentIy to the 
Latin text and its English translation. 
A few supplementary remarks about my translation are in order. I have 
striven for comprehensible literality . Though my translation is not, in any 
way, de verbo ad verbum, it is as c10se to the original as feasible without 
impairing the understanding. This is how it should be, I feel, since, 
with Dr. Johnson I believe that "it is not the business of a translator 
to be better than his author". The Perspectiva is not a work of great 
literary merit: nor is Wite lo a graceful writer. He is, rather, graceless, 
I would say: verbose, repetitive, didactic, and overbearingly dry. (lt is 
some of those very features that made his treatise so successful as an 
optical encyclopedia). Any translation that fails to convey to the reader 
(especiaUy the one whose Latinity stands in need of some refresher course 
or, worse yet, the more common kind who has no Latin at aU) the 
stylistic, syntactic, and lexical idiosyncrasies of Witelo's opus magnum is 
a bad translation. A good translation, contrariwise, whatever other charac- 
teristics it might display, must present not just the disembodied contents 
but also the genuine, historicaUy inseparable, linguistic vestments in which 
they appear. This, I believe, my translation does. Thus, to use only one 
example, when Witelo's terminology is nonuniform, unnecessarily rich and 
varied, my rendering of it will reflect this. Hence resentments at this 
procedure addressed to the translator are misdirected. 
The variant readings include aU significant variations in the manuscripts 
and printed editions, "significant" having been construed with unusual 
largesse. To give an inkling of my liberality in these matters (about which 
more will be said belo w), the foUowing sprinkling of iIIustrations is furnished: 
"optica" and "obtica", "coroUarium" and "correlarium", "girationis" and 
"gyrationis", "quidditatis" and "quiditatis", "diafanitatem", "diaphanitatem", 
"diafonitatem", and "diafaneitatem" have aU been included amongst the 
variant readings. So have variations in the notation of geometrie al diagrams 
as weU as any other idiosyncrasies that should help the reader with no 
access to the manuscripts to gain a veridical image of their makeup. The 
copious references to Euclid's Elements, on the other hand, have been uni- 
formly given according to the mol d "per. sam vt", this meaning "by the 
fifth proposition of the sixth book", and the various readings of the manu- 
scripts ("per S 6", "per S VI", "per sam VI", "per S p 6", etc.) have 
not been included in the critical apparatus. This uniformization, however, 
has not been undertaken when it comes to references to the various books 
ofthe Perspectiva, which references always include the label "huius". Misleading
>>>
37 


diagrams, diagrams requiring reconstruction, alternative diagrams needed to 
facilitate understandirig have all been called attention to, corrected, and 
supplied in the Notes and Commentaries. 
The critical apparatus used in recording the variant readings is a "ne- 
gative" one, i.e., it provides, whenever ambiguity is not an actual danger, 
only readings for those manuscripts and editions that diverge from the 
critical textual reading. Thus "9 diffundentis: dividentis v.. t' means that 
in line 9 of the given page, manuscript v.. substitutes "dividentis" for the 
preferred reading "diffundentis", all other manuscripts and editions agreeing 
with the preferred reading. Moreover, when the chosen reading and the 
first given variant are unmistakably c10se in form, so that the note can 
be taken to refer to only one word in the specific line of the text. the 
lemma has been omitted. Thus instead of "17 sicut: sicud O", the apparatus 
will contain "17 sicud O", if line 17 contains only one word of which 
"sicud" could be a variant. When a certain word appears more than once 
in a certa in line, a superscript number following the lemma indicates which 
instance of the word is meant in the variants. So "7 Item 3: Idem P" 
means that instead of the preferred third "Item" in line 7 of the page 
in question, P has "Idem". . 
Some additional examples should c1arify the system employed in the variant 
readings: 
"8 ante linea I hab R 7 et E mg inser 7" means that in line 8 of the 
specific page R has written "7" before the first "linea" and E has inserted 
in the margin the same "7". 
"6 post diversitas scr et dei v" inpressis [?] et mg inser inpressionis" means 
that in line 6 of the page concerned v" has written and deleted after 
"diversitas" an uncertainly deciphered "inpressis" and has inserted in the 
margin "inpressionis". 
"7 gyrationis co est ex graitioris in v,," means that in line 7 of the 
particular page the preferred "gyrationis" has been corrected by the scribe 
from the initially wrongly written "graitioris", 
"2 LN: LM I. co est ex LM in v,," means that against the preferred 
reading "LN" in line 2 of the specific page, I has "LM" and that the 
preferred reading "ł..N" was corrected by the scribe in v" from "LM". 
"3 stili: sili [?] E stilli (co est ex silli) v.." means that in line 3 of the 
given page. instead of the preferred "stili", E has the unclearly deciphered 
"sili", while v" has 'stilli' which has been corrected from "sil li" . 
"I sequens: secans inser mg et dei sequens O" means that in line I of 
the page in question. instead of the preferred "sequens" which it originally 
had, O has deleted it and inserted instead in the margin "secans". 
"4 similiter : sic v", in v" co est ex super" means that in line 4 oC the 
given page, instead of the preferred "similiter" v.. has "sic", while v" has 
"similiter" which has been corrected, however, from the originally written 
"super". 
"5 diafanitatem: diaphanitatem R diafonitatem I dyaphaneitatem v,," means
>>>
38 


that in line S of the particular page the preferred reading is "diafanitatem" 
which C, P, v", E and O have, while R has "diaphanitatem", I has 
66diafonitatem", and v., has "dyaphaneitatem". 
It should be elear from these examples that my critical apparatus is 
meant to provide exhaustive information about the actual appearance of the 
manuscripts, ineluding, erasures, corrections, interlinear and marginal in- 
sertions, etc. Note also that all italicized words, letters, and abbreviations 
belong to the critical apparatus used to express textual variations, while all 
non-italicized words or letters are actual readings of the collated text or the 
individual manuscripts and printed editions. 
The following abbreviations have been employed in the variant readings: 
add. = addidit or addiderUnl = has. or have. added 
a/m. = alia manu = in another hand 
('orr. ex = correxi ex = I have corrected from 
co. est ex = correctum est ex = has neen corrected (in the manuscript) from 
hab. = hahet or habent = has or have 
inser. = inseruit or inseruerunt = has, or h ave, inserted (always interlinearly) 
mg. = margine = in the margin 
obs. = ohscura1'it = has obscured 
om. = omisit or omiserunt = has or have omitted 
rep. = repetivit = has repeated 
ser. et de/o = scripsit et de/evit = has written and deleted 
sup. = super = above 
tr. = transposuit oc transposuerunt = has, or h ave, transposed (the mutual 
Ehange of position of two words).
>>>
II. ENGLlSH TRANSLATlON OF BOOK TWO 


Having previously introduced the mathematical truths of this branch of 
knowledge '. we are (nowl. as promised. setting out. in this second book. to de al 
with the passage of light through one (or more) transparent medium(a). 
over various shapes of bodies. as well as with the casting of shadows 
and the sharing of light falling through windows. placing Ihoweverl first 
some naturaI preambles to the universal action of sensible forms. namely such 
[preambles] without which it would not be convenient to consider the discus- 
sio n of visible forms. as will become obvious shortly while we proceed. 
And these are in fact the things known to the senses with which we 
begin. 


[DEFINITIONS] 


Every body that is suffused by its own light is called a 'umil1otls hody. 
A body through which the passage of light is free and unimpeded is 
called a transparent body. A body through which the passage of light 
is obstructed is caned opaque (shadow casting). That light which produces 
secondary light is caned primary, as is the light entering a house through 
the windowand illuminating the rest of the house; where it enters it 
is caned primary. while in the corners of the house it is caned secondary. 
Light which. when undergoing division. is understood as not having anymore 
(left] of the act of light is caned minima' Iifi!ht. A luminous line is called 
a ray. A line by means oC which the diffusion oC Corms is perCormed 
is caned, a radial line. A line the parts oC which contain an angle is 
caned a r
rracted line. A pyramid (cone) whose basis is in the surCace 
or the body spreading out its form and whose vertex is in a point 
of any other body is caned the pyramid (cone) ol radiation. That IpyramidJ 
whose vertex lis] in a point of the lummous body and whose basis lis] 
in the surface of the illuminated body is caned the pyramid oj i/umination. 


[POSTU LATES] 


Now we claim these [things] as wen known intrinsicany to perception: 
Compressed light is stronger than expanded light. Likewise. stronger light
>>>
40 


iIluminates more vehemently and spreads itself longer [than weaker light). 
Likewise, in the absence of light: shadow is made. Likewise, when light 
is imparted. the shadow secedes. Likewise. any shadow lis similar) at its 
extremity to a needle and is limited by a point. Likewise, light is spreading 
out equally to all various positions. Likewise, light passing through colored 
objects is colored by the colors of those [objects), as is obvious of light 
passing through glass windows, which is informed by the colors of those 
glasses [so as) to be carried together with the forms of those colors over 
[various) corporeal objects. Likewise, [we c4aim) that nature does nothing 
in vain, [precisely) as it does not leave undone anything [that is) necessary. 


[PROPOSITIONSI 


[Proposition) I. Allluminous rays, as well as the multiplications of forms, 
stretch forth in straight lines. 
What is proposed here can be made known not by a demonstration 
but rather instrumentally; indeed the diversity [of the attempts), of the 
ancients to prove this made use of a variety of instruments, while we 
are using that which we describe below, which we believe to harmonize better 
and more legitimately with what is proposed here. 
Let, therefore, a round, sufficiently thick bronze vessel (like the mother 
of an astrolabe) be taken, the width of whose bottom be one cubit or 
greater, and let the height of its edge be equal to the width of two 
inches [and be) perpendicular to the base of the vessel: and let there be 
in the middle of the back of this vessel a certain perfectly round column-like 
body. erected [there) perpendicularly, the length of which be equal to the 
width of three inches, while its width be less than one inch. And let the 
vessel be placed according to its middle points in a lathe, and let it 
be turned until its periphery be truly round bot h extrinsically and intrin- 
sically; and let its piane surfaces be levelled and let the column-like 
body, which is in the middle of the back, be [also) made round. 
Then let two orthogonally intersecting diameters be marked in the interior 
surface of the bottom of that vessel and let them be AB and CD [Fig. q. 
It is obvious that these diameters pass through the centre E of the bottom 
cirele. Then let a point F be marked in the base of the edge of this 
vessel, which is cirele A BCD, at a distance from the extremity of any of 
the [two) drawn diameters, say that of diameter AR, according to the width 
of one inch [from the extremity A?]; and from this point, let a third 
diameter be traced through center E, namely FG. And from the two ends 
of this diameter FG let two lines be drawn on the inside surface of the 
vessers edge, which will necessarily be perpendicular to the surface of 
the bottom plate, for the surface of the edge in which these perpendiculars 
are drawn [contains only lines which) are erected [perpendicularly) over the 
bottom surface. as is elear from above. Also. let those perpendiculars 


ił
>>>
, 
.
 


41 


[Fig. 1] 


be FH and GK: and on any of these lines. say FH. let three equidistant 
points. nameły L. M. N. be marked according to the quantity of half 
a grain of barley. the first of which. L. be the c10sest to the base 
of the vessel and to the same point F from which it be distant according 
to the quantity of half a grain of barley. And then let the vessel be 
brought back to the lathe, and let three parałłel circ1es be marked in 
it, passing through those three points L, M. N, which circ1es wiłł divide 
line GK. [which is) opposite to this divided line FH. proportionałły to 
Ithat] divided formerly. according to the 17th Iprop.] of the Xłth Ibook 
of the Elements) I. And let the divisions of line GK be [at) the points 
O. P, Q; and in any one of these three circ1es, two opposite points shall 
be produced. which are the extremities of some diameter of those circ1es. 
as [for example] point O in line G K is opposed to the point of division 
of line FH, which is point L. and [thus) line LO becomes a diameter 
of a concentric circ1e to circ1e ABCD. And similarly line M P becomes 
a diameter of another [concentric) circ1e. and line NQ becomes the diameter 
of the third [concentric] circle. And so let the middle one of these circles 
be divided in 360 parts and, should it be possible, [also) into minutes. 
Then let a round opening be driłłed on line FH, one of the two per- 
pendicular lines, FH and GK, over the middle point M. and let half 
the diameter of the opening be [measured) according to the distance 
of the [concentric) circIes, which is line M L. Consequently that opening 
touches both extreme circIes, and the middle of the [t h ree) circ1es will 
divide the circIe of the opening equally, since it passes through the centre 
of the opening. 
Then Id a pIane, somewhat thick, bronze plate be taken [Fig. I A), 
and let its thickness be as [that) of the edge of the instrument and its 
length two i!1ches, just as the edge of the vesseI.. and its width be near
>>>
42 


to this, and let it be [made] of parane I surfaces. And let it be levened 
to the point that the common section of the surfaces of its width and 
thickness be a straigtht line. say RS. and let it be divided in two equal 
Iparts] according to the I ()th Iprop.] of the 1st Ibook of the Elemell[s]; 
and from it.. midd'e point. ..ay T. 'et a straight 'ine ne drawn perpendicu'ar'y 


v 


G 


R 


s 


[Fig. lA] 


to the same line RS in the surface of the width, and 'et it be TV. And 
this [line] , as obvious from what precedes and by the 29th [prop.] of 
the 1st [book of Euclid], will necessarily be parane' to both longitude lines 
dividing the surface of the p'ate equally2. And on this perpendicu'ar 'ine 
TV. starting from the side of line RS over which it stands. let three points 
ne marked equally distant from one another. according to the quantity 
of half a grain of barley, namely X, - Y, Z, and in the middle of these 
points, which is Y, let the plate be bored through with a round opening; 
and let the periphery of the opening to uch the other two points, Iso 
that] this open ing will be equal to the opening LM N previously made in 
the edge of the vessel. 
Then let the semidiameter of the bottom of the vessel (which is FE) 
be divided in two equal [parts], [namely that semidiameter] over whose 
extremity. [laying] in the edge of the vessel. stands one of the perpendi- 
cu'ar 'ines. which is FH. and 'et the point of division be T. And from 
thi, midd'e point T. 'et a perpendicu'ar 'ine to 1he same diameter be 
drawn, namely RTS. Then let the base of the litt'e p'ate be p'aced over 
this line, till the line which is the com mon difference of the width and 
depth of the plate. namely RTS. is placed over that perpendicular line 
Iwhich was previously] drawn to the diameter land] which is also RTS. 
And let point T. dividing the plate line which is the common difference 
of the surface of width and depth. coincide with point T taken in line 
FE, the semidiameter of the vessel. Then let the little plate be fastened 
to the bottom of the vase. Next the opening XYZ. which is in the little 
plate RVS. will be directly opposite to the opening LMN. which is in the 
edge of the vessel. And the straight line M Y, connecting the centers of these 
openings. will be in the surface of the middle of the three circles previously 


......
>>>
43 


marked, [Le., the one] whose diameter is line MP. And line MY will 
be parallei to the diameter of the vessel, which is FE. 
Then let a part intercepting two diameters cutting one another ortho- 
gonally be cut otf from the edge of the vessel, which part be nearest 
a quarter lof a circle. Le..] cutting a quarter of that in which there is the 
open ing. to which the opening of the plate is opposed. and which. [open- 
ing] is (placed] in circle ABCD. corresponding to arc AD, and let the 
place of section be made level until it be one surface with the surface 
of the bottom of the vessel. And let the drawn quarter of a circle, which 
is AD. be divided according to the quantity of the edge-circle into 90 
degrees. and let the degree be dividied into minutes. And to this vessel. 
thus formed and moulded. we give thereafter the name of "the instrument". 
Then Jet a hwnze quad"rangular ruler he taken. the length of which 
be of one cubit, and let the four surfaces containing the same [ruler] 
be of two inches width. and let its surfaces be equalized till they be 
equal to [the surface] of a rectangle. Then. let there be ffidde in the 
middle point of the length of the ruler and in the middle of any of those 
surfaces a round opening whose size be fit for the body which is in 
the back of the instrument. And let there be an opening perpendicular 
to the surface of the ruler, passing through to the other slde ot' the 
opposite surface, and let it be such that [the ruler could] be turned around 
in the same instrument with some etfort of rotation, and let the instrument 
be placed over the ruler inserted into the body which is in its back. 
until the surface of the instrument be joined to the surface of the ruler; 
and the length of the ruler will be equal to the diameter of the instrument. 


o 


[Fig. lB]
>>>
44 


And let tWQ little strips [Fig. I B] of the width and thickness of the 
ruler be made, but of a length more than one inch, and let them be 
consolidated over the extremities of the ruler such that their preponderance 
over the extremities of the ruler be of one inch. or alittle more or less, 
and let those consolidated little stnps he over the nonperforated surface of the 
ruler. And since the width of the ruler is of two inches, while the height 
of the body at the back of the instrument is of three inches, let that 
third inch by which the body surpasses the ruler be perforated, as in an 
astrolabe, and let a pin be inserted holding together the ruler with the 
instrument. 
Then let another hronze ruler he taken. the width of which be double 
its thickness. while the thickness be equal to the diameter of the opening 
located in the edge of the instrument and its length be equal to half 
a cuhit; and let this ruler be truly straight and its lopposite] surfaces 
equal and paralleI. Then let that ruler be cut obliquely in one of its parts 
until the end of its length should contain an acute angle with the end 
ot" its width so that it be capable of being moved with ease. In its other' 
part, however, let the end of its width be perpendicular to the end of 
its length. Then let the line of its width be divided in two equal [parts], 
and from the point of section let a line be drawn, paralleI to the line 
of length, which, by 1.29 [E/ements], will be perpendicular to the line of 
width. And so since this ruler will have been placed over the bottom 
surface of the instrument. in such a manner that its thickness be erected 
orthogonally over the botoom ot" the instrument and the surface of its 
width be applied to the surface of the bottom of the same instrument, 
then its upper surfaC?e will be in the surface of the middle circle of 
the three circles drawn in the edge of the instrument, the diameter of 
which is line M P, since the thickness of the ruler is equal to the diameter 
of the open ing, and the diameter of the opening, which is. N L, is equal 
to the perpendicular line going out from the center of the open ing to 
the pIane surface of the instrument, namely line M F. to which adjoins 
the line of thickness of the ruler [which is] equal to it 3 . 
And 50, since it pleased us to show the proposed concłusion experi- 
mentally, let the instrument described above be placed opposite the solar 
body, or any other luminous body, perhaps even a candle, and let the 
center of the aperture of the instrument, nameły point M, be set opposite 
to the luminous body, in the best possible manner, and (t hen] the luminous 
ray will pass through the centers of bot h extant opposed apertures, of 
the one in the edge of the instrument and of the other in the perforated 
tablet, namely M and Y; and a luminous circle will be described in 
that part of the edge of the instrument which is directly opposite the 
aperture LMN all around diameter MP; and the center of that luminous 
circle will be in point P, which can easily be seen if the parts of the 
circumference of the luminous circle Iying between point P and each side
>>>
45 


of the circumference of the middle cirele (among the three cireles divided 
into degrees and minutes) are computed: indeed, equal numbers will be 
found in each direction. Therefore point P is the center of that luminous 
cirele. 
And so line M P, according to which the ray passing through the center 
of the circle of each aperture and through the center of the luminous 
cirele fali s, is entirely in the piane surface of the middle cirele amongst 
those three cireles and is the diameter of that circle 4. It is therefore 
a straight line. And if some body, colored throughout by a strong co lor, 
for instance green or red, be placed outside the aperture of the edge of 
the instrument. so that the light of the sun. or of another body, passing 
through that [colored] body should afterwards fali through the apertures 
of the instrument and pass through them, then, as was elear by the last 
of the [specifically optical] premised suppositions 5 , around point P in the 
edge of the instrument a cirele of colored light will be created, Inamely] 
of that [same] color. Consequently, color mixed with light spreads its form 
in straight lines precisely as light by itself. It is obvious, therefore, that 
any luminous rays, as well as the multiplications of forms, stretch forth 
in straight lines. And this is the proposed thing. 


[Proposition]2. Unimpeded light must be carried instantly through a whole 
medium that is [throughout] analogous to itself [Le., a homogeneous me- 
dium]. 
L.et there be a line [which is] proportional to the transmission of strong 
light. as iso in [the case ot] the sun's light, the diameter of the world. 
which may be [designated by] line ABCD. and let the very strongly lu- 
minous body be in point A [Fig. 2]*. Therefore. if it be said that light 


A 
I 


B 
I 


c 


D 
I 


[Fig. 2] 


is carried in time and not instantly over line ABCD. then, in a part 
of that time it .is carried over line AB. and in the smallest sensible time 
it will be carried over the smallest sensible part of line AB; for if in per- 
ceptible time it were delivered over imperceptible space, it would follow that 
perceptible space is composed of imperceptible [parts], as the time measuring 
motion over that space [AB] consists of perceptible times as its parts I; 
therefore, it wilI be carried in the smallest sensible time over the smallest 
perceptible space. But in the same time, the form of a weaker luminous body 
than that stronger luminous [one] would be carried over the same space, 
since no perceptible space is less than the minimal sensible space, precisely 
as no perceptible time is less than the minimum sensible time. Stronger 
and weaker light. therefore, will be of equal power. which is impossible,
>>>
46 


since contradictions are impłied [by this conelusion]. It is therefore impossible 
that light be carried in time over a medium analogous to itself. It is 
therefore necessary that the transmission be made instantaneously. which 
is the proposed [thing]. Indeed other natural reasons of Aristotle's can 
be devoted to this [prooł], which whoever might so desire can cover 
[by himselł], since for us this one way of reaching the result. however 
unsuitable. suffices. 


[Proposition]3. Every line along which light from a luminous body 
reaches to an opposite body is a natural sensible line, having a certain 
width. within which a mathematical łine is to be assumed imaginarily. 
Indeed łight does not proceed except from a body, since it does not 
exist except in a body. From which it is elear that in the minimai light 
that can be taken there is width; for we cali minimai light that which. 
when divided, has no further acting efficacy of light, because it wiłł not 
be visible, but [its] every part will be extinguished. since none of its parts 
wiłł be light, nor will it appear to the sense [of sight]. Therefore, there 
is a certain width in the radial line by which the diffusion of light 
takes place, by virtue of which perceptibiłity appertains to it, and in the 
middle of that line there exists an imaginary mathematical line to which 
all other mathematical łines within that natural line will be paralleJ. And 
since minimai light proceeds toward the minimai part of a body that 
light can occupy. it is necessary that its passage take place according 
to a mathematical line that is in the middle of a sensible line and [also] 
according to extreme lines [that are] parallei to the middle [mathematical] 
łine. Nor does minimaI light fali on a mathematical point of the opposite 
body, but on a sensible point corresponding to all the indivisible mathe- 
matical points to which the mathematical lines [attending] that sensible 
line can be terminated. And on account of this, we shall make use, 
in Ifuture props.] to be [yet] demonstrated about the occurrences of light, 
of the fancy of mathematical lines in action. 


[Proposition]4. Transparent bodies are fit for the penetration of light 
and color without [undergoing] any essential transformation. 
Indeed these bodies have the property not to prevent the forms of 
light and color from penetrating them. Nevertheless they are not changed 
by lights or colors. nor are they aJtered by them with a fixed aJteration, 
rather the diffusion of light and color takes place through them along 
straight łines. by the first [prop.] of this [book] ; some of which rlines] 
are paralleJ. some intersecting and some of diverse position. And the distin- 
ction of all these lines is done according to the difference of position 
of the luminous body from which the diffusion of that light and color 
is performed. And so the forms of light and color coming from diverse 
bodies to the same transparent [body] are extended (any of them) in 


.......
>>>
47 


straight lines and pass through [it] to the opposite bodies. Indeed the 
transparent body is not soaked by lights or colors. but only entered 
into; neither indeed do slIch hodies lose their forms becallse of the lights 
and colors [penetrating t hem]. nor are they soaked by lights or colors 
with a permanent dye. for the forms of light or color do not remain 
in them after the departure of light or color from their [place] opposite 
Ithe transparent body]. Therefore. those bodies are not transformed in 
any essential manner by lights or colors, which is the proposed [thing]. 


[Proposition]5. Lights and colors do not blend in transparent bodies, 
but penetrate [them] separately. 
The reason for this thing is to be shown experimentalły. Let many 
locally distinct candles be placed in a certain place and let them alł be 
opposite to one aperture leading to a darkened place. and let a certain 
non-trasparent body be placed opposite to the aperture in the darkened 
place. And so the lights of the candles appear over that body separately 
according to the number of the candles, and any of those [Iights] appears 
opposite to one candle. along the straight line passing through the aperture 
and through the middle of the light of the candle. And if one candle 
is covered completely. only one light [which is] opposite to that candle 
will be destroyed. and should the candle be uncovered. the light returns. 
And so it is obvious that in the middle of the aperture. where all. or 
many. [Iights] intersect in one point. the lights do not blend in the same 
point. bul are separate according to their essences; and on account of 
this. when extended later. they are distinguishable localły, according to their 
diversity. by the places in which they fali. And since light traversing 
colored things gets colored by those colors. as was supposed. it is obvious 
that if light penetrates separately. the colors too, which are carried with 
the light. will penetrate separately. What was proposed is therefore elear. 


IProposition]6. The ratio of the power of a whole luminous body to 
the whole luminous body [itself] is like [that] of a determinate part of 
the power to the part of the body proportional to it. 


A 
G 


B 
D 


[Fig. 3J 


Let there be a certain luminous body AB [Fig. 3]. I say that the 
ratio of the power of the whole body AB to the whole body AB is 
like the ratio of a part of the power, say A. to the [corresponding] 
part of the body, which is A. If in fact their proportion is not the same, 
then [one of the ratios] will be either greater or smaller [than the other]. 


--
>>>
48 


First let it be greater and let the power of the whole body AB be 
designated by line GD; and let G be the power of [that] part of the 
body which is A, and let D be the power of [that] part of the body 
which is B. Therefore, whatever the ratio of G to A. the same is 
the ratio of D to B; consequently, by V. 18 [Euclid], componendo, (t he 
ratio] of GD to AB will be like that of G to A l. If, therefore, the ratio 
of G to A is greater than the ratio of GD to AB, the ratio of GD 
to AB will also be greater than (that] of GD to AB2. which is impossible. 
Indeed there cannot oe two ratios of one 1hing to another. of which 
one would be greater than the other. The same impossible [consequence] 
from the [given] data would also occur should the ratio of part of the 
power. G. to part of the body, namely A. be smaller than [the ratio] 
of power GD to body AB. If indeed the ratio of G to A is smaller 
than Uhat ot] GD to AB and (the ratio ot] G to A is the same as 
(that ot] D to B (by the third (prop.] of the first (book] of this (treatisd])3, 
then. by V. 18 IEuclid] componendo. the ratlo ot" the whole power, which 
is GD, to the body AB shall be. smaller than the ratio of GD to AB4, 
which is impossible. Consequently the ratio of G to A is like (that ot] 
GD to AB. And this is the proposed (thing]. 
And (this] is universally lthe case]. unless perhaps something [else than 
the concerned luminous body] should contribute to one [partial] power, 
since a united power is always stronger than [when it is] itself divided. 
From which (it follows that] our proof holds when the part s, unseparated 
from the whole, act in that same whole without being actuaJly distinct; 
for if in fact they are distinct from the whole, then they are not [anymore] 
part s, since the name of part, by common acceptance, designates that 
which signifies potentiality not actuality. And on this [matter] there has 
been a more complete discussion in other places. 


[Proposition]7. The action of any luminous [and] unchangeable body in 
respect of form and place in another body (that is] also equal and homo- 
geneous, either lundertaken] directly or through a uniform facing medium, 
is always equal and uniform. 
Indeed let A be the power (virtue) of some given luminous body [Fig. 4] l 
and let BG be an equal and homogeneous body opposed to it; and let 
the impression of virtue A in body BG be designated by C. I say that 
A impresses always in body BG the imprint C. which is always equal 


D 
C 
A 
B G 
[Fig. 4]
>>>
49 


to itself and lalso] uniform. For if in fact it were given that Asome1imes 
impresses in BG an imprint which is C, while at some other time it 
does not impress C, but another limprint] greater or smaner than that 
very C. say D. then, as the imprinted body is always homogeneous and 
uniform. the diversity of the impression would stem not from the passively 
suffering body BG. but from the inherently diversified virtue A. which 
iso however. impossihle. since the luminous body was posited unchangeable 
in i1s fOl.m and l'la'..e. Therefore. its action is always equal and uniform 
in the body immediately lexposed] to it or through the uniform facing 
medium. And this is the proposed Ithing]. 


IProposition]8. It is necessary that the boundary of the length of any 
shadow be a luminous ray. 
What is proposed here is sufficiently elear according to the premised 
principles. For, indeed, Isince] by the third postulate (supposition) only 
in the absence of light is shadow made, and by the fourth postulate 
when light is imparted the shadow secedes, then it is absolutely necessary 
that shadow be caused in a space [exactly] by the amount of secession 
of light: and where light reaches there the shadow grows weaker. There- 
fore. as the lon
itudinal boundary of any shadow is a line. it is ohvious 
that it IS necessary for that line to be luminous. Consequently, that line 
is a luminous ray by the definition of a ray. The proposed rthing] is 
therefore elear. 


IProposition]9. The intersecting lines produced from the ends of the paranel 
al1itudes of a higher luminous body and a lower umbrageous (opaque) 
body are Irespectively] proportional to their altitudes; from which it is 
obvious that the same al1itude of the umbrageous body throws a longer 
shadow from a lower light 1han from a higher light. 


o 


K 


[Fig. S] 


4 - Wit.lonia P.npectivllC...
>>>
50 


Let line AB be the altitude of a certain umbrageous body [Fig. 5] 
and let there be another altitude of a luminous body, namely DE, parallei 
to it. And let line DE be greater than line AB. And let lines EB and 
DA be drawn which Iwhen] extended. will intersect on some side lofthe parralel 
lines], in point G, by the sixteenth [prop.] of the first [book] of this 
[treatise] ł. J say that the ratio of line GB to line GE and [that] of 
line GA to line GD will be like [that] of line AB to line DE. 
Indeed since line BA is parallei to line DE by hypothesis, it is therefore 
plain, by 1.29 [Elements]. that angle GBA is equal to angle GED and angle 
GAB fis] equal to angle GDE. Moreover angle BGA is common to both 
triangles DGE and AGB. Therefore, by VIA IElements], the ratio of line 
GB to line GE is like [that] of line BA to line ED. Consequently, by 
the 5th [prop.] of the first [book] of this [treatise], inversely, the ratio 
of line GE to line BG will be like [that] of line ED to line AB2. The 
proposed [thing] is therefore plain, since it can be proved in the same 
manner about lines GA and GD. And from this it is elear that the 
same altitude of the umbrageous body throws a longer shadow from a lower 
light than from a higher light. 
In fact let it be that some luminous body is situated in point H. 
And let ray HA fall in a point of line EG, which is K. And, as previously, 
the ratio of EK to BK will be like [that of] HE to AB. But, by V.8 
[Elements]. the ratio of HE to AB is smaller than [that] of DE to AB. 
Therefore, by V.Il [Elements], the ratio of EK to BK is smaller than 
[that of] EG to BG3. Hence shadow BK has increased a lot with respect 
to shadow BG, as is elear by V.1O [Elements] and by the fourth [prop.] 
of the first [book] of this [treatise] 4. And on this account it happens 
that the lunar shadows are always longer than the solar shadows; and 
such is lthe case] concerning any other higher and lower luminous bodies. 
The proposed[thing] is therefore elear. 


[Proposition] 10. Ił is necessary that any luminous ray eKtended through 
one transparent medium across the vertex of any umbrageous body be 
one straight line. 
Let the whole arrangement of the immediately preceding [prop.] remain 
[unchanged] and let point G be the end of the shadow [Fig. 6] ł. And 
so since the boundary of any shadow is a luminous ray, as is elear by 
the 8th [prop.] of this [book], J say that that ray limiting the shadow 
is a straight line, as is [the case] in the proposed figure [with] line DAG. 
Jf in fact line DAG is not straight, then, since line DA is straight, 
by the first [prop.] of this (book], as there is no cause for rany] hindrance 
in [its] progress, and line AG, similarly, is [also] straight for the same 
Ireason], flet us] therefore [assume that] lines GA and DA are being conjoined 
angularly in point A. Consequently a [certain] base may be stretched somehow 
underneath that angle from points D and G; and let the Uoining] line
>>>
51 


o 


E B 
[Fig. 6] 
DUG be straight, and let line A V be drawn forth or cut otP. And so 
triangle EDVG is divided by line BV (which is] paralleI to line ED. 
Therefore. by 1.29 [Elements]. triangles EDG and BVG will be equiangular. 
Hence. by VIA [Elements], the ratio of line GE to line GB will be like 
[that] of line ED to line BV. But. by the immediately preceding [prop.], 
the ratio of line GE to line BG is like [that] of line DE to line BA. 
Therefore, by V.II [Elements], the ratio of line DE to each of the lines 
BV and BA is the same, which is against V.8 [Elements]. and [therefore] 
impossible: in fact the ratio [of a third magnitude] to the smaller [of 
two other] is greater and to the greater lof the two] is smalIer; or 
[in our case] it would folio w that the greater ]ine is equal to the smalJer, 
by V.9 IElemenrs]. This is indeed impossible. It is nefessary therefore 
that ray DAG be one straight line, which is the proposed [thing]. 


[Proposition] I I. Ali dense non-transparent bodies cast a shadow on the 
side facing away from the luminous body [re ach ing] aU the way to the 
incidence of the (limit ing] ray drawn through the extremity of the dense 
object. 
Indeed since in dense non-transparent bodies the nature of translucency 
and transparence is hampered through the admixture of opaque, earthy 
bodies (in fact all such [opaque bodies] are of earthy nature by God's 
will). it is necessary therefore that (these dense bodies] hinder the passage 
of ligbt. Hence, by a postulate I, in the absence of light, [dense bodies] 
produce shadow in that part [of theirs] in which the access of light through 
them is impeded. This is indeed [the case] in the part not facing the 
luminous body. Now let there be one of thosc umbrageous bodies [and] 
let its altitude from the horizon be AB and its vertex A [Fig. 7]. And 
let there be a hnninous body. higher than line AB, of which some point 
D be the highest. And so the rays falling on line AB in its entirety 
are hampered from passing through because of the opacity of the body.
>>>
52 


D 



 


1
 
'\\ \ 
 
\\ \" 
 
\\\" 
'..:: A 
\\\",,'..:: 
\\ \ "- 
\\\",1 
\'\'1 
\ \ \1 
\ \ 'j 
\ \
 
\ 


c 


B 


[Fig. 7] 


Indeed let ray DC Ibe the one to] fali the e10sest over ray DA. Hence, 
since this ray is not impeded. it passes over [and] beyond body AR; 
hence it brings Iwith it] light in its Ipoint of] incidence, which is C. 
The shadow, therefore. secedes and the proposed Ithing] is elear. 


IProposition] 12. Of two umbrageous bodies of equal altitude the one 
c10ser to the luminous body Iwhich is] more elevated than it casts a smaller 
shadow. 
Let the highest point of the luminous body. whose aItitude above the 
surface of the horizon is line AG [Fig. 8], be G land] let [the luminous 
body] be higher than [each ot] the two umbrageous bodies. And let the 
equal heights of the two umbrageous bodies be erected over line AR. 
produced in the same surface of the horizon. land] let them be DE and 


G 


B 


[Fig. 8] 


-
>>>
53 


ZH, of which let DE be the eloser to the luminous body AG and ZH 
the more remote. And let ray GET be drawn through the top of body 
DE, which [ray] will be one line by the tenth [prop.] of this [book]; 
and let ray GHB be drawn through the top of body ZH. 
By the previous Iprop.], the shadow of body DE will be DET and 
the shadow of body ZH. ZHB. I say that shadow DET is smaller than 
shadow ZH B. Indeed let a parallei line to line ET be drawn from point 
H. by 1.31 I Elements]. [and] let it be H K. And it is obvious. by the second 
[prop.] of the first [book] of this (treatise]. that line H K will intersect 
line AB, with which its paralleI. which is line ET. [also] meets. And since 
lines H B and ET meet in point G. the highest point of the luminous 
body. therefore. by the second and 14th [props.]2 of the first (book] of 
this [treatise], point K will falI between the two points T and B. Hence 
let line EH be joined, which. by 1.33 [Elements] and by hypothesis. will 
be equal and parallei to line DZ. But, by 1.34 [Elements]-' lines EH and 
TK are equal. Hence lines TK and DZ are [also] equal. If. then. line ZT 
is added to each. line DT will be equal to line ZK. Therefore, by VI.l, 
shadow ZH K is equal to shadow DET, since they are of the same altitude, 
by hypothesis. But shadow ZH K is smaller than shadow ZH B. since it is 
only a part of it. Therefore shadow DET is smaller than shadow ZHB. 
The proposed Ithing] is therefore elear. 


[Proposition] 13. The shadow of a straight line opposed perpendicularly 
to a luminous body [and which is also] implanted in the surface of a dense 
body is naught; [when the line] is elevated [above the dense surface], 
however, (its shadow] is linear. but appears punctual. 


If indeed. by the third postulate, in the absence of light shadow is 
made. then it is elear that should a mathematical line, infixed in the surface 
of a natural body, happen to meet perpendicularly a luminous body. 
only a single radial line will be prevented from passage [which is not 
the case] with the other radial lines which pass to the surface of that 
[opposite] body. In fact no other radial line is hampered because of the 
interposition of that line. For otherwise it would happen that two or more 
radial lines meet In one point with one perpendicular line interposed to 
the very same (rays]. which is impossihle, since indivisibles exceed themselves 
in nothing '. Indeed since a ray is not anything but a luminous line. as 
is plain by definition 2, it is obvious that a ray intersects the surface 
of a body in the manner of a line, in a point, therefore it is also hampered 
in a point. But when light is conveyed. the shadow decreases, by the 
fourth postulate. Therefore, since a single ray is impeded and it intersects 
[the dense body] in a point, it is plain that no shadow whatever remains. 
When. however. the line is elevated over the surface of the dense body, 
wheresoever under the line the dense surface be placed. a shadow is to
>>>
54 


be found. And if the descent [of łight] was done through various points 
[of the interposed perpendicułar łine]. it is płain that a łinear shadow woułd 
be cast, since between any two points one can draw a connecting łine. 
[The present shadow]. however. appears ałways punctuał in its intersection 
with the surface of the dense body, since it commingłes there only with 
the shadow of the surface's density3. What was proposed is therefore c1ear. 


[Proposition] 14. The shadow of a pIane surface of any shape Iwhich 
is] perpendicular to the surface of the luminous body [and is] implanted 
in a dense body is naught; [if it is] e1evated, however. Ithe shadow] is 
two-dimensional but appears as a straight line. 
This is obvious by the preceding [prop.]. It beftts us indeed to draw 
a line opposed perpendicularly to the luminous body, [re ach ing] to an 
arbitrary point of the line terminating any given surface .opposed perpen- 
dicularly to the luminous body. It follows that the shadow of any such 
line, when the proposed extant surface is implanted in a dense body, 
is naught. ConsequentIy, nor will the sbadow of the entire surface be 
aught. Should the proposed surface, however. be elevated above that dense 
body, the shadow of any of those lines is punctual, by the preceding 
proposition; when taken together moreover, such points are seen to con- 
stitute a line J. Therefore the shadow of such an elevated surface appears 
linear . 
And since circular surfaces [too] do not intercept but points of the 
shadows by means of their diameters, or other perpendiculars produced 
to the luminous body, which [points] run together into a straight line 
below (since [dense surfaces] prevent the passage of a straight line), their 
shadow results into a straight line. Indeed shadows are not caused by the 
shape las such] of any objects whatever, except inasmuch as the passage 
of Iight is impeded [by the objects]. Consequently of whatever shape 
the proposed surface will have been [its] shadow will have always appeared 
as two-dimensional, but will be seen as linear because of the previous 
causes. The proposed [thing] is therefore c1ear. 


[Proposition]15. The shadow of any dense body whose base is equal to 
or greater than the surface opposite to it [and which is] perpendicular 
to the facing luminous body [and] impłanted in a dense body is naught; 
when elevated however [its shadow] is three-dimensional but seen as two- 
-dimensionaI. 
For ex ample, let there be a cylinder, or another body [and] let its. 
base be equal to or greater than the surface of that same body opposed 
to its base (assuming the surface of that same body, does not end in 
a point, as is [the case] with a pyramid or cone), [and] let it be implanted 
in the surface of any solid body and be opposed perpendicularly to the 
luminous body. I say that what is proposed is true.
>>>
55 


Indeed should that body be a cylinder or another body whose base 
is equal to the surface opposed to the base, and [should the base] be 
opposed to the luminous body, it is elear that, since the luminous rays 
from all sides reach the base by lines of longitude, no shadow is produced. 
And the same is obvious, should that body be pyramidal, or should 
[its] base be greater than the surface opposed to it [which is itself] opposite 
the luminous body; then, indeed, light is not impeded in any way, which 
would in fact happen if the surface opposite to the luminous body were 
. 
greater than the same base of the umbrageous body; for then the passage 
of light having been impeded, a shadow would be caused. 
But whatever the. shape of the given body, should the same be elevated, 
away from the other body to which it was [previously] implanted, there 
will appear a surface-like shadow. Indeed the surfaces cutting the body 
and perpendicularly incident to the surface of the luminous body constitute, 
by what precedes, a linear shadow. And since the entire surface of the 
body opposed to the luminous body is exhausted by such surfaces I, such 
lines when taken together constitute a surface. Ił is plain that the shadow 
of any body arranged it this manner appears as a surface. Moreover 
that shadow will necessarily be corporeal 2 since it is measured by the 
dimensions of a body, which can be shown as before. The proposed 
[thing] is therefore elear. 


[Proposition] l 6. The longest lluminous] ray reaching a sphere, or the 
circ1e of a cylinder or of a cone. is much the same as a tangent line. 
Let there be a great cirele of a sphere, or cylinder, or cone, namely 
DG I Fig. 9]*, [and] let its centre be point A and diameter GD. And 
since light spreads itself [equally] in all directions, as is obvious by the 


Z 
[Fig. 9]
>>>
56 


sixth postulate. let Z be that point of the luminous body whose light may 
spread over cirele DG and let line ZA be drawn from the point of 
the luminous body to the centre of the illuminated cirele, and let a cirele 
be described with diameter AZ (which] is cutting cirele DG in points E 
and B. And let rays ZE and ZB be drawn. J say that rays ZE and 
ZB are touching the sphere or any of the other (mentioned] bodies I and 
that no longer rays than those can reach to those bodies. 
Let indeed lines AE and AB be drawn from the centre of cirele GD, 
\\hich is point A. to the points of intersection B and E. Ił is obvious, 
therefore. by III. 30 2, that the two angles ZEA and ZBA are righL Hence. 
by 111.153, it is elear that lines ZE and ZB are tangent to cirele GD. 
If extended. then. they will not cut cirele DG. And so lines ZE and ZB 
are longer Ithan any other] lines that can be drawn from point Z to 
those bodies. Jf indeed it were given that some other longer rays can 
be drawn from point Z to those bodies. it is elear. by 111.8. that those 
would not intersect arc EB4. Hence. when extended. the same would cut 
lines ZE and ZB before they reach arc EG and BD5. And so lit would 
follow] that two straight lines enelose a su r face. which is impossible. 
And this. certainly. is demonstrated not only concerning bodies to be il- 
luminated. but it can also be demonstrated in the same manner about 
luminous bodies. since. lin this case] too. the largest ray from them inci- 
dent to opposite bodies is tangent to the same luminous bodies. The 
proposed Ith ing] is therefore elear. 


!proposition] '7. It is impossible that 'ight coming forth from a luminous 
body emerge only from the centre of the luminous body; from which 
it is elear that it is necessary that luminous rays be spreading out from 
any point of the surface of the luminous body. 
If indeed it is elaimed that luminous rays come forth only from the 
centre of the luminous body. let the luminous body be cirele AB (Fig. 
10]. of which the centre lis] G. and let the illuminated body be circle 


v 


z 


[Fig. 10]
>>>
. 


57 


DE. And from centre G of the luminous body let there come forth the 
two largest rays that can reach to the body to be illuminated from that 
point G. which. by the previous [prop.], will be the two lines touching 
the extremities of the illuminated body. which lIines] are GDV and GEZ. 
And let the points of contact. which are E and D. be joined by line 
DE. and. by 1.3 I. let line VZ be drawn paralle1y to it. 
And the part of the illuminated body over which light falls will be 
part DHE. and the obscure part, over which light do es not fali. DC£. 
And since the part over which no [Iuminous] ray fali s is not illuminated. 
therefore the part contained under the boundaries VDCEZ is shaded. 
being obscured by paralleI lines DE and Vz. And so. by 1.29. triangles 
VGZ and DGE are equiangular. because angle DGE is common to both 
triangles. By VIA. therefore, the ratio of line GE to line GZ is like [that] 
of line DE to line Vz. But line ZG is greater than line EG. Therefore 
line VZ is greater than line DE. Consequently the shadow of all bodies, 
whatever might be the ratio of their diameters to the diameters of the 
luminous body, is always greater than the opaque (umbrageous) body 
litselt] and is always increased inasmuch as [the rays from the cent re] 
are extended Ifarther and farther] beyond the opaque body, the contrary 
of which is known by the senses I. From which [also what] was supposed 
in the beginning [of this book. name1y that] any shadow lis similar] in 
its extremity to a needle and [that] it is ended in a point [would also 
be contradicted]. The proposed [thing] is therefore plain. 
And since laccordingly] light comes forth from the [entire] luminous 
body and not only from the centre, as we have shown, the corollary 
is manifest. since it is necessary that [rays] emerge from any point of 
the surface of the luminous body land spread] to the bodies to be illuminated. 
Indeed Ihe IUlllinous h()d
 is homo!!eneolls throllghollt. whence lit is elear 
that] whatever reason will be given that Iight is to spread from one point 
of its surface. the very same reason [can and] will be given about any 
other of [its] points. The proposed [thing] is therefore elear. 


[proposition] 18. It is impossible that rays emerging from the surface 
of the luminous body be solely paralIe1y incident to the body to be il- 
tuminated. 
If indeed this were said to be necessary, then there would follow 
[something] manifestly impossible. For in stance, let there be a luminous 
body whose diameter lis] AB [Fig. II] and an ilIuminated body DG. And 
let the two largest rays be drawn from the luminous body, which. by 
the 16th [prop.] of this [book], will be two lines touching the boundaries 
of body GD. [and] let them be AGE and BDV, which are paralleI by 
hypothesis I. In fact let the illuminated part, over which light falIs. be 
GZD and let the part over which the shadow falls be GHD. Hence the 
shadow is contained by the two lines EG. DV which are paralIeJ.
>>>
58 


G 


A 


E 


H 


v 


[Fig. II] 


Consequently. should there correspond to the [part of the] body to be 
i1Iuminated a part of the illuminating body equal to it, then indeed the 
rays would cut across [space] only by paralleI lines, according to 1.33 2 . 
It is elear. therefore. that [in such a case] every shadow would be equal 
in each of its parts to [the corresponding part s] of the opaque body. 
Consequently the shadow would neither increase nor decrease, but would 
always merely be extended [equally] to infinity, which is against a postulate 3 . 
In fact any of the shadows has a sharp end. Hence this [assumed alter- 
native] is impossible; therefore the opposite is necessary. And this is the 
proposed [thing]. 
[proposition] 19. Every point of the luminous body iIIuminates that part 
of the opaque body to which straight lines can be produced from the 
same point; from which it foIlows that one land the same] point of the 
luminous body does not iIIuminate the entire opaque body. 
Indeed luminous bodies are of one kind in [ali] their parts l. Hence 
the effect of their part s is not diversified, nor is it possible [for them] 
to iIIuminate from one part and not from another. Ali the same, straight 
lines cannot be produced from one point of the luminous body to any 
arbitrary point of the opaque body. And on that account, one point does 
not iIIuminate everything, but opaque bodies are iIIuminated by diverse 
points of the luminous body. Indeed let the luminous body be circle 
AR [Fig. 12] which line DG touches in [one] point, A. by 111.16, and 
let the iIIuminated concave body be Irepresented by] arć EV, and let line 
DG cut it in two points Z and H. I say that it is possible for the 
whole arc ZH to be iIIuminated by point A of the luminous body since, 
as obvious, it is possible for a straight line to be drawn to point A
>>>
59 


G 


D 


B 


(Fig. 12] 


from every point of arc ZH. On the other hand it is impossible for 
any [straight] lines to be drawn to point A from arc ZE and from arc 
H V, by 111.16 2 , since it is impossible for any other straight line to be 
intercepted between line G D, the tangent to the cirele, and that same 
circle AR. 
Hence if any other line is drawn from any of the points of those 
[two] arcs to point A, it will necessarily cut the circle, as 'for instance], 
line V A [which] cuts cirele AR in point T before it may reach to point 
A. And in like manner it is [possible] to conelude [the same] about all 
lines produced from any point of arcs VH and ZE to point A. Indeed 
all cut circle AR in some other point than the same point A. before 
they may reach to point A. And so a ray emerging from point A does 
not illuminate bot h arcs VH and ZE, but only arc HZ; but nothing 
prevents those arcs from being illuminated by other points of the luminous 
body, [Le.,] of cirele AR. from which straight lines can be produced to 
the same arcs. And [the case] is similar with any other illuminated bodies, 
for - if concave bodies - of which more is seen [than] what can be il- 
luminated from one point - are not illuminated by one point of the luminous 
body, then 'this is much more the case with] rectilinear solids having 
many piane surfaces [as their boundaries]. or spherical bodies, or other 
convex [bodies which] can be illuminated considerably less by one point 
of the luminous body. The proposed [thing] and its corollary are therefore 
elear . 


[Proposition]20. Light spreads itself out from any point of the luminous 
body along any straight line that can be drawn from that point to the 
opposite surface ; [moreover there is] only one line [which is] falling per- 
pendicularly to the surface of the opposite body. From which it follows
>>>
60 


that the light from any point of the luminous body is spread out ac- 
cording to a pyramid (cone) of iIIumination. 
That in fact the light of any point of the luminous body is spread 
out along any line that can be drawn from that point to all the various 
positions over the surface of the opposite body is elear from the preceding 
Iprop.]. Moreover. that only one of the lines drawn from any point of 
a luminous body to the surface of the opposite body is perpendicular 
[to that body] is elear by the 20th [prop.] of the first [book] of this 
[treatise] ł. Therefore a unique line intersects perpendicularly the surface 
opposite to it; indeed all other lines produced from that point cut [the 
surface] obliquely. 
It is elear. therefore, from this that light from any point of the luminous 
body spreads itself out according to a pyramid (cone) of ilumination. of 
which the vertex is in the point of the luminous body and the base on 
the surface of the opposite body; and this, furthermore, is elear instru- 
mentally, by the first [prop.] of this [book]. Indeed when light passing 
through the aperture of the instrument. the centre of which is point M 
Wig. I], has spread itself in the opposite part of the edge of the instru- 
ment in the shape of a cirele the centre of which is point P, it can 
actually be seen that cirele P will be greater than cirele M by computing 
in different directions the parts in the edge of the instrument which inter- 
cept the circumferences and the centres of those cireles. The proposed 
Ithing] is therefore elear. 
IProposition]21. The part of an opaque body to which light is incident 
from more part s of the luminous body is iIIuminated more than the part 
to which lIight is incident] from fewer [Iuminous] parts; from which it 
follows that every [part ot] an opaque [body which is] around the ray 
fali ing perpendicularly to it is illuminated more (than others]. 
Let the luminous body be cjrcIe ABG [Fig. 13] of which the centre 


L 


Wig. 13]
>>>
-----4. 


61 


tS point D; and let the arc of its convexity, namely ABG. facing the 
body to be iIIuminated. be bisected in point B. and let line ZE. be 
drawn tangent to the cirele in point B. by 111.16. And let line IK be 
tangent to the cirele in point G and line TH in point A. and let the 
opaque body be [represented by] arc KZTIEH. Furthermore let line DBL 
be drawn from the centre of the luminous body to the opaque body. 
And this [Iine] will be perpendicular to line EZ, the tangent to the cirele 
in point B. by III. 17 I. Consequently each of the part s of arc HT is 
inuminated by point A of the luminous body, by the 19th Iprop.] of 
this Ibook]. Hence point L is iIIuminated by point A. And similarly arc 
KI is iIIuminated by point G. Therefore both point L and the entire 
arc ZE are inuminated by point B; therefore point L too. And so point 
L is iIIuminated by three points of the luminous body, namely points 
A. B. G and the entire arc TI is common to the iIIumination of the three 
points A. B. G. 
Moreover arc El is common only to the two illuminations of points 
A and B. Furthermore arc ZT is similarly common only to the two 
iIIuminations of points B and G. since it is common to the arcs ZE and 
KI [which are] iIIuminated by these two points; in fact arc HE is inuminated 
only by one point. A. and arc ZK only by one point G. Therefore the 
i1Iumination of arc Tl has the triplicate lamount ot] light while arcs ZT 
and El have [only] the double and arcs EH and ZK have lonly] the simple 
lamount]. Arc TI, which is around the perpendicular line LD. is therefore 
iIIuminated in a much higher degree than all the other arcs. And the 
iIIumination of the two arcs ZT and El is equal since one. as well as 
the other. is iIIuminated by just as many points of the luminous body. 
In fact the inumination of both is greater than the illumination of the 
two arcs EH and ZK. 
The measure of the amount of illumination wilI always be according 
to the number of points of the iIIuminating body facing a [certain] part 
of the inuminated body. And so from these [considerations] it is elear 
that that [part] which is eloser to the perpendicular is inuminated more 
strongly than that [part] which is more remote from the same perpendicular, 
for truly more light fans on that same [body] which is inuminated by 
more luminous parts. Indeed what was demonstrated now about arc KH 
occurs similarly in any other bodies; on the other hand. we iIIustrated 
this in a concave body because such [a body] ought to be iIIuminated 
more uniformly. The proposed [thing] is therefore elear. 


[Proposition]22. Any opaque body [which is] eloser to the luminous 
point is iIIuminated more strongly than a body more distant from that 
point. 
Let the luminous body be in point A [Fig. 14] and let the inuminated 
body be [situated] at line BG. And let Iines AB and AG be joined
>>>
62 


A 


o 


v 


z 


E 


[Fig. 14] 


And so the power of body A iIluminating body BG also iIluminates the 
intermediate air which is contained in triangle ABG; and let line DE, 
parallei to line BG, be drawn by 1.31 1 . And let line BG be closer than 
body DE to the luminous body placed in point A. I say that body BG is 
illuminated more strongly than body DE. For let ray AB fali in point 
D and ray AG in point E, and from point B, by 1.12, let a perpendicular 
line, namely BV, be drawn to line DE, and from point G [another] 
perpendicular, which let be Gz. 
By 1.34 line VZ will be equal to line BG and line BV equal to line 
ZG. And so let lines V A, ZA be drawn. These [lines] will cut line BG, 
by the second [prop.] of the first [book] of this [treatise] 2. Therefore let 
line V A cut the same in point H and line ZA in point T. Hence. since 
the power imparting light in body BG is spread throughout the entire 
triangle ABG, while the power illuminating body VZ, equal to body GB, 
is spread only throughout triangle AHT, and since, by VI.I, triangle ABG 
is greater than triangle AHT, because base BG is greater than base HT, 
therefore the diffusion of light in triangle ABG is more than in triangle 
AHT. Moreover light is equally diffused on any point of these triangles. 
Therefore light incident to the body placed in line VZ illuminates that 
body more weakly than body BG, because less light fali s on it. 
In fact the ratio of the light's power incident to line HT to its impres- 
sion in body VZ is smaller than the ratio of the power incident to line 
BG to its impression in body VZ, by V.8, because, as is obvious from 
the preceding, the light incident to line BG is greater [in quantity] than 
the light incident to line HT. Moreover, the ratio of the power incident 
to line HT to its impression in body VZ is like the ratio of the power 
incident to line BG to its impression in body BG, by the sixth (prop.] 
of this [book]. Hence, by V.16, alternando, the ratio of the power reaching 
line HT to the power reaching line BG will be like (the ratio] of the 


.......
>>>
- 


63 


impression made in body VZ .to the impression made in body BG. But, 
by what was said before, the light reaching line HT is weaker than the 
light reaching line BG. Therefore the impression coming from line HT 
into body VZ 1s weaker than the impression coming from the power of 
the light incident to line BG in to body BG. And so the body [which 
is) cIoser to the luminous body is more strongJy illuminated than the 
[body) which is more remote from the same. And this is the proposed 
[thing]. 


[Proposition)23. Rays [coming) from more points of the luminous body 
are incident to the point more remote from the luminous body than to 
the cIoscr point. 
Let there be a circIe ABC [Fig. J 5]* of the luminous body, the centre 
of which fis) D, and let perpendicular DG be drawn I in which two points 
may be marked. G the farther [from) and H the cIoser [to the luminous 


G 


[Fig. 15] 


body). I say that rays from more points of the luminous body are incident 
to the more remote point, which is G, than to the very point H. For 
let the longest rays from the luminous body to point G be drawn and, 
similarly, Jet the longest rays from the luminous body to point H [also) 
be drawn. And so, by the 16th [prop.) of this [book), those rays will 
be tangent to the sphere. Accordingly. let the rays coming to point G 
touch [the sphere) in points A and B. and let the rays IOcident to pOlOt H 
touch the sphere in points E and F; and it is plain by the 60th [prop.) 
of the first [book) of this [treatise) that the points of contact E and F 
will falI inside the points A and B2.
>>>
64 


And so smee point H is irradiated soleły by the points of are ECF 
and not by others, while point G is irradiated by the points of are ACB. 
whieh is greater than are ECF. the proposed [thing] is elear. But whereas 
point G will be illuminated evenly by the surfaee of the luminous body 
whieh are ACB euts otf. and point H will be illuminated evenly by the 
surfaee of the luminous body whieh are EC F euts otf. nevertheless, beeause 
of the strength of the rays [involved], whieh is a [direet] eonsequence 
of their shortness, point H will be illuminated more strongly by fewer 
rays than point G by more. Indeed the multiplicity of light in the more 
remote point stems from the meeting of many rays ineident obliquely3 
and enfeebled, while in the eloser point light is strengthened on aceount 
of the brevity of the rays aeeording to whieh more [luminous] strength 
is dispatehed Ithere] by the luminous body. 


[Proposition]24. Every luminous body illuminates a smaller place from 
which it does not eseape more strongly than a greater spaee. 
What is proposed here is sufficiently elear by means of an example. 
Indeed a smali eandle illuminates a smalI room more strongly than a house 
or a greater room. Nontheless the same ean be demonstrated by means 
of a Igeometrieal] figure. Indeed let it be [assumed] that A is a point 
of a certain luminous body !Fig. 16]. from which rays AG. AB. AD may 


A 


B 


D 


[Fig. 16] 


be spreading out aeross a large space in whieh there is a line BG and 
let ray AD be perpendieular to line BG. And so the entire spaee BG is 
illuminated by these lines Iwhieh are] ineident to it from point A. Henee 
let line AE be eut otf from line AB. however it may have pleased lus], 
and from line GA line AF will be eut otf equal to line AE. And let the 
produced line EF cut the perpendicular line. which is AD. in point H.
>>>
65 


If then the space would be terminated in line EH F so that light could 
not pass through beyond [it], that space would be smaller than the space 
boulJd by line BGD, by VI.2. Indeed all rays reaching line BG [also] re ach 
line EF. Consequently the rays gather together more [efficiently] in space EF 
than in space BG; therefore they are [thereby] made stronger, since they 
are more united in [their] power. They are therefore more active than 
in space BG in which they are more diffused. Therefore the smaller space 
is illuminated more. when the power of light is terminated at its boundanes, 
than the space [which is] greater than it, and this is the proposed [thing]. 
I 


[Proposition]25. The axis or diameter of any opaque body facing; non- 
-perpendicularly the surface of the spherical luminoys body is parallei 
to some diameter of that [Iuminous] body. 
Indeed let the axis or diameter of the opaque body be line AB [Fig. 
17] I, facing non-perpendicularly the surface of the spherical luminous body, 


D 


A 


B 


E 


[Fig. 17] 


the centre of which is point C. I say that line AB is parallei to some 
diameter of body C. Indeed let line AC be drawn from the end of line 
AB to the centre of the luminous body, and at point C, the' end of 
line AC, let an angle be constructed equal to angle BAC, by 1.23, [and] 
let it be DCA, line DC having been extended in such a manner that 
angles BAC and ACD are made alternate. Consequently, lines DC and AB 
are parallei to one another, by 1.27. And since line CD is drawn from 
the centre of the luminous body, it is elear that it is part of the diameter 
of that spherical body. Hence when produced to [becomd] diameter DCE, 
it IS elear that the same is parallei to fine AB, and this. is the proposed 
[thing]. 


[Proposition]26. When the diameter of the given spherical luminous body 
is equal to the diameter of the body to be illuminated, only half of 
the latter is illuminated and the [boundary of the ] shadow is made equal 
to that [of the opaque] object extended indefinitely. 


s - Witeloni. Penpectivae...
>>>
66 


B 
E 


H 


[Fig. 18] 


Let the diameter of the iIIuminating body be AG [Fig. 18] I, and let 
ABG be that part [of the ]uminous body] facing the body to be iIIuminated. 
Moreover let DV be the diameter of the body to be iIIuminated, Iwhich 
is] by hypothesis and the preceding [prop.] equal and parallei to diameter 
AG, and let the iIIuminated surface be DEV. I say that DEV is half 
the surface of the body to be iIIuminated. 
For let rays AD and GV he drawn. And so since diameter AG is 
equal and parallei to diameter D V by hypothesis and by the preceding 
[prop.], it is obvious that rays AD and GV are [also] parallei and equal, 
by 1.33. Therefore they will never meet when extended indefinitely. Hence 
no other part of body DEV is iIIuminated beyond diameter DV. That is, 
only half of this body is Jlluminated. Indeed the shadow of the diameter 
is extended indefinitely, [remaining] equal to the diameter of the body, 
and it is spread out between Iines DV and VH, and line ZH is equal to 
line DV. And so the portion of arc DFV, which is half of the entire 
surface of body DE V, and lines DZ and VH contain a shadow equal 
lin its boundary] to the opaque objeet. which Imoreover] is extended inde- 
finitely. The proposed [thing] is therefore elear. 


[Proposition]27. If the diameter of the given spherical luminous body 
is greater than the dJameter of the spherical body to be dluminated, more. 
than half of the [opaque] body is iIIuminated and the base of the shadow
>>>
. 


67 


E 


[Fig. 19J 


is smaller than a great cirele of the illuminated body [and the shadow] 
is running together in a point at the back of the body. 
Let the luminous body be contained by cirele AR [Fig. 19]*. and 
Jet tlle opaque body to be ilIumonated he contained hy cirele GD. and 
let the diameter of AR be greater than the diameter of GD. And let 
there be [drawn] the incident rays AG and RD. which will necessarily 
meet beyond body GD. Should they in fact not meet, then they would be 
paralleI. It would then be necessary that diameters AR and GD be equal, 
which is against the hypothesis. And so let them meet in point E. It 
is elear, therefore, that rays AG and RD do not pass through the ends 
of a diameter of cirele GD. For if they passed, it is plain, since those 
rays would be tangent to cirele GD by the 16th [prop.] of this [book] I, 
that angles EGD and EDG, would be right. by III. 17 2 . Therefore, there 
would be two right angles in triangle GDE, which is impossible and against 
I. 32. And 80 it is plain that rays AE and RE do not pass through 
the ends of a diameter of cirele GD, but [that] they touch the surface 
of the body to be illuminated beyond those (ends]; therefore. more than 


.......
>>>
68 


half of the [opaque] body is illuminated. And since ą smaller cirele of 
that spherical body contains the shadow, it is elear that the base of 
the shadow is smaller than a great cirele of the illuminated body, which 
is the proposed [thing]. 


[Proposition]28. If the diameter of the given spherical luminous body is 
-_- smaller than the diameter of the spherical body to be illuminated, less 
than half [the opaque body]. is illuminated and the shadow extended inde- 
finitely is considerably greater than the iluminated body. 
Let there be a luminous body whose great circle is DG and a body 
to be illuminated whose, great cirele is AB [Fig. 19].. AIid let the diameter 
of cirele DG be smaller than the diameter of cirele AB; and so the rays 
GA and DB will meet beyond the luminous body GD because of the 
premised size of the diameters. Let [them] meet therefore in point E beyond 
the diameter of body DG. Therefore these rays do not touch the extremities 
of the diameter of cirele AB, since if they were [touching, it would folIow], 
as in the preceding (ptop., that] triangle ABE has two right angles, by 
III. 16 I and III. 17 which is impossible. Consequent1y ,less than half the 
body AB is illuminated. And since a great cirele of body AB fali s inside 
the shadow, and the shadow extended beyond that [circle] is always expanding, 
[and] since it is impossible, by the' 14th [prop.] of the first [book] of 
this [treatise]2, for the rays GA and DB to meet on that side [where 
the shadow expands], it is elear that the shadow extends indefinitely. And 
this is what is proposed. And according to these preceding [conelusions], 
[the same] can be demonstrated, entirely in the same fashion, [to be the 
case] in cylinder s and cones; indeed the manner of proof is the same 
in those [bodies]. 


E 


ł [Fig. 20] 


Il1o.....-
>>>
69 


[Proposition]29. Ił is necessary that a pIane surface erected [perpendicularly] 
over the middle of the shadow divide equany the opaque body and the 
luminous body. 
Let there be a luminous body AB, the centre of which lis] C, and 
let the opaque body be DE, the centre of which lis] F [Fig. 20]*. And 
let there be a point in the middle of the shadow, namely G, and let 
line CFG be joined. And so line FG will falI in the middle of the shadow. 
Hence the surface erected [perpendicularly] over the middle of the shadow 
will necessarily be erected over line GF; therefore that surface will pass 
through the centre of the opaque body as wen as the centre of the lu- 
minous body. ConsequentIy it wilI necessarily divide those bodies equally, 
according to those [conelusions] which were exhibited in the fust [book] 
of this [treatise] I. The proposed [thing] is therefore elear. 


[Proposition]30. Ił is necessary that the piane surface bisecting the luminous 
body and the opaque body be erected [perpendicularly] over the middle 
of the shadow; from which it is elear that there are as many shadows. 
of the same opaque body as the number of luminous bodies to which 
it is opposed. 
Let there be a body over which light faUs [and] which is contained 
by cirele AB [Fig. 21] I, the centre of which is point G, and let there 


[Fig. 21] 


be one of the luminous bodies contained by circ1e DE, of which the 
centre is V. And let there be another luminous body contained by cirele 
ZH. the centre of which is T. And so the shadow opposed to the luminous 
body DE, the middle point of which [shadow] is M, will be seen as 
contained by lines AK, BL. Therefore since any surface [which] will have 
bisected the luminous body and the opaque body will necessarily pass 
through line VGM. that Isurface] will also bisect that shadow, since Ithe
>>>
70 


surface] erected perpendicularly passes through the centre G of that same 
body. Furthermore, the surface dividing equally both bodies ZH and AB 
passes through line TG drawn through the centres of those bodies; but 
the same traverses the centre of the shadow contained under Iines AN 
and BS, i.e., the middle point of NS, which is Q. 
Hence that surface dividing bodies ZH and AB in two halves will 
also divide the shadow in two equal [parts]. And since the pIane surfaces 
bisecting the opaque and luminous bodies in various directions are distinct, 
it is elear that the shadows too are counted according to their [number]. 
The proposed [thing] is therefore elear. Indeed, in general, the shadows of 
thc same opaque body will be just as many as [the numbers] of luminous 
bodies opposed to it. 


[Proposition]31. The shadow of the more remote opaque body from the 
luminous body is a weaker shadow, [that] of the cIoser [opaque body] 
certainly stronger. 
Indeed, as follows from the 22nd [prop.] of this [book], since any 
opaque body [which is] eloser to the luminous body is iIIuminated more 
strongly than a more distant body, it is elear that the shadow of the 
eloser body prevents more Iight from gett ing through. Moreover the rays 
bounding it are [comprised] of stronger light; therefore the shadow between 
those rays appears darker and obumbrates more, because the rays bounding 
those shadows are more luminous, for which reason, likewise, there appear 
more shadows in their presence. Furthermore the shadow of the more 
remote body from the luminous body prevents less light from getting through. 
Also the rays containing that shadow are [comprised] of weaker light; 
therefore the shadow between those rays appears weaker [and] will. therefore, 
obumbrate less. The proposed [thing] is therefore elear. 


[Proposition]32. Every strengthened shadow obumbrates (darkens) more. 
Indeed let it be that one opaque body is opposite to many luminous 
bodies. Hence it is plain, by the 30th [prop.] of this book, that there 
will be as many shadows of the same opaque body as the [number ot] 
luminous bodies to which it is opposed. And so should it happen that 
the shadows intersect, I say that the strengthened shadow obumbrates more. 
Indeed any of the shadows takes away some Iight; therefore an increased 
shadow will remove more Iights which [otherwise] remain in other parts 
of the medium in which the shadow is not increased but remains simply 
an [unaugmented] shadow. Therefore that [simple shadow] is merely imbued 
with some light which does not reach [however] the increased shadow. 
ConsequentIy the increased shadow obumbrates more, since the place of 
that shadow is deprived of more light. The proposed [thing] is therefore 
elear .
>>>
71 


[Proposition]33. It is necessary that two bodies, of which one casts 
a shadow along the other's half, stand in the same surface erected Iper- 
pendicularly] over the luminous body; and [conversely] if they stand in 
the same surface related [as above]. to both, one will cast a shadow 
along the other's half. 
This [prop.], inasmuch as its first part is conceroed, is elear by the 
30th [prop.] of this [book]. Indeed since a piane surface bisecting the 
luminous body and the opaque body is erected [perpendicularly] over the 
surface of the luminous body and also over the middle of the shadow 
of the opaque object (the shadow in fact falling over the middle of the 
opaque body), it is, therefore, necessary that that body [which is] darkened 
along its half be in the surface erected [perpendicularly] over the surface 
of the lumlOOus body. 
And from this the second part of this theorem is also elear: for if 
two bodies elose to one another lie with their middle part in the same 
surface erected [perpendicularly] over the surface of the luminous body, 
one of them will cast a shadow over the other [as stated]. because the 
[body which is] more remote from the light, when elose [enough] to that 
[one] which is nearer to the light, will fali within the shadow of that 
[body] which is nearer to the light, as [happens] when the same [Iuminous] 
ray passing through the extremity of the eloser [body] passes also through 
the extremity of the more remote, or [through] some other point which 
may be higher than it [1]. The proposed [thing] is therefore elear. 


[Proposition]34. The paralleTism of radial lines or their intersection does 
not stem entirely from the nature of the rays but [also] from the ratio 
of the diameter of the luminous body to the diameters of the [various] 
opaque bodies. From which it is elear that light spreads itself uniformly 
through the surrounding air. 
This is obvious by the 17th and 18th [props.] of this book and can 
be shown thus by means of an example. For let the luminous body be 
cirele AB [Fig. 22]1 and let line AG be one of the radial lines emerging 
from it, and BG another line, and let those lines meet in point G. Let 
there also be a line EV and another DZ and and let EV and DZ be 
paralleI. And let there be a body over which the light falls, the diameter 
of which is smaller than the diameter of the luminous body, [and let 
it be] placed between the two [lines] AG and BG which intersect [and] 
let its great cirele be T/; and let line BG touch that cirele in point I and 
line AG in point T. 
And let another body over which the light falls [and which is] equal 
to the luminous body be placed between the two parallei lines EV and DZ 
[which are] tangent to that body, the diameter of which is KL, and let 
it be touched by line EV in point K and by line DZ in point L. And 
so the shadow arising from body TI is being diminished and is terminated,
>>>
-72 


G 


L 


z 


v 


[Fig. 22J 


and becomes pyramidal, by the 27th [prop.] of this [book], because the 
rays touching body T/, which are AG, BG, meet in point G. Therefore 
the shadow of body T/ is contained by the two lines /G and TG and by 
. the surface of body T/ that is on the side of G. Therefore the shadow is 
terminated at point G. Moreover the shadow of body KL extended between 
parallei Iil}.es LZ and KV, does not terminate at any point, as is c1ear 
by the 26th [prop.] of this [book], because those lines containing the shadow 
do not meet when extended indetinitely. 
Furthermore should body T/ be moved outside of lines AG and BG 
and be placed between lines EV and DZ, lines EV and DZ would meet 
and the shadow would be moditied [with respect to that] previously contained 
by the same [Iines], according to the diversity of the ratio of the diameters 
of body T/ and. body KL to the diameter of the luminous body BA. 
And from this it is c1ear that [Iuminous] rays a:r:e by themselves neither 
regular nor irregular lines, neither parallei nor concurrent, but that their 
characteristic feature occurs as a function of their appearance relative to 
bodies upon which they light; and parallelism and concurrence happen 
to them according to the ratio of the diameters of opaque bodies to the 


........
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73 


diameters of the luminous body. Therefore light spreads itself out uniformly 
throughout the entire amhient air in such a manner that every point 
of the air from which straight lines can be drawn to some point of 
the luminous body is illuminated by the light of the luminous body, as 
is elear from the 19th Iprop.] of this [book]. The proposed [thing] is 
therefore elear. 


IProposition]35. Rays proceeding from one point of the luminous body 
approach more the state of being perceptibly paranel as the length of the 
[luminous] rays [increases]. 
Let it be so that the equal rays AB and AG [Fig. 23] emerge from 
the center point of the ]uminous body. which is A: moreover Jet base 


G 


M S B 


[Fig. 23] 
BG be joined and ]et Jine DE be drawn, parane] to base BG, by I. 10 
and I. 3], cutting triangle ABG short of the midd]e of its side AG I. And 
]et ]ine AZ be extended from point A perpendicularly to the base BG, 
by I. ] 2, and ]et it cut ]ine DE in point V; and ]et ]ine EG be divided 
in two equa] [parts] at point H, by I. 10. and line DB in point T, and 
let ]ine HT be drawn. Therefore Jine HT will be parane] to base GB, 
by VI. 2; it wilJ cut, therefore, line VZ, by the second [prop.] of the 
first [book of this treatise]2, [and] ]et K be tlie point of section. A]so 
]et perpendicular lines to base BG be drawn from points E, D, H, T, 
and Jet them be EL, DM, HN, TS. And perpendicu]ar EL wilJ cut Jine 


......
>>>
74 


HT; let the point of section be Q, and let the point of section oflines 
DM and HT be F. 
Hence, line Q F will be equal to line ED. by I. 34. It is elear there- 
fore that line HT is greater than line DE. And so since triangles AVE 
and EHQ are equiangular. by I. 29. their (respective] sides will be pro- 
portional. by VI. 4. Hence. as became elear above, since line AE is greater 
than line EH3, line EV will, therefore. be greater than line HQ. But 
line HT is greater than line ED. as shown beforehand; therefore, by the 
9th (prop.] of the first Ibook] of this (treatise]. the ratio of line EV to 
line ED is greater than Ithat] of line QH to line HT 4. Indeed the ratio 
of line EV to line ED is like Ithat] of line H K to line HT, by VI. 4 
and by v. 16 and V. 18 5 . But line HQ is part of line HK; therefore, 
by V. 8, the ratio of HQ to HT is smaller than (that ot] HK to HT. 
Consequently the ratio of line HQ to HT is smaller than (that ot] EV 
to ED. And it can be demonstrated in the very same manner that the 
ratio of line GN to line GB is smaller than Ithat] of line HQ to line HT. 
And 50 the excess of base GB over base HT is smaller than the 
excess of base HT over base DE6; and according as the bases are more 
remote from point A of the luminous body, the excess of the more remote 
bases over the eloser bases is correspondingly diminished 7 . It is plain there- 
fore that when the distance Ifrom A] is more remote, the rays proceed 
almost in a paralleI manner; and when the quantity of the excess of the 
bases (to one another] is of imperceptible amount. then the radial lines 
will be almost paralleJ. Indeed whereas line BG does not exceed line HT 
sensibly, thereupon rays HG and TB will be al most paralleI according 
to the senses. And this is the proposed Ithing]. 
And the .property of the rays. according to which (a ray]. inasmuch 
as possible. always draws near to its perpendicular Iposition], contributes 
a great deal to this; because of this, the rays from all points of the 
entire luminous body meet always in any point of the body to be iIlumi- 
nated and thus constitute a radial pyramid. 


IProposition]36. When light is falling through a window onto an opposite 
solid body. the perimeter of the light will be ampler than the perimeter 
of the windowo 
Let there be a luminous body, the centre of which (is] A [Fig. 24], 
and (whose] great cirele lis] DEG and let the diameter of the window 
be Be. And let line TZ be in the surface of the solid body opposite 
to the light, to which (body] the ray is incident. Also let there be produced 
radial lines touching the periphery of the window, which are EB and Ge. 
And so these lines will intersect in some part of the medium; let the 
point of intersection be F. And these lines when extended would be incident 
to the surface of the body (which is] opposite the light, and let line EB 
fali in point Z and line GC in point T. 



 


......
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75 


z 


T 


[Fig. 24] 


Accordingly since in triangle FTZ side TZ is greater thaJ'! side BC, 
because triangle FTZ is greater than triangle BCFł, and since intersecting 
rays fali in this manner through each point of the window's periphery, 
because the emission of light is made from any point of the luminous 
body through the entire windowo by the 19th Iprop.J of this Ibook]2. it 
is plain that the perimeter of the light incident to the solid body Iwhich 
isJ opposite the window is greater than the perimeter of the windowo And 
this lis whatJ what was proposed. 


[PropositionJ37. When the ray from the center Df the luminous body 
is perpendicularly incident to the centre of a circular aperture, the light 
in the surface of the dense body Iwhich isJ parallei to the surface of 
the aperture is truly circular. 
Let- the circIe of the aperture whose centre lisJ E be ABGD (Fig. 
25J.. to which the surface of the solid body FH KL is paralleJ. And let 


[Fig. 25]
>>>
76 


line EZ be erected perpendicularly from centre E to the surface of cirele 
ABGD. And so the centre of the luminous body is somewhere in a point 
of line EZ. I say that the light incident to the surface FH KL is truly 
circular. For it is plain. by the 65th [prop.] of the tirst Ibook] of this 
[treatise]. that all lines ZA, ZB, ZG, ZD drawn from pole Z to the cir- 
cumference are equal] and make equal angles with line EZ. by I. 8 2 . 
Accordingly, let line ZE be extended beyond point E to the surface pa- 
rallel to the aperture cirele which is FH KL; and [this line] will be incident 
perpendicularly to that [surface] by XI. 14 3 . Let it be incident in point M. 
and let line ZB be extended to surface FH KL. [cutting it] in point K, 
and [similarly] line ZA Icutting it] in point F. and line ZD in point H. 
and line ZG in point L. 
And lines AF, BK. DH. GL will be equal. by the 25th Iprop.] of 
the tirst [book] of this [treatise]. because of the concentricity of the surfaces 
and the equality of the angles 4. Therefore the entire line ZF will be equal 
to the entire line ZH and ZK [will be] equal to line ZL5. Also let 
lines FM, H M, KM. LM be drawn. Accordingly in triangle FMZ. base 
FM will be equal to base HM of triangle HMZ, by I. 4. In the same 
manner lit can be coneluded that] line KM will be equal to line H M 
and line LM equal to line KM. It is obvious therefore, by III. 9. that 
surface FH KL is circular 6 and it is this same Isurface] at which the lu- 
minous rays coming in through window ABGD are terminated. since the 
same demonstration is (true] about all other (like] lines. The proposed 
[thing] is therefore elear. 


IProposition]38. If the luminous ray (entering] through the center of 
a circular aperture lis] obliquely incident to the surface of the opaque 


L 
[Fig. 26]
>>>
77 


body laying beneath the surfaee of the aperture. the ineident light will 
be shaped as a eonie seetion. the longer diameter of whieh will be in 
the surfaee ereeted perpendieularly to the surfaee of the windowand to 
the surfaee of the underlaying body. 
Let there be a eireular aperture ABCD [Fig. 26]. the eentre of whieh 
lis] E, to whieh surfaee H M KL is paralleI. and let F be the eentre of 
the luminous body. First let it be that line FE falls obliquely over the 
surfaee of eircle ABCD; and so, when extended. this [line] will be, similarly. 
obliquely ineident to surfaee H M KL, beeause of the parallelism of the 
[two] surfaees. by the argument of the 23rd [prop.] of the first [book] 
of this ItreatiseJ'. Aceordingl}. let i1 fall in point G. and let line AEB. 
the diameter of the eircłe, be drawn. And so angle AEF is aeute; therefore, 
by I. 13, angle BEF will be obtuse. 
And sinee the square of line FA amounts to less than the two squares 
of line EF and EA, by 11. 13, and the square of line BF is greater than 
the square of line FE and the square of line BE. by II. 12. while the 
square of line BE is equal to the square of line AE, sinee they are both 
semidiameters, and the square of line FE is eommon, it is elear that 
the square of line FB is greater than the square of line FA. Therefore 
line FB is greater than line FA. And lines F A and FB having been 
produeed to surfaee H M KL, should line FA eut [that surfaee] in point 
M and line FB in point L, line FL will be greater than line FM. by 
the same [reasoning] as before. And lines LG and MG having been joined 
to point G. to which the ray passing through the eentre of the window's 
aperture is 'alsol ineident. then. by VI. 2 and V. II. the ratio of line 
LG to line BE will be like [that] of line GM to line EA. beeause any 
of these [two] ratios is in tum like [the ratio] of line GF to line FE. 
Therefore, by V. 16, the ratio of line LG to line MG is like [that] of 
line BE to line EA. 
But line BE is equal to line EA. Therefore line LG is equal to line 
GM. Then let diameter CD be drawn orthogonally to diameter AB and 
let lines FC, FD be eontinued and extended to surfaee H M KL up to 
points H and K, and let line HGK be drawn. And sinee the surfaee 
. in whieh lines FE and AB are is the only one ereeted [perpendieularly] 
to the eirele of the windo w: sinee all other surfaees in which line FE 
lays eut that [windo w] surfaee obliquely - for this is how we to ok line 
AB2 - therefore surfaee AFB will be ereeted [perpendieularly] to the surface 
of the circłe of the windowo It is obvious therefore that angle FED is 
equal to angle FEC. Consequently, by I. 4, line FD is equal to line Fe. 
Henee, as earlier, line HG will be equal to like GK and line FH equal 
to line FK, but also FG is eommon [to bot h triangles involved]. And 
sinee line HK is perpendieular to line ML and to line FG3, it is plain, 
by XI. 4, that line HG is perpendieular to the surfaee in which lines 
FG and MG4 lay. Therefore, by XI. 18, surfaee HMKL will be ereeted
>>>
78 


Iperpendieularly] to surface FMG; henee surface FMG is also ereeted 
Iperpendieularly] to surface H M KL. 
And S{'t let it he ima!!ined that ahout roint G. the extremity {'tfaxis 
fG. a CltcJe i,.. dra\\n ar{'tund the !.'one of illummation. hy the 102nd Irrop.] 
of the first Ibo{'tk] of this Itreatise]5. and theref{'tre. by Iprops.] 100 and 89 
{'tf the first Ibook] of this Itreatise]. axis FG will be perpendieular to that 
circ1e while the same is {'tblique to surface HMKL6. Henee, by the 103rd 
Ipwp.] of the first Ibook] of this Itreatise]. line H M KL will be a eonie 
seetion. the greater diameter of whieh will be in surface FML Iwhieh is] 
perpendieular to surface H M KL 7. The proposed Ithing] is therefore elear. 
And if the surface of the eireular window were the base of the eone 
of illumination, so that the eentre of the luminous body were the pole 
of the eirc1e of the windowo and the axis were perpendieular to the surface 
of the windowo Iwhile] in faet the surface of the solid body emitting the 
luminous rays will not have been parallei to the surface of the window, 
still the shape of the light would be a eonie seetion, whieh ean be demon- 
strated las] in the previous manner. 
Indeed should a surface parallei to the surface of the window be drawn 
from point L. the extremity of the lon
r ray, whieh is FL, by the 102nd 
Iprop.] of the first Ibook] of this Itreatise]. it is elear. by the 100th [prop.] 
of the first Ibook] of this Itreatise]. that that surface would eut the eone 
of illumination aeeording to a eirc1e, whieh is LPQ. Therefore, surface 
HMQL euts (he same aeeording to a eonie seetion. The pwposed [thing] 
is therefore elear. 


fProposition]39. Any light entering la spaee] through angular apertures 
IS Inevertheless] rounded otr. 
What is proposed here is c1ear by the 35th Iprop.] of this (book]. 
Indeed sinee all rays proceedinp from a point of the luminous body eome 
c10ser to the state of being pereeptibly parallei as the length of the (luminous] 
rays inereases, it is elear that the ineident rays eonvey themselves thwugh 
those same angles of the angular apertures aeeording to the disposition 
of parallelism to the ray lentering] perpendieularly. or approximately so, 
into the surface of the ineident aperture I. Therefore Ithese rays] withdraw 
from angularity and thus the light ineident to the opposite surface of the 
aperture begins to round itself otr. And sinee. as is elear by the 20th 
fprop.] of this [book]. light from any point of the luminous body spreads 
itself out aeeording to lines whieh ean be drawn from that point to the 
opposite surface, all those rays too meet in an arbitrary point of the 
medium. It is elear that the same Irays] interseet in any point and [that] 
the rays from the lower points of that same luminous body eut the 
other rays from the higher points (also] in point s belonging to the edges 
of the aperture and [that] these rrays] are extended beyond Ithe aperture]. 
And thus light pass ing through openings of this kind is rounded otr.
>>>
79 


which would not quite happen if rays entering the opening would emerge 
only from one point of the luminous body2. The proposed Ithing) is 
therefore c1ear. 


[Proposition)40. Jf the luminous ray is incident perpendicularly to the 
middle point of a square aperture, the light falling on the surface of 
the body Iwhich is) parali el to land behind) the surface of the aperture 
is lalso) square lin shape). approaching Ihowever) some circularity. 
Let the centre of the luminous body he E iFig. 271 and let the square 
aperture be ABC D. to the center point of which. i.e.. F. ray E F is incident 


[Fig. 27] 


perpendicularly; and let there be the surface of the opaque body paralleI 
to the surface of the aperture. which [body) is GH KL. J say that the 
light fali ing on that surface will be of square shape. 
Jndeed two pyramlds are Ithereby) created. having one vertex. point 
E, of which the greater base is GHKL, while the smaller base is ABCD. 
and their bases are paralleI ; they are, therefore similar. by the 99th IProP.) 
of the first [book) of this [treatise) I. Hence, since base ABCD is square by 
hypothesis, it is c1ear that base GHKL is also square. And this is the 
first proposed [thingJ2. 
Moreover since, by the 35th [prop.) of this [book). the larger the rays 
[the morel they tend to a certain paralIelism. this shape too tends to 
a certain circularity, because of the compression of the rays or because 
of their intersection at points on the lines bordering the openings, as we 
stated in the preceding [prop.). The proposed lthing] is therefore c1ear. 


[Proposition]41. If the [Iuminous) ray fe nter ing] through the middle of 
a square opening is obliquely incident to the surface of the opaque body 


......
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80 


laying underneath the surface of the aperture, the incident light wilI be 
of a shape [having] one part longer [than the other], [while] its angles 
wilI be equally bent. 
Let point E be, as in the preceding [prop.], the centre of the luminous 
body [Fig. 27] and [Iet] the periphery of the square aperture [be] ABCD, 
to the middle point F of which let ray EF come. And let GH KL be the 
surfac
 of the opaque body underlying that aperture, to which [surface], 
similarly, the ray is obliquely incident. I say that the shape of the light 
in the underlying surface wilI be longer in one of [its] sides I. 
Indeed since those surfaces are not the base of the pyramids of iIIumi- 
nation but only cutting those pyramids obliquely2, it is elear, by the 99th 
[prop.] of the first [book] of this [treatise]. that both figures ABCD and 
GH KL, be they parallei or not, are of a longer shape on one [of their] 
side[sjJ: for those [two] figures which bound the pyramids according to 
the points [taken in the previous prop., and] to which proposed surfaces 
axis EF is [now] obliquely incident, are both squares, while the other 
[present surfaces, which are] obliquely incident to the axis, according to 
those point s, are both longer in one of Itheir] parts. The first proposed 
[thing] is therefore elear 4. 
And since as follows from the 35th Iprop.] of this 'book], the longer 
rays tend. as it were, to a certain paralIelism, it is elear that the angles 
of that light-shape are bent some way or another. as was also shown 
in the two previous Iprops.] 5. And this is the proposed 'thing]. 


[Proposition]42. A perpendicular ray drawn from the center of the 
luminous body to the surface of the opposite body always penetrates through 
the midst oC the second transparent [medium] without being reCracted. 
The proof of this proposition rests more on an instrumental endeavor 
than on other demonstration. Therefore when someone want s to experience 
the mode of refraction of luminous rays in a second transparent medium 
,which is] more dense than the first, for instance in water which is denser 
than air, let a vessel with straight edges be taken, of whatever material 
or shape may be desired. Now indeed let the height of the edges be 
greater than half a cubit, and let the diameter of its width be not less 
than the diameter of the instrument which was previously required to be 
made in the first [prop.] of this [book]. And let the edges of that vessel 
be evened out till the surface passing through its edges be equally piane, and let 
a certain little visibly colored body, be placed on the bottom of the vessel, 
something like a coin or an object painted in a distinct co lor, then let 
the vessel be filled with elear water. 
Hence, as the water's motion shall have quietened down, should the 
beholding sight be extended perpendicularly to the middle of the coin 
or of the painted object, it would find the shape and the color and their 
position and the arrangement of the pr rts the same way as they are disposed
>>>
81 


by themselves, had they been seen in air I. Consequently, let the experimenter 
eonsider the position of this body, be he standing or sitting, and his 
distanee from the vessel, and the position of the same vessel, and all 
the circumstanees of that act of vision. And so let this vessel. fuli of elear 
water, be plaeed in a spot where the sun shines, and let the vessel be set 
up in sueh a manner that the surfaee of the circumferenee of the vessel be 
paralleI to the horizon; this ean indeed be earefully inferred from this eon- 
sideration, [namely] if the water's surface be paralleI to the periphery of the 
vessel. 
Then let the [refraction] instrument be plaeed into this vessel in sueh 
a manner that the little fins existing over the extremities of the ruler 
be set over' the edgCl of the vessel on each side. Then. therefore, the 
middle of the instrument with the entire ruler will be inside the vessel. 
After that let water be taken away till the water's surfaee would eut the 
eentre or the instrument 2, and let the instrument be turned in the eireuit 
of the vase till the edges above the water overshadow the others under 
the water; and then, while the ruler is restrained with one of the 
hands, let the instrument be turned with the other hand in a circuit 
about its eentre, till the sun's light may pass through aperture LM N [Fig 
I]. which is in the edge of the instrument. and the aperture of the 
square plate, and come to the water's surfaee, for light passing through 
a round aperture is always widened. by the 36th [prop.] of this [book]. 
AIso let the instrument be plaeed in su ch a manner that the light 
falling over the rIale of the seeond arerture. whieh is XYZ. have an equal 
position 3 . And then. having withdrawn [his] hands from the instrument, 
let the experimenter inspect the bottom of the water from the side of the 
quarter of the instrument the edge of which is cut off, which is AD. 
according to all positions and manners in which he had previously beholden 
the coin. And he wił1 diseover light passing through from the two aper- 
tures, over the surface of the other edge [of the instrument] which is 
inside the water. and [that] light [will be situated] between the two extreme 
eireles of the. three eireles drawn parallelwise. perhaps [Ioealizable only] 
by adding somewhat to the distance of 1hose cireles. but (this] addition 
WIIl he equal on hoth sides of 1he eirc1es. From whieh it is elear that 
the middle point of this light falls in some point of the circumference 
of the middle of those three cireles. say. in point P4. 
Next having brought into e10se eontaet a smali iron- or wooden-jt1eedle 
with the interIor slde or the aperture of the edge of Uie instrument, so 
that it would pass diametrieally through the middle of the aperture, thereupon 
the shadow of the needle will appear to the inspecting eye as before, 
in the middle of the opposite light, by the I I th [prop.] of this [book], 
dividing it [also] equally. 
Next let the needle be withdrawn till its sharpened point be in the 
middle of the aperture and. [in this ease]. the shadow of the needle's 


6 - Witclonis Pcrspcctivac... 


......
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82 


extremity will be in the middle of the light which is in the surface of 
the water and of that which is inside the water. And generally, according 
to the ratio by which the needle will have cut off the periphery of the 
aperture (as a chord) by the very same ratio the shadow of the needle 
will cut off the periphery of the existing light in the surface of the water 
or under the water, while, when the needle will have been entirely withdrawn, 
the light will have returned [in its entirety]. 
It is plain, therefore, from these [considerations] that the point which 
is in the middle of the existing Iight inside the water 5 issues from the middle 
point of the existing light in the surface of the water 6 , and that the middle 
point of this light issues from the light which is in the centre of the 
upper aperture 7 . ConsequentIy the light which comes to the centre of the 
Iight existing in the surface of the water is extended in a straight line 
passing through the two points M and Y which are the centres of both 
apertures. And this line is in the surface of the middle of the three circles 
and is a part of the diameter of that circle, which is M P, since it is 
paralleI to the diameter of the circle present in the base of the instrument, 
which is FEG. Therefore the point which is in the middle of the light 
existihg in the surface of the water is [also] in the surface of this middle 
circle. But point P too, [which is] in the middle of the existing light 
inside the water, is in the circumference of the middle circle. Consequently 
these two points will be in the surface of the middle circle. Therefore 
that entire line will also be in the surface of the middle circle, by XI. 1. 8 
But if the light which is in the surface of the water will not have 
been manifest, let the smaller ruler be dispatched into the water, and 
let its surface, in which a line was drawn dividing the surface of its 
width equally, be applied to the surface of the water, in such a way that 
it become one surface with it, and let its other surface be applied to 
the surface of the base of the instrument 9. 
It is obvious, therefore, from what precedes, in the first [prop.] of 
this [book], that the line which is in. the surface of the ruler is [also] 
in the surface of the middle circle, passing through M and Y, the centres 
of the two apertures; and [in this way] the light which is in the water's 
surface will [also] appear over tłie surface of the ruler, and the middle 
of that light [will be] over the line which is in the middle of the ruleT. 
And should the needle have been placed over the middle of the hi8her 
aperture, the line which is in the middle of the ruler will have been 
shaded. And should the sharpened point of the needle be placed over 
the centre of the aperture, the shadow of the needle's sharpened point 
will fali in the middle of the light which is over the ruler, and when 
the needle is removed, Iight returns. In this way, therefore, that light 
falling over the surface of the water will appear with a manifest presence, 
and it will become obvious that. light incident to the centre of the higher 
aperture is itself laying alongside the line pass ing through the centres of 


......
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83 


the two apertures. And since the water's surface passes through the centre 
of the instrument. and the ruler's (upper) surface is one with the water's 
surface. therefore the ruler's surface will pass through the instrument's 
center. Hence the remoteness of the light's centre from the instrument's 
centre will be equal to half the width of the ruler, which [width) is equal 
to the perpendicular falling from the centre of the aperture to the surface 
of the instrument's base. 
Therefore the centre of the light which is in the surface of the ruler 
or of the water will be the centre of the middle circle. Let therefore 
the ruler he turned till its acute angle may pass through the instrument's 
centre 10 and let the lower part of the line bisecting that angle be in the 
centre of the light which is inside the water. ConsequentIy, the upper 
edge of the ruler II will pass through the centre of the middle circle. 
Hence a point of the line of the upper surface of the ruler 12 which is 
in the water's surface is the centre of the middle circle and of the light 
which is in the water's surface, and that line will be the semidiameter 
of the middle circle. 
Now let a long needle be immersed in water in such ą manner that 
its extremity be in the point of the angle of the ruler; and so the needle's 
shadow will cut the light which is inside the water. and the shadow of 
the needle's point will be at the end of the ruler, which is in the middle 
of the light. And if. while keeping the needle's point fixed, the needle should 
be moved. the needle's shadow will change position to various part s of 
the light. while the point's shadow will not have been moved from the 
middle of the light. When the needle was totally withdrawn. however, 
total light would return. Indeed the same happens when the needle's point 
should have been placed in any point of the line which is in the surface 
of the ruler. 
From which it is cle ar that the light existing in any point of the 
light within the water proceeds from a point corresponding to it in the 
light which is in the water's surface, and that from the middle point 
of the light which is above the water to the middle point of the Iight 
inside the water, a [Iuminous) ray is extended according to the straight 
line which is the middle of the ruler. From which it is clear that the 
passage of light through the body of water is by straight lines, by XI. I. 
And this is what we intended to show experimentally about the intended 
proposition. 


(Proposition)43. The refraction of oblique rays in.a second transparent 
medium which is denser that the first transparent (medium) is performed 
from the anterior surface of the second transparent (medium) toward the 
perpendicular (issuing from the point of refraction) to the surface of 
the second body. 
This proposed theorem can also be shown experimentally (to be true). 


--
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84 


After the upper aperture of the same instrument has been set obliquely 
opposite the same solar body, so that the [luminous] ray be obliquely 
incident to the edge of the instrument [that is itselt] opposed to the aperture. 
and after the centre of the light which is inside the water has been 
throughly examined in the same manner as in the previous [prop.]. let 
that [centre] be marked in the surface of the same instrument by means 
of a dent [made with] hardene
 iron; and that centre will not be found 
on line GK [Fig. I], [which is] erected perpendicularly over extremity G 
of the diameter, opposed to line FH, in which there is the aperture of 
the edge of the instrument, but will deviate from that line toward the 
side in which the sun iso And between this centre of light and point P. 
which is the common difference of line GK, the perpendicular to the end 
of the instrument's diameter. and of the circumference of the middle cirele 
passing through M and Y. the centres of the apertures. there will be 
a perceptible distance. 
And 50 let the ruler be immersed in water, and let it be applied 
to ihe surface of the plate, so that the broader end of the ruler be over 
the centre of the plate; and let the ruler be moved until its edge be 
perpendicular to the water's surface according to the senses. And so the 
centre of the light which is inside the water will be between the edge 
of the ruler and line GK, the perpendicular to FG. the diameter of the 
instrument's base. It is elear. then. from this that the refraction takes 
place toward the si de of the perpendicular issuing forth from the re- 
fraction spot perpendicularly over the water's surface. 
And so this having been continued. let a fixed sign be marked. by 
means of a puncture achieved with hard iron, in the circumference of 
the middle cirele of the three marked cireles, over the extreme point of 
the perpendicular issuing forth from the centre of the same cirele, [and 
which is drawn] perpendicularly to the water's surface. 
And since it became elear by the previous [prop.] that. when the instru- 
ment is directly opposite to the sun and the ray of the sun lis] per- 
pendicularly incident to it, the light which reaches the center of the light 
which is inside the water is the light extended according to the straightness 
of the line joining the two centres of the apertures. which line comes 
to the centre of the middle cirele, [which is] paralleI to the surface of 
the base of the instrument and is its diameter, [therefore] if this line 
were imagined to be extended according to [its] straightness inside the 
water, till it would reach the edge of the instrument, then it would be 
drawn in its entirety paralleI to the instrument's diameter and would come 
to line GK. the perpendicular to diameter FG in the interior side of 
the instrument's edge. 
And since the center of the light which is now inside the water, is 
not on that perpendicular line produced in the instrument's edge, it is 
elear, then, that the light extended from the middle of the light which 


.....
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85 


is in the water's surface is not extended to the middle of the light 
which is inside the water according to the straightness of the line passing 
through the centres of both apertures, but that it is refracted [away] from 
it. Indeed it was shown, by the first [prop.] of this [book]. that this 
ligh1 is extended straight from the middle of the light which is in the 
water's surface to the middle of the light which is inside the water. There- 
fore the refraction of this light takes place at the water s surface. which 
is the proposed lthing]. 


[Proposition]44. A [Iuminous] ray larriving] perpendicularly incident to 
the surface of the opposite body from the centre of the luminous body 
penetrates the second transparent medium lwhich is] rarer than the first 
without being refracted. 
The proposed theorem can be similarly shown by instrumental experience. 
Indeed let there be taken [some] pieces, cubical in shape. of c1ear glass 
or crystal. the length of which lis] double the diameter of the aperture 
of the instrumenł's edge. and let the pIane surfaces of the same 'pieces] 
be made equal and paranel, and let their sides be straight and let them 
be thoroughly polished. Next let a straight line be marked in the middle 
of the instrumenł's base. by the carving of a hard iron. passing through 
the centre of the same, which is E [Fig. I]. [and] perpendicularly [erected] 
to the same diameter. which is FG, over the extremities of which are 
drawn in the edge of the instrument two perpendiculars. FH and GK; 
and let that line be extended on each side of the surface of the circ1e's 
base. and let it be ZEX. 
And so let one of these glasses be placed on the surface of the in- 
strumenł's base. and let one of its sides be applied to the Uust] drawn 
perpendicular, which is ZEX. in such a manner that the middle of the 
side of the glass be exactly over point E, the centre of the instrument. 
And let the whole body of the glass be on the side of the apertures, 
nameły betwen the apertures of the edge and the slate, and the centre 
of the instrument, which is E. Consequent1y. the sa id diameter of the instru- 
ment, which is FG, passes through the middle of the surface of the glass 
[which is] superimposed on the base of the instrument. 
Accordingly, let the glass be applied to the instrumenł's base by means 
of a strong joining with stiff bitumen. in such a way, however. that it 
can be removed when desired. Next let the second [piece ofj glass be 
placed beyond the first. namely on the same side of the apertures. and let one 
of its surfaces be applied to the surface of the first 'piece ofj glass. and 
let it be applied to the instrumenł's base with a stable joining. Next let 
a third [piece ofj glass be applied to the second and let its surface be 
made level with the two surfaces of the sides of the second glass, and let 
it be applied to the instrument's base. and thus let it be done with many 
[other pieces ofj glass tiłł they reach to the other perpendicular [drawn] 


--
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86 


to the surface of the instrument's base I, or nearby, to be sure towards 
point T. 
And so since the [pieces ot] glass were applied to the surface of the 
instrument's base in the previously described manner, it is plain that the 
former diameter of the instrument. which is FG, will pass through the 
middle of all the surfaces of the [pieces ot] glass superimposed on the base 
of the instrument. And the height of all [pieces ot] glass is dciuble the 
diameter of the aperture, while the aperture's diameter is equal to per- 
pendicular M F2, issuing forth from the aperture's centre, to the base surface 
of the instrument and to its diameter FG. Hence every one of the per- 
pendiculars issuing forth from the centres of the [base) surfaces of the 
Ipieces ot] glass, [which are) perpendicular to the diameter of the instru- 
ment's base 3 , is equal to line MF, that is to the perpendicular issuing 
from the aperture's center to the surface of the instrument's base. Therefore 
the line that passes through the centres of bot h apertures will [also) pass 
through the centres of the glass surfaces Iwhich are) perpendicular to the 
base surface of the instrument 4. 
Now let the slender ruler be taken, the shape of which we set out 
before and let it be raised over the edge of the instrument [and placed) 
in the surface of the instrument's base, and let the surface of the ruler. 
in which a line was drawn. be placed on the [lateral) side of the first 
[piece ot] glass which is over E, the centre of the instrument's base. And let 
the ruler be placed near to the glass, and let it be applied so that the 
line which is in the surface of the ruler be [also) in the surface of the 
middle circle and Iso that] the straight line passing through the centres of 
hoth apertures and through the centres of the surfaces of the Ipiel:es ot] glass 
will cut perpendicularly the width line of the ruler. and Ithe ruler] will 
proceed to point GS. 
Next, then, let the instrument be placed in the previously described 
vessel, empty of water, and let the vessel be set in the sun, directly 
opposite to the centre of the sun, so that it may receive the perpendicular 
ray. Indeed this can be done if the instrument be moved till the sun's 
light would pass through both apertures and equally distributed light be 
made [to appear) at the second aperture. And let the opposite surface 
of the ruler be facing the glass, and [thus) the light issuing. forth from 
both apertures of the same instrument will be seen extended over the 
surface of the same ruler, and that opacity which surrounds the light in 
the surface of the ruler will be darkened by 
he shadow of the instrument's 
edge, .and the centre of the same facing [Iight) image will be on the line 
which is in the ruler's surface 6 . 
Next let a slender needle be placed over the upper aperture so that 
the needle's extremity be perpendicular to the aperture's centre, and then 
the shadow of the needle's extremity will fali over the light's centre in 
the line which is in the surface of the ruler. And so, then. Jet the point of that 


......
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87 


shadow be marked delicately with inko and let the needle be removed from the 
upper aperture, and let its extremity be set over the centre 'of the lower 
aperture; and the shadow of the needle's extremity will once more fali 
over the point marked in the ruler's surface, while, when the needle has 
been removed the light returns. From which it is elear that the light 
which is over the point that is in the ruler's surface passes through the 
centres of both apertures. 
Next let a black sign be marked with ink in the point [which is] 
in the middle of the surface of the glass. on the side of the ruler- 
indeed that point can be found, by the 40th [prop.] of the fi.rst [book] 
of this [treatiseJ7, because that point is the common intersection of the 
two diagonals of the surface of the glass - and then, looking at the light 
which is over the ruler 8 , [the observer] will find the shadow of the point 
which is in the middle of the glass over the [corresponding] point which 
is in the surface of the ruleT. It is therefQre elear from this that the 
light which passes through the centres of both apertures passes [also) 
through the point which is in the middle of the glass. 
Next let the first Ipiece ot] glass, which is over the instrument's centre, 
point E, be pulled away. And let the middle point be marked in the 
surface of the second [piece ot] glass as was previously done in the surface 
of the first [piece ot] glass, and let the instrument be put together aga in, 
and let it be moved till the light may pass through both apertures; and 
the light passing through the centres of both apertures will reach the 
centre of the light which is in the surface of the ruler 9 . Accordingly it 
is elear from this that the light passing through the centres of both 
apertures passes lalso] through the point which is in the middle of the 
surface of the second [piece ot] glass, and that the light that passes through 
the centres of both apertures in the first experiment passes also through 
the point which is in the middle of the second [piece ot] glass. 
And so let the second [piece ot] glass be withdrawn and let the third 
be exposed [and inked as before], and thus [Iet it be done] wit h the others, 
until the last. And it is generally elear that light, passing through the 
centres of both apertures [and] reaching to the surface of the ruler, passes 
also through the centres of the surfaces of all the [pieces ot] glass placed 
over the surface of the slate, and that all centres of all glass surfaces 
are in one straight line joining the centres of bot h apertures. Accordingly, 
the light passing through the centres of the apertures, bot h in the body 
of the glass as well as outside [that] body, in the air, is extended according 
to the straight line joining the centres of both apertures, and that is line 
M p [which is] perpendicular to the surfaces of all the [pieces ot] glass 
[which are] opposite to the aperture, by XI. 14 10 . 
Indeed that line M P is paralleI to line FG, the diameter of the slate, 
which is perpendicular to the surface of the [pieces ot] glass, since it 
is perpendicular to the common section of the surface of the glass and 


......
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88 


of the surface of the plate. And if, while all [pieces ot] glass. or any 
of the same, are disposed in the previous manner over the bottom of the 
instrument, water is poured into the vessel up to the concavity of the 
surface of the glass 1 I, the same thing as before will happen. because the 
perpendicular ray penetrates always without being refracted. 
Likewise nobody should think that the straightness of perpendicular 
rays is assisted by the cubic shape of the [pieces ot] glass. Let half 
a sphere of elear glass or erystal be taken. the semidiameter of which 
is smaller than the distance between point C and the centre of the plate, 
which is point E, and let the centre of its base be found. over which 
let a slender line be marked with inko Next let a line be marked otf 
from this line, on the si de of the center of the sphere. equal to line LN. 
the diameter of the aperture in the instrument's edge. Hence this line 
will be equal to the line which is between M, the centre of the aperture 
in the instrument's edge. and the surface of the plate 12. 
Next let a perpendicular [to the diameter] be drawn at the end of 
this line marked otf from the diameter, on each side of the spherical 
surface, which can be done by I. II, and let the glass sphere be cut 
along that line. And let the surface of the cut glass be levelled tilI it 
be quite even, and let it be erected perpendicularly to the pIane surface 
of the hemisphere, which can be measured by [means ot] a right angle [made 
ot] copper; therefore the common section of this erected surface and of 
the surface of the base of the sphere will be a straight line, to which 
the line previously produced from the centre of the sphere will be per- 
pendicular; it will also. therefore. be perpendicular to the erected surface. 
Next let a mark be made with ink in the middle of that line which 
is the common section; then let this exceedingly well polished glass on 
its cut surface be placed over the surface of the instrumenł's plate, in such 
a way that its convexity faces the apertures, and let the middle of the 
line which is the common section of both pIane surfaces of the glass be 
applied to the centre of the plate, and let the glass be fixed over the 
plate so that it would not falI otf. Next let the slender ruler be placed 
over the surface of the instrument's plate, as [was done] in the experiment 
with the cubic [pieces ot] glass, so that the surface of the ruler in which the 
straight latitude line lies be on the side of the glass and near to it 13. 
Next let the instrument be placed in the previously described vessel, and 
let the vessel be set in the sun, empty of water, and let the instrument be 
moved until the sun's light may pass [through] both apertures; and the 
light will falI over the surface of the ruler. Next let the extremity of 
a needle or of an iron stilus be placed over the centre of the upper aperture; 
and the shadow of the needle's point will falI over the centre of the 
light, while, when the stilus has been removed, the light return s to its place. 
And the very same thing happens when the needle's point is being placed 
over the center of the second aperture.
>>>
89 


Next Iet the needle's point be placed over the glass sphere 14. and the 
shadow of the needle's point will fali over the centre of the light. From 
which it is elear that the light passing through the centres of bot h apertures 
passes also through the centre of the glass sphere 15 and through the middle 
of the surface of the light. which is in the convexity of the glass. It is also 
elear from these (considerations] that the light passing through the body of 
glass extends itself according to the straightness of the line passing through 
the centres of bot h apertures and that line is (also] the semidiameter of 
the sphere. 
But the perpendicular issuing from the centre of the base of the glass 
to the plate is equal to the diameter of the aperture and to the line 
issuing from the centre of the aperture perpendicularly to the surface of 
the plate. And since these two perpendiculars fali over the diameter of 
the plate. it is plain that the line passing through the centres of bot h 
apertures, when extended straightforwardly. comes to the centre of the glass 
sphere. Therefore, the diameter of th is glass sphere lies in that line; it is, 
therefore, perpendicular to the surface of that sphere. by the 72nd Iprop.) 
of the first [book) of this (treatise) Iti. Indeed since it passes through the 
centre of the sphere, it is elear that the same is perpendicular to the 
convex surface of the sphere. as became elear above in Ithe case ot] 
cubicallv rshaped) glasses. 
And so, let the slender ruler applied to the surface of the plate be 
removed, and let the instrument be placed again in the vessel as before, 
and let it be moved until (Iight) may pass through both apertures. And 
let the (presence ot] light be ascertained over the edge of the instrument. 
and let the (Iocation of the) center of the light be ascertained in point P, 
which is common to the circumference of the middle cirele and to line CK, 
the perpendicular [drawn) in the edge of the instrument; this is in the 
extremity of the diameter of the middle cirele. which is M P. passing through 
the centres M and Y of both apertures. From which it is elear that 
light passing through the body of the glass and reaching its center and 
[then) emerging in the body of [surrounding] air is a continuation of the 
[same] line which was extended through the body of the glass. 
Indeed since the straight line passing through the centres of bot h 
apertures is perpendicular to the surface of the glass, it is c1ear that the 
same is [also) necessarily perpendicular to the surface of the air [that is) 
tangent to the surface of the glass. Accordingly, if water were poured 
into the vessel, the glass remaining in its position, till the water would 
overflow the center of the glass, the centre of the light would still be 
found on the extremity of the diameter of the middle circ1e; and if the 
glass sphere were turned around so that its convexity were situated at 
the second aperture and the pIane surface at the instrumenł's centre, that 
is point E, whether water would be poured over [it) or not, yet all other 
[phenomena) that happened in the previous situation would happen [aga in],
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90 


since the ray passing through the centres of both apertures would always 
pass through the sphere's centre also. 
From aU these [considerations] with cubical and spherical glasses. it 
is elear that. irrespective of the second transparent medium being den ser 
or rarer [than the first]. as long as tJ1e line along which the Iluminous] 
ray is extended remains perpendicular to the surface of the second (trans- 
parent] body, that light is extended in the second body along the [very 
same] straightness of the line through which it was extended in the first 
body. The proposed [thing] is therefore elear; indeed a body of glass is 
of a denser transparency than the hody of air and even than the hody 
of water. 


[Proposition]45. The refraction of obliquely incident rays is made in a second 
transparent medium [that is] rarer than a first transparent (medium start ing) 
from the posterior surface of the second transparent [medium and] away 
from the perpendicular issuing from the point of refraction to the surface 
of the second body. 
What is now proposed here is to be shown by instrumental experience 
in conformity with the preceding [prop.]. Indeed let that spherical glass 
be taken which was already put to use in the immediately preceding 
theorem, and let it be placed over the instrument's plate so that a pIane 
surface of the same Iglass] may face the apertures, and that the middle 
of the straight line that lies in the same [surface] I be over the centre 
of the plate, and the line which is the common section of the pIane 
surfaces of the glass may faU obliquely over the diameter of the plate, 
with any obliquity whatever 2 . It is plain, therefore, that the line passing 
through the centres of both apertures is oblique to the pIane surface of 
the glass.
. 
And so let the glass be joined firmly to the plate of the instrument 
in this position. and let the instrument be set in the vessel and the vessel 
in the sun. and let the instrument be moved until the light may pass 
through the two apertures. And the light will faU in the interior of the 
instrument's edge, and the light's centre will. be on the circumference of 
the middle cirele. but outside that point P [Fig. I] which is com mon 
to the circumference of the middle cirele and to the line standing in the 
edge of the instrument, which is GK. and its deviation [from P] will be 
toward the side where the sun is; hence it will be to the side of the 
perpendicular. issuing from the spot of refraction, to the spherical surface 
of the glass. 
And since this light is extended in air according to the straightness 
of the line passing through the centres of both apertures, as is elear by 
.he first (prop.] of this (book]. and [since, furthermore]. this line, in this 
position. reaches to the centre of the glass sphere and is oblique to the 
pIane surface of the sphere. it is plain. therefore. that the boundary of
>>>
1 


91 


that light's extension is in the centre of the glass. Hence light is extended 
in the body of the glass along the straigh1 line issuing forth from the 
centre of the sphere to the circumference. which line. because it is a diameter, 
is obviously. by the 72nd Iprop.] of the first Ibook] 4 of this [treatise], 
perpendicular to the spherical surface of the glass and. consequently, lalso] 
to the concave surface of the air containing the sphere of glass. Therefore 
it is not refracted in the second air or. likewise, in the first. but neither 
is it reflected in the body of the glass. nor in the convexity of the same; 
therefore it is refracted at the centre of the glass. since it was oblique 
to its pIane surface. in which the centre of the gIass lies. 
Accordingly from these experiments IfolIows] that what was shown above 
is also plain lin this case too], namely that when Iight has been extended 
in the more subtIe body, obIiquely incident to the surface of the denser 
body, it will be refracted by it, and its refraction will be toward the 
side of the perpendicular issuing from the spot of refraction to the water's 
surface. wlthout the retractlon reaching the perpendicular. 
Now if the glass were placed contrariwise, namely so that its convex 
spherical surface may face the upper aperture, and [it] the middle point of 
the line that is the common section of the piane surfaces. which Ipoint] 
is [under] the centre of the glass sphere, may be Ilocated] over the in- 
strument's centre, and [it] this line may falI obliquely to the diameter 
of the plate 5 , Ithen] let a line be drawn in the same surface of the plate. 
from the pIate's centre. Iwhich line is] perpendicular to the line which 
is the common section of those piane surfaces, [and] which will necessarily 
be perpendicular to the pIane surface of the glass erected [itself perpendicul- 
arfy] to the pla1e's surface 6 . 
And so let the instrument be placed in the vessel without water and 
let it be moved until the light may pass through the two apertures; 
and the light's centre will falI on the circumference of the middle circle, 
away from point P, which is common to the middle circle and to line GK, 
the perpendicular to the surface of the plate drawn in the edge of the 
instrument. which point P, [moreover], is the extremity of the diameter M P 
of the middle circle. And the deviation of the light will be toward the 
opposite side to that in which lies the perpendicular to the piane surface 
of the glass, drawn from the place of refraction. Indeed this light is extended 
in the glass according to the straightness of the line passing through the 
centres of both apertures, since that line, as it passes through the centre 
of the glass sphere, lies in that diameter of the glass sphere. Accordingly 
the refraction of light is made at the centre of the glass sphere, because 
the light pass ing through the centres of both apertures is made oblique 
to the pIane surface of the glass and to the surface of the air touching 
the gIass. 
And if water is poured into the vessel until it rises above the centre 
of the instrument, the centre of the light will still falI in the circumeference 


........ 


-"
>>>
92 


of the middle circle. outside the extremity of its diameter. obliquely [but] 
to the other side. [of that] over which the perpendicular falls. And since 
air is subtler than water. and water subtler than glass. the distance of I 
the light's centre from the extremity of the diameter of the middle circle 
will be made greater in air than in water. But if the glass is placed 
differently in the surface of the plate, namely so that the line which is . 
the common section of the two pIane surfaces of the same glass be over 
the line cutting perpendicularly the diameter of the plate 7 , without its middle 
point. which is (under] the centre of the glass sphere, being [however] 
over the centre of the plate, and [it] the convexity of the glass is turned 
toward the apertures. and the slender ruler is fixed over the surface of 
the plate, Ibeing] erected over its edge. and [it] its surface. in which there 
lies the [previously drawn] line, be on the side of the glass. and the boundary 
of the ruler cut the diameter of the plate perpendicularly, [then] it is 
plain that the line passing through the centres of both apertures does 
not pass through the centre of the sphere but through some other point 
of the pIane surface of that same glass and [that] it will be oblique to 
the spherical surface. by the 72nd [prop.] of the first [book] of this 
[treatise)8. 
And so let the instrument be placed inside the vesseI. and the vessel 
in the sun. and let the instrument be moved until the Iight may pass 
through the centres of both apertures. And the light will not fali directly 
over the surface of the ruler. nor will the centre of the light fali on 
the line which is in the surface of the ruler, but it will turn away obli- 
quely. outside the line which passes through the centres of both apertures, 
toward the side in which the centre of the g1ass lies, that is, toward 
the opposite side of the perpendicular issuing from the place of refraction 
perpendicularly to the spherical surface of the glass. And the line passing 
through the centres of both apertures will be perpendicular to the pIane 
surface of the glass, by XI. 81), because that line is paralleI to line FG, 
the diameter of the plate, which is, by hypothesis, perpendicular to the 
pIane surface of the glass. 
Hence if light were to pass through the centres of both apertures and 
extend rectilinearly to the pIane surface of the glass, it is plain that in 
such a case it would [also] extend rectilinearly in the air. But since the 
centre of the light which is in the ruler does not fali on the extension 
of this line, it is clear that the light is not extended rectilinearly to the 
pIane surface of the glass; it is therefore refracted light, but it is not 
being refracted in the air, nor in the body of the g1ass. Accordingly 
it is refracted at the spherical surface of the glass; indeed it is obliquely 
incident to the spherical surface, because the line passing through the 
centres of both apertures does not pass through the centre of the glass, 
and this light, issuing forth from the pIane surface of the glass, is re- 
fracted more because it is obliquely incident to the air. 


,......
>>>
93 


But if the glass should be arranged contrariwise. so that its piane surface 
be opposed to the first aperture in such a way that the common section 10 
be over the line cutting perpendicularly the diameter of the plate, and 
the middle point of that line be away from the centre of the plate, then, 
necessarily, the line pass ing through both apertures does not pass through 
the centre of the glass but through some other point of that piane surface 
and is perpendicular to that surface. Accordingly. let the instrument be 
moved in the sun until the light may pass through both apertures. And 
the centre of the light. which fans in the interior side of the edge of 
the same instrument. in the periphery of the middle circle, will fall outside 
point P, which is the extremity of the diameter of the middle circle, that 
is line M P, but will deviate towards the side in which lies the centre 
of the glass sphere. 
And the .line which. in (our] imagination, issues forth from the centre 
of this sphere to the spot of refraction is perpendicular to the surface 
of this sphere; it is, therefore, [also] perpendicular to the surface of the 
air containing the surface of the glass sphere. And so this refraction is 
toward the opposite side to that in which is located the perpendicular 
issuing from the spot of refraction to the surface of the air containing 
the sphere. In fact the light passing through the centers of both apertures 
traverses also straightly the body of the glass, since it is perpendicular 
to the piane surface of the glass, but it is not perpendicular to the convex 
surface. because it does not pass through the centre of the sphere. Hence 
this light is also not perpendicular to the surface of the air containing 
the convexity of the glass. And since this light is found to be refracted, 
it is being refracted. therefore, at the convex surface of the glass sphere. 
Now, then. should water be poured into the vessel below the center 
of the plate, the light will also be found refracted towards the side in 
which the centre of the glass is located. But this is towards the opposite 
side to that in which is located the perpendicular issuing from the spot 
of refraction. which is extended in the body of air, (being] perpendicular 
to the concave surface of the same air (that is] touching the convex 
surface of the glass. And this is the proposed (thing]. 


lProposition]46. It is necessary that every incident and refracted ray be 
situated in the same piane surface. 
But even that which is now proposed can be shown experimentally. 
Indeed, all having been ordered as in the 43rd (prop.] of this (book]. 
since the light that is incident to the centre of the light which is in 
the water's surface, which is the centre of the middle circle, and (that 
comes] from the centre of the existing light above the water's surface 
to meet the centre of the existing light inside the water, which is in 
the circumference of the middle circle. passes through the centres of both 
apertures. which are similarly in the surface of the middJe circle. it is 


-.......
>>>
94 


plain that the line according to which light is com ing to the water's surface 
through the medium of air. and Ithat] according to which it is refracted 
in the medium of water are in the same surface. since each of the same 
lJines] is in the surface of the middle cirele of the three alIotted 
cireles. Indeed this refraction takes place in the solar ray when the solar 
ray passing through the centres of the apertures was oblique to the water's 
surface. not when it was perpendicular. 
And by reason of the obliquity of the position of the instrument with 
respect to the centre of the sphere of water ł. this radial line may never 
be made perpendicular to the water's surface. unless the sun were Iilluminating] 
perpendicularly above the zenith of the head; when the existing sun is in 
fact beyond or below the zenith of the head. thls test is sufficiently evident 
each time. Therefore what was proposed is elear. And this surface lin 
which the two rays lie] we calI the surface of refraction. 
Accordingly it is elear from all these five preceding propositions that 
any light traverses any 1ransparent bodies according to straight lines. And 
as long as the lincident] lines are perpendicular to the surfaces of bodies. 
however ditferent they may be in their transparency. lJight] is always extended 
according to the straightness of the same lincident] line and is not refracted. 
In fact in bodies of diverse transparency. all light obIiquely incident to 
the surface of the second body is refracted along straight lines ditferent 
than those according to which it was incident in the first body; which 
lines. moreover, will always be in the same surface. which can be imagined 
to cut any of those Itwo] bodies; and this surface turns out to be. by 
examining the instrument. the middle cirele of the three cireles marked 
in the interior side of the instrument's edge. the diameter of which is 
line MP IFig. I]. 
In fact should the oblique light have exited from the subtIer body 
to the gros ser, it would have been refracted toward the side of the per- 
pendicular issuing forth from the refraction spot. which is perpendicular 
to the surface of the second denser body. And should the oblique light 
have exited from the denser body to the subtler. it would have been 
refracted toward the opposite side of the perpendicular (drawn in the pre- 
vious way) to the surface of the second body, Le., the subtler. 


[proposition]47. IAlthough] the perpendicular ray penetrating any trans- 
parent body lis not refracted], a ray obliquely incident to a second denser 
transparent medium is refracted towards the perpendicular drawn from the 
point of incidence to the surface of the second transparent [medium]; 
and in a second Ibut] rarer transparent medium, it is refracted away from 
the same [perpendicuJar]. 
That which was proved thus far instrumentally by means of specific 
triaJs we intend to enhance by a natur aJ demonstration. Indeed all naturaJ 
motions which are made aJong perpendicuJar lines are stronger, because
>>>
95 


they are intensified by the universal celestial virtue flowing into every body 
lying beneath according to the shortest line I. For the impulses of pro- 
jections made perpendicularly are stronger than those made obliquely. And 
simiłarly percussions which are made perpendicularly are stronger than any 
- oblique percussions. and among all the oblique ones. those are stronger 
which come cłoser to perpendicularity. 
Accordingly - since the density of any body prevents the passage of Iight. 
it is necessary that light be imagined as being repelled in lits] passage 
by the resistance of the dense body. and more Irepelled] by the resistance 
of the denser body. And by this resistance of the passive quality. that 
is density. to the active quality. that is light. we understand a certain 
mode of motion of light through the medium of resisting bodies. which 
(bodies] are more or less capable of Ireceiving) the luminous impression, 
(and] not that in the change of place of the same light there is a certain 
(real] motion. as is c1ear by the second Iprop.] of this (bookP. but Irather] 
that light compresses or scatters itself more in the same instant. according 
to the diversity of the media. And this lis what] we call here the motion 
of the same Iight. 
Accordingly any light traversing a transparent body traverses it with 
an exceedingly fast and imperceptible motion. in such a manner. moreover. 
that the motion is performed faster through more transparent Imedia] than 
through less transparent (ones] 3. Indeed any transparent body resists more or 
less the penetration of Iight according to its Ibeing] a lesser or greater 
participant in transparency; for the grossness of body is always resisting 
the penetration of light. Therefore should light have traversed a certain 
transparent body obliquely and should it have encountered another body of 
grosser transparency, then the grosser body would resist light more vehemently 
than the first rarer body resisted lit]; it iso therefore. necessary that. because 
of the resistance of that denser body, the motion of the light be chan{!ed. 
And if the resistance were strong. then that motion would be refracted 
in the opposite direction 4 for. as it does not resist strongly Ideviation]. 
then Iight would not (be allowed to] proceed in the direction in which 
it was (originally] moved. 
If indeed the resistance were weak, because of the greater rarity of 
the more transparent body. then the incident light would not be refracted 
in the opposite direction, nor could it proceed (however] along that (same] 
line in which it started. but would be changed in direction while. if it 
were perpendicularly incident to any transparent bodies, whatever their 
diverse transparency, it would not be changed, but would penetrate Ithem] 
all directly, because the perpendicular is stronger than all (others], and the 
oblique [lines which are] cłoser to the perpendicular are stronger than aU 
more remote oblique [lines]. 
Accordingly, when light is obIiquely incident to the denser transparent 
body, it is extended [after refraction] along a straight line aproaching the
>>>
96 


perpendicular issuing from the point in which light encounters the surface 
of the dense 
ransparent body (and] produced to the surface of the denser 
body. because motion is easier along the perpendicular line. Therefore if 
a ray of Iight would be incident along the perpendicular line, it would 
pass on straight because of the strength of motion over the perpendicular 
and if the ray would fali obliquely. then it could not pass through Idirectly] 
because of the weakness of motion over oblique lines. Therefore it happens 
that it is bent towards the direction through which the passage is easier 
(rather] than through that direction along which it was ,moved along the 
incident line 5 . Indeed (that] motion is easier and more assisted by the 
celestial influence which [takes place] along the perpendicular line, moreover 
Ithat] which is c10ser to the perpendicular is of easier passage than the 
more remote from it. 
Accordingly let it be that rays fali from point A of the luminous 
body 
Flg. 281". some of which Ipass] throu!!h medium AB. to the surface 


A 


E 


B 


K Q LG F 


[Fig. 28] 


of another transparent body in which line BCDE lies, and let line BF 
be the depth line of that body, and let line AB be perpendicular to that 
surface. 
And so ił is plain.. according to the previous reasoning concerning 
the strength of the perpendiculars, and by (various] instrumental triaIs, 
Ispecifically] by the 42nd and 44th [props.] of this [book]. that the ray
>>>
97 


[which is]. incidcnt pcrpcndicularly along linc AB pcnctratcs thc wholc body 
BEF. Furthcrmorc. thc ray incident along linc AC, werc i1 to pass through 
hod} REF dirceII} Iwithollt deviationJ, thcn thcre wOllld n01 hc lany] 
divcrsity in thc transparcncy of bodies ABE and BEF. which is against 
thc hypothesis. Accordingly linc AC will not he a countinous line because 
of thc diversity of the resistance. But if it werc movcd freely through 
thc Icss rcsistant nody along hnc AC. it could not hc movcd along thc samc 
linc in a morc or Icss rcsisting body. Thcrcforc, should body BEF bc 
dcnscr than body A BE, it is c1car from thc prcccding that thc passage 
through it is morc difficult. 
Aceordingly if line AC were refracted [away] from the perpendicular 
line drawn from point C to the surfaec of body BC DE, whieh [pcr- 
pendicular] is CG. it would be weakened so that its effect would eome 
to nothing.; thcrefore it would be incident [to BEF] in vain. But nature 
does nothing in vain, as "as postulated at the beginning lof this bookF. 
Therefore line AC, as is indeed obvious experimentally by the 43rd [prop.] 
of this [book]. is nceessarily refraeted toward the side of pcrpcndieular 
CG. so that its action may be fortified. Moreovcr [the case] with the rays 
ineident along lines AD and AE is similar. 
Now if the body in whose surfaee lies line BCDE were of rarer 
transparency than is [that ot] body ABE, still, beeause of the strength 
of 1hc aetion [involvcd]. thc pcrpcndieular ray, whieh is AB. pcnetrates 
without being rcfractcd. whilc thc ray travcrsing thc denser body along 
linc AC, and [which] is incidcnt to thc surfaee of thc rarer body in point C, 
does not eneountcr the [samc] rcsistance as beforc". And sinee thc eharac- 
teristie of the forms is to cxtcnd thcmselves always according to the amplitude 
of matter's cntirc eapaeity [of rceeiving thcm]. it is c1car that ray AC 
does not proeced along line AC, beeau3c in these eircumstanees the arran- 
gcment of thc transparent bodies aeeording to [thcir] "rcsistanee to the 
rceeption of light would be uniform. whieh is against the hypothcsis. 
Thcrcforc ray AC is refraetcd. but not towards pcrpcndieular CG. beeausc 
that refraetion is not madc on aeeount of thc matter's resistance, but [ra- 
ther] on aeeount of the victory of the agcnt's form ovcr a more widely 
distributed matter than was prcviously the ease; whcnee [it follows that] 
the form cxtcnds itself by its propcr virtue from [its] beginning advance 
along linc AC and [thcn] toward the side oppositc to that samc pcr- 
pendieular CG and its paralleI BE. And in like manncr it is [to be eoncluded] 
about all othcr oblique rays, as AD and AE. 
Aceordingly thc motion of thc obliquely incidcnt ray along line AC 
in thc sceond, dcnscr transparent body, namcly BEF. is eomposcd of the 
motion on the side of pcrpcndicular AB, travcrsing body BEF. in which 
thcre is [unimpcdcd] motion. and of the. motion donc along line C B. which 
is pcrpcndieular to line CG. Furthermorc sinec the passagc of thc perpen- 
dieular is thc strongest and easicst of motions. and [sinee] the body's 


7 - Wit.loni. P.rspcctiva.".. 


.......
>>>
98 


density acts in opposition to the end point of motion toward which it 
was directed, line AC will necessarily be moved toward perpendicular CG. 
issuing from point C where ray AC meets the surface of the denser body. 
Furthermore since there is opposition to that [initial] motion, because of 
the density of the medium. and also [since], by the nature of the other 
[component of the] motion, which is along line CB. it is not entirely 
destroyed but only hindered by the resistance of the medium. therefore 
the light will deviate toward point B, always approaching to perpendicular 
ABF8. 
And so in the second tł:ansparent medium [that is] denser than the 
first medium, the refraction of ray AC takes place along line CL, [which 
is] eloser to perpendicu]ar CG, issuing from point C, in which [AC] meets 
the denser body, than line AC, along which [th
 ray] was incident to 
the surface of that body. [and which when] extended beyond point C 
[comes] to point Q: [CL] wilI have been eloser to the same perpendicular 
extended beyon,d point C to point H, such that angle ACH is greater than 
angle LCG: indeed lCL] wilI not meet with perpendicular BF in the direction 
of point F. but [rather] in the direction of point A. by the 2nd [prop.] 
of the first Ibook] of this Itreatise]lJ, because it meets wit h its paralleJ. 
line CG, in point C. 
Now should ray AC emerge from a denser body into a subtler. then. 
since it has [sudden]y] less resistance, its motion will be faster and [its] 
spread greater. And since the resistance of the denser medium impels always 
oblique light so that it may be joined to the perpendicular line drawn 
from the point of incidence to the surface of that body. namely CG. 
it is elear that in the medium of rarer transparency that resistance will 
be smaller than in the [denser] first. Therefore the lighCs motion is made 
toward the side from which the greater motion was repelled by the [greater] 
resistance. Hence light is moved in the rarer transparent body more toward 
the contrary side to the side of the perpendicular, so that angle GCK 
may be greater than angle ACH. Thus the motion of [the ray] of light AC 
in its bending by the second body of rarer transparency than the first is 
always made between lines CG and CE, because, as angle GCE is right, 
angle GCK can never be made right. The proposed [thing] is therefore 
elear. 


I 


[Proposition]48. It is impossible that the refraction ofall the rays incident 
to the pIane surface of a transparent body be made to some unique 
point. 
Indeed since, as follows. from the preceding [props.], in any transparent 
body the refraction is always made either toward the same perpendicular 
drawn from the points of incidence of the rays [com ing] to the surface 
of the transparent body from which the refraction takes place, or away 
from those perpendiculars (however, in fact, this may happen), it is elear,
>>>
99 


since those perpendiculars to the pIane surface are paralIel, by XI. 6 1 , 
that it is not possible that the refraction of all rays incident to that pIane 
surface take place to a single point 2 , whether the refraction be made toward 
the same perpendiculars or away from the same. The proposed [thing] 
is therefore clear. 


[Proposition]49. No ręfraction changes the [mutual] posltlon of the 
parts of the refracted form, but only increases or diminishes [its) shape. 
Indeed since, as is obvious by the 47th [prop.] of this fbook], any re- 
fraction takes place in a second transparent med.ium, and in the rarer 
[medium) away from the perpendicular, while in the denser towards the 
perpendicular, it is plain that, always, the right ray remains to the right 
and the left to the left. And simiIarly about the other differences of 
position. Therefol'e the sites of the part s of the refracted form are not 
changed [mutually], but indeed remain always in [their] own [respective] 
fashion: for when the bending takes place away from the perpendicular, 
the form is increased due to [angle] enlargement; and when the refraction 
takes place toward the perpendicular. [the form) is decreased, because the 
angles containing it are made narrower. The proposed [thing] is, there- 
fore, c1ear. 


[proposition)50. In any simiłar surfaces of equal transparency, the rays 
incident according to equal angles are refracted according to equal angles; 
and if the angles of incidence are greater, "the [corresponding] angles of 
refraction are [also] greater, and if smalIer.. smalIer. 
Indeed either the mode of refraction ensues from the part [played by] 
the [shapes ot] the surfaces of the bodies in which the .refraction takes 
place. (for there is a different refraction made from a spherical surface 
and a different [refraction] from a pIane [one]). or from the role [played 
by] the nature of the [various) transparent [bodies) (for there is a dif- 
ferent J:efraction from a rarer transparent [body and) another from a denser 
[one), as is clear by many propositions of th.is book), or it ensues from 
the role [played by) the angles of incidence; it is [therefore) always clear 
that when the existent angles of incidence are equal l , there is no c10se 
at hand cause for a diversity of the mode of refraction. in accordance 
with the [above) proposed manner. Therefore the refraction will always 
take place [in such a situation) according to equal angles. And this is 
the first proposed [thing). 
And an example of this is [providedJ when radial lines AF. BF. CF. 
DF, EF [coming) from a point F I Fig. 29J. of the luminous body would 
be incident to a spherical transparent body denser than the same air, 
in whose surface there is a circle ABCDE, the centre of which is P; 
furthermore let ray FC be perpendicularly incident [to the circle) and the 
others oblique. Ił is clear. by the 47th [prop.) of this fbook), that all
>>>
100 


x 


F 


rays obliquely incident will be refracted in the surface of that transparent 
body. Therefore let it be [assumedJ, for the sake of the ex ample and 
the brevity of representation and nam ing of the lines, that all those refracted 
rays meet in point G, and let lines be drawn perpendicularly to the sur- 
face of the body, and let them be PDQ and PBR and PAX and PEX2. 
I say that if the angle of incidence, which is FDQ, be equal to angle 
FBR, that [then] angle GDP will be equal to angle GBP, which is obvious 
by the preceding, because of the uniformity of all premised conditions 3 . 
Moreover I say similarly that if angle FDQ be greater than angle FAX, 
that [t hen] angle PDG will be greater than angle PAG. lndeed let there be 
made ahout point A. the end of line XA. an angle equal to angle fDQ. 
by I. 23. which angle is HAX; and let ray HA be refracted in point A and 
let it meet with line FG in point K. and. by the first part of this [prop.], 
angle PAK will be equal to angle PDG. But angle PAK is greater than angle 
PAG: indeed it is not equal. for. [if it were]. then it would follow from the 
preceding that the angles of incidence must be equal 4 , which is against the 
hypothesis. So they are taken to be unequal, but not fso that PAK is] the 
smalIer. because this way refraction would be made irregular. which is against 
the 43rd and 45th [props.] of this [book] 5. It is, therefore, greater. Conse- 
quently angle PDG is also greater than P AG. Furthermore the same can 
be demonstrated easier should angle FEZ be made equal to angle FAX, 
by HI. 8 6 , inasmuch as [this is] possible if arcs AC and CE are taken 
equal: indeed then angIes P AG and PEG will be equal by the preceding.
>>>
101 


In fact angle PDG is smaller than angle PEG. which is the case even 
if the angles of refraction are assumed to be equaI. Indeed concerning 
this matter we are speaking lhere] briefly. because we shall pursue the 
same better in the tenth book of this ltreatise], where it has its proper 
place. The proposed [thing] iso therefore, elear. 


[Proposition]51. To know the size of a given height by [its] shadow. 
the sun being visible. 
Let the given heighl. the size of which we desire to know. be AB. 
the sun heing in sight I Fig. 30]*. If. then. that height is erected over the 


A 


B 


z 


D 


(Fig. 30] 


surface of the horizon, let line BD be drawn in that surface perpendicular 
to the extremity of altitude AB. Le.. B. And let the solar ray [coming] 
to the vertex of AR. which is A. be incident to the same point D, and 
let [that ray] be AD. Therefore. by the I I th lprop.] of this lbook], line 
BD will be the shadow of the same height AB. 
And Jet a known Jine EZ be erected between the shadow BD and 
the ray AD, parallely to altitude AB, as if. lsay,] ZE were a stick of 
known size. Therefore traingle DZE will be, by I. 29 1 , equiangular to 
triangle ABD; hence, by VI. 4 2 as well as by the 9th [prop.] of this 
[book]. the ratio of DZ to ZE will be like lthat of] DB to BA. But the ratio 
of DZ to ZE is known, since, as ZE is assumed known. the line of its 
shadow, which is ZD, can also become known by a slight measurement. 
Therefore. the ratio of DB to BA is known; but DB can become known 
by having been measured. Therefore AB will also be known. which is 
the proposed lthing], as if line AB were. lsay]. the altitude of a certain 
tower or wall, that can be approached so that the distances of the shadows 
are measurable.
>>>
III. ENGLlSH TRANSLATlON OF BOOK THREE 


In the preceding books, we assigned to the beginning the mathematical 
and natural principles by means of which we intended to show, in the 
measure of our possibility, the naturai succession of our [weB argued] 
claims. Indeed want ing to pursue the actions of natural forms in vision 
under [their] triple aspect, namely of that which takes place by simple 
vision, and that which [happens] by reflection, and [finally] that which 
[happens] by refraction, we are pursuing in this third book the mode of 
simple vision and the proper arrangement of the organ of sight. Indeed 
we assume [here] these [principles]. which follow [either from] what was 
shown in other places or [can be seen] as known by themselves. 


[POSTU LATES] 


That sight is not to be completed save only by the arrival of the 
visible form at the soul. Likewise that there are only two [entities which 
are] visible by themselves, namely light and color; for light is seen by 
itself and is itself the hypostasis of color. The other [optical phenomena] 
visible accidentally are 20, namely remoteness, magnitude. position, corporeity, 
shape, continuity, separation or division, number. motion, rest, asperity. 
smoothness, transparency, density, shadow. obscurity, beauty. deformity, 
similitude, and diversity I. Indeed these ar
 not understood by sight alone 
but also by means of other senses. Likewise we postulate that strong 
light damages the eye contemplating [it] for a long time. Likewise that a thing 
of greater size than is the eye can be seen by the eye. Likewise that 
the observed object is seen according to [its] position, shape, and the order 
of its parts. Likewise that the eye can see simultaneously various visible 
objects. Likewise that one (and the same object] is seen simuItaneously 
by both eyes. Likewise that color is not the motive power of sight, save 
by means of an act involving light. Likewise that vision cannot take 
place without contact, nor [for that matter] can any natura' action. Like- 
wise that the visual power is finite and cannot be extended indefinitely.
>>>
103 


[PROPOSITIONS] 


[Proposition] I. When light does not share actually m the visible object, 
it is impossible for it to be seen. 
Indeed those [entities] which .are, as postulated, visible by themselves. 
are light and color. For light is not visible save by itself. And furthermore. 
as light is the hypostasis of colors, it is not possible for colors to be 
seen without light. But the form of color is a weaker form than is the 
form of light, since color is a certain light incorporated in mixed bodies. 
Therefore the eye does not receive the form of co lor of the visible object. 
except by means of light mingled with the form of color. And this is 
why the colors of many things change at the eye, due to the alteration 
of the day's light [fali ing] on the same [objects]. And if color, which 
is visible by itself, is not the motive power of the same sight, except 
by means of an act involving light, it is elear that when light is not 
actually participating in any [potentially] visible [object], it is impossible 
for the same [object] to be seen. The proposed [thing] is, therefore, elear. 


[Proposition]2. It is necessary that straight lines be producible between 
any point of the surface of the visible thing and a certain point of the 
surface of the eye for the thing to be actually seen; from which it is 
elear that vision takes place only when the visible thing is opposite to 
the eye. 
Indeed vision takes place either because rays emerge from the eye [and 
fali] over the points of the visible object, or due to the fact that the 
iorms of the points of the visible object re ach to the surface of the organ 
of sight by means of radial lines. It is, [hence], always necessary that 
straight lines couJd be produced between any point of the surface of the 
visible thing and a certain point of the eye's surface for the things to be 
actually seen. Whence [it is elear that] when these lines can be produced, 
in any of the proposed ways, vision takes place, unless perhaps sight 
were prevented due to the hindrance of some other impediment. 
And so shouJd the eye be opposite to the visible thing, it will see 
the same, and should it be removed from its opposition, it will not perceive 
the same [thing]. and when it will be restored to [its] opposition. the 
perception will be restored. since lines cannot be produced from other 
parts than from [those] directly opposite, [namely] from the visibJe objects 
to the points of the surface of the eye. The proposed [thing] is, therefore, 
elear . 


[Proposition]3. It IS necessary for the organ of visual power to be 
spherical. 


a.....
>>>
104 



. 


Indeed should it not be spherical, I say that vision would be hindered - 
as when. say. it were [shaped] like a pIane surface. For then it will not 
see at a glance. save [somethingJ equal to itself. Indeed whether the rays 
issue forth from the eye land reach] to the visible thing. or 1he form s 
of the points of the visible thing reach by radial lines to the surface 
of the visual organ. it is elear that the perpendiculars are always shorter. 
by the 21st lprop.] of this first Ibook] I; whence the object approaches 
so much more to the eye by means of those lperpendiculars]. since the 
visible object is seen directly by means of the same perpendiculars land] 
not by other oblique lines which are. refracted. Since. as is elear by the 
48th Iprop.] of the second Ibook] of this Itreatise]. the refraction of the 
forms to some single point cannot take place in the pIane IsurfacesJ of 
Itransparent] bodies. because in su ch Ibodies] no point is commonly related 
to all lothers]. therefore only those Ithings] can be seen by a visual organ 
llimited by] a pIane surface that reach the same directly without refrac1ion. 
These moreover are reaching the eye by means of perpendicular lines. 
Accordingly let there be a pIane sight surface in which there is a line 
A B. and let straight line C DE /Fig. I] be in the pIane surface of a certain 


A 


B 


c 


D 
[Fig. I] 


E 


visible object Iwhich is] parallei to the eye and to line AR. And let a per- 
pcndicular be drawn from point C to the surface of the eye. by XI. 11 2 , 
and let it cut lthe eye] in point A and be C A. And from point DIet 
lanother] perpendicular. which is DB. be- drawn similarly to the surface 
of the eye. Accordingly since lines AC and BD are parałłel and equal, \ 
by the 23rd-
 and 25th -ł Iprops.] of 1he first lbook] of this Itrea1ise]. 1herefore. 
by I. 33 5 . line AB will be equal to line CD. And since line AB is 
equal to line CD. but line CDE is greater than line CD. therefore line 
C DE is not seen simultaneously in its entirety. because in sllch an arrangement 
the visible object cannot exceed the size of the eye's surfacetJ. And since 
this is false and against a notion which is elear to the Ivisual] sense. 
because it is possible for a thing grea1er than the cye itself lo be seen. 
it is plain that it is not possible for the surface of the visual organ 
to be pIane: bUl neithcr of ano1her shape than spherical. since /otherwiseJ
>>>
105 


there would always happen impossible inequalities of vision. By necessity. 
then. the surface of the visual organ will be sphericał. in the centre of 
",hich the meeting of radial lines may take place Iwhich mayaiso belong 
to objects] of a greater magnitude by far than is the same organ of 
sigh.. The proposed Ithing] iso therefore. elear. 


!proposition]4. The eye is the spherical organ of visual power constituted 
of three humors and four tunics. issuing from the substance of the brain 
land] arranged spherically. 
In what manner the eye may be the organ of visual power. we leave 
to the labor of another part of Inatural] philosophy. That indeed it is 
spherical. is necessary by the preceding proposition and also Ifollows] from 
Ithe fact] that it is of a wat ery nature. the property of which it is always 
to hecorne wunded. as is shown elsewhere. Moreover the assidunus concern 
nf the anatomists has taught lusl 1horoughly that the eye is constituted 
of three humors and four tunics. Accordingly the first of 1hese humors 
is called the crystalline or glaciał. since it is the proper organ of the 
visual power. and it is positioned in the middle of the eye; and it is a smali 
white. humid sphere. of a humidity retentive of the visible forms. in which 
there is a not strongly intense transparency. since there is in it a certain 
thickness. whence its transparency makes it similar to the transparency 
of a crystal or nf ice. and because of this it is called the crystalline 
or glacial humor. 
In fact lit is 1he case) that the transparency of this humor is changed 
in its posterior part toward the brain. from which part the whole eye 
receives nutriment. land] that before Ithis part) is perfectly united to the 
crystalline humor. which it is mai nI y intended to nourish. land though] 
not yet fuli of substantial and accidental forms assimilated to it. it is 
Inevertheless). by necessity. of another transparency than that lofthe crystalline]. 
And on this account it is called another humor and it is named vitreous. 
because it is similar. as it were. to broken glass. And since in everything 
that is nourished the pure is always separated from the impure. that which 
is lexcreted] by the nourished crystalline humor is separated. as unbefitting 
to its purity. to the opposite part to the nutrimental part. that is to 
say. Ithe nutrimental part) flows onwards to the anterior part of the crystalline 
humor. And though Ithe anterior part] is lalso] transparent land] somehow 
similar to the crystalline humor. though not of its perfect makeup in 
density because it is overflowing with the denser body of nutriment, it 
is elear Inevertheless] that it is by necessity transparent land] liquid; whence 
it is called the albugineous humor, since it is similar to the egg's albumen 
in tenuity and transparency. Indeed it is a white. thin. elear. transparent 
humor; and Ithe eye] has 11his albugineous] humor at the anterior part. 
precisely las it has] the vitreous humor at the posterior part. for guarding 
the crystallinc humor. so that it may not suffer exceedingly from external 


--
>>>
106 


as well as internal eireumstanees, and [eaeh part of the eye] is subjeet 
by duty to (the overall funetioning] of the visual organ and has [appropriately] 
redueed the keenness of nature. 
Moreover an exeeedingly tenuous and subtIe web eontains the first two 
humors, namely the erystalline and the vitreous, separating them from the 
albugineous and surrounding them both, a eertain part of whieh web, 
deseending through the (surrounding] medium, separates the erystalline from 
the vitreous; and this web, beeause of [its] fine strueture is ealled aranea. 
[i.e.]. the spider's web. lndeed sinee the albugineous humor is liquid, in 
itself not eonsistent. it was neeessary for it to be retained by something 
solid for the sake of guarding the eye; eonsequently nature has surrounded 
it with a viseous. very strong. not greatly transparent skin. whieh by its 
density may retain better, and by its warmth may temper, the albugineous 
humor. so that the erystalline would not eongeal and beeome inappropriate 
for the reeeption of visible forms. 
And sinee the visible forms eould not reaeh the erystalline humor on 
all sides where it is surrounded by this tunie. beeause of the density and 
viseosity of sueh atunic, therefore nature has ineised this tunie. in the 
anterior part of the eye. where the place of reeeption of visible forms 
is found. and produeed a round aperture the diameter of whieh is almost 
equal to the side of the eube that ean be inseribed in that (oeular] 
sphere. or [equal] to the side of the inseribable square in the great eircJe 
of that sphere. And this opening is round on the aeeount that it may be 
better suited for the aeeeptanee of all the forms [and] it is penetrating 
all the way to the eoneavity of the same tunie; and this tunie is ealled 
the uvea, beeause it is similar in appearanee to a grape. And this tunie 
is very blaek. frequent1y even green and sometimes bluish grey, and the 
body of that tunie is somewhat dense, not rare. 
In faet, in order that the albugineous humor may not eseape from 
the open ing of the uvea and so that the operation of the visual power 
may not be impeded, it was neeessary for nature to underlay the open ing 
of the uvea with a veil-like eovering. transparent [and] solid like white, 
c1ear horn. and this tunie is ealled cornea. In faet where this tunie is 
joined to the other part s surrounding .the body of the eye. there [its] 
transpareney eeases and it becomes a tu nic of a different arrangement. 
more solid than the eornea, nontransparent, completing, however, with the 
same eornea one sphere, whieh is the sphere of the whole eye and the 
posterior part of that sphere is made into another nontransparent but 
rather t1eshy tunie. 
And this [new tunie] is ealled the conjunctiva, or the consolidativa, because 
it joins together the eye and consolidates the same with the parts of the 
neighboring body. Henee the eorneal tunie, the albugineous humor, and the 
g1aeial humor, and the vitreous humor will be following one another, and 
all these are transparent for the sake of better reception of the visible
>>>
107 


forms. The humors and the tunies of the eye eome forth from the substanee 
of the brain, seeing that from the anterior portion of the brai n, from 
the two .parts of the same, there develop the two eoneave, similar optieal 
nerves, having two tunies sprung from the two webs of the brain, and 
these nerves proeeed to the middle of the anterior part of the brai n, 
where they become one optieal nerve, whieh in [its] advanee is again 
divided into two similar and equal optieal nerves, whieh, having their 
position interehanged sueh that the right one beeomes the left one and 
the left the right, proeeed to the eonvexity of the two eoneave bones 
eontaining the eyes. seeing that in the eentres of these two eoneave bones 
there are two opemngs, equally bored. whieh are ealled the eireular openings 
of the eoneave nerves; and sinee those two openings are round, the middle 
point of any of those openings is ealled the eentre of that opening. 
Henee those two nerves enter these two openings and exit them at 
the eoneavity of the previously deseribed bones, and these are snread out 
and enlarged, and the extremity of eaeh of them is made into [a shape] 
quite like the instrument for putting wine in barrels, that is [something] 
very mueh like [the shape] of a eoneave eone; and eaeh of the eyes 
is built around one extremity of this nerve and is eonsolidated with the 
same. 
Similarly and from the tunies of these nerves rise the tunies of the 
eyes, for the eorneal tunie rises from the external tunie of the two [extant] 
tunies of this nerve and the uveal tunie rises from the internal tunie 
of the two tunies of the two nerves. And inside this uveal tunie is set 
the erystalline humor. over the extremity of the eoneavity of the nerve, 
dividing in the middle the vitreous humor, beeause both [tunies ultimately] 
rise from the medullary substance of the brain. And between these humors 
and the uveal tunie is interweaved the spider web, out of the most subtle 
threads of the uveal tunie, whieh othe,rs eall the retinal tunie, beeause 
it is entwined in the manner of a net. 
The humors and tunies of the eye are fixed in spherieal ehannels. 
Indeed sinee the uveal tunie does not attain to a eomplete sphere inside 
the eye, sinee, as was premised, there is a round opening in its anterior 
part. whieh is eovered by the eorneal tunie, therefore the sphere of the 
eorneal tunie interseets neeessarily the sphere of the uvea and the eommon 
seetion of their spherieal surfaces is the cireumferenee of that opening, 
and it is a eireular line by the 80th [prop.] of the first [book] of this 
[treatise] I. Moreover in the anterior [part] of the erystalline humor. for 
the sake of better reeeption of the forms, there is a smali superfjeial 
compression of smaller eurvature than is the surfaee of the eornea containing 
it. indeed the sphericity of the surface of the crystalline humor is made 
similar to the compression of the surface of a lentii, as is obvious from 
the (eoncIusions ot] those consiclering the eye's anatomy. Hence the anterior 
surface of the same is a portion of a surfaee of a greater sphere than
>>>
108 


is the uveal sphere contammg the same. And this ftattening is equally 
deftected when opposing the opening as it is in the anterior part of the 
uvea. because its position with respect to it is symmetricaI. 
Indeed precisely as the round aperture that is in the anterior part 
of the uvea is directly Opposi1e to the extremity of the concavity of the 
nerve over which the eye is arranged
. so too in the posterior part of 
the concavity of the uvea there is a round open ing which is [set] over 
the extremity of the concavity of the nerve. and the opening which is 
in the anterior [part] of the uvea is opposed to the opening of the concavity 
of the nerve. [This is sol because the optic nerve intersects the conjunctiva 
and the uvea and penetrates all tunics of the eye up to the crystalline 
sphere. which intersects the nerve's cone. precisely as the vitreous humor, 
which is set up in the concave conical optical nerve, and so the common 
section of the conical surface of the optical nerve and of the crystalline 
sphere is a circ1e. by the I 10th [prop.] of the first Ibook] of this [treatise)3. 
Accordingly. the glacial sphere is constructed on the extremity of the 
concavity of the optical nerve and in the round posterior of the opening 
of the uvea. Therefore the extremity of the nerve contains the middle 
of the glacial sphere. and that concave nerve is bringing down within 
itself the visual spirit from the brain to the eye. and through its smali 
veins nutriment comes to the eye and is spread in the same through 
the roads of nutriment. And in the intersection of this nerve [with the 
brain] in the anterior part of the brain. there is a visual virtue. perceiving 
and discerning all visible things; and the uvea is joined with the glacial 
in the circ1e containing the round opening Iwhich is] in the posterior Ipart] 
of the uvea. 
Moreover these two spheres, name1y the glacial and the vitreous, necessarily 
intersec1. because the convexity of one hinders the convexity of the other; 
indeed precisely as 1hey are of various nature and transparency. thus they 
are portions ot d Ifferent spheres cutting one another. Accordingly the common 
section of these spheres is a circ1e, by the 80th [prop.] of the first [book] 
of this [treatise] 4. Therefore the same circ1e is the base of the cone of the 
optical nerve and of the intersection of the same cone and of the crystalline 
sphere. and of the consolidation of the śphere of the uvea with the crystalline 
sphere. and. necessarily, of the intersection of the same [two] spheres. 
In fact the body of the consolidativa contains the conical part of the nerve, 
which is inside the open ing of the bone (through which passes the nerve), 
and lalso] inside the circumference of the glacial sphere. and it lalso] contains 
the uveal sphere. 
And so it is cIear from these [considerations] that the glacial humor 
is the proper organ of visual power. for its transparency alone is retentive 
of the visible forms. and it is located in the middle of all humors and 
tunics; and should a lesion occur to any other tunic or humor. the glacial 
humor remaining unharmed. the eye would always receive a cure with 
the help of medicine and it would be healed. even sight would be restored.
>>>
M 


CONCAV ITY OF 
THE o PT (CAL 
NER VE 


Ul 
::- lu 
a: UJ ::- 
Ul  Q: 
:z a:: lu 
UJ Z 
:z 
... -' 
« ...J { 
U  u 
..... u = 
Q. ..... Q.. 
Q Q. e 
e 
..... "- 
Ul Q ..... e 
? '-' Q lu 
a:. 
 ::- 
Ul Z UJ Q: 
-z. ? z Z lu 
..... Q :) 
 
... U I- 
« a:. -' 

 Q .... Q: { 
l- Q £ u 
o- a:. Q: i:: 
Q Ul II) UJ Q.. 
'i x I- e 
.....  
 
o "- 
e 
u 

 v 
? 
 
I- ;:) 
'- 
a:. 
o Q: 
«. e 
Ul Ci 
l- ł:!! 
')oL 
Ul 
 
lu 


[Fig. 2] 


109
>>>
110 


while, should the same [glacial] be corrupted. the entire eye would be 
wasted without hope of restoration with the help of a medicinal cure. 
Accordingly, the crystalline or glacial humor is principally the organ of 
visual power, for which reason it is diligent1y preserved by nature. 
And nature has constituted two eyes for the sake of perfecting the 
excellence of vision and its fultillment. Hence it is thus c1ear that the humors 
and tunics of the eye are tixed in spherical channels, and the statement 
of the proposed detinition of the eye is elear. according to the experience 
of all those who have written thus far about the anatomy of the same. 
Now all these [things] which precede about the structure of the eye, in 
this fourth proposition of this third book of our Perspectiva, are, of course, 
now to be exemplitied compendiously by means of a mathematical tigure 
[which] we have brought forward [and] which is of this kind. 


X 
albugineous 
body 
glacial 
humor 
spider S 
web vitreous 
a humor 
cavity ot 
the opticol 
nerw 



 

 
u 


G 
[Fig.2A] 


-....
>>>
111 


Jndeed leI the centre of the eye be point A [Fig. 2 and 2A]:". and the 
convex surface of the same glacial [humor] arc BCD, .and the convex surface 
of lhe same vitreous [humor] arc BED, and let arc BO D be the spider 
web covering the glacial frontward, and let the spider web between the- 
body of the glacial and of lhe vitreous [humors] be a straight or curved 
line. name1y BD, and let the web covering the same vitreous rearward 
be BQD, and let the exterior tunic of the optical nerve be GK on the 
right and GH on the left, and let the interior tunie of that nerve be 
GD on the right and GB on the left. Moreover, let there be the surface 
of the uvea. the centre of which is N, and in which are the arcs TMV 
and BLD. and let there be its opening, the diameter of which is M L 
and its centre point F. Furthermore, let the albugineous body be BLDQ. 
and let the internal surface of the same cornea be arc H FK, and let 
the external surface of the cornea be arc HXK. Therefore the middle 
point of the common nerve will be G and the axis of the cone of the 
whole optic nerve will be line GAF, in whieh there will be the centres 
of all humors and tunics of the same eye. 
Accordingly this is the figure of the entire eye which. should it be 
opportune. we wiU be using later. 


[Proposition]S. It is impossible for the eye to be joined with visible 
objects by means of rays issuing forth from the eyes. 
Jf indeed some rays issue forth from the eyes, by which the visual 
power is joined to outside things, [then] either those rays are corporeal 
or incorporeal. Jf corporeal. then, since the eye sees the stars and the 
heaven, it is necessary that that corporeal something going out from the eye 
fiU up the whole space of the universe that is between the eye and the 
seen part of the sky, without the diminution of the same eye, which 
cannot be done, and, moreover, be done so quickly, the substance and 
the size of the eye remaining preserved safe1y. 
If, moreover, it were given that the rays are incorporeal, since sensation 
may not occur except in a corporeal thing, then the same rays would 
not feel the visible object; therefore nor could a corporeal eye perceive 
by means of this incorporeal, nonperceiving [entity]. Nor indeed do such 
incorporeal [entities] bring something back to the eye, by means of which 
the eye could grasp the visible thing, since sight could not take place 
but through contact of the eye with the visible form. because there is 
no action without contact. 
Therefore. if the rays proceeding from the eye do not bring back anything 
to the eye. then there cannot be vision by means of the same. If in 
fact they do bring something back to the eye, these [entities] must be 
lights or colors. which are seen by themselves, and which are multiplied 
to the eye with the rays. Therefore, the [supposed] rays are not the cause 
of the joining of the eye wit h the visible thing, but [rather] something
>>>
112 


else. that multiplies itself to the eye. is by itself the cause of VISlon. 
Hence it is impossible for the [postulated] rays to be by themselves the 
cause of vision uniess. perhaps. by rays we mean the lines described by 
the p()ints of the forms multiplied from the surfaces of visible things to 
the eye: hecause. as is elear hy the 2nd Iprop.) of this book. it is necessary 
tha1 straight lines be producible between any point of the surface of the 
visible thing and some point of the surface of the eye for the thing to 
be actually seen. However such rays do not issue forth from the eyes. 
The proposed Ithing) is therefore c1ear. 


IProposition]6. Vision occurs as a result of the action of the visible 
form on the eye and on account of the eye's suffering from Ithe action 
ot] this form. 
Tha1 visible forms act on the eye is obvious from [our) postulate I. 
Indeed the eye suffers from stron g light as. Ifor example). at the sight 
of the solar body or of another strong light. say the light reflected to 
the eye from a polished body or from another exceedingly white body. 
Indeed in [wat ch ing) these things. the eye is disabled so that it diminishes 
in its operation until it may be restored [to its normaIcy) through lits) 
. intrinsic, natural virtue. And yet the eye [always) suffers from [t he action 
ot] the sensible forms, as. indeed. it retains in it their strong' impressions. 
For after the eye had inspected for a long time a strong light or color, 
should it then face a dark place. or a place with weak light. it would 
discover within itself, strongly visible. [and] with [ali) its light. color, and 
shape. that which it had previously inspected. And whenever a strong 
color impressed on 1he eye is mingled together with the colors of visible 
objects in the dark. then those objects will be seen as colored by another 
mixed color. for example when an eye. strongly [affected by the color) 
green. makes white objects. seen afterwards in a more obscure place. appear 
mixed with greenness; and if the eye is c1osed, nevertheless the previously 
seen form will occur to the eye. Therefore the visible forms act on the 
eye and the eye suffers on their account 2. 
And since what is visible by itself are light and color, and light is 
the hypostasis of colors, while light - is always spreading in a sphere to 
all and sundry positions. it is plain, therefore. that colors too are spread 
in the same manner. Accordingly. when the eye is opposite a certain colored 
illuminated object. then the light is multiplied. either by itself or with 
the color of that opposite object. to the eye. and [it is) reaching to the 
surface of the eye and acts on the eye, and the eye suffers from thaL 
And so since light and color arrive simultaneously at the eye's surface 
and act in it, and the eye suffers on their account, and the soul's power 
comes to be a knowing [power) by virtue of the union of the visible 
forms with its organ. consequently vision takes place on account of the 
presence of visible forms acting on the eye. And this action and passion 


-
........
>>>
113 


take place in the guise of other natural actions, because the whole agent 
acts in any passion and (it does it] indivisibly, and the whole ex tent (of 
the eye] sutfers from any point of the agent. Therefore the forms of light 
and color. which are in any of the points of the visible thing. reach 
to the entire surface of the eye. and the forms of all points of the surface 
of the visible. thing come to one point of the surface of the eye, and 
thus is performed the action and the passion among these-\ 
For the action of the visible forms on the eye does not take place 
unless the visible form be potent lenough] to act and I be] of complete 
letfectiveness in constituting] the hypostasis Istemming] from the presence 
of light. and unless the medium external to the eye and to the visible 
1hing be actually transparent. and unless the organ of sight be receptive 
of the forms through (its] middle tunics. and the transparent humors Ihe] 
of appropriate transparency. Indeed if the part of the corneal tunic (placed 
over the uveal opening) which is the first to be joined to the external 
air. and the albugineous humor filIing up the open ing of the uvea should 
undergo privatión of (their] proper transparency. for example when their 
proper quality was changed. or when another impediment occurred. or even 
when the same glacial humor was atfected by an excessive congelation 
or was prevented from receiving the forms in any other way. Ithen] there 
would not he vision. hecause the sensihle form could not be impressed 
upon the visual organ 4. 
And so the visible form. coming from the visible thing through the 
transparent medium all the way to the surface of the eye, passes through 
the transparency of the eye's tunics and reaches to the visual power from 
the aperture which is in the anterior Iside] of the uvea, and comes to 
the glaciał. and traverses it according to the manner of its transparency; 
and by reason of this. nature has ordered all the transparent tunics of 
the eye so that they may be atfected by the sensible forms having the 
power to act through light. In fact the eye is ordained to sutfer from 
the visible forms. For it is not tainted by the form of light or color 
after the departure of the presence of the lighted or colored body, as we 
have shown this passion to harmonize universally with any transparent 
body. by the 4th (prop.] of the second (book] of this Itreatise]. 
And whenever a certain impression is made in the eye on account 
of the strength of light and color. and a change is allowed (to happen] 
throul!h those lights and colors. that Ichange]. moreover. does not remain 
in the eye hut for a short time: therefore such a change is not permanent. 
And so the eye is not tinged with a permanent taint by the sensible 
forms acting on the eye through either the colors or the forms of light. 
The proposed (thing] is, therefore elear. 


IProposition]7. It is necessary that the centre of the sphere of the whole 
eye. and the centre of the glaciał. and the centre of the exterior and 


8 - Witelonis Perspectivae...
>>>
114 


interior surfaces of the cornea, and the centre of the convex surface 
of the albugineous humor, be [ali] the same. From which it is elear that 
the internal surface of the cornea is concentric to its external surface. 
Having recovered the figure of the eye [Fig. 2] that we set out before 
in the 4th [prop.] of this [book], I say that what is proposed here is 
true, because point A is the common centre of the proposed spheres. 
For if it were given that the centre of the sphere of the whole eye, 
which is point A, is not the centre of the glacial sphere, it is plain, 
by the 75th [prop.] of the first [book] of this treatise I, that the straight 
lines perpendicular to the surface of the sphere of the eye would not be 
perpendicular to the surface of the glacial sphere, save only the one that 
passes through the centres of both, while the others which will be per- 
pendicular to the surface of the eye will be oblique to the surface of 
the glaciał. Therefore if the glacial understood the forms of visible things 
according to the incidence of these lines which are perpendicular to the 
surface of the eye and obliquely incident to the surface of the glacial 
then. by necessity, the glacial would understand all forms of visible things 
as awry and bent away from their [normal] position and shape, which 
they have outside [the eye] in the surfaces of visible things, which is against 
a postulate advanced in the beginning of this book 2 . 
And since the torms incident along nonperpendicular lines to a second 
transparent medium den ser [than the first] are refracted toward the per- 
pendicular. as is obvious by the 47th [pro p.] of the second [book] of 
this Itreatise], while the substance of the humors and tunics of the eye 
is denser than the surrounding air and [made] of substance[s] of varying 
transparency among themselves, as is elear by the 4th [prop.] of this [book], 
it is plain that in the same surface of the glacial there would take place 
another refraction than in the surface of the cornea. Therefore the glacial 
could not distinguish [accurately] anything in visible things on account of 
the refraction of forms taking place in its surface 3 . 
Indeed it is manifest that lines obliquely incident to the surface of the 
eye are [even] more oblique in the surface of the glacial because the glacial 
is of another transparency than the cornea or the albugineous humor 4. 
Furthermore there is in the glacial - a certain transparency on account of 
which it receives the forms, and a certain thickness prohibiting the passage 
through [unimpeded] of the forms, and on this account the forms are 
fixed in its surface and body. Therefore, [given our assumption], the glacial 
cannot comprehend any of the visible forms according to the position 
and shape they have outside the eye. But this is impossible because it 
is cIearly manifest. hy a postulate. that the glacial grasps the forms of 
visi/11e things according to the position and shape they have in the things 
outside lor the eye]. It iso therefore. necessary that lines which are per- 
pendicular to the eye's surface be lalso] perpendicular to the surface of 
the glaciał. Therefore the surfaces of the eye and of the glacial will be 


- .....
>>>
I 


115 


surfaces of concentric spheres, [i.e.], (having the same centre), and the extremity 
of all lines 5 that can be imagined produced from any point of the surface 
of the visible thing perpendicularly to the surface o( the eye meet in 
this centre, by the 74th [prop.] of the first [book] of this [treatise]6, and 
they are [also] perpendicular to the surface of the glacial, by the 72nd 
[prop.] of the first [book] of this [treatise]7. 
And since the anterior surface of the cornea completes the spherical 
surface of the eye and constitutes wit h it one spherical surface, it is elear 
that the centre of the eye is [also] the centre of the cornea. by the definition 
of a sphere, and so it is 'e1ear that the centre of the eye and the centre 
of the glacial and the centre of the cornea are [one and] the same centre. 
Therefore since the centre of the eye, which is [also] the centre of the 
exterior surface of the same cornea, and the centre of the glacial sphere 
are one with the centre of the entire eye consi
ting of all its humors 
and webs, it is more convenient for nature that the centre of the gl aci al 
be the same centre of the inside surface of the cornea. so that the centres 
of all the surfaces opposite to the opening of the uvea be one common 
point. and the concave surface of the sphere of the cornea be concentric 
wit h its convex surface. Thus indeed, by the 72rid and 74th [props.] of 
the first [book] of this [treatise], all lines issuing from the centre to the 
surface of the eye will be perpendicular to all the surfaces [which are] 
opposite to the opening, and the excellence of vision will be increased, 
and the whole eye will be round because of the unity of the centre of 
the cornea with [that ot] the whole eye. 
And since, by the 73rd [pro p.] of the first [book] of this [treatise] 8, 
the internal surface of the cornea is paralleI to the external surface of 
the same, since the centre of both is the same, while the albugineous 
humor touches the concavity of the cornea with its own convexity, as was 
previously established by the experience of anatomists, in the 4th [prop.] 
of this third [book], the convex surface of the albugineous humor will 
be. by the 79th Iprop.] of the first [book] of this [treatise]9, part of a spherical 
surface touching with its convexity the concave sunace of the sphere 
of the cornea. It is elear, therefore, by the 73rd [prop.] of the first [book] 
of this (treatise], that [the centre] of the convex surface of the albugineous 
humor and [t he centre] of the concave surface of the cornea is [one and] 
the same centre. And this is the proposed [thing], and the corollary 10 
is obvious. 


(Proposition]8. It is necessary that the uveal sphere be eccentric to the 
whole eye, and that its centre come e10ser to the anterior [part] of the 
eye. whiJe the centre of the eye be set much deeper. From which it 
is elear that the centre of the uvea must be considerably elevated wit h 
respect to the centres of - all tunics and humors of the anterior part of 
the eye.
>>>
116 


Indeed. as is elear by the 4th Iprop.] of this Ibook] and by the preceding 
[prop.]. since the corneal sphere is continuous with the lexternal] surface 
of the whole eye. along its [own] surface, and [since] part of the same 
Icorneal] sphere together with the whole eye constitutes a greater sphere 
than the uveal sphere. because it contains inside it the greatest cirele of 
the uveal sphere. Itherefore] it is elear. by the definition of spheres inter- 
secting internally I, that the surface of the corneal sphere is greater than 
the surface of the uveal sphere. Accordingly it is plain from the definition 
of a greater sphere that the semidiameter of the cornea is grea1er than 
the scmidiameter of the uvea. 
And since the inside surface of the cornea set over the opening of 
the uvea is a concave spherical surface concentric with the apparent. li.e., 
external]. surface of the same cornea, because the entire cornea is of equal 
thickness, as is conspicuous in the preceding Iprop.J. for the centre of 
the internal surface of the cornea is identical with the centre of the 
apparent convex surface of the same cornea. but the concave surface of 
the cornea cuts the surface of the uveal sphere over the circumference 
of the opening which is in the anterior part of the uvea. as was previously 
stated in the 4th [pro p.] of this book and shown by the 80th Iprop.] 
of the first [book] of this Itreatisep, therefore. by the 84th Iprop.] of 
the first Ibook] of this [treatise]\ it is necessary that .the centre of the 
corneal sphere containing the uveal sphere be more remotely [situated] 
in the depth [of the eye] than the centre of the uveal sphere. 
.Jt is elear. therefore, that it is necessary that the uveal sphere be eccentric 
to the whole eye. and that its centre come eloser to the anterior I part] 
of the eye, while the cen1re of the eye be set much deeper. which is 
the main proposed Ithing]. And from this the corollary is also elear. because. 
as the sphere of the uvea -ł is not in the centre of the consolidativa but 
in front of it. toward the side of the apparent surface of the eye, and 
as (he apparent surface of the same eye is part of a greater sphere. it 
is plain. as stated. that its centre will be more remotely Isituated] in the 
depth. than the centre of the uvea. 
Jt is in fact manifest. Igiven] the external convex surface of the same 
cornea of the eye to which the internal concave surface of the same is 
concentric. that, as a consequence. the centre of the concave surface as 
well as of the convex surface of the same cornea is set much deeper 
in the eye than the centre of the uvea. And since the concave surface 
of the cornea touches the surface of the albugineous humor which is in 
the front of the opening of the uvea and is superimposed to it. it is 
elear from the preceding and by the 73rd [prop.] of the first [book] of 
this [treatise] that the convex surface of the albugineous humor is a spherical 
surface. the centre of which is the centre of the surface superimposed to it. 
Therefore the convex surface of the cornea and the concave surface 
of the same. and the convex surface of the alhugineous humor touching 


" ....
>>>
117 


the concavity of the cornea. being all spherical surfaces of paralleI spheres. 
it is plain. by the 73rd Iprop.) of the first Ibook) of this [treatise). that 
the centre of all of the same [spheres) is one point which is set much 
deeper than the centre of the uvea. And since the anterior surface of 
the glacial is spherical [and] concentric to the whole eye. by the preceding. 
and also because the convex surface of the glacial sphere cuts the surface 
of the uveal sphere internally. it is clear. by the 84th [prop.) of the first 
(book) of this Itreatise]. since the surface of the glacial is part of a greater 
sphere than the surface of the uveal sphere. that the centre of the glacial 
is set much deper than the centre of the uvea. Accordingly. the centre 
of the uvea is considerably more elevated than the centres of all tunics 
and humors of the eye that are (a portion] of the anterior part of the 
eye. facing the external part of the air. which is the entire proposition. 


IProposition]9. The line produced he1ween the eentre of the eye and the 
eentre of the uvea will neeessarily pass through the eentre of the circle 
of seetion of the uvea (and cornea] and Ithrough] the middle of the eon- 
cavity of the optical nerve. 
It is obvious. by the 7th [prop.) of this Ibook]. that the centre of 
the whole eye is the same as the centre of the cornea, but the line that 
joins the two centres of the cornea and the uvea. which is line AF in 
the previous figure of the eye /Fig. 2). in the 4th [prop.) of this (book]. 
when produced, reaches the centre of the common circle of their section. 
by the 82nd Iprop.] of the first (book) of this (treatise) I, that is point 
F. the centre of the circle of the aperture of the uvea. along the periphery 
of which those spheres interseet. 
Indeed the concave surface of the cornea and the convex (surface) of 
the uvea are two spherical surfaces cutting one another along the periphery 
of the opening of the uvea. as is clear by the 4th Iprop.] of this (book). 
It is also plain. by the 86th Iprop.] of the first (book) of this (treatise]2, 
that 1he same line extended [further] reaches the two bisecting points of 
the two surfaces of the cornea [which are] concentric (and] set over that 
aperture of the uvea, the periphery of which aperture is the circumference 
of the circle of section. And since the aperture which is in the anterior 
[part] of the uvea is directly opposite to the aperture which is in the 
posterior part of the uvea. which is (in its turn) the extremity of the 
concavity of the nerve. .it is plain. by the Illth [prop.) of the first (book] 
of this Itreatise]\ that the same extended line will necessarily pass through 
the middle of the concavity of the optical nerve, and this is the centre 
of the base circle of the cone of the concave optical nerve. The proposed 
Ithing] is therefore cle ar. 


IProposition] 10. The straight line produced between the centres of the 
glacial and uveal spheres will neeessarily eome to the eentre of the clrcle
>>>
118 


that the glacial and vitreous spheres have in common with the uvea, and 
wiII be perpendicular to the surface of that cirele. 
It has become obvious from the preceding [remarks) in the 4th [prop.) 
of this (book) that the glacial sphere cuts internally the uveal !Ophere. 
Therefore the line passing through the centres of these spheres, i.e., line 
AN [Fig. 2), will be perpendicular to the centre of the common cirele 
of section of the same, by the 82nd (prop.] of the first [book) of this 
[treatise). In fact this circ1e of section is either the circ1e marking the 
boundary of the mutual consolidation of these spheres I or is parallei 
to it. For the surface that is in the anterior part of the glacial is opposite 
to the opening that is in the anterior part of the uvea and its position 
with respect to it is a like position throughout. as became elear in the 
4th [prop.] of this [book]. Therefore the boundary of this surface, which 
is the circ1e of section between the two surfaces of the glacial and the 
vitreous spheres, is either itself the cirele of consolidation of these spheres 
with the uvea. or is parallei to it 2 . 
Hence if the circ1e of section between the two surfaces. name1y of the 
glacial and vitreous spheres, were itself the circ1e of consolidation of the 
same with the uvea. then this cirele would be the cirele of section between 
the surface of the glaciał and of the uvea, and then, as before. the pro- 
posed thing is elear, by the 82nd [prop.] of the first [book] of this [treatise]. 
But if the cirele of section between the surface of the głacial sphere 
and the surface of the vitreous sphere were not itsełf the cirele of consołidation 
of the crystalline and vitreous spheres with the uveal sphere, but were 
paralleł to the cirele of their consolidation with the uvea, then. if the 
surface of the crystalline sphere is imagined with the mathematical intellect 
to be extended beyond the naturał form of extension of its sphere. it will 
cut the sphere of the uvea over a cirele paralleł to this circ1e of section 
of the glaciał and vitreous spheres. 
Because this circ1e has an equal position with respect to the circum- 
ference of the sphere of the uvea, and because this cirele is parallei to 
the circ1e of consolidation 3 . the circ1e of section between the surface of 
the głaciał and the uveał sphere will necessariły be either the same circ1e 
of consolidation or parallei to it. Now if this cirele were the same circ1e 
of consolidation, it is płain, by the 82nd (prop.] of the first [book] of this 
[treatise]. that the łine passing through the centre of the glaciał and through 
the centre of the uvea will pass perpendicułarly through the centre of this 
circ1e 4, because this circ1e is the circ1e of section between those two sphericał 
surfaces 5 . But if this cirele were parallei to the circ1e of consolidation, 
and [since) it is also parallei to the circ1e of section between the surface 
of the głacial and the surface of the vitreous, therefore it woułd [ałso] 
be in one spherical surface with the circ1e of section between the surface 
of the glacial and of the vitreous. whicb is the surface of the glaciał 
and [which] is parallei to the cirele of the said section.
>>>
t19 


But if two cirCles were parallei in a certain sphere, the line passing 
perpendicularly through the centre of one will necessarily pass perpendicularly 
through the centre of the other, as is elear by the 68th 6 and 66th 7 [props.] 
of the first [book] of this [treatise]. Therefore the line that will pass through 
the centre of the uvea and through the centre of the glacial passes through 
the centre of the cirele of consolidation of the glacial and vitreous spheres 
with the uvea, according to aU arrangements of the spheres and of those 
cireles. Hence the line is perpendicular to the surface of that cirele, by 
the 66th [prop.] of the first [book] of this [treatise]. which is the proposed 
[thing]. AU the same, these three cireles are necessarily one cirele, albeit, 
even if they were different but parallei cireles, the same proposed [things] 
would happen to all. Indeed the glacial and the vitreous intersect along 
the same cirele and both of those cut the uvea and are consolidated 
with it according to the same cirele, and that is the cirele of base of 
the concavity of the optical nerve; and thus that one cirele fulfills the 
role of four cireles. 


[Proposition] II. It is necessary that the vitreous sjJhere be eccentric 
to the glacial sphere, and that the centre of the vitreous come closer 
[than that of thę glacial] to the anterior [part] of the eye
 
Indeed because the surface of the glacial sphere l and the surface of 
the vitreous sphere are two spherical surfaces cutting one another, therefore 
the centre of the anterior surface with respect to the outside of the eye 
is more remotely situated in the depth [of the eye] than the centre of 
the posterior surface, by the 84th [prop.] of the first [book] of this [treatiseJ2, 
while the posterior of these two is the surface of the same vitreous, as 
was previously shown in the 4th [prop.] of this (book]. The proposed 
[thing] is therefore elear. 


[Proposition] 12. It is necessary that the line passing through the centre[s] 
of the glacial and the uvea also pass through the centre of the vitreous 
and the middle of the concavity of the optical nerve. 
As is elear by the 10th [prop.] of this book, [it is obvious] that the 
straight line passing through the centre of the glacial and [that] of the 
uvea, which is line AN in the previous figure of the eye (Fig. 2], [when] 
extended to. the centre of the cirele of consolidation of the glacial with 
the uvea, is perpendicular to the surface of the cirele of consolidation 
of the glacial and vitreous with the uvea. Indeed the cirele of intersection 
of the glacial with the vitreous is either the same with this circle or 
parallei to it. Moreover, whichever of these possibilities is realized, the 
PTeviously mentioned line will always be perpendicular to the cirele of 
section of the glacial sphere with the vitreous; therefore, it is plain, by 
the 83rd [prop.] of the first [book] of this [treatise] I, that the same lIine] 
passes through the centre of the vitreous sphere.
>>>
120 


Therefore since this line passes through the centre of the vitreous. it 
is elear. by the 82nd Irrop.] of the first Ibook] of this !treatise], that 
the same will necessarily pass perpendicularly through the centre of the 
cirele of consolidation. It is therefore extended through the middle of the 
concavity of the optical nerve over which the eye is constructed. because 
the cirele of consolidation is the base and boundary of the concavity 
of the optical nerve, as is elear from the 4th [prop.] of this [book]. 
In truth since it is obvious las seen] above, by the 9th [prop.] of 
this Ibook]. that the line drawn between the centre of 1he eye and the 
centre of the uvea passes necessarily through the centre of the cirele of 
section of the uvea land cornea] and [through] the middle of the concavity 
of the optical nerve. since one cannot produce from the same point. say, 
the centre of the optical nerve, to the same surface many perpendiculars, 
as is elear by the 20th [prop.] of the first Ibook] of this [treatiseJ2, it 
is plain that the same line is passing through the centre of the cirele 
of section of the sphere of the uvea and of the glaciaP. and the cen1re 
of the uvea and of the eye. and of 1he srhere of the glacial and of the 
vitreous. and 1hrough the centre of the circle of consolidation. 
And so it is elear from the preceding that one and the same line. 
name1y QAf. passes through the centre of concavity of the optical nerve 
and through the two centres of all the tunics facing the opening of the 
uvea, and Ithat] the same iso by the 74th [prop.] of the first [book] of 
this Itreatise]4, perpendicular to the surfaces of all tunics facing the open ing 
of the uvea: and it is perpendicular to the surface of the aperture of 
the uvea. and perpendiClllar to the surface of the cirde of consolidation. 
and it is extended through the centre of concavity of the optical nerve 
over which the eye is constructed. And the same is the axis of the entire 
eye. which. in the above displayed figure, is line GAF. 


, 
, 
I 


IProposition] 13. The eye does not grasp the visible things without the 
existence of an intermediate transparent body. 
Indeed since. as is elear by the 6th [prop.] of this Ibook]. vision cannot 
take place except through the action of the visible form coming to the 
eye from the visible thing, in trulh the forms are not extended except 
through transparent bodies of like transparency, in which the extension 
of light and [its] forms is made according to straight lines, as is elear 
by the 1st Iprop.] of the second [book] of this [treatise]. Therefore when 
some intermediate non-transparent body does not cut otf the lines produced 
from the visible things to the eye. then the Ivisible] forms reach the eye 
and vision is completed; but should some non-transparent body interfere, 
the multiplication of forms to the eye is impeded. The proposed Ithing) 
is therefore elear. 


"\ 


IProposition]14. There can be no VISlon when the extant visible body 
is of like transparency with the medium. 


) 


--
>>>
121 


Indeed if the visible body is transparent, then it is not colored. nor 
does it have the form of light but only Ithat] of a perspicuous body; 
therefore it is not seen, because, as is elear by the 4th Iprop.] of the 
second book] of this Itreatise], light is not fixed in transparent bodies 
in such wise that it tinges them or that it evinces for them the act of 
visibility. Therefore when the transparency of the visible body might happen 
to be similar to the transparency of air. then its disposition will be like 
the disposition of air and it is not apprehended by sight. precisely as 
air is not leither]. 
. And the same is the case with any other I transparent] medium: for 
none of these is seen, since the transparency of the visible object will 
not have been thicker than the transparency of the intermediate body. 
Jf in truth the visible body will have been transparent. but less than the 
medium. as for instance a crystal compared to air, then the visible thing 
will be seen through the intermediate air because it has some color compared 
to its thickness. just as a colored objeet; because when light shines over 
the same, it is fixed in it with a certain fixity. namely in accordance 
with that which is thickness in it. and it will traverse it according to 
its transparency, and it will persist in that form in the air. according 
to the color and Iight which are in its surface. and when that form will 
have reached the eye. it will have an effect in the eye and the eye will 
perceive the visible object. The proposed Ithing] is therefore elear. 


} 


IPropositionj 15. It is necessary that there be an average distance between 
the visible lobject] and the eye. 
Indeed the eye does not gra sp the visible thing, except when there will 
have been in it some average light. by the first Iprop.] of this Ibook]; 
moreover this is not Ithe case] except through some average distance. There- 
fore whcn the eye will have been superimposed on the object without 
an intervening Idistance]. then the same will not be seen. For things 
luminous by themselves cannot be applied immediately to the surface of 
the eye. Indeed such are the stars and fi re. which cannot be applied 
immediately to the eye. because as a result of their application. the cor- 
ruption lof the capacity] of seeing would follow. Furthermore the remaining 
non-Iuminous bodies. should they be applied Idirectly] to the eye. would 
not be seen without light; therefore some intervening distance is required 
between these bodies and the surface of the same eye. in which the 
forms of those bodies could multiply themselves by means of light. 
And even when the visible bodies have been applied to the same 
eye d irectIy. then the body of the eye is prohibited from the visual 
operation by its position. Now since vision does not take place except 
in the side opposed to the aperture of the uvea. as is elear by the 4th 
Iprop.] of this Ibook]. therefore. if the eye would grasp the visible thing 
by direct application. it would not grasp it except according to the part 
applied to the opening of the uvea. and it would not grasp the remainder
>>>
122 


of the visible object. And if the visible thing were imagined to be moved 
over the surface of the eye until the eye would touch that whole thing, 
on account of this the discernment would not be by sight lalone). but 
rather by touch lalso); nor. for that matter, does the visible form, which 
is a form multiplied outside the sensible thing, act in the eye in this 
manner but [rather) the thing itself [acts) I. Therefore vision will not occur 
except when there be a certain intervening distance between the visible [thingJ 
and the surface of the eye. and this is what was proposed. 


[Proposition)16. Vision does not take place without pain and suffering 
endured by the substance of the eye. From which it is elear that the 
eye ought to be of an adequate disposition in [its) health in order to 
prosecute completely the [process ot] vision. 
Indeed since the glacial receives the form of light and color, and light 
and color toil in the glaciał. that work will necessarily be not without 
pain, however often that pain may not be felt. simply because it is not 
strong enough. In truth strong lights narrow the [pupil of the) eye and 
hurt the same manifestly, as is elear in the sun's light. or in light re- 
tlected by polished bodies to the eye. And since the operation of all 
light in the eye is of one genus, not diversified (but) according to more 
or less. and the greater [degree 00 operation of any light in the eye 
belongs to the genus of pain, and these I degree s) do not vary in this 
except according to more and less. accordingly even when pain is sometimes 
sparing to the same sense [of vision), still that suffering. however insensible, 
is always endured by the substance of the eye. 
Therefore it is elear from this that the eye ought to be of adequate 
disposition in its sanity in order to prosecute completely the [process ot] 
vision. because the grasping of visibles by the eye is always [performed) 
according to the strength of the eye. for the sense of sight of the eyes 
is diversified according to the vigor and weakness of the same; indeed 
the humid [parts) of the eye are hurt faster by Iights and colors and the 
dry [ones) less. And this we wanted to show. 


IProposition117. Distinct vision takes place solely alonl! perpendicular 
lines drawn from the points of the visible thing to the surface of the 
eye. From which it is elear that any visible form is to be arranged in 
such a manner in the surface of the eye as it is arranged in the surface 
of the visible thing. 
Indeed one is at liberty [to assume), as is obvious in the 6th [prop.) 
of this [book). that the whole form of the visible thing acts on the eye 
and [moreover) in any point whatever of the eye's surface. Ali the same, 
since. by the 20th [prop.) of the first [book) of this [treatise). the form 
of only one point of the entire surface of the visible thing opposite to 
the eye is perpendicularły incident to any one point of the eye's surface,
>>>
I I 


123 


and the forms of all the remammg points of the surface of the visible 
object come to that same point of the surface of the eye along oblique 
lines, by XI. 13 J, and [since] the forms of all the points that are in 
the surfaces of all visible objects facing the eye at a given time pass 
through any point of the eye in the very same time, because it is postu- 
lated at the heginning of this Ibook] that the eye sees simultaneously 
various visible things 2 , while only the form of the point that is perpendicularly 
incident to that point of the eye's surface passes straight through the 
transparency of all tunics of the eye, by the 47th [prop.] of the secon"d 
[book] of this [treatise]. whereas the forms of all other points are refracted 
and pass through the transparency of the eye's tunics along lines oblique 
to the surface of the eye, and [since]. moreover, only one perpendicular 
to the eye's surface exits from any point of the surface of the glacial, 
because, as the centre of the glacial and [that] of the entire eye are the 
same. as is elear by the 7th [prop.] of this [book], any line that will 
have been perpendicular to one surface will [also] be perpendicular to the 
other surface, by the 74th [prop.] of the tirst [book] of this [treatise]\ [it 
follows] agai.n [that] just as an indetinite [number ot] lines that are oblique 
to the surface of the eye issue from the same point of the surface of 
the glacial sphere to the surface of the eye, assuming that rays emerge 
from the eye. in the same way, from a certain point of the surface of 
the glacial from which there emerges a perpendicular to the surface of 
the eye and [which perpendicular] passes through the open ing of the uvea, 
there issue forth an indetinite [number ot] other lines passing through 
the open ing of the uvea and reaching the surface of the eye obliquely4. 
And just as the rays 5 imagined to issue forth from the eyes, when 
conceived to be refracted according to the kind of difference in the transparency 
of the cornea as compared to the transparency of the air, by the 4th 
[prop.] of the second [book] of this [treatise], reach at the same time 
to various places and to various points in the surfaces of visible things 
facing the eye, and none of these lines goes toward the point that lies 
in the extremity of the perpendicular 6 , just so too, according to our 
position that rays do not issue forth [from the eye], but that the forms 
are [rather] being radiated toward the eye, the forms of the visible points 
which are at the extremities of these lines are extended along the straightness 
of these lines and reach to the surface of the eye and, by the same 
47th [prop.] of the second [book] of this [treatise], are refracted to the 
same point of the surface of the glacial [humor]7. 
Moreover only one point which lies at the extremity of the perpendicular 
is not refracted, but [its form] is always extended according to the straightness 
of the perpendicular and comes to that point of the glacial 8 . And so if 
the glacial would perceive according to non-perpendicular lines, then the 
points that are in the surfaces of visible objects would never be arranged 
in the sense [of sight] according to their mo de of arrangement in the
>>>
124 


surface of the visible thing. because forms Ithat are) mingled together out 
of many diverse forms and out of diverse colors would meet in the same 
point and nothing would be distinguishable in theml}. 
But if the glacial senses only according to perpendicular lines. then 
the points that are in the surfaces of visible things will be distinguishable 
in it. nor will there be any difference of location and' arrangement between 
the visible forms in the surface of the glacial and in 1he external visible 
things. Now since. according 10 our postulate. the forms of visibles come 
to 1he eye with the I same) shapes they have in the outside things III. it 
is elear that vision takes place soleły according to perpendicular lines; 
indeed only then is the visual form ordered in such a manner in the surface 
of the eye as it is ordered in the surface of the visible thing. The pro- 
posed Ithing) is therefore elear. 
Accordingły all 1he lines of diffusion II of any of the visible forms that 
are perpendicular to the surfaces of the eye's tunics are contained in a cone 12, 
the vertex of which is the centre of the eye and the base of which 
is the cirele of the aperture of the uvea. or a part of the surface of 
that cirele. And the more this cone is extended and removed from the 
eye. the more it is amplified. all the forms of objects falling inside that 
cone are extended alon
 the straightness of radial Iines and pass through 
the tunics of the eye unrefracted. and this cone we cali the cone of 
radia' ion. 
In fact the forms of visible things "hich are outside of thls cone are 
never incident along any of those perpendicular lines. but it happens by 
chance that the same are extended along straight lines that lie between 
the same Ipoints) B and the surface of the eye. opposed to the opening 
of Ihe uvea. and those forms are refracted by the transparency of the 
eye's tunics and do not arrive well ordered to the visual power; whence 
distinc1 vision does not take place along those. and yet it happens that 
those refracted forms can be seen somewhat. but indistinctly. nameły at 
their meeting Ipoints) with the perpendicular Iines drawn from the centre 
of 1he eye outside the cone of radiation H. 
Furlhermore we now calI surface oj (he eye that part of the eye's 
surface that is opposite to the surface of 1he aperture of the uvea. That 
the eye. however. may grasp sometimes leven] those Ithings) that are outside 
the cone of radiation is elear experimentalIy. lndeed the extremity of a needle 
Ol' of a delicate straw having been placed at the very boundary of the 
eye. for instance between the eyelids Ol' in the lachrymal part. will Inever- 
theless) be seen when 1he eye is quiescenł. even 1hough that extremily 
may be outside the cone of radiation. In a similal' way. moreover. the 
forefinger Ol' ano1her finger having been placed about the eye in the same 
places. outside 1he cone of radiation (which is exceedingly sublle. because 
its conical shape is not ample. whence nothing belonging to it reaches 
lo the places that surround 1he eye). 1he surface of the same index finger 
Ol' ()f fhe o'her tingel' will. however. he scen.
>>>
125 


Accordingly the form of these visible objects comes to the surface of 
1he eye along oblique Iines that are outside the cone of radia1ion. Ił is 
elear. therefore. that the forms of things placed in this fashion with respect 
to the cone of radiation reach the surface of sight 15 through refraction 
undergone in Icrossing (wer] the surface of the eye from the air. 1he trans- 
parency of which is rarer than Ithat] of the tunics of the same eye. 
That moreover the refrac1ion of forms obliquely inciden1 to the eye takes 
place in the surface of the same eye is also elear in 'the case of] those 
lobjects] whose forms. unless prevented 'from it] would faIl inside the cone 
of radiation. ff indeed a needle. or another subtle smali thing. directly 
opposite to the opening of the uvea. were interposed between the eye 
and a white wall. nevertheless the form of the entire wall would be seen, 
even though actually the form of Ithat] part of the wall directly opposed 
to the needle and to the eye cannot come directly to the surface of 
the same eye. but lafter all] does reach ił. as is elear from the fact that 
it is seen. 
Ił is plain. therefore. that it reaches Ithe eye] through refraction undcrgone 
in 1he surface of the same eye. Nevertheless all these Ithings] are seen 
indistinctly. whence. when they have been brought back within the cone 
of radiation. and any in1erposed body has heen removed. their forms will 
be seen directly and better than before. Therefore distinct vision takes 
place only along perpendicular lines drawn from the points of the visible 
thing to the surface of the eye. while indistinct vision takes place along 
non-perpendicular lines. and in this manner indistinct vision assists the 
distinct. 


IPropositifln] I 8. Oistinct vision flf all visible fflrms takes place accflrding 
tfl a cone. the vertex flf 
hich is in the centre of the eye. while lits] 
base lis] in the surface of the visible thing; from 
hich it is elear that 
every1hing that is seen Imust] be seen under an ,mgle. 
Since. by the 6th IPwp.] of this Ibook]. aIl vision takes place through 
the action of the visible form in the eye. and Isince] any part of the 
visihle form and lany] point multiply themselves to the entire surface of 
the eye through the external medium. and the entire surface of the visible 
thing Imultiplies itselfl to lany] nne point of the eye. land] since. moreflver. 
the eye's tunics are of another transparency than the external air. Itherefore] 
those Iines of Ithe visihle] form s produced from the surface of the visible 
thing to the surface of the eye which. Iwhen] extended. pass through the 
centre of the eye. as being perpendicular to the eye's surface. are not 
refracted in the transparent medium of the same cornea. as is elear by 
the 72nd Iprop.] of the first ,book] of thls 'treatise] I. and by the 47th 
Iprop.] of the second Ib
)ok] of this Itreatise]. and by the preceding; in 
truth all other lines are refracted because they are obliquely incident. from 
which lit foIlows! 1hat Idistinct] vision does not take place along those 
Ilincs].
>>>
126 


Moreover since only the glacial 'humor] is the proper organ of sight. 
and not the surface of the eye that is part of the sphere of the cornea. 
it must happen necessarily that the lines along which vision ought to 
take place come to the glacial [humor]. And since it is not possible for 
the sight to grasp the visible thing according to its [true] being, except 
when it apprehends the form of one point of the visible thing by means 
of only one point of its own surface, because, as is elearly shown in 
the preceding [prop.], the whole form of the visible thing is thus arranged 
in the surface of sight as it is ordered in the surface of the visible 
thing. therefore it is not possible for the glacial to comprehend the visible 
thing according to its [true] being except when it grasps the color or 
the form com ing to it from one point of the visible object by means 
of only one point of the surface of sight. 
And since the centre of the eye and the centre of the glacial sphere, 
as is elear by the 7th [prop.] of this [book], are one [and the same] 
point. it is necessary that aH lines drawn perpendicularly from the points 
of visible lobjects] to the transparent surface of the eye meet in the centre 
of the glacial; and furthermore they will be diameters in the surfaces 
of the eye's tunics [and] perpendicular to the same tunics of the eye 2 . 
And any perpendicular cutting the surface of the cornea in one point 
will also cut the surface of the glacial in one point. And only one per- 
pendicular passes through any point of the glaciał. from the centre of 
the cornea through the very surface of the cornea opposed to that point 
of the glacial, that is [also] perpendicular to the surface of the visible 
thing, because, by the 20th [prop.] of the first [book] of this [treatise], 
only one perpendicular can be drawn from a certain point to one surface. 
From which [it is obvious that] when the surface of the visible object 
will have been paralleI to the surface of the same eye, that line will be 
perpendicular to the eye's surface and to the surface of the visible object, 
by the 23rd [prop.] of the first [book] of this [treatise]3, while alI other 
lines are oblique to the surface of the visible object, even though [when] 
drawn to the centre of the eye they would be perpendicular to the eye's 
surface and to the surface of the same glaciał. Therefore the form of any 
point of the surface of a visible- thing, moved to the eye along one 
perpendicular line drawn from the point to the eye's surface, cuts the eye's 
surface in one point in which no form of any of the other points of 
the visible thing 4 cuts it. Hence lines having been produced from any 
point of the surface of the visible thing to the centre of the eye, it 
is plain that these produced lines will cut the spherical surface of the 
eye in diverse points of the eye and that they will alI meet in the centre 
of the eye, since alI these lines are contained, as it were, in one "continuous" 
body 5, because, [coming] from quite contiguous points as it were of the 
surface of the visible thing, they are [alI] terminated at one point. which 
is the centre of the eye.
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127 


lt is płain. therefore. that alI these imaginabłe łines are. in a certain 
cone having litsl vertex in the centre of the eye and the base in the 
surface of the visibłe thing; indeed the form of any point of the surface 
of the visible thing will be extended along the straightness of the line 
łying between that point and the vertex oC the cone that is the centre 
of the eye. And alI the tunics of the eye and the surfaces of the humors 
cut this cone. because the forms pass through them; and, on account 
of the fact that the convex surface of the gładal cuts the cone so to 
speak paralIeły to lits] base, a quite new "cone" is formed in that surface 
of the glaciał. the base of which is in the same surface of the głaciał, 
and Iwhose] vertex lis] where lit was] previously. and the bases of those 
"cones" are made quite simiłar, as is elear by the 99th and IOOth Iprops.] 
of the first Ibook] 0'- this ltreatise]ti. And from this it is elear that alI 
that is seen is to be seen under an angłe lnameły that angle] which the 
radiał lines meeting in the centre of the eye contain. The proposed lthing] 
iso therefore, elear. 
And so the straight łine passing through alI the centres of the eye's 
tunics toward the płace of gyration 7 of the nerve's concavity, over which 
the eye is constructed is calIed the axis of the cone of radiation. since 
that łine, as is elear from the preceding and the ł2th lprop.] of this 
Ibook], passed through the centre of the eye, and through the centre of 
the aperture that is in the anterior of the uvea. and through the centre 
of the same uvea. [and] is extended through the middle of the cone of 
radiation; the other lines of this cone. moreover, are calIed radiał lines. 


IProposition]ł9. The visibłe body ought to be of a certain size with 
re gard to the surface of the eye for it to be actualIy seen. 
Indeed now it is evident, by the preceding Iprop.], that vision takes 
płace always by means of a cone the verte" "f which is in the cent re 
of the eye and [whose] base lis] in the surface of the visibłe thing, and 
that this cone marks off from the surface of the sensing organ a smalI 
part in which the form of the visibłe thing is ordered. as is elear by 
the 17th [prop.] of this [book]. Consequentły, in the case of exceedingly 
smali things. the cone will be smalI. and the part marked off by it from 
the convex surface of the glaciał. which is the first sensing organ, will be 
ałmost like a point, or exceedingły smalI. But the sensing organ does not 
perceive the form except when the part of its surface to which the form 
arrives will have been of a sensible size with respect to the entire eye, 
because the powers of the sense are finite and they do not extend indefi- 
nitely: from which lit is elear that] they must be matched up with a certain 
limit to which the sensitive power lof the eye] can reach. 
Hence since the part of the sensory organ to which the form comes 
is not of a sensible size when compared to the entire sensing organ. therefore 
the organ does not sense the action that the form of the visible thing
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128 


performs in that part. because of the smallness of the same. for which 
reason it Icannot] comprehend the form of su ch a smali thing. And so 
only those objects "re actually perceptible. the cones of which. between 
the object and the centre of the eye. mark otf in the surface of the glaciał 
a certain part of sensibłe size with re gard to the whołe surface of the 
głaciał; therefore those things ought to be of a certain Iminimał] size 
with respect to the eye's surface. And this is the proposed Ithing]. 


I Proposit ion]20. Vision i,s not compłeted except when the arrangement 
of the form received in the surface of the głaciał will have reached to 
the common lopticał] nerve I. . 
Indeed since. as is elear in the 4th Iprop.] of this Ibook]. the visual 
power Ithat] perceives and discerns all that is visihłe is constituted when 
bl)th optical nerves meet in the anterior part of the brain. for which 
reason there prevaiłs unity in the sense of sight in seeing one Ithing], 
a umty from which there happens tha1 one and the same thing is to 
be seen simultaneously by both eyes. Itherefore] it is elear that vision 
will not be completed except when 1he visibłe form would be united to 
the sensing power. which is łocated in the concavity of the common nerve 2 ; 
indeed the knowabłe ought to be ałways united to the same knowing 
lorgan]. In fact since. by the 17th Iprop.] of this Ibook]. the arrangement 
of visibłe forms in the same surface of sight is performed according to 
Itheir] arrangement in the surface of the visihłe thing. and. by a postulate 
of this Ibook]\ the visibłe thing is seen according to lits] position. shape. 
and the order of its parts. it is necessary. therefore. that the arrangement 
of the form in the same common nerve be made according to the manner 
of lits] arrangement where it is received in the surface of the głaciał. 
and otherwise vision will not be completed. The proposed Ithing] iso therefore. 
elear. 


IPropositlon]21. It is necessary that the transparency of the vitreous 
humor be ditferent from Ithat] of the głaciał. 
Indeed if the transparency of these two hodies. namely the glaciał and 
the vitreous humor. were the same. then (as it is elear from the first 
Ipwp.] of the second Ihook] of this Itreatise]. and fwm the 17th Iprop.] 
of this Ibook]. and from the 72nd Iprop.] of the first Ibook] of this Itreatise])'. 
lit is manifes1] that the visihle forms received in the surface of the glaciał 
ałong radiał lines IwiII] not be refracted land] will meet in the centre 
of the eye. because of the łikeness in transparency. and. while intersecting 
there. would Igo on] spreading heY(lOd Ithe centreJ. 
Now. as is elear fwm the preceding Iprop.]. vision is not compłeted 
except after the arrangement of the form which is received in the surface 
of the głaciał reaches 2 to the common nerve. whiłe the position of the 
parts ()f the form. according to its heing on the surface of 1he glaciał.
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129 


cannot reach to the com mon nerve except through its extension.
 in 1he 
concavity of the nerve over which the glacial sphere is set up. since 
it is impossible for it to reach [to the nerve] ditferent1y; in fact the form 
cannot be extended from the surface of the glacial to the concavity of 
the com mon nerve along the extension oP straight lines and [still] preserve 
the position of its parts according to its [true] being, unless it encounter 
the nature of another elearer transparency before reaching to the centre 
of the eye. because. if the łintervening] medium were not of another trans- 
parency. all these lines would meet at the centre of the eye and one 
[common] point 5, as it were, would be yielded Iby them]. 
And since this centre of the eye is before the place of union of the 
optical nerves. it is elear. by the 91 st [pro p.] of the first [book] of this 
Itreatise]6. that if those Iines ought to be extended beyond the centre 
of the eye. the intersection of those lines will necessarily be in the centre, 
and after the centre a new cone will be created. the lines of longitude 
of which will have an inverse relation with regard to position and order 
to [those] of the previous cone. Therefore the entire position of the figure 
of the visible thing will be turned around, [that is the position] it has 
on the surface of the visible thing and in the surface of the glaciał. in 
such a way that what is right in the surface of the glacial would be 
left at the sense and vice-versa, and the higher would be lower and vice-versa; 
nor would anything of the form come directly to the common nerve. 
except only one point that is on the extremity of the axis of the cone. 
Therefore every thing would be seen in an opposite manner to its natural 
position. which is against a postulate and manifestly against that which 
happens in perception. 
It is elear, therefore. that it is necessary that these humors be of dif- 
ferent transparency, which is the proposed [thingJ. 


[Proposition]22. It is necessary that the surface of the common section 
of the glacial and vitreous spheres be located before the centre of the 
eye, and tha1 the vitreous humor and the visu al spirit be of quite the 
same transparency. and that each of them be more transparent than the 
glacial humor. 
Since. as obvious by the 20th. łprop.] of this [book], every form of 
a visible thing comes to the common nerve preserving its position, shape, 
and the arrangement of its parts. it is plain. as is obvious in the preceding 
łprop.], 1hat it is necessary that there be some refraction before the arrival 
of the form to the centre of the eye. hecause even if there were a re- 
fraction after the passage through the centre. the forms would necessarily 
be turned around. as even then. by the 91 st [prop.] of the first łbook] 
of this [treatise] I, the arrangement of the parts of the form would be 
changed. 
In fact since refraction takes place only towards the perpendicular or 


9 - Witelonio Peropeclivae...
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130 


away from the perpendicular. as is elear by the 47th [prop.] of the second 
Ibook] of this Itreatise]. it is plain that the arrangement of the parts would 
not be changed 2 , but only Ithat] the shape lof the object] would increase 
or diminish. by the 49th [prop.] of the second Ibook] of this [treatise). 
Moreover since the glacial. to which the forms come in Icomplete] straightness, 
is wholly of one transparency. while refraction does not take place except 
by means of another transparency, it is plain that the refraction of the 
forms cannot take place except at the vitreous humor. the body of which 
is of a different transparency than the body of the glacial. as became 
manifest in the preceding Iprop.]. Therefore this humor necessarily comes 
before the centre of the eye. so that the forms may be refracted by it 
before reaching to the same centre of the eye. which is identical with 
the centre of the !!Iacial humor. by the 7th Iprop.] of this Ibook]: for 
otherwise the meeting of all the radial 'ines \ would take place in thaf 
centre. hy the 72nd Iprop.l of the first Ihook] of this Itreatise) '. 
I Moreover] since those lines are all perpendicular to the surface of 
the glacial. a change of position would have happened to those forms 
on their advance beyond Ithe centre]. by the 9'st [prop.] of the first Ibook] 
of this Itreatise] 5, as was established before. and since this is impossib'e. 
it is. therefore. elear that the vitreous humor comes before the centre 
of the glacial. Accordingly, ever so much as the glacial. in which resides 
the principle of 'perception. stands in need of radia' lines extended in 
accordance with [their] straightness (for it is impossible that the form of 
the visible thing be arranged in the surface of sight but by means of 
these lines through which the comprehension of the visible thing in accordance 
wit h its being is completed. because of the magnitude of the visible thing 
and on account of the unity of the surface of the body of sight) still. 
the arrival of the forms to the ultimate sentient power does not require 
only the extension of the forms in accordance with the straightness of 
these lines. because the reception of the forms in the sentlent member 
is not altogether similar to the reception of the forms in a transparent 
body: indeed the sentient member receives these forms on account of its 
diaphanousness and senses them on account of its sentient power 6 . And 
thus it receives the forms in accordance with a sentient reception. while 
other transparent bodies 7 receive the forms mainly to exhibit the same 
to the sight and not to sense [them]. Hence the kind of reception of 
forms in the vitreous humor along refracted lines is due to its difference 
in transparency from the body of the glacial and to the qua'ity of sensible 
reception that is not complete in the glacial humor 8 . 
But it is necessary that the subtle body that is in the nerve's concavity 
between the vitreous humor and the common nerve. which body is called 
the visible spirit 9 : since in it the visible spirits run through for the first time. 
be transparent. because when the forms of visible things arrive in the 
body of the vitreous humor. the rcapacity for] feeling is extended from 


... 


.......
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131 


it in the sentient body spread thn'ughout the c('ncavity of the nerve joining 
the eye to the anterjor Ipart] of the hrain. and the forms. ordered according 
to its disp,'sition. are extended in accordance with the extension of this 
sense lU. II is elear. therefore. that the Iproper] arrangement of the parts 
of the h()dy sensing the forms. as well as the strucfUre pf the sentient 
powet litselfL are equally ne','essar} in the vitreous body and in any subtle 
body extended in the concavity of the nerve. 
Indeed. as the form arrives at some point ('f the vitreous sur face. it 
is directly e\tended and its rosition IS not changed in the concavity of 
the nerve in which the sentient body is extended. and the forms of all 
points will remain mutually arranged in the same manner. And so the 
sentient body that is in the concavity of the nerve will necessarily be 
transparent on account of the reception of visible forms. and its transparency 
will be quite the same with the transparency of the vitreous humor. so 
that the forms may not turn aside Ol' be monstrous at their arrival at 
the terminal surface of the vitreous bordering on the body that is in 
the concavity of the nerve. Therefore the forms traverse this subtle body 
by reason of lits] transparency and manifest themselves before the sensitive 
power by reason of the consistency of the same body. 
And the ultimate sentient power II that is !located] in the common 
nerve comprehends light on account of the illumination of its body. and 
color on account of its coloration. because the light forms traverse lit] 
and are fixed in the same Ibody]. Moreover the refraction of forms takes 
place at the vitreolls humor as much on account of the lintrinsic) variety 
111 1he quality of reception of the sense as on account of the difference 
in the transparency of the glacial and vitreous humor. And if the trans- 
parency of their bodies were alike. the form would be extended in the 
vitreous body according to the straightness of radial lines Iprecisely) because 
of the likeness of transparency. and would be refracted (only] on account 
of the diversity of the linherent] quality of sense between these two bodies; 
and the form might have been made either monstrous Ol' there would have 
been two forms I
. 
In fact when refraction takes place on accollnt of different transparency. 
and the difference in the quality of the senses strengthens that refraction 
Ol' obliquity 13. then the form will be. after the bending of 'refraction. 
one form. ordered according to the position of its parts. shape. and the 
arrangement that the form possesses in the outside thing, and the sensitive 
power senses the form of the visible thing by means of the whole sentient 
body Ithat is] extended from the first sensing surface of the eye (that 
is] receiving the forms 14 all the way to the concavity of the common 
nerve. that is the ultimate sentient body. because in it is constituted the 
sensitive power. 
And so the vitreous humor and the body that is in the concavity 
of the nerve are of quite the same transparency. because no sensible re-
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132 


fraction to speak of is undergone between the same. but [on the contrary, 
the form] is led through the unity of sensitive power to the oneness of 
the simple extension of the form aft er [its] refraction in the surface of 
the vitreous IS. And since the [proper] advance of the forms beyond the 
cen1re of the eye takes place in both of these bodies 16, it is elear that 
that [earlier] refraction was made away from the perpendicular erected 
at the point of refraction to the surface of the glacial 17; 1herefore any 
of those bodies, [i.e.. the vitreous and the optic nerve]. is more transparent 
than the body of the same glaciał. by the 45th and 47th [props.] of the second 
[b00k] of this [treatise]. The proposed [thing] iso therefore. elear. 


IProposition]23. Ił is necessary that the surface of the common section 
of 1he glacial and vitreous spherels] be pIane, or, [more accurately], be 
part of a sphere greater than the sphere of the glacial, and eccentric 
to the surface of the eye. 
The surface of the common section of these spheres, namely of the 
glacial and the vitreous, is necessarily pIane. or rather of such a nature 
as is proposed. because it is necessary that the surface of this section 
be of even arrangement. so that its ex.tremities may be ordered at an 
en1ircly similar and identical distance from the centre of the eye. in order 
that no monstrous forms appear after refraction. Furthermore a surface 
of even arrangement is either pIane or spherical. Bul this surface cannot 
be [part] of a sphere concen1ric with the eye. for otherwise the radial 
lines that are perpendicular to the surface of 1he glacial would also have 
to be perpendicular to it. by the 74th [prop.] of the first [book] of this 
[treatise] I, and there would be no refraction of the forms, but they would 
meet in the centre [of the eye], and las a result] they would become 
monstrous forms. as is manifest by the preceding [prop.]. Therefore that 
surface, should it be part of a sphere. would necessarily be eccentric to 
the eye. 
Hence it cannot be [part] of a sphere smaller than a sphere concentric 
with the eye. because, [if it were], by reason of the different centre. 
the forms would have to meet before their arrival at the centre of the 
eye; but the diameter of a smalter sphere is [itself] smaller inasmuch as 
this is of the nature of sphericity. and. on account of the greater trans- 
parency of the vitreous sphere over the glaciał. which [was) manifest in 
the preceding [prop.]. the forms would be refracted away from the same 
perpendicular to which they are incident, by the 47th [prop.] of the second 
[book] of this [treatise], by reason of the rarer transparency; however, 
should it be part of a smaller sphere. the forms would be refracted in 
the surface of common section toward the perpendicular 2 . In this manner, 
then, monstrous forms would be created, since they would proceed toward 
the perpendicular (by reason of its [being] a perpendicular to the spherical
>>>
133 


surface. which perpendiculars likewise pass always through the centre. by 
the 72nd (prop.] of the first [book] of this (treatise])3 and Iyet] they should 
be refracted away from the perpendicular. 
Therefore this surface is either piane or sphericaI. namely part of a sphere 
of a certain good size. so that its sphericity match the arrangement according 
to which the [actual] size of the refraction away from the perpendicular 
takes place on account of the nature of another transparency. Therefore 
alI forms arriving at the surface of the glacial are extended through 
the body of the glacial along the straightness of radial lines until they 
arrive at this sur face ; then they are refracted at the same according to 
lines of like arrangements. cutting the radial lines. And so the form arriving 
at some point of the surface of the glacial is always extended along the 
same incident line to the same point of the surface of sight and to the 
same point of location of the common nerve. Therefore two forms 4 
are extended to the same point in the common nerve from any two 
points of like position with regard to the two nerves, until a perfect unity 
of the forms is achieved. 


(proposi1ion]24. Ił is necessary that amongst alI the lines of the cone 
of radiation. only the axis pass ing through the centre of the opening of 
the uvea he perpendicular to the common surface of the glacial and the 
vitreous and to (he posterior surface of the vitreous. 
Indeed should this axis not be perpendicular but oblique to any of 
those surfaces. the result would be a diversification of the arrangement 
of the forms coming to that surface, and the orderings of those forms 
would be changed on account of the deelivity of the axis. Hence only 
then when the axis would have been perpendicular to the surface of the 
g1acial would the form of the visible thing arrive at the surface of the 
glacial arranged in accordance wit h the order of the parts of the surface 
of the visihle thing. and the form of the point that lies in the extremity 
of the axis in the surface of the visible thing will arrive at the point 
that lies on the axis in the surface of the glacial. as is elear by the 17th 
[prop.] of this [book]. 
And since the radial axis is perpendicular to the surface of the glacial, 
it is plain, by XI. 18 l. that alI piane surfaces issuing from the axis and 
cutting the surface of the glacial will be perpendicular to this surface. 
And since the surface of the vitreous humor facing the same glacial sphere, 
which [surface] is the common section of the glacial and vitreous sphere[s]. 
as is elear by the preceding [prop.], is either a piane or a spherical surface, 
and its centre is not the center of the eye, if. then, the radial axis is 
obliquely inelined to the same surface and is not perpendicular to the 
same. there wilI not issue from the axis a piane surface perpendicular 
to (his surface. except only one surface, namely that which forms the 


.......
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134 


greatest inequality of the angles. which is elear hy thc 39th [pro p.] of 
the first Ibook] of this [treatise]2; and aII remaining surfaces issuing from 
the axis will be oblique to the same surface of the vitreous. 
rf indeed two or more surfaces issuing from the axis were perpendicular 
to the said surface\ since those surfaces intersect by necessity and their 
common section is the axis of the cone of radiation. this axis will be 
perpendicular to the same surface. by XI. 194; but it was given that 
it was oblique
. Accordingly, let the centre of the eye be point C [Fig. 3.]; 


A 


c 


I Fig. 3] 


and in the surface of the eye, or in the same surface of the glacial, 
which. by the 7th [prop.] of this [book] and by the 73rd [prop.] of the 
first [book] of this [treatise] 6, is paralIel to the surface of the same eye, 
let there be line BAD. and in the surface of the vitreous humor facing 
the glacial humor, let there be line EGF, and let the axis or the cone 
of radiation be line AC. 
Consequently let us imagine surface A BCD, issuing from the axis and 
perpendicular to the surface of the glacial, passing through the centre of 
the eye, which is C, and this surface is also perpendicular to the surface 
of the vitreous humor which is CGF7. And let line BAD be the common 
section of this perpendicular surface ABCD with the same surface of the 
glaciał. and let points B and D be equally distant from point A. which 
is the end of the axis of the visual cone. And let its common section 
with the surface of the vitreous humor be line EF. Moreover let two 
lines go out from centre C, and let them be CB and CD; consequently 
these two lines CB and CD will be, together with axis CA, in the common 
surface perpendicular to surface EGF, by Xl. 1 8 , because all points C, 
B, D are in that surface. 
And the two angles ACB and ACD will be equal by hypothesis, which 
is elear by l. 8 9 , if the chords BA and DA are stretched underneath 
the arcs BA and AD 10. And lines CB and CD are cutting line EF, which 



 


........
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135 


is the common section of the said [perpendicular] surface and of the surface 
of the vitreous, in two points E and F. and let axis CA cut the same 
line EF in point G. If, then. the surface that is the common section 
of the glacial sphere and of the vitreous is pIane, the common difference 
which is EGF will be a straight line. And ifaxis AC we
e oblique to 
the surface of the vitreous, and the same is erected [perpendicularly] in 
surface BCD over surface EGF. then it will be necessary that axis CA 
be oblique to line EF. Therefore angles EGC and FGC will be unequal, 
because the line drawn perpendicularly from point G to line EGF will 
make, by I. II 1 I, equal angles with line EF. 
And so since angles EGC and FGC are unequal, whereas angle CGF, 
is, say, smaller than angle CGE, and the two angles ACB and ACD are 
equal. the two lines EC and CF will be unequal, by I. 2412; indeed 
line CF is shorter than line EC. If however those lines were equal, since 
angles ECG and FCG are equal and line GC common to both triangles
 
angle EGC and FGC would be equal by I. 4 I3, which is against the 
given. because axis AC is [taken to be] oblique to line EF. Hence let 
line CH be equal to line CE, and let line HG be drawn, which, by 
I. 4 and from the preceding, will be equal to line EG; and let perpendicular 
GL be drawn from point G to line CH, by I. 12 14 . Consequently, from 
the penultimate [prop.] of the first [book of the Elements] 15, side GH 
[which is] opposed to a right angle in triangle HL G, is greater than 
side GL; therefore, by the 19th [prop.] of the same first [book of the 
Elements] Iti, line GH will be greater than line GF. 
Since moreover angle GFH is an external angle [with respectJ to the 
right angle GLF, it is plain that angle GFH is obtuse 17; it is, therefore, 
greater than any of the angles of triangle FGH. Therefore 'ine EG, which 
is equal to line GH, is greater than 'ine GF. ConsequentIy the two points 
E and F will be at different distances from point. G, and these two 
points E and F are those to which arrive the forms of the two points 
of the surface of the glacial, namely B and D, which are equally distant 
from the. axis. And so points [that are] equally distant from the axis 
in the surface of the glacial are at unequal distances from the point of 
the axis [that is] incident to the surface of the vitreous, so that, this 
being the case, it is plain that the arrangement of the fo
m (which would 
have arrived from the surface of the glacial to the surface of the vitreous 
humor) would not be according to the being that it possesses in the surface 
of the glacial nor according to its being in the surface of the visible 
object. 
Therefore when the axis would have been oblique to the pIane surface 
which is the common section of the surface of the glacial and of the 
vitreous, thereby the line that is the common section of any surface issuing 
from the axis [and] perpendicular to the surface of the vitreous, and of 
the surface of the same vitreous, will be making with the axis two unequal 


.......
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136 


angles. except in only one sur face. which cuts according to right angles 
the surface passing through the declivity of the axis, because only the 
common section of this surface will contain with the axis right angles. And 
since the previously referred to angles will have been unequal, and the 
angles about the centre of the glacial equal. the two parts of the common 
section that lies in the surface of the vitreous will be unequal 18 . Therefore 
the forms arriving at the surface of the vitreous from these points that 
are in the extremities of these sections will have different distances from 
the point of the axis that is in this surface. But since the points of 
these lines [which are] in the surface of the glacial are equally distant 
from the point of the axis. the forms will be seen in the same surface 19 
not in accordance with their arrangement in the surface of the glacial 
and in the surface of the visible object. 
Furthermore [the claim] is to be demonstrated similarly if the surface 
of the vitreous were spherical and the axis were oblique to it: indeed then 
the axis will not pass through the centre of the vitreous but will, all 
the same, pass through the centre of the glaciał. Therefore the lines that 
go out from the centre of the glacial 10 the poin1s \\ hose distance from 
the point of the axis Ithat is] in the surface of the glacial is equal make 
\\ith the axis equal angles at the centre of the glacial. And since the 
centre of the glacial is not the centre of the vitreous, as is elear by 
[the 11th prop.] of this [book], these lines will cut orr unequal arcs from 
the surface of the vitreous. Indeed since line EC, as was said before 
is greater than line CF [Fig. 3A], let line CH be equal to the line CE, 


A 


[Fig. 3A] 


and let line GH be drawn. on which the described part of circle EGF. 
which is GH. will he equal to the part EG. hy m. 2P'. hecause chord 
EG is equal 10 chord GH, by I. 4
 I. Hence perpendicular GL having heen 
produced. chord GH will be. as before. greater than chord G,[. Therefore 
arc GH will be greater thaJ1 arc GF. by III. 23. Therefore straight line 
EG. which is equal to line GH. will also he greater 1han straight line 


........
>>>
137 


GF. Therefore arc EG is not equal to arc GF. by III. 28

. ConsequentIy 
no lines mak ing with the axis right angles, and issuing together with line 
AC in the same surface, cut off two equal arcs from the surface of the 
vitreous. except only the two lines that are in the surface cutting orthogonally 
the surface Ithat is itself] perpendicular to the surface of the vitreous. 
Hence should the axis be oblique to the surface of the vitreous. the 
forms arriving at the surface of the vitreous would be diffe ren t in Itheir] 
arrangement Ifrom that in the visible object]. whether the surface of the 
vitreous be pIane or spherical. In facl. should the axis be perpendicular 
to the surface of the vitreous. it would lalso] be perpendicular to all the 
tcommon] sections of any pIane surface passing through line AC and of 
the slIrface of the same vitreous. And any two lines going out from the 
centre of the glacial. which is a point of the axis. and containing equal 
angles wit h 1he axis. wilI cut off two equal parts from 1he common sectlon 
that lies in the surface of the vitreous. whether that surface be piane 
or spherical. 
And the forms are understood by the sense lof sight] in accordance 
\\ith their arrangement in the surface of the glacial and in the surface 
of the visible object. And since the comprehension of the forms takes 
place in this manner,. as is elear by a postulate. it is plain that the axis 
of the visu al cone is always perpendicular to the anterior and posterior 
surface of the vitreous humor. the reason being the same. and It is to be 
demonstrated in the same manner. Moreover all other lines will be oblique 
to these surfaces
\ because 1hey proceed just as if they can cut the axis in 
the centre of the glacial. and none of them passes through the centre 
of the vitreous, assuming it is spherical except the axis alone. by the 7:!nd 
Iprop.] of the first Ibook] of this I treatiseF ., because it alone is perpendicular 
to the same. The proposed Ithing] iso therefore. elear. 


IProposition]25. The motion of the actual eye is possible lonly] as a whole; 
it is not possible Imoreover] for the Imutual] position of its parts to be 
changed. 
It was manifest in the 4th Iprop.] of this Ibook] that there exists 
an opening in the concavity of the bone through which the optic nerve 
passes. But between this opening of the bone and the circumference of 
tha glacial Iwhich is] joined with the uvea. there is a linIe space. and 
the optic nerve is extended in this space. from the end of the opening 
all the way to the circumference of the glacial. in a conical shape. and 
it is Icontinually] widened until it reaches to the circumference of the 
glacial sphere with which it is joined. Therefore when this nerve is turned 
aside. its bending win be al the opening of the concavity of the same 
bone. And since the concavity of the bone contains the whole eye. should 
the nerve be deflected in this manner. the eye itself would be moved 
as a whole in this concavity: indeed the consolidativa which is joined 


........
>>>
138 


\\ith those Iportionsl of the nerve and of the remalmng tunics that are 
in the anterior Iregion} of the eye is always preserving position. 
Therefore. the deflection of the nerve when the eye moves does not 
take place except at the bad of the whole eye. Hence it is not possible 
for the Imutual] arrangement of the eye's parts to be changed. because, 
as hecame elear by the 7th fprop.] of this Ibook}, the centre of the surface 
of the eye's tunics opposed to the aperture of the uvea and of the cornea 
is the same with the centre of the eye. Therefore just as the eye will 
be moved, the centre of the eye will not be changed, because. when some 
sphere IS slightly moved, the position of its centre is not lalso] moved 
as a result of this. Thus nor is the centre of the surfaces of the tunics. 
opposed to the aperture of the uvea changed. Therefore the position of 
the eye's tunics iso also not changed. 
Indeed since the hne passing through the centres of all tunics and 
humors of the eye passes through the middle of the nerve's concavity 
Ibein
} orthogonally erected to the base of the nerve's cone. as is elear 
hy the 9th Iprop.} of this Ibook}, and Isince] the line that passes orthogonally 
through the centre of the base circle of any cone necessarily traverses 
the vertex of the cone, by the 89th Iprop.} of the first Ibook) of this 
Itreatise] I, land since] moreover when the eye is moved. the verte\ of 
the cone of the optic nerve in [this) concave cone is not changed. it is 
necessary [thereforeJ that the motion of the eye [take place] as a whole 
[and that the position ot] its patts not be changed in any manner because 
the line that passes through the centres of those part s passes [also] through 
the middle of the concavity of the optic nerve, by the 9th [prop.] of 
this [book]. From which it is c1ear that the [mutual position of the] parts 
of the eye cannot be changed in any manner. Indeed the oblique inelination 
of the conical part of the nerve to the surface of the cirele of consolidation 
is always the same 2. Therefore the parts of the eye are not changed when 
it comes to their mutual layout. And this is the proposed [thing]. 
And since both eyes are of like arrangement in their tunics and parts, 
and in the shapes of their tunics, and in the position of any of the tunics 
with regard to the whole eye, it is elear that there is no difference between 
them concerning that which is being proposed here about the change of 
position of their parts, when the same [two] eyes have been moved. Indeed 
the position of both lines passing through the centres of the eyes' tunics 
in each of the eyes is always a Iike position in all arrangements of the 
eyes. And so that which was proposed is elear. 


[Proposition]26. Having moved one eye, it is [always] necessary that 
the other be [concomitantly] moved in confonnity with the first. 
Indeed since the position of the eye's parts does not change in any 
of the eyes, and [since] the motion of one eye takes place through the 
motion of the optic nerve in the centre ot" the bone s opening, while
>>>
139 


the motion of the individual nerve proceeds from the point of the common 
nerve, because always that which is moved in some of [its] part s is moved 
about something fixed, therefore the motion of the [individual] partial nerve 
begins in the point of the com mon nerve of both optic nerves of both 
eyes. in which exists the sentient and moving power of the soul. 
And since that power is indivisible and uniform, and [since] the principle 
to the end of which it moves at first is a naturaI body, indivisible 
according to its natural form. it is plain that when one eye is to be 
moved, the other moves also. Nor is the ratio by which one eye move
 
greater than that by which the other Imoves]; and so one eye havin! 
been moved. both eyes are being moved, and one is moved in conformity 
with the other. as if the motion of both begins from the same principle. 
[and] thus both motions are terminated at the same boundary, and precisely 
as they begin from one indivisible, thus are they terminated at one divisible I. 
That which was proposed is therefore elear. 


[proposition]27. When the two eyes are directly opposed to one visible 
[objeet], it is necessary for two cones to be shaped. the common base 
of which is the surface of the visible thing, and Iso that] the axis of 
each passes through the centre of aperture of lits] uvea and through the 
centre of its eye. 
Indeed since. as is elear by the 17th [prop.] of this [book], the position 
of the parts of the surface of the visible thing arrives at the surface 
of each eye and is fashioned in that [eye] according to perpendicular Iines 
drawn from all points of the surface of the visible thing to the surface 
of that eye, the meeting of all of which, according to their points of 
incidence, aims at the centre of the eye, to the surface of which [each 
perpendicular] is incident, and not till after refraction does each of those 
shapes arrive at the midd1e point of the common nerve, therefore the 
meeting of bot h of those forms takes place in the middle point of the 
common nerve to which they are incident. 
And sa since the centres of the twa eyes are twa, it is plain that 
in the vision of the same thing by 
oth eyes, twa visual cones are fashioned 
in the proposed manner. Indeed the surface of the visible thing will always 
be the base of each cone proceeding from each of the eyes, on account 
of the equal multiplication of the form of each point of the surface of 
the visible thing to the eye, and the axis of any of them passes through 
the centre of aperture of the uvea to the centre of its eye. 
Indeed. by hypothesis, precisely as the visible [objeet] is Iplaced) opposite 
one eye, thus it is also directly opposite the other [eye]. And since both 
eyes are moved equally in order that anything be seen, by the previous 
[prop.], it is elear that, always, in seeing one thing, the middle point 
of the visu al surface of the eye is opposed to the middle point of the 
surface of the visible things. or c10se to it; moreover the middle point
>>>
140 


of the surface of sight or of the eye is the centre of the aperture of 
the uvea " by the 4th [prop.] of this [book]. Therefore the form of that 
middle point of the surface of the visible thing, or of a point c10se to 
it. arrives at the centre of its eye through the centre of aperture of the 
uvea. And this is the proposed [thing]. 


[Proposition]28. It happens that only a single form of one thing is 
to be seen by the two existing eyes. 
Indeed since, as was previously stated repeatedly, the form received 
in the surface of the glacial passes through the body of the glaciał. after 
which it is extended through the subtle body that is in the optic nerve " 
and comes to the anterior (part] of the brain in which the uItimate sentient 
body is located, which is the sensitive power comprehending sensibles. of 
which power the eye is the instrument. receiving the forms of things and 
returning them to the ult;mum sent;ens. in such a manner that vision is 
completed only then [when the forms arrive] at the common nerve of 
both eyes, the position of which nerve is similar in all respects with regard 
to both eyes, therefore it is allowed that two forms arrive in the two 
eyes from one visible thing. . 
Furthermore both of these forms meet when they arrive at the common 
nerve and become one form, and through the union of these forms the 
ultimum sentiens understands the form of the visible thing, and thus it 
happens that only one form is seen of [each] individual thing; unless perhaps, 
due to some intervening circumstances, it would happen that the forms 
received by the two eyes would not be united, because they do not meet 
in the union of both optic nerves. Then indeed it would happen that two 
forms would be seen. as when the beholding [individual] would have moved 
the position of one eye toward the front. while the other eye would have 
remained unmoved 2 . 
However when the position of the two eyes would be natural, then. 
since the position of the same with regard to one visible thing is an 
entirely like position, the form [stemming] from one visible thing will arrive 
at two places of entirely like position; and when the position of ronly] 
one of the eyes would be turned aside, then the position of the rtwo] 
eyes with regard to that visible thing would be different. and thus two 
forms of different layout of that visible thing would arrive rat the two 
eyes]; but this is not naturally characteristic to the eye, as rit happens] 
only on account of violence brought about intentionally, or ron account 
ot] natural weakness rbefalling] nature's usage. 
And so when the position of the eyes would be n atu raI. then it always 
happens that ronly] one form of one thing is to be seen by bot h eyes, 
which is the proposed rthing]. Therefore the two forms of a visible point 
are impressed in the two centres of the two surfaces of both eyes, and any 
other point of the visible form is impressed in two rother] places of entire1y 


.110....-
>>>
141 


like position in the two eyes. After that the two visual forms arrive at 
the concavity of the common nerve, and the two forms which are in the 
point that is in the two axes of those cones of radiation according to 
which vision takes place arrive at the point that is in the common axis, 
and are made into one form. And any two forms that are in two points 
of entirely like position with respect to the two eyes arrive at the same 
point among the points surrounding the point that lies in the common 
axis. In this manner, then. the two forms of the whole visible object 
are superimposed to one another and become one form, and in this way 
one visible thing is being grasped 3 . 


[Proposition]29. Ił is necessary that any point of the [visual] form [that 
is] incident to the surfaces of the eyes along the radial axes reach to 
the centre of the gyration-aperture of the concave nerve. 
Indeed since any of the axes passes through the centre of aperture 
of the uvea [go ing] to the eye's centre, as is cIear by the 27th [prop.] 
of this book, therefore it also passes through the centre of the same 
uveal sphere, by the 8th [prop.] of this [book] I. In fact any straight line 
drawn between the centre of the eye and [that] of the uvea will necessarily 
penetrate the centre of the circ1e of section of the uvea and the middle 
point of the nerve's concavity, by the 9th [prop.] of this [book]. It is 
plain. therefore. since the perpendicular remains always unrefracted, by the 
47th [prop.] of the second [book] of this [treatise], that any point of the 
form incident to the surfaces of the eyes along the radial axes must necessarily 
reach to the gyration centre of the com mon nerve. Furthermore from this 
point the form is extended to the middle point of the common nerve, 
and since there is only one middle poin1 of the com mon nerve, it is 
plain that the axes of bot h eyes meet always in one point of the common 
nerve. The proposed [thing]' is, therefore, cIear. 


[Proposition]30. Jf two straight lines are drawn to the centre of the 
common nerve from the ends of the line joining the two gyration centres 
of the apertures of the concave nerves. it is necessary that, in the constituted 
triangle, the base angles be equal; from which it is cIear that those drawn 
lines are equal. 
Let the two centres of the gyration apertures of the concave nerves 
be R and T [Fig. 4], between which let line RT be drawn, and let the 
middle point of the com mon nerve be A, and let the triangle RA T be 
formed. J say that angle ART is equal to angle ATR. Indeed since the 
position of the two nerves with respect to the concavity of the common 
nerve is an entirely similar position, because the concavity of one nerve 
is altogether similar to 1he concavity of the other, by the 4th [prop.] 
of this [book]. therefore the centre of concavity of one is entirely similar 
to the centre of concavity 'of the other. whence the axis of one nerve 


.......
>>>
142 


A 


R 


T 


[Fig. 4
 


is equal to the axis of the other nerve. But, by the same 4th Iprop.) 
of this Ibook). the position of the two nerves (in the centre of which 
nerves lines RA and TA are produced as axes) with respect to the two 
apertures is an entirely like position. 
It is plain. therefore. that the position of the two lines RA and TA 
at line RT is a position similar in all respects. But this is impossihle 
unless angles ART and ATR be equal. because from the inequality of these 
angles would follow the inequality of position of the middle axis of the 
same concave nerves and. consequent1y, of the nerves themselves. Therefore 
those angles at the base lof the triangle] are equal. Therefore. by I. 6 I. 
those drawn lines are equal. namely line AR to line AT. The proposed 
Ithing) is therefore elear. 


IProposition131. While Ithe form of) one point of the visihle ohject 
IS incident perpendicularly to the surfaces of both eyes. it is necessary 
that the radial axes be refracted angularly in the centres of the gyration 
apertures of the concave nerves. 
lndeed since. as is elear by the 27th Iprop.) of this [book), any of 
those axes passes through the centre of the aperture of the uvea and the 
centre of the eye. furthermore Isince) the motion of any of the eyes takes 
place in the centre of the gyration-aperture of the optic nerve, it is elear 
that those radial axes are changed in accord with the motion of the eyes; 
in those axes there are always the same semidiameters of the eye, namely 
[those) which are extended from the centres of the same [eyes) to the 
centres of the apertures of the uvea. But the superior part s of those axes 
to which the forms arrive from the centers of the gyration-apertures of 


.......
>>>
143 


the concave nerves to the middle point of the common nerve. remam 
always abiding in the same manner. 
And so since some parts of those axes are always immobile and others 
always mobile, [and] since one (and the same] point is seen through them. 
it is elear, by XI. II, that those lines are not one line. For inasmuch 
as the form of point B I Fig. 5] may be seen along both axes BR and 


A 


R 


T 


B 


[Fig. 5] 


BT (and, as it was done in the preceding (prop.]. let lines RA and TA 
be drawn to the middle point of the com mon nerve. which is A), (then] 
it is elear. by XI. I, that lines BR and RA are not one [and the same] 
line: for (otherwise] a part of it would happen to be above (t he pIane 
and] a part in the pIane, which is impossible. 
It is elear, therefore, that they are joined angularly which is the proposed 
(thing]. And it is allowed (then] that [when] the axes are refracted in the 
previously described manner. the formation of the visual cone takes place 
accordingly, but if the axes would arrive unbroken to the vertex [A], 
(then] no diversity would occur to the eye from that (situationJ2. 


[Proposition]32. It is necessary that the axes of the visual cones of 
both eyes passing through the centres of the apertures of the uvea be 
always joined in one point of the surface of the visible thing, even while 
the eyes have been moved over the [en t i reJ surface of the visible thing. 


.......
>>>
144 


Indeed when the behoJding (subject] will Jook at a certain visibJe thing, 
then each eye will be in opposition to that visibJe thing, by the 2nd 
[prop.] of this (book], and any of the pupiJs will be directed toward that 
thing in an equaJ direction, on account of the equaJity of the eyes, by 
the .4th (prop.] of this (book]. Hence Jet the two centres of the eyes be 
E and G I Fig. 6] and Jet point A be the middJe point of the common .nerve; 


A 


B 


F 


Q 


x 


c 


V. 


D 


[Fig. 6) 


and Jet BCDF be the surface of the visibJe objeet, and Jet it be, for 
instance. paralleJ to the Jine connecting the centres of the eyes, which 
is EG. 
It is pJain, therefore, that the perpendiculars drawn from the centres 
of the eyes to the same surface BCDF, which are EQ and GX, are 
paralleI, by XI. 6 1 . And so Jet a point, which is V, be marked in this 
surface BCDF. I say that, on account of the equaJity of bot h eyes in 
all their arrangements. if one eye were moved so that point V couJd 


..........
>>>
145 


be seen. the remaining [eye] will also be moved instantly so that the same 
point V could be seen, in such a manner that the axes of both visual 
cones passing through the centres of the apertures of [each] uvea will 
be joined in point V, when one of the same is reaching there. 
Indeed if [when] one of those axes is fali ing in point V the other 
would fali in another point (Jet that point be Z), then there would be 
two axes EV and GZ, between the ends of which let line ZV be drawn. 
And since the axes drawn in this manner from the two eyes do not 
meet in any of the points of line ZV, precisely as they would not meet 
if vision would take place only along perpendicular lines, like EQ and GX, 
it is plain that none of the points of line ZV would be seen [simultaneously] 
by both eyes. but only by one. Hence any one of the eyes is seen to 
be superftuous, as lonly] one of the eyes can traverse imperceptibly by 
means of its axis all points of line ZV. But nature has constituted two 
eyes on account of the perfect excellence of vision and its fulfillment, 
in order that the united power of the same be stronger, as is clear 
by the 4th [prop.] of this [book]. 
If. then, the visual axes do not meet in any one point of the line ZV, 
it follows either that nature acts superftuously or that it operates in a weaker 
fashion than it actually can. either of which [aIternatives] is impossible. 
For nature does nothing in vain, nor does it fail in [do ing] necessary 
things, as is elear by a postulate 2 . And so something impossible happens 
if the axes fali only on different points of the visible surface. while no 
impossible thing ever happens if they fali in the same poinł. Accordingly 
it is plain that it is always necessary that the axes of both visual cones 
fali in the same point, because the functioning of bot h eyes is uniform. 
Moreover should the eye be moved over the visible thing, then any 
of the eyes will be moved over it and the axes united in one point of 
the surface of the visible thing will both be moved simuItaneously to 
another point on the surface of that visible thing, when [one] eye has 
been moved. Indeed both eyes are equal in all their characteristic arrangements 
and there is one common nerve to both eyes. And since the motion of 
the eyes proceeds from one virtue, it is necessary that [this] motive power 
proceed through the unity of the nerve. Hence when this [virtue] has moved 
one eye. both eyes will be moved [by it), as is elear by the 26th (prop.) 
of this [book). Accordingly the activity and the passivity of the eyes is 
always equal and entirely alike; and if any of the eyes were moved to 
see something, the other will be moved instantaneously [and] with the same 
motion in order to see the same thing; and if any one of the eyes is 
quiescent, the remaining [one] will also be quiescenł. 
For it is impossible for any one of the eyes to be moved and for 
the other to be al rest. except when one would have been impeded, as 
is elear by the 26th [prop.] of this [book]. And yet, as has been shown 
by the 18th [pro p.) of this fhook). the surface of the visibJe thing will 


10 - Witclonil Pcnpectiv.c... 


.......
>>>
146 


always be the base of each cone proceeding from any of the eyes, because 
then the position of the point in which both axes are united is an entirely 
like position. since it is opposite to the two centres of both eyes. The 
proposed Ithing] is, therefore, elear and we cali the meeting point of both 
axes in the surface of the visible thing the point oj union. 


IProposition]33. If a straight )ine is drawn from the middle point of 
the common nerve to the centre of the )ine connecting the centres of the 
gyration-apertures of the concave nerves, it is necessary that the produced 
Iline] be perpendicular to the divided l)ine], and, the visible point having 
been joined with the axes, the produced )ine [also] bisects the triangle 
contained by the axes and by the divided )ine. 
What is proposed here is elear by the preceding [prop.] and by the 
31 st Iprop.] of the fi.rst [book] of this Itreatise] ł. However, so that it 
may be proved more specifically, let all be arranged as in the 30th [prop.] 
of this Ibook) and let )ine RT IFig. 7) be bisected in point S. And let 


A 


R 


T 


D 


c 


[Fig. 7] 


a certain visible object be opposite to both eyes, and let it be BC, in 
whose middle point, which is B, the same radial axes, which are RB 
and TB. meet, by the preceding [prop.) ; and let line AS be produced 
from point A, which is the middle point of the [common) nerve's concavity, 
to point S. 
I say that line AS is perpendicular to line RT. Indeed since angle 
ART is equal to angle ATR, by the 30th [prop.) of this [book], line 
AR is also equal to line A T; but )ine AS is equal to itself, hence, by
>>>
147 


I. 8 2 , triangles ARS and ATS are equiangular3. Therefore angle AST is 
equal to angle ASR. Consequently, by the detinition of a perpendicular 4 , 
line AS is perpendicular t.o line RT. Likewise let line AS be extended 
all the way to the meeting point of both axes, which is point B5. I say 
that line SB bisects triangle RBT. But this is elear from the preceding 
and by I. 15 6 and I. 4 7 . Indeed the partial triangle SRB will be equal 
to the partial triangle STB. The proposed Ithing] is, therefore, elear. 
And from this it is elear that the entire line AB is not changed, 
no matter to which visible point it is incident land] howsoever the radial 
axes may have been changed 8, but [that] it always takes up lits] position 
in their middle; we can, therefore, cali that IlineJ the common a\:is, since 
it is always drawn equally to the meeting point of both axes in the surface 
of the visible thing, from the point that is in the middle of the Icommon] 
nerve's concavity, in which [point] intersect the two lines drawn through 
the two centres of the concavities of the two nerves. In fact there is 
always one such unchangeable point, and furthermore point S is always 
one tixed [point], through which this line AB passes always. Consequent1y 
it itself is always tixed, even though it is allowed that the other axes may 
be changed at one time or another [in relation] to that common axis 9 . 


[Proposition]34. When the common axis together with the radial axes 
are incident to a point of the visible thing, the line joining the centres 
of the gyration-apertures of the concave nerves, and the lines drawn from 
these centres to the middle of the common nerve, and the common axis, 
and both radial axes must [ali] be located in the same surface. 
Let the arrangement be as in the immediately preceding [prop.J. I say 
that it is necessary that line RT [Fi
. 7], and the two lines RA and 
T A, and the common axis which is AB, and the two radial axes, namely 
RB and TB be always located in the same sur face. Indeed the two axes 
TB and RB pass through the centres R and Tł, by the 29th [prop.] 
of this [book]; and so they pass through the centres of the gyration- 
-apertures of the two concave nerves. And since they meet with the common 
axis in the point oj union, by hypothesis, they will necessarily be in the 
same surface with the common axis, by XI. 2 2 . But line RT, joining 
the centres of the gyration-apertures of the nerves, also cuts these two 
radial axes in points R and T and the common axis in point S. Moreover 
lines RA and TA cut lines RT and AB in [thoseJ points in which they 
meet with the same, and since all these lines are straight, it is plain, 
by XI. I, that any of the same is in one surface. Ił is elear, therefore, 
by XI. 2, that all are in the same surface, and this is the proposed 
[thing]. 


[Proposition]35. Ił is necessary that the radial axes meet with the common 
axis in a point whose distance from the eye is a muąiple of the line 
joining the centres of the eyes; secondly, that the parts of the axes in ter-
>>>
148 


cepting the point oj union and the surfaces of the same eyes be equal 
and equally incident to the surfaces of both eyes, and, besides, to the 
anterior surface of the same vitreous, and [that they cut both the eyes 
and the vitreous) according to equal angles. 
Let there be again. as in the 30th [prop.) of this [book), the two 
centres R and T of the gyration-apertures of the concave nerves [Fig. 8). 


A 


[Fig. 8J 


Consequently, since the eye is moved as a whole not in part, as is elear 
by the 25th [prop.) of this [book), it is plain that points R and T are 
at the back of the eye. Hence let the two eyes be shaped, quite touching 
points R and T, about the centres O and P, and let the axes proceed 
from a certain point of the surface of the visible thing, which is B. to 
the centres of the eyes, and let them be extended beyond to points R 
and T. And so it is plain that axes RB and TB will traverse the entire 
eye: hence let axis RB traverse the anterior surface of its eye in point 
N and let axis TB traverse the anterior part of its eye in point Q, and
>>>
II 


149 


let line NQ be drawn. Consequently points Q and N are those point s 
of the surfaces of the eye onto which the form of the point oJ union 
of the axes, which is B, is impressed. And since axes RB and TB are 
equal. by the preceding [prop.], I say that the parts of the axes, which 
are BN and BQ, are equal and that they are incident to the eye according 
to equal angles. 
Indeed since lines RN and TQ are equal, because they are diameters 
of equal eyes equally distant from points R and T, it is necessary, when 
those are cut off from equal axes. that what is left over be equal; therefore, 
line BN will be equal to line BQ. And since line NQ is paralleI to line 
RT, by VI. 2 I, because the sides TB and RB are divided proportionally 
by line NQ, therefore, by I. 29 2 , angle BNQ will be equal to angle 
BQN3. But angle BRT is equal to angle BTR, because line BS divides the 
triangle RTB in two equal parts and (this i neludes] its base RT, as is 
elear from the preceding [prop.]. It is elear, therefore, that the radia I 
axes are equally incident to the surfaces of the eyes and [cut them] according 
to equal angles. 
And if they are incident to the surfaces of the eyes in such a manner 
that they pass through the centres of the eyes. it is plain that they are 
orthogonal to the surfaces tangent [to the eyes] in points N and Q; hence 
they are equally incident to the surfaces of the eyes, cutting [t hem] according 
to right angles. And because of this, in any arrangement of the eyes, 
in motion or at rest, their two axes are always equal, or else there is 
not in them any sensible difference that might cause a certain dissimilarity 
of vision [that would be] maximaI when the visible thing would not be 
very elose to the eye, but when its distance from the eye would be average. 
Indeed when the visible thing would be very near to the eye, so that 
the line that lies between the two centres of the eye. which are O and 
P. would have a ratio of equality, or of excess, or of slight decrease 
to the radial axis. then the axes would be sensibly unequal and would 
make unequal angles [with the eyes]; but otherwise they wiJI always be 
sensibly equal and form sensibly equal angles lwith the eyes], because, 
on account of the unity of the eyes and the uniform reception of the 
forms. [the form ot] any point is multiplied uniformly to each eye, for which 
reason, likewise, all lines equally distant [rom the axes make equal angles 
Iwith the eyes] and all of them are sensibly equal. Furthermore it can 
be proved in exactly the same manner that the angles that are formed 
by the axes in the same vitreous surface in which refraction takes place 
are equal. The proposed Ithing] is, therefore, elear. 


IProposition]36. The refraction of all oblique lines of the cone of radiation 
[which are] in eloser proximity t
 the axis takes place according to smaller 
angles, and [that ot] the more remote [lines] along greater angles, while 
(that] of equally distant [Iines happens] according to equal angles. 


.......
>>>
150 


A 


[Fig. 9] 


Let there be a cone of radiation, the vertex of which 'be] A I Fig. 9], 
and let lits] base diameter, which. by the J 8th fprop.] of this 'book], 
is the surface of the visible thing, be BCDEF, its axis DA, and Jet Jines 
C A and EA be obJique radial lines, in greater proximity to axis DA, 
and let BA and F A be more remote. I say that lines CA and EA will 
be refracted according to a smalIer angle and lines BA and FA according 
to a greater angle. Indeed let alI these Jines be understood to meet in 
point A, which is the vertex of the cone, and let line GH IKL be the 
line in the surface of the vitreous to which lalI] these lines are incident. 
Consequently this line will be either straight or curved, by the 23rd 'prop.] 
of this 'book] ; let it first be straight. 
And let line BA cut that line in point G, and line CA in point H, 
and line DA. the axis. in point l, and line EA in point K, and line 
FA in point L. Therefore, since angle GIA is right, by the preceding 
Iprop.]. it is plain. by I. 32 1 , that angle GHA is obtuse, therefore, by 
I. 19 2 , Jine AG is greater than line AR. And since two Jines. which are 
AC and AB, go out from point A to the base of triangle AGI, which 
is GHl, therefore angle AHI is greater than angle AGI, by I. 16 3 . Therefore 
since angle AHI together with angle CHI amounts to two right [angles], 


......
>>>
II 


151 


by I. l3 4, and similarly angle BGH together with angle AGH amount 
to two rights. it is plain that angle CHI is smaller than angle BGl. 
Therefore, by the penultimate lprop.] of the second [book] of this [treatise] 5, 
the angle of refraction of line CH is smaller than the angle of refraction 
of line BG. 
It is elear, therefore, that line CH is refracted according to a smaller 
angle than line GB, and the same is the case with lines EK and FL. 
And since lines equally distant from axis AD, as are for example lines 
AC and AE, make equal angles in the surface of the vitreous in accordance 
with the previously described manner, such as are CH l and EKl. it is 
elear, by the penultimate lprop.] of the second [book] of this [treatise], 
that the angles of refraction are equal. The proposed [thing] is, therefore, 
elear, because when line GHIKL is a circular line, lthe claim] will have 
to be demonstrated in the same manner, by the 50th [prop.] of the second 
Ibook] of this [treatise]. 


[proposition]37. Ali the forms of point s surrounding equally the points 
that fali on the surfaces of the eyes along radial axes arrive in an entirely 
like manner at points surrounding equally the middle point of the common 
nerve I. 
Let all the rest be arranged as in the 35th [prop.] of this [book], 
and let two points, V and X be marked in the surface of the eye 
Wig. JO], the centre of which is point O, on each side of point N, and 
in the surface of the eye, the centre of which is P, let two points, Y 
and Z, be marked, on each side of point Q. And let there be the surface 
of the visible thing opposed to the eyes, in which let there be a straight 
line, namely GBC, the middle point of which is B and the extreme points 
G and C. And let the radial axes, which are RB and TB, meet with 
the common axis, which is AB, in the same point B, which is the point 
oj union of all three axes. And let two straight lines, which are VG and 
XC, be drawn from points V and X of the surface of the eye with centre 
O to points G and C of the surface of the visible thing, and let lines 
ZC and YG be drawn from points Y and Z of the surface of the eye 
with centre P. 
I say that the forms of points G and C of the surface of the visible 
thing, which fali in the surface of eye O in points V and X and in 
the surface of eye P in points Y and Z, do not arrive at the middle 
point, A, of the common nerve, but are located about the same point A, 
in a similar arrangement to the way points C and G were arranged 
about point B in the same surface of the visible thing, so that the point 
which is at the right of point B, which IS the point oj union of the 
axes in the surface of the visible thing, is reaching to the right of point 
A, and Ithe point] at the left of the same point B is reaching at the 
left of the same point A; and the same lis the case] with other differences 


'Il1o.....
>>>
152 


[Fig. 10] 


of position, so that what is above point B would be above point A 
and what is below point B would be below point A. 
Indeed let line l M be drawn in each of the eyes, either straight or 
curved, separating the surface of the vitreous from the surface of the glacial; 
and this line should (indeed) have been either straight or curved, one or 
the other being necessary by the 23rd [prop.) of this [book). Moreover 
the angles of incidence 2 will always be equal. by the 35th [prop.) of this 
(book), since the proof concerning them is also the same. But the angles 
of refraction are also made equal by the previous [prop.) and also because, 
due to the conformity of the eyes and to the equal distances of points 
G and e from point B by hypothesis, it follows that triangles YGV 
and xez are equiangular. Therefore angles GYV and exz are equal; 
but the shapes of the eyes are also entirely similar and (their) transparency 
is alike. Hence the refraction of lines ex and GY in the surface of re- 
fraction will be made conformably, and similarly the refraction of lines 
GV and CZ will be made conformably and according to equal angles.
>>>
II 


153 


Therefore any of the same is equally refracted reckoning from the per- 
pendicular 3. 
Hence let line ex be refracted to point F and !ine GV to point H, 
which are points of the gyration-aperture of the nerve about point R. 
Moreover let line GY be refracted to point L and line CZ to point E 
of the other aperture, which is about point T. And since all the points 
of the forms are refracted away from perpendicular NR along the shortest 
straight lines, it is plain that they do not meet with the same but (that], 
spreading themselves directly to the points of the common nerve, they 
receive an entirely similar position and arrangement to those they have 
in the surface of the visible thing, which is the base of the visual cone 4. 
Therefore line X F, which comes from point C of the visible thing, is 
refracted to a certain point of the nerve, which is D, other than point A. 
And line VH, which comes from point G of the visible thing, is refracted 
to another point than point A, namely to K. 
And since both eyes are of one arrangement and the distance of the 
eyes is moderate, as is elear by the 4th (prop.] of this (book], and the 
lines drawn to such points from both eyes are equal, and the angles 
of incidence (are] equal, by the 35th (prop.] of this (book], while the angles 
of refraction are equal by the preceding (prop.], it is plain that line YL, 
which is the form of point G, will be refracted to point K where the 
form of the same point G, coming along line VH has fallen. Furthermore 
line ze, which is the form of point C, will be refracted to point D, 
where the form of point C, coming along line XF, is fali ing 5. And (this] 
is to be demonstrated in a similar manner about any two points of the 
surface of the visible thing [which are] equally distant from the point 0/ 
union which is B. Therefore all the forms of the point s of the visible 
thing [which are] equally located around points that are incident to the 
surfaces of the eyes along radial axes arrive in an entirely similar fashion 
at points equally located around the middle point of the common nerve; 
and the shape and arrangement of the entire surface of the visible thing 
is retained in its parts and in [its] remoteness from the point that is 
in the axis, in accordance with the type of distance and the inelination 
of the points, the forms of which are received there reckoning from the 
point 0/ union in the surface of the visible thing, in keeping with the 
arrangement of the angles of refraction in the surface of the vitreous. 
And two forms which are impressed in two points of like position 
on the surfaces of the two eyes arrive at that very same point of the 
concavity of the common nerve and are superimposed to it in that point 
and will become one form. Furthermore lines obliquely incident to the surfaces 
of the eyes, which are refracted in the surface of the same eye can (also] 
arrive at the same arrangement of the form. The proposed [thingJ is therefore 
elear. 


..... 


-
>>>
154 


[proposition]38. It is necessary that both radial axes, meeting with the 
common axis in the surface of the visible thing, make equaI angles on 
all sides with the line [that is] paralleI to the line joining the centres 
of the eyes, as well as with the whole surface [of the objeet]. 
Indeed both eyes are of equal arrangement, by the 4th [prop.] of this 
[book]. It is elear even to the senses that they have a very moderate 
distance from one another and [that] only one line in any eye, the axis, 
passes always through the centre of the aperture of the uvea and the 
centres of all tunics, arriving at the centre of the gyration-aperture of the 
concave nerve, as is obvious by the 29th [prop.] of this [book]. Conse- 
quentIy, let it be that line BFC [Fig. II] be paralleI to line EG joining 


A 


[Fig. 111 


the centres E and G of the eyes, and let A be the middle point of the 
com mon nerve, and let it be that the form of point F of the surface 
of the visible thing arrive along axes FE and FG to the centres E, G of 
the eyes, which are connected through line EG, and let it arrive at point 
A, which is the middle point of the common nerve. 
And let the common axis AF be incident to the surface of the visible 
thing in point F according to right angles, because, the surface in which 
were marked all the lines representing the axes and the points [in question] 
is, by the 34th [prop.] of this [book], perpendicular to the surface of the 
visible thing l, and the com mon axis is directly incident, [Le., perpendicular 
to BFCj, by the 33rd [prop.] of this [book] and by I. 29 2 , because the
>>>
155 


line connecting the centres of the eyes is parallei to line RT [ef. Fig. 
10] connecting the centres of the gyration-apertures of the concave nerve, 
and therefore [also] to the line or surface parallei to it, by I. 30 3 . 
Therefore since, by the 33rd [prop.] of this [book], angle AFE is equal 
to angle AFG, the remaining of the two right angles contained by the 
axis and by line BC, which is the common section of the surface of the 
visible thing and of the surface of the axes, will be, as a consequence, 
equal to one another 4 . Hence the radial axes are incident to the surface 
of the visible thing along equal angles. And this is the proposed [thing], 
because angle EFB is equal to angle GFC. 


[Proposition]39. From the point oJ un;on, to draw a parallei line to the 
line connecting the centres of the eyes, in the surface of the visible thing 
[which is itself] parallei to that [connecting line]. 
Let the centres of the two eyes be points E and G [Fig. 12], and 
let line EG be drawn, and let the surface of the visible thing b.e BCD£, 


E 


R 


G 


B 


F 


z 


A 


c 


D 


(Fig. 12] 


from a given point of which, A, a parallei to line EG ought to be produced. 
And so let line EG be bisected in point R. by I. 10 ł, and let line AR 
be drawn from point A to point R. Let lines EA and GA be drawn, 
which are the visual axes meeting in point A of the surface of the visible 
thing; it is cłear, therefore, that axis EA is equal to axis GA. by the 
35th [prop.] of this [book], and line ER is equal to line GR, and line 
RA is common: therefore. by I. 8 2 , angle ERA wiłł be equal to angle 
GRA, and both are right. Therefore line AR will be perpendicular to 
line EG. by the definition of a perpendicular line; and let two paralIeis
>>>
156 


to Jine RA be drawn from the centres E and G of the eyes, by I. 31 3 , 
which are lines EZ and G Y. ConsequentJy these are equal and paralleJ 
to one another, by the 25th [prop.] of the first [book] of this [treatise] 4, 
and are in the same surface, by the first [prop.] of the first [book] of 
this [treatise)5. And since the common section of this surface and of the 
surface of the visibJe thing passes through point A and is, by I. 33 6 , 
paralleJ to line EG, it is plain that the same line ZA Y is the line that 
is sought. Consequently that which was proposed was done. 


[Proposition]40. Ali the lines drawn from both eyes to the same point 
of the line making right angles with both axes of the cones of radiation 
are necessarily eq uaI. 
Let there be, for example, as above, in the immediateJy preceding [prop.], 
points E and G, the centres of the two eyes, and let the surface of the 
visible thing be BCDF, in whose point A let axes EA and GA meet 
[Fig. 13]. And from point A let one line be drawn to each side l, and 


B 


F 


K 


L 


c 


v 


D 


[Fig. 13] 


let it be ZA V, making right angJes with each of the axes, and let lines 
EV, GV, EZ, GZ be drawn from the centres of the eyes. I say that 
line EV and GV are equal to one another and Jines EZ and GZ [are 
also] equal to one another. 
Indeed since the axes of the eyes are equal, by the 35th [prop.] of 
this [book], it is plain that axis EA is equal to axis GA and angle EA V 
fis] equal to angle GA V, because each of them is right by hypothesis. 
But line A V is a com mon Jine in triangles EA V and GA V. Therefore. 
by I. 4 2 , base EV will be equal to base GV, and similarly base EZ will
>>>
157 


be equaI to base GZ; and [this] happens in the same manner in all points 
of line zv. Therefore what was proposed is pIain. 
This can aIso be demonstrated differentIy. Indeed Iet a paralleI line 
be drawn to line EG, which is between the two centres of the eyes, from 
point A of the surface of the visibIe thing in which the axes meet, [wich 
can be done] by the preceding [prop.]. and Iet it be line KL. And that 
line KL will be in the surface of the visibIe thing; moreover let perpendicular 
line ZA be drawn to line KL, by I. 12 3 . And likewise Jet a Jine be 
drawn from point A orthogonally to line EG, and Jet it be line AR, 
and line AR wiłł bisect Jine EG in point R, by the 3 J st [prop.] of the 
first [book] of this [treatise] 4, and by the 35th [prop.] of this [book], 
and by I. 55. 
Indeed since axes EA and GA are equal, the angles at the base wiłł 
be equaJ, and Jine RA [is] common to both triangles EAR and GAR, 
and the angJes about point R are equal, as right [angles]. Therefore, by 
I. 32 6 and by VI. 4 7 , Jine ER wiłł be equaI to Jine RG; and let line 
RZ be drawn. Consequently, by I. 29 8 , Jine RA will be perpendicular 
to Jine KAL. And since by the 34th [prop.] of this [book], lines EA, 
GA, and RA are in the same surface, and line ZA is perpendicuJar to 
lines EA and GA, as is cłear by hypothesis, therefore, by XI. 4 9 , line 
ZA is erected perpendicularly to that surface in which lie lines EA, GA, RA, 
and, therefore, [also] to line RA. Hence, likewise by XI. 4, line KA wiłł 
be perpendicuJar to surface RZA. Therefore, by XI. 8 10 , line ER wiłł 
be perpendicular to the same surface RZA. 
Consequently from the definition of a Jine perpendicular to a surface II, 
Jine ER wiłł be perpendicular to Jine RZ. Therefore, since the two angJes 
ERZ and GRZ of the two triangJes ERZ and GRZ are equal, because 
right, and line ER is equal to line RG. and side RZ [is] common, Jine 
EZ wiłł be equal to line GZ, by I. 4 12 . And it is to be demonstrated 
in the same manner about any other points of line zv. The proposed 
[thing] is therefore cłear. 


[Proposition]41. All lines drawn from both eyes to the same point of 
a Jine mak ing unequaJ angJes with the two axes I are necessarily unequal. 
Let the entire arrangement be as above in the preceding [prop.]. I say 
that all Jines [drawn] from both eyes to the same point outside of Jine 
VZ (which aJone makes right angles with bot h axes) are aJways unequal. 
Indeed Jet two points be marked in line KL [Fig. 13], howsoever it may 
cut Jine VZ, start ing from point A, at whatever distance one pJeases, 
and let them be M and N, and let Jines EM and EN, GM, GN be 
drawn. I say that lines EM and GM are unequaJ and Jikewise Jines EN 
and GN [are] unequal. 
For Jet a Jine be drawn from point R to point M. which is RM. 
Therefore, since angJe ERA is right. as became cłear in the preceding
>>>
158 


lprop.], it is obvious that angle ERM is smaller than a right langle]; 
hence angle GRM is greater than a right langle]. by I. 13 2 . Consequently 
in triangles GRM and ERM side RM is common and line ER is equal 
to line GR, and angle GRM is greater than angle ERM; therefore, by 
I. 243, side GM will be larger than side EM. And it is to be argued 
in the same manner concerning all other points outside of line vz. The 
proposed lthing] is, therefore, elear. Furthermore the inequality of those 
lines is less perceivable, when the points deviating Ifrom line VZ] happen 
to be rather elose to the point oj union. 


IProposition]42. Ali lines drawn from alternate eyes to equidistant points 
from the point of union of the axes, Iwhich points lie] on a line making 
non-right angles with the two axes, are necessarily equal and form equal 
angles with lany ot] the lines Imaking non-right angles withthe two axes]. 
Let the entire arrangement be as above in the two preceding lprops.], 
and let M and N be points of line KL Wig. 14]'" making non-right angles 


F 


L 


[Fig. 141 


with both axes, land let them be] equally distant from point A, which 
is the point of union of the axes, so that line MA is equal to AN. 
I say that lines drawn from alternate eyes, as EN and GM, and EM 
and GN are equaI. 
Indeed since axis EA is equal to axis GA, by the 35th lprop.] of 
this lbook], and the angle of incidence ofaxis EA, which is angle EAM, 
is equal to the angle of incidence ofaxis GA, which is angle GAN. because 
angles RAM and RAN are right, while angles RAE and RAG are equal, 
as these lconelusions] became elear from the previously demonstrated Ielaims] 
in the two preceding propositions, therefore, there remain the equal angles
>>>
159 


EAM and GAN. But the axes EA and GA are also equal, and line MA 
is equal to line NA by hypothesis. Therefore line GN will be equal to 
line EM. by I. 4 1 . and angle GNA equal to angle EMA. And as a consequence, 
base EM is equal to base GN in triangles EMN and GNM, by the same 
I. 4. And it can be similarly demonstrated concerning all similar points: 
mdeed lines GB and EF. GF and EB, GK and EL, GL and EK, GC 
and ED, GD and EC, all, as thus called and as they are drawn from 
aIternate eyes to points equidistant from point A. are necessarily equal. 
The proposed Ithing] iso therefore, cIear, no matter how many other lines 
might be drawn in the same manner. 


IProposition]43. The comprehension of forms by the eye is accomplished 
wit h certainty along all the lines of the cone of radiation, with greater 
Icertainty], however, along lines cIoser to the axis, and with greatest Icertainty] 
along the axis passing through the centre of aperture of the uvea. 
Indeed only the axis of the leye] is extended rectilinearly till it reaches 
the place of the gyration-Iaperture] of the concave nerve, and all other 
lines are obliquely incIined I, as is cIear by the 24th Iprop.] of this [book]. 
Therefore. the form of tIle visible object opposed to the center of the 
eye's surface will arrive at the glacial and vitreous [humors] along [the 
same rectilinear] extension Ithat goes on] all the way to the place of the 
gyration-Iaperture] of the concave nerve, while the forms which arrive along 
other lines are bent [on their way]. 
And since the quality of bent forms is not like the quality of forms 
extended Ithroughout] straight, because the bending alters the same necessarily 
with a certain alteration in the certainty of comprehension. therefore the 
point of the form arriving at the spot of the gyration-Iaperture] of the 
concave nerve, which is extended along the straightness of the axis, is 
better authenticated than all the points of [other] forms. And since the 
bending of the lines cIoser to the axis is smaller and [that] of the more 
remote greater, because the angles which are made by the lines along 
which the forms arrive and by the perpendiculars drawn to the axis in 
the surface of refraction are more acute [in the case] of lines cIoser to 
the axis and less acute [in the case] of the more remote, as is cIear by 
the 36th Iprop.] of this [book], assuredly, the forms. the bending of which 
is smalIer, manifest themselves more than the forms the bending of which 
is greater. 
Therefore the point which lies on the axis arriving at the spot of 
the gyration-Iaperture] of the concave nerve is more manifest than all other 
points and of more trustworthy comprehension. And that which is cIoser 
to it is more manifest than the more remote from it. And it is the same 
with the form arriving at lany other point ot] the common nerve through 
which the sensitive power comprehends the forms of things. The proposed 
Ithing) is, therefore, cIear.
>>>
160 


IProposition]44. When the point ol un;on lies on the common aXłS, 
vision is achieved with the greatest certainty: moreover. when cło ser to 
that ax.is. lit is]. for this very reason, more certain, and (when it is] more 
remote. less certain. 
Let there be a line joining the centres of the apertures of the uvea, 
name1y AR [Fig. 15]. and the line CE be the com mon ax.is: moreover 


H 


[Fig. I S] 


Jet D be the point of union in the same Jine CE, in which [point] Jet 
ax.es AD and RD meet, and let the middle point of the concavity of 
the com mon nerve be point H. I say that when point D Jies in line 
CE then vision is achieved with the greatest certainty. 
Indeed the visual forms arriving at the eye's surface are then more 
alike, because, as the ax.es are passing through the centres of the apertures 
of the uvea which are marked by points A and B, the forms of the 
points encircłing point D are incident about those centres in a distinct 
and completely alike [manner]. And since the common axis. which is EC, 


.......
>>>
161 


bisects line AB in point C, by the 33rd [prop.] of this [book] and by I. 29 1 , 
because the line joining the centres of the apertures of the uvea is paranel 
to line RT connecting the centres of the gyration-apertures of the concave 
nerves, as is elear from the preceding and by the 4th [prop.] of this [book], 
whence, by the 31st [prop.] of the first [book] of this [treatiseJ2, it is 
elear that line HC bisects line AB and is perpendicular to it. 
It is, therefore plain, by I. 43, that axis AD is equal to axis BD 
and angle DAC Cis] equal to angle DBC. But, by the 30th [prop.] of 
this [book], angles HAC and HBC are also equal. And since the common 
axis, which is EC, reaches at H, the middle point of the concavity of 
the common nerve, to which the forms from points A and B are extended, 
it is plain, by I. 26 4 , that angles CHA and CHB are equal. Moreover 
the same happens in an points to which are incident radial lines elose 
to the same axes AD and BD, which are quite equal according to sense, 
as is elear by the 40th [prop.] of this [book]. Indeed these radial lines 
are quite equany incident to equal[ly situated] points of the surface of 
the common nerve, by the 37th [prop.] of this [book]. 
Apd so the forms of the points seen in this manner are exceedingly 
alike, whence again, vision takes place in a more 'certain fashion. But 
when the point oj union would have been slightly outside the common 
axis, as, say, in point F, or if its remoteness would be to the left side 
or to the right, above or belo w, or in any other way, then still the two 
forms which are impressed in the two eyes would not be greatly dissimilar. 
From which [it is obvious] to which point the forms to which the two 
axes are impressed [arrive, namely,] to the same middle point H, that 
is to the [central] point of concavity of the incident, [Le., common] nerve s , 
[while] the remaining points of the form of the visible thing [which are] 
incident along radial lines elose to the same visible axes are united in 
the concavity of the common nerve around point H, but not in line with 
the perfection of the prior arrangement; and so the object is seen then 
too with certain vision, but not in the degree of the prior certainty. 
Furthermore when the point oj union will have been [rather] remote 
outside the common axis (which is CE), as in point G, [or] at whatever 
[other] changed position this might happen, then, for this purpose, the point 
of the visible thing in which the two axes meet would [still] be impressed 
upon the same point H. But the forms of the remaining points of that 
.visible thing [which are] impressed in the surroundings of point H will 
not receive a quality similar to the prior tw0 6 , nor will the vision of 
those points be wen certified, but it remains [rather] less certain. The 
proposed [thing). is, therefore, elear. 


[Proposition]45. Every visible object is seen with greater ,!:ertainty in the 
point of conjunction of the two visu al axes than [in] that [point which is 
seen] along rays elose to the axes, and the degree of certainty decreases 


II - Wilclonia Pcrapoctivac... 


.... 


.
>>>
162 


with the remoteness from the axes; from which it is elear that the points 
of the surface of 1he visible object [which are] equally distant from the 
point oj union are exhibited in like manner to the visual power. 
lndeed since. as is elear by the 43rd [prop.] of this [book]. the comprehen- 
sion of the visible form by the eye is made with certainty along all the 
lines of the visual cone, but with greater [certainty] along lines c10ser 
to the axis, and with greatest [certainty] along the axis passing through 
the centre of the aperture of the uvea (for the two axes meet in the 
point oj union. by the 32nd Iprop.] of this tbook]). it is plain, therefore, 
since a doubled power is stronger than its half. that vision is accomplished 
in a surer way in the point oj union [wherever it be] in the whole 
surface of the visible thing. which is the base of both cones of vision, 
and according to the ratio of double to double, which is tequal] to [that] 
of simple to simple I. 
Furthermore vision along radial lines that are elose to the axes is 
accomplished in a less certain manner than along the axes. because the 
forms of the points arriving at the sensitive power Inot along the axes] 
do not arrive directly to the middle of the common nerve, from which 
lit is obvious that] their discernment does not take place in as perfect 
a manner as [happens] with the forms arriving along the same axes. Truly, 
the degree of certainty of vision decreases wit h the remoteness of those 
lines from the axes, because [it is the case] with the parts of the surface 
of the visible thing to which the axes are incident and [with] the parts 
elose to them [that] they are being seen more manifestly, by the 43rd 
[prop.] of this Ibook]; Imoreover] the certainty of vision is weakest for 
the remote parts of that surface to which are incident the extreme lines 
of longitude of the radial cone; and for other middle parts, the nature 
of the certainty [achieved] is [also] average. depending on how much nearer 
to the axes or removed from them they are. The proposed [thing] is, 
therefore. elear. 
And from this the corollary is taIso] elear, because the degree of certainty 
of vision in points of the surface of the visible thing eqirally distant from 
the point oj un;on is the same on all sides, since the forms of those 
[points] are always fashioned in the same manner in the surface of the 
same eye and, consequently, [also] in the surface of the common nerve. 
Therefore all that was proposed is elear. 


(proposition]46. Every visible object in which the two visual axes, or 
the rays c10se to them. come together is always seen as one tthingJ. 
lndeed since the forms arriving at the eye along radial axes are equally 
incident to both eyes, by the 35th [prop.] of this [book], and [since] they 
arrive equally at the middle point of the nerve's concavity, by the 30th 
[prop.] of this [book], therefore bot h of those forms come together in 
one point, and one of them, is superimposed to the other, and they become
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. 


163 


one form. And since all visible objects customary to us are always opposite 
both eyes, and bot h eyes behold any of those visible things, on account 
of which the two axes of the two eyes meet always in one point of 
those visible things. by the 32nd Iprop.] of this (book], and (s i nce] the 
position of. the remaining rays which fali around the common point of 
the same laxes] is an entirely like position, by the 37th Iprop.] of this 
Ibook, this likeness being] maximaI when [the points in question] do not 
differ in [their] remoteness from "the two axes to any great extent, therefore 
any of the customary, visible objects is seen by both eyes as one object. 
And so, as was premised. it is elear, by the 37th lprop.] of this lbook], 
that all the forms of points equally surrounding the points that are incident 
to the surfaces of the eyes along radia I axes arrive in an entirely like 
manner at points equally surrounding the middle point of the common 
nerve. Furthermore the radial lines elose to the visual axes, not being 
very obliquely incident to the eyes and, therefore, not being refracted very 
obliquely either, because their refraction is along smaller angles (than otherwise], 
by the 36th lprop.] of this [book], arrive therefore more directly to the 
concavity of the nerve and lthere] arrange themselves around the middle 
point of the nerve's concavity, and order themselves accordingly and become 
one form. And this is what was proposed. 


[Proposition]47. Any visible object in which the common axis and one 
of the visu al axes meet is always grasped as one [thing]. 
Indeed the com mon axis assists the certainty of comprehension and 
the unique visual axis imprints a single, regularly arranged form onto 
the middle point of the com mon nerve. Therefore only one form is seen, 
since then no refraction of another form takes place to some part of 
the nerve. distinct according to the part or according to the remoteness. 
The proposed lthing] is, therefore elear. 


[Proposition]48. No visible object is seen simultaneously equally well 
10 its entirety. 
. Indeed since any visible object lies either in the common axis or outside 
of it. (and since] a point to which the visual axes are incident is always 
seen with greater certainty than points to which [only] more [or less] elose 
rays are incident, and those [latter] points are seen with greater certainty 
than the points to which remoter rays are incident, by the 45th [prop.] 
of this [book], it is elear that no visible object is seen simultaneously 
equally [well] in [its] entirety. Truly [only] when all the points of the same 
[object] will have been seen covered by all three axes together, or, at 
least by two visual [axes], when the eye would have been moved. only 
then would the whole object [be seen] equally [well], because [only] then 
is the form of any of its points infixed in the middle point of the nerve's 
concavity, £lnd a new arrangement of the whole form will always be about 


_.a.--"""""-
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164 


that point; therefore the more uniformly pondered [be] the mutual- equality 
of the parts in all their arrangements, [the more] uniformly, then, will 
the whole thing be seen. 
Furthennore no motion is in an instant, but only in time. It is plain. 
therefore, that no visible [object] is seen simultaneously equally well in 
[itsJ entirety. But it is quite possible for the same [objectJ to be seen 
simultaneously in [its] entirety unequally, because all the points of the form 
opposite the eye from which straight lines can be produced to the eye 
are multiplied simultaneously to the eye, howsoever they may be seen 
according to the diversity of the angles [and] according to the difference 
of the various parts: thus smali bodies having smali diameters are seen 
more evenly than bodies of greater diameters, while the remoter parts 
from the point oj union are not well certified to the extent that the eloser 
[part s are], by the 45th [prop.] of this [book]. And if the object be of 
uniform color, less of an unevenness occurs in it than if it will have 
been of many colors, or if there be in it [some] drawing, or picture, 
or other subtle intentions; then indeed the form of the' boundaries will 
be more doubtful and not well certified. These indeed are comprehended 
by radiallines remote from the axis. The proposed [thing] is, therefore, elear. 


[Proposition]49. It is impossible for many [things] to be seen simultaneously 
equally well. 
. Indeed let, at any time, the eye be simultaneously opposed by howsoever 
many visible things of diverse colors, between any of which and the eye 
straight lines can be drawn in the intermediate air connecting between 
them and the eye, and let the forms of light and color which are in 
the visible things arrive at the surface of the eye at the same time, and 
the form of any of the same [would arrive] at any part of the eye's 
surface, on account of their direct opposition. And it is allowed that the 
seeing [subject] may see at the same time visible objects of diverse colors 
opposed to the eye, and thus there are in the whole surface of the eye many 
diverse Iights and many diverse colors, each of which fills the surface of the 
eye opposed to it, whether it is incident perpendicularly or obliquely. 
All the same, as is elear by the 17th [prop.] of this [book], vision 
is not accomplished distinctly except along perpendicular Iines only, drawn 
from points of the visible thing to the surface of the eyes. And the forms 
are discriminated in accordance with this by the differentiation of the part s 
of the eye's surface in which only the perpendiculars fall, and it is thus 
allowed that mixed forms of diverse lights and colors may arrive at the 
eye's surface [and thatJ, nonetheless, the eye comprehends all forms according 
to their characteristic. 
It is, consequently, not possible for it to see many [thingsJ simultaneously 
equally well, but [only] unequally and indistinctly. For it is allowed, as 
is elear by the 17th [prop.] of this [book], that the glacial humor senses 


-
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165 


the form of a visible object according to its being and that the shape 
[is] arranged in its surface according to the arrangement it possesses in 
the surface of the visible thing. Besides it will also be able to sense in 
that arrangement the forms of other visible things beyond that [one] visible 
thing, by means of cones separating from its surface other parts [cor- 
responding] to the same thing[s]. And it will be able to sense the form 
of any of those visible things according .to its being, and to sense [also] 
the mutual position of those [forms]. however not equałly well, but more 
perfectly of that which it sees according to a cone, the axis of which 
Is incident through the centre of the circle of the uvea to that very centre 
of the eye, and less perfectly other [objects] the axes of which cones are 
incident to the surface of the said circle in other points, as is clear by 
the 43rd [prop,f of this [book] ; indeed all the axes of those [cones] are 
longer, even if they proceed from the same distance. 
And so }¥hen the observer wiłł have been opposite many visible objects, 
and his eye will have been at rest, he will have ascertained the thing 
opposed to the centre of his eye more manifestly than those which are 
laterally, on the fłanks of that centre, and what is cIoser to the centre 
will be more manifest, and what is more remote will be less manifest, as 
all these [conclusions] are obvious by the 43rd [prop.] of this [book). It 
is, therefore, impossible for many [objects] to be seen simultaneously equally 
weB. since it is impossible for the axis of the visual cone passing through 
the centre of the uvea to be simultaneously incident to many points, let 
alone surfaces, by the 20th [prop.] of the first [book] of this [treatise] l. 
The proposed [thing] is, therefore, clear. 


[Proposition]50. Whenever various visible objects have been interposed, 
vi sio n of the more remote ones is somewhat impeded. 
For example, let the two points N and M be the centres of the two 
eyes [Fig. 16], and let R be a point of a certain visible thing, which 
is LO, [and which is] more remote from both eyes than is the visible 
thing which is BKC, in whose 
oint Klet b9th visu al axes, which are 
M K and N K, meet. And let point R be placed in such a manner that 
when the axes N K and M K have been extended, N K to point Q and' 
M K to point H, the same [point R] is intercepted between the axes and 
nothing of that [object] is taken hold of, due to the interposition of 
the visible object, which is Be. Moreover let the visible thing ED be more 
remote than is the same BC and [yet] close to point R. [and let it] be 
placed in such a way between the two axes that the extended lines NB 
and MC, meeting in point P, do intercept a certain part of it, which 
is FG. Furthermore lines MP and NP, intersecting in point P, extended 
touch the periphery of the body in which point R lies in points L and O. 
In fact let A be a certain object close to the eye [and] falling between 
the axes M K and N K. I say that, when the eye grasps at the same hour
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166 


o 


(Fig. 16] 


[and) simultaneously the forms of the visible o bject s, which .are BC and ED 
and R, that. as [this happens), the vision of the same ED is somewhat 
impeded, because it is hampered with respect to its part. which is FG, 
which. as it is hidden from sight by the interposition of object Be. makes 
it elear that the form of that part will not arrive at the eye. nor will 
it be retained in the common nerve. Moreover. [concerningJ the form of 
the more remote object. which is LO, in which point R lies. as the same 
falls between lines NB and MC intersecting in point P, which, [when) extended 
beyond point P. are incident to its extremities L and O, it is elear that 
it will arrive at the eye unhindered by object BC 
Nevertheless since the visual axes do not meet in any of its points, 
its form will be seen as confused with respect to the position of the part s 
of that same form, which will not be directly superimposed to the axes, 
as was obvious, in the 37th [prop.) of this (book). Therefore they will 
be unordered with respect to their remoteness from the middle point of 
the common nerve. which remoteness will be unequal in various directions. 
on account of the diversity of incidence of the same lines along which
>>>
. 


167 


arrive the same point s of the forms. as are, [say], lines ML and NL 
with respect to the form of point L. and lines MO and NO with respect 
to the form of point o. However that aspect of the universe [of vision] 
having to do with considerations of 1I0cation] of the parts of that form 
to the right or to the left, above or below is not changed. 
Indeed since object BC is smaller than object LO in which lies point R, 
when the two axes M K and N K come together in point K of object BC, 
then the form of object BC possesses an entirely like position in the two 
locations of the two eyes, while the form of object LO will be diversified 
in accordance with the position of the parts of its form and due to its 
unequal remoteness from the middle point of the common nerve. since 
there is great diversity in the angles of refraction of its partial forms 
as weU as in the angles of incidence of the same. as this can be elarified 
by the 36th [pro p.] of this [book]; however there will not be error concerning 
the aspect of the object as a whole. because the forms of the parts will 
be arranged in the same order as they are in the object and [only] one 
thing wiU be seen, which does not happen in [the case ot] the form of 
an objeet, namely of the same A. that is eloser to the eye. if it be 
of small size and lit] there would not be in the position of those [exceedingly 
small] bodies any sensible diversity, so that, [say]. body A would fali between 
the axes MK and NKI. 
Accordingly when both eyes grasp both visible things, in which R and 
DE are, and when .the two axes are fixed in object BC, the forms of 
those visible things DE and LO are set up in the two places of the two 
eyes in accordance with [their] non-screened regions, and they are made 
of like position in the
r overall aspect but not in [their] remoteness from 
the middle point of the common nerve, or rather not aU of their parts 
wiU be of an entirely like position in ltheir] remoteness from the two 
axes. nor will their form be weU certified. Furthermore concerning object 
A. which is elosest to the eyes. as it falls between the axes MK and NK 
and is eloser to the eye. and because of this Ivery] c10seness the axes 
are not impressed in it, its position with respect to both eyes can be 
made diverse [concerning] the part lit occupies] in the same universe, so 
that Inow] it would not be seen at the left (and] now at the right. because 
its form. in and of itself, is not ordered at any part of the universe 
with respect to the middle point of the same concave nerve to which 
the visual axes are incident 2 . 
Therefore, the extant eye having been fixed in this way, whenever diverse 
objects have been interposed Ire1ative] to it, the vision of the more remote 
lobjects] is somewhat impeded, as is elear. However when the eyes will 
have been moved and the axes will have been joined in any of the objects 
apprehended simultaneously, then the forms of all objects will be grasped 
simultaneously in both eyes, alike in respect to part s and remoteness, and 
the forms of any object whatever will be understood according to the manner
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168 


of its [appropriate] certainty. For the whole reason of this thing is this, 
that the certainty of vision takes place along the axes, and vision occurs 
through the multiplication of the visible form in the eye, which is, then, 
sometimes hampered by an interposed body, when, the line of multiplication 
of the form touches somehow another opposite surface of an intermediate 
body. And this is what we wanted. 


[Proposition]51. Ali vision takes place either by simple sight or by diligent 
intuition. 
First we cali simple s;ghl that act by means of which the form of 
the visible object is received for the first time directly in the surface of 
the eye, while we cali inlu;l;on that act by means of which the eye inquires 
diligently and thoroughly after the comprehension of the form of the objeet, 
not [being] content with the mere reception but [striving for] a profound 
examination. 
Accordingly the eye grasps by simple sight the manifest intentions which 
are in things, but does not certify them; however through intuition it 
L 
looks elosely at all the visual intentions of the parts of the form hidden J'- 
from sight and certifies all the qualities of that visual form. And since 
simple sight can be without intuition, although intuition cannot be without 
simple sight, it is elear that all vision takes place either by means of 
one of these ways or by the other, and this is the proposed [thing]. 


[proposition]52. Simple sight is possible along the entire extant visu al 
cone, [while] intuition takes place only according to the incidence of the 
axis of the visu al cone. j 
Indeed since, as is obvious from the preceding [prop.] simple sight 
consists solely in the reception of the sensible form in the eye's surface, 
it is plain that the same takes place along the entire visual cone. For 
any of the perpendiculars or radial lines making up that cone conveys, 
by the 17th [prop.] of this [book], a certain form of a point of the surface 
of the visible thing which the eye beholds, while, in fact, intuition certifies 
the truth of the grasped forms. Truly the certification of all visible forms 
is performed rather through th
 axes of visual ;ones than through other J 
lines of that cone, by the 43rd [prop.] of this [book]. 
It is elear Ihat inluitian takes place anły Ihrougb Ihe incidence ar ' 
that axis. Therefore when the eye will have been stationary opposite a certain 
visible object that will have been of a certain [minimal] size, and that 
[part] of the visible thing, which is opposite the centre of the eye will 
have been over the visual axis or elose to it, then that which lies in 
the axis, or which comes elose to the axis, will be more manifest [to 
the eye] than the remaining parts of the visible thing. And so if the 
observer will have wanted to obtain a certification concerning the form ł \ 
of the whole visible objeet, he will have moved both eyes until thejT J
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169 


centres may be opposite to any parts or points of the surface of the visible 
thing opposed to it, and then, since both radial axes will be incident 
to any of the points, by the 32nd [prop.] of this [book], the complete 
intuition of the entire form will be achieved in this manner. 
Indeed should the eye be opposite to the visible thing, then the sensing 
[subject] would comprehend the entire form with a certain comprehension, 
by the 43rd [prop.] of this [book], but it would comprehend the part 
that is at the extremity of the axis wit h true comprehension; then. the 
axes having been moved to another point, that same point will immediately 
be understood more truly, and thus, with this [procedure], the entire pre- 
viously comprehended form will be understood [more fully] for the second 
time and even that point in which the axes will have been previously 
fixed. 
And when the axes are moved to a third point, a third comprehension 
of the entire form will be accomplished and even of those points to which 
the axes were previously incident and thus the comprehension of the entire 
form is accounted for in accordance with the num ber of points to which 
the axes are incident; however, the point to which the axes are incident 
is always understood with greater certainty than the other points. In this 
manner, then, the intuiting [subject] comprehends through the motion of 
the axes the certainty [of the reality] of any point whatever of the visible 
thing, and moreover repeats the frequency of understanding of the entire 
form according to the number of points to which the same axes are incident. 
Therefore everything that can appear in the form of that visible object 
appears at such a time to the eye and the form of the visible thing 
will not be certified except after the motion of the eyes through their 
radial axes over all parts or points of the surface of the visible object; 
nor indeed do the subtle intentions that are in the visible thing appear 
to the eye except through the motion of the eye and through the transit 
of the axis or of the radial lines which are elose to the same, over any 
of the parts of the visible thing. And even if the object would be at the 
border of smallness and not opposite the eye, the eye would not intuit it 
with perfect intuition, except until, the eye having been moved, the radial 
axes will have passed through all the particles or points of that thing. 
In this way, then, intuition takes place only along the incidence of 
the axis of the cone of radiation, while simple sight is accomplished along 
all radial lines of the entire cone of radiation. The proposed [thing] is, 
therefore, elear. 


. [Proposition]53. The radial axis remains always fixed in its position 
throughout the motion of the same eye, because that motion of the eye 
is of imperceptible I velocity. 
(\ Indeed the motion of the axis over the parts of the visible object 
is not [accomplished] through the displacement of the axis from the place 


...-i1.
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170 


of the centre of the same eye, but through the motion of that (eye] by 
itself over the part s of the visible thing. For it is elear. by the 24th 
and 12th (props.] of this (book], that the line of the axis is extended 
straight all the way to the place of gyration of the (optic] nerve over 
which the eye is constructed and that its position is not changed with 
respect to the eye. but that it is moved wit h the who le eye in opposition 
to the visible thing; and (that when] the middle of the eye, in which 
resides the sense of sight. is opposite some of the parts of the visible 
thing, then the axis passes through any of the parts of the visible thing. 
And in this manner the entire form of any part of the visible thing 
is always extended to the eye according to the straightness of the axis, 
and the turning of the axis will be immutable from the standpoint of 
its place with respect to all the part s and tunics of the eye, but the 
axis will be turned around in the concavity of the bone together with 
the motion of the eye as a whole. And should the eye want to intuit 
the visible thing and should it have begun to intuit (it] in the extremity 
of the visible thing, then the end of the axis would be (Iocated] over 
the extremity of the visibJe thing. And in this arrangement. the greater 
part of the entire visible object will (have its form situated] in a part of 
the eye's surface deviating from. or even oblique with respect to, the axis, 
and wilI be (positioned] in another part [of the eye] than the part containing 
the axis, because the form of that (latter part] will be in the middle 
of the eye and in the place of the axis. whiJe the remaining [part] of 
the form will be obliquely inelined to another part [of the eye] away 
from the axis. 
And since the eye will be moved after that arrangement over some 
other diameter of the visible thing, the axis will be transferred to the part 
following that (earlier] part of the visible thing, and the form of the first 
part will (now] become obliquely inelined to another opposite place to which 
the axis is moved, and the form will not cease deviating as long as the 
axis is moved over that diameter, until the axis may arrive at the end 
of that diameter of the visible thing which is another part of the visible 
thing. And in this way the form of the entire visible thing will (eventually] 
be in this oblique arrangement to the eye and to the end point of the 
same axis, to which it was already oblique previously while the radial 
axis (in its sweep] will have been incident in diverse points, beyond [the 
arcs belonging to] the last part (attended] and (eventually only] the last 
(part] of the same visihle object will remain (in the end] over the aXlS 
and the middle of the eye. And (yet] in this entire motion, the axis wilI 
be fixed in its place as much as possible in order to achieve a uniform 
penetration of all the tunics of the eye. Therefore that which was proposed 
is elear. 


(proposition]54. In the motion attending the intuition, the axis never 
becomes the base of an angle at which the surface of the visihle thing
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171 


looks back, nor does it always cut an angle at which any of the diameters 
of the visible thing looks back. 
Indeed since it was already obvious in the preceding theorem that the 
axis remains always fixed during the entire motion of the eye to intuit 
[objects], therefore, if the axis were made into the base of an angle that 
the surface of the visible thing looks back at,.it would become necessary 
that the lines containing that angJe remain immobile and that the axis 
be moved. This, however, would not be possible except when the axis 
were moved by itself. while the entire eye would have remained quiescent. 
And since this is impossible by the preceding Wrop.], [it fo110 ws] indeed 
[that] the 'entire eye is moved when intuiting [objects], and the axis is 
moved on account of that motion, and the axis having been moved, all 
the lines containing the angle of the cone are moved, and the axis having 
been changed, the whole cone is changed. 
Indeed when the radial axis is incident to various points of the surface 
of the visible object, it is allowed that the vertex of the cone would 
remain the same and even the base stays the same; however when the 
axis has been changed. a new cone would come into being, even though 
it would always be seen as one [and the same] because the motion of the 
eye is of imperceptible velocity. And so, on account of this motion the 
eye grasps any point of the surface of the visible thing in its middle, 
nameJy in a point of the axis, and in this manner the form of the visible 
thing is moved to the same surface of the eye, and the part of the eye's 
surface in which the form has previously been is changed, because the 
form of the visible thing will be, on accoUllt of the motion of the axis, 
in one part of the eye's surface, after another part of the eye's surface. 
For as often as the sentient power will have grasped the part of the 
visible thing that is by the extremity of the axis, o ust] as often it will 
have grasped with it the entire surface of the visible thing. and that whole 
part of the eye's surface, in which the form of the entire visible object 
arrives, will grasp [it], which [part, moreover,] changes continually. 
And so as long as the axis falls in any of the points of a diameter 
of the visible object not limiting the same diameter, then the axis divides 
the angle at the centre of the eye to which that diameter is subtended; 
but when it is incident to the end point of the same diameter, then that 
axis becomes one of the lines containing that angle. Hence it does not 
always divide that angle, which is the proposed [thing]. 


[Proposition]55. It is necessary that all vision that takes place by simple 
sight be accomplished in an instant. 
Indeed if simple sight would take time, however smali that time might 
be. it would be part of a greater time, and since it is not given that 
vision is accomplished in time except on account of the distance of the 
object from the same eye, it is plain, then, that the time would be in- 
creased in accordance with the extension of the distance of the object 


--
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172 


A 


B 


c 


D 


[Fig. 17] 


from the eye. Accordingly let line ABCD be drawn [Fig. 17] and let the 
eye be in point A, and let any. object be at point B. And so, since,_ 
as was said and shown in the 6th [prop.] of this [book], the form of 
point B is multiplied to the eye, should this happen in some time, even 
strongly imperceptible, [and if] there be another visible object in point C 
and space AC be a multiple of space AB, then the time in which the 
form of point C is multiplied to the eye A will be a multiple of the 
time in which the form of point B is multiplied to eye A. 
And if this time is not quite sensible, let it be in a further visible 
point D [which is] more remote from eye A than is the same C, and 
let space DA be a multiple of space CA; therefore the same [DA] will 
be a greater multiple of space BA. Accordingly, the form of point D will 
be multiplied to eye A in a time [which is] a multiple of the time in 
which the form of point C arrives at the eye. But [in reality] in the 
passage of the form of point D over the same space AD, no more time 
is required for that visual operation than Cis required] for [covering] space 
AB. Indeed the eyes having been opened, remote and e10se [things] are 
seen equally fast, nor, surely, is there a sensible ditference of time during 
which a e10se thing is seen but not any of the fixed stars, whose distance 
is quite in accordance with the world's semidiameter, which is the greatest 
of entities [in the domain] of natural lines. 
It is therefore impossible that vision occurring by simple sigbt be accom- 
plished in time, but [on the contrary] it is necessary that any such vision, 
inasmuch as it is by simple sigbt, be accomplished in an instant and 
suddenly; and so its beginning does not ditfer from its end. And this 
is the proposed [thing]. 


[Proposition]56. Ił is necessary that any intuition be accomplished in 
time, and the time [requ ired] for the intuition of visible intentions ditfers 
according to the diversity of the intentions of the intuited forms. 
Indeed since, as became elear in the 51st [prop.] of this [book], intuition 
is the act of visual power by means of which the eye inquires diligently 
and thorougbly after the true comprehension of the form of the visible 
thing, and [since] dur ing that same intuition the radial axes are moved 
always across all the points of the surface of the visible thing, as was 
shown by the 52nd [prop.] of this [book], moreover since any perceptible 
motion is performed in a perceptible time, because, as we have shown 
elsewhere, time is proportional to motion, it is plain that it is necessary 
for any intuition to be accomplished in a perceptible time.
>>>
173 


Also the time [required] for intuition differs according to the diverse 
intentions of those visible forms which somebody intuits, an example of 
which is: when the eye should comprehend that a long animaI with many 
smali feet moves, then it comprehends first, by a modicum of intuition, 
its motion, and through the motion it grasps [later] that the same is 
an animaI. After that it will comprehend, by a modicum of intuition [focused] 
on the feet, that the same [animaI] has many feet, [reaching this conelusion] 
from an understanding of the distance between the feet, but without knowing 
[yet] the number of those feet. And then, by looking at it more diligently, 
it will leam the number of the feet by more intuition and by an effort 
requiring more time. 
Therefore the comprehension of its animality will be [achieved] in a smalIer 
time and the comprehension of the multitude of the feet will be in a greater 
time than that prior time in which it became known that the same was 
an animai ; moreover the [grasping of the] num ber of feet will, as yet, 
be in a [still] greater time than any of those [prior] times. Indeed the 
eye ought to look on any of those feet and count them; furthermore 
the amount of time [required] for the intuition of the feet will be according 
to the num ber of multitude or paucity of the feet, and this is also elear 
concerning the diversity of the other visible intentions. Thus the time of 
intuition of the visible intentions of the forms, one of which is number, 
varies according to the diversity of the intentions of the intuited forms. 
The proposed [thing] is, therefore, elear. 


[Proposition]57. The eye cannot comprehend the true form of the visible 
object by means of a first simple sight, but [only] after diligent intuition. 
Indeed as the forms of objects are composites of many particular intentions, 
of which some of the extant ones, presenting themselves to the first glance, 
[are rather] rough, while some [are] rather subtle (as are the smalllineaments 
and colors scattered in little bits, and similar [features] which cannot exhibit 
themselves immediately to the first sight, which is instantaneous by the 
55th [prop.] of this [book], from which [it is elear] that they stand in 
need of time in order to be seen), therefore [such subtle features] will 
be seen [only] after having been intuited diligently and not before. 
For the eye does not grasp the true form of the object except through 
grasping of all the true particular intentions which are in that form. It is 
therefore elear that the form of the object in which there are subtle intentions 
is not grasped by the eye, in conformity with the truth of the object's 
being. by a first glance, but [only] after a diligent intuition. And since 
even in forms in which there are no subtle intentions, the eye cannot 
assess correctly their absence at a glance, therefore even then there is 
also acting by intuition, for [the eye] cannot certify the truth of the form, 
except after the diligent intuition of any part of the form of the object. 
And so it is plain that the eye can never grasp the true form of the 


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object at first glance, but only after diligent intuition, and this IS what 
we have proposed. 


[proposition)58. Repeated looks [at objects) are imprinting and certifying 
better [their) sensible forms remaining in the soul. 
Indeed when the eye grasps a certain object and its form will have 
been certified at the [uItimate) sentient [body) then the form of that object 
remains in the soul and is shaped in the imagination of the same observer, 
as was shown in [the treatise on) the natural passions of the soul.. And 
if the grasping of the object will be repeated, then its form will be better 
imprinted in the soul than the. form of the object seen [only) once, because 
the eye rarely comprehends perfectly a thing seen once, but on account 
of the iteration of vi sio n, the form arrives always anew at the soul and 
the previously seen form is refurbished in the soul; and if any of the 
intentions of that form was surrendered to oblivion, it is [thus) restored, 
and if it was not previously seen, it is reelaimed. 
For the soul is reminded through the second form of the first form, 
and when the occurrence of the same intention is repeated many times 
in the soul. the soul will be remembering more that intention, and thus 
that form will be more [strongly) impressed in the soul, but also better 
certified. because in the first vision, in which the form of the object arrived 
at the soul, the soul did not comprehend adequately all the intentions 
which are in that form, nor will it have certified the same. And when 
the form will have come back for the second time, the soul will have 
grasped out of it something that it did not grasp the first time, and 
the more the [imprint ot] the form will be repeated on the soul, the 
more of it will become manifest that did not appear previously, and when 
the soul will have grasped the more subtle intent.ions of the forms, it will 
certify to itself more of the being of the entire. form. Ił is elear. therefore. 
from these things that repeated looks [are)... and so forth, as was proposed. 


[Proposition)59. No visible object is understood by the sense of sight alone, 
except only light and colors. 
Indeed since only these are' visible by themselves, as was premised in the 
postulates of this book I, it is elear that the same are prior to all other visual 
qualities, from which [it follows) that the same are exhibited to sight without 
other [visual entities), as without position, shape, and size, and similar [things). 
For others are not pres
nted to sight without those, for [when), visible entities 
do not actively participate in light, it is impossible for anything to be seen, 
as is elear by the fust [prop.) of this [book) ; therefore no other operation of 
the soul takes place amongst light and color except only the sensation of sighL 
For the light which is in an iIIuminated body is grasped by the eye 
according to its being and by itself through [the capacity) of the same 
sense, while the light and color whic.h are in a colored and iIIuminated 



 


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body are comprehended and mixed together by the eye at the same time; 
furthermore any of these is comprehended by the sense of sight alone. 
For first light is comprehended by the eye from the iIlumination of the 
sentient body. which is [fashioned] out of the substance of the eye, and 
color from the alteration of the form of the same sentient body and o( 
its coloration with the admixture of Iight, which is the hypostasis of color. 
Indeed just as with the arrival of the form of fust light, the sentient 
body comprehends only the light, thus, with the arrival of the form of 
color, it comprehends colored light. Therefore these two are comprehended 
by the sense of sight alone without other power and operations of the 
soul. which does not happen in [the case ot] any other visual entities, 
because those. as multiform, are being sensed by many senses. And if 
any of the same were sensed by the sense of sight alone and not [also] 
by other particular senses, this would occur either out of some participation 
of these [entities] or [out ot] privation of the sarne 2 , as is [the case] with 
transparency and opacity, darkness and shadow, in which the contributing 
reason is necessary all around, which is not necessary in the comprehension 
of light and co lor. The proposed [thing] is, therefore, elear. 


[Proposition]60. Every visible object is grasped either simply by sight, 
or with [accompanying] reason and discrimination. 
Indeed. as is elear by the preceding [prop.], sight alone comprehends 
simply [and] by itself light and co lor ; moreover there are rnany other 
[features] which were postulated concerning the number of visual phenomena 
which sight comprehends indeed not directly [and] by itself, but by means 
of other contributing actions of the soul. And there are many such visible 
phenomena the comprehension of which is not [reached] by the pure sense 
of sight. because when the eye comprehends two individuals of the same 
species and form at the same time, then it will comprehend them as 
individuals and it will comprehend that they are similar. But the sirnilarity 
of the two forms is not [itselt] the same two forms, nor one of those, 
neither [is it] a third form proper to [their] similarity, but it consists 
of those two forms com ing together in some thing. 
Consequent1y the similitude of the two forms will not be grasped but 
from a comparison of one of the same to the other. Therefore the comprehension 
of the similitude is not achieved by sight alone, but [also] as a consequence 
of the soul's power, which we cali reason, by an act, [then], of ratiocination 
comparing the various visual forms to one another. And likewise when 
the eye sees two white colors, one of which is whiter than the other, 
it will grasp the whiteness of both, and that one of them is of a stronger 
whiteness. Hence it will understand the similitude of the two whites in 
their whiteness and their difference in strength and weakness. 
In fact the distinction between those two whitenesses is not itself pertaining 
to the sensation of whiteness, because the sensation of whiteness sterns
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from the whitening of the eye's surface, which is accomplished by any 
whiteness, while the distinction of those [two] whites is achieved on account 
of the difference of action of those two whites in the same eye. Hence 
that distinction is not [stemming] from the sense alone, but it is [also 
a result stemming] from another power of the soul, which we cali the discerning 
[power]. And it is the same with the comparison and discrimination of 
other sensible forms; indeed nothing of those [forms] is perceived by sigbt 
alone, but by the collaboration of reasoning and the discerning virtue. 
For sigbt by itself has no discerning power, but [only] the discerning power 
of the soul distinguishes all those [things] by means of sigbt. The proposed 
[thing] is, therefore, elear. 


[Proposition]61. Out of the frequently intuited intentions of individual 
forms, there remains in the soul the fixation and certification of the universal 
form, [which is] for the extant eye the [fundamental] principle by means 
of which all the individuals of the same species are to be known. 
Indeed [it is the case] that any of the individual optical phenomena 
has a form and shape in which come together all the individuals of that 
species which differ only in the particular intentions understood by the 
sense of sigbt, and there will be in all those ind iv idu aIs a strong color 
of one kind, as [happens] quite universally in individual birds, say with 
a swan, a raven, a jay, and a jackdaw, and simiłar [individuals], in which 
there is uniformity of color fitting for an entire species, or even for more, 
since we actually see [bot h] a white raven and a white bear. Accordingly 
if the form, and shape, and co lor, and all the intentions from which 
the form of any individual of the species is composed is the universal 
form of the entire species, and the eye grasps that form, and shape, and 
color, and the intention of all those [individual forms] which are suitable 
to that species, then the soul will assess that particular object to be an 
individual of that species. 
Ali the same, on this account [the eye] will not recognize one individual 
as distinct from another indiyidual of the same species, untił it will have 
also grasped the particular intentions througb which the individuals differ, 
and until those will have settled in the soul and in the same imaginative 
power. Then indeed any intention of that universal form which is of the 
species of the previously seen individuals (appearing before the same eye 
througb the intention of individuals of that species), [and] whose form is 
in the soul, will be repeated by the eye together with the diversity of 
particular forms of those individuals. And when that universal form will 
be compared in the soul with the intention of another individual of the 
same species, then it will be fixed in the soul and become settled. 
Thus the soul will grasp by intuition the diversity of individuals of 
the same species, out of the diversity of the particular forms arriving 
at the eye with the universal forms, and, througb the coming together
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of visible accidents in diverse individuals, it will grasp that the form in 
which all individuals of that species come together is the universal form 
of all of them. Thus, then, the universal form remains in the soul and 
in its imaginative power, and that form is for the eye the [fundamental] 
principle by means of which all the individuals of the same species are 
to be known, inasmuch as it pertains to that [element] in them which 
[co me s] from the individuated universal intentions and any particular sensible 
intentions whatever. The proposed [thing] is, therefore, elear. 


[Proposition]62. Any true comprehension of the visible forms is [accom- 
plished] either thro\lgh intuition alone or through intuition together with 
foreknowledge. 
The comprehension of visible objects by intuition alone is accomplished 
when external objects are grasped [directly], as when the eye grasps a visible 
thing. which it did not perceive before. either as such or in its species; while 
through diligent intuition [the soul] acquires all the qualities lof the objeet] 
and its true form. For it cannot recognize its form. since it has not perceived 
it before. nor can it remember it; in this manner. then. that form will 
be grasped with true comprehension by intuition alone. 
Moreover the true comprehension of visible forms one from another. 
which should be [usually] accomplished solely by intuition. may sometimes 
be accomplished by intuition together with foreknowledge, as when the eye 
grasps the form of a certain visible object which it also knew before and 
the intention of whose form is [already] in the soul. either all or a certain 
part of it; then indeed the eye will gra sp at once, in glanc ing at that 
thing, its form. And after that, by a modicum of intuition, it will gra sp 
its entire form, which is the universal knowledge of its species, and it 
will recognize the universal form which it will grasp in that object in 
view of the comprehension of the form in the soul. especially through 
the recollection of that objeet. And then intuiting the remaining intentions 
which are in that object, it will certify the particular form of that lobject] 
appropriate to the same individual object. And should it remember that 
particular form, as previously comprehended by the eye, then it would 
recognize that individual form. And since no object is grasped with true 
comprehension except by some of these modes, therefore the proposed 
Ithing] is elear. 


[Proposition]63. Visual comprehension through recogmtlon. [Le., fore- 
knowledge.] is always accomplished by a certain manner of participation 
of the reasoning [power]. 
Indeed recognition is the comprehension of the similarity of two forms, 
namely of the form which the eye grasps with lits] knowledge when it 
feels itself as knowing the thing that it sees, and of the previously understood 
form quiescent in the soul. From which lit follows] that no visu al recognition 


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takes place except through recollection, because if no form were quiescent 
in the soul in this manner and lhence] available to memory, the eye would 
not recognize the visible object. Accordingly recognition is always accomplished 
as a result of the comparison of the form quiescent in the soul to the 
form seen outside thereafter, whether the quiscent form be the form of 
the species or of the individual to be known. 
And so sight grasps many things by recognition: indeed it recognizes 
a man as being a man, and a horse to be a horse, and Socrates to 
be Socrates, and it recognizes the animals to which it is accustomed, and 
trees, and plants. and stones which it has previously seen, and it recognizes 
the Ithings] simi1ar to those; and aU the intentions to which it is accustomed 
in visible things, and the quantities of ałl things to which it is habituated, 
which are not recognized by sight alone, by the 59th lprop.] of this IbookJ. 
For sight does' not recognize ałl that it has previously seen except 
when it wiłł have been recołlecting the previously seen forms. Therefore 
recognition is not visual comprehension by sense alone. but through reason 
comparing the present form of the object to the previously seen and 'now] 
quiescent form in it. Indeed recognition can never take place except through 
the comparison of the quiescent form in the soul to the external visual 
form. In this manner, then. it is elear that visual comprehension by re- 
cognition always takes place through some manner of the reasoning 'power] 
making lits] contribution. The proposed 'thing] iso therefore. elear. 


IProposition]64. Ił is necessary that every visual comprehension by re- 
cOlmition take place in time, but in less than is the time 'required] for 
comprehension through intuition alone. 
Indeed since. as was previously established in the preceding proposition, 
every visual recognition is accomplished through intuition and/or the quiescent 
form in the soul lbeing] remembered and applied to the form now being 
perceived through diligent observation, and since every intuition takes place 
in time, by the 56th [prop.] of this Ibook], and any recołlection of 1he 
previously seen form is accomplished in much greater time. because it is 
achieved through the soul's traversing the forms that it possesses in its 
imagination (which. if it were to occur instantaneously to the searching 
soul. it would not be a recollection but an ongoing memory). since. then, 
both of these lprocedures], namely intuition and recoUection. or any of the 
same, take place in time, it is elear that any visual comprehension by 
recognition also takes place necessarily in time; but in less than is the 
time lrequired] for comprehension by intuition alone. because the intentions 
extant in the soul [and] present to the memory are not needed for the 
identification of ałl the intentions that are in the forms of the recognized 
things, out of which they are composed as a matter of facto but the 
comprehension of any intention appropriate to them suffices for their com- 
prehension. 


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Consequently when the discerning virtue will have grasped in the form 
coming to it a certain intention appropriate to that form. it will be reminded 
of the first form and it will recognize all the forms coming to it. because 
any intention appropriate to some form is setting an imprint over those 
forms, as when the sight beholding Socrates grasps the feature of a human 
hand (and] comprehends immediately that it is a man, and (this] before 
it grasped the feature of his face or of other parts. Therefore from the 
comprehension of some intentions which are appropriate to the human form 
it grasps that the same object is a man. without the need of comprehension 
of other part s which it grasps only by means of previous knowledge 
(drawn] from the forms residing in the soul. and through the comprehension 
of some intention proper to that individual. as when it will grasp the 
intentions of that whole individual through the brightness of (his] eyes, 
or the roughness of the mouth. or the curvature of the eyebrows. or 
similar things. 
And in a like fashion it will recognize a horse by me ans of a certain 
spot in lits] forehead or elsewhere in (its} body, and a scribe recognizes 
from some (superficial] comprehension of the letters all the parts of speech 
or of language which he sees frequently and continually. And since the 
comprehension which is gained only by intuition takes place through the 
consideration of all the parts of the object and of all the intentions that 
are in it. while the comprehension by recognition takes place through 
consideration of only some intentions which are in that form. it is plain 
that vision by recognition is (achieved] in a shorter time than is vision 
by intuition alone; and this is why the eye grasps customary objects faster, 
ILe.,] in shorter time, (since they are. as it were] quasi latent in the sense. 
and (grasps] fastest those (objects] which it was habituated to know from 
its very beginning, or with which it persisted carrying on for a long time. 
Therefore that which was proposeą is elear. 


IProposition]65. Vision by foreknowledge (and] brief sight. (i. e.. a modicum 
of intuition.] does not produce sure comprehension of the form of the 
object. 
Indeed (it is the case] that vision by previous cognition is not lachieved] 
except superficially concerning the totality and comprehensiveness of the object. 
and (only] roughly, and due to certain external signs of that object. and 
the discerning virtue grasps the particular intentions which are in that 
object in the manner in which it recognized the objects by the first form 
of that object extant in the soul. But all the particular intentions of 
optical phenomena that are in corruptible things are changed with the 
passage of time. Moreover the eye does not gra sp the change of intentions 
of the visible thing by means of the previously customary form. because 
the change will not have been obvious or compreh-ensible to the eye at 
first sigh t. 


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Therefore previous cognition does not result in the true knowledge of 
the thing, as when in the previously known neat face of a man there 
would occur afterwards aspot, or a not so obvious scar; for when. after 
this, having seen that man for a long time, the observer will not recognize 
him according to his face, which he previously retained from memory, 
nor indeed will he grasp the spot or that scar in his face, except after 
diligent beholding performed at that spot or scar, and then it will grasp 
his form according to his being. And it is the same if the spot will 
have been always in the face of that known (individual]. but it will not 
have been manifest to the eye for a long time. Indeed. then. it is allowed 
that the observer may have at his disposal the nonspotted form of that 
(individual], (and] yet he will not apply it to that spotted face and he 
will not recognize that [individual] except after the intuition of many other 
particular intentions. And it is the same with other individual objects and 
with their diverse intentions. For in all those [cases], vision by previous 
cognition [and) by a modicum of intuition does not result in the sure 
comprehension of the form of the object. The proposed [thing) is, therefore, 
elear. 


[Proposition)66. The quiddity of no thing is visible by itself, but (only) 
by accident, by means of [its) sensible intentions, which are seen by themselves. 
Indeed since, as was postulated in the beginning of this book I, vision 
is not completed except with the arrival of visible forms at the soul. all 
of which [forms) belong to the genus accident, as is elear in the singular 
examination of the same [forms), it is plain - since the quiddity of no 
substance is of the genus accident - that no quiddity is visible by itself. 
Furthermore the quiddity of corporeal substances is perceived by the 
eye accidentally, namely by means of the comprehension of their visible 
intentions, which are visible by themselves. In this way, then, the comprehension 
of the quiddity of a substance is not accomplished except through the 
intrinsic cognition of the soul. which takes place by the comparison of 
one form grasped later to another form grasped earlier [which is) quiescent 
in the imagination. Consequent1y the comprehension of the quiddity of the 
seen substance, for instance [that] of a man. or of a dog, or of any 
other substance, is not [achieved) but as a result of the comprehension 
of the comparison of the form of the object to one of the universal 
forms quiescent in the soul and fixed in the imagination, which the eye 
had grasped previously. 
And rit is also obvious] that the discerning virtue which resides in 
the soul [and) through which the soul assesses the ditferences of things, 
for instance that a man is not a dog and vice versa, compares naturally 
those visible forms, namely those seen afresh, to the visible and fixed 
forms in the imagination. Therefore, when the eye will have grasped a certain 
object, the discerning virtue will instantaneously begin searching for its 


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Iikeness among the forms extant in the imagination. and that Ilikeness] 
once found, it will recognize through it the object and grasp its quiddity: 
and if it will not have found among the quiescent forms in the soul 
the similar form to the form of that object. it will not recognize that 
object, nor will it grasp its quiddity. 
In this way, then. no quiddity of any substances is grasped by itself 
by sight [aJone], but only accidentally, as was proposed. Indeed if any such 
quiddity were grasped by itself by the eye, then the quiddity of the sub- 
stance of any object whatever would also be comprehensible by the eye 
[alone]. as is elear in lights and colors, and [in such a case] the existing 
substances, inasmuch as they [are] indivisibJe 2 to the sense and [its] sensible 
operation, would [also] be seen according to their quiddities, which is 
not true. 
Indeed it is necessary that the visible body be of a certain [minimal] 
size with respect to the surface of the eye, for it to be actually seen. 
as is elear by the 19th [prop.) of this [book]. Furthermore it is similarly 
elear about the quiddities of aIl other things whatsoever: indeed the quiddity 
of any composite [body) is aJways [itself] composite, and the eye cannot 
grasps by itself its composition. And if the eye would have identified a certain 
quiddity as quiddity, then the eye would have [thereby potentially] identified 
any quiddity, many of which are certainly invisible, [and] as aIl of the 
same are inteIligible by themselves, and as this [identification by sight 
alone) is impossible. the proposed [thing] is elear. 


[Proposition]67. The first thing that the discerning power grasps from 
among the intentions appropriate to the visible form is the quiddity of 
light and color. 
Indeed inasmuch as light and color are the first [optical phenomena] 
and visible by themselves, surely their quiddities and essential differences 
cannot be grasped by the sense of sight alone. For the quiddity of light 
is not grasped by sight alone, except when the soul's power, which is 
an identifying Ipower] is cooperating. because the eye recognizes the sun's 
light and distinguishes between it and the moon's light, and the light 
of fire, by a previously made cognition and by means of the form retained 
in the soul. 
Also in the same way the quiddity of color is not grasped by the discerning 
power, except by recognition, when the color of the object will have been 
among the customary colors. But that distinctive cognition takes place 
[as a result] of the comparison of the form of color seen now to the 
similar form to that color grasped previously: for the eye cannot comprehend 
the reddish color and that it is reddish except because it recognizes the 
same, because, Ithat is.] in the soul of the observer. its form. as previously 
seen. has endured. 
For if the eye had never previously seen a reddish color, it could 


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not now identify such an object but it would assimilate it to the colors 
e10se to it land] known to the eye, as it does daily in the new mixing 
of any colors. And so when the discerning power grasps the diversity of 
light lscattered] over visible objects and the diversity of colm, it also grasps 
lthereby] the diversity of the quiddity of light land its ditference] from 
the quiddity of colors, inasmuch as the form that the eye grasps is mixed 
out of the form of light and color, lboth ot] which reside in the object. 
And since light and color are the first visible phenomena by the participation 
and help of which all the others are seen, therefore, it is necessary that 
the first which the discerning virtue grasps from lamong] the intentións 
appropriate to the visible form be the quiddity of light and color. and 
just as visual comprehension is due to that first [phenomenon], so the 
first operation of the discerning power is also due to those quiddities 
by themselves, as to those the prim presence of which shines back in 
the visual organs 'and to which] all lother optical phenomena] approach 
more or less in transparency. The proposed lthing) iso therefore. elear. 


tproposition]68. The grasping of color as such is prim to the comprehension 
of the quiddity of color; from which it is elear that the comprehension 
of all vlsible phenomena as sucho which are visible in ltheir] own genus, 
is prior to Ithe comprehension] of their special quiddities. 
Indeed the eye grasps color and senses that it is color before it may 
sense the kind of that co lor, as is elear in 'the case ot] strong colors 
positioned in a not too well lighted place. For there indeed the eye grasps 
the colors only indistinctly; but they would be Iproperly] distinguished 
by the arrival of a greater light or by a more lasting beholding. Therefore 
the first thing that the eye grasps from the form of color is the change 
of the sensing member and its coloring. because the eye is colored with 
the advent of the form in the eye, which, sensing itself colored. immediately 
senses the color, and after that. understands the quiddity of color from 
the distinction and comparison of that Icolor] with the colors known to 
the eye. 
Therefore the comprehension of color as such is 'achieved] before the 
comprehension of the quiddity of the same co lor, which is not accomplished 
by the sense of sight atone. but through cognition. when the same co lor 
had been previously grasped by the eye and its form had been retained 
in the soul's memory. And if the eye would grasp an external color which 
it had never [before] seen. then it would comprehend that it is a color, 
and yet it will not know what kind of color, but, having compared the 
same to other colors, it will liken it to the e10ser color similar to itself. 
and, as it happens, many observers Iwatching] that color at the same time 
in the same light will Iiken it to diverse colors. as happens with a color 
prepared from the dissolution of a mixed body from copper and silver. 
Indeed some will liken that [co lor] to the greenish 'tint] that stems
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from copper and somebody leIse) to the bluish color which comes from 
the silver. It is elear, therefore. through these test s that the comprehension 
of color as such is prior to the comprehension of the quiddity of color. 
And since color is the first visible lphenomenon) after light, it is elear 
that the comprehension of all visible Iphenomena) as such is prior to 
Ithe comprehension ot] their special quiddities: indeed position in general 
is grasped in the sense of sight before any species of position, and shape 
in general before any special shape. And should it happen for the species 
to be extricated in the. eye, (our elaim) still remains Itrue in) general. 
both (concerning] that which is of the first order lof generality) and that 
which is of the second order. And this was proposed. 


IProposition)69. The comprehension of different visible intentions through 
reason and discernment is accomplished at once, in an instant. while lthat) 
of similar lintentions) takes time. 
Indeed shape, and size. and transparency, and many 10ther) similar 
Ithings). when they are grasped at first glance. which always takes place 
in an instant of time. by the 55th Iprop.) of this Ibook). present themselves 
to sight forthwith (and) are comprehended through reason and discernment 
in the same instant, [they) and all the intentions which are in them, on 
account of the velocity of the reasoning [power). lndeed the discerning 
power does not prove by means of the composition and structuring of 
propositions in a syllogistic form. Therefore just as in the intellect, which 
is the necessary condition of lthe existence ot] principles, there is no need 
of any time for the actual understanding of universal propositions and 
Isuch as are) manifest by themselves, so indeed no time is needed, in 
apprehending particular conc1usions from those, because with the under- 
standing of the universal proposition. [reason) accepts at once the con- 
elusion which follows immediately from it, since the human soul was born 
suitable to argue without difficulty and labor. 
From which (it is elear) too Ithat) a man does not perceive that com- 
prehension, which is accomplished through reasoning and discernment, is 
performed through argument. just as aliule child separating and choosing 
the more beautiful out of two beautiful (things) does not perceive that 
this was done by way of argumentation and the consideration of things 
to be chosen. And so in a manner similar and conformable to this, 
inasmuch as this is possible. is accomplished the instantaneous comprehension 
of aU visible intentions by reasoning and discernment. 
Indeed the separation and argumentation of the distinguishing power 
takes place at once with the arrival of forms at the middle of the common 
nerve, because the entire (ocular) body, extended from the first surface 
of the eye receiving the forms I, all the way to the middle of the common 
nerve is sentient and transparent. and the passage of forms through it 
takes place in an instant for straightaway beyond the substance of the 


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eye 2 is the transparent visual SplfIt through which the sensltlve power 
is carried to the whole transparency of all humors and tunics of both 
eyes.\ Indeed all those transparent media are iIluminated by light and are 
colored by one color, or by various [colors] according to the diversity 
of colors of the sensed body, and the body which is in the concavity 
of the common nerve is the last body to which light and color arrive. 
Therefore since the form is extended from the first surface of the sentient 
body all the way to the middle of the common nerve, any part of the 
sentient body will sense the form, and when it will have arrived at the 
concavity of the com mon nerve, then it would be understood by the ultimum 
sent;ens, and then would the separation of forms be achieved. Thus there 
is no temporaI ditference between the act of distinction (separation) and 
the act of first sight, because, just as light multiplies itself across the 
diameter of the world in one instant, on account of the transparency 
of the intervening body, just so, as is obvious by the 55th [prop.] of 
this [book], the sensible forms reach in an instant across the medium 
of any transparent body to the middle of the common nerve, where they 
are sensed. understood, and distinguished by the soul's power. 
And as the soul's power is indivisible, all this takes place simultaneously, 
in a single instant. However when the visible intentions are made very 
much alike, as is [t he case with] the green Icolor] of rue [when compared] 
to the green [co lor] of mint, then their separation is not achieved in that 
instant in which their greenness is grasped by the eye, but [only] after 
the comparison of one to another, [Le.,] arter the fact of the lfirst] grasping; 
it is, therefore, accomplished in another instant, and thus between the 
instant of the first simple sight and the instant of distinction by comparison, 
it is necessary for an average time to be assumed. Therefore that which 
was proposed is elear. 


[Proposition]70. Ił is necessary that the comprehension of the quiddity 
of color be achieved in time; from which it is elear that the comprehension 
of the quiddity of all similar visible [phenomena] does not take place 
except in time. 
Indeed the comprehension of the quiddity of color is accomplished after 
the comprehension of oolor as such, as is elear by the 68th [prop.] of 
this Ibook]. And since color as such cannot be understood by simple 
sight except in an instant, by the 55th [prop.] of this [book], since, moreover, 
the comprehension of the quiddity of any color is composed of the com- 
prehension of color as such and additionally of another distinctive [and] 
ensuing comparison by which the quiddity of one co lor is distinguished 
from the quiddity of another color, because all mixed colors have an 
essential harmony in the act and hypostasis of light, and, in addition, 
some of the same have a maximum agreement with one another in the 
proximity of mixture, it is plain that that distinction of the quiddity of
>>>
185 


the same colors is completed in another instant of time than [that] in 
which it is grasped by [first] sight; but bet\\'Cen any two instants there 
is an average time. 
Accordingly. as the comprehension of the quiddity of color is accomplished 
through the separation of one color from another, it is plain, by the 
preceding [prop.]. that that separation is completed in time; therefore the 
comprehension of the quiddity also, necessarily. takes place in time. Moreover 
the eye does not comprehend the quiddity of color except through intuition. 
because. if color were not in some surface, so that the visual axes could 
be impressed in it in a sensible time, the eye would not comprehend 
the quiddity of colors. From which [it follows that] in things moved fastly, 
the quiddity of color is not distinguished; but if there are many colors 
in a thing moved rapidly, all will be seen indistinctly as one mixed co lor, 
as is obvious in a bali of diverse colors moved rapidly by strong throwing. 
It is therefore elear that it is necessary for the comprehension of the quiddity 
of the same color to be accomplished in time; and from this it is elear 
that the comprehension of the quiddity of all visible forms is not achieved 
except in time. 
Indeed if the eye does not grasp except in time the quiddity of color. 
which is comprehended by the sense of sight alone, it is plain that more 
time is required for the intentions of other visible [things], which are com- 
prehended as a resuIt of exceedingly [greater] distinction and cognition. 
Accordingly. the comprehension of all the quiddities of the visible intentions 
is accomplished in time. even though it is allowed that that time may 
sometimes be very smali. And this is what was proposed. 


[Proposition]7I. In individual forms, sight grasps specific intentions 10 
a shorter time than individual lones]. 
Indeed when sight grasps a certain individual man, it grasps that he 
is a man before it grasps his particular form. and it grasps that he is 
a man casually. by means of the intentions of the human form, or by 
same other proper agreement with the human form. as, for ex ample, from 
the uprightness of the body and the arrangements of the limbs of the body, 
while it may not grasp the lineaments of his face. However the individuality 
of the visible object will not be grasped except as a result of the com- 
prehension of all the particular intentions characteristic of that individual, 
or [at least.] of some of them; and these cannot be grasped except after 
the comprehension of all, or same, of the universal intentions which be- 
long to the genus or species of that individual. 
But the comprehension of a partial form is [achieved] in a smaller 
time than [the comprehension] of the entire form. And as individuality 
adds something over [and beyond] specific characteristics, it is elear that 
individuality iso as it were. somdhing like a totality with respect to the 
characteristics of the species. Therefore the comprehension of the species 


.......
>>>
186 


of the object is [accomplished] in less time than the comprehension of 
(its] individuality, and this is what was proposed. 


[Proposition]72. The specific and individual intentions of certain customary 
objects are comprehended in less time than other specific and individual 
intentions. 
Indeed some of the usual species of visible objects are not like other 
species, as. [for instance,] the human species, which, on account of the 
body's uprightness, is not Iike any of the other animals, and some are 
like other species. as [for instance], the horse species which is like many 
10ther] animals in its overall shape. Consequent1y the time during which sight 
grasps the species of an individual man and understands it to be a man, 
is shorter than the time during which it understands a horse to be a horse, 
and (t he time required is] maximaI when it grasps any of these at great 
distance, because the sight grasping an individual man moved locally will 
comprehend him immediately to be an animaI on account of motion, and, 
on account of the body's uprightness it wilI comprehend the same to be 
a man; but [even though] it is allowed that on account of motion it 
can also comprehend that an individual horse is an animaI, and [that] 
due to the number of four legs it may comprehend the same to be a beast, 
stiII it wilI not comprehend on account of this the same to be a horse, 
because the equine intentions which are perceptible from a remote distance 
from the eye reside in many quadrupeds, as [for ex ample] in a mule and in 
other [animals], which are like a horse in many essential and accidental 
intentions. 
Accordingly if the eye does not comprehend some of the intentions 
proper to a horse, it does not comprehend that [animaI] to be a horse. 
And so as the time in which sight comprehends the upright position of 
the human body is not like the time in which it comprehends the shape 
of a horse together with the particular intentions through which a horse 
is distinguished from other beasts, as is the ont1ine of its face, and the 
length of the neck, and the velocity of motion, and the amplitude of 
(its] paces, therefore the comprehension of the human species is [achieved] 
in less time than the comprehension of the equine species. Indeed even 
though those two times are [both] smali. stiII one of them is greater than 
the other, in accordance with all its dispositions. 
And similarly as the garden rose is not like any other flower in its 
species' shape, or even in the intensity of its redness, therefore the eye 
understands its species, on account of [its] rose-like redness, in less time, 
than the species of the rue on account of its greenness, to which many 
grasses are similar. And generally the quiddities of all the species which 
can simulate others are not comprehended as fast by the eye as the quiddities 
of all species that are similar to few [others] or to none. And the same 
is also (the case] with individuals. because an individual that is unlike
>>>
187 


any other is comprehended by a modicum of intuition and by [characteristic] 
signs, while that individual that is like another individual ought to be 
comprehended by a lot of intuition. What was proposed is, therefore, 
elear . 


[proposition]73. The sensltIve power grasps the size of the angle in 
the centre of the eye facing the surface of th
 visible object only as 
a result of comprehending the part of the eye's surface in which the 
form of the visu al object is fashioned. 
Indeed albeit the viewpoint of pure mathematics may be [appropriate] 
in this matter, since the size of the part s of spherical surfaces subtending 
lany] angles may be known from the size of those angles, because just 
as the centre is the principle of constitution of the entire sphere, so the 
parts of the eight solid angles which are about the centre of the sphere 
or about any point of the universe, are the distinctive principle of all 
the parts of the sphere's surface, by the 87th [prop.] of the first [book] 
of this (treatiseJ I, still [we must say more, as] in the experience of this 
sensible science, which is intermingled with the condition of natural things, 
the sensitive virtue comprehends a posteriori via the senses from the grasp 
of the parts of the eye's surface in which the form of the visible thing 
is fashioned. the size of the angle at the centre of the eye facing the 
surface mentioned before 2 . 
Indeed the sense of sight grasps naturaUy that surface in which the 
form of the object is fashioned, by distinguishing the light and co lor which 
fali by themselves in that distinct part of the eye from other surfaces. 
And when it will grasp the size of that part, then it pictures to itself 
the angles which correspond to those parts and it grasps their sizes at the 
centre of the eye in accordance with the size of the parts of the surface 
of the eye subtending those angles: for the angles are not certified then, 
except due to the motion of the beholding eye over the diameters of the 
visible thing. or over the space the magnitude of which the eyes desire 
to know. The proposed (thing] is, therefore elear 3. 
And it is admitted that the radial lines do not meet in the centre 
of the eye, because the intersection of the visual axes comes to pass in 
the middle point of the common nerve, as became elear in many preceding 
theorems. Ali the same the parts of the surface of the same eye are 
informed as if the radial lines might meet in the centre of the same 
eye. were [this not] prevented by their refraction in the second transparent 
medium 4, as is elear by the 22nd (prop.] of this (book). And this deserves 
being noted. as we shall make use of the centre of the eye in the following 
(booksJ. as if the radial lines do meet angularly in it. because in this 
manner aU vision is informed. 


......
>>>
IV. NOTES AND COMMENT ARIES 


Book II 


PROLOGUE 


I In book I of his treatise. 


PROPOSITIONS 


Proposition I 
I Prop. XI. 17 proves that two straight Iines intersected by parallei planes are divided 
according to the same ratios. 
2 This last sentence contains obscure elements, as Witelo does not state which "Iongitude 
lines" he is referring to. nor does citation of I. 29 Ełement.f seem entirely appropriate. 
Stm the meaning is obvious: TV II to the lines which are .L to RS. 
3 Here ends this typically Witelonian, phatic description of the construction of "the 
instrument" and begins the proper experimental-instrumental "proof' of the enunciated pro- 
position. r 
4 In other words, Witelo establishes that the centers of the two apertures (M. Y) 
and the center of the image circłe, P, are colinear, or that Y lies on the diameter of 
the middle circłe. M P. 
5 The last "supposition" (postulate) will not do; cłearly, and this is the reason for 
Risner's reading. Witelo intends his penultimate postulate as the warrant for his concłusion. 


Commentary 
This very long-winded. but quite plain, proposltlon requires anything but further elabo- 
ration. For a succinct and simpJified description of the instrument and its functioning, 
see A. C. Crombie, Rohert GrOS.fe1e.fte and 1he Origins oJ Experimen1a/ Science IlOO-1700 
(Oxford. 1953), pp. 220- 223. For a detailed modern reconstruction of the instrument. in- 
cłuding an attempt to cope with the medieval units of measurement. see Appendices Nos. 
l and 2, prepared by Andrzej Bielski and Witold Wróblewski in the Polish version of 
the present book. Witelo substantially copied this proposition from Alhazen's De a.fpectibus 
VII. 2 (cf. the Risner edition, Op1. The.f. A/h., pp. 231 - 233). Alhazen's proposition was 
translated and commented on by D. C. Linberg in Ed. Grant, A Source Book in Mediella/ 
SciRnce (Cambridge, Mass.. 1974). pp. 420- 422. The reader should also consult this translation 
to acquire an easier understanding of the construction of the instrument. 


Proposition 2 
· The Jine in this Figure is horizontal ID the MSS and vertical in the Risner edition 
(The editor). 
I I.e.. just as the total time required to cover AD ensues from the summation of 
the partial times, the total space is a result of the summation of the partial imperceptible 


--
>>>
189 


spaces corresponding to those partial times. Ergo. perceptible space is composed of impercep- 
tible part s, which is absurd, according to Witelo, and hence light "will be carried in the 
smallest sensible time over the smallest perceptible space". 
Commentary 
In this instance, Witelo contradicts the modern view, according to which the speed of 
light is finite and agrees with Aristotle's position, whom he cites by name, as presented 
in De anima II. 7. 418b 20-26 and De sensu 6. 446b 27-447 a 3. The proof is as 
good as (or better than) they come during the Middle Ages, although. of course, it 
is not clinching. It is the kind of logico-mathematical argumentation by reduetio ad absurdum 
reminiscent, somehow, of Parmenides' and Zeno's style in their arguments against change 
and motion. 
The question of the speed of light aroused sizeable interest during the Middle Ages. 
A careful survey of the various views appears in D. C. Lindberg, "Medieval Latin Theories 
of the Speed of Light", in Roemer et ła lIitesse de la łumiere (CNRS, Colłection d'histoire 
des seiences, 3) (paris, 1978), pp. 45-72, reprinted as article IX in D. C. Lindberg, Studies 
in the History ol Mediellal Optics (London, 1983). 


Proposition 3 
Commentary 
It is on account of the claim this prop. makes that geometry is pertinent to optics and 
physical rays can be dealt with mathematically. as if they were mere unidimensional lines. 
The prop. is taken over from Alhazen IV. 16 (Op/. Thes Alh.. p. 112), where it is said. 
inter aUa: "Omnis linea. per quam movetur lux a corpore lumjnoso ad corpus oppositum. 
est linea sensualis, noQ. sine latitudine. Lux enim non procedit. nisi a corpore. quoniam 
non est, nisi in corpore: sed in mjnore luce, quae sumi potest. est latitudo, et in linea 
processus eius est latitudo: et in medio illius lineae sensualis, est linea intellectualis. et 
aliae eius lineae sunt aequidistantes illi". Still. it seems to me that. in this case, Witelo's 
exposition is both more lucid and shorter than Alhazen's. 


Proposition 4 


Commentary 
This is hardly a proof, but rather a rhetorical elaboration of the second definition. 
It is fair to assume, remembering Witelo's phatic style, that he himself, probably, saw 
the prop. as simply an actual, physical justification of that definition. In any case, the 
content is appropriated from Alhazen I. 28 (Opt. Thes. Alh., p. 17), where it is seen 
as a sheer explanation of what occurs when light and colors penetrate transparent bodies 
("Et remanet modo explicare quaestionem..."); since Alhazen 's Perspecti\la contains no de- 
finitions. postulates, or even the enunciation of propositions as such (what appear as enun- 
ciations in Opt. Thes. Alh. stem from Risner), his procedure is not open to the criticism 
levelled against Witelo. 


Proposition 5 


Commentary 
This is one of those experimentally grounded pro ps. that abound in book II (as well 
as in some other books). For an assessment of Witelo's alleged empiricism. by Crombie 
for instance. see S. Unguru. "Mathematics and Experiment in Witelo's Penpectił'a". in 
Edward Grant and John Murdoch, eds., Mathematics and lts Applications to Science and 
Na/ural Philosophy in /he Middle Ages (Cambridge University Press, 1987, pp. 269- 297. The 
prop. is taken over from Alhazen I. 29 (Opt. Thes. Ałh., p. 64). 


--
>>>
190 


Proposition 6 
I This would follow not from V. 18 but rather from V. 12. 


G GD 
2 This does not follow. What does follow. from Witelo's premises, is that if - - 
A AB 


G D D GD 
and - = - then -  -. 
A B' B AB 


3 The words included in round parentheses, though appearing in the fjrst edition and 
in all collated manuscripts (this being the reason for their preservation in the text) clearly 
represent an jncoherence in the text. Prop. I. 3. Perspectilla shows how to construct a fourth 
proportional to three given lines. (Cr. my edition of Book I in Studia Copernicana XV, 
pp. 49- 50, 217). 
4 Notes l and 2 above are a1so pertinent. mutatis mutandis, to the unwarranted con- 
clusion of this sentence. . 


Commentary 
This is a very curious and original (i.e., not coming from Alhazen) proposItIon. The 
"proof', in addition to being circular, contains inexplicable mistakes and inadvertences, 
as we saw in the notes above. Thus, if we denote by Px the "power of x", then what 
Witelo sets out to prove is that PAs:AB: :PA:A. He takes PAS = GD, PA = G, and Ps = D. 
Then, he claims, it follows ("Therefore") that G:A: :D:B, which is the same as P A:A: :Ps:B. 
But this does not folIowand it is nothing but another form of the original claim; hence, 
the circularity. The import of the proposition is apparent: it calls for the light of any 
luminous body to be uniformly distributed throughout the body's matter. 


Proposition 7 
I This Figure is based on the MSS. A somewhat different Figure, based on Risner's 
edition, is reproduced in the Polish version (The editor). 


Commentary 
Even though Witelo does not say SOf it is apparent that he could not have drawn 
his conclusion without the truth of the previous proposition which ensures uniformity 
of distribution of light through matter. 


Proposition 9 
I In I. 16 Witelo says: "If straight lines are drawn through the ends of two parallel 
and unequallines, it is necessary that they meet on the side of the smaller line" (S. Unguru, 
Wilelonis Perspectillae Liber Primus, p. 56; my edition of Book I will be quoted in the 
future as Wit. Persp.). See a1so ibid., p. 223 for the Latin text. 
2 Cr. ibid.. pp. 50- 51. 171. 218. Why Witelo performed this gfatuitous inversion, when. 
without it, the proportion he obtained corresponded exactly to his own statement in the 
diorismos above, is not elear. 
3 This follows from the previously established fact that DE: AB : : GE: GB. 
4 Strictly speaking, V. 10 Ełements does not apply since it deals with three magnitudes, 
one of which is taken as the standard against which the ratios of the other two are 
assessed; on the other hand I. 4 Witelo clearly applies: "An addition of equal lines to 
two unequal lines of known ratio having been made. the ratio of the longer to the shorter 
is diminished [thereby)" (ibid., p. 50; p. 217 for the Latin text).
>>>
191 


Proposition 10 
I That is, as in the case of prop. 9, AB /I DE-:'AB -:: DE, and the former is umbrageous 
while the latter is luminous. 
2 The ambiguity between U and V is intentioual, as the opening of angle DAG 
may be towards either U or V. the reasoning being unałfected. This also explains why 
AV can be either "drawn forth" or "cut off". 


Proposition II 
I The reference is to the third postulate of this book. 


Proposition 12 
I As will become elear in the course of the proof, "smaller", and not "shorter", is the 
proper translation, in this context, of "breviorem". At issue is not just one dimension 
but rather the entire size of the two shadows. 
2 The pertinence of I. 14 (cf. Wit. Persp., pp. S5, 222) seems unwarranted. It is apparent, 
however. that the conelusion follows from I. 2 and by some additional elementary argumenta- 
tion (by reductio ad ahsurdum. for example). 
3 Hereinafter, "( Elements)" will be dropped, &0 that references of this type (say "x. 15") 
from the critical text will always involve the Ełements. 


Proposition 13 
I It is not entirely elear what Witelo has in mind with this statement. The purport 
is, however, apparent. Considering all the perpendicular rays emerging from the luminous 
body (and only those), .there can be only one such ray issuing from each point, i.e., 
the interposed straight line coincides with only one ray, all other rays ("indivisibiles") 
emerging from other points. 
2 This is the sixth delinition of Book II. 
3 Again Witelo's last statement is obscure; the implication. however, is elear: between 
the end point of the elevated line and the underlying surface stretches the extension of 
that line. which is a shadow line. Ali that shows, however, on the dense surface is a punctual 
shadow. where the extension intersects the surface. 


Proposition 14 
I This needs some justilication. Keeping in mind. however, that Witelo presupposed 
the knowledge of Euelid's Ełements (cf. my Wit. Persp., pp. 169-170 and "Witelo and 
Thirteenth-Century Mathematics: An Assessment of His Contributions", [sis, vol. I (1972). 
Part 63, no. 219, pp. 496- 508), it is fair to assume that he relied tacitly on XI. 3, 
which proves that the intersection of two pIane surfaces is a straight line, thus providing 
the required justilication. 


Commentary 
The truth of the prop. follows, indeed, quite directly from the previous prop., as 
Witelo himself points out. Suffice it here to say that the justilication of the conelusion 
for the case of the elevated perpendicular pIane surface is the same, mutatis mutandis, 
as that given in n. 3 to prop. 13. 


Proposition I 5 
I This is, of course, mathematically non-rigorous. It is, however, an assumption that 
even the early creators of the integraI and differential ca1culus made without qualms; in 
a sense, they were "vindicated" by non-standard analysis.
>>>
192 


2 When "aU" the paralleI surfaces Iying between the surface of the shadow and the 
elevated base of the umbrageous body are taken into aceount. 


Commentary 
This is obviously the generalization of the set-up of the prevlous two props. to the 
three-dimensional case. 


Proposition 16 
· The present Figure corresponds, in the Polish version. to Fig. P3 (The editor). 
I The construction of lines ZE and ZB reproduces indeed the steps of a weU-known 
construction for drawing the two tangents to a given circle from a given point. 
2 The correct reference according to the Heiberg eóition is III. 31, which proves, among 
other things, that the angle in a semicircle is right. 
3 Again, the correct reference is III. 16 (Porism). which concludes that a line drawn 
at right angles to the diameter of a circle at its extremity touches the circle. 
4 Because, by III. 8, aU Jines faUing between E and B. on the convexity of circle 
DG as seen from Z, are shorter than the two equal tangents. 

 If there are longer straight Jines from Z lo the circle than the tangents. they would 
have to fali on the eoncave are BE (as seen from Z). and this would imply that such 
lines must intersect the tangents between Z and the points of tangency. This is Witelo's 
claim and it makes perfect intuitille sense; indeed. by III. 16, no straight line can be drawn 
between the arc of the circle and the tangent. from which it "follows" for Witelo that 
the posited longer lines would have to cut the tangents between Z and the points of 
tangency, which is, of course, absurd. as it would mean that there is more than one 
line between two point s. orf as Witelo says in the next sentence, "two straight Jines enclose 
a surface". (III. 16 proves that the tangent faUs "outside the circle". This implies, then, 
that lines drawn on the outside of the tangents wiU not even touch the circle, and so 
they are excluded by Witelo.) 


Proposition 17 
I We have left. then, the realm of mathematics. 


Proposition 18 
I This implies that AB = DG. 
2 According to I. 33, straight lines joining equal and paraUel straight lines In the 
same direction are themselves equal and paraUeJ. 
3 This is the fifth postulate. 


Proposition 19 
I This would appear to be warranted by prop. II. 6 above. 
2 As obvious from the eritical apparatus. the first edition as well as all coUated MSS 
refer the reader to I II. 1 5. 


Proposition 20 
I "Only one perpendicular can be drawn from a given elevated point to any piane 
or convex surface Iying under it" (Wit. Persp., p. 58); cf. also, ibid., p. 225 and p. 176). 


Proposition 21 
· In the Polish version whieh follows Risner, point E, Fig. 13 is marked as C (The 
editor).
>>>
193 


I One can. conceivably. vindicate this reference (III. 17 gives the construction of a tangent 
to a cirele from an external point and uses the fact that the tangent is perpendicular 
to the radius at its extremlty.) Risner's refcrenee. III. 18, is more appropriate. h()wever. 
(Of cI'urse. III. 16 w()uld iIIsI' do.) 


Commentarv 
This proposition stems from AI-Kindi's Dl' A.
pl'("lihll.. prop. 14. Cf. A. Bjornbo 
and S. Vogl. "Alkindi. Tideus, und Pseudo-Euklid: Drei optische Wer
e". Ahlrandlul/gen 
zur Geschichte der mathemati.
chen Wissenschaften. vol. 26. pt. 3 (1912). pp. 1- 176. AI-Kindi's 
treatise appears on pp. 3-41: for an assessment of AI-Kindi's role in the development 
of Islamic opties. see A. Mark Smith. Witelonis Perspectil'ae Uher Quintus. pp. 23-25 
and lindberg's Tlreorie.
 o" Vi.ioll, chap. 2, pp. 18-32. 


Proposition 22 
I Sinee I. 31 deals with drawing a straight line pilrallel to a given straight line through 
a given point. one must assume that one of the two lines. AB and AG. is extended 
(say AB to D) and through its extremity (D) the required paralIel, DE. is drawn. 
2 "A line drawn from a point of one of (two] parallei lines. in the same surfaee 
[as those parallei lines]. must necessarily intersect the other. if [t he laUer] is indefinitely 
extended" (Wit. Penp., p. 49): cf. illso ihid.. pp. 216-2'7 ilnd 170- '71. 


Commentary 
The crucial steps of Witelo's proof, if P" represents the light's power in X and ly its impression 
in y, are: 
P HT P BG 
(1) -(-, by V. 8, since PBG)P HT 
lvz lvz 
P HT P BG 
(2) - = -, by his own 11.6 
lvz lBG 
P l 
(3) 
=
, by V.16 
P BG lBG 
Now, since PHT(P SG :.I..z(ISG' q.e.d. 
It is important, in order to avoid confusion and reach absurd conelusions on Witelo's behalf, to 
distinguish between mere mathematical dimension and its physical signification in eontcxt. Thus. 
although YZ = BG, lvz "ł' lBG. Lastly, it should be stated that we, too, believe something akin to 
Witelo's c1aim to be the case, i.e., that there exists an "inverse-square relauonship" between 
distance from light-souree and surface iIIumination. 


Proposition 23 
· In Fig. 15 of the Polish version. which follows Risner. Jines EF. AB and the diilmeter 
paralle' to those lines have been omitted, since they are not ineluded in the proof (The 
editor). 
I DG is perpendicular to the pIane of cirele ABC belonging to the sphere of the 
luminous body. 
2 In his I. 60. Witelo says: "If between two tangent lines drawn to a cirele from 
one point. two other lines tangent to the same cirele are drawn [f rom another point]. 
the points of tangency of the interior [tangents] will fali between the points of tangency 
of the exterior Itangents]: and if. thereafter. the arcs between the points of tangency [on 
each side of the diameter] are equal. the meeting-points of each [pair of tangents] will 
always be in the same extended diameter of the cirele; moreover lit] the interior [tangents 
are] extended on each side they will necessarily intersect the exterior [tangentsJ" (Wit. Persp., 
p. 92): er. 'illso p. 253 for the latin tex!. 
.\ II is not elear which obliquity Witelo has in mind. 


13 - Witcloni. Penpcctivae... 


...... 


..J
>>>
194 


Proposition 24 


Commentary 
This is hardly a proof in the mathematical sense of the word, so WJtelo could have 
finished his attestation of the claim with his ex ample, without any real loss to the reader. 


Proposition 25 
I The present Figure corresponds to Fig. P 4 In the Polish version (The editor). 


Proposit ion 26 
I This Figure In the MSS has circles instead of Risner's ellipses, see Polish version 
(The edi(or). 


Commentary 
Risner remarks at the end of the enunciation. "Aristarchus Samius in libro de magni- 
tudinibus et intervaUis solis et lunae" (Opl. Thes. Wit.. p. 71). There seems to be no 
record of a Latin translation of Aristarchus's treatise before George Valla's of 1488. Further- 
more. Pappus's comments on the treatise. contained in book VI of his Collectio Mathematica. 
for a possible translation of which I have argued ("Pappus in the Thirteenth Century 
in the Latin West", Archille for History of E:t:act Sciences. vol. 13, no. 4 (1974), pp. 307- 
324). compris; no specific information as such which Witelo could have taken over directly 
for his proposition (cf. Hultsch's edition of Pappi Ałexandrini Collectionis Quae Super.vunt, 
3 vols. (Rerlin. 1876-78. VI. pp. 554.6-560.10: an English translation appears in T. L. 
H eat h. Ari.Vlarchus ol Samo.v The Ancient Copernicus (Oxford. 1959), pp. 412- 414). Proposition 
I of Aristarchus's treatise could have served to obtain Witelo's conclusions (cf. Heath. 
op. cil.. pp. 354-359). On the translations into Latin. see R. Weiss. "The Translators 
from the Greek of the Angevin Court of Naples". Rina.vcimento. vol. I (1950). pp. 195- 
226. A. A. Bjorn bo. "Die mittelalterlichen lateinischen Obersetzungen aus dem Griechischen 
auf dem Gebiete der mathematischen Wissenschaften", Archiv fii die Geschichte der Nalur- 
wissenschafien und der Technik, vol. I (1909). pp. 385-394, C. H. Haskins, Sludies in 
Mediellal Cułture (New York, 1958), ide m, Studies in the History of Mediellal Science (Cam- 
bridge, 1924), J. G. Wenrich. De auctorum groecorum lIersionihus et commentariis Syriacis Ara- 
bicis Armeniacis Persicisque (Leipzig, 1842), and aU the other pertinent items appearing in the 
Bibliography of S. Unguru. Wit. Persp.. pp. 317-325. 


Proposition 27 
· In the Polish version the present Figure corresponds to Fig. P 5 (The editor). 
lAssuming, that iso that these are "t he longest rays" reaching the opaque body. so 
that II. 16 Witelo clearly applies. 
2 Cf. n. I to prop. 21 above. 


Commentary 
As in the case of the former propoSItion. Risner mentions again. and in the same 
words. Aristarchus's treatise. Concerning this proposition. however, proposition 2 in Aristarchus 
coułd have clearly served as a direct model: "If a sphere be iIIuminated by a sphere 
greater than itself. the iIIuminated portion ofthe former sphere will be greater than a hemisphere" 
(op. cit., p. 359). The proof in Aristarchus is more to the point (ibid., pp. 359-361) 
and both more elegant and less wordy. On the status of Risner's citations of sources, 
cf. Wit. Persp.. passim. 


.........
>>>
195 


Proposition 28 
· See slightly different Fig. 19A in the Polish version (The editor). 
t "III. I S" as appears in all MSS and in the first edition is clearly wrong and was, 
consequently. replaced by "III. 16", which is also Risner's reading; on the other hand, 
"III. 17" was preserved in this edition, in light of the remarks appearing in n. 1 to prop. 
21 above. 
2 I. 14 Witelo states: "If a straight line fali ing on two straight lines makes the alternate 
angles unequal, or two interior (angles on the same side] less than two right (angles], 
or an exterior (angle] unequal to the interior (and opposite angle on the same side], those 
two lines must meet on the side of the smaller angles, (this being] impossible on the 
other side; and (reciprocally] if the lines meet. the said angles must exhibit themselves 
in any of the proposed manners" (Wit. Persp., p. S5); cf. also ibid.. p. 222 for the Latin 
text and p. 175 for a commentary; also, S. Unguru. "A Thirteenth-Century »Proof« 
of the Parallei Postulate", Historia Mathematica. vol. S (1978). pp. 20S- 210. 


Proposition 29 
· In Fig. 20 of the Polish version, which follows Risner, lines AC and BC have been omitted, 
since they are irrelevant for the proof (the editor). Here E and G should be interchanged. 
I Possible candidates are several. One cannot know exactly which props. Witelo had 
in mind. I. 15. could conceivably be one such prop. (cf. Wit. Persp.. p. 102). 


Proposition 30 
ł In the Polish version. following Risner, Fig. 21 tS slightly different (The editor). 


Commentary 
Clearly the first part of this proposition is the converse of the immediately preceding 


one. 


Propositions 31, 32 


Commentary 
Even .tJ1
ugh Witelo does not say so explicitly. the "message" of these two props. 
is an immediate consequence of his first four postulates and. specifically. of the stance 
that sees in light and darkness two contraries. 


Proposition 34 
I In the Polish version. following Risner. the notation in this Figure has been symmetrically 
changed from right to left and left to right (The editor). 


Proposition 3S 
I I. 10 gives the Euclidean method of bisection of a segment of a straight line. Its 
mention here can only mean that Wite lo selects point E. which "falIs short of the middle 
of its side AG', after he has established in principle by means of I. 10 where exactly 
the bisection point of AG lies; I. 31 shows how to draw a straight line paralleI to a given 
straight line through a given point. 
2 Cf. n. 2 to prop. 22 above. 
3 This is 50. presumably, because AE is ałmost half of AG (see n. I above). 
4 In I. 9 Witelo says: "If four quantities (are given so that] the first is greater than 
the third and the second less than the fourth. (then) the ratio of the first to the second
>>>
196 


will be greater ,han ['hat] of the ,hird to the fourth" (Wit. Persp.. p. 52). Cr. also ihid.. 
p. 219 for ,he La,in 'ex' and p. 172 for the commentary. 

 The proposi'ion is obviously 'rue; why Witelo selected just these Euelidean propo- 
si,ions in no' elear. 
/ t\J!ain Wilelo's conclusion IS 'rue. This is easy to see. Wha' Witelo has shown is 
GH HQ EV 
--- 
GB HT ED 
Now, if we cali 'he lenghts of 'he consecutive paralleI lines (like those appearing in the 
dCl1omina'ors of the three raUos aboveJ Al' A2' ...A.,... then Witelo has shown that 
A.-A._ I A.- I -A.- 2 
 
A. A.-I 
(ac'ually. Wi'elo's ra'ios are one half of thoseJ. This leads to: 
l A._I l A.-2 
-- -- 
A. A.- 1 


A.- 1 A.- 2 
-- 
A. A.- 1 


A. A.- 2 
-- 
A._I A._I 


Q.E.D. 
7 Up 'o Ihis point, there is no'hing wwng geometrical/y with this. The conclusions that 
follow. however. are another matter altogether establishing the absurdity of the elaim con- 
tained in ,he enunciation; bu' morc on 'his in the following commentary. 


Commen'ary 
II iso of course. a non sequilur. 'hough a rather "charming" and attractive one in 
light of the enuncia,ion. ,hat as "the excess of the more remote bases over the cIoser 
bases is... diminished". ,he diverging rays AR and AG will start proceeding in a paralleJ 
manner! If anything. ,he proof has established ,hat though AR and AG increasingły di- 
verge. the amount by which "successive bases" exceed one another.decreases monotonously! 
However. without defending Wi'elo's blunder in any way, I feel it stems. basically. from 
Iwo sources: (I) The confusion between some correc' geomelrica/ conclusions and the wrong 
sen.
ihle (or perceptih/e) implica'ions drawn from ,hose conclusions. (2) The need to find 
some geometrical justification for ,he very difficult problem of pinhole images. As to (I). 
it was obvious 'o Witelo (too) that solar rays proceed for all practical purposes as if 
they were paralleI. Prop. II. 35 was meant among other things, to provide a geometrical 
jus'ification for what is essentially an empirical phenomenon. Concerning (2) more will 
be said in dealing with props. 39-41. Finally. though Witelo does no' say a word abou' 
ił. ,he prop. is "horrowed", down lo diagram including no'ation (and enlarged). from the 
Pseudo-Euclidean De specu/is, "an Islamie compilation of 'heorems drawn primarily from 
Hero __ ("a/op/rico. Euclid's Op/ico. and ,he Pseudo-Euelidean Catoptrica of Theon of Alexandria. 
This De specu/i.
. which circulated widely in the West during the ,hirteen,h cen'ury, is 
of uncer'ain date..... (David C. Lindberg, "The Theory of Pinhole Images from An'iquity 
to 'he Thirteenth Cen'ury", Archil'e Jor History ol Exact Sciences. vol. 5 (1968). pp. 154- 
176. a' 159). On Dl' speclI/i.,.. see A. A. Bjornbo and Schaslian Vogl. "Alkindi. Tideus 
und Pseudo-Euklid. Drei optische Werk e", Ahhand/ungen zur Geschichte der mathemati.
chen 
Wissen.
dra.ften. vol. 26. pt. 3 (1912). pp. 1-176 and S. Vogl, "Ober die (Pseudo-) Eukli- 
dische Schrift » De speculis«", Archill lur die Geschichte der Naturwissenrchaften und der 
Technik, vol. I (1909). pp. 419-435. 


- 


.....
>>>
197 


Proposition 36 
I Witelo speaks loosely. Since. however. all indications seem to be that triangles FBC 
and FTZ are similar. Witelo's conelusion is correct. 
2 Actually II. 19 Witelo. as applied to the case at hand. shows that it is npt the 
case that rays from any point of the luminous body can be drawn to the penphery 
of the window; once more. Witelo speaks loosely. 


Proposition 37 
.. In Fig. 25 of the Polish version. which follows Risner, lines EA. ED. EG and EB 
have been pmilted. sinee they are irrelevant fpr Ihe proof (The editor). 
I I. b' Wltelo says: "Att slraijl:ht lines drawn from a pole to Ihe peripher) of its 
cirele are equal" (Wit. Penp.. p. 96); cf. also iNd.. p. 256 for the Latin text of the prpposition 
and pp. 183-1
4 for the commentary. 
2 I. 8 Euelid elearly applies since all triangles ZEA. ZEB. ZEG. ZED are congruent. 
having their corresponding sides respectively equa!. 

 The appeal to XI. 14 is unwarranted, as it proves that planes to which the same 
line is orthogonal are paralIe!. What Witelo needs is the converse of XI. 14. sinec he 
knows. to begin with. that the piane of the aperture and of surface FHKL are paralIe!. 
4 I. 25 Witelo states: "Ali perpendicular lines drawn between parallei lines or surfil(''eS 
are [themselves) paralleI and equal; and if straight lines interseet parallei lines or surl
lces 
at equal angles, they are equal" (Wit. Persp.. p. 61); er. also iNd., p. 228. II is elear 
that since ZA. ZB. ZG. ZD make equal angles with the surface of the aperture. they 
(or their extensions) also make equal angles with surface FH KL which is paralleJ to Ihe 
aperture surface. From this Witelo's conelusion follows. 

 Actually ZF = ZH = ZK = ZL and this could have shortened the proof considerably. 
6 Strictly speaking, an appeal to 111. 9. the way Witelo has it. is unwarranted. since 
in III. 9 a cirele is gillen and the center is decided upon as that point from whieh more 
than two equal straight lines can be drawn to the circumference! II would have been. 
however. quite straightforward to show that an arhitrar.v point of surface FHKL connected 
with M. has its distance to M equal to the constant distance FM = H M = KM = LM. 
and this would have established the truth "f the proposition. by definit ion I. 15 F.lcl/lcnts. 


Proposition 38 
I I. 23 Witelo: ..[It] a straight line interseeting two piane parallei surfilces is per- 
pendicular 10 one of them il will also be perpendiclllar lo (he pther" f Wit. PCł"'!'.. p. 6'); 
er. also iNd.. p. 227. Clearly the 'conelusion at hand \\ould follow from the cited proposition. 
by a I'edllctio ad ah.ml'dlll/l proof. for inslanec. 
2 Ali this means is that AB is the Jine of inlersection of a pIane perpendicular 10 
the circular aperture and the pIane of that aperture. 
.1 This follows from the parallelism of surfaces HLKM and ABCD and the line con- 
structions previously deployed. 
4 Which is the same as M L. 

 I. 102 Witelo: "From a given point either in the axis or in the curved surface 
of the cone or cylinder to draw a cirele in the surface of the cone or cylinder (Wit. Persp.. 
p. 119); cf. also ihid.. p. 276. 
6 l. 100 Wite lo: "The com mon seetipn of any pIane surface. intersecting a cone or 
a cylinder across the axis parallei to the base. and of the curved surface of the cone or 
cylinder. is a cirele: and if that section is a cirele. the intersecting surface is parallei to the 
base: from which it follows that any pIane sllrface intersecting a cone or a cylinder parallei to 
the base forms a new cone or cylinder" (Wit. Penp.. p. /17): cf. also ibid., p. 274. I. 89 Witelo: 
"Ali łhe hnes of longitude li.e.. generator linesl of a cone are equal and form eQual but acute 
angles wiłh the radii of the base: from which ił is elear that the apex of any cone is a pole 


......
>>>
198 


of the circłe of its base; and that each line of longitude is in the same surface with the 
axis; moreovcr that the axis itself falls orthogonally upon the center of the circłe of base" 
(ibid.. p. 110); cf. also ibid.. p. 268. 
1 Relying on his I. 103 is tantamount to not having proved the present proposition. 
Indeed. I. 103 says: "The common section of any surface intersecting a cone or a cylinder 
across the axis, not parallei to the bases, and of the curved surface cannot be a circłe" 
(ibid.. p. 120); cf. also ibid., p. 277. In its "proof' Witelo merely enumerates, but does 
not prove. the properties of the ellipse (or the "acute-angled section"). Moreover, he 
cites during his "proof' his I. 98: "The common section of any piane surface. intcrsecting 
a cone not through the apex. and of the surface of the cone cannot be a triangular 
tigure" (ihid., p. 115); cf. also Jbid.. p. 273. But I. 98 itself, in which Wttelo merely 
baptizes the three conic sections (parabola. ellipse and hyperbola) without providing any 
proofs whatever of their basic geometrical properties. is unsatisfactory as a mathematical 
proposilion worthy of the name. (See my commentary in ibid.. pp. 191 - 192). In concłusion. 
then. the present proposition too remains unproven. in spite of Witelo's aUegation: "The 
proposed [thing] is therefore elear". For a three-dimensional representation of the situation 
described in II. 38, see tig. K3 prepared by A. Bielski and W. Wróblewski in the Polish 
vcrsion of the present book. 


Proposition 39 
I Ali this says is .that, after their (probably lengthy) travel through the external medium, 
the increasingly perceptibly parallei luminous rays - in accordance with prop. II. 35 above - 
have, somehow, their sensible parallelism increased by the passage through the edges of 
the aperture. becoming practically parallei to the cent. al ray through the aperture. 
2 To what was said in n. I above must now be added that, according to Witelo 
(prop. II. 20), due to the pervading and all-encompassing diffusion of light, luminous rays 
intersect at all points of the medium and go on beyond the intersection points until stopped 
by opacity. He seems to say that it is due to this plethora of intersecting rays. primarily 
within the contines of the aperture, that the tendency towards rotundity is generated. This 
is an interesting, but undeveloped, suggestion, that in the hands of others, primarily John 
Pecham, will provide an appealing, if deticient, explanation of the circularity of images 
produced through non-circular apertures (cf. Lindberg. "Pinhole Images" (complete reference 
in Commentary to prop. 35), pp. 167-176). 


Proposition 40 
I "The common section of any piane surface, intersecting a pyramid or a prism across 
the axis, parallei to the base, and of the pyramidal or prismatic surface is similar to the 
periphery of the base; and [conversely] if that section is similar to the periphery of the 
base, the intersecting surface is paralleI to the base of the pyramid or the prism" (Wit. 
Persp., p. 116); cf. also, ibid., p. 274 for the Latin text. 
2 Up to this point there is nothing problematic about the proof. 


Commentary 
The last paragraph of the proof is the bewildering one. One can leel. I think, Witelo's 
unease with (and uncertainty about) it. Once more, as in prop. 35 above, the tension 
is between the straightforward, ruthless, impeccąble, "straight lines" geometry, on the one 
hand. and, on the other hand. the empirically grounded necessity of, somehow, eliciting 
from it, the impossible, Le.. the curving of the straight lines in order to obtain the pre. 
posterou8, but necessary, degree of circularity. 
In speaking of this proposition, Lindberg says: "AJso evident is Witelo's hesitancy to 
commit himself unambiguously regarding .. compression ( or .. intersection« or both as the 
cause of circularity and his unwiHingness or inability to elaborate on the exact meaning 
of .. intersection «. It is surprising that the fundament al contradiction in proving. in the 
tirst place. that by the rectilinear propagation of rays the image must be square and.
>>>
199 


in the second place, that by the rectilinear propagation of rays... the image becomes rounded. 
seems to have escaped Witelo. Perhaps the redeeming feature o, his theory. from our 
standpoint, is its reference to intersection' as a possible cause of circularity" (/oc. cit., p. 167). 
Ali right. However, beyond the slight1y retrospectively-anachronistic final assessment Ol 
Witelo's theory, I feel this is somewhat too harsh and undeserved an outburst against 
our author. whose predicament was not easy and whose "solution". involving a patent 
contradiction. of which Witelo was not entirely unaware - if I am right in my reading-, 
is not unheard of in the history of science. What was he to do except hedge? And how 
could he have helped discerning the glaring contradiction between straightness and roundness? 
But he needed them both and knew no better way out. 80, like his rays, he swerved 
(or, perhaps, dodged). Part of Lindberg's dismay stems probably from his overgenerous 
impression that Witelo "was an exceptionally competent geometer" (ibid., p. 166. n. 31). 
He was not. He was quite competent and exceptionally learned in mathematics for his 
time. He was also, at times, as Lindberg rightly point s out, "uncritically ec1ectic" (ihid.) 
For an exceptionally competent discussion of the history of the problem of pinhole. images, 
Lindberg's articles are a must. In addition to the one cited above cf: also "A Reconsidera- 
tion of Roger Bacon's Theory of Pinhole Images" and "The Theory of Pinhole Images 
in the Fourteenth Century", Archille Jor History oj Er:act Sciences, vol. 6 (1970), pp. 214- 
223 and 299- 325. 


Proposition 41 
I I.e., it will be a quadrilateral. but not a square. 
2 For Witelo, by definition, all cones (and pyramids of i1Iumination) were right. 
3 This is c1early absurd. at least with respect to ABCD, which cannot be. at one 
and the same time. both square (as it was taken by hypothesis) and non-square (as 
c1aimed now); I. 99 Witelo would also seem to contradict the c1aim with respect to 
GHKLIIABCD. 
4 In this terribly contorted and almost incomprehensible justification, one can rightly 
see, I think, a pelitio principii. 
5 The previous remarks and comments concerning prop. I. 35 and the following props. 
having a bearing on radiation through non-circular apertures (39, 40) apply in this case 
too and make anv further discussion unnecessarv. 
General Comment 
What we have encountered in Book II so far are. in addition to the definitions and 
postulates, a series of propositions drawn from Alha
n and a few others that do not 
appear as such in Alhazen's Perspectilla, but which Wite10 introduced. it seems to me, 
spurred on by what I have called elsewhere his systematic bent of mind. It turns out 
that this bent toward systematization that was. at least in part, responsible for the existence 
of Book I. is also at work in the optical books of. his Per.rpectil'o and this explains. in 
some measure. the great popularity (and excessive length) of the treatise. Additionally. there 
seems to have been at work in. Witelo's optical writing the penchant for devoting special 
and independent propositions to light phenomena, the properties of which can be handled 
exc1usively or overwhelmingly ;"ore geometrico. It is not rare. therefore, to see Witelo ex- 
patiating in separate propositions on matters that are either taken for granted by Alhazen 
(as self-evident or easily reachable) or considered derivative and dealt with as sucho en 
passant, in other propositions. Clearly, some of the propositions encountered so far iIIustrate 
pregnantly this characterization. 
Proposition 42 
I Which, in an intromission ray theory, should prove (even though Witelo does not 
say so) the rectilinearity of perpendicular rays (Le., no refraction) as they pass from a 
denser (water) into a rarer (air) medium. 
2 This is the situation appearing in Crombie's figure, in op. cit., p. 222 (cf. "Commentary" 
to prop. II. I ahove).
>>>
200 


.1 Ab(luI Y. the cenler (lf the aperlure. i.e.. be equally dislribuled ab(lut the cenIer. 
4 Since Ihe experimenlal sel-lIp in\'(,lves checking Ihe irrefrangihility (lf perpendicular 
rays 10 Ihe inlerflce of Ihe IW(I mcdia. Ihe sun musI be rising l'n Ihe h(lriz(ln and the 
ligh. involved is Iheref(lre practically parallei t(l Ihe waler's surface. 

 This is poinl P. 
" This IS Ihe p(linl bisecling diameter MP. i.e.. the center (lf Ihe middle circle. 
7 This is poinl M. 
K This is line MYP. 
Ił Ali Ihis is d.\ile merely in (lrder 1(1 provide a c(lnlrasl surface f(lr the incident 
light. It d.'es 01'1 atfect in ,my way Ihe previ(lus descripli(ln and reas(lning. 
III I.e.. E in Ihe acc.'mplnyin!l li gll rc. "hich. needless h' slale. is nol drawn lo scale. 


L. 


V I W 


E 


II I.e.. VW in Ihe ligure in n"'e '0 ahove. 
12 I.e.. p(linl l. 
C(lmmcnlar) 
In spile of its lenglh and l'ften aimless verhosily. Ihe meamng and experimenlal pr(l- 
cedure of Ihe prop.'sit i(ln ..Iwuld bc apparent. It inv(llves sh(lwing ("experimenlally") Ihal 
a s(llar ray enlering Ihe wa'er perpendicularly from Ihe rising sun (such a ray is aClually 
parallei 1(1 Ihe water's surface) penelrates Ihe denser medium wilh(lul being refracled. Once 
m(lre I refer the weary reader 1(1 Cwmbie's brief. but accurate, descripli(ln (lf the inslrumenl 
and ils mode (lf (lperalion (see n. 2 ab(lve) (lr. better still. 1(1 Appendices N(ls. I and 
2. in Ihe P(llish versi(ln (lf the presenl b(l(lk, especially 1(1 Fig. Z 7. 
The pwp(lsiti(ln c(lmes from Alhazen (Ihis is VII. 3 in Risner's numbering. Al/r. Persp.. 
pp. 233-235). where it is aClually longer Ihan Wilel(l's versi(ln! This is h(lw Risner enun- 
ciates il: "Radius medi(l densi(lri perpendicullris. irrefraclus penelral" (ihicl.. p. 233). II fullills 
an (lbligalion Alhazen 1(l0k up0n himself earlier lin b.\.,k I. prop. 17) when he merely decllred 
in R isner's summation without any pmof: "Lux perpendicularis penetrat per qualibet di- 
versa mcdia: .,hliqul refrin",ilUr" li"itI.. p. I}). HI\ ing descrihcd slIccinclly Ihc silll,ui.'n 
which ohllins. Alhazen commils himself lo an empirical corrohoralion in Ihe f.'lIo"lllg 
w(lrds: "El h(lc nos decllrlhimlls in sern1l'ne de refrlcli.'ne I.his "iII hecome ollr VII. 31. 
et oslendemus viam. per quam poterit quis experiri istam dispositionem: et apparebil sensu i, 
et cldet sllper ipsam cerlitudo"fihitl.). VII. 3 ilself. d.'ublless Wilel(l's model. refers specilically 
\() Ihe rising sun (..... si quis v(llueril experiri Iransilum lucis: eligat I(lcum. super quem 
orilUr lux solis. in qUI' ponal vas. el obsenet..:' (ihid.. p. 233). which eases Ihe under- 
stand ing considerably. but is nOI. bey.'nd Ihal. sensibly clearer (lr easier t(l f(lllow Ihln 
the imilall'r's lexl. 


PWposili(ln 43 
Cl'mmenlary 
There is a 101 of verhl'sily in Wilelo's descrip.ion. hUI Ihe meaning is apparent. Once 
more Ihe proposition comes from Alhazen. "here. in VII. 4. it j.s said "Radillm medio 
densi....i ohliquus. refringilur Id perpendiclllarem a refraclionis punc\() excilallm" Ulh. Pcn/'., 
p. 235): and. lS wilh his VII. 3. Alhazen carries (lUI I p...'mise undertaken in his I. '7 
(cf. c.'mmenlary 1(1 Ihe previous prop(lsilion ab.'ve). There is nOlhing in Wilelo's propo- 
siti"n Ihal does n(ll slem from Alha7en. . 


- 


....
>>>
201 


Proposilion 44 
I This is FH. 

 This is so because MN = LF. by consln,clion. 
3 And limiled by diamcler .łol P of Ihe middlc circle. 

 Because Ihe side of Ihe cubic pieces of glass is taken lo be double the diameter 
of Ihe aperlure. 

 Ali Ihis call be underslood as follows: The cubic pieces of glass co'ver conlinuously 
the p0rli0n Ef of Ihe base diameler lo a highl equal lo FH. The portion EG of Ihe 
same diameler is covered by Ihe ruler. whose "width line" coincides wilh the lalerai surface 
of Ihe first piece 0f cubic glass (\\hich is ilself limiled by line ZEX and Ihe base line 
of which is bisecled al E) and wh0se heighl is equal lo FM. The top portion of Ihe ruler 
has. as before. drawn 0n il a slraighl line parallei lo EG. 
fi This proves. Ihen. gm.uo modo. Ihal lighl emerging perpendicularly from glass (Ihe 
den ser medium) inl0 air (Ihe rarer) proceeds slraightforwardly. wilhout refraclion. 
7 "The diag0nals of all parallelogramic surfaces divide one another inlo equal parts; 
from which il is clear Ihal Ihe poinl of inlerseclion of the diagonals is the middle point 
of Ihal same surface" (Wit. Persp.. p. 73); cf. also ihid.. p. 237 for Ihe Lalin text. 
x Having reslored. that iso Ihe original arrangemenl of Ihe pieces of glass and Ihe 
ruler. 
j Including the shadow of the poinl marked in Ihe second piece of glass. which will 
also reach Ihe corresponding poinl in the surface of Ihe ruler. as before. 
10 AClually. Ihe converse of XI. 14 would be more appropriale. since the planes in 
queslion are paralleJ. by C('nslruclion. (XI. 14 establishes thal planes lo which Ihe same 
slraighl line is orthogonal are paralIel). 
II The lerm "concavity" is inadequale: what is meant iso presumably. Ihe upper surface 
of Ihe glass (?). 
12 I.e.. LN = MF. 
13 Even Ihough Ihe descriplion is awkward. il is obvious what Witelo does: he "covers" 
Ihe aperlure Ihrough which Ihe incidenl lighl comes "ilh Ihe CUI glass hemisphere (instead 
of Ihe cubic grass pieces). leaving Ihe posilion of Ihe marked ruler unchanged as in Ihe 
fpllo" in!! figure. 


Ruler 
Equal to EG 


Equal to FE 


G 


Light Ray 
extends along 
diameter FEF'G 


A similar tigure laken from Risner is presented in Ihe Polish version. Fig. 28 (The editor). 
14 Al any place along line FE.
>>>
202 


15 Unlike Alhazen, Witelo did not actually place the needle's point in the center of 
the glass sphere to see what happens and (strictly speaking) to verify his conclusion. Alhazen, 
on the other hand, includes among his triaIs the following: "Poste a ponat extremitatem 
stili apud centrum basis vitri (quod est centrum sphaerae) et inlueatur lucem. quae est 
super regulam: inveniet umbram extremitatis stili super centrum lucis" (A/h. Persp., p. 237). 
At times. then. Witelo tried to be shorter than his master. It is apparent that the centre 
of the hemisphere enjoys no special status from the standpoint of this analysis, and Wite lo 's 
procedure was, therefore, bot h correct and more economical than Alhazen's. 
16 "To draw a piane surface tangent to the spherical surface at a given point lon 
it]; from which [procedure] it follows that every line passing through the center of the 
sphere is perpendicular to its [spherical] surface : and if it is perpendicular to the spherical 
surface. it necessarily passes through the center of the sphere" (Wit. Persp.. p. 100); cf. 
also ibid., p. 259 for the Latin text and p. 185 for the notes and commentary. 


Commentary 
This proposition is taken over from Alhazen's Perspeetilla, VII. 6, where it reads: 
"Radius medio rariori perpendicularis, irrefractus penetrat" (Alh. Persp.. p. 235) and where 
all the elements of Witelo's proof (and more) are contained. Actually. Alhazen's proof 
is sensibly longer than Witelo 's. łn part this stems from the lack of any notational reference 
in the body of Alhazen's proof, necessitating rhetorically detailed descriptions of the various 
geometrical elements occuring in the proof: in part it is a result of Alhazen's greater 
descriptive care and of the additional information contained therein. 


Proposition 45 
1 What is meant is point E in the figure of n. 13 to proposition 44. which - IS. as 
it were, an "idea)" middle of a sphere having EF as radius. 
2 The new position of the glass piece is obtained from that appearing in n. 13 of 
the previous proposition by a 90 o rotation, so that it now lies flatly on the portion 
of its smali cirele and is facing the apertures with its other flat surface. namely that 
of the portion of its jZ1"eat circle. 
3 This is how Alhazen puts it. in his VII. 7, in more pregnantly accurate terms: "Necesse 
est igitur, ut perpendlculans, quae egreditur a centro vitri. quae est super superficiem vltn 
perpendicularis, quae extenditur in corpore vitri; obliqua sit a linea transeunte per centra 
duorum foraminum ad partem, in qua sunt duo foramina" (Opt. Thes. A/h., p. 238). 
4 "To draw a pIane surface tangent to the spherical surface at a given point [on it]; 
from which [procedure] it follows that every line passing through the center of the sphere 
is perpendicular to its [spherical] surface: and if it is perpendicu1ar to the spherical surface, 
it necessarily passes through the center of the sphere" (Wit. Persp., p. 100); cf. also ihid.. 
p. 259 for the Latin text and ibid., p: 185 for the notes and commentary. 
5 I.e., to FG in Fig. I. 
6 This is so, as the drawn 'ine in the surface of the plate lies in a pIane per- 
pendicular to the other pIane surface of the glass piece. 
7 This is line XEZ in Fig. I. 
8 Cf. n. 4 above; it is elear that it follows from this proposition that on/y diameters 
are perpendicular to the "spherical surface of the sphere". 
\I XI. 8 Euclid proves that, of two paralleI lines, if one of them IS perpendicular 
to a piane, then the other too will be perpendicular to the same pIane. 
III Of the two pIane surfaces. 


Commentary 
As was already intimated above, in n. 3, this proposition originates in Alhazen's VII. 7, 
where it is proved. in Risner's words. that "Radius medio rariori obliquus, refringitur a per- 


....
>>>
203 


pendiculari a refractionis puncto excitata" (Opt. Thes. Ałh., p. 238). Alhazen's proof extends 
over slightly less than Ił pages (compared to Witelo's slightly less than Ił pages, both 
in Ihe Risner edition). So again Witelo is more concise, but hardly elearer. And, as in 
the case of the previous proposilion. there are good reasons for Alhazen's more long-winded 
exposition. among them being, as before, the lack of notational reference, the additional 
information provided. but also the greater care in drawing and justifying inferences and 
the attempt at synthesis, summation. and exhaustive treatment of aU cases, incłuding the 
case of what he caUs "lux accidentalis", lacking entirely in Witelo's shorter proposition 
(cf. ibid., p. 240). 


Proposition 46 
I Is this an aUusion to Aristotelian cosmology? It would seem so. 


Commentary 
This is proven by Alhazen in his VII. 5 (Opt. Thes. A/h., p. 235), where, according 
to Risner's enunciation, it is shown that "Radij incidentiae et refractionis sunt in uno 
plano" (ihid.). 


Proposition 47 
I Perpendiculars, being the shortest Jine from a point to an underlying surface, are 
aided in their natural effects by the celestia] powers, influencing everything Iying beneath 
the heavens likewise along perpendicular Jines. This view. according to Lindberg. seems 
to introduce something new into discussions of reftaction (cf. Lindberg's translation of 
II. 47 Witelo in Edward Grant, ed., A Source Book in Mediellał Science, (Cambridge, 
Mass.. 1974). p. 432. 
2 If light is "carried instantly", as argued in proposition II. 2. then, cłearly, change 
of place as such (Iocal motion) cannot be involved in the transmission of light. 
3 The tension between instantaneousness and temporality is elear here. In this connection 
this is what Lindberg has to say: "This appears to contradict previous arguments on the 
instantaneous propagation of light. Wbile it might be maintained that Witelo's earJier arguments 
suc{'('ed in demonstrating merely Ihal the time required for the propagation of light is 
insensibly smaU. it appears that his intention was lo demonstrate that the oropagation of 
lighl over a finite space does not require any time at aU" (Grant. Source Book. p. 433. 
n. 181). 
4 This is actuaUy reflection which is, however, seen by Witelo as a case of extreme 
refraction. 
, Lindberg's remark here: "It is apparenl that Witelo 's explanation of the direction 
of refraction is essentiaUy the same as Bacon 's" (loc. cit.. n. 184). 
6 Fig. 28 corresponds to Fig. 29 in the Polish version. (The editor.) 
1 This is the last postulate. 
M II is elear that what we have here is an analysis of refraction in lerms of the 
decomposition of light into perpendicular components a la Descartes, a remarkable and original 
idea indeed in the realm of optics. Lindberg's remarks in this context are: "Here Witelo 
attempls an explanation of refraction in lerms of the perpendicular components of the incident 
and refracted rays. Apparently, the ray assumes a direction eloser to the perpendicular 
in the denser medium because (or partly because) its component paraUel to the interface 
is 'weakened (or slowed?). It is not cłear. however, what is happening to the perpendicular 
component at the same time. Nevertheless, the resemblance to Descartes' later analysis 
is noteworthy and suggests some reservations regarding Descartes' originality" (/oc. cit., 
p. 434. n. 186). Fine, but I believe it is quite apparent. from everything Witelo has said 
up lo this point about the unimpeded aClion of the perpendicular. thal nothing is happeninJ!
>>>
204 


to the perpendicular component: it penetrates the second medium unaffected. Le.. without 
devia,ion and s(' it is necessarily unimpeded! 
\I "A Ijne drawn from a point of one of [twoI parallei lines, in the same surface 
[as those paralle' lines]. must necessaruy mtersect the other.- if [the latter] is indetimtely 
extended" (Wit. Pers p.. p. 49); cf. also. ibid., pp. 216-217 for the La,in text and ibid., 
pp. 170- 171 for the notes and commentary. 


Commentary 
The origins of this proposition lie squarely in Alhazen's VII. 8. where it is proved 
that "Radius medio perpendicularis. irrefractus penetrat. obliquus refringitur : in densiore 
quidem ad perpendicularem: in rariore vero a perpendiculari e refractionis puncto excitata" 
(Opt. TIres. A/h., p. 240). The remarkable and original idea spoken of in n. 7 above 
stems also. needless to say, from Alhazen. where it is stated with greater precision. clarity 
and pregnance warrant ing therefore fuli citation: "Et motu s in corpore. in quod transit, 
si fuerit obliquus super superticiem iIIius corporis. componitur ex motu in parte per- 
pendicuiaris [sic] transeuntis in corpus. in quo est motus. et ex motu in parte Iineae, 
quae est perpendicularis super perpendicularem. quae transit in ipsum. Cum ergo lux fuerit 
m(l\a in c(lrpore diaphan(l gnsso super lineam (lhliquam: 'unc transitus eius in ill(l cnrp(lre 
diaphano erit per motum compositum ex duobus praedictis motihus. Et quia grossities 
corp(lris resistit ei ad ver' ical ionem. quam intendebat. et resistentia eius non est valde f(lrtis: 
ex quo sequere'ur. quod declineret ad partem. ad quam facilius transiret: et motus super 
perpendicularem est facilimus motuum: necesse es' ergo. ut lux. quae extenditur super lineam 
obliquam. moveatur super perpendicularem. exeunlem a puncto. in qU(l motus eius est wm- 
positus ex duohus motibus, quorum alter est super lineam perpendicularem super superticiem 
corporis grossi. et reliquus super lineam perpendicularem super prependicularem hanc: et 
m(l\Us c,'mpositus. qui est in ipso. non (Imnino dimittitur. sed solummod(l impeditur: ne- 
cessc csl ut lux dec1inel ad partem faciliorem parte. ad quam prius m(wehalUr remancnte 
in ipso motu composito: sed pars facilior parte. ad quam movehatur remanente motu 
in ips('. es' illa pars. quae est vicinior perpendiculari. Unde lux. quae extendi'ur in corpore 
diaphalll'. si ,'ccurril c(lrpori diaphano gwssiori corp('re. in qlll' c:l.ist il: rcfnl1!!CIUr per 
Iineam propinquiorem perpendiculari. exeunti a puncto. in quo occurnt corpori grossiori, 
quae extenditur in corpore grossiore per aliam lineam quam sit linea per quam movebatur. 
Haec ergo caussa est refractionis splendoris in corporibus diaphanis. quae sunt grossiora 
corporibus diaphanis. in quibus existunt: et ideo refractio proprie est inventa in lucibus 
obliquis... Causa autem. quae facit refractionem lucis a corp(lre grossiore ad c(lrpus subtilius 
ad partem contrariam parti perpendicularis. est: quia cum lux mota fuerit in corpore diaphano. 
repellet eam aliqua repulsi(lne. e' corpus gwssius repellat eam maiore repulsi(lne. sicut 
lapis. cum movetur in aere, movetur facilius et velocius, quam si moveretur in aqua: eo 
quod aqua repellit ipsum maiore repulsione quam aer. Cum ergo lux exierit a corpore 
grossiore in suhtilius: tunc motus eius erit velocior... Cum ergo lux exiverit a corpore 
grossiore. et perveneri' ad corpus subtilius: tunc resistentia corporis subtilioris facta luci. quae 
est in parte, ad quam secunda exit perpendicularis, ent mmor prima resistentia: et sil motus 
lucis ad partem. a qua resistehatur. maior. Et sic est de luce in corpore suhtiliore ad partem 
contrariam part i perpendicularis" (ihid., pp. 241- 242). 


Prop(lsition 48 
I XI. 6 claims that if two straight lines are perpendicular to the same piane. they 
are parallei to one another. 
2 Actually Witelo does not say explicitly why this is so. but it is apparent what he 
must have had in mind: parallei incident rays. for instance. will be refracted in a parallei 
manner Uhat .is not to a unique p,'int). while rays incident to the same p(linl (lf the
>>>
205 


interface of the two media will also be refracted in such a way that their angle of inter- 
section is preserved. This. in facto establishes the correctness of Witelo's claim 


Pwposit ion 50 
.. Fig. 29 corresponds to Fig. 30 in the Polish version. 
I The refracting surface being the same. 

 Risner has PEZ. but most MSS have PEX; since no confusion can arise f[{lm using 
X twice (PAX and PEX), I have chosen to preserve the MSS reading while adding X in 
parenthesis. 
-' Sml.'e there are only three factors intwducing variely in refraction (shape and nawre 
of the refracting surface and angle of incidence), and all three are identical in this case, 
the conclusion follows. 
4 I.e.. that FAX = FDQ. which contradicts the hypothesis. 

 Perhaps. proposition II. 49 should also have been mentioned. 
6 In III. 8 Euclid proves: "If a point be taken outside a circle, and from the point 
straight lines be drawn through to the circle, one of which is through the centre and 
the others are drawn at random. then. of the straight lines which fali on the concave 
circumference. that through the centre is greatest. while of the rest the nearer to that 
through the centre is always greater than the more remote. but. of the straight lines fali ing 
in the convex circumference. that between the point and the diameter is least. while of 
the rest the nearer to the least is always less than the more remote. and only two equal 
strai!!ht lines will fali on the circle from the point. one on each side of the least" (T. L. Heath, 
ed. and transl.. Tlle Thirteen Books ol Eue/id's Elements. 3 vols.. (Cambridge, 1956). vol. 
2. pp. '7-18). 


Proposition 51 


.. Fig. 30 corresponds to FijZ. 31 in the Polish version. 
I "A straighl line łalling on parallei straight lines makes the altemate angles equal 
to one another. the exterior angle equal to the interior and opposite angle. and the interior 
angles on the same side equal to two right angles" (Heath. Eue/id. vol. I. p. 311). 

 "In equiangular triangles the sides about the equal angles are proportionaI. and those 
are corresponding sides which subtend the equal angles" (ihid.. vol. 2, p. 200). 


Commentary 
Risner's reference at the end of the enunclatlon is to "Euclides 18 theo. opticorum" 
(Opt. TlIes. Wit.. p. 84). There was indeed available in the Xlllth century a Xllth century 
Latin translation of Euclid's Optica (cf. Wit. Persp.. pp. 176. 180). Moreover. prop. 51 
is manifestly. but for its wordiness. a faithful replica of the Euclidean model, which it 
follo\NS in all its essentiol (and not so essential) steps. Among the latter figures. for instance, 
Euclid's diagrammalic notation which. exduding a minor interchange. Witelo wholly adopts 
(cf. J. L. Heiberg. ed.. Euclidis Optica. Opticorum Recensio Theonis Catoptrico. Cum Scholiis 
Anti"l/i
. beinl! vol. VII of Eue/idis Opera Omnia. eds.. J. L. Heiberg and H. Menl!e (leipzijZ. 
1895). pp. 26-29). The actual reasoning steps of pwp. 51. however. are closer to Theon's 
consideralions in his recension of Euclid's Optica (Cf. ihid.. pp. 174-177). 
If I am right. then. in assuming the availability to Witelo of Theon's recension of 
the Optics. then Heiberg is clearly wrong when he says: "Per totum igitur medium aevum 
sola Optica genuina in manibus hominum :ra.t" (p. Xl, ibid.).
>>>
Book III 


POSTULATES 


I The twenty "accidental" optical phenomena (or intentions) are somewhat haphazardly, 
though traditionaUy. strung together; thus, whiJe "remoteness" obviously also covers. c10seness 
and "magnitude" stands for both "greatness" and "smalIness", "asperity" and "smoothness" 
are separated, as are "transparency" and "density", "shadow" and "obscurity", etc. Clea1"ly, 
the disjunction of "motion" and "rest" is historicaUy legitimate and in another category 
altogether, point ing to the differing ontological status of the two phenomena. 


PROPOSITIONS 


Proposition I 


Commentary 
Drawn as it is from Alhazen I. 39, Witelo's "proof' is. for a change, shorter. crisper, 
and more direct than that of his archetype. Cf. Opt. Thes. Ałh., pp. 22-23. 


Proposition 2 


Commentary 
What is established in this prop. is that for VISIOn to take place it is necessary that 
contact be effectuated between the visible object and the eye. and a precondition of this 
is what Witelo caUs "opposition". Witelo's model iso as pointed out by Risner. Alhazen's 
I. 21 (Opt. Thes. Alh.. p. 13), where it is shown that "VisibiJe visui oppositum videtur" . . 
(ihid.). "I 


Proposition 3 
I "The perpendicular is the shortest of aU Jines drawn from the same point to the 
same pIane or convex surface" (Wit. Persp., p. 59). Cf. also ibid., p. 226 and Opt. TIres. Wit., 
p. 10. 
2 "From a given elevated point to draw a straight line perpendicular to a given pIane" 
(Heath, Euclid, voI. 3. p. 292). 
3 "[II] a straight line intersecting two pIane paralleJ surfaces is perpendicular to one 
of them it wiłł also be perpendicular .to the other" (Wit. Persp.. p. 61); cf. also ibid., 
p. 227, for the Latin text. and Opt. Thes. Wit., p. 11. 
4 "AlI perpendicular lines drawn betw'een paraUel lines or surfaces are (themselves] pa- 
raUel and equal; and if straight lines intersect paralleJ lines or surfaces at equal angles, 
they are equal" (Wit. Persp.. ibid.); cf. also ibid., p. 228 for the Latin text. and Opt. 
Thes. Wit.. p. 12. 


.
>>>
207 


5 "The stTaight lines joining equal and paralleI straight Jines (at the extremities which 
are) in the same directions (respectively) are themselves also equal and paralIel" (Heath, 
Euclid. vol. I. p. 322). 
6 As it was already implied above that "only those things can be seen by a visual 
organ limited by a pIane surface that reach the same directly without refraction". 


Commentary 
Risner thinks Alhazen. I. 35 to be the source of this proposltJon. Perhaps. As it stands. 
however. Alhazen's account has very linie in common with Witelo's "pToof'. This account, 
summed up by Risner as "Oculus est globosus" (Opt. Thes. A/h., p. 21). contains no 
geometrical demonstration whatever of the eye's sphericity (the word sphere and its cognates 
never make their apperance in Alhazen's statement) and concentrates mainlyon convincing 
the reader that. while the eye is round. it is so because of the perfection and the ease 
oJ motion of rotundity. The emphasis throughout is on roundness as enabling the speedy 
mOlion of the eye which is, so Alhazen thinks, necessary for accurately fast visual per- 
ception. The other element s of Alhazen's discourse deal with the role of the eyelids and 
the eyelashes in preserving the eye from damage caused by light, smoke, dust, etc. (ibid., 
pp. 21-22). 


PToposition 4 


lens 


COJ"l1eO 


iris 


selera 


charoid 


ret ino 


axis of the II bali 


[Fig. 2B] 


I "If a sphere intersects [another] sphere. the common section of the mutually inter- 
secting spherical surfaces will be the perip
ery of a circle" (Wit. Persp., p. 103); see also 
ibid.. p. 262. for the Latin text and Opr. Thes. Wit.. p. 31. 
2 This view, though standard anatomical fare in the seventeenth century, is, of course, 
wrong. 
3 "If a cone inlersects a sphere and its conical surface is not intersected by Ihe surface
>>>
208 


of the sphere. the common section of the surfaces of the sphere and of Ihe cone will 
be Ihe circumference of Ihe circle of base of Ihe cone" (Wit. Per.
p.. p. 126); see ihid.. 
p. 2R2. for Ihe Lalin lexl and ..IsI' Opt. The.{. Wit.. p. 42. 

 Cr. n. I ..1",\.:. 
Fig. 2 presents a drawing of the eye accordinll. to manuscripts of the Perspectiva; 
Fig. lA IS a somewhat simplified modern reconstruction of this drawing. while Fig. 2B inc1uded 
in tue present notes presents a modern cross-section drawing ol thc eye. (The editor.) 


.Commenlary 
In Ihis lenglhy pnp,,
iti"Il. Witc!0 merely enlar!!.:
 UP"1l his immcdiate s"urce. Alhazen 
l. 4 (q.\!. at Opt. Thes. Alh.. pp. 3-4). His view of ocular anatomy. then. is hardly original. 
This tradilional knowledge is explained c0mpelenlly and succinclly by D.C. Lindberg in 
his Theories oj Vision Jrom AI-Kindi to Kep/er (Chicago and London. 1976). pp. 67-69 
and need not delain us here any more. Suffice it merely lo add that Ihe few elements 
of Witelo's ace0unt going beyond Ihe inherited opinion of Alhazen are exhausted by Ihe 
nominal reference to a "relinal lunic" as a synonym for Ihe aranea. or Ihe arachnoid 
membrane. by S0me furlher minor anal0mical delails. and by an addilional smattering of 
geometrical information. e.g., the dimensions of Ihe pupil in terms of the cube inscribed 
in Ihe ocular sphere. elc. 


Proposili0n 5 


Commentary 
Another English annotated translation of this propo SIt 10 n. by D. C. Lindberg. appears in 
Grant's Source Book. p. 407 (fuli reference in n. I to Witelo's 11. 47 .wpra). Witelo's 
telling arglJmenls against visual rays (a position. in which. lo believe Lindberg. he stood 
alone in the Ihirteenth century). all slem from Alhazen. Risner calls attention lo the 23rd 
proposition of Books I and II of Alhazen's Perspectil'a (cr. Opt. Thes. Alh.. pp. 14-15 
and 38- 39). In these. according lo Risner, Alhazen shows. respectively. that "Visio non 
fit radijs a visu emissis" (p. 14) and "Visio non fit radijs ab oculo emissis" (p. 38). A careful 
reading of these propositions shows convincingly that (I) there is nothing in Witelo's pro- 
position that is not in Alhazen and (2) there is a lot more in Alhazen that is not in 
Witelo. The latter has mainly to do with the mechanism of visual perceplion by means 
of perpendicular rays, issuing from the object to the eye and with the completion 0f vision, 
beyond the stage of sensory perception. by means of "cognilio" and "distinclio anlecedens" 
(Opt. Thes. Alh.. p. 39). 


Proposition 6 
I What is meant. most likely. is the third postulate stating that "stron g Iight damages 
the eye contemplating lit] for a long time" (cf. supra). 
2 This is the well-known argument for inlromission from the existence of Ihe after-image. 
appearing in Alhazen and summarized by Risner in these words: "Lux per se. et color 
illuminallls feriunt oClll0s" (Opt. The.{. Alh.. p. '). Alhazen himself slarls his Plnpeuiva 
with the statement: "Invenimus quod visus. quando inspexeril luces valde fortes. fortiter 
dolebil ex eis. et habebil nocumenlum" (ihid.). 
.' Here, as in other places. what Witelo means by the "surface of the eye" is actually 
the surface of the glacial humor. which is the part of the eye sensitive to lighl. according 
to the theory he propounds. 
4 I.e.. the glacial humor. Ihal iso Ihe crystalline lens. 


.......
>>>
209 


Commentary 
The suhslanlive delails of this pwposltlon come ff(lm A'hazen. Indeed Risner gives 
Ihe following PWposilions of Bl'ok I as Ihe source: I. 2. 3. and 14 (cf. Opr. Thcs. Alh.. 
pp. 1.2-3.7-8). 


Pwp,',il il'n 7 


I "lf two spheres have different centers. it is impossible for lines perpendil'lllar to 
Ihe surlace 11 one lo he perpcndlcular lo Ihe surlace of the olher. exccpl for l'ne such 
iline). which passes Ihwugh Ihe centers of both Ispheres)'" (Wit. Penp.. p. '02): see ibid.. 
p. 2(,0. for the lalin text and also Opr. T11e.
. Wit.. p. 30. 
! This is Ihe poslulate staling that "the observed object is seen aceording to litsJ po- 
sitioll. shape. and the order of its part s" (see .mpra). which precludes refraclion before 
the imprint of the visible form on the glaciaJ. Refraction not only weakens Ihe rays of 
light hUI also leads to the dish'rtion of posilion. shape. and mutual parl-ordering of real 
ohjecls. 
.I. Ali this on assumplion of the non-perpendicularity I'f Ihe rays entering Ihe pupil 
lo Ihe various lunics and humors constiluting the eye. 
-ł And nonperpendicular rays meeting it would he further refracted. 

 Mmhenrmical lines. not lighl rays carryin!! Ihe visible forms. since the latter are 
acHlally refracted at Ihe poslerior surface ef the glacial humor. situaled hefore Ihe center 
l,f I he eye. in order to prevent image inversion and reversion. had I hose rays come lo 
an apex in Ihe cenIer of (he eye and continued beyond. (See lindber
. Theorie.
 (
f !"i,itm. 
p. HO), 
ło "'f Iwo spheres are paralleJ. or louching ene anolher according te Iheir wh"le sUl'face. 
any line perpendicular te (he surface of one will alse be perpendicular lo (he sllrface 
I,f the other" (Wir. Persp.. p. 102); see iNd,. p. 260. for Ihe latin text and alSI' Opt. 
TI/('.
. Wir.. p. 30. 
7 "Te draw a pIane surface tangent to the spherical surface al a given poinl lon 
it); fwm which IpwcedureJ it fl'lIows thal every line passing thwugh Ihe cenler of (he sphere 
is perpendicular lo ils Ispherical) surface: and if i( IS perpendicular to the spherical 
surface. it necessarily passes thwugh Ihe cenler of the sphere" (Wir. Penp.. p. 100); see 
iNd.. p. 259. fer the latin text and also Opr. T//('s. Wir.. p. 29. 
K "The center of aU spheres whose convex surfaces are paralle' er touch l'ne anl'lher 
according to Ihe whole is necessarily Ihe same" (Wit. Penp.. p. 10'); see iNd.. p. 259. 
for Ihe latin lex t and also Opt. Thes. Wil.. p. 30. 
9 "If Ihe concavity of any sphere touches another surface at all its pein(s. it is nccessary 
Ihal the 10uched-lIpen surface be part ef a smallcr sphere" (Wir. Persp.. p. 103): see 
iNd.. p. 262. for the lalin text and alse Opr. The.
. Wit.. p. 31. 
III Namely that (he two surfaces of the cernea are cenccntric. 


Commenlary 
In Alhazen Ihe c1aim of Ihis pwposlllon is deall with succinctly in I. 12 (Opt. Thes. 
Al".. p. 6) l'n the basis of I. 6 and 8 (according lo Risner). For some additional remarks 
l'n these Iwo last pwposilil'ns see my cl'mmenlary on the following PWposilion. 


PWposilien 8 
I "We cali spheres » internally inlersecting« if the larger part of one is contained in 
the other" (Wit. Pers p.. p. 47); the latin reads: "Speras inlrinsecus se intersecantes dicimus 
quorum maeir pars unius in altera cOlllinelur" (ihid.. p. 215). See also Opr. The.
. Wit.. 
p. 4. 


14 - Witeloni. Per_pecti.ae...
>>>
210 


2 "If a sphere intersects lanother) sphere, the common section of the mutually intersecting 
spherical surfaces will be the periphery of a cirele" (WiJ. Persp.. p. 103); see ibid.. p. 262, 
for the Latin text. as well as Opt. Thes. Wit., p. 31. 
.\ "If a sphere intersects łanother) sphere internally, it is necessary that the centers of 
those spheres be farther apart with respect to the position of their contact, according to 
the ma8nitude of the periphery of the cirele. which is the common section of their surfaces; 
and lit is necessary) that the center of the containing sphere be deeper" (Wit. Persp., p. 106); 
see ibid.. p. 26S for the Latin text and also Opt. Thes. Wit., p. 32. 
.. I.e.. its center. 


Commentary 
Risner refers the reader to I. 8 Alhazen as the place where the latter treats the same 
issue. This is correct. There are however matters dealt with by Witelo in this proposition 
that are actually taken up by Alhazen in his I. 6. Between them these two propositions 
contilin all of Witelo's claims. The way Risner enunciates. however. the claims contained 
in Ihe
c IWO rroro
itions is quite misleadin
 heeause of the terminology he uses. Thus. 
ab('ut I. 6 Risner says: "Oculus totus et sphaera uvea centris differunt: et oculi centrum est 
altius" (Opt. TIres. A/lr.. p. 4). "Altius" here means "deeper" (and not "higher") as it 
is meant to tally with "remotius in profundo centro uveae" (ibid.). Similarly. in summing 
up I. 8. Risner says: "Centrum sphaerae uveae est inferius centris reliquarum oculi partium" 
(ibid.. p. 5). which seems to c1ash with Witelo's and Alhazen's elaims in their respective 
propositions. We are .Iready familiar with Witelo's statements. As to Alhazen's, they can 
be iIIustrated by the following sentence: "Centrum igitur earum Ii.e., of the two surfaces 
of the cornea and of the convex surface of the albugineous humor) est unum punctum 
commune. et est remotius in profundo centro uveae" (ibid.). It is elear, then, that Risner's 
use of "inferius" above is deceptive /Ind must be understood. consistent with the views 
of both Alhazen and Witelo. as meaning "c1oser to the anterior part of the eye" and 
"more elevated", which can be achieved if one takes, against the usage of Witelo and 
Alhazen. "deeper in the eye" to mean "hi!!her" and "eloser to the eye's surface" to mean 
"Io\\cr". 


Proposition 9 
I "If a sphere intersects lanother) sphere, the line which passes through the centers 
of those spheres must pass through the center of the clrele of periphery of the common 
section and be perpendicular to its surface" (Wit. Persp., p. 104); see ibid.. p. 263, for 
the Latin text and also Opt. Thes. Wit.. p. 31. 
2 "A line drawn orthogonally from the center of a sphere through the center of a cirele 
intersecting the sphere is necessarily connected to the middle Ipoint) of the portion Iwhich 
has been) cut off łby the piane of the circ1e)" (Wit. Persp., p. 108); see ibid.. p. 266, 
for the Latin text and also Opt. Thes. Wit.. p. 33. 
3 "If a cone intersects a sphere so that the circ1e of base of the cone is paralIel, 
in the surface of the sphere, to a great cirele of the sphere. the diameter of the sphere, 
which is perpendicular to that great cirele. must necessarily pass through the center of 
the circ1e of base and be perpendicular to it; from which it is manifest that when the 
diameter of the sphere and the axis of the cone have been joined. they form one line" 
(Wit. Persp., p. 127); see ibid., p. 282, for the Latin text and also Dp'. TIres. Wit.. p. 42. 


Commentary 
The proposition and its proof are quite straightforward and are dealt with by Alhazen 
matter-of-factly and more briefły than by Witelo in I. 7 (Opt. Thes. Alh., p. 4), the obvious 
source of Witelo's theorem: "Et recta linea. quae continuat duo centra, scilicet centrum 
superficiei c('rne:le. et centrum uveae. quand(' c'\ilrahitur reete. rervenit ad centrum foraminis
>>>
211 


quod est in anteriori uveae. et ad duo media duarum superticierum corneae aequidistantium: 
superticies enim concava corneae et convexa uveae sunt superticies sphaericae secantes se: 
et linea quae continuat centra earum. transit per centrum circuli sectionis, et est perpen- 
dicularis super superticiem eius. transit per centra duarum sphaerarum". 
The only ascendancy that Witelo's proof possesses over Alhazen's is its more formally 
geometrical style of presentation and the explicit justitication of its intermediate conelusions. 
reached on the basis of previously demonstrated ela'ms in Book I. the mathematical book 
of Witelo's Penpc'c"il"ll. 


Proposition 10 
I With the vitreous sphere. 
2 These alternatives are theoretical. mathematical ones and of course do not all tit 
the actual physical situation. In his discussion of the matter in I. 9 (Opt. Thes. Ałh., p. 5), 
which represenłs the source of Witelo's proposition. Alhazen makes even further distinctions. 
stemming from his differentiating between the anterior and posterior surfaces of the glacial 
and the various possible mathematical outcomes growing out of the mutual location and 
position of the glacial. uveal and vitreous spheres. 
.\ Of the three spheres. 
4 Of consolidation of the three spheres. 

 I.e.. the gI;.cial and the vitreous. 
li "The centres of paralleI cireles in a sphere must necessarily lie in the same diameter 
of the sphere; from which ił is elear that all parallei circles in a sphere have the same 
poles. And if lcireles in a sphere] have the same poles they are paralIel" (Wit. Persp.. 
p. 97); see ibid.. p. 257, for the Latin text and also Opt. Thes. Wit.. p. 28). 
7 "Every line connecting the center of a sphere with the center of a smali circle of 
tha' sphere is perpendicular to the surface of that circle" (Wi,. Persp.. p. 96); see ihid., 
p. 256. for the Latin tex' and also Op'. The.f. Wi,.. p. 28. 


Commentary 
By means of an appropriate drawing of the eye, embodying all the elements of its 
anatomy. as they come to the fore in the third book. łhe proposition becomes pretty 
obvious. The relevant drawings are meant to illustrate faithfully Witelo's (i.e., Alhazen'5) views on 
the matter and to set them against the modern view embodied in a contemporary simplified 
drawing of the eye. Pertinent to the claims made in this proposition are Fig. 2 (p. 109), drawn from 
the MSS and Fig. 2A drawn from the Polish translation of Books II and III of Witelo's Perspeetiva 
by A. Bielski and W. Wróblewski appearing as Studia Copernicana, vol. XXIX. 
For the sake of comparison, a modern simplified drawin& of the eye is al50 provided in Fig. 2B (p. 
207). 


Proposition II 
I The intent is to the anterior surface of the [dacial. 
2 Cf. n. 3 to prop. 8 supra. The conelusion obtains because the anterior part of 
the glacial. the crystalline humor. is part of a greater sphere than the posłerior part. i.e.. 
the vitreous humor. and "contains" the larger part of the vitreous. (See the detinition 
of 'in'ernally intersecting spheres' in n. I to prop. 8. supra). 


Commentary 
This is dealt with by Alhazen in his I. 10 in the following manner; "Et iterum super- 
ticies anterioris glacialis, et superticies residui glacialis, sunt duae superticies sphae'ricae se- 
cantes se: centrum ergo superticiei anterioris. est remotius in profundi centro superticiei 
posterioris". The d,fferences he'ween the styles of master and apprentice are pregnant.
>>>
212 


Pwposit','n 12 


I "If a sphere inlersecls lanother] sphere. Ihc linc passing Ihwugh Ihe cenlcr "f Ihe 
circlc of periphcry of Ihe common secli.'n. en(ering perrendicularly ", ils sllrfa,'c. musI 
pa" Ihwugh Ihe cenlcrs "f holh spheres" (II i/. P/'np.. p. 1(16): sec iMci.. p. 
t4. f()l" 
(hc I alin 'e\' and al", Op/ 7111'1. Wil. 1'. 3
 
1 "Onl) ,'ne pcrpcndicular can he dra\ln fwm a given elcvalcd PUII11 10 any piane 
or C('nvex ,urfacc Iying undcr il" (Wi/. P/'/".Ip., p. 58); see iMci.. p. 225 for Ihe Latin 
lexl and also Opl. 711es. Wil., p. 10. 
I This follows fwm ł- 1/1 WiliI". 

 "If IWo sphercs are paralleI. .'r touching one anolher according tCI Iheir whole surf.lces. 
any linc perpendicular lo Ihe surlace ,., one \lIII also be perpendicular to Ihe surface "f Ihe 
olher" (Wil. Per.lp.. p. 102): see for Ihe La,in text iMd., p. 260 and also 01'1. Thes. Wi,.. 
p. 30. 


Commentary 
The same Ihing is proved by Alhazen. wilh more directness and concision 111 his I. II 
(01'1. T"(-.
. Ali,., p. 5). 


Pwposil ion 13 


Commenlary 
According to Rlsner, Alha7en covers Ihe same lopic in his I. 22 and 41. This is 
so. cven tlwugh Ihe true cOllnlerpart tCI Wilelo's proposition is ł- 4] and not ł- 22. which 
is much more long-winded Ihan Wilelo III. 13 and lakes up olher mallers 100. e.g., Ihe 
crucial wie of Iransparency n. humidily of the intervening medium in achieving proper 
vision of cXlcrnal ohjccls. Cf. Op'. 711L'.
. Al".. pp. 13. 25. 


Proposil ion 14 


Commenlary 
This proposition is sl/mmarb-d by Wilelo from Alhazen I. 42 in a quite acceplable 
manner. But for Alhazen's consideration of .
ccołlcla,.y forms of light and color reaching the 
eye from duly illuminated objects 101'1. T"es. Al".. pp. 23- 24) and thus mak ing those 
objecIs visible, Witelo 's summary inc1udes in its thirteen lines all the essential elements 
of Alhazen's Ihirty-three line proposilion: one instance. Ihe first I am aware of. in which 
his objeclionable statement concerning "taedium verbositatis arabicae" in his dedicatCIry 'elier 
to William of Moerbeke (Opl. T"c.f, Wil.. p. I) may be "justified". 


Propositlon 15 
I Thal iso ID direcl contact. il is the object itself. and not its visua' form. that acts 
on Ihe eye. 


Commentary 
As stated by Risner. there is an cquivalent proposition in Alhazen. I. 37 (Op" T"es. Al".. 
p. 37). which is however both more systemalic and more complete than Witelo's. Witelo's 
language leaves no doubt Ihal his source is indeed Alhazen. 


Pwposil ion 16 


Commentary 
Thal vision is in a sense a passive pn'cess involving Ihe experience of pain. follows. 
almo
1 inevitahlv. from the intn'mission theory of visi,'n adopled by Witelfl from Alha7cn.
>>>
213 


The latler. in keeping wilh Risner. covered Ihe same lopic in his I. I. 2 and 26. This 
is S0. but. obviously. he covered much more Ihere. as is elear even from Risner's formulations 
of Ihe conlenls of Ihe involved proposilions: ooQuod lux per se. 1.'1 colores iIIuminati operenlur 
in visum aliquam operalionem.... I. Lux per se. el color iIIuminalus feriunl oculos... 2. lux 
vehemens obscurat quaedam visibilia. quae lux debilis iIIuslrat: et conlra" (Opt. T"e.
. Al".. 
p. I). "26. Visio est ex eorum numero quae doll'rem faciunt" (ihid.. p. 15). Furlhermore. 
it is perhaps worlhwhile menlioning Ihat Wilelo chose not lo appeal al all lo any of 
his poslulales. some of which mighl have come in handy. Thus one may Ihink of Ihe 
firsl postulalI.' and. cerlainly. l'f Ihe one requiring "Ihal slrong lighl damages Ihe eyc con- 
templaling Iii) for a long lime" (see .mpm). 


Proposil ifm 17 
I ooFwm the same poinl Iwo straighl lines cannol be set up al right angles to Ihe 
same piane on Ihe same side" (Healh. Euclicl. voł. 3. p. 295). 

 00(...] Ihe eye ean see simullaneously various visiłle objecis" (.
upra). 

 See n. 4 lo prop. 12 slIpru; I. 74 Wilelo is aPPf'sile. sinel'. of course. cl'ncenlric 
spheres are "paralIel". Wilelo's "awkward" definilion of "paralIeJ" sphe..es reads: "We 
cali spheres or ci..cJes conlaining one anolher 'paralleJ' if Ihe segments d..awn belween 
Ihem lalf'ng radii leading] ff()m Ihe cenler of Ihe larger. from Ihe oUlside of Ihe 
maller 
lo Ihe inside of the larger. are (ali] equal" (Wit. Persp.. p. 47); cf. also my commenlary 
in ihid.. p. 168. For Ihe lalin lexl of Ihis definilion. see ibid.. p. 215 and also Op" 
TI/('.
. Wi,.. p. 4. 
ł Thi
 lengthy bUl inm'cuous senlence. translaled as closely lo Ihe Wilclian lext as 
possible. gives Ihe reader who has no lalin a feeling of Wilelo's slyle. 

 These are imapinary visua' rays issuing oh/il/lle
l' Ihrough Ihe cornea from one ilnd 
Ihe same poinl in Ihe glaciał. 
fi This is Ihe perpendicula.. Ihrough Ihe cornea issuing. imaginarily. from Ihe same 
point. as in n. 5 above. in Ihe glaciał. 
7 This is an awkward way of saying Ihal one may reverse Ihe imaginary direclion 
of Ihe f()fmerly assumed visual rays and Ihe same geomelrical silualion would oblain. 
H "That" point of Ihe glacial where Ihe perpendicular ray inlersecls it. 
.. One-lo-one correspondence between Ihe points of Ihe object and Iheir impression 
on Ihe glacial is achievable only if Ihe obliquI.' rays can be discounled and Ihe perpen- 
dicular rays illl'ne carrying Ihe punclual forms of Ihe object's poinls are allowed lo leilve 
their impressions on Ihe glacial humor. Ihus achieving elear vision ralher Ihan blurry chaos, 
as Wilclo will say in Ihe immedialely following lines. 
1(1 And since only pe..pendicularity preserves shape. order. pl'silion. arrangemenl of parls. 
as well as elarily of pe..ceplion. 
II O.. ....ildialion... 
I
 Or "pyramidOO; Ihis is me..ely a malhemalicalcone.aslhe (perpendicularj rays never 
quile reach Ihe cenler l,f Ihe eye. as we sawo lo avoid inversion and reversion. bUl are, 
rai hel'. refracled at Ihe poslerior surface of Ihe glacial humor. i.e.. the vilreous. 
LI Relonging lo Ihe f,bjecls Iying oulside of Ihe cone of radialion under considerillion. 
14 Ali Ihis seems lo say is Ihal peripheral vision is possible. Ihl'ugh weak and indislinct. 
due W Ihe impression refracled rays. entering Ihe eye obliquely. may leave on Ihe anlerior 
slIrfilee of Ihe glaciał. al pf'inls Iying oUlside Ihe co nI.' of radialion of elear and dislinct 
vision; al such poinls. of course. one can draw Ihe perpendiculars ff()m Ihe cenler of 
Ihe eye. 
That dislincl vision lakes place s01ely along perpendicular rays of lighl slems. firsl. ff()m 
Ihe nl'ed lo c
lah"
h a f'ne-h'-I'ne ('orres!''ndence łel\\'een Ihc I'f,jnls I,f Ihe f,hjel'l and 
their impres,jf'n on Ihe glal'ial humor and. second. because f,f Ihe sI/'/'ng'" f,f Ihe per- 
pendiculars as compared lo Ihe \\eakness of ołlique rays. 
The queslion of f,hlique liglll and ils impression on Ihe eye w.\s a vexed ol1e in 
Alha7en s (.lI1d Wilelo'" il1ll"1'missiol1 Ihcnr
 f,f I i,jf'l1. nell'r quile ,olled ,alisfac\('ril
 unljl
>>>
214 


Kepier. It led Alhazen to a geometrically indefensible position with respect to the vision 
of refracted light that contradicted the cłaims he made in the tirst book of his Perspectilla 
(De aspectibus) which Witelo appropriated /Ind advanced. among other places, in this pro- 
posilion. For a thorugh discussion of this issue see Lindberg. Theories 0/ Vision. pp. 7S-78. 
I
 Clearly here Ihe placial is meant rather Ihan Ihe whole eye or its exlernal surface. 
Commentary 
Risner gives Alhazen's I. 1 S and 18 as places in which the latter deals with the 
same issue as Witelo in his III. 17. It must be said. however, that not all that Witelo 
covers is tackled in the propositions mentioned by Risner. To achieve congruence, one 
has to incłude I. 19 and VII. 37 of Alhazen's De aspectibus too. (Cr. Opt. Thes. Ałh., 
pp. 8. 9-10. 268- 270). 
As Ihe role of this proposition in Witelo's theory of vision has already been considered 
in the Introduclion. we shall nnt dwell further on it. 


Proposilion 18 
I Cr. n. 7 to prop. III. 7 supra. 
2 They will be diameters when extended throughout the globe of Ihe eye and, as such, 
perpendicular lo the concentric tunics of the eye. 
3 "lU] a straighl line intersecting two pIane paralleI surfaces is perpendicular to one 
of them it will also be perpendicular to the other" (Wit. Persp., p. 61); see ibid., p. 227, 
for Ihe Latin text and also Opt. Thes. Wit., p. I l. Strictly speaking, since the surface 
of the eye is not a piane, appeal to I. 23 Witelo is unwarranted. 
4 Moving along the perpendicular. 
, One can imagine the cone created by "ali" the perpendiculars and its largely conti- 
nuous lateral surface. 
1\ I. 99 reads: "The common section of any pIane surface, intersecting a pyramid or 
a prism across the axis, paralleI to the base, and of the pyramidal or prismatic surface 
is similar to the periphery of the base; and Iconversely] if that section is similar to the 
periphery of the base. the intersecting surface is paralleI to the base of Ihe pyramid or 
the prism" (Wit. Persp., p. 116). 
I. 100 reads: "The com mon section of any pIane surface, intersecting a cone or a cylinder 
across the axis parallei to the base, and of Ihe curved surface of the cone or cylinder, 
is a circłe; and if that section is a circłe, the intersecting surface is paralleI to the base; 
from which it follows Ihat any pIane surface intersecting a cone or a cylinder paralleI 
to Ihe base forms a new cone or cylinder" (ibid., p. 117). For the Latin texts of the 
two propositions see ibid., p. 274 and Opt. Thes. Wit., pp. 37, 38. Of course the tigures 
Witelo refers to are not actually cones as their bases are not circłes. 
7 This is the round opening in the bone Ihrough which the optical nerve enters the 
globe of the eye. 


Commentary 
Th,", '"'''Illents should he elear in 'iphl of Ihe Inlroduction and what precedes. Risner 
n:lers to "Eudides 2 h)pothe. 01'1." and to "Alhazen 19 n l". (Opt. Thes. Wit., p. 93). 
Indeed the second pl'slulale l,f the Oplica (the Greek term Heiber, uses is "OPOI, de- 
finilions) reads in Latin: "el sub visibus contentam tiguram conum esse verticem quidem 
in oculo habentem. basim vero ad terminos conspectorum" (Euclidis Optica, p. 3; fuli re- 
ference in commentary to Witelo II. SI supra). Also Alhazen I. 19 is highly germane 
lo Witelo's concerns in this proposition (cr. Opt. Thes. AIh., pp. 10- 12), as it contains 
essentially all that Witelo says here and in other propositions .of book III, e.,., III. 21 
and 22. Risner's summalion of the contents of Alhazen I. 19 reads: "Visio tit per pyramidem, 
cuius verlex est in visu. basis in visiblis" (/oc. cif.. p. 10).
>>>
215 


Proposition 19 


Commentary 
This proposition is entireły taken over from Alhazen's I. 40 (Opt. Thes. A/h., p. 23). 


Proposition 20 
I Or the optic chiasma. 
2 Aftcr refraction of the visual forms at the interface of the glacial and vitreous, these 
forms continue their travel, properly arranged, through the vitreous to the optic nervc, 
which is filled with "sensing power" ("sentient body" or "sentient member" or "visual 
spirit"), an entity of equal transparency to the vitreous, thus preserving further the proper 
arrangement of the forms, corresponding to their arrangement in the visible objects, IInd 
through this sentient spirit to the optic chiasma (or the common nerve) where the identical 
visual forms from the two eyes are united to be pcrceived by the ultimate sensing body, 
the ultimum sentiens. 
3 "Likewise that the observed object is seen according to [its] position, shape, and thc 
order of its parts" ([Postulates], supra). 


Commentary 
According to Risner, in I. 25 Alhazen deals with the same issue: "25. Visio perficitur, 
cum forma visibilis crystallino humore recepta, in nervum opticum pervenerit" (O pt. Thes. 
Alh.. p. 15). This is only approximately accurate, as there is no mention of the common 
nerve in the entire proposition, which is given wholly to a discussion of the properties 
of thc glacial and the fixation cum transmission of visual forms in/through it. 


Proposition 21 
I Ali of these proposltlons together warrant the rectilinearity of transmission of light 
rays and the lack of refraction under the given circumstances (cf. props. II. 1, supra, 
III. 17, supra, and I. 72 in n. 7 to prop. III. 7, supra). 
2 Unchanged. 
3 Without mutual disturbance of the parts. 
4 The same unbroken. 

 Were the forms to come to an apex in the center of the eye, and end there, they 
would, in a sense, be annihilated as preservers of the "position, shape and the order of 
the parts" of the visible object, which is patent1y absurd. The visual forms cannot, therefore, 
terminate in a single point of light. 
fi "The lines of longitude of any cone or pyramid intersect one another only in the 
apex. (In the axis: liO extended. however. they ori
nate another similar c(lne lor pyra m id] 
whose lines of longitude relate to [those 00 the prior cone lor pyramid] in an opposite 
mode with re gard to position and place" (Wit. Persp., p. 112); for the Latin text see 
ibid.. p. 269 and also Opt. Thes. Wit., p. 35. 


Commentary 
To avoid inversion and reversion of optical forms, leading to unacceptable consequences 
in the perception of thc real world, the intersection of the light rays in the center of 
the eye and their continuation beyond must be preventcd. This is done by nature as a result 
of the diłferent transparencies of the glacial and vitreous, so that at their interface precisely 
the rcquired refraction is produced, enabling the visual forms to mirror faithfully the outside 
world in the common nerve; see also the next proposition. 
Alhazen deals with the same issue in his II. 2 (Opt. Thes. Alh., p. 25). His treatment 
is clearer. more complete. and more systematic than Witelo's.
>>>
216 


Pwpl\
ition 22 
I See n. 6 to pwp. III. 21 .mpra. 
Merely by refraction. 
Being perpendicular to the tunics of the eye. 
4 Cf. supra. n. 7 to prop. III. 7. 

 Cf. sllpra. n. 6 to pwp. III. 21. 
f Like Alhazen. his model. Witelo thinks that the tran
mlSSlon of forms thwugh the 
vitrcous sphere and the visual spirit. filIing the optic nerve. is not merely a consequcnce 
l\f Ihcir diaphanous character (identical in both). but also of their inherenl receptil'ity or 
,
enticl/t P(JII"('r (what Lindberg calls the "equality of receptivity" (Tlteorie.
 (ll Vi.fioll. p. 83)). 
This rccepl ivity the tunics of the eye and the other two humors (t he !.dacia] and the alłmgineous) 
do not possess. Cf.. Iwwevcr. the commentary to pwp. III. 37 illfru. 
7 l.c.. the other constitlltive part s l\f the eye. 
M This would seem to illlply that the glacial humor tOl\ has some kind l\f imperfect. 
residll..1 receptivity. 

 I am translating literally. 
III The rectilinearity of the transmission of light forms ceases when they reach the optic 
nerve: in their motion thwugh the selllinrt lIIelllher (l\r the "i.flltll .fpiril) the form
 prescrve 
oni} their mUlual arrangemen!. In generał it is the case that U.fil'r their passage Ihl"l'ugll 
the vitreous. the forms do not travel in straight lines but Tilther al'Cl\rding to the dispo- 
sit ion and structure of the visual spiri!. 
II This is the ultimlllll selllien.
. 
11 That iso the difference in ;lIlterel/l receplil'i/l' alone betY,leen the glaciał and Ihe "ilreous 
wi\lIld cause the refraction of light forms at their interface. bUl. wilhout the refr..ction 
due 11\ their different transparencies. this might not be enough to avoid nwnstwus or 
mllltiple forms. 
LI Without disturbing the arrangement I\f the forms. as is the case with the pro(;ess 
1"\c(;lIITing belween the glacial and the vitreous. 
14 And this would be. from what precedes. the vitreous itself. 
I
 Ali this says is that the last refraction of light forms in Ihe eye takes place at 
the bl\rder between glacial and vitreous and that. afterwlrds. the homogeneity of the .n-milil'(, 
pOIl'C/' (t he "visual spirit'") assures the mere transmission (h or the simple extension") of the 
form al the w..y to the lIlIilIlIIIII .fl'IIIil'IIS. 
In I.e.. the vitreous and the optic nerve l'r. ralher. the ,';..uu/ .
pir;t filIing it. 
" To "vI\id Illlltual cros
ing over l\f the forms and. Ihus. invcrsion and reverSJon. 


Cl\llllllentary 
The vic\\s prcsenlcd In Ihis pwposilion all stcm fl"l'm Alha.lcn. AI:Cl\l"ding h' Risner. 
A lilii ze n trealed Ihe subiect in his I. 30. II. 4. 5 and c.. Largcly speaking this is Cl'rrec!. 
Cf. Opt. TItC.f, Alb.. pp. 17 - 18. 26- 27. 


Propositlon 23 
I Cf. n. 4 ti\ prop. III. 12. .mpra. 
1 This is pltently absurd. When light enters a rarer medium. as proved by Witelo. 
it is refracted (/\l"lIy fwm the pcrpendicular. What IIIU)' happen is that the !łDgle of re- 
fraclion may nol be large enollgh to avoid the intersection and. hence. the crossing over 
of the forms. Icadin!! thus to "monstrosity". Ali that Alha7en (f rom whom Witelo appropriated 
Ihis proposition) says in this connection is: "Neque pOlest esse Icommunis sectio cryst..lline 
et vit rceJ ex sphaera parva: quoniam. si fuerit ex sphaera plrva. quando forma rcfringetur 
lb COl. ct elongabjtur ab ea. fict monstruosl" (Opl. Tlte.
. A/fr.. p. 25): thls comes from 
prop. II. .l 


--
>>>
217 


3 Cr. n. 7 to prop. III. 7. .
upra. 

 From Ihe Iw() eyes. 


C()mmentary 
This is h()w Lindherg sums up Ihe process described in Ihis propositi()n: (He deals 
wilh "The C()mpleli()n and Cerlificati()n of Visual Perceplion" in Alhazen. but. as the ingredienls 
of Ihe d()Clrinc ()f vision of Ihe Arab masler and Ihe Polish discipie are identical. Lindherg's 
words may servc as a fitting descriplion ()f Wilel()'s views t()o). "Having been pwperly 
refracled. Ihe f()rms are direcled Ihwugh the vilreous hum()r to Ihe ()Plic nerve. which 
is Iilled wilh visual spirit «)r Ihe 'sentienl b()dy') and c()nducts Ihe f()rms I() Ihe ()ptic 
chiasma Ol' c()mm()n nerve. There idenlical f()rms com ing from the tW() eyes lin ile. H()wcver. 
Ihc pwper uniI ing of Ihe f()rms lo produce a single image requires that Ihere be a fixed 
correspondence belween poinls on Ihe glacial humor and points in Ihe C0mm()n nerve Ii.e., 
Ihal fl'rms arriving al a particular point on Ihe surface ()f Ihe glacial humor are always 
Iransmitted lo Ihe same poinl in the Cl'mmon nerve). that this corresp()ndence he idenlical 
for Ihe IWO eyes. and Ihal the visible p()inl from which Ihe forms originale be similarly 
silllated wilh rcspecI I() bOlh eyes. Upon being united in Ihe comm()n nerve. Ihe f()rms 
are perceived by Ihe lIlIilIlIIIII selllies. Ihe ultimale senlienl power. and Ihe acl ()f sight 
is I hus compIcIed" IT"('orie.
 uf f 'i..ioll. p. 81). 


PWp,.siUl'n 24 


I "If a slraighl line be at righl angles to any pIane. aU the planes Ihrough il wiU 
ais,. he ;II righl angles lo Ihe same piane" IHealh. Euditl. vol. III. p. 302). 
! "IIł] from In elevalcd poinl a line lis drawn] perpendicularly lo an underlying piane 
surface land] anolher lline] inlersecls 'Ihe same surface] obliquely. and Ihe slraighl line 
hetween Ihe poinls of incidence lof Ihe perpendicular and Ihe oblique line] is drawn in 
Ihe same surface. (I hen] Ihe angle conlained belween Ihe oblique line and Ihe line drawn 
Ibetwecn Ihe ablwe menlioned poinls of incidence] is Ihe smallesl of aU angles conlained 
hel\\Cen Ihul ,'blique Ijne and any olher line drawn in Ihe underlying surface; and every 
angle ncarer lo Ihal IsmaUesl anglc] js smaUer than Ihe more reml'le and Ihe IWO langles 
form,." rc
rc,.ti\"cly hy h\"() Iillc
 \\hich] appn'ach /Ihal 
mallcsl ;mgle] equally on Ihe Iwo 
sides [and by Ihe line joining Ihe ab()ve menlioned points of incidence] are equal" (Wit. 
Penp.. pp. 71 -72); for Ihe Latin text, see ibid.. pp. 235- 236 and also Opl. Thes. Wit.. 
p. 16. 
Now the meaning of Ihe somewhal curious and hermetic phrase "which forms Ihe 
greatest inequality of Ihe angles". by which I translated "que transit per inequalilatem maximam 
angulorum". becomes quile elear. From an arbitrary poinl of the assumed oblique axis. 
one can draw only one perpendicular to the common section of the glacial and vilreous. 
The axis and the perpendicular determine the only piane passing through the axis which 
is perpendicular lo Ihe common section glacial-vitreous. Furthermore this unique perpendicular 
pIane is such, by Wilelo I. 39. thal the line joining the poinls of inlersection of the 
axis and the perpendicular with the pIane of common section glacial-vilreous, forms Ihe 
smaUesl possible angle with Ihe axis out of aU possible angles between Ihe axis -tlnd any 
line coming to it in the underlying piane of common section. This means. elearly. Ihal 
Ihis unique perpendicular pIane to the common seclion glacial-vitreous "forms Ihe grealest 
inequality of angles" with the Ijnes drawn in the underlying surface. 
.1 I.e.. the surface of common section between the glacial and Ihe vilreous. 
4 "If two planes which cut one another be al right angles lo any pIane, Iheir common 
seclion will also be al righl angles lo the same piane" (Healh. Euclid, vol. III. p. 304). 
And so no olher piane can be perpendicular "lo the said surface". 
fi Cf. n. 8 to prop. III. 7 .
upra. 
7 Because il passes Ihrough a linc. Ihe axis. which is perpendicular lo it. 


--
>>>
218 


8 "A part of a straight line cannot be in the pIane of reference and a part in a pIane 
more elevated" (Heath, Euclid, voI. III, p. 272). 
9 "If two triangles have the two sides equal to two sides respectively, and have also 
the base equal to the base, they will also have the angles equal which are contained by 
the equal straight lines" (Heath, Euclid, voI. I, p. 261). 
10 CB = CD as radii, CA is common, and AB = AD as chords subtending equal ares. 
II "To draw a straight line at right angles to a given straight line from a given 
point on it" (Heath, Euclid, ibid., p. 269). Actually, appeal to delinition I. 10 would 
have sufficed. 
12 "U two triangłes have the two sides equal to two sides respectively, but have the 
one of the ang1es contained by the equal straight lines greater than the other, they will 
also have the base greater than the base" (Heath, Euclid. ibid., p. 296). Strictly speaking, 
then, I. 24 does not apply to the case at hand; stiIl the conclusion is true. 
13 "If two triangles have the two sides equal to two sides respectively, and have the 
angles contained by the equal straight lines equal, they wiU also have the base equal to 
the base, the triangle will be equal to the triangle, and the remaining angles will be equal 
to the remaining ang1es respectively, namely those which the equal sides subtend" (Heath, 
ibid., p. 247). 
14 "To a given inlinite straight line, from a given point which is not on it, to draw 
a perpendicular straight line" (ibid., p. 270). 
15 This is prop. I. 47, the so-called theorem of Pythagoras (cf. ibid., p. 349). 
16 "In any triangle the greater angle is subtended by the greater side" (ibid., p. 284). 
Strictly speaking, the citation of I. 19 is inappropriate. Still, the concłusion is obviousIy 
true, and could have been inferred properly from what immediately follows in the next 
paragraph. 
17 As GFL is obviously acute. 
18 It is already proved above. 
19 Of common section of the glacial and vitreous. 
20 "On the same straight line there cannot be constituted two simiłar and unequal 
segments of circles on the same side" (Heath, ibid., voI. II, p. 52). 
21 Cr. n. 13 supra. The reasoning is the same as in the case of a pIane vitreous, 
above. 
22 "In equal circles equal straight Iines cut off equal circumferences, the greater equal 
to the greater and the less to the less" (Heath, ibid., vol. II, p. 59). 
23 Of the vitreous. 
24 Cr. n. 7 to prop. III. 7 supra. 


Commentary 
It is, as Risner correctly points out, in his II. 7 that Alhazen proves that "Axis 
pyramidis opticae solus ad perpendicułum est communi sectioni crystallinae et vitreae sphaerarum" 
(O pt. Thes. A/h., p. 27). The import of the views contained in this proposition in the 
general context of the theory of vision advanced in the Perspectilloe of Witelo and Alhazen 
has already been discussed in the Introduction. 


Proposition 25 
I "Ali the lines of longitude [Le., generator lines] of a cone are equal and form equal 
but acute angles with the radii of the base; from which it is cłear, that the apex of 
any cone is a pole of the circle of its base; and that each line of longitude is in the 
same surface with the axis; moreover that the axis itself falls orthogonally upon the center 
of the circle of base" (Wit. Persp., p. 110); for the Latin text see ibid., p. 268 and Opt. 
Thes. Wit.. p. 34.
>>>
219 


2 This "oblique inclination" ("declinatio") is determined by the constant angle between 
the generator Iines and the circle of base of the cone, in this case, the circle of con- 
solidation of the glacial and vitreous sphercs. 


Commentary 
The proposition is clear. It is, as almost all else in Book III, taken over from Alhazen, 
who dealt with the topic in his I. S and 13. The former is concisc, sharp, and precise. 
It ends with the conclusion: "Declinatio ergo nervi, super quem componitur oculus, non 
est, nisi apud foramen, quod est in concavitate ossis, sive moveatur oculus, sive quiescat" 
(O pt. Thes. Ałh., p. 4). It contains most of the issues covered by Witelo and practically 
all (there are a few exceptions) matters tackled anew in the latter, I. 13 (ibid., pp. 6-7), 
which is thercfore repetitive.. verbose and strikingly otiose. Risner's lapidary resume of the 
two emphasizes this fact eloquent1y. It reads: "In totius oculi seu motu seu quiete, situs 
partium stabilis permanet" (ibid., pp. 4, 6). It is identical for both. 


Proposition 26 
I Ali this may mean is that though the beginning of ocular movement is rooted in 
the indivisible virtue of the sentient and moving power of the soul, its termmation, being 
spatially localized in the world of objects, is divisible, apportionable, severable. 
The whole "proof' smacks of the order of ipse dixit. 


Proposition 27 
I This should merely be taken as a nominal dełinition of "the center of the aperture 
of the uvea". 


Proposition 28 


J But not before refraction at the interface glacial-vitreous. located in front of the 
ccnter of the eye. 
2 Ali this, needless to emphasize, is pathological or artiłicially, premeditatedly obtainable. 
It can happen, but, normally, it does not. Alhazen explains this beautifully in his I. 27, 
the source of Witelo's proposition. Some of Alhazen's statements deserve auotation in extenso: 


Qtiod autem ultimum sentiens non comprehendat formam. nisi post adunationem 
duarum formarum : est: quo d quando aspiciens mutaverit situm oculi unius, et alius 
fuerit immotus, et motus unius oculi mutati secundum situm. fuerit ad anterius, vide bit 
de re una opposita duas, et si aperuerit unum oculum, et cooperuerit alterum. non 
videbit nisi unum. Si ergo sentiens comprehendisset unum, quia unum, deberet ipsum 
comprehendere semper unum: et si venissent ad ipsum scmper duo formae ab uno 
viso, comprehenderet semper unum visum, duo. Et cum ultimum sentiens non comprehendat 
visum, nisi ex forma veniente ad ipsum. et aliquando comprehendat unam rem visam. 
duas, et aliquando unam: est signum quod id quod venit ad ipsum, quando comprebend.t 
ipsum duo, cst forma duplex: et quando comprehendit unam rem visam, unam, quod venit 
ad ipsum, est forma una. Et cum in utraque dispositione perveniunt ab uno viso ad duos 
oculos duae formae: et iIIud quod redditur ultimo sentienti, aliquando est duplex forma, 
aliquando una: et forma, quae redditur, ultimo sentienti; non redditur nisi a visu: tunc 
iIIud quod redditur ultimo sentienti ex duabus formis, quae perveniunt ad duos oculos ab 
uno viso, quando comprehenderit ipsum unum, est una forma. Et cum ita sit, duae ergo 
formae praedictae extenduntur a duobus oculis, et concurrunt, antequam comprehendat ip- 
sas ultimum sentiens, et post concursum inter se, comprehendet senUens ultimam formam 
adunatam ex eis. Et duae formae, quae perveniunf ad duos oculos ab uno visu, auando
>>>
220 


ultimum sentiens comprehendit ipsum duo, extenduntur a duobus oculis, et non concurrunt, 
et perveniunt ad ultimum sentiens, et su nt duae formae. Et comprehensio unius visi, 
quod apparet aliquando unum, aliquando duo, significat quod visio non est per oculum 
solummodo; quoniam si ita esset apud comprehensionem visi, quod unum apparet, 
comprehenderent duo oculi ex duabus formis pervenientibus ad eos, unam et eandem 
formam: et si ista esset, comprehenderet semper ex duabus formis unam formam. Et 
cum unum visum comprehendatur aliquando unum, aliquando duo, et in utraque dis- 
positione sint in duobus oculis duae formae: significatur quod illic est aliud sentiens, 
praeter duos oculos, ad quod perveniunt ab uno viso, quando comprehenduntur per 
unum, duae formae unum, et apud quod comprehenduntur duae formae, quando com- 
prehenduntur, duae: et quod sensus non completur, nisi per illud sentiens tantum, non 
per oculum tantum... 
Duae ergo formae extenduntur ab oculo in nervo extenso inter oculum et 
cerebrum, quousque perveniant ad ultimum sentiens. stae ergo duae formae extenduntur 
a duobus oculis, et concurrunt in loco concursus duorum nervorum. Et significatio 
manifesta, quod formae rerum visarum extenduntur in concavo nervi, et perveniunt ad 
ultimum sentiens, et post perventum compleatur visio: est: quod quando fuerit oppilatio 
in isto nervo, destruitur visio, et quando destruitur oppilatio, revertitur visio. Et ars 
medecinalis testatur hoc. Quare vero aliquando concurrant duae formae, aliquando non: 
est: quia quando situs duorum oculorum fuerit naturalis, erit situs eorum ab uno viso 
situs consimilis: et sic perveniet forma unius visi in duo loca consimilis situs: et cum 
fuerit declinans situs unius oculi, diversabitur situs oculorum ab iIIo viso: et sic per- 
veniant duae formae iIIius visi diversi situs. Et iam praedictum est in forma oculi... 
quod situs nervi communis a duobus oculis, est situs consimilis: et sic erit situs duorum 
locorum consimilis, situs a duobus oculis ab eodem loco nervi communis situs consimilis, 
et ex duobus nervis concavis fit unus, in quo uniuntur duae formae visus (Opt. Thes. 
Alh, p. 16). 
3 This last part of III. 28 covers matters taken up by Alhazen in his III. 9 (ibid., 
p. 79). Both his I. 27 and III. 28 have been summarized by Risner in essentially the 
some words: "27. Utroque visu una visibilis forma plerunque videtur" (ibid., p. 16) and 
"9. Utroque visu visibile unum plerunque videtur" (ibid., p. 79). 


Proposilion 29 
I This is a most inappropriale citation: whal fils IS ralher III. 9. as Wilelo says 
correctly immedialely Ihereafter. 


Proposition 30 
I "If in a triangle IWO angles be equal to one anolher. lhe si des which sublend Ihe 
equal angles will also be equal lo one anolher" (Healh. ElIclid. vol. I. p. 255). 


Commen'ary 
This can hardly be called a proof since il is lrue "by God's will" so lo speak. 11 
should be laken. ralher. as an explanalion of Ihe facl it states. Alhazcn provides his ex- 
planalion in III. 6 (DpI. T"e.
. Al".. p. 78). 


Proposition 31 
I Cr. n. 8 lo prop. III. 24, .mpra. 
1 I do nol undersland Ihe phrase "no diversity would oeeur lo Ihe eye from Ihal". 


Proposilion 32 
I "If IWO straighl lines be al righl angles lo 'he same piane. Ihe straighl lines will 
be paralkI"' /Jłeath. Elle/iti: vol. 111. p. 210). 


.......
>>>
221 


2 This is precisely the last postulate of Book II: "Likewise. (we claim] that nature 
does nOlhing in vain. (precisely] as it does not le ave undone anything (that is] necessary" 
(cr. .
upra). 


Commenlary 
Rlsner gives III. 2 Alhazen as the corresponding proposition m De aspectihus. From 
the standpoinl of the main claims contained in it, this ascription is correct. However, 
even though there is an accompanying diagram to the text (identical to Witelo's). there 
is no reference to it at all during the "proof'. The "proof' itself. then, is considerably 
less geometrical than Witelo's and much more heavily slanted IOward considerations of 
anatomy. and Ihe actual reciprocal operation of the two normai eyes (Cf. Op'. Thes. Alh., 
p. 76). 


ProPl,sition 33 
I "A pcrpcndil"lllar dra"n li.om Ihe lapex) angle of an isosceles Itriangle) to Ihe hase 
divides the isosccles (triangle] into Iwo partial similar triangles: from which it follows that 
thal perpendicular necessarily reaches to Ihe middle point oflhe base" (Wit. Persp., pp. 65-66): 
for the Latin text see ihid.. p. 231 and 01'1. Thes. Wit.. p. 13. 
2 cr. n. 9 to prop. II I. 24. supra. 
.1 This follows from their congruence. granled by I. 8. 
-ł This is definition I. 10 Elements: cr. Heath, Euclid, vol. I. p. 153. 

 This is infelicitously stated as. at this juncture, it is not actually geometrically war- 
ranted that the eXlension of AS would really come to B. (Of course it would. but this 
requires proof. which is hoth ohvious and immediate!) Witelo could have formulated the 
sentence slighlly differently and thus skirted this minor infelicily. 
(, "If two slraight lines cut one another, they make the vertical angles equal to one 
another" (Healh. Euclid. vol. I. p. 277). 
7 cr. n. 13 lo prop. III. 24 supra: the proof is obvious and immediate. 
H As a result of a change of posilion of Ihe eye a.f a "'1101('. following. say. from 
the ohserver moving his/her he ad or allcring olherwise hisfher po
ition. 
Ij One feels aculely Ihal the lasl senlence could have been clearer: what is meanI. 
most likely, is thal as point B can. in principle. be any point of the object similarly 
situated with respect to both eyes to which the common axis comes, each such point 
will have "its own" radial axes. 


Commentary 
In prop. III. 7, from which Witelo borrowed and enlarged his own proposition. Alhazen 
shows. wilhoul sufficient iuslilicalor\" apparatus (the littlc thal exists hclongs to Risner), 
thal. "Si recta linea sit a medIo nervi communrs admedium I.\"ic) reclae Imeae connedcntls 
cenIra loraminum gyri nervorum cavorum: erit ad ipsam perpendil:ularis" (Opl. The.
. A/lr.. 
p. 78); Ihls lormulatlon. of cour
e. is part of Risner's edilonal apparalus. 


Proposition 34 
I Of the gyration-apertures of the concave nerves of the two eyes. 
2 "lf two straighl lines cut one another. Ihey are in one piane. and every Iriangle 
is in one piane" (Heath. £uclid. vol. III. p. 274). 
3 cr. n. 8 to prop. III. 24. supra. 


Commcnlary 
III. 8 Alhazen (Op'. Thes. Alh.. p. 79) is Ihe model and Ihe source of inspiration 
for Witelo in Ihis case.
>>>
222 


Proposition 35 
I "If a straight line be drawn paralleI to one of the sides of a triangle, it will cut 
the sides of the triangle proportionally; and if the sides of the triangle be cut proportionally, 
the line joining the points of section will be paralleJ to the remaining side of the Iriangle" 
(He at h, Euclid, vol. II, p. 194). 
2 "A straight line falling on paralleI straight lines makes the alternate angles equal 
to one another, the exterior angle equal to the interior and opposite angle, and the interior 
angles on the same side equal to two right angles" (Heath, ibid., vol. I, p. 311). 
3 Some steps are condensed here: BNQ = BRT, BQN = BTR, but BRT = BTR, hence 
BNQ = BQN. 


Commentary 
Crucial for a proper understanding of this proposition are a number of elements. First, 
it must be stated that the term "multipIe" ("multipIex") in the enunciation turns out to 
mean merely "greater than". so that Wilelo's claim is that the distance of the poinls of 
Ihe o
ject from Ihe eye musi be llTealer Ihan (and of an appropriate value to) Ihe distance 
of the centers of the two eyes for accurate vision to take place. Furthermore. and Ihis 
is exceedingly imporlanl. the position of Ihe visihlc ohjecl wilh respect to Ihe 1"0 eyes 
musI be "an entirely like position" otherwise the enlire analysis collapses, as the Iriangles 
involved in the discussion would not be isosceles triangles, the rays entering the eyes would 
not be visual axes. ele.. elc. In proposition III. 2 of Alhazen. on which we touched briefly 
in the commentary to prop. III. 32 supra, and which Risner also cites as the equivalent to 
Witelo's III. 35, this point is made with great care and unusual trenchancy. As ł believe 
Alhazen's words are helpful in understanding our proposition and, at the same time, make 
abundantly clear the limiling conditions of applicability of the Alhazenian - Witelian intromission 
theory of vision, I shall cite them here: "Et dictum est etiam... quod unum visum. quod 
comprehendilur doubus oculis simul. non comprehenditur unum, nisi quando positio eius in 
respectu duorum oculorum fuerit positio consimilis: et quod si positio fuerit diversa: tunc 
comprehendetur unum duo. Sed unumquodque visibilium assuetorum. quae semper compre- 
henduntur a duobus visibus, semper comprehendetur unum. Unde oportet nos declarare, 
quomodo unum visum comprehendatur a duobus visibus unum in maiore parte temporis [my 
emphasis] et in pluribus positionibus: et quomodo positio unius visi ab ambobus ocu1is in 
maiore parte temporis [my emphasis], et in pluribus erit consimilis. Et declarabimus etiam 
quo modo positio unius visi ab ambobus visibus eril positio diversa. et quomodo accidat hoc. 
Et iam diximus hoc... et declarallimus ipsum unillersałiter, non determinate [my emphasis] 
(/oc. cit., p. 76). 


Proposition .36 
I "In any triangle, if one of the sides be produced, the exterior angle is equal to 
the two interior and opposite angles and the three interior angles of the triangle are equal 
to two right angles" (Heath, Euclid. vol. I., p. 316). 
2 "In any triangle the greatef ang1e is subtended by the greater side" (ibid., p. 284). 
3 "In any triangle, if one of the sides be produced, the exterior angle is greater than 
either of the interior and opposite angles" (ibid., p. 279). 
4 "If a straight line set up on a straight line make angles, it will make either two 
right angles or angles equal to two right angles" (ibid., p. 275). 

 Cr. prop. II. 50, supra. 


Commentary 
In his II. 9 Alhazen treats, among other things, the topic of this propoSItion. His 
treatment is dogmatically declarative and unaccompanied hy any geometrical diagram or
>>>
223 


geometrical reasoning. As a matter of fact, it is inDocent of aDY kind of reasoning whatever. 
As such it enjoys the elarity and pregnancy of some well- formulated administrative ordi- 
nances. The greatest part of Alhazen's proposition deals with the quality of visual per- 
ception of variously refracted rays and with the connection between visual perceptions and 
variously structured visu al fields (cr. Opt. Thes. Ałh., pp. 29- 30). 


Proposition 37 
I I.e., the arrangements of the forms on the surfaces of the eyes (and of the glacials) 
are preserved in the common nerve. 
2 Of the axes BO and BP with the eye. 
3 To the points of incidence. 
4 This is exactly what needs to be proved! 
s This is hardly a satisfactory proor. 


Commentary 
Witelo set out to show the complete homology between the structures of the forms 
on the two surfaces of the eyes and the glacial humors and their counterparts in the 
common nerve. In this belief he is the ever faithful disciple of Alhazen. In the course 
of his entirely inadequate proof. however, (1 do not believe it is an exaggeration to see 
it plainly for what it is, a simple petitio principi,), he is shooting for bigger game. relying, 
against his master, on the rectiłinearity of transmission of the forms through the optic 
nerves. His whole argument is predicated upon this rectilinear transmission. In this he 
goes, certainly, beyond his model. 
There is no correlate proposition to this one in Alhazen's De aspectibus, and this 
is no mere happenstance. For Alhazen the propagation of forms through the optic nerve, 
whose sentient power has the same transparency as the vitreous humor. is not. and cannot 
be, rectilinear: "Et omnes formae pervenientes in superficiem glacialis, extenduntur in corpore 
glacialis secundum rectitudinem linearum radialium, quousque perveniant ad istam super- 
ficiem [Le., communis sectio crystallinae et vitreae sphaerarum), et cum pervenerint ad super- 
ficiem istam: refringuntur apud ipsam secundum Iineas consimiJis ordinationis, secantes li- 
neas radiales. Lineae ergo radiales non iuvant ad ordinationem formarum rerum visibilium, 
nisi apud glacialem tantum: quoniam apud membrum istud [i.e.. humor vitreusJ principium 
est sensus. Et deelaratum est... quod impossibile est. ut forma rei visae sit ordinata in 
superficie visus cum imagine rei visae et parvitate rei sentientis. nisi per istas lineas. Istae 
ergo lineae non sunt. nisi instrumentum visu s, per quas completur comprehensio rerum 
visarum secundum suum esse. Perventus autem formarum ad ultimum sentiens non indiget 
extensione secundum rectitudinem istarum linearum" (Opt. Thes. A/h., pp. 2S-26). It is 
this fact that entitles Lindberg to state: "... Alhazen admits the inapplicability of the law 
of rectilinear propagation to transmission within the vitreous humor or optic nerves. Only 
as far as the glacial humor are straight lines required... Thereafter, all that is required 
is the preservation of the proper arrangement of forms. .. Indeed it is elear that forms 
issuing from a single point in the visual field cannot pass through the two eyes and reunite 
in the optic chiasma without deviating from rectilinearity" (Theories oj Vision, p. 83). And 
yet it is this very impossibility that Witelo denies in his argument and the accompanying 
diagram. 


Proposition 38 
I This is an awkward way of putting it, as the surface in question is perpendicular 
to the object precisely because it passes through AF, which is perpendicular to the surface 
of the object, in accordance with prop. III. 33, supra. 
2 cr. n. 2 to prop. III. 3S. .fUpra.
>>>
224 


"Slraight lines parallei lo Ihe same slraighl line are also pantllel lo I\ne another" 
IHealh. E
d. VI'J;..1. p. 314). 
4 I.c. BFE = CFG: as BFC is Ihe l"f'mm.\n s
'Cli.\n .\1' planc AEFG and nr .he "urface ol' 
Ihe oojeel. it rollows Ihal Ihe laner is also a piane surrace. 


Proposilion 39 
I "To bisect a given tinite straight line" (Euclid. iMci.. p. 267). 
2 cr. supra. n. 9 lo prop. 111. 24. 
3 "Through a given poinl lo draw a straight line paralleI lo a given slraighl line" 
(Eudid. iMci.. p. 315). 
4 "Ali perpendicular lines drawn belween paralleJ lines or surfaees are Ilhemselves) pa- 
rallel and equal; and if slraighl lines inlerseel paralleJ lines or surfaces al equal angles. 
they are equal" (Wil. Persp.. p. 61); for the lalin lexl see ihicl.. p. 228 and Opl. Thes. 
Wil.. p. 12. 

 "Ali parallei lines neeessarily lie in Ihe same surface" (Wil. Persp.. p. 48): for Ihe 
lalin lexl see ihicl.. p. 116 and Opl. Thes. Wil.. p. 5. 
fi Cr. n. 5 lo prop. III. 3. .mpra. 


ProPl,silion 40 
I Ol' Ihe Iwo opposing sides ol' BCDF. 
2 CI'. n. 13 to prop. III. 24. supra. 
.' More would need lo be said lo show Ihat ZA is indeed Ihe perpendicular drawn 
from Z lo KL: I. 12 reads: "To a given intinile slraight line. from a given poinl whieh 
IS not on it. lo draw a perpendicular straighl line" IEuclid. vol. I. p. 270). 
-I Cr. n. I lo prop. III. 33 .
upra. 

 "In isoseeles Iriangles Ihe angles at Ihe oase are equal lo one anolher. and. if Ihe 
equal slraight lines be produced further. the angles under the base will be equal to one 
anolher" (Euc'lid. iMd.. p. 251); it is not elear lo me why Witelo invokes Ihis Euclidean 
proposilion here. except lo surmise that he is feeJing eontidenl enough wit h Ihis materia I 
to flex. as il were. his malhemalical museles (quite flabby by their nature) by giving. 
atypieally. alternalive proofs and invoking. not always poinledly. a richness of Euclidean 
proPI\silions. 
fi Cr. supra. n. I to prop. 111. 36. 
7 "In equiangular triangles the sides about the equal angles are proportional and Ihose 
are eorresponding sides which subtend Ihe equal angles" (Heath. Elldid. vol. II. p. 200); 
Ihe remark I made in n. 5 above applies to Ihe citalion ol' this Euclidean proposilion 
too. 

 cr. .
upra. n. 2 lo prop. III. 35. 
II "II' a straight line be set up at right angles to two straighl lines whieh eut one 
anolher. at Iheir common point of seelion. it will also be al right angles to Ihe piane 
throullh them" (Euclicl. vol. III. p. 277). 
10 "II' two straight lines be paralleJ. and one ol' them be al right angles to any piane. 
the remaining one will also be at right angles lo the same piane" (ihid.. p. 287). 
II This is der. X I. 3: "AsIraiK'" line is al riglll angles lo a piane. when it makes 
right angles with all Ihe straighl lines whieh meet it and are in Ihe piane" lihid.. p. 260). 
12 Cr. supra n. 13 lo prop. III. 24. 


Commentary 
Alhazen covers Ihe issue deall with in Ihls tIToposltlon in a very brief diagramless 
expository aceounl in III. 3 10pl. Thes. Alh.. p. 3).
>>>
225 


Proposition 41 
I Al their point of intersection. 
2 Cr. supra. n. 4 to prop. III. 36. 
J hIf two triangles have the two sides equal to Iwo sides respectively. but have the 
one of Ihe angles contained by Ihe equal straight lines greater than the other. they will 
also have the base greater Ihan the base" (Euc/id. vol. l. p. 296). 


Commenllry 
This lime Alhazen's proof of prorosilion III. 4. the source of Wilelo's, conlains a diagram 
(the one appearing in Wilelo's former prorosilion. III. 40) and is Ihoroughly argued. covering 
much more Ih.m ll'relrs in Wilelo Id
 01". T/w,. ,"h.. r. 77). 


Proposilion 42 
· The diagram. which is Risner's. has no counlerpart in Ihe MSS. II has been re pro- 
duced here because il eases the reader's understanding of Witelo's exposi'ion. 
I Cr. supra. n. 13 lo prop. III. 24. 


Proposilion 43 
I Ajier refraction at the interface glacial-vitreous. 


Com ment ary 
The problem dealt with in this proposJtlon is the problem of authenlicati"n or 
erli- 
ficatlon. covered by Alhazen in his II. 8 (the correlate of this proposition. accordi:l!! 'l' 
Risner (cf. Opl. Thes. Alh.. p. 29). hUI also in II. 7. 9. 64. 65. 66 and III. 10. . 5. ! 6. 
This is Ihe rrohlem of reliahilih' of visua' peJ"l'ep'i.'n. Accordiny lo Wilelt' l1l(
 he is 
not original in Ihis. Ihe highest reliabillly is achieved by Ihe unrefracted form arriving al 
the optic nerve along the lxis of the visu al pyramid. Forms travelling on lines closer 
to Ihe I"is łchie\e !!reUer c
.rtainl
 in Ihe darily of Ihen I'erl'erlion Ihln f.'rms Iran:lling 
on lines remoler 'o ,he axis. This is so because evcn Ihl'Ugh all rays of Ihe c"lle of 
radialion (excepl the axis. Ihal is) are refracted at the vitreous. and Ihus weakened. Ihe 
former are less refracled th.1Il the latter. 
There are further aspecls of Ihe problem of cerlification with which. however. I resolved 
not to deal here. merely because Wilelo does not dcal with Ihem in his proposition and 
because they have been covered in Ihe Introduction. See also my commentary to the following 
propositions. For a ,l!ood discussion oflhe issue. see Lindberg. Theoril's of Vi
ion. pp. 84-85. 


Proposil ion 44 


I Cr. n. 2 to prop. III. 35. supra; again there IS an hembarras de richesses" ID citing 
relevanl and irrelevanl proposilions. 
2 Cr. supra. n. I to prop. III. 33. 
J Cr. supra n. 13 to prop. III. 24. 
-ł hlf two triangles have the two angles equal to two angles respectlvely. and one 
side equal to one side. namely either the side adjoining the equal angles. or that subtending 
one of Ihe equal angles. Ihey will also have the remaining sides equal to the remaining 
sides and the remaining angle to the remaining angle" (Euc/id. vol. I. p. 301). 

 That this is so is obvious. due lo the fact that the axes of Ihe visual pyramids. 
by definition and nature. pass unrcfraCled to the gyration-apertures of the optic nerves. 
and are. Iherefore. a/l1'ays arriving lo the midpoint .of the common nerve. in this case H. 
ń Cases considered. 


IS - Wit.loni. P.rspcclivac...
>>>
226 


Commentary 
This proposition too deals with the issue of certification. as does the next one. Its 
correlate. according lo Risner. is Alhazen's 111. 10 (cf. Opt. Thes. Ałh., pp. 80-81). The 
import of these proposilions is elear and can easily be gTasped by selective gleanin
. from 
Alhazen. although. once aga in. he deals in this case too with more than does Witelo in 
his imilation. though he does it without the aid of an accompanying geometrical diagram: 
"Sed lamen cum visum fueril in axe communi, et duo axes concurrerint in puncto ipsius, 
qU0d esl in axe communi. tunc duae formae istius puncti erunt magis consimiles... Quapropter 
fl'rma visi. quod est in communi axe. cum fuerit infixa in concavitate communis nervi, 
erit ma!!is certificata. Sed cum visum fuerit extra communem axem, et remolio non fuerit 
maxima : tunc suae duae formae. quae infiguntur in duobus visibus, non maxime different. 
Quapropter formae eius, quae infiguntur in concavitate nervi communis. non erunl duae. 
Cum veTO visum fuerit extra communem axem, et maxime fuerit remotum ab ipso: et 
axes duorum visuum concurrerint in aliquo puncto ipsius: tunc forma eius infigetur in 
concavitale communis nervi una forma: et forma puncti eius, in quo duo axes concurrunt. 
infigelur in puncto communis centri : sed tamen forma eius non erit verificata sed dubitabilis... 
Quapropter forma totius visi infigetur una in omnibus dispositionibus: sed tamen forma 
extremorum. et iIIorum. quae remola sunt a puncto conctu'sus. erit non certificata... Cum ergo 
duo axes fuerin' moti super omnes partes huius visi: tunc ,'ertificahi,ur forma eius" (wid., 
p. 80. my emphasis); cf. also the next proposition in Witelo. 
The underscored phrase points to an important ingredienl in Ihe certification apparatus 
of Ihe eye. Since the besl certified portion of the object arrives at the common nerve 
unrefracted al0nł/ the tW0 axes 0f the pyramids 0f vi sio n (and. al times. al0nłl Ihe C0mm0n 
axis). Ihe eyes will normally sweep the whole ohject with their axes in very s"ift motion. 
achieving in this manner utter cerlificati0n and hence clear percepli0n. f0r all the c0nsti- 
lutive .parts of the objects' surface. Cf. also prop. 111. 4H i/!/;'a. 
Proposition 45 
I The intenti0n of Ihe last words of the senlence is rather I'hseure. 


Commentary 
The counterpart to this prop0sition appears in III. 15 De a
peL"ihus (ef. Opt. Thes. Alh., 
p. 85). 


Proposition 46 


Commentary 
Roughly Ihe first half of the proposltJon carries very little conviction; the second 
half makes it a mere Irivial consequence of III. 37. The correlate in Alhazen is III. 14 
(Op'. The.\'. Ali,.. pp. 83-85). a mammoth theorem, almost eighl times as long as Witelo's, 
and c.'\cring Ihcrcf.'rc l'l'nsiderahly mere 
round than Witel0. Risner sums it up as "Vi,ihile. 
in quo concurrunl axes optici. aut radij his propinqui: videtur unum" (ihid., p. 83). In 
il Alhazen cl'nside..
 a great variety of visu al siluatjons in which Ihe trulh of the elaim 
is vindicated and anolher spate in which. due to the "extraordinary cireumstances prevailing", 
it is no' the case thal the object would be seen as one thing. In the course of his argument. 
Alhazen appeals profusely lo bolh experience and experimentation and attempts quite sueeessfully 
to deal exhaustively with the topic. Of course. I shall not analyze here Alhazen's procedure 
and modll,f ar
lIendi. 


Propesition 48 


Commentary 
In his III. 16 (Op'. Th".f. Ali,.. p. 85- 87). Alhazen deals wilh what Risner refers 
te as "Visihilc małlnum sil11l1l t.'tum aequahiliter non \'idelllr" liMll.. p. 85). It elearly 


.......
>>>
227 


contains the elements of Witelo's argument and much more and, again. states its conclusions 
on the basis of a quasi experiential-experimental set-up. I shall only quote the first sentence, 
as it makes the central point with utter clarity: "Amplius apparet. quod visus non compre- 
hendit rem visam. quae est remotarum diametrorum, vera comprehensione, nisi moveat radialem 
axem super omnes eius diametros. et super omnes eius partes. sive comprehensio sit ambobus 
visibus. sive uno" (wid.). 
In addition to calIing attention to Alhazen's III. 16. Risner also gives "Euclides in 
praefatione et I the. opticorum" (DpI. Thes. Wit.. p. 107) as a correlate to Witelo's pro- 
position. Indeed this is 50. The postulates contain statements from which the claim of the 
proposition clearly follows. I mention these postulates as there is no introduction ("praefatio") 
in Euclid's Oplica. There is, however. a rather lengthy introduction in Theon's recension 
to the Oplica and it contains statements consistent with Risner's remark (cf. pp. 146- 
147 (lf Heiber
's edilion. EIIClidif Opl;cO). Furlhermore. prop(lsiti(ln I (lf Ihe 01'1;('0 reads 
precisely Oóotv tcDV 6poj.ltvrov d
a 6).ov 6pdtUl (ibid., p. 2). which in Heiberg's Latin 
translation strikes one as quite close to Witelo's enunciation: "Nullum visorum simul videtur 
totum" (ibid., p. 3). 
Needles to insist Ihal the arguments (lf Witell' and Euelid differ dramalically as Ihey 
stem from two diametrically (lpposed Iheories of vision. In Euelid the argument is based 
on the issuance of visu al rays from the eye and (In the "discreteness" (lf the rays leading 
to Ihe exislence (lf "empty". unseen spaces helween c(lnsecutive rays. 
Prop(lsition 49 
I "Only (lne perpendicular can be drawn from a given elevaled p(linl lo any piane 
or Cl'nvex surface Iying under it" (Wil. Persp.. p. 58); f(lr Ihe Lalin lexl see ihid.. p. 225 
and 01'1. Thr.
. W;I.. p. 10. 


Proposilion 50 
I 11 is n(ll entirely elear what is meant here. Perhaps some clarification could be achieved 
by quoting Alhazen on the same point: "Et etiam forma eius. quod propinquum est illi 
viso. si fuerit parvae quantilatis; instituelur in duobus locis duorum visuum. jnler quorum 
positiones non eril differenlia scnsibilis" (01". Thc'.f. Alh.. pp. 77 - 78). This Cl'mes from 
III. 5, a short proposition that is clearly the souree of Witelo'9, couched in generaJ. non- 
-specifically ge(lmclrical lerms (there iso thou
h. a diagram. idenlical 10 Wilelc"s bUl unrelaled 
lO Ihe lex I of Ihe "Iheorem"; cf. ihid.. p. 78). making precisely the same poinls as Wilelo 
and summarized by Risner as follows: "E pluribus visibilibus ordinalim intra opticos axes 
disp(lsitis: remotiMa incerte videnlur" (ihid.. p. 77). 
2 This d(les not seem lo be entirely consistenl wilh Alhazen's understanding of the 
siluatic'n in n. , .mpro. 


Pwp(lsiti(ln 5' 
Commenlary 
One can either see superficially - aspectus - or profoundly. intuilio. For the latter. clearly. 
Ihe axis l,f the visual pyramid must pass over the surface of Ihe l'bjecl (cf. next Pfl'Pl'silion). 
As Alhazen pUls il in his II. 64 (DpI. Thes. Alh.. p. 67). the source of Wilelo's pro- 
posilion and a repelitive and prolix exp(lsilion: "... comprehensi(l visibilium erit secundum 
duos m(ldos. qui sunl cl'mprehensio superficialis. et comprehensi(l per intuiti(lnem. quae 
profundum aspicit... Comprehensi(l autem per primum aspeclum. est comprehensio non certi- 
ficala: el c(lmprehensio per inluili(lnem. est comprehensi(l. per quam certificantur f(lrmae 
visibilium" (ihid.). 


Pwposition 52 
C(lmmenlary 
The impl'rl of 'his pWposlllon is elear enough nol \(I require further comment. Suffice 
it t(l say thal its s(luree is Alhazen III. 65 (Op'. Thrs. Alh.. pp. 67-69). another enorm(lus 


........
>>>
228 


PWPl'sllll'n. earmarked by pn'lixi'y and liresome repetlllousness. conlammg little more Ihan 
Wilelo's abridgemenl .'f it. by way I'f s.'me discussi.'n .'f the wIe .'f Ihe l';rfU.
 di.
/;",.,iva 
in Ihe pwcess l'f cerlificalil'n. 


Proposili.'n 53 
I Becausc very fast; as Alhazen PUlS il. in his II. 42. Ihe clear-cul m.'del l,f Ihis 
pwpl'silion. "Et axis in 'l't.' ist.' ml'lu erit fixus in suo silu. el erit isle ml'lus valde 
velos. et in mail'ri parle esl insensibilis pwpler velocitalem" (Opl. Thes. Alh.. p. 57). 


Pfl'p.'siti.'n 54 


C.'mmentary 
Clarily and c.'nciseness are nI'l the Irade-marks .'f this pwp.'siti.'n. II is an expalialion 
.'n All'azen's II. 43 {()p'. T/Jes. Al".. p. 57) which iso this time. succincI and t.' Ihe point: 
"Axis aulem Dl'n suppl'mtur in su.' ml'lu lerminus anguli. quem respicit illa res visa apud 
cenrrum visus. neque seca' laliludinem anguli. quem respicil aliqua diamelwrum rei visae: 
qUl'niam h.'c non eril. pisi quando axis fuerit motus per se. et 1.'lus .'culus quieveril. 
qUl'd esl imp0ssibile: to: us enim llculus m0velur apud intuili.'nem. et axis movetur per 
m"lum eius" (ihid.). II is as if ',le 'eacher and his pupil c.'mpele wilh l'ne anOlher. 
from time w 'ime. fI'[ Ihe lilIe .'f champi.'n of verb.'sily. 


Prnp.'silion 55 


C.'mmenlary 
An'lher ins!ancc of word-wastefulness Ihis lime in Ihe guise .'f a quasi-j!eomelrical 
dem.'nslrali.'n in which Ihe ge.'mclrical cl'nsiderati.'ns are wholly irrelevanl tl' Ihe actual 
"pro.'f' based l'n cxperience and .'bservali.'n. There scems lo be no specific cl'unlcrparl 
lo Ihis gem in Alhazen. 


Pwpl'silion 56 


C.'mmentary 
This prl'pOSililln i
 a slighl c0mpressi.'n. bUl a ralher faithful imitali.'n (even the example 
llf Ihe many-Iel!ł!ed animai is nOI original and appears in Alhazen I.'gelher wilh olher 
examples) .'f Iwo Ihel'rems in Alhazen. II. 70 and 74 (Opl. T"es. Al".. pp. 71. 73). The 
f.'rmer is very brief shl'wing. in Risner's words. Ihat "obtulus fit in lempore" (p. 7'). 
Ihe latter is much Il'nger. c.'nlaininł! Ihe above-menlioned example. and is summed up by 
Risner under the lilie "Tempus .'blulus pr.' specierum visibilium varielale varial" (p. 73). 


PWp.'silion 57 


CI'mmenlary 
The s.'urce .'f this pWposltl.'n is !he lasl pn'p.'slll.'n .'f b.'.'k II .'f Alha7.en's PenpI'CI;\"U. 
nl'. 76 (01'1. The.
. Alh.. pp. 74- 75). Alha7en is more Ihorough in hi
 di
cllssion Ihan 
I'is imilalN. dislinguishing bclween IWl' kinds of prinrll.
 aspeclus and hH' kinds nf ;I/II/;Iio: 
"El visil. ljuac esl in pri:'ll' Hspeclu: quandl'que esl s.'lum phanlaslice: el qlland"qllc 
cum cogniiione praecedente... El visio per inluitionem erit secundum duos modos. scilicel 
visio s.'la intuitione. el visio per intuilinnem cum praecedente cl'gnilionc" (iNd.. p. 74). 


Pwplsition 58 
I This work is I.'st. Ali we know If it is Ihis reference (cf. Wil. Persp., pp. 20- 23. 
f.'r a discussi.'" of .'Iher pnssibly II'sl works by nur aUlhor).
>>>
229 


Commentllrv 
The c.'nlcnls of Ihis pmposltion are sublimated fmm Alhazen's 11. 66 (Opt. TIres. 
Al".. p. 69). where pwlixily and repelitiousness are Ihe ('rder (,f the day. and enable Alhazen. 
among olher things. to include in his slalement many more speeific examples than appear 
in Wilek'. These examples. however. do not add anything significanl to the basic point 
of Ihe proppsition. formulated by Risner in Ihe following words: "Obtutus iteratio altius 
jmprimit formas visibiles animo. certioresque efficit" (ihid.). Finally. I should not omit to 
cali explieil attenlion lo Ihe remarkable facI Ihal. in a sudden fil of laconism. Wilelo 
ends his argumcnt I'n an alypieal DI',e pf /'Irevily and laeilurnily. 


Prpposit ;I'n 59 
1 This IS ,he secl'nd pI'slulalc ICf. \/II'/"U) reading: ..... Ihere are only Iwo (enlilies 
which are] visi/'lle /'Iv Ihemselvcs. n:lmcly 'i!!ht and wlor; for light is seen by itself and 
is itself the hypostasis of cI,lor". 

 In Ihe phcnl'mena of lighl and eolor. 'hal iso Ihal would pwvide the observer wilh 
some kind I'f \ i,"a' gra'p. CH'n in I hc a/'lsencc I'f par,ieiJ1al il'n I'f "'her senses. a part i- 
eipalion Ihat iso however. required for a fuli grasp of the cnlity in question. 
Commentary 
The inspiration for this pwposJllon eomes from II. 18 Alhazen (Opt. TIres. Alh.. p. 35) 
where. in a brief. direct statement. Alhazen actually covers only half of Witelo's claim. 
namely. the pereeption of light and color by sight alone; the pereeption of other entities 
than Ihose is not tackled here hy A'hazen. hill ralher in his II. lO (ibid.. pp. 30-.1 I). 
summed up by Risner in the words: "Visihile percipitur aut solo visu: aut visu et syllogismo: 
aut visu et anrieipata nolione" (iNd.. p. 30). Cf. also the next pwposition and ,he ('om- 
men'al) I'n il. 


Pwposition 60 


Commentary 
II should be evident by now that there is no fuIl one-t o-one eorrespondence between 
Wilclo's imitative pwpositions and their sources in Alhazen, as the former divided, separated. 
developed. shrunk. glued together and otherwise modified what he appmpriated fmm the 
latter. sometimes bringing together things which are separate in Alhazen and. on oecasions. 
severing ('ne and the same proJXIsition into a number of parts to be distributed among 
a few of his own Witelian propositions. 
A ease in point is presented by propositions III. 59 and 60 that have drawn element s 
ffl\m Alhazen's II. 18 and 10 (ef. comm., supra. to III. 59) and from II. 10 respeetively. 
Alhazen's II. 10 is more complete. thorough, and morc riehly iIIustrated than Witelo's III. 60, 
hut there exists no doubt whatever that Witelo copied extensive parts of his argument. 
down to Iypieal turns of language. from Alhazen; furthermore. II. 10 also supplied Witelo 
wit h building elements for his III. 63 (cf. in/raI. 
It is interesling how Alhazen and Witelo avoid the famous trap of multiplying forms 
beyond neeessity poinled out by Aristotle in his "third man argument" against the Platonic 
theory of forms. Speaking of similar forms. they stress that similarity (bet-ween two white 
colors. say) is not aspecial. third form. but rather the outcome of the comparison of 
the two individual forms by the ,'irtus distinct;\'a. It is precisely this comparison that the 
unarmed eye is not fil to perform. 


Proposition 61 


Commentary 
The proposed may he indeed elear to the pwponent but it is far from elear and 
dislinl:1 II' ,he rcader who feels disappI'inled hy ,he Jack of sharpness and precision of
>>>
230 


-- 


terms. by the intermingling of individual and universal forms. indeed by the appearance of 
"individuated universal intentions". etc. Nor is it particularly helpful to appeal to the source. 
in this case. Alhazen's II. ł4 and 6 (DpI. Thes. Ałh.. pp. 33-34. 69-70). which. though 
perhaps less woolly, say between them precisely the same things, incłuding the rem ark 
that what the eye sends to the virtU.f distincliva are not merely individual but also uni- 
versal forms. Still the generał import of the proposition and its modum argumentationis 
are obvious enough to make them reasonably palatable to the curious and discering reader. 
What one is left with is the generał cłaim that. as Risner puts it in summ ing up identically 
the two Alhazenian propositions. "E visibili saepius viso remanet in animo generalis notio: 
qua quodlibet visibile simile percipitur et cognoscitur" (ibid.. pp. 33, 69). This is a quite 
familiar Aristotelian doctrine. (Cr. in/raf commentary to prop. III. 64. in which proposition 
one can distinguish some Platonic influences: Witelo's ecłecticism. spoken of in my edition 
of Book I of the Perspectillo. ("Introduction". passim) and in "Witelo and Thirteenth-Century 
Mathematics" - fuli reference in n. I to prop. II. ł4 .fUpra - . comes again to the fore). 


Proposit ion 62 


Commentary 
The direct inspiration for this propOSJtlon is Alhazen II. 69 (Opt. Thes. A/h., pp. 70- 
71). The model is somewhat cłearer. more systematic and philosophically more careful and 
astute (in distinguishing between the individual form and the form of the species, for instance) 
than the imitator. This is how Alhazen end s his exposition (and it constitutes a good 
recapitulation of the cłaim): "Comprehensio ergo omnius visibilium secundum intuitionem 
erit duobus modis: sola intuitione. et comprehensione per intuitionem cum sdentia praecedente. 
Cognitio autem talis et scientia quandoque erit secundum speciem tantum. quandoque secundum 
speciem et individuum simul" (ihid.. p. 71). 


Proposition 63 
Commentary 
Risner gives II. II Alhazen as the counterpart to our prOposltlOn. It is indeed the 
case Ihat part of Witelo's proposition cI)mes.fwm II. II (Opt. TIII'.f. A/lr.. p. 31); but 
that is not all. since some elements are copied I'erhatinr from II. 10. providing us with 
another iIIustration of the fact mentioned in the commentary to prop. III. 60. supra about 
the lack of agreement between the Witelian propositions and their correlates Ił la Risner. 
It seems sometimes as if Witelo was trying to cover his tracks by doing such a thorough 
job of putting things together. as it were, by means of cards. paste and scissors. 
Substantially. Alhazen. of course, says everything Witelo does and says it better. with 
additional enlightening detaiI. Among the significant additionał elements in Alhazen there 
is the distinction between "cognitio per signum" and "cognitio per intuitionem": "Cognitio 
ergo non est, nisi modus rationis. Sed ista ratiI) distinguitur ab omnibus rationihus: quoniam 
cognitio non erit per inductionem omnium intentionum. quae sunt in forma. sed per signa. 
Cum ergo visus comprehendit aliquam intentionum. quae sunt in forma. et fuerit memor primae 
formae, statim cognoscet formam, et non est ita omne, quod comprehendit per rationem: 
quoniam plura eorum, quae comprehenduntur per rationem, non comprehenduntur, nisi post 
inductionem omnium intentionum. quae sunt in eis... lIIud ergo quod comprehenditur per 
cognitionem. comprehendetur per sij!num: et non omne quod comprehenditur per rationem, 
comprehenditur per signum" (iNd.). And this is how Alhazen ends his proposition: "Et virtus 
cogmtloms est coniuncta virtuti sensus; et non completur comprehensio visibilium nisi per 
cognilionem. Cognitio autem non est solo sensu. Intentiones ergo quae comprehenduntur 
a sensu \ i,u I.juil.:dilm compr.:henduntur solo sensu. quaedam per cognitionem. quaedam per 
rat lonem et distinctionem" (ihid.). 


--
>>>
231 


Pwposil ion 64 
Commentary 
Risner gives Ił. 13 and 71 in Alhazen as the counlerparts to this proposItIon. His 
choice further strengthens my remarks in Wit. Persp.. pp. 28- 29 that Risner's references 
were not meant to pinJ!oinl the actual sources of Witelo's idea. but rather to cali attention 
lo other places in which similar. or related. ideas were discussed. Indeed there is very 
linie in II. 13 directły related to our proposition; it is an import ant proposition for an 
understanding of Alhazen's psychology of cognition and, as sucho clearly germane to the 
issue tackled by Witelo here (the solution of which is superficially reminiscent of the Platonic 
anamnesis). but its main concerns are much more general and its actual elaboration quite 
foreign to the specifics of III. 64 Witelo (Cf. Op'. Thes. A/lr.. pp. 32- 33). II. 71 on 
the other hand is jndeed the actual source of Witelo's proposition. as it contains all the 
elements of Witelo's argument. sometimes the very same word ing. and even the order of 
the exp,'sili,'n is largely the same (iNd.. pp. 71-72). 


Pwposit ion 65 


Commentary 
The source of this proposition is Ił. 75 Alha7en (Op'. Thes. Ałh.. pp. 73-74). summari:zx:d 
by Risner as follows: "Visio per anticipatam notionem et brevem obtutum. est incerta" 
(ihid.. p. 73). It is richer in examples than its imitation. crisper and clearer in its exposition. 
and. as a result. the argument is somewhat more convincing and better structured than 
Witelo 's. The gist of the argument is given in the following sentences : "Et cum omnia 
visibilia sint praeparata mutationi. quae possit comprehende a visu: nullum ergo visibile. 
quod visu s comprehendit modo. et erat prius comprehensum: certificalum est apud comprehen- 
sionem secundam a visu. scilicet. quod visus sit securus secundo. quod non fuerit mulat um, 
cum mutatio sit possibilis in omnibus visibilibus. Cum ergo visus comprehenderit aliquam 
rem visam. quam ante comprehendit: et intuitus fuerit ipsam: et certificaverit formam eius: 
et fuerit memor suae formae apud comprehensionem. cognoscet ipsam... Comprehensio ergo 
visibilium per cognitionem praecedentem. et per signa. et per modicam intuitionem. non 
est vera comprehensio" (ihid.. p. 74). 


Proposition 66 
I This is the first postulate: "That sight is not to be completed save only by the 
arrival of the visible form at the soul" (cr. supra). 

 I.e.. unanalysable. 


Commentary 
This proposition is taken over from Alhazen II. 68 (Opt. Thes. Alh.. p. 70) where 
both the statement of the claim and its substantiation are clearer and more to the point. 
As an illustration of this assessment. I shall only give the beginning of Alhazen's statement 
that. somehow. contains a distillation of the central argument: "Et !ustentatio sentientis 
in comprehensione quidditatis visibilium non est. nisi super formas pervenientes in animam : 
quoniam comprehensio quidditalis visibilium non erit. nisi per cognitionem: et cognitio non 
est. nisi ex comprehensione formae. quam visus comprehendit modo ad formam secundam. 
quae est in imaginatione ex formis visibilium. quas visus comprehendit ante: et ex comprehensione 
considerationis formae comprehensae modo ad aliam formarum pervenientium in imaginationem. 
Comprehensio ergo quidditatis rei visae non est, nisi ex comprehensione assimilationis formae 
rei visae alicuius formarum quiescentium in anima. fixarum in imaginatlone" (ihid.). 
This time it mayaIso be worthwhile to quote Risner's enunciation of the propo- 
sition as it hits the nail on the head in extracting its very essence: "Essentia visibilis 
percipitur e speciebus visibilibus. beneficio formae in animo residentis" (ibid.). 


--
>>>
232 


Proposil il'n 67 
Commenlary 
This propl'sition Witelo copied. wilh slighl modifications. from Alhazen's II. 17 (Opt. 
Thes. Alh.. p. 35). 


Pwposil ion 68 
Commentary 
Wilelo copied. somewhat creatively. this prop0silil'n from Alhazen's II. J 9 (ihid.. p. 36). 
'Is meaning is quite elear and need not delain us here beyond what was already said 
on the matter in the Inlroduclion and in these Nl'tes and Commenlaries. The "crealivity" 
referred lo above involves his allowance IlIal Ihe eye may in S(lme cases be able to unravel 
a special quiddity (he does not eleborale which specifically) bUl that. even in such a case, 
Ihere would stm be left essences of Ihal parlicular phenomenon. of al leasl two levels 
of generality. that would not be availab!e fl'r pH'per identificatil'n excepl to the \'irlUS 
dislinclil'U, This apparenl concession g:ranted Il' the eye to extricate by ilself the quiddity 
l'f 
"ll1e 
pecies. AlIlazen c"\p'icilly dcnics: ""'!lit en.." lJllI'd eomprehendil vi
lIs sol" sensu. 
non esl. nisi color In eo. qul'd esl col..r. el lu\ in eo. qll"d e
1 lux: el praeler isla 
nihil cpmprchendil 
1,11' sensu. sed per dislinclionem. et argumcnlalionem el c"!!nilionem" 
(ihid. I. 


Proposition 69 
I Thal wOllld be Ihe anleripr surface "f Ihe !!Iacial. 
2 AClually Ihe glacial humor. 
3 This elearly could have been stated more precisely. 


1- 


Cl'mmentary 
The correlates of this proposItIon. according lo Risner. are three proposltlons in book 
II of Alhazen's De aspectihus: 13. 15 and 71. Indeed between them they conlain all that 
Witelo says here and considerably more. I have already referred lo the first and lasl in 
connection with prop. III. 64. su pro. Concerning II. 15 Alhazen (DpI. Thes. Alh.. p. 34) 
I shall limit myself to stating thal its main concern is Ihe enumeralion of Ihe Iwenty 
two visible intentions, appearing in Witelo amon!! Ihe postulates. and an argumenl to the 
etfect that all other optical phenomena fali lInder pne l'r am'lher of Ihese Iwenly Iwo. 


Pwposition 70 
Commentary 
The claim dealt with in this proposltll'n comes from Alhazen's II. 20 (DpI. Thes. 
Allr.. pp. 36-37). I find Witelo's treatmenł. though. more general and abstract than Alhazen's 
whl' spends mosl of his time on considerations following from pbserving the mixing of 
colors on a disc moved fast. (In this connection. it is perhaps worth mentioning that 
Wilelo changed Alhazen's disc (trochus) for a bali (pila). making. however. the same point 
in much shorler compass and in a broader setting. Neither Alhazen nor Witelo. of course. 
have any inkling Ihat while light is a comppsile "f the various spectral colors:) 


Proposition 7' 


Commentary 
This proposition is copied from Alhazen's II. 72 (DpI. Thes. A/h.. p. 72) formulated 
by Risner in the folIowing words: "Generales visibilis species citius percipiuntur singularibus" 
(ihid. I. 


1
>>>
233 


Proposition 72 


Commenlary 
This proposition is copied, with very slight abridgments. from Alhazen's II. 73 (Opt. 
Thes. Alh.. pp. 72-73) formulated by Risner in the foUowing words: "E visibilibus com- 
munibus alia alijs cilius percipiunlUr" (ibid.. p. 72). 


Proposili()n 73 
I "T he ratio of parl of asphencal surface I() Ihe whole surface of ils sphere musI 
necessarily be equal to Ilhe ratio] of Ihe solid angles. faUing in the same [part of the 
spherical surface] from the center of the sphere. to eight right solid angles" (Wit. Persp.. 
p. 109); for the Latin text see ibid., p. 267 and Opt. Thes. Wit.. p. 33). 
:! This is another way of saying that optics. or perspective (in the medieval sense) 
is more than mathematics. 
.1 There is very little real proof here, but rather an explanation of the Ihing meant 
In the enunciation. 
-ł That of the vilreous humor. 


Commentary 
This proposition is laken over from Alhazen's II. 44 (Opt. Thes. A/h., p. 58). In 
Alhazen the proposition is c1earer and more to the point. It does not involve the gratuitous 
reference to Ihe eight right solid angles "surrounding" any point in space and amounts 
to slating .that although the sentient power cannot comprehend the angle at the center 
as sucho it can imagine it from Ihat part of the surface of the eye (or Ihe g1aeial) covered 
by Ihe hght forms arriving from the object along perpendicular lines. That is aU. Risner's 
enuncialion of Alhazen's proposition reads: "Visus percipit magnitudinem anguli optici e parte 
superficiei visus. in qua formatur rei visibilis forma" (ibid.). 


.......
>>>
LA TIN TEXT AND V ARIANT READINGS 


[Liber secundus] 


5 


Universalibus huius scientie axiomatibus mathematicis premlssls, in hoc 
secundo libro, ut promisimus universali actioni sensibilium formarum quedam 
preambula naturalia premittentes de modo proiectionis luminis per medium 
unius dyafani vel plurium super diversas figuras corporum et de proiectione 
umbrarum et figuratione lucis cadentis per fenestras agredimur tractatum 
ut de hijs sine quibus sermonem visibilium formarum agredi conveniens 
non fuit. prout in processu postmodum patebit. Que vero premittimus ut 
nota sensui sunt ista: 


IDIFFIN ITIONES] 
Corpus luminosum dicitur omne corpus quod est sui luminis diffusivum. 
Corpus dyafanum dicitur corpus per quod lumini patet transitus. Corpus 
umbrosum dicitur corpus per quod lumini non patet transitus. Lux prima 
5 dicitur iIIa que elIkit secundam, sicut lux intrans domum per fenestram 
et iIIuminans domum residuam, in loco cui incidit dicitur prima, in angulis 
vero domus dicitur lux secunda. Lux minima dicitur que si dividi intelligatur 
non habebit amplius actum lucis. Radius dicitur linea luminosa. Linea radialis 
dicitur linea per quam fit diffusio formarum. Linea refracta dicitur linea 
10 cuius partes angulum continent. Pyramis radialis dicitur pyramis cuius basis 
est in superficie corporis suam formam diffundentis et vertex in puncto 


R (p. 61): Vitellonis Filii Thuringorum et Polonorum Opticac Liber secundus. I (p. 38v): Liber 
secundus Perspectivae Vitellionis. C (fol. 25r): Incipit prologus secundi libri magistri Witelionis. De 
modo proiectionis lineis per medium unius dyafoni vel plurium et cetera. 


l [Proemium] 2 mathematicis om. Rl 3 l
 E hab. premisimus 3-4 quedam 
preambula tr. O 6 ante figuratione inser. Rl de I aggrediamur l egrodiamur 
 
l [Diffinitiones] om. CPV.
EO 2 ante Corpus hab. RE l I diITusuum V. 3 ante 
Corpus l hab. RE 2 linter dicitur et corpus inser. Rl omne I ante Corpus 2 hab. R 3 4 ante 
Lux hab. R 4 5 sicud O 6 residua V. residuum 
 7 ante Lux hab. RE 5 8 ante 
Radius hab. RE 6 I linea l ser. et dei. et mg. add. 
 I linea luminosa tr. E I ante Linea 2 hab. 
R 7 et E mg. inser. 7 9 linea 1 0m . E I ante Linea 2 hab. R 8 et E mg. inser. 8 10 ante 
Pyramis hab. R 9 et E mg. inser. 9 11 suam: smea f?] V. I diITundentis: dividentis 
V. I puncto: punctis l
>>>
235 


alterius corporis cuiuscunque. Pyramis illuminationis dicitur illa cuius vertex 
in puncto corporis luminosi et basis in superficie rei iIIuminate. 


IPETJTIONES] 
Petimus autem hec ut per se sensui nota: Lucern compressam fortiorem 
esse luce disgregata. hem lucern fortiorem vehementius iIIuminare et longius 
se ditfundere. hem in absentia luminis umbram fieri. Item in allatione 
luminis umbram deficere. Ttem aliquam umhram in sui termino acui et ' 
ad punctum terminari. hem lucern ad omnem rositionis ditferentiam equalitel 
ditfundi. hem lucern res coloratas pertranseuntem ilIarum coloribus colorari 
ut patet de luce transeunte vitreas fenestras que iIIorum vitrorum coloribus 
informatur secum formas iIIorum colorum super obiecta corpora deferendo. 
Item quod natura nich il frustra agit sicui nec deficit in necessarijs. 10 


[PROPOSJTIONES] 
[propositio] l. Radij quorumcunque luminum et multiplicationes formarum 
secundum rectas lineas protenduntur. 
Hoc quod hic proponitur non demonstratione sed instrumentaliter potest 
declarari; diversitas tamen antiquorum ad hoc probandum pluribus usa est , 
diversis instrumentis. nos vero utimur isto quod hic subscribimus. quod 
regularius huic proposito credimus convenire. Assumatur itaque vas eneum 
rotundum convenienter spissum, ad modum matris astrolabij. cuius fundi 
latitudo sit unius cubiti. vel maiol', et altitudo hore eius sit equalis la- 
titudini duorum digitorum, perpendicularis super basem vasis; et in medio 10 
dorsi huius vasis sit perpendiculariter erectum aliquod corpus plurimum ro- 
tundum columpnare, cuius longitudo sit equalis latitudini trium digitorum, 
latitudo vero eius sit minor uno digito, et ponatur hoc vas secundum sui 
puncta media in tornatorio, et tornetur quousque periferia eius sit extrinsecus 
et intrinsecus vere rotunditatis. et adequentur piane superficies ipsius, et ., 
corpus columpnare quod est in medio dorsi fiat rotundum. 


12 ante Pyramis hab. R 10 et E mg. inser. 10 I post vertex hab. RIPV.EO est et V b ser. et dei. est 


I [Petitiones] om. PV.v"EO I post Petitiones inser. C sunt VII 2 hec: hic V.E I post 
nota hab. R l 3 ante Hem hab. R 2 I lungius pv. 4 ante Hem l hab. R 3 I ante Item 2 
hab. R 4 I 5 ante Item hab. R 5 6 ad l: a [1] V. I ante Item hab. R 6 7 ante Item hab. 
R 7 8 patuit E 9 informantur lCv"O I formas om. E et scr. et dei. v" I colorum: 
colores CpV.VbO 10 ante Item hab. R 8 I sicud E 


I [Propositiones]: Theoremata Rom. OICPV.v"E 2 Propositio [1] om. RPV.v"EO et l hab. 
Theorema 3 post protenduntur add. RAIhazen 2 n 7 5 post pluribus inser. RlC et 6 
diverses C 5-6 diversis usa est instrumentis RIC I vero: ergo V b I hic om. PV.EO 8 
ad om. v" I astrolabij co. est ex astrolabiis in v" 9 maiori v" 10 basi m R 13 latitudo: 
longitudo O I vero rep. V. I minor: maior v" [co. est ex minor] 14 post media scr. et dei. 
V b ut tornatorio co. est ex tornatoio in E 14-15 intrinsecus et extrinsecus RI 15 
intrinsecus: extrinsecus E I rotunda E I ipsi E
>>>
236 


Signentur itaque in interiori superficie fundi huius vasis due dyametri 
orthogonaliter se secantes. que sint AB et CD [Fig. I]. Palam quoniam 
iIIe dyametri transeunt per centrum circuli fundi quod sit E. Deinde signetur 
20 in basi hore istius vasis. qui est circulus ACBD, in distantia extremitatis 
alterius dyametrorum productarum. ut dyametri AB. secundum latitudinem 
unius digiti punctum. quod sit F; et ex hoc puncto tertia trahatur dyameter 
per centrum E. que FG. Et a duobus terminis istius dyametri FG ducantur 
due linee in intrinseca superficie hore vasis. que necessario erunt perpen- 
2
 diculares super superficiem fund i lamine. ideo quod superficies hore in qua 
perpendiculares iste producuntur sunt erecte super superficiem fundi. ut patet 
supra. lite quoque perpendiculares sint FH et GK: et in altera istarum 
linearum. lit in FH. si!!nen1ur 1ria punda equedistantia secundum quantlta1em 
medietatis grani ordei. que sint L. M, N. quorum primum. quod est L. 
30 sil propinquius basi vasis et ipsi puncto F, a quo distet per quantitatem 
medietatis grani ordei. Et deinde reducatur vas ad tornatorium. et signentur 
in ipso tres circuli equedistantes, transeuntes per ilIa tria puncta L, M. N, 
qui circuli divident lineam GK isti divise linee, que est FH. oppositam, 
proportionaliter prius divise per 17 am XI'. Sintque divisiones linee GK puncta 
35 O. P, Q: et fient in unoquoque istorum trium circulorum duo puncta 
opposita que sunt extremitates alicuius dyametri iIIorum circulorum. ut 
puncto divisionis linee FH. quod est punctum L. opponitur in linea GK 
punctum O. et fit linea LO dyameter circuli equedistantis circulo ABCD. 
Et similiter linea MP fit dyameter alterius circuli. et linea NQ fit dyameter 
40 circuli tertij. Dividatur itaque medius istorum circulorum in partes 360, 
et si possibile fuerit per minuta. Deinde, super lineam FH alteram duarum 
linearum perpendicularium. que FH et GK. punctum medium. quod est M, 
perforetur foramen rotundum, et sit medietas dyametri foraminis secundum 
quantitatem distantie circulorum. que est linea ML. Attinget ergo foramen 
45 ilIud ambos circulos extremos, et medius circulorum dividet circulum foraminis 
per equalia, quoniam transit centrum foraminis. 
Deinde accipiatur lamina enea pIana aliquantulum spissa (Fig. I AJ. et 
sit eius spissitudo sicut hore ipsius instrumenti. et eius longitudo sit duorum 


17 itaque: utique E I in om. v.. I vasis om. v.. et E hab. vas I duo l 18 post sint scr. et 
dei. E AC 19 inter iIIe et dyametri inser. E due 20 que RE I ABCD R ACD l 21 
productorum v.. 22 hoc om. E 23 post que hab. Rl sit er v" ser. et dei. sit 24 in om. 
pv" I post superficie hab. v.. istius I vasis:basis E 25 fundi om. v.. 26 producantur 
l 28 equidistantia RlC 29 hordei Rl 30 sit: sint P sic v" I basis V b 31 medietati 
lV b I hordei Rl 32 aequidistantes R linter equidistantes et transeuntes inser. v.. 
et I puncta co. est ex punctum in v". 33 istae l I opposita PEO 34 inter 17 et Xli hab. 
R p I divisionis CPv..EO 35 in rep. R I post in v" ser. et dei. uno I unoquoque co. est 
mg. ex quoque in v" I duo om. E 36 ut: in Iv.. 37 puncto l 38 fit: sit E I dyametri 
v.. I aequidistantis R I ACBD RPv..EO 39 fit l: sit E 39-40 fit dyameter alte- 
rius... Dividatur itaque medius istorum circulorum om. v.. 39 NQ: NK PE 40 partes 360 tr. 
Rlv" 42 post que add. RlV b sunt 43 inter peńoretur et foramen inser. E et sit I foramini 
PO 46 post transit hab. Rlv" per 47 accipitur Iv" 48 ore RE 


--
>>>
237 


digitorum. sicut et hora vasis. et eius latitudo sit prope hoc, et sit equedistantium 
superficierum. Planeturque adeo ut communis sectio superficierum sue latitudinis so 
et spissitudinis sit linea recta, que sit RS. dividaturque in duo equalia 
per 10 primi; et ab eius medio puncto, quod sit T. ducatur linea recta 
perpendiculariter super ipsam lineam RS in superficie latitudinis, que sit 
TY. Et hec. ut patet ex premissis et per 29 8m li. necessario equedistabit 
i.l11l'"'abus lineis lonl!itudinis. dividens surerflciem tabule per eQualia. Et in ss 
hac linea perpendiculari. que est TY. aparte linee RS. cui superstat. in- 
cipiendo signentur tria puncta equaliter distantia ab invicem secundum quanti- 
tatem medietatis grani ordei. que sint X. Y, Z. et a medio istorum 
punctorum. quod est Y. perforetur lamina foramine rotundo; sit que foraminis 
periferia ad alia duo puncta pertingat. eritque hoc foramen equale foramini 60 
LMN prius facto in hora vasis. 
Deinde in duo equalia dividatur semidyameter fundi vasis, que est FE, 
cuius extremitati in hora vasis superstat una linearum perpendicularium 
que est FH. sitque punctus divisionis T. Et ab hoc puncto medio T. 
ducatur linea perpendicularis super eandem dyametrum. que sit RTS. Deinde 6S 
ponatur basis parve lamine super hanc lineam. donec linea que est differentia 
commutris latitudinis et profunditatis lamine. que est RTS. superponatur 
linee isti perpendiculari. ducte super dyametrum. que similiter est RTS. 
Sitque punctus dividens lineam lamine" que est communis differentia super- 
ficierum latitudinis et profunditatis, qui est punctus T, superpositus puncto 70 
T signato in linea FE. semidyametro vasis. Deinde, consolidetur parva 
lamina fundo vasis. Erit quoque tunc foramen XYZ, quod est in parva 
lamina. que est RYS. directe oppositum foramini LMN. quod est in vasis 
hora. Et erit linea recta, que est MY. copulans centra istorum foraminum 
in superficie circuli medij trium circulorum prius signatorum, cuius dyameter 7S 
est linea MP. Eritque linea MY equedistans dyametro vasis. que est FE. 
Deinde. resecetur ex hora vasis pars interiacens duas dyametros orthogonaliter 
se secantes. que sit pars quarta proxime sequens quartam iIIam. in qua est 
foramen, cui foramen lamine opponitur, et est in circulo ABCD, correspondens 
"ITui AD. et planetur locus sectionis. do nec fiat una superficies cum superficie 80 


49 et I UIII. C lora Rl E I aequidistantium RE 50 ut: inser. V b I post ut sa. et de/. nec 
v" I latitudinis: longitudinis E 51 RS: es 1 RG C RESv" co. est ex ES 52 post 10 hab. 
R p 53 RS: RG lC KS v" co. est ex RG 54 et per om. C I post 29 hab. 
R p I aequidistabit RE 55 amhabus om. E I linea EO I post lonjotitudinis hab. E utrique 
56 post TV hab. n;,v" et I RS: KS v" co. est ex RS . 58 hordei Rl 59 quod: que 
l V h I sicque Rl 60 pertinget R ()I ora Rl E 62 fundi vasis tr. Rl V b 63 extreminati 
f?] co. est ex extremitati in V b I Ofa RE 64 medio om. l 64-65 medio T ducatur mg. 
inser. v" 65 KTS lCv" 65-68 que sit RTS. Deinde ponatur..,perpendiculari ducte super 
dyametrum om. V. 66 basis: vasis V b 67 supponatur R supponitur l 70 que PEO 71 
consolidatur l consolidetur co. est ex consolit in V b 73 lamina:linea O I foramen LMA 
l I LMN co. est ex LMA in v" I quod: que lV.v"E 740ra RlE I que est om. l 75 
trium om. V. 76 equidistans RE 77 Deinde co. est ex dinde in v" lora Rl I duos 
10 78 se om. E I sequens: secans inser. mg. et dei. sequens V b 79 foramenz: foramini 
V. I ABeD v"
>>>
238 


fundi vasls. Et ducta quarta circuli, que sit AD. secundum quantitatem 
circuli hore dividatur per 90 gradus. et dividantur gradus in minuta. Et 
isti vasi taliter informato et figurato. deinceps damus nomen instrumenti. 
Deinde acciriatur regula enea quadran!!ula. l'uius longitudo sit unius cubiti, 
85 et sinI quatuor superficies ipsam continentes. la1itudinis duorum digitorum, 
et adequentur supertleics eius. donec tlant equales rectangule. Deinde. in medio 
puneto longitudinis regule. et in medio alieuius illarum superficierum. fiat 
foramen rotundum, cuius amplitudo sit capax corporis quod est in dorso 
instrumenti. Et sit foramen perpendiculare super superfkiem regule. transiens 
90 ad aliam partem superficiei opposite, fiatque taliter quod revolvatur in ipso 
instrumentum. non levi revolutione, ponaturque instrumentum super regulam 
immisso corpore. quod est in eius dorso. in foramen regule. do nec super- 
fieies instrumenti cOnlungatur surerficiei repule : eritque longitudo regule equalis 
dyametro instrumenti. Fiantque due pinnule (Fig. l B] latitudinis et spissitudinis 
95 regule. sed longitudinis plusquam unius digiti, que consolidentur super ex- 
tremitates regule. ita quod ipsorum preminentia super extremitates regule 
sit unius digiti. vel parum plus, vel minus, et ille pinnule consolidate sint 
super superficiem regule non perforatam. Et quia latitudo regule est duorum 
digitorum. altitudo vero corporis in dorso instrumenti est trium digitorum, 
100 ille tertius digitus quo corpus preminet regule perforetur, sicut in astrolabio, 
et immitatur cuspis continens regulam cum instrumento. 
Deinde as.mmatur alia regula enea cuius latitudo sit dupla sue spissitu- 
dini. srissitudo vero sit equalis dyametro foraminis quod est in hora in- 
strumenti. et longitudo eius sit equalis medietati cubiti; fiatque hec regula 
105 recta et vera. et eius superficies equales et equedistantes. Deinde secetur 
illa regula in una sui parte oblique. donec finis longitudinis eius contineat 
cum termino latitudinis angulum acutum, ut facilius valeat moveri. In parte 
vero altera sit finis latitudinis eius perpendicularis super finem longitudinis. 
Deinde dividatur linea eius latitudinis in duo equalia. et a puncto sec- 
110 tionis ducatur linea equedistans lineis longitudinis. que erit rerrendil'lllaris 
super lineam latitudinis, per 29"m I'. Cum itaque hec regula fuerit super- 
posita superficiei fundi instrumenti. taliter ut eius spissitudo sit orthogonaliter 
erecta super fundum instrumenti. et superficies latitudinis applicetur super- 


82 orae R I per: in R 83 post deinceps rep. et dei. deinceps P 84 accipitur l linter 
cuius et longitudo scr. et dei. O latitudo 8S continentis J 86 adequatur l adequetur 
CPv.VbEO I eius om. E 90 superficiei: superficie v" 92 in misso V.V"P 93 coniun- 
gatur: transmutatur f?] V. I regule om. V b I eritque longitudo mg. inser. v" 96 ipsarum 
E I praeeminentia RIO permanentia C 97 parvem v" I iIIe pinnule tr. Rlv.Vb I con- 
solidatem v" 98 peńoranie f?] v. 99 in dorso instrumenti; instrumenti in dorso 
PEO 100 quo: cuius V. t praeeminet R pinni et l I post preminet rep. et dei. v" 
preminet I sicud O I astrilabio CP astralabio O 101 in mittatur v.v" 102 sit: est 
v" 103 foraminis: formans v" lora RlE 104 meditati V b lOS aequidistantes 
Rl 106 continuat l contineat co. est ex continuat in V b 107 termino: tertio l 109 eius 
latitudinis tr. E 110 aequidistans Rl HI post 29 hab. R p 


\ 


1liiio....-
>>>
239 


ficiei fundi IpSIUS instrumenti. tunc erit elUs superior superficies In super- 
ficie circuli medij trium circulorum in hora instrument i protractorum. cuius 115 
dyameter esl linea MP. ideo quia spissitudo regule est equalis dyametro 
foraminis, et dyameter foraminis que est NL est equalis linee perpendiculari 
exeunti a centro foraminis. super superficiem planam instrumenti. que est 
linea MF, cui adiacet linea spissitudinis regule equalis illi. Cum itaque 
propositam conc1usionem experimentaliter placuerit dec1arare, opponatur in- 120 
strumentum premissum corpori solari. vel alteri corpori luminoso cuicunque. 
vel etiam candele. et applicetur centrum foraminis instrument i quod est 
punctllm M. opposito corporis IlIminosi. secundllm qllod melius fllerit 
possibile. transibitque radius luminosus centra amborum oppositorum fora- 
minum unius in hora instrumenti et alte
ius in tabella perforata existentium. 125 
que sunt M et Y; describeturque circulus luminosus in parte hore in- 
strumenti opposita foramini LMN directe per dyametrum MP. eritque centrum 
illius circuli luminosi in puncto P, quod faciliter patere potest si a puncto 
p ad utramque partem periferie circuli medij illorum trium circulorum, 
. secundum gradus et minuta divisi. partes interiacentes luminosi circuli periferiam 130 
computentur: invenientur enim equales numer i hinc inde. Est ergo punctum 
p centrum iIIius circuli luminosi. Linea itaque MP. secundum qua incidit 
radills transiens reI' centrum clrculi utriusque foraminis et per centrum circuli 
luminosi. tota est in superficie pIana circuli medij illorum trium circulorum. 
et est dyameter illius circuli. Est ergo linea recta. Et si aliquod corpus 135 
forti colore medio coloratum. ut viride vel rubeum. ponatur extra foramen 
hore instrumenti. ita ut lumen solis vel alterius corporis transiens per iłłud 
corpus. postmodum incidat foraminibus instrumenti. et transeat per iłła. 
tunc. ut patuit per ultimam premissarum suppositionum. circa punctum P 
in hora instrumenti describetur circulus luminis co lorat i iłło colore. Color 140 
ergo mixtim cum lumine diffundit formam suam secundum Iineas rectas. 
sicut et ipsum lumen. Patet ergo quoniam radij quorumcunque luminum 
et multiplicationes formarum secundum lineas rectas protenduntur. Et hoc 
est propositum. 


[propositioJl. Lumen non impeditum. per totum sibi proportionatum 
medium in instant i necessarium est deferri. 
Sit linea proportionata delationi IlIminis fortioris. lit est in lumine solis 
114 ipsius: ipsuis C I erit om. lV b I post superior add. Y" pars 115 circuli medij tr. 
v" lora RIE 118 post exeunti ser. et dei. v" cuius I a centro mg. inser. v" I planam 
instrumenti tr. v" 123 oppositam v" I post opposito hab. CPY"EO in 125 ora 
RIE I existentium: exeuntia 'v" 126 in: ex lV b I ore RE 127 opposito l 129 
utranque RlC periferie circuli tr. E 130 post divisi hab. Y" in 131 equalis nuneri C 132 
luminose Y" I quam E 134 pIana circuli tr. V w 137 orae Rl E 138 post modia [1] 
v" 139 ultimam: 7 R I premissarum co. est ex premissorum in V b 140 ora 
RlE I colore: corpore v" 141 mixtum lCy"V b 142 sicud E I quoniam: quod 
RIV b I quorumcum E 143-144 Et hoc est: patet ergo E 
1 Propositio [2] om. RlPYwVbEO I 2 mg. hab. EO 3 post fortioris ser. et dei. V b voeabulum 
illectum
>>>
240 


mundi dyameter. que linea sit ABCD. et sit corpus fortiter luminosum in 
5 pum:1O A IFig. 11. Si ergo dil:atur quod lumen in tempore defertur per 
lineam ABCD et non in instanti, ergo in parte illius temporis defertur 
per lineam AB. et in minimo tempore sensibili feretur per minirnam 
partem sensibilem linee AB; quoniam si in tempore sensibili feretur per 
spatium insensibile, contingeret spatium sensibile ex insensibilibus componi, 
10 sicut tempus mensurans l?] motum per iIIud spatium compositum [?] ex 
temporibus sensibilibus ut suis partibus; feretur ergo in tempore minimo 
sensibili per minimum spatium sensibile. Sed in eodem tempore feretur 
per idem spatium forma luminosi corporis debilioris illo corpore fortiori 
luminoso, quoniam minimo spatio sensibili non est aliquod spatium sensibile 
15 minus, etiam minimo tempore sensibili non est aliquod sensibile tempus 
minus. Equalis ergo virtutis erunt lumen fortius et debilius. quod est 
impossibile quoniam implicantur contradictoria. Est ergo impossibile lumen 
in tempore per proportionatum sibi medium diffundi: Necesse est ergo 
quod diffusio fiat in instanti. quod est propositum. Ad hoc etiam deser- 
20 viunt alique naturales rationes Arist)tilis quas. qui voluerit. percurrat. quia 
suftkit nobis hoc unum inconveniens secutum. 


[Propositio]3. Omnis linea qua pervcnit lux a corpore luminoso ad corpus 
oppositum est linea naturalis sensibilis. latitudinem quandam habens. in 
qua est linea mathematica ymaginahiliter assumenda. 
Lux en im non procedit nisi a corpore. quoniam non est nisi in corpore. 
5 Unde patet, quia in minima luce que sumi potest est latitudo; quoniam 
minirnam lucern dicimus que. si dividatur. non habet amplius actum lucis, 
quia non erit visibilis sed utraque pars extinguetur quia neutra pars eius 
erit lux neque apparebit sensui. Est ergo in linea radiali secundum quam 
fit diffusio luminis aliqua latitudo. propter quam inest ei sensibilitas, et in 
10 medio illius linee est linea mathematica ymaginahilis. cui omnes alie mathe- 


4 mundi dyameter tr. E I linea sit: est linea R tr. nI" om. linea Y" S dicatur co. est ex ducatur 
in JI" I deferetur C 8 post AB hab. P mg. alm. certam notam 8-12 quoniam si in tempore 
sensibili.. .per minimum spatium sensibile om. PY"EO 10 mensuratum RJ mensuram f?] 
C I motum mg. inser. JI" I per: post Rl I componitur JI" I I post 'ut scr. et dei. JI" 
in I ut: in Rl 13 idem: i\1ud JI" I ante illo ser. l minus et JI" scr. et dei. minus I corpori 
PEO IS etiam: et CPY"JI"EO 17 implicantur: multiplicantur O 18 per om. Y" I sibi 
mg. insa. JI" 19-20 deserviunt alique tr. Rl 20 Aristotelis Rl aristoteles Papostolicis f?] 
Y" I rationes aristotilis mg. inser. JI" I post naturales ser. et dei. V h 20 P f?] intentiones I quas 
om. E I procurat V h 21 hic V. 
I Propositio [3]" mg. hab. C 2 quadam P 3 linca om. Y" I post assumenda add. 
RAIhazen 16 n 4. 4 precedit JI" I quoniam: quia E S quia: quod Y"JI"E I in: ex 
JI" I sum i potest tr. E I est om. Y" 6 post lucem scr. et dei. JI" diITerentis f?] dicimus mg. 
inser. V h 7 utroque Y" I utraque co. est ex extraque f?] in JI" I neutra co. est ex ventra in 
JI" I eius: cuius JI" 8 neque: nec Y" 9 propter: per Y" I propter quam: qua propter 
E 10 linea: linee f?] v" I ymaginalis Iv" 10-11 mathcmatice linee tr. v"
>>>
241 


matice linee in illa linea naturali equedistantes erunt. Et quoniam lux minima 
procedit ad minimam corporis parlem quam lux occupare polest. necesse 
est quod processus eius sit secundum lineam mathematicam que est in 
medio linee sensibilis et secundum lineas extremas equedistantes linee medie. 
Neque cadit lux minima in punctum mathematicum corporis oppositi. sed 15 
in punctum sensibilem correspondentem omnibus punctis mathematicis indi- 
visibilibus ad quos linee mathematice ipsius linee sensibilis possunt terminari. 
Et ob hoc utemur in demonstrandis passionibus lucis figuratione linearum 
mathematicarum in processu. 


IPropositio]4. Corpora dyafana sunt apta penetrationi luminis et coloris 
sine essentiali sui transmutatione. 
Hec enim corpora proprietatem habent ut non prohibeant formas lucis 
et coloris se penetrare. Attamen non immutantur a lucibus vel coloribus. 
nec alterantur ab eis alteratione fixa. sed fit per illa ditfusio lucis et coloris 5 
secundum lineas rectas per primam huius; quarum alique su nt equedistanles. 
alique secantes se. et qucdam diversi situs. Et omnium istarum linearum 
dislinctio fit per dislinclum situm corporis luminosi a quo fit ditfusio illius 
lucis vel coloris. Forme itaque lucis et coloris extense a corporibus diversis 
in eodem dyafano extenduntur. quelibet ipsarum. secundum lineam rectam 10 
et pertranseunt ad corpora opposita. Corpus vero dyafanum non tingilur 
per luces vel colores, sed solum penelratur; neque en im talia corpora propter 
luces et colores perdunt suas formas. neque tinguntur per luces vel colores 
tinctura fixa. quia in eis non remanent forme lucis vel coloris post recessum 
lucis vel coloris ab ipsorum oppositione. Non ergo transmutantur illa corpora 15 
essentiali transmutatione per luces vel colores. quod est propositum. 


/Propositio]S. Luces et colores in corporibus dyafanis non admiscentur 
adinvicem sed penetrant distincti. 
Huius rei exprimentaliter declarande causa. Ponantur in loco aliquo candele 
mulle localiter distincte. et sint omnes opposite uni foramini pertranseunti 


II post linea ser. et dei. V b vocabulum iIIectum I post linea hab. E lata I equidistantes 
RE 12 corporis partem tr. v.. 13 est i om. E 14 aequidistantes R I Iinee medie Ir. 
v.. 16 sensibile Pv..EO I punctis: praedictis l co. est mg. ex predictis f?] in 
 17 ipsi PEO 
co. est mg. ex iIIius in V b I possunt rep. et dei. 
 18 utemur co. est ex utimur in 
 I post 
utemur ser. et dei. 
 de 
I Propositio [4] C I penetrationi co. est ex penetrationem in 
 2 essentiali om. 
E I post transmutatione add. RAIhazen 28 n l. 4 ante non hab. v.. et I mutantur Rl in 
mutantur v.. V b S nec: neque CPV.. 
EO I sed: si C 6 aequidistantes R 8 fit i: sit 

 9 corporibus: coloribus l mg. co. est ex coloribus in 
 II pertransibunt l pertransiunt 

 I opposita co. est mg. ex opposta f?] in 
 I intingitur 
 12 colores vel luces 
 13 
et: vel E Iluces vel colores: colores vel luces v.. I vel: et Rl
 14 post tinctura ser. et dei. 

 tura I quia: que V b I non om. 
 15 ipsarum l 16 vel: et Rl co. est ex et in 

 I quod: et hoc E 
I Propositio [S] C 2 post distincti add. RAIhazen 29 n l. 3 ante causa ser. et dei. 
 
exempli 


16 - Wiielonis Perspect;Vk.
>>>
242 


s ad locum obscurum. et opponatur foramini in loco obscuro aliquod corpus 
non dyafanum. Luces itaque candelarum apparent super illud corpus distincte 
secundum numeru m candelarum. et quelibet iIlarum apparet opposita uni 
candele. secundum Iineam rectam transeuntem per foramen et per medium 
luminis candele. Et si cooperiatur una candela. destruetur unum lumen 
10 oppositum iIli candele tantum, et discooperta candela revertitur lumen. Palam 
itaque quod luces in medio foraminis, ubi se intersecant omnes vel plures 
in puncto uno, non admiscentur in eodem puncto, sed sunt distincte per 
sui ipsarum essentias: et ob hoc, cum ulterius protenduntur, tunc secundum 
locorum quibus incidunt, diversitatem localiter distinguuntur. Et quoniam 
H lux res coloratas pertransiens,. iIlarum coloribus coloratur, ut suppositum 
est, palam, si lumen penetrat distinctum, et colores qui feruntur cum lumine 
penetrabunt distincti. Patet ergo propositum. 


[Propositio]6. Proportio virtutis totius corporis luminosi ad totum corpus 
luminosum est sicut determinate partis virtutis ad partem corporis sibi 
proportionalem. 
Sitcorpus aliquod luminosum AB Wig. 3]. Dico quod proportio virtutis 
s totius corporis AB ad totum corpus AB est sicut proportio partis virtutis, 
que est A. ad partem corporis, que est A. Si enim non est istorum 
eadem proportio, aut ergo maior, aut minor. Sit primum maior, et sit 
virtus totius corporis AB signata per lineam GD; sitque G virtus partis 
corporis, que est A et D sit virtus partis corporis que est B. Que est 
10 ergo proportio G ad A eadem est proportio D ad B; ergo, per 18 8m Si, 
erit coniunctim GD ad AB sicut G ad A. Si ergo proportio G ad 
A est maior proportione GD ad AB, erit quoque maior proportio GD 
ad AB quam GD ad AB. quod est impossibile. Non en im poterunt esse 
unius rei ad aliam due proportiones, quarum una sit maior' alia. Idem 
H quoque accidit impossibile danti, quod minor sit proportio G, partis virtutis, 
ad partem corporis, que est A, quam GD virtutis ad AB corpus. Si enim 
minor est proportio G ad A quam GD ad AB. et que est G ad A 


6 candelarum co. est ex candelalerum in v" 8 transeunte Iv" 9 luminis: suis v.. I post 
luminis hab. l lumen 10 ille v" I discooperata V u I revertetut PE revertemur O I post 
revertetur add. E unum 12 non: mimine E I distincti v.. 12 per inser. v" .13 sui: 
suarum [1] v.. et om. v.. I huc Rl 14 distinguntur v" 16 distinctum co. est ex 
distnctum in v" I que V b . 
l Propositio [6] C I 6 mg. hab. E 2 est rep. E I determinate co. est ex determinante in 
v" I virtutis co. est ex virtis in P 3 proportionabilem l 5 corporis om. v.. l est: et 
v" 6 A) om. v.. I non: istorum v.. I est istorum: non v.. 7 aut minor om. E' 8-9 
partis corporis tr. v" 9 partis om. V u 10 eandem v" I post proportio ser. et dei. 
O partis I proportio om. lCVuv" I post 18 hab. R p 11 coniunctum [1] v" I erit om. 
E I ad) rep. et dei. C I sicud O 12 maior): nHnor O 13 quam GD ad AB mg. inser. 
v" I non: tunc [1] v" I potuerint l potuerunt v" 14 post aliam hab. V b rei 15 sit in
er. 
v" 16 quam: quae Iv"
>>>
243 


eadem est D ad B. per tertiam primi huius, erit ergo, per 18 am 51, coniun- 
ctim proportio totius virtutis, que est GD, ad corpus AB minor proportione 
GD ad AB, quod est impossibile. Est ergo proportio G ad A sicut GD 20 
ad AB. Et hoc est propositum. Et est universale, nisi forte aliquid conferat 
unio virtuti. quoniam virtus unita semper est fortior se ipsa divisa. Unde 
tenet nostra demonstratio quando partes non divise a toto agunt in ipso 
toto non actualiter distincte; cum enim distincte sunt a toto, tunc non 2S 
sunt partes, quia nomen partis, id quod dicit signat potentiam non actum. 
Et de hoc completius in alijs sermo fuit. 


[Propositio]7. Orimis corporis luminosi intransmutabilis secundum formam 
vel situm in corpus aliud equale et omogeneum eidem immediate vel per 
medium uniforme oppositum est semper actio equalis et uniformis. 
Sit enim dati alicuius corporis luminosi virtus [Fig. 4] A, et sit corpus 
equale et omogeneum eidem oppositum BG; et sit impressio virtutis A , 
in BG corpus signata per C. Dico quod Asemper imprimit in corpus 
BG impressionem C, que est semper equalis sibi ipsi et uniformis. Si en im 
detur quod A quandoque imprimit in BG impressionem que est C, quandoque 
vero non imprimit C, sed aliud maius vel minus ipsa C, ut D, tunc, 
cum corpus obiectum sit semper omogeneum et uniforme, erit diversitas 10 
impressionis non a corpore BG patiente, sed a virtute A diversificata in 
se; hoc autem est impossibile. cum corpus luminosum positum sit intransmu- 
tab ile secundum formam et situm. Est ergo ipsius actio semper equalis 
et uniformis in corpus eidem immediate vel per medium uniforme oppositum. 15 
Et hoc est propositum. 


[Propositio]8. Necesse est terminum longitudinis cuiuslibet umbre radiu m 
luminosum esse. 
Quod hic proponitur satis patet per premissa principia. Quoniam en im 
per tertiam suppositionem solum in absentia luminis fit umbra, et per 
quartam suppositionem in allatione luminis umbra deficit. tunc necessario , 


18 18 rep. v" om. E I post 18 hab. R p 18/19 coniuncti v" 19 post virtutis ser. et dei. 
C coniunctim I AB: ad [1] b v" 20 sicud O 21 est universale tr. E I nisi forte rep. 
O I aliquid conferat tr. E 22 divisa: disspersa E 23 in ipso tr. E 25 distincte sunt: 
sunt directe v.. I enim distincte mg. inser. v" 26 post dicit add. R philosophus 27 
completus R conplectius v..v" completus [1] E I fuit: fuerit Iv" 
I Propositio [7] C I 7 mg. hab. O I Omnio E 2 homogeneum R omogenium 
EO I immediate: in mei est V. 3 actio equalis tr. v" 4 A co. est ex d [1] in E 5 
homogeneum R omogenium O I inpressio CPV..V" 6 corpus 1 : corpora l 7 in inser. 
v" I post in hab. R corpus 8 post quandoque l ser. et dei. v" in ipsum 9 vero mg. inser. 
v" I inprimit mg. inser. v" I alius V b 10 semper om. Rl I homogeneum R omogenium 
O I post diversitas ser. et dei. v" inpressis [1] et mg. inser. inpressionis II diversificata co. est 
f!X diversificate in v" 12 positum om. v.. 14 situm: actum v.. J equaJe v" 
l Propositio [8] C I 8 mg. hab. EO 3 premissam CPV.v" 4 abstentia v" I post 
umbra scr. et dei. v" in tanto 5 in allatione luminis umbra mg. inser. v"
>>>
244 


oportet in tanto spatio umbram causan In quanto lumen deficit; et ubi 
lumen accedit. ihi umhra deficil. Terminus er!!n Ic)ngitudinis cuiuslibet umbre 
cum sil linea. palet quod oportet ut illa linea sit lumim)sa. Est ergo illa 
linea radius lumim)sus per diffinitionem radij. Palet ergo pwpositum. 


IPwpositio]9. A termims equedistantium aItitudinum corporis luminosi 
alt inris et corporis umbrosi bassioris producte linee concurrentes sunt suis 
aItitudinibus pwrortionales; ex quo patet quod eadem altiludo .corporis 
umbrosi ex lumine bassiori long:iorem proicit umbram quam ex lumine 

 alt iori. 
Sit altitudo corporis umbrosi cuiuscumque linea AB I Fig. 5]. et sit 
aItitudo alia illi equedistans ipsius corporis luminosi. que DE sit. Sitque 
linea DE malor quam linea AB. Producanturque linee EB et DA que 
protracte concurren1 ad aliquam partem in puncto G. per secundam primi 
10 huius. Dico quod erit proportio linee GB ad lineam GE. et linee GA 
ad lineam GD. sicut linee AB ad lineam DE. Quia enim linea BA equedistat 
linee DE ex ypothesi. palam ergo. per 29 am Ii. quoniam angulus GBA 
est equalis angulc) GED. et angulus GAR equahs angulo GDE. Angulus 
quoque BGA communis est amh(lbus trigonis DGE et AGB. Ergo. per 
H 4 am 6' est prc'pNtic.' lineee GB ad lineam GE sicut linee BA ad lineam 
ED. Ergo. per 5,'m rrimi ł uius. erit econtrarin proportio linee GE dd 
lineam BG sieuI linee ED ad lineam AB. Palam ergo est propositum. 
quoniam endem mc'dc) demonstrari pc'test de lineis GA et GD. Et ex 
hoc patet. quc'mam eadem aItitudo corporis umbrosi ex lumine bassiori 
20 longiorem rr0Jcit umhram quam ex lumine aItiori. Esto enim quod aliquod 
corpus lummosum sit in puncto H. Cadatque radius HA in punctum linee 
ECI. quod sit K. Entque. per premissum modum proportio EK ad BK 
sieuI HE ad AR. Sed. per 8 am vi. proportio HE ad AB est minor quam 
DE ad AR. Er,.w. per 'lam 5 i . proportio EK ad BK est minor quam 
H FG ad RG. Multum ergc) excrevit umhra BK respectu umbre BG. ut 


6 oportct: apparet f?] E I post oportet ser. et dei. v" est per 4 I in tanto inser. v" I quarto 
v" 7 accedit co. est ex accidit in V b I ibi co. est ex in in v" 8 oportet: optet P I ut scr. et 
dei. v" 9 pust per add. R 6 I definitionem R I radij om. R 
I Propositio [9] C I 9 mg. hab. O I A om. E I equidistantium REO 2 et ser. et dei. 
v" 3 eadem co. est ex adem in v" 7 aequidistans R I ipsi v" I corpori v" I post que 
scr. et dei. v" sit I DE sit tr. Rl I sit om. Cv.EO 9 aliquam: aliam v" I secundam 
primi: 16t l R 10 lineam om. E I linee 2 : linea IV. co. est ex linea in v" )) ad lineam GD 
om. V b I GD:GB V. I aequidistat R 12 post 29 hab. R p 13 post GED scr. et dei. V b et 
an I angulus l : angula V. I post GAB add. V.E est 14 communis est tr. E 14/15 per 4 am 
6 i est proportio linee GB ad mg. inser. v" 15 post 4 am hab. R p I lineam l mg. inser. 
v" I post lineam l add. v" et I sicud O 16 post 5 am hab. R t 17 est om. V. 19 
bassioris v" I post bassiori ser. et dei. v" longitudinem 20 longiorem mg. inser. v" 21 
Cadatque co. est ex caditque in v" 23 AB I co. est ex B in V. I post 8 am hab. R p I post 
proportio hab. lE linee et v" ser. et dei. linee I HE 2 om. EO 24 post AB add. R sed 
proportio de ad ab est, sicut proportio eg ad bg, ut patuit: I post 11 am add. R p I est minor 
rep. et dei. C
>>>
24S 


patet per 10 am S' et per 4 am l' huius. Et ex hoc accidit quod umbre 
lunares semper sunt 1{'Ogiores quam umbre solares: et ita est de alijs eor- 
porihus luminosis altioribus et bassioribus quibuscunque. Patet ergo propo- 
sił UIll. 
IPn)positio) 'O. Omnem radium luminosum per medium unius dyafani 
trans verticem alicuius corporis umbrosi protensum necesse est esse lineam 
unam rectam. 
Remaneat to1alis dispnsitin proxime precedentis. et sit punctus G finis 
umbre !Fig. 6]. Quia itaque. ut patet per 8 huius. cuiuslibet umbre terminus 5 
est radius luminosus. diw qund ilłe radius terminans umbram est linea 
reeta. ut est in propnsita figura linea DAG. Si enim non est recta linea 
DAG. tunc. eum DA linea sit recta per primam huius. ideo qund nulłam 
habet causam impedimenti in pwgressu. et linea AG similiter est recta 
propter idem. cnniunguntur ergo linee GA et DA angulariter in puncto A. 10 
Subtendatur ergo ilłi anguln. utcunque eon1ingat. basis a punctis D et G: 
et sil linea DUG recta. et pwtrahatur vel abseida1ur linea A V. Trignnum 
itaque EDBG dividitur per lineam BV equedistantem linee ED. Ergo. per 
29 am 1 i emnt trignni EDG et BVG equianguli. Ergn. per 4 i1m 6 i . erit 
pwportio linee GE ad lineam GB sieut linee ED ad lineam BV. Sed. 15 
per prelllissam pro\imam. est pn'p()rti() linee GE ad lineam RG sicut 
linee DE ad lineam HA. Est er
o. per II "m S'. eadem rrnpnrtin linee DE 
ad amhas lineas BV et BA. q uod est c{)ntra 8"'" S' et impossibile: ad mi- 
norem enim maior et ad mainrem minor est pwportio. vel sequetur maiorem 
lineam esse equalem minnri. per 9 am si. Hoc autem est impossible. (p{)rtet 20 
ergo ul radius DAG sit linea una reeta. quod est propositum. 


IPwposlti())' '. Omnia cnrpora densa non dyafana in parłem Iuminoso 
corpori aversam umbram proieiunt usque ad incidentiam radij per rei .knse 
vert icem product i. 


26 post lO'"m hab. R p I post 4 0m hab. R p 27 quam om. E I umbre solares: umbris 
solaribus E I est om. lCV w inser. V b 28 post bassioribus scr. et dei. J-;, 
cuiusque I quibuscunque mg. inser. J-;, 28/29 post propositum add. E istius 
I Propositio [10] C I 10 mg. hab. E 2 est esse tr. E 3 unam rectam tr. J-;, 4 
totalis: tota V w 5 itaque: utique lVwV b 6 est 1 : et J-;, I post iIle add. Eterminus 
scilicet I post est 2 scr. et dei. v" et 8 cum om. l I sit: fit l I quod: que RPv. quia 
E I nullo C 9 habet causam tr. E I post impedimenti ser. et dei. J-;, et I in inser. 
J-;, 10 propter: per R1v.v" I coniungitur Iv" I GA et DA: DA et GA Rlv"O II 
subtenditur E subtendantur O I ergo iIli tr. lV b I utcunque co. est ex cunque in 
J-;, I contingit E 12 alescindatur Rl I A V: AB Rl Pv.EO I post linea ser. et dei. 
E recta I AB rep. et dei. V. 13 itaque: igitur E I EDBG:EDVG Cv.J-;,E I BV:DV 
l co. est ex DV in J-;, I aequidistantem REO I Ergo om. J-;, 14 post 29 0m hab. R p I post 
4 0m hab. R p 15 GB: GV l co. est ex GV in J-;, I ED rep. O I BV: DV l I Sed: si 
V. 16 premissam proximam tr. RłJ-;, 17 post 11 0m hab. R p I eandem J-;, 18 post 8 0m 
hab. R p 20 esse: est V. I post 9 0m hab. R p I posl 5 i add. V. huius 
I Propositio [II] C I 11 mg. hab. E Iluminosi O 2 adversam Rlv" radij: rady
>>>
246 


Quia enim in corporibus densis non dyafanis natura dyafaneitatis et 
5 transparentie est impedita per admixtionem corporum opacorum terreorum 
(sunt enim omnia talia nature terree a domino). necesse est ergo ut transitum 
luminis impediant. Ergo, per petitionem. in absentia luminis umbrositatem 
efijciunt in ea parte in qua per ipsas luminis accessus impeditur. Hoc 
autem est in parte aversa corpori luminoso. Sit autem aliquod talium 
10 umbrosorum corporum cuius altitudo ab orizonte sit AB. et eius vertex A 
[Fig. 7]. Et sit corpus luminosum altius quam linea AB. cuius aliquis 
supremus punctus sit D. Radij itaque in toto linea AB incidentes impediuntur 
a transit u propter corporis opacitatem. Cadat vero radius DC proximus 
super radium DA. Hic ergo radius, quia non impeditur. transit ultrn 
u corpus AB; in sua ergo incidentia, que sit C, affert lumen. Deticit ergo 
umbra, et patet propositum. 


[Propositio]12. Equalium altitudinum corporum umbrosorum. quod fuerit 
corpori luminoso se altiori propinquius, breviorem facit umbram. 
Sit supremus punctus corporis luminosi G. quod sit altius duobus corpo- 
ribus umbrosis, cuius aItitudo a superficie orizontis sit linea AG /Fig. 8]. 
5 Sintque duo rum corporum umbrosorum equales altitudines erecte super Iineam 
AB, productam in ipsa superficie orizontis, que sint DE et ZH. quorum 
DE sit propinquior corpori luminoso AG. et ZH remotior. Ducaturque 
per verticem corporis DE radius GET qui erit linea una per 10 huius; 
et per verticem corporis ZH ducatur radius GHB. Erit itaque, per premissam. 
10 corporis DE umbra DET et corporis ZH umbra ZHB. Dico quod umbra 
DET est minor quam umbra ZHB. Ducatur enim a puncto H linea eque- 
distans linee ET, per 31 8m 1'. que sit HK. Palamque, per 2 8m t huius, 
quoniam linea HK concurret cum linea AB, cum qua concurrit eius equedi- 
stans, que est linea ET. Et quoniam linee HB et ET concurrunt in puncto 
u G, supremo puncto corporis luminosi, cadet ergo punctum K, per 2 8m et 
14 8m Ii huius, inter duo puncta T et B. Copuletur ergo linea EH que. per 
33 8m Ii et ex ypothesis, equalis et equedistans erit linee DZ. Sed. per 


v.. 4 enim om. v.. 5 amixtionem mg. ser. V b co. est ex amixtionem 6 omnia talia tr. 
v" 7 luminis: lucis E I impediant co. est ex impendiantur in v.. I post per add. R e 8 
quas v.. I post per ser. et dei. v.. suis [1] lipsas inser. v.. I impeditur: inpedit v.. 9 
adversa Rl v.. I aliquod om. v.. 10 vertex: vertice v.. I A : BA v.. 12 supremus: soprenus 
[1] v.. I sit inser. v.. I Radij: rady v.. 13 vero: ergo v.. 14 super: supra R I ergo om. 
v.. 15 aufert v.. 
I Propositio [12] C I 12 mg. hab. EO I fuerit: fuit v.. 3 supremus co. est ex 
sopremius in v.. I post corporis rep. et dei. v.. corporis 7 remotiore l 9-10 ducatur 
radius GHB. Erit itaque per... et corporis ZH umbra ZHB om. v.. II post enim ser. et dei. 
O a linea 11/12 aequidistans R 12 post 31 0m add. R P I HK om. v.. I post 2 0m add. 
R t 12-13 Palamque per..,quoniam linea HK mg. inser. v.. 13 post linea 2 scr. v.. 
GB I post AB inser. v.. concurret [1] I cum qua om. v.. I qua: quo v.. 13-14 
aequidistans R 14linee: linea v.. 15 coporis R I cadat E I ante per inser. v.. et I post 
et add. Iv.. per 16 post 14 0m add. R t 17 post 33 0m add. R p I et om. Iv.. inser. 
v.. I aequidistans R esdistans V. I DZ: DE E De O
>>>
34 am 1'. linee EH et. TK sunt equales. Linee ergo TK et DZ su nt equales. 
Addita ergo linea zr utrobique. erit linea OT equalis linee ZK. Ergo. 
per lam VI'. umbra ZHK est equalis umbre OET. quoniam sunt eiusdem 
altitudinis. ex ypothesi. Sed umbra ZHK est minor quam umbra ZHB. 
quoniam est pars eius. Ergo et umbra OET est minor quam umbra ZHB. 
Patet er!!o propositum. 


IPropositio]13. Umbra linee recte perpendiculariter corpori luminoso op- 
posite infixe superficiei corporis. densi nulla est; elevate vero est linearis, 
apparet autem punctalis. 
Si enim per suppositionem 3. in absentia luminis fit umbra. tunc patet 
quod si lineam mathematicam naturalis corporis superficiei infixam accidat 
luminoso corpori perpendiculariter offerri, non impedietur. nisi umca linea 
radialis a transitu cum alijs lineis radialibus. que transeUQt ad superficiem 
iIIius corporis. Nulla vero aliarum linearum radialium impeditur propter 
obiectum iIIius linee. Alias en im accideret duas vel plures lineas radiales 
cum una linea perpendiculari ipsis obiecta in uno puncto concurrere, quod 
est impossibile, quia indivisibilia in nullo se excedunt. Cum autem radius 
non sit aliud quam linea luminosa, ut patet per diffinitionem, palam quod 
radius ad modum linee incidit superficiei corporis secundum punctum, ergo 
et impeditur secundum punctum. Sed in allatione luminis umbra deficit, 
per 4 suppositionem. Quia ergo unicus radius est impeditus et ille incidit 
secundum punctum, palam quod non manet aliqua umbra. Cum vero linea 
elevatur super corporis densi superficiem. ubicunque sub linea ponatur densa 
superficies. umbra invenitur.. Et si per diversa puncta fiat descensus, palam. 
quia umbra proicitur linearis, eo quod inter quelibet duo puncta est lineam 
mediam ducere. Apparet autem semper punctalis in concursu sui cum superficie 
corporis densi, quia ibi solum cum umbra densitatis superficiei commiscetur. 
Patet ergo illud quod .proponebatur. 
[PropositioJI4. Umbra superficiei pIane cuiuscunque figure perpendicularis 
super superficiem corporis luminosi, infixe corpori denso nulla est; elevate 
vero est superficialis. sed apparet linearis recta. 


18 post 34 am add. R P I EH co. est ex EK in P I DZ: DT O 19 utrique R 20 post lam 
add. R P 22 quoniam: quia E 
1 Propositio [13] C I 13 mg. hab. EO 2 corpori lJ.b 3 punctualis Rl puntalis v" 
punctalis co. est ex punctualis in V b 43: primam E ł abscentia J.b I fit: sit lV b 5 accidit 
lJ.b 6 impedietur co. est ex impeditur in J.b I unita l I linea om. v" 7 post transeunt scr. 
et dei. J.b a superficiebus 8 vero: ergo J.b I linearum om. v" 10 perpendicularis 
J.b I obiecta: quoniam v" 12 post linea scr. et dei. V. radiosa I post per add. R 6 13 
post ergo add. v" que 15 4 suppositionem: positionem quartam E 16 secundum: super 
O I palam: palamque v" 17 corporis densi tr. RlV b 19 proijcitur Rl I iotra l 20 
punctualis Rl punctalis co. est ex punctaalis in J.b 21 comisseos v" comissentur J.b 22 iIIud 
quod om. E I propositum E 
1 Propositio [14] C I 14 mg. hab. E 2 est om. J.b 


247 


20 


s 


10 


15 


20
>>>
248 


Hoc patet per precedentem. Ad quemlibet enim punctum linee terminantis 
5 quamcunque datam superficiem corpori luminoso perpendiculariter oppositam. 
contingit ducere lineam perpendiculariter oppositam corpori luminoso. Umbra 
ergo cuiuslibet iIlarum linearum. superficie proposita existente infixa corpori 
denso. nulla est. Ergo neque umbra totius superficiei fit aliqua. Elevata 
vero superficie proposita ab iIlo denso corpore. umbra cuiuslibet iIlarum 
10 linearum per precedentem propositionem est punctalis; aggregata vero talia 
puncta videntur lineam constituere. Apparet ergo umbra superficiei taliter 
elevate umbra linearis. Et quoniam superficies circulares ex suis dyametris 
vel alijs perpendiculariter super corpus luminosum productis non accipiunt 
nisi puncta umbrarum. que ad lineam rectam inferius concurrunt qUla im- 
15 pediunt transit um reete linee. fit ipsarum umbra linearis ree1a. Non enim 
causantur umbre a hgura quorumlibet obieetarum nisi secunuum quod transitus 
luminis impedltur. Cuiuscunque ergo figure fuerit proposita superficies. umbra 
apparens semper erit superficialis: videbitur autem linearis. propter premissas 
causas. Patet ergo propositum. 


!propositio] 15. Omnis corporis densi. cuius equalis vel amplior est basis 
contraposita sibi superficie perpendiculariter corpori luminoso opposito infixi 
corpori denso. umbra nulla es1: elevate vero est corporalis. videtur au1em 
superficialis. 
5 Verbi gratia. sit columpna rotunda. vel aliud corpus. cuius basis sit 
equalis vel amplior superfieie illius eiusdem corporis contraposita ipsi basi. 
si ipsius corporis superficies non terminetur ad unum punctum. ut est 
in pyramide. quod infigatur superfieiei alicuius corporis solidi. et perpen- 
dieulariter opponatur corpori luminoso. Dieo quod verum est quod proponitur. 
10 Si enim iIlud corpus sit eolumpna rotunda. vel aliud corpus, euius basis 
sit equalis superficiei contraposite basi. et adverse corpofl luminoso. patet. 
quoniam radij luminosi ex omni par1e seeundum lineas longitudinis perveniunt 
ad basem. nulla ergo fit umbra. Et idem patet si illud corpus sit pyramidale 
vel si basis sit maior sibi contraposita superficie adverse corpori luminoso; 
15 tunc enim lumen nullatenus impeditur. qllod tamen aeeidcret si sllperficies 


4 Hocc V b I quemlibet: quodlibet l quamlibet v" 7 cuilibet V. I proposita: opposita 
v" 8 neque: nec E I fit: erit E 9 vero: nisi l I post vero add. V. sit I post superficie 
ser. et dei. C ab 10 punctualis Rlv" I talia: taliter a E 12 umbra om. E 13 vel: ex 
'V. 15 transitu V b I post ipsarum ser. et dei. O ipsa 16 umbra V. obiectorum RE 17 
umbra: apparet E 18 apparens: umbra E I semper erit om. E 
1 Propositio [15] C I 15 mg. hab. E 2 sibi inser. v" I post sibi ser. et dei. v" cui 
f?] I infito E 5/6 sit equalis rep. E 6 iłłius mg. inser. V b om. E 7 superficiei E I non 
om. l I terminantur V. terminentur v" 8 post infigatur inser. V. alicui I post quod scr. et 
dei. v" est f?] figura linfigatur inser. mg V b I post corporis add. E sumpti 9 opponatur co. 
est ex opponitur in v" I post luminoso ser. et dei. v" quod add. PV.O quod I proponebatur 
l proponibatur co. est ex proponebatur in v" 11 conposite V. I basis R 12 longitudum 
f?] V b 13 Et idem patet: Patet ergo idcm V. 14 ante superficie add. PV.V"O sibi I ad versa 
RE adversi l I corporis HI" Iluminosi l 15 ante si hab. v" et
>>>
249 


adversa corpori luminoso esset amplior ipsa basi corporis umbrosi; tunc 
enim. impedito transitu luminis. causaretur umbra. Sed quacunque figura 
corporis existente. si ipsum elevetur ah alio corpore cui fuit infixum. apparehit 
umhra superficialis. Superficies enim secantes corpus. et perpendiculariter 
sllperticiei l'orl'oris IlImino
1 inciden1es. umhram l."onstitlllln1 linearem per 20 
premissam. Et quia tota superficies corporis opposita lumiJ1()so corpori per 
tales superficies exhauritur. linee vero tales coniuncte superficiem constituunt. 
Palam omnis corporis sic disposite umbram superficialem apparere. Erit 
autem illa umbra necessario corporalis. quoniam erit dimensionata dimensio- 
nihus corporis. quod potest dec\arari ut prius. Pat et ergo propositum. 2
 


IPropositio] 16. Longior radius ad speram vel circulum columpne vel 
pyramidis rotundarum perveniens. quasi linea contingens est. 
Sit circulus magnus spere. vel columpne. vel pyramidis rotunde qui 
OG I Fig. 9]. cuius centrum sit punctum A. et dyameter GO. Et quoniam 
lumen ad omnem differentiam positionis se diffundit. sicut patet per 6 am 
 
suppositionem. sit punctum corporis luminosi Z. cuius lumen se diffundat 
super circlIllIm OG. ducatur que linea ZA a puncto corporis luminosi 
ad cenIrum illuminati circuh. et secunaum dyamt:l1"um AZ descrihatur circulus. 
secans circulum OG in punctis E et B. Et copulentur radij ZE et ZB. 
Oico quod radij ZE et ZB sunt contingentes speram vel aliud il\orum 10 
corporum et quod nulli radij longiores illis possunt ad illa corpora per- 
venire. Oucantur en im a centro circuli GO. quod est punctum A. ad puncta 
sectionum B et E linee AE et AB. Palam ef!!o. per 30. m III'. quoniam 
duo anguli ZEA et ZBA sunt recti. Ergo. per IS am III'. palet quod linee 
ZE et ZB contingunt circulum GO. Producte ergo non secabunt cir- 1
 
culum OG. Sunt itaque linee ZE et ZB longiores linee que a puncto 
Z ad iIIa corpora duci possunt. Si enim detur quod aliqui longiores radij 
duci p0ssint a punct0 Z ad illa C0rpora. patet. per 8 am III'. qU0d ille 


16 basis PE 17 transitu luminis tr. E I post transitu ser. et dei. v" corporis 20 corpori 
v" I constituunt linearem tr. PEO 21 luminoso corpori tr. E 22 tales coniuncte tr. 
V. 23 superficiem E ante apparere add. E punctantem f?] 24 aut V b I corporalia 
p dimentionata P dimensionita V. 24-25 dimentionibus P 25 potest: patet v" 
1 Propositio [16] C I 16 mg. hab. O I circularem V b 4 post dyameter hab. V. BG 5 
lumen: Iineam V. 6 ante sit ser. et dei. O et I sit mg. inser. V b I lumen se tr. V. I difTundit 
lV b 8 post dyametrum hab. E circuli 9 et 2 om. Rł inser. v" 10 aliud: aliam 
E lilIorum: aliorum RIC II rep. et dei. et mg. inser. v" possunt I ad inser. alm. P 13 
sectionem f?] E I E inser. JI. I AE et AB: AB et BE E I ergo om. V. I 30 0m : 31 
R I ante I1I i hab. R p 14 ZEA et ZBA: ZBA et ZEA E I 15 0m : ab R I ante IW hab. 
R p 15 ZE et ZB: ZB et ZE E I post ZB ser. et dei. v" contingunt circulum GD producte 
ergo non secabunt circulum DG su nt itaque linee ZE et ZB 18 possunt E I possint mg. inser. 
v" et add. possunt I a puncto Z ad iIIa corpora: ad iIIa corpora a puncto Z E I patet inser. et 
rep. et dei. v" I post 8 am hab. R p I post III i scr. et dei. V. duci possunt I patet per 8 am IIIi: 
per 8 3 1 patet E
>>>
250 


non cadent in arcum ER Ipse ergo producte secabunt lineas ZE et ZB 
20 prius quam perveniant ad arcus EG vel BD. Due itaque linee recte incIudent 
superficiem. quod est impossibile. Et hoc quidem non solum demonstrabile 
est in corporibus iIIuminandis. sed etiam per eundem modum. demonstrari 
potest de corporibus luminosis. quia et ab iIIis longior radius ad obiecta 
H corpora incidens, ipsa eorpora luminosa est eontingens. Patet ergo propositum. 


IPropositio)17. Impossibile est ut lumen egrediens a corpore luminoso 
egrediatur tantum a centro corporis luminosi; ex quo patet quod necesse 
est a quolibet puncto superficiei corporis luminosi diffundi radios luminosos. 
Si enim dicatur quod radij luminosi tantum egrediuntur a centro corporis 
5 luminosi. sit corpus luminosum circulus AB I Fig. 10). cuius centrum G, 
. sitque corpus iIIuminatum circulus DE. Et a centro G corporis luminosi 
egrediantur duo radij longissimi, qui possunt ab iIIo puncto G corpori 
iIIuminando incidere. qui, per premissam, erunt due linee eontingentes fines 
corporis illuminati. que sin t GDV et GEZ. Et puncta contactuum. que 
10 sint E et D. copulentur per lineam DE, et ei equedistanter ducatur linea 
Vz. per 31 am t. Eritque pars corporis iIIuminati, super quam cadet lumen, 
pars DHE, et pars obscura. super quam non cadit lumen, que DTE. Et 
quia pars supra quam non cadit radius non iIIuminatur. ergo pars contenta 
sub terminis VDTEZ est umbrosa, obscurans lineas DE et VZ equedistantes. 
15 Sunt itaque, per 29 3m li. trigoni VGZ et DGE equianguli, quia angulus 
DGE est communis ambobus trigonis. Est ergo per 4 am VI'. proportio 
linee GE ad lineam GZ sieut linee DE ad lineam VZ. Sed linea ZG 
est maior quam linee EG. Ergo linea VZ est maior quam linea DE. Umbra 
ergo corporum omnium. cuiuscunque sint proportionis ipsarum dyametri 
20 ad dyametros corporis luminosi. semper est maior corpore umbroso et s
mper 
augmentatur secundum modum quam elongantur ultra corpus umbrosum, cuius 


19 cadent: cadant v" I in om. v" 20 EG vel BD: ED vel BG R I linee recte tr. E 22 per 
inser. v" I ante eundem ser. et dei. v" in 23 ad: in R om. l et pv. O inser. v" 24 ipsa mg. 
inser. v" patet: quod est O 
l Propositio [17] C I 17 mg. hab. O I ut om. V u I egredies v.. 2 egrediatur co. est ex 
egredit in v" 2 - 3 lumioosi ex quo patet. . . est a quolibet mg. inser. v" 3 puncto om. v" 4 
dicatur co. est ex ducatur in V b 6 Et om. Iv.. inser. v" 7 egrediuotur l egrediaotur co. est ex 
egreditur in V b I possunt rep. mg. v" 8 illuminato E 9 et 1 0m . l inser. v" 10 E co. est 
ex D in v" et D co. est ex E I DE co. est ex DEZ in v" I equidistanter REO equedistantur co. 
est ex ęQuedistantur in P ex eQuedistanster in v" I ductabur C I post linea hab. v.. v 11 
post 31 0m add. R p I illuminata V. I quem PO I cadit EO l2 quam: 'quem P I DCE 
Rl 13 quem P om. V u 14 UDCEZ R UDCPEZ l co. est ex VDTIEZ in P VDT v.. co. est ex 
terminens f?] in v" I obscuritatis V u I post obscurans scr. et dei. v" ritatis I aequidistantes 
R 15 post 29 0Dl add. R P I quia: quod v.. mg. inser. v" 16 post 4 0m add. R p 17 ZG co. 
est ex GZ in v" I post linea inser. O ut 18 linea om. E 19 ipsa V. co. est ex ipsa in 
v" I dyametri: dyameter v..v" l 21 augmentantur Rlv" augmeotant V b I quam: quo RE 
quem f?] O I e10ngatur v..v" 


.....
>>>
contrarium notum est sensui. Unde fuit suppositum in principio aliquam 
umbram in sui termino acui et ad punctum terminari. Palam ergo est 
propositum. Et cum lumen egrediatur a corpore luminoso. et non solum 
a centro. ut ostendimus. manifestum est correlarium. quoniam a quolibet 
puncto superficiei corporis luminosi necesse habet egredi ad corpora illu- 
minanda. Corpus enim luminosum secundum quod- [libet aspectum] huius 
[ materie] omogeneum est, unde qua ratione dabitur ab uno puncto sue 
superficiei lumen difTundi, eadem ratione dabitur de quolibet aliorum punc- 
torum. Patet ergo propositum. 


lPropositio] J 8. Impossibile est ut a superficie corporis luminosi egrediantur 
radij solum equedistanter corpori illuminando incidentes. 
Si en im hoc dicatur esse necessarium. tunc sequitur evidens impossibile. 
Sit en im corpus luminosum cuius dyameter AB I Fig. J J]. et corpus ilIu- 
minatum DG. Et producantur a corpore luminoso duo radij longior
s qui. 
per 16 8m huius, erunt due linee contingentes fines corporis LiD. que sint 
AGE et BDV. et sint equedistantes ex ypothesi. Pars quoque illuminata, 
super quam cadit lumen. sit GZD et pars super quam cadit umbra sit 
GHD. Umbra ergo continetur a duabus lineis EG. DV que sunt equedi- 
stantes. Si ergo unicuique corpori ilIuminando correspondeat equalis sibi 
pars corporis illuminantis, tunc enim solum secundum lineas equedistantes 
radij incidunt, per 33 8m li. Patet ergo quod omnis umbra in omni sui 
parte equalis erit sue rei umbrose. Igitur non augebitur umbra neque minuetur, 
sed protendetur semper in infinitum. quod est contra suppositionem. Habet 
enim aliqua umbrarum terminum acutum. Est ergo hoc impossibile; oppositum 
ergo est necessarium. Et hoc est propositum. 


[Propositio] I 9. Omnis punctus corporis luminosi eam partem corporis 
umbrosi ilIuminat ad quam ab eodem puncto rectas lineas possibile est 
produci; ex quo patet quod unus punctus luminosi corporis non ilIuminat 
omne umbrosum corpus. 
Sunt enim corpora luminosa unigenea in suis partibus. Non ergo diver- 
sificatur effectus suarum partium, neque est possibile ut ab una parte 


24 et 2 inser. V b 26/27luminanda E 27 quodlibet R quod lCPY.v"EO I huius: sui Rom. 
y. huiusmodi EO 28 in loco [materie] hab. R punctum I omogeneum: unigeneum Rl 
homogeneum E 29 lumen diffundi tr. y. lumen diffundere E 
l Propositio [18] C I 18 mg. hab. E 1 equidistanter RE 3 hic y. I sequeretur 
Rl I evidnes y.v" 4 iIIuminosum y. 5 luminoso: luminosi l 5 que E 7 AGU 
R I BDE R BDH l BCL V u co. eSl ex BDL in v" I equidistantes RE I quoque: quorum 
.Y" 8 supra E 9 ergo: quoque O lEG: ED R I post EG hab. Rlv" et I sint l sunt co. 
est ex sint in v" 9-10 equidistantes Rl 10 correspondeant v" I sibi om. E 11 
aequidistantes R 12 post 33 am add. R P 14 semper om. E super l I suppositum v" 16 
ergo est tr. Rl 
l Propositio [19] C I 19 mg. hab. E 2 eodem: eo v" 3 perduci y'v,,0 I ex quo: 
quare E I iIIuminat co. est ex iIIuminant in v" 5 lumino unigenia E 6 effectus mg. co. est 
ex estuns f?] in v" 


..... 


251 


25 


30 


5 


10 


tS 


5
>>>
252 


illuminent et non ab aha. Non tamen ab un0 punct0 corp0ris luminosi 
ad quodlibet punctum umbrosi c0rporis possunt recte linee produci. Et 
0b hoc unus punctus n0n i1Iuminat omnia. sed illuminantur corp0ra umbrosa 
10 a diversis punctis C0rporis luminosi. Sit enim C0rpus luminosum circulus 
AB I Fig. 12] quem C0ntingat linea DG super punctum A per '6 m II( 
sitque corpus iIIuminatum C0ncavum arcus EV et secet ipsum linea DG 
super duo puncta Z et H. Dico quod p()ssibile est om nem arcum ZH 
ilIuminari a puncto A C0rporis luminosi. quoniam ut patet possibile est 
lS ut ah omni punCI) arcus ZH ducatur linea recta ad punctum A. Sed 
ab arcu ZE et ah arcu HV aliquas lineas duci ad punctum A est imp0- 
ssihile per I hOlm I. qU0niam inter lineam (J D t'0ntingentem .:irculum et 
mter ipsum cireulum AB aliquam lineam rcetam intel"Cipi est impossihile. 
Si erg0 aliqua linea ab aliqu0 punctorum iIIorum areuum ducatur ad 
20 punctum A. ilIa necessari0 secabit circulum. sieut linea V A secat circulum 
AB in punct0 T prius quam perveniat ad punctum A Et similiter est 
de 0mnibus lineis a quoeunque punet0 areuum VH et ZE ad punctum 
A produetis. Omnes enim secant circulum AB in aliqu0 alio punet0 ab 
ipS0 puncto A. prius quam perveniant ad punctum A Radius itaque exiens 
2S a punct0 A non iIIuminat amb0s arcus VH et ZE. sed s01um arcum 
HZ; sed illos arcus ab alijs punctis luminosi C0rp0ris circuli AB. a quihus 
ad e0sdem areus recte p0ssunt produci linee. nichil prohibet illuminari. 
Et similiter est de alijs quibuscunque corp0ribus illuminatis. quoniam si 
corpora concava. de quibus plus videtur quod possint ab uno puncto 
30 ilIuminari. non illuminantur ah uno puncto corp0ris luminosi. ergo multo 
minus corpora reeta plures planas superficies habentia. vel corpora sperica. 
vel alia convexa p0ssunt ab uno punct0 lumin0si corporis illuminari. Patet 
ergo propositum et eius correlarium. 


IProp0sitio]20. A puncto cuiuslibet corporis lumin0si lumen diffunditur 
seClmdum 0mnem lineam reetam que ah ilIo punct) ad opp()sitam superficiem 


7 iIIuminetur l 8 punctum umbrosi corporis possunt: punctum umbrosi possunt corporis v.. 
corporis umbrosi punctum possunt E I possunt inser. v" et rep. et dei. 11 qucm: quod 
Iv.. I post DG scr. et dei. v" et I 16: 17 R I ante 111; hab. R p 12 concavum: 
contactuum v" I EV: EB lv..v" I ipsa l ipsam v.. 13 possibile est tr. v.. I post possibile 
scr. O e 14 A: AL l 15 sed: et l inser. v" I post sed scr. et dei. v" et 16 HV co. est ex 
HB in V b 17 16 0m : 15 lCPv..VbEO I post 16 0m add. R p I inter: minus V v 17-18 et 
inter ipsum circulum om. l 18 rectam om. E I post rectam ser. et dei. V b et I impossibile co. 
est ex possibile in v" 19 punctorum iIIorum tr. v.. I ducantur v..v" 20 post secabit scr. v.. 
lineam 21 perveniat co. est ex perveniant in v" 23 aliquo om. R et inser. v" 23 alio 
puncto tr. v.. 23-24 ab ipso puncto om. E 25 ambos: alios V b I arcus: arcu v.. 26 
alijs: illis f?] v" 27 recte: item f?] v" I possunt inser. et rep. et dei. v" possunt produci linee: 
linee produci possunt E 28 illuminatis co. est ex illuminantis in v" 29 possunt E I ab: ad 
v" I post ab ser. E aliquo 30 ergo om. v.. 31 recta om. O 32 post convexa scr. et dei. v" 
possunt ab uno puncto luminosi corporis iIIuminari et mg. inser. alm. possunt ab uno illuminari 
puncto luminosi corporis 
l Propositio [20] C I 20 mg. hab. EO 2 post secundum scr. et dei. P iIlectum vocabulum 
(Iincamen f?]) lomnem inser. alm. P I lineam rectam tr. Rl I rectam om. E
>>>
253 


duci potest: unica tantum linea perpendiculariter superficiei obiecti corporis 
incidente. Ex quo patet lucern cuiuslibet puncti corporis luminosi secundum 
pyramidem illuminationis ditfundi. 5 
Quod enim lux cuiuslibet puncti corporis luminosi ditfundatur secundum 
omnem lineam ducibilem ab illo puncto super superficiem corporis obiecti 
ad omnem positionis ditferentiam. hoc patet per premissam. Quod autem 
unica tantum linearum ab aliquo uno puncto corporis luminosi productarum 
ad superficiem unam corporis oppositi sit perpendicularis. hoc patet ex 10 
20" primi huius. Unica ergo linea perpendiculariter incid,t superficiei sibi 
opposite: omnes vero alie linee ab eodem puncto producte incidunt oblique. 
Patet ergo ex hoc quod cuiuslibet puncti corporis luminosi lumen secundum 
pyramidum illuminationis ditfunditur. cuius vertex est in puncto corporis 
luminosi et basis in superficie corporis obiecti: et hoc quidem instrumentaliter 15 
patet. per primam huius. Lumine enim transeun1e foramen instrumenti. 
cu ius centrum est punctum M Wig. I]. et ditfuso ipso in partem opposita;n hore 
instrumenti secundum circulum. cuius centrum est punctum P. erit circulus 
p maior circulo M. quod sensibiliter potest videri computatis hinc inde 
partibus in hora instrument i que inter.acent periferias iIIorum circulorum 20 
et centra. Patet ergo propositum. 


IPropositio]21. Corporis umbrosi pars. cui a pluribus partibus corporis 
luminosi lumen incidit. plus illuminatur quam pars cui a paucioribus: ex 
quo patet unumquodque umbrosum circa radium sibi perpendiculariter 
incldentem plus illuminari. 
SIt corpus luminosum circulus ABG I Fig. 13]. cuius centrum sit punctum 5 
O: sitque arcus sui convexitate respiciens corpus illuminandum. qui ABG. 
divisus per equalia in puncto B. et ducatur linea ze contingens circulum 
in puncto B. per 16 am III'. Et in puncto G contingat circulum linea IK. 
et in puncto A linea TH. sitque corpus umbrosum arcus KZTIEH. Du- 
catur quoque linea OBL a centro corporis luminosi ad corpus umbrosum. 10 
Eritque hec perpendicularis super lineam EZ. contingentem circulum in 
puncto R. per 17"m II ( Unaqueque igitur partium arcus HT illuminatur 


6 lux inser. P I puncti om. 1 inser. V b I post lux add. E a quolibet puncto Ilux 8 post 
differentiam add. v.. et I per om. E I Quod: quoniam v.. 9 alico v.. I uno puncto tr. 
v" 11 post 20" add. R t 11-12 sibi opposite tr. O 12 oblique: alique v" 15 basi 
Iv" I quodam ł 16 per primam huius patet E 17 diffusio IV. co. est ex diffusio in 
v" I post diffuso hab. l in et v" ser. et dei. in I in partem rep. et dei. v" I ore RE 18 ante 
p hab. v.. d 19 ante computatis ser. et dei. v" computans et computatis mg. inser. 20 ora 
RE I interiacet v" 
I Propositio [21] C I 21 mg. hab. EO 2 lumen incidit tr. E I incidit co. est ex cecidit 
in v" 5 punctum om. 1 mg. inser. v" 6 concavitate łCPv..v"EO 7 ZE lCPv..VbEO 
I ducatur linea co. est ex ducantur linee in v" 7-8 circulum in puncto co. est ex in puncto 
circulum in v" 8 post B scr. et dei. V b et ducantur linee I 16: 17 R I ante III i hab. 
R p I post Et ser. et dei. v" a I in: a l inser. v" I post G scr. v.. certum vocabulum 
i/lectum 9 puncta E I KZTICH Rl KZTEIH v.. KZTLlEH f?] E 10 PBL l 11 CZ 
Rl ł2 17: 18 R I ante III i add. R p I unamqueque PO ergo v..
>>>
254 


a puncto A corporis luminosi. per 19 huius. Punctus ergo L iłłuminatur 
a punc10 A. SlInrliterque arcus KI illuminatur a puncto G. Ergo et PUIlCtuS 
15 L totusque arcus ZE illuminatur a puncto B; en'o et punctus L. Punctus 
itaque L iłłuminatur a tribus punctis corporis luminosi. scilicet punctus 
A. B. G. et totus arcus TI est communis illuminationi trium punctorum 
A. B. G. Arcus vero El est communis duabus tantum iłłuminationibus 
punctorum A et B. AI"Cus quoque ZT est similiter communis duabus tantum 
20 illuminationibus punctorum B et G. quoniam est communis arcubus ZE 
et KI ab iłłis duobus punctls illuminatis; arcus vero HE iłłuminatur tantum 
ab uno puncto A et arcus ZK ab uno tantum puncto G. lIIuminatio 
ergo arcus TI triplicatum habet lumen "quod arcus ZT et El habent duplum. 
et quod arcus EH et ZK habent simplum. Magis ergo omnibus alijs 
2S arcubus iłłuminatur arcus TI qui est circa lineam perpendicularem. que est 
LD. Et illuminatio duorum arcuum zr et El est equalis. quoniam a totidem 
punctis corporis luminosi iłłuminatur unus ut alius. Ipsorum vero amborum 
iłłuminatio maior est illuminatione duorum arcuum EH et ZK. Eritque 
semper proportio excessus iłłuminationis secundum numerum punctorum 
30 corporis illuminantis, respicientis partem corporis iłłuminati. Patet itaque 
ex hijs o.uoniam semper id quod est propinquius perpendiculari fortius 
iłłuminatur illo quod est remotius ab eadem perpendiculari; super ipsum 
namque plus luminis cadit quod a pluribus luminosis partibus iłłuminatur. 
Quod enim nunc demonstratum est in arcu KH. similiter accidit in alio 
35 corporum quocunque: exemplificavimus autem istum in corpore concavo, 
quoniam iłłud videtur plus uniformiter debere iłłuminari. Patet ergo pro- 
positum. 


IPropositio]22. Omne corpus umbrosum punc to luminoso propmqulUs 
iłłuminatur ab iłło puncto fortius corpore plus distante. 
Sit corpus luminosum in puncto A I Fig. 14] et corpus iIluminatum 
sit ap ud lineam BG. Et copulentur linee AB et AG. Virtus itaque corporis 
5 A iłłuminans corpus BG illuminat etiam aerem medium qui continetur 


13 L: B l 14 similiter E I ergo om. lE 15 ZC Rl I totusque arcus ZE iIIuminatur: 
iIIuminatur totus que arcus ZE E 17 iIIuminatio Vwv,,; in V b co. est ex iIIuminationum 18 El: 
CI Rl I duabus tantum tr. E 19 duobus V w 20 B: L l v" H v" E G O I. ZE: ZC 
Rl 21 KI co. est ex KZ in v" I punctis mg. inser. v" I HE: HC Rlv" [7] 22 ab uno 
tantum puncto: tantum etiam ab uno puncto E 23 lumen co. est ex lumem in v" I post 
lumen scr. et dei. O quam [7] I arcus ZT tr. O 24 EH: CZ Rl ZH v" EZ v" 25 TI mg. 
inser. v" 26 El: CI Rl I El co. est ex ER in v" 27 iIIuminatur co. est ex iIIuminatąs in 
v" I ut: et E 28 EH: CH Rl 29 iIIuminationis om. E I secundum numerum: ex 
numero E 32 iIIuminatur iIIo: fortificatur in lumine et magis illluminatur idea E lipsam 
Iv" 33 lumini v" I a inser. v" 34 post KH ser. et dei. V b a citum [7] similiter mg. inser. 
v" 35 quocunque: quousque v" I exemplificavimus co. est ex exemplificamus in P 36 
inter illlud et videtur inser. v" plus I post debere add. v" plus 
I Propositio [22] C I 22 mg. hab. E 2 iIIuminat v" 4 sit apud Jineam om. O 5 
etiam: in lv"v" (inser. rep. et dei.)
>>>
255 


In triangulo ABG. et ducatur linea DE equedistans linee BG. per 31 am 
Ii. Sitque linea BG propinquior corpori luminoso in puncto A existenti 
quam corpus DE. Dico quod corpus BG fortius illuminatur quam corpus 
DE. Sit en im ut radius AB cadat in punctum D et radius AG in punctum 
E. et a puncto B ducatur super lineam DE linea perpendicularis que sit lO 
BV. et a punc10 G perpendicularis que sit GZ. per 12 am Ii. Erit ergo, 
per 34 am I'. linea VZ equalis linee BG et linea BV equalis linee ZG. 
Ducantur itaque linee V A. ZA. Hee ergo secabunt lineam BG. per secundam 
primi huius. Secet ergo ipsam linea V A in puncto H et linea ZA in puncto T. 
Quia ergo virtus imprimens lumen in corpus BG est diffusa per totum IS 
triangulum ABG. virtus autem illuminans corpus VZ equale corpori GB 
est diffusa solum per trigonum AHT. et quia per IBm VIi, triangulus ABG 
est maior triangulo AHT. quoniam basis BG est maior base HT. plus 
itaque luminis diffusum est in trigono ABG quam in trigono AHT. In 
quolibet enim istorum triangulorum puncto est lumen equaliter diffusum. 20 
Lumen ergo incidens corpori existenti in linea VZ illud corpus debilius 
illuminat quam corpus BG. quia paucius sibi lumen incidit. Proportio en im 
virtutis luminis incidentis linee HT. ad impressionem suam in corpus VZ 
est minor proportione virtutis incidentis linee BG ad impressionem suam 
in corpus VZ. per 8 am vi, quoniam ut patet ex premissis lumen incidens 2S 
linee BG est plus lumine incidente linee HT. Proportio vero virtutis inei- 
dentis linee HT ad impressionem suam in corpus VZ est sicut proportio 
virtutis incideotis linee BG ad impressionem suam in corpus BG, per 6 am 
huius. Ergo. per 16 Bro vi, erit permutatim proportio virtutis pervenientis 
ad lineam HT. ad virtutem pervenientem ad lineam BG sieut impressionis 30 
facte in corpus VZ ad impressionem factam in corpus BG. Sed per premissa 
lumen perveniens ad lineam HT est debilius lumine perveniente ad lineam 
BG. Ergo impressio perveniens a linea HT in corpus VZ est debilior 
impressione perveniente a virtute luminis ineidentis linee BG in corpus BG. 
Corpus itaque propinquius corpori luminoso fortius illuminatur quam remotius 3S 
ab eodem. Et hoc est propositum. 
[Propositio]23. Puncto remotiori a corpore luminoso incidunt radij a plu- 
ribus punct is corporis luminosi quam puncto propinquiori. 
6 equidistans RE I BG: BGE l co. est ex BGE in Vr, I post 31 am add. R P 8 fortius co. est 
ex portius in Vr, 9 cadit v.. I puncto lVr, I radius: arcus lVr, 10 DE: BE RIVr, II 
post G ser. et dei. v.. certum voeabulum illectum I post 12 am add. R P 12 34: 14 Vr, I post 
34 am add. R P 13 Ducatur v.. I linee inser. alm. P I post V A inser. Rl et Vr, scr. et dei. 
et I v.. rep. et dei. V A I Hee: hec lCPv.. Vr,0 I post secundam add. R t 14 igitur E 16 
GB: BA lCPv..Vr,EO 17 AHT: IHT v.. I post lam add. R p 18 basi R 21 corpora 
Vr, 23 virtutis om. v.. 25 post 8 am add. Rp 26 est: et Vr, I vero: ergo E 27 VZ: VC 
l 27-28 VZ est sicut proportio..,suam in corpus mg. inser. V b 29 post 16 am add. 
R p I permutatim om. E 30 perveniente Vr, I impressionis co. est ex in premissis in 
J'b 31 premis8am v... 32 ad. om. E I lineam.: linee E I lumine: Iinee lVr, I ad 2 : in 
lCP v.. Vr,EO 33 igitur PEO I in pressione v.. 
I Propositio [23] C I 23 mg. hab. EO 2 punto v.. I PUDcto propinquiori tr. J'b
>>>
256 


Sit corporis luminosi circulus ABC (Fig. 15]. cuius centrum D. et ducatur 
perpendicularis DG. in qua signentur duo puncta. G remotior et H propinquior. 
s Dico quod puncto remotiori. qui est G. incidunt radij a pluribus punctis 
corporis luminosi quam ipsi puncto H. Ducantur enim radij longissimi 
a corpore luminoso ad punctum G et similiter ducantur radij longissimi 
a corpore luminoso ad punctum H. Erunt itaque. per 16 huius. iII i radij 
contingentes speram. Contingant itaque radij incidentes puncto G in punctis 
10 A et B. et radij incidentes puncto H cClntingant speram in punctis E et 
F: palamque. reI' hO primi huius. quoniam puncta contin
entie E e1 F 
cadent intra puncta A et B. Quia itaque punctum H solum irradiatur 
a punctis arcus ECF et non ab alijs. punctum vero G irradiatur a punctis 
arcus ACB. qui est maior arcu ECF, pat et propositum. Quoniam punctum 
lS G iIIuminabitur a superficie corporis luminosi quam per equalia dividit 
arcus ACB et punctum H iIIuminabitur a superficie corporis luminosi. quam 
per equalia dividit arcus ECF. tamen. propter radiorum fortitudinem. que, 
consequitur ipsorum brevitatem. fortius iIluminabitur punctum H a paucioribus 
radijs quam punctum G a pluribus. Multiplicitas enim luminis in puncto 
20 remotiori est ex concursu radiorum multorum Clolique incidentium et deoilium. 
sed in puncto propinquiori fortificatur lux ex brevitate radij. secundum quem 
a corpore luminoso immittitur plus virtutis. 


IPropositio]24. Omne corpus luminosum minus spatium a quo non egreditur 
fortius iIIuminat quam spatium maius iIIo. 
Quod hic proponitur. satis patet per exemplum. Una enim candela 
parvam cameram fortius iIIuminat quam domum vel cameram maiorem. 
s Potest tamen idem figuraliter demonstrari. Esto enim ut sit punctus aliquis 
corporis luminosi A !Fig. 16], a quo per spatium magnum. in quo sit 
linea BG. diffundantur radij AG. AB. AD. et sit radius AD perpendicularis 
super lineam BG. IIIuminatur itaque spatium totum BG secundum has lineas 
a puncto A sibi incidentes. Abscidatur itaque a linea AB linea AE, ut 
10 placuerit. et a linea GA aoscidetur linea AF equalis linee AE. Productaque 


3 corporis luminosi: corpus łuminosus et E I pOSl circulus add. E eius 7 G: H l 7-8 et 
simiJiter ducantur . . . ad punctum H om. l mg. inser. alm. V h 9 contingentes: contingentis 1 V. co. 
est ex contingentis in v" 11 pałam quia Iv" I post 60 add. R t I F: EF PO 12 cadant 
E linter PEO I A: D l I iradiatur P radiatur V. 13 post alijs ser. et dei. v" punc- 
tis 14 rep. et dei. v" ACB qui est maior arcu 16-17 mg. inser. V b ACB et punctum 
H ... quam per equalia dividit arcus 17 ECF: ECK Iv" CEF V. 18 consequitur: sequuntur 
l I ipsarum C 19 punctis O I Multipłicatas co. est ex multiplicatos in v" 20 est: et 
v" 21 fortificabitur Iv" I quam Rłv" quod V. 
I Propositio [24] C I 24 mg. hab. EO I egreditur co. est ex egrediatur in v" 3 satis 
inser. V b I patet om. V. 7 AG, AB tr. E I AD): A O I AD2: AB 1 8 iIIumindntur 
PO I itaque: que ita V. I mg. inser. v" et partiale il/ecta itaque f?] spatium totum f?] 9 
incidentes: incidens l co. est ex incidens in v" I Absciondatur Rl E abscindatur co. est ex 
abscidantur in v" a om. V. 10 GA: GE l I abscindetur Rl abscindatur E
>>>
linea EF secet lineam perpendicularem que est AD in puncto H. Si ergo 
in linea EHF terminetur spatium. ne lumen ultra pertranseat, erit illud 
spatium minus spatio terminato per lineam BGD, per 2 8m VI'. Omnes 
. autem radij pervenientes ad lineam BG perveniunt ad lineam EF. Plus 
ergo agregantur radij in spatio EF quam in spatio BG; fortiores ergo 
fiunt. cum sint virtutis plus unite. Magis ergo agunt quam in spatio BG, 
in quo sunt diffusiores. Plus ergo ilIuminatur spatium minus, cum ad eius 
terminos virtus luminis terminatur, quam spatium maius iIIo, et hoc est 
propositum. 


[Propositio]25. Omnis axis vel dyameter corporis umbrosi non perpendi- 
culariter respiciens superficiem corporis sperici luminosi alicui diametro ilIius 
corporis equedistat. 
Sit enim axis vel dyameter corporis umbro si linea AB [Fig. 17], non 
perpendiculariter respiciens superficiem corporis luminosi sperici, cuius centrum 
sit punctum C. Dico quod linea AB equedistat alictJi dyametrorum corporis 
C. Ducatur enim linea AC a termino linee AB ad centrum corporis lu- 
minosi. et super punctum C, terminum linee AC, fiat angulus equalis 
angulo BAC, per 23 8m t, qui sit DCA, producta linea DC taliter, ut 
anguli BAC et ACD fiant coalterni. Linee ergo DC et AB equedistant 
adinvicem, per 27 8m t. Et quoniam linea CD est ducta a centro corporis 
luminosi. patet quod ipsa est pars dyametri sperici illius corporis. Producta 
ergo dyametro DCE, patet quod ipsa equedistat linee AB, et hoc est 
propositum. 


[Propositio]26. Dyametro corporis luminosi sperici existente equali dyametro 
corporis illuminandi, tantum eius medietas ilIuminatur et umbra fit equalis 
rei in infinitum protensa. 
Esto corporis illuminantis dyameter AG [Fig. 18] et eius pars aspiciens 
corpus ilIuminandum sit ABG. Dyameter vero corporis illuminandi sit DV, 
equalis ex ypothesi et per premissam equedistans dyametro AG, et super- 


II Iineam perpendicularem tr. v" I in om. O 13 spatium om. E I post 2 0m add. R p 14 
perveniunt: pervenant Iv" 16 fiunt: sunt E I virtutes O I unice v" I ergo: autem 
E 17 illuminantur V b I minus: unius Y.. I cum: tamen v" 18 luminis om. Y.. I ter- 
minetur Y.. terminantur v" I spatium: in spatio Y.. I maius om. E 
I Propositio [25] C I 25 mg. hab. E 2-3 iIIius corporis tr. E 3 aequidistat R 6 
a\:quidistat R 7 AC: C PEO I a co. est ex ad in E 8 C: D Y.. I terminum Iinee AC om. 
Y.. 9 post 23 0m add. R P I que Iv" 10 aequidistant R 11 post 27 0m add. R P 13 
diameter l I equidistat RE 
l Propositio [26] C I 26 mg. hab. E Iluminosi co. est ex lumini in v" I sperice Y.. 2 
iIIuminatur l 3 post protensa add. R Aristarchus Samius in libro de magnitudinibus et 
intervallis solis et lunae. 4 dyametro Y.. I et eius: cuius Rl et cuius PY w et e cuius 
EO I inter et et cuius inser. P vocabulum ilIectum I et ser. et dei. v" I eius co. est ex cuius in 
v" 5 sit: lit v" I DV: DB l co. est ex DB in Y..v" 6 equidistans RE 


17 - Wilclonis Pcrspeclivae... 


257 


15 


5 


10 


5
>>>
258 


ficies illuminata sit DEV. Oico quod OEV est medietas superficiei corporis 
ilIuminandi. Oucantur enim radij AD et GV. Quia itaque dyameter AG 
est equalis et equedistans dyametro DV. per ypothesim et per premissam, 
10 palam quod radij AD et GV sunt equedistantes et equales, per 33 8m 
Ii. Ergo in infinitum protracti nunquam concurrent. Non ergo illuminatur 
aliquą pars corporis DEV ultra dyametrum OV. Eius ergo corporis tantum 
medietas illuminatur. Protenditur en im umbra in infinitum equalis dyametri 
cum dyametro corporis, et est extensa inter lineas DZ et VH. et est linea 
ZH equalis linee OV. Portio itaque arcus DFV, que est medietas totius 
15 superficiei corporis DEV, et linee DZ et VH continent umbram equalem 
rei umbrose que protenditur in infinitum. Patet ergo propositum. 


(Propositio]27. Dyametro corporis luminosi sperici existente maiore dya- 
metro corporis sperici illuminandi, plus medietate corporis illuminatur et 
basis umbre est minor magno circulo corporis illuminati, concurrens ad 
punctum unum retro corpus. 
5 Sit corpus luminosum contentum circulo AB [Fig. 19], et sit corpus 
umbrosum illuminandum contentum circulo GD. et sit dyameter AB maior 
dyametro GD. Et sint radij incidentes AG et BO; hij ergo radij necessario 
concurrent ultra corpus GD. Si en im non concurrant, tunc equedistabunt. 
N.ecessąrium ergo erit dyametros AB et GD esse equales, quod est contra 
10 ypothesim. Concurrant itaque in puncto E. Patet ergo quod radij AG 
et BD non transeunt terminos dyametri circuli G et O. Si enim transeant, 
palam. cum illi radij per 16 huius circulum GD contingant. quia anguli 
EGD et EDG erunt recti. per l7am III'. In triangulo ergo GOE sunt 
duo anguli recti, quod est impossibile et contra 32 3m Ii. Palam itaque 
15 quod radij AE et BE non transeunt per terminos dyametri circuli GD, 
sed ultra illos contingunt superficiem corporis illuminandi; magis ergo me- 
dietate corporis illuminatur. Et quia minor circulus illius sperici corporis 
continet umbram, patet quod basis umbre minor est magno circulo corporis 
illuminati, quod est propositum. 


. 
7 DEV 1 . 2 : DEB l co. est ex DEB in v" I superficiei om. v.. 8 GV: GB l co. est ex GB in 
v" I post GV add. Iv.. et et v" ser. et dei. et 9 aequidistans R I DV: DB v.. 10 et 2 inser. 
v" I post et l scr. et dei. V b ad I GV: DU RIPV u DV EO I equidistantes RE I post 33 am 
add. R P II post ergo ser. et dei. v" g I numquam v.. 12 ultra: intra v.. 13 diameter 
l dyametro E 14 intra lV b 15 DEV: DEB l co. est ex DEB in v" 
I Propositio [27] C I 27 mg. hab. EO I existente om. v.. 2 mediate co. est ex 
immediate in V b I iIluminantur v" 3 iIluminanti V b 4 post corpus add. R Aristarchus 
Samius in libro de magnitudinibus et intervallis solis et lunae. 6 dyametros Clpv..v" I post 
dyameter add. R circuli 7 post dyametro add. R circuli 8 equidistabunt RE II traseunt 
l I et om. R I post enim add. l non I transeunt co. est ex transeant in v" 12 contingunt 
v.. 13 17: 18 R I post 17 am add. R p 14 post 32 am add. R P I itaque: ergo R. om. 
l 15 quod: quia v.. I GD: GG O 16 magis: maius E 16-17 mediate v" 17 
iIluminatur inser. rep. et dei. v" I et: sed E I iUius sperici corporis circu1us E
>>>
IPropositio]28. Dyametro corporis luminosi sperici existence minore dya- 
metro corporis iIIuminandi sperici. minus medietate iIIuminatur et est umbra 
multo maior corpore iIIuminato in infinitum protensa. 
Sit corpus luminosum. cuius maior circulus sit DG [Fig. 19], et corpus 
illuminandum, cuius maior circulus sit AB. Et sit dyameter circuli DG minor 
dyametro circuli AB; concurrent itaque radij GA et DB ultra corpus luminosum 
GD propter premissam dyametrorum portionem. Concurrant ergo in puncto 
E ultra dyametrum corporis DG. Hij ergo radij non attingunt terminos 
dyametri circuli AB. quia si sic erunt ut in premissa, per 15 8m et 17 8lD 
lIIi. trigoni ABE duo anguli recti, quod est impossibile. Minus ergo medietate 
corporis AB illuminatur. Et quoniam magnus circulus corporis AB cadit 
intra umbram, et umbra ultra iIIum protensa semper dilatatur; cum per 
14 primi .huius, radios GA et GB ad iIIam partem concurrere sit impo- 
ssibile, patet quod umbra extenditur in infinitum. Et hoc est quod pro- 
ponitur. Et per hec premissa penitus similiter in columpnis et pyramidibus 
potest demonstrari; idem enim in illis est demonstrandi modus. 


[Propositio]29. Superficiem planam super medium umbre erectam, corpus 
umbrosum et corpus luminosum per equalia dividere est necesse. 
Sit corpus luminosum AB. cuius centrum C, et corpus umbrosum sit 
DE, cuius centrum F [Fig. 20]. Sitque punctum in medio umbre quod 
sit G. et copuletur linea CFG. Cadet itaque linea FG in medio umbre. 
Superficies itaque erecta super medium umbre necessario erit erecta super 
lineam GF; transit ergo iIIa superficies centrum corporis umbrosi et centrum 
corporis luminosi. Necessario ergo dividet iIIa corpora per equalia, per 
ea que ostensa sunt in primo huius. Patet ergo propositum. 


[Propositio]30. Superficiem planam corpus luminosum et corpus umbrosum 
per equalia dividentem, super medium umbre eri
i est necesse; ex quo 
patet tot esse umbras eiusdem umbrosi corporis quot ipsum opponitur 
corporibus luminosis. 
Sit corpus super quod cadit lumen quod continetur a circulo AB [Fig. 
21], cuius centrum est punctum G, est sit unum corporum luminosum 


l Propositio [28] C I 28 mg. hab. EO 2 mediate V b 6 DB: DV v" 7 propter: per 
Rl I portionem: proportionem Rv"O 8 attingunt: contingunt RIO tangunt E co. est ex 
contingunt in v" 915: 16 R I 17: 18 R I et 17 om. Iv" I post 17 8m add. R p 10 ABE: 
ABC lCE 11 post circulus ser. et dei. P ab 12 ultra: intra 10 I protentensa l I cum: 
ut v" 13 post 14 add. R p I primi huius tr. v" I GB rep. et dei. C Iillam: aliam O 16 
deqtonstrandi modus: demonstrandum f?] et eadem modo E demonstrandum modis O 
l Propositio [29] C I 29 mg. hab. E I super rep. O 2 et om. O 3 post centrum scr. 
et dei. v" est I c: E l est v" om. v" I et rep. v" 4 punctus R I quod: qui R que 
PEO 5 CFG: FG R EFG Iv"VbEO I medium R 6 super medium: in medio E 9 
primo: principio Rl 
l Propositio [30] C I 30 mg. hab. E I corpus luminosum et corpus umbrosum: corpus 
umbrosum et corpus luminosum E 6 punctum om. l 


259 


5 


10 


15 


5 


5
>>>
260 


10 


contentum a circulo DE, cuius centrum est V. Sitque aliud corpus lu- 
minosum contentum a circulo ZH. cuius centrum est T. Videbitur itaque 
umbra opposita luminoso corpori DE, contenta a lineis AK, BL cuius 
medius punctus sit M. Cum ergo aliqua superticies diviserit corpus luminosum 
et corpus umbrosum per equalia, iIIa necessario transibit per lineam VGM, 
secabit ergo per equalia ipsam umbram, quia perpendiculariter erecta transit 
per ipsius corporis centrum, quod est punctum G. Similiter quoque super- 
ticies dividens per equalia ambo corpora ZH et AB transit per Iineam 
TG ductam per centra iIlorum corporum; sed eadem pertransit centrum 
umbre contente sub lineis AN et BS secundum punctum medium ipsius, 
qui sit Q. IIIa ergo superticies dividens corpora ZH et AB in duo media, 
dividet etiam umbram per duo equalia. Et quoniam superticies pIane se- 
cantes corpora umbrosa et lu minosa hinc inde per equalia sunt divise, 
patet quod secundum ipsas numerantur etiam et umbre. Patet ergo propositum. 
Universaliter en im tot erunt umbre eiusdem umbrosi corporis quot ipsum 
opponitur corporibus luminosis. 


., 


20 


5 


lPropositio]31. Corporis umbrosi remotioris a corpore luminoso umbra 
minus umbrescit, propinquioris vero magis. 
Quoniam enim, ut patet per 22 h uius, omne corpus umbrosum corpori 
luminoso propinquius iIIuminatur fortius corpore plus distante, patet quod 
umbra corporis propinquioris plus privat luminis. Radij quoque ipsam 
terminantes su nt fortioris luminis; umbra ergo inter iIlos radio s apparet 
nigrior et plus umbrescit, quoniam radij terminantes iIlas umbras sunt plus 
luminosi, propter quod etiam plus apparent umbre in presentia iIlorum. 
Corporis vero remotioris a corpore luminoso umbra minus privat luminis. 
Radij quoque continentes ipsam umbram sunt debilioris luminis; umbra ergo 
inter iIlos radios apparet debilior. minus ergo umbrescit. Patet ergo propo- 
situm. 


10 


[Propositio] 32. Omnis umbra multiplicata plus umbrescit. 
Esto enim ut sit unum corpus umbro sum obiectum pluribus corporibus 
luminosis. Palam ergo; per 30 huius, quoniam tot erunt umbre eiusdem 


7 contentorum v.. I est V. sitque om. l 7-8 DE cuius centrum est V...corpus luminosum 
contentum a circulo mg. inser. v" 8 ZH: ZB l I videbiturque ita v.. 14 ZH: ZA l 15 
TG: GG R v.. I centra: centrum v.. , mg. inser. V b lilIorum: vocabulum illectum in 
v" I corporum sed mg. inser. v" I eadem centrum: vocabula illeeta in v" 16 AN et BS co. 
est ex A V et VS in v" I BS: US l BG C 17 in: et v" 18 etiam: et l co. est ex et in 
v" I per: in E 19 divise: diverse CP (co. est ex diiverse) v.. divisi et diverse E 20 etiam: et 
O I et om. V. 21 quot: qyod v" 
l Propositio [31] C I 31 mg. hab. E . 2 vere E 3 per om. E I huius om. V. 4 
luminoso: illuminoso co. est ex luminoso in v" I propinqus O S post corporis ser. et dei. v" 
umbrosi I quoque ipsam tr. V. 9 post corpore ser. et dei. V. umbroso I umbra inser. V. om. 
E 10 quoque inser. P I lumine V. 11-12 Patet ergo propositam om. E 
l Propositio [32] C I 32 mg. hab. E 2 sit om. E 3 post luminosis add. E sit post 
umbre add. E unius 


.......
>>>
umbrosi corporis quot ipsum opponitur corporibus luminosis. Si itaque 
accidat ut umbre se intersecent, dico quod umbra multiplicata plus umbrescit. 
Quelibet enim umbrarum aufert aliquod lumen; multiplicata ergo umbra 
plura auferet lumina que remanent in alijs partibtis medij in quibus umbra 
non multiplicatur. sed remanet simpliciter umbra. Ergo ille simpliciter per- 
funditur aliquo lumine quod ad umbram multiplicatam non pertingit. Multi- 
plicata ergo umbra plus umbrescit, quoniam pluri lumine privatur locus 
illius um bre. Patet ergo propositum. 
[Propositio133. Duo corpora quorum unum obumbrat reliquum 
ecundum 
SUI medium, in eadem superficie, erecta super corpus luminosum consistere 
est necesse; et si in eadem superficie, propinqua adinvicem conslstunt. 
unum reliquum secundum sui medium obumbrabit. 
Hoc, q UiiutUIIl ad primam partem, patet per 30 huius. Quomam emm 
superficies piana corpus luminosum et corpus umbrosum per equalia dividens 
est erecta super superficiem corporis luminosi, et ipsa erigitur super medium 
umbre rei umbrose. umbra vero cadit super medium corporis obumbrati, 
ergo oportet quod illud corpus obumbratum secundum sui medium sit 
in superficie erecta super superficiem corporis luminosi. Ex hoc etiam patet 
secunda pars presentis theorematis: quoniam si duo corpora propinqua 
adinvicem secundum sui partes medias in eadem superficie erecta super 
superficiem luminosi corporis consistunt, unum ipsorum reliquum obumbrabit, 
quoniam remotius a lumine, quando fuerit propinquum iIIi, quod plus accedit 
ad lumen, cadet in umbra iIIius quod est propinquius lumini, ut quando 
idem radius transiens verticem propinquioris transit etiam verticem remotioris, 
vel punctum aliquod quod sit altius iIIo. Patet ergo propositum. 
[Propositio]34. Equedistantia linearum radialium vel ipsarum concutsus 
non est taliter per se ex natura radiorum, sed ex proportione dyametri 
corporis luminosi ad dyametros corporum umbrosorum. Ex quo patet quod 
lumen ditfunditur uniformiter per aerem circumstantem. 


4 umbrosi corporis tr. RPEO I quot: quod v.. I post ipsum scr. et dei: 
 corpo- 
rum I opponitur: opponunt lPv..C I corporibus luminosis tr. l 5 se om. E I 
umbrescet C 6 enim umbrarum tr. v.. I aliquod lumen om. v.. I post multiplicata add. v.. 
ad lumen 7 plura auferet tr. v.. I auITeret co. est ex auITert in 
 I in 2 om. E 8 
multiplicabitur f?] E I simpliciter l : simplex Pv..EO I simpliciter 2 : simplex RlPv..O 8-9 
profunditur Rlv..
 9 multiplicantem l I pertingit: contingit nec pervenit E 10 plurimo 
R plurinium l 11 umbra E umbet f?] sive umbret f?] v.. I itaque CE 
l Propositio [33] C I reliqum C 2 eadem: ipsa 
 3 neccesse est Rl 40bumb- 
rabit co. est ex umbrabit in 
 5 Hoc quantum co. est ex sit qua in 
 I post partem hab. 
pertinet v.. 7 est om. l et inser. alm. v" I ipsam C I medium: lumen Rl 9 quod: ut 
v.. I illud: idem v" 10 etiam om. Iv.. et inser. alm. 
 11 Theoramatis v.. theoreumatis 
 
E 13 luminosi: illuminosi l v" Iluminosi corporis tr. PEOC I consistunt co. est ex 
conslstant in v" RIJW.EO inser. v" 14 remotioris v.. I propinqum C I accidit V b 115 quod: 
quoniam v..V b 16 verticem: virtutem l I etiam: ad l et V. etiam ad v" (etiam inser) 
1 Propositio [34] C I 34 mg. hab. EO I Equidistantia RE 


ł-o..... 


261 



 


10 



 


10 


15
>>>
262 


5 Hee patet per 17 et per 18 huius et potest sie exemplariter dec1arari. 
Sit enim eorpus luminosum eireulus AB [Fig. 22] et una linearum radialium 
ab ipsa egredientium sit linea AG, et alia linea BG, et eoneurrant iIIe 
in puneto G. Sit item una linea EV et alia DZ, et sint EV et DZ 
equedistantes. Sitque eorpus unum, euius dyameter sit minor dyametro 
10 eorporis luminosi, super quod eadit lumen. positum inter duas AG et 
BG se eontingentes. euius maior circulus sit TI: et eontingat ipsum linea 
BG in puncto I. et linea AG in puncto T. Et corpus aliud equale corpori 
luminoso super quod eadet lumen sit positum inter duas lineas equedistantes 
EV et DZ illud corpus contingentes. cuius dyameter sit KL, contingaturque 
15 a linea EV in puncto K et a linea DZ in puncto L. Umbra itaque proveniens 
a eorpore TI minuitur et terminatur, et fit pyramidalis per 27 huius, ideo 
quia radij eontingentes eorpus Tł, qui sunt AG, BG concurrunt in puncto G. 
Umbra ergo corporis TI continetur a duabus lineis IG et TG et superficie 
eorporis TI que est a parte G. Umbra ergo finitur apud punctum G. 
20 Umbra ergo eorporis KL protensa inter lineas equedistantes LZ et KV, 
ut patet per 26 huius, non terminatur ad aliquod punctum, quoniam iIIe 
linee continentes umbram in infinitum protracte non concurrunł. Si vero 
eorpus Tł motum extra lineas AG et BG ponatur intra lineas EV et DZ 
eoneurrent linee EV et DZ et variabitur umbra ab ipsis prius contenta 
25 seeundum diversitatem proportionis dyametrorum eorporis TI et corporis 
KL ad dyametrum eorporis luminosi BA. Et ex hoc patet quod radij 
per se non sunt linee neque regulares, neque irregulares, neque equedistantes, 
neque eoneurrentes, sed aeeidit eis lineatio per aspeetum ad eorpora quibus 
incidunt: et equedistantia et.eoncursus aeeidunt eis per proportionem dyametro- 
30 rum eorporum umbrosorum ad dyametros corporis luminosi. Diffunditur 
ergo lumen uniformiter per totum aerem eireumstantem ita ut omnis punetus 
aeris, a quo possibile est produei lineam reetam ad aliquod punctum eorporis 
luminosi illuminetur a lumine eorporis luminosi, ut patet per 19 huius. 
Patet ergo propositum. 
[Propositio]3S. Radij ab uno puneto luminosi eorporis proeedentes seeundum 
linearum lo ngitudinem ad equedistantiam sensibilem plus aecedunł. 
S Hoc RlV.O I per 2 om. Rl I exemplariter dec1arari tr. v" 8 ser. et dei. v" tunc post sit 
item: tunc IV., inser. v" I DZ 1 : HZ l BZ CPV.Vb (co. est ex HZ) EO DZ 2 : HZ n;.v" BZ 
CPEO 9 aequidistantes R 10 duas: duo l 13 aequidistantes RE 14 DZ: UZ l BZ 
CPV.v"EO I contingaturque: contingatque V. IS in puncto K mg. inser. v" I DZ: BZ 
lCP V. v" EO 16 a: ex l co. est ex ex in v" 18 IG: LG l co. est ex LG in v" 20 
aequidistantes R 21 ad rep. O 22 continentes: contingentes lV b 23 extra: intra 
v" I AG: AB Iv" I DZ: BZ lCPV.EO LZ v" 24 concurrent linee EV et DZ om. O I DZ: 
OZ lCPV.v" E 26 luminosi om. l mg. inser. v" 27 aequidistantes R 28 sed: si 
v" I lineatifi: linatio V. linea d V b I aspectum: respectum Rl aspectam C I post corpora 
add. IV. in v" scr. et dei. in 29 aequidistantia R I post equedistantia ser. et dei. C ac- 
cidit I post eis ser. et dei. C sus [1] 31 circomstantem E I omnes punctibus V. 33 
lumine corporis tr. V. '" post corporis ser. et dei. V. luminis I 19: 29 [1] V. 
1 ProposItlo [3S] C I 35 mg. hab. EO 2 linearum co. est ex lineam in v" I aequidistan- 
tiam R
>>>
263 


Esto ut a puncto medio corporis luminosi, quod sit A. egrediuntur 
radij AB et AG [Fig. 23] equales; copuletur quoque basis BG et ducatur 
linea DE secans trigonum ABG citra medium sui lateris AG equedistanter 5 
basi RG. per 10 8m et 31 0m I'. Protrahaturque a runcto A linea AZ per- 
pendiculariter super basem BG, per 12 8m Ii, que secet lineam DE in 
puncto V; dividaturque linea EG in duo equalia in puncto H, per wam 
Ii. et linea OB in puncto T. ducaturque linea HT. Linea ergo HT erit 
equedistans ....asi GR. rer 2 8m VI': secabit ergo Iineam VZ. rer secundam 10 
primi huius. sit punctus sectionis K. Ducantur item a punctis E, D, H, T 
linee perpendiculares super basem BG. que sint EL, DM, HN, TS. Secabit 
quoque perpendicularis EL lineam HT; sit punctus sectionis Q. et punctus 
sectionis linearum DM et HT sit F. Erit ergo linea QF equalis linee ED, 
per 34 8m t. Patet ergo quod linea HT est maior quam linea DE. Quia 15 
itaque trigona AVE et EHQ su nt equiangula, per 29 8m Ii, erunt, per 
4 8m vt. latera ipsorum proportionalia. Quia ergo, ut patuit supra, linea 
AE est maior quam linea EH. erit ergo linea EV maior quam linea HQ. 
Sed linea HT est maior quam linea ED, ut preostensum est; ergo, per 
9 primi huius. maior est proportio linee EV ad lineam ED quam linee 20 
QH ad lineam HT Est enim proportio linee EV ad lineam ED sicut 
linee HK ad lineam HT, per 4 8m vt et per 16 8m et per 18 8m Vi. Sed 
linea HQ est pars linee HK; ergo, per 8 8m Vi, minor est proportio HQ 
ad HT quam HK ad HT. Minor est ergo proportio linee HQ ad HT 
quam EV ad ED. Eodemque modo demonstrandum quod linee GN ad 25 
lineam GB minor est proportio quam linee HQ ad lineam HT. Excessus 
itaque basis GB super basem HT est minor excessu basis HT super basem 
DE; et quanto bases su nt remotiores a puncto A corporis luminosi tanto 
excessus remotiorum basium super bases viciniores plus minuitur. Palam 
ergo quia in remotiori distantia radij quasi ad equedistantiam plus proeedunt; 30 
et eum quantitas excessus basium sit quantitatis non sensibilis, tune linee 


3 quod sit A om. E 5 citra: erit l I aequidistanter R 6 post 31 0m add. R p 7 basim 
Rl E I post 12 0m add. R P 8 10 co. est ex 20 in V b I post 10 0m add. R P 9 linea ergo HT 
om. V. 10 aequidistans RE I GB: GL IV. I post 2 0m add. R p I post secundam add. 
R t II post Ducantur add. E recte I post D hab. l et, v" ser. et dei. et 12 perpendiculares 
co. est ex perpendicularis in v" I basim R I TS: TG C 13 Q: K V. 13 -14 Q et 
punctus sectionis mg. inser. v", om. l 14 sectionis co. est ex sectioniis in E 15 34: 23 
V. I post 34 0m add. R p I quod: quia V. I post linea 2 ser. et dei. V b HT 16 post 29 0m add. 
R et 32 p 17 post 4 0m add. R P I proportionabilia l I patet l 19 post HT add. V. 
etiam 20 post 9 add. R t I EV: EB l 22linee om. E I post 4 0m add. R P I 16: aa R 26 
E I per 3 om. Rlv. E I 18: 16 R I post 18 0m add. R P 23 pars linee HK: linee HK pars 
E I post 8 0m add. R P 24 ad 2 co. est ex et in v" I est ergo fr. C 25 ad ED inser. alm. in 
l, inser: mg. in V b I modo om. V. 26 GB: GV V. I proportio quam tr. V b I quam om. 
V. 27 GB co. est ex 8GB in v" I post GB scr. et dei. V b et I basim 1.2 R lexcessus 
PEO I HT 2 om. V. 28 tanto co. est ex tanta in v" 29 remotior V. I minuentur 
R minuuntur l 30 quia: quod PEO I equidistantiam RE I procedunt: accedunt E 31 
cum: quia E I quantitas V b 


--
>>>
264 


35 


radiales erunt quasi equedistantes. Quoniam enim linea BG sensibiliter non 
excedit lineam HT, tunc erunt HG et TB radij quasLequedistantes secundum 
sensum. Et hoc est propositum. Et forte ad istud multum cooperatur 
proprietas radiorum, que semper ut potest approximat sue perpendiculari; 
propter quod radij omnium punctorum totius corporis luminosi semper 
concurrunt in quolibet puncto corporis illuminandi. et sic constituunt py- 
ramidem radialem. 


5 


/Propositio]36. LUmine incidente per fenestram super corpus oppositum 
solidum erit luminis perimeter amplior perimetro fenestre. 
Esto corpus luminosum, cuius centrum A [Fig. 24]. et circulus magnus 
DEG. et sit dyameter fenestre Be. Sitque linea TZ in superficie corporis 
solidi opposita lumini cui incidit radius. Producantur quoque linee radiales 
tangentes periferiam fenestre, que sint EB et GC. Hee itaque linee secabunt 
se in aliqua parte medij; sit punctus communis sectionis F. Et hee linee 
producte incidant superficiei cbrporis oppositi lumini, cadatque linea EB 
in punctum Z et linea GC in punctum T. Quia itaque in trigono FTZ 
latus TZ est maius latere Be, quoniam trigonum FTZ maius est trigono 
BCF, et quoniam per omnem punctum periferie fenestre sic incidunt radij 
se secantes, ideo quod a quolibet puncto corporis luminosi in totam fene- 
stram fit missio luminis, per 19 huius. palam quoniam perimeter luminis 
incidentis corpori solido opposito fenestre est maior perimetro fenestre. 
Et hoc proponebatur. 


10 


15 


5 


[Propositio]37. Ad centrum circularis foraminis radio a centro corporis 
luminosi perpendiculariter incidente, lumen in superficie densi corporis eque- 
distante superficiei foraminis est vere circulare. 
Sit circulus foraminis ABGD, cuius centrum E [Fig. 25], cui sit equedistans 
superficies solidi corporis FHKL. Et erigatur a centro E linea EZ perpen- 
diculariter super superficiem ABGD circuli. In quocunque itaque puncto 
linee EZ sit centrum corporis luminosi. Dico quod lumen incidens 
superficiei FHKL est vere circulare. Palam enim. per 65 primi huius. quoniam 


32 aequidistantes R I post BG scr. et dei. v" non 33 TB: TU l TV v.. et TB co. est ex TU in 
v" I aequidistantes R 34 est om. v.. 35 potest: patet v" I approximat: appropinquat 
f?] E I post potest ser. et dei. O appropinquat 37 post concurrunt scr. et dei. v" ab 
aliquo I in: a l mg. inser. v" in quolibet 
l Propositio [36] C I 36 mg. hab. EO I super co. est ex sic in v" 2 solidum om. 
E I perimeter co. est ex perimmeter in E 4 sicque v" 6 sint: sit PEO I et om. linser. 
v" I GC: GT C GE f?] O et GC co. est ex GE in v" 7 hee om. E 8 oppositae R 9 
puncto v.. GC: GT C GE O I puncto f?] v.. I FTZ: FCZ Iv" 10 TZ: CZ l I BC: BT 
l v" I FTZ: FCZ l 11 BCF: BEF O I omne l 13 missio : mistio f?] v" I 19: 20 R 10 
l 14 est inser. v" I post maior add. E et amplior 
l Propositio [37] C I 37 mg. hab. O 2 in cidente v.. 2-3 aequidistante R 3 
foraminis inser. v" 4 cui om. l I aequidistans R 5 superlicies om. E superliciei O 6 In 
co. est ex a in v" I itaque puncto tr. E I itaque co. est ex ita in O 8 65: 64 l I post 65 
hab. R t
>>>
omnes linee ZA, ZB. ZG, ZO ducte a polo Z ad circumferentiam sunt 
equales et equales angulos continent cum linea EZ, per 8 8m t. Producatur 
itaque linea ZE ultra punctum E ad superficiem equedistantem circulo fora- 
minis que est FHKL; incidet que perpendiculariter super illam, per 14 8m 
Xli. Sit ut incidat in punctum M, producaturque linea ZB ad superficiem 
FHKL in punctum K, et linea ZA in punctum F, et linea ZD in punctum 
H. et linea ZG in punctum L. Eruntque linee AF, BK, OH, GL, per 
25 prim i huius, equales propter equedistantiam superficierum et equalitatem 
angulorum. Tota ergo linea ZF erit equalis toti linee ZH et ZK equalis 
linee ZL. Ducantur quoque linee FM. HM. KM, LM. In trigono itaque 
FMZ, basis FM erit equalis basi HM trigoni HMZ, per 4 8m Ii. Eodemque 
modo erit linea KM equalis linee HM et linea LM equalis linee KM. 
Palam ergo, per 9 8m lIIi, quoniam superficies FHKL est circularis et ipsa 
est ad quam terminantur radij luminis incidentis per fenestram ABGO, 
quoniam de omnibus alijs lineis eadem est demonstratio. Patet ergo pro- 
positum. 


[Propositio]38. Per centrum circularis foraminis radio luminoso oblique 
incidente superficiei densi corporis substrate superficiei foraminis, lumen 
incidens erit figure sectionis pyramidalis, cuius maior dyameter erit in super- 
ficie erecta super superficiem fenestre et super superficiem corporis sub- 
strati. 
Esto foramen circulare ABCD [Fig. 26], cuius centrum E, cui sit super- 
ficies equedistans HMKL, et sit F centrum corporis luminosi. Sitque primo 
ut linea FE oblique cadat super superficiem circuli ABCD; hec itaque 
producta incidet superficiei HMKL similiter oblique propter equedistantiam 
superficierum, argumento 23 primi huius. Incidat itaque in punctum G, 
et ducatur linea AEB dyameter circuli. Sit itaque angulus AEF acutus; 
erit ergo, per 13 8m t, angulus BEF obtusus. Et quia quadratum linee 
FA valet minus duobus quadratis linearum EF et EA. per 13 8m II'. et 
quadratum linee BF est maius quadrato linee FE et quadrato linee BE, 
per 12 8m IIi, quadratum vero linee BE equale est quadrato linee AE, quia 


9 sunt: semper v" lO angulos mg. inser. v" I post 8 0m add. R P 11 equidistantem 
RE 12 post w m add. R p 13 incidet v.. I in om. O IZB: ZV v.. 14 ZA: ZH f?] 
PZHEOZV v.. I postpunctum 2 scr.etdel.Ed ł5AF:FHKL V. I BK:DG v.. I DH: 
DG CPEO om. I V. I post per scr. et dei. v" numeri i1Iecti 1625 mg. inser. v" I post 25 
add. R t I propter: per O I aequidistantiam R 17 post toti ser. et dei. V. super- 
ficiei I toti Iinee tr. E 18 KM om. v.. I LM inser. P 19 FM co. est ex FMZ in 
V b post 4 0m add. R p 20 modo erit linea: linea modo erit v" 21 9 co. est ex e in 
v" post 9 0m hab. R p quoniam superficies om. E 
1 Propositio [38] C I 38 mg. hab. EO 2 substracte Pv..VbO 3 sectionis inser. 
v" 4-5 substracti PV..V"O 7 equidistans RE 8 circuli om. Iv.." inser. v" 9 aequidis- 
tantiam R lO post 23 add. R t I incidatur l incidat co. est ex incidatur in v" I puncto 
v.. E 11 AEF: AEB l 12 post 13 0m add. R p I 13: 14115 CPv..v"EO I post obtusus add. 
R ducantur ergo Iineae FA. FB. 13 post duobus scr. et dei. v" lineis I post 13 0m add. 
R P 14 Iinee 1 0m . E I FE et quadrato linee om. E 15 post 12 0m add. R P 


'110...... 


265 


10 


lS 


20 


s 


10 


lS
>>>
266 


sunt equales semidyametri, et quadratum linee FE est eommune. patet 
quia quadratum linee FB est maius quadrato linee FA. Ergo linea FB 
est maior quam linea FA. Productis que lineis FA et FR ad superfkiem 
HMKL. si linea FA incidat ad punctum M et linea FB ad punctum 
20 L. erit linea FL maior quam linea FM. per eadem que prius. Copula- 
tisque lineis LG et MG ad punctum G. eui ineidit radius transiens centrum 
foraminis fenestre. erit quoque. per 2 am Vi et per I ł-am Vi proportio linee 
LG ad lineam BE sieut linee GM ad lineam EA. quoniam utrarumque 
iIIarum proportio est adinvieem sieut linee GF ad lineam FE. Est erg,. 
2S per 16 8m Vi. proportio linee LG ad lineam MG sieut linee BE ad lineam 
EA. Sed linea BE est equalis linee EA. Ergo linea LG est equalis linee GM. 
Dueatur tune CD dyameter super AB dyametrum orthogonaliter et eonti- 
nuentur linee FC, FD produeanturque ad superficiem HMKL in puneta H 
et K. et dueatur linea HGK. Et quoniam superfieies in qua sunt linee FE 
30 et AB sola est ereeta super eireulum fenestre. quoniam omnes alie superfieies, 
in quibus est linea FE, ineidunt iII i superfieiei oblique - sie enim aeeipimus 
lineam AB - erit ergo superfieies AFB ereeta super superficiem cireuli fe- 
nestre. Palam ergo quia angulus FED est equalis angulo FEe. Est ergo, 
per 4 8m Ii, linea FD equalis linee Fe. Ergo, ut prius. eril linea HG equalis 
3S GK et linea FH equalis linee FK. sed et FG est eommunis. Et quia 
linea HK est perpendieularis super lineam ML et super lineam FG. palam, 
per 4 am Xt. quod linea HG est perpendieularis super superfieiem in qua 
sunt linee FG et MG. Ergo, per 18 8m Xt. erit superfieies HMKL ereeta 
super superfieiem FMG; ergo et superfieies FMG est ereeta super super- 
40 fieiem HMKL. Ymaginetur itaque a puneto G termino axis. qui est FG, 
eireumduei pyramidi iIIuminationis eirculus. per 102 prim i huius; eritque, 
per 100 et 89 prim i huius, axis FG ereeta super iIIum eireulum et ipsa 
est obliqua super superficiem HMKL. Erit ergo. per 103 priini huius, linea 
HMKL sectio pyramidalis, cuius maior dyameter erit in superficie FML 
4S erecta super superficiem HMKL. Patet ergo propositum. Et si superfieies 


16 semidyametri: dyametri V. 17 quia: quod Rlv" I FA: KA l et FA co. est ex KA in 
v" 20 FL: FB l 21 et om. l inser. et dei. v" 22 post 2 0m add. R. p I per om. E I post 
11 0m add. R P 23 utnunque l utrunque V. 24 post adinvicem ser. et dei. v" sicut linee gm ad 
lineam ea quoniam utrumque illarum proportio est ad invicem linee inser. P 25 post 16 0m add. 
R P I Vi: huius V. 26 EA 2: BA CP V. v" (co. est ex EA) EO I post linea ser. et dei. BG 
V. 27 tunc: item CPEO 27-28 continentur v" 28 FC: FE IV. I FC co. est ex FE in 
v" I H: F V. 30 post quoniam ser. et dei. C corporis ex ypothesi 31-32 in quibus est 
linea FE.. .lineam AB erit ergo superficies om. V. 33 quia: quod PO 34 post 4 0m add. 
R p I FD: FC [1] V. I post FD hab. CV.v" (inser.) est I FC: FE f?] E I HG: BG 
Iv" I post equalis add. E linee 35 et linea FH om. V. I sed om. R I et 2 om. V. I FG: 
TFG V. I et 3 om. l inser. v" 36 post lineam 1 ser. et dei. V. MK 37 post 4 0m add. 
Rp 37-38 quod linea HG esL. FG et MG Ergo per 18 0m XII mg. inser. P 38 post 18 0m 
add. R p 39 et om. E 40 itaque mg. co. est ex ergo in V b I termino: tertio l I qui: quae 
l 41 pyramidis l v" I post 102 add. R t I primi mg. inser. v" I eritque: erit Rl co. est ex 
erit in v" I post eritque hab. Rlv" ergo 42 post et 1 hab. PV.EO per I post 89 add. 
R t I ipse R 43 obliquus R I post 103 add. R t I 103 rep. V b I primi huius om. V.
>>>
fenestre circularis sit basis pyramidis iIIuminationis. ita quod centrum corporis 
luminosi sit polus circuli fenestre. et axis erectus sit super superficiem fe- 
nestre. superficies vero solidi corporis excipientis radios luminis non fuerit 
equedistans superficiei fenestre. adhuc erit figura luminis sectio pyramidalis. 
quod est premisso modo demonstrandum. Ducta enim. per 102 primi huius. 
a puncto L. termino longioris radij qui est FL, superficie equedistante 
superficiei fenestre. patet. per 100 prim i huius. quod iIIa superficies secabit 
pyramidem illuminationis secundum circulum qui sit LPQ. Ergo superficies 
HMKL secat ipsam secundum pyramidalem sectionem. Patet ergo propositum. 


IPropositio]39. Omne lumen per foramina angularia incidens rotundatur. 
Quod hic proponitur patet per 35 huius. Quoniam e
im omnes radij 
ab uno puncto luminosi corporis procedentes secundum linearum longitu- 
dinem ad equedistantiam sensibilem plus accedunt. patet quod radij secundum 
foraminum angularium dispositionem ipsis angulis incidentes se applicant 
equedistantie radij perpendiculariter vel circa hoc superficiei foraminis inci- 
dentis. Retrahunt ergo se ab angularitate. et sic lumen superficiei foramini 
obiecte incidens incipit rotundari. Et quoniam, ut patet per 20 huius. a puncto 
cuiuslibet corporis luminosi lumen diffunditur secundum omnem lineam, que 
ab illo puncto ad oppositam superficiem duci potest. omnes quoque iII i 
radij in quolibet puncto medij concurrunł. Patet quod ipsi in quolibet 
puncto se intersecant et radij inferiorum punctorum ipsius corporis luminosi 
in punctis linearum fenestre alios radios superiorum punctorum s;cant. et 
ultra protenduntur. Et sic lumen huiusmodi fenestras pertransiens rotundatur, 
quod non adeo accideret si solum ab uno puncto luminosi corporis egre- 
derentur radij fenestram penetrantes. Patet ergo propositum. 


47 post fenestre scr. et dei. v.. superlicies circularis 48 solidi corporis co. est ex corporis solidi in 
v" I excipientis: exipientis PVbO I fuerit: fuit l 49 aequidistans R 5(1 enim: tamen V u 
cum v" I per om. v" I post 102 add. R t 51 termino. tertio 'v" I longioris co. est ex 
longitudinis in v" I aequidistante R 52 post 100 add. R t 53 qui: quae l 54 Patet ergo 
propositum om. E 
I Propositio [39] C I 39 mg. hab. EO I angularia co. est ex angulariter in 
v" I rotundatur incidens per foramina angularia v.. 2 post per scr. et dei. O tertiam 51 3 
luminosi corporis tr. E I precedentes v"EO 4 aequidistantiam R I sensibile C I post 
quod add. l anguli 5 post foraminum scr. et dei. O diss f?] I angularium co. est ex angularum 
in v,,: angularum l 6 aequidistantiae R I radij perpendiculariter: radio perpendiculari 
E I cirta C I hoc om. R huius lV b f?] hic OE f?] 6-7 incidenti P 8 obiecte: oblique 
v.. I patet om. v.. inser. v" I ut om. E 9 secundum: super Iv" 10 omnis l I quoque: 
enim Rlv" II concurrunt co. est ex concurrint in v" 12 intersecant co. est ex intersecent in 
v" I ipsius om. l inser. v" 13 alio Iv" I radio l, radios co. est ex radio in v" 14 
huiusmodi: hoc l huius V b I fenestram l, fenestras co. est ex fenestram in V b I post fenestram 
scr. et dei. v" illectum voeabulum 15 non: nunc PO I post non hab. v.. illecta littera [c 
? t?] I adea: ab eo Iv.. I si: sibi f?] V b , si co. est ex superficies in v.. I solum ab uno puncto: 
ab uno puncto solum E I post luminosi ser. et dei. v" illectum vocabulum 


267 


SO 


s 


10 


tS
>>>
268 


[propositio]40. Radio luminoso medio puncto foraminis quadrati perpen- 
diculariter incidente, lumen superficiei corporis equedistantis superficiei fora- 
minis incidens est quadratllm ad circularitatem aliquam accedens. 
Sit centrum corporis luminosi E Wig. 27] et foramen quadratum sit 
s ABCD. cuius puncto medio qui sit F incidat perpendiculariter radius EF; 
sitque superficies corporis densi equedistantis superficiei foraminis que est 
GHKL. Dico quod lumen incidens iIIi superficiei erit figure quadrate: Fiunt 
enim due pyramides unam verticem habentes punctum E. quarum maioris 
basis est GHKL, minoris vero basis est ABCD. et earum bases su nt eque- 
10 distantes; sunt ergo similes, per 99 primi huius. Quia ergo basis ABCD 
ex ypothesi est quadrata. patet quod etiam basi s GHKL est quadrata. 
Et hoc est propositum primum. Quoniam vero. per 35 huius. radij longiores 
ad aliquam equedistantiam accedunt, accedit et hec figura ad aliquam cir- 
cularitatem. propter compressionem radiorum vel propter ipsorum inter- 
1 s sectionem in punctis linearum terminantium fenestras. ut diximus in premissa. 
Patet ergo propositum. 


[Propositio]41. Per medium quadrati foraminis radio oblique incidente 
superficiei densi corporis substrate superficiei foraminis, lumen incidens erit 
figure altera parte longioris suis angulis equaliter arcuatis. 
Esto, ut in premissa. centrum corporis luminosi punctum E [Fig. 27] et 
s periferia quadrati foraminis ABCD. cuius medio puncto qui sit F oblique incidat 
radius EF. Sitque superficies corporis densi substrati iIIi foramini que GHKL, 
cui similiter oblique incidat radius. Dico quod figura luminis in substrata 
superficie erit altera parte longior. Quoniam enim iIIe superficies non sunt 
bases pyramidum iIIuminationis, sed solum secantes iIIas pyramides oblique, 
10 patet per 99 primi huius quoniam ambe figu're et ABCD et GHKL, sive 
earum superficies equedistent sive non equedistent, sunt figure altera parte 
longioris: quoniam ille figure que secundam illa puncta. quibus axis EF 


I Propositio [40] C I 40 mg. hab. EO I post luminoso ser. et dei. v" in 2 aequidistan- 
tis R 4 corporis luminosi tr. O 5 medio: in eo l I. incidit y'PEO 6 sitque: sit haec 
l I aequedistans R aequedistanti l 7 Fiunt: sunt R 8 pyramidens E I unum 
RE I verticem: a vertice V M 9 est om. y. 9-10 aequidistantes R 10 post 99 add. 
R t 11 etiam: et Rl I bases E I est 2 om. v" 12 post vero scr. et dei. C radij 13 
aequidistantiam R 14 compressionem co. est ex compermissionem f?] in v" 15 feneslrae 
l fenestras co. est ex fenestrae f?] in v" 
I Propositio [41] C I 41 mg. hab. EO I oblique incidente tr. E 2 substracte 
v" I post foraminis ser. et dei. O luminis 3 figura Rly'O, figure co. est ex figura in 
v" I longior Rl I equaliter: aliqualiter Cv" (co. est ex equaliter) 4 premissa co. est ex 
prima in v" I post luminosi scr. V M centrum corporis 5 F: E R 6 subtracti V b I que om. 
y. 7 similiter oblique tr. O I incidat co. est ex incidit f?] in v" I subtracta v" 8 enim 
om. E 9 pyramidum co. est ex pyramidis in v" 10 post 99 cuM. R t I primi huius tr. 
y. I et l om. R1 inser. v" 11 equedistent l : aequidistent R aequedistant l I aequidistent 2 
R I sive non equedistent om. y. 12 longiores RlC
>>>
269 


propositis superficiebus oblique incidit. pyramides terminant sunt ambe 
quadrate, relique vero oblique secundum iIIa puncta axi incidentes sunt 
ambe altera parte longiores. Patet ergo propositum primum. Et quoniam. 15 
ut patet per 35 huius, longiores radij quasi ad aliquam equedistantiam 
accedunt, patet quod anguli illius figure luminis aliqualiter arcuantur sicut 
etiam in duabus premissis dedaratum est. Et hoc est propositum. 


lPropositio]42. Per medium secundi dyafani densioris primo radius per- 
pendicularis ductus a centro corporis luminosi super superficiem obiecte 
corporis semper penetrat irrefractus. 
Huius propositionis probatio plus experientie instrumentorum innititur 
quam alteri demonstrationum. Cum ergo quis experiri voluerit modum frac- s 
tionis radiorum luminosorum in medio secundi dyafani densioris primo, 
ut in aqua. que est densior aere, assumat vas rectarum horarum qualis- 
cunque voluerit materie vel figure. Dum tamen sit altitudo horarum maior 
medietate cubiti. et dyameter latitudinis eius sit non minor dyametro in- 
strumenti, ut faciendum premisimus in prima huius. Et planentur hore 10 
illius vasis donec superficies per eius horas transiens sit equalis piana, et 
ponatur in fundo vasis aliquod corpusculum coloratum visibile, ut aliquod 
numisma vel res picta diversi coloris, deinde impleatur vas aqua dara. 
Cum ergo quieverit motus aque, si aspiciens visum perpendiculariter proiecerit 
super medium numismatis vel picture, inveniet figuram et colorem et ipsorum 15 
situm et partium ordinationem eo modo quo sunt secundum se ordinata 
si in aere viderentur. Consideret ergo experimentator iIIum sui corporis 
situm sive sit stans sive sedens. et sui distantiam a vase, et situm ipsius 
vasis, et omnia circumstantia iIIam visionem. Ponatur itaque vas istud plenum 
aqua dara in loco in qua splendet sol. et sistatur vas taliter ut superficies cir- 20 
cumferentie vasis sit equedistans orizonti; hoc autem potest perpendi ex 


13 propositis co. est ex proportio in v" lobIique: aliquae l I incidunt l I terminant om. 
l lambe: abe l 13-14 incidit pyramides terminant ... relique vero oblique mg. inser. 
v" ł6 longiores radij tr. Rlv" I aequidistantiam R 18 etiam: et Rl om. E 
I Propositio [42] C I 42 mg. hab. EO I primis v" 3 post irrefractus add. RAIhazen 
3 n 7. 4 proportionis v" I probati l I innititur: immitatur v" 5 quam: quoniam 
O I demonstrationi PEO demonstrationem v" I motum v" 7 que: qua v" lorarum 
Rl 8 post voluerit ser. et dei. v" me I tamen co. est ex tam in P, om. E I sit altitudo 
horarum maior: altitudo orarum sit maior E lorarum Rl 9 minor: maior l 10 ut: quod 
R, om. C I premissis R I orae Rl II illius: eius E I Ofas RIE 12 ut aliquod om. 
l mg. inser. v" 13 numisma: minisma v" nunmisma f?] EO I res: tres l I diverisi [1] 
E I inpleatur V.v" 14 queverit v.. I asspiciens E 15 super medium mg. inser. 
v" I nunmismatis EO I vel: ut l I picture: figure seu picture v.. I figuram et colorem: 
colorem et figuram v.. I calorem co. est ex colorem in v" 16 post partium ser. 
E tium 16-18 secundum se ordinata... sui corporis situm sive sit om. v.. 16 ante secundum 
hab. E per se et 17 Consideret co. est ex considerit in v" 18 stans sive sedens: sedens sive 
stans O I vase: base l 19 vasis: vasia v.. I illam visionem om. l I ilIam: ipsam 
E I visionem co. est ex vasionem in v" I itaque: ut a E 21 aequidistans R I horizonti 
Rl I potest: patet J
>>>
270 


hoc si superficies aque sit equedistans periferie vasis. Deinde imponatur 
instrumentum in hoc vas ita quod pinnule super extremitates regule exis- 
tcntes superponantur hore vasis ex utraque parte. Tun
 ergo mediet;ls 
2S instrument i cum tota regula erit intra vas. Deinde auferatur aqua do nec 
superficies aque secet centrum instrumenti. et revolvatur instrumentum in 
circuitu vasis donec hore super aquam obumbrent alias sub aqua; et tunc 
retenta regula cum altera manuum. revolvatur instrumentum cum reliqua 
manu in circuitu sui centri, donec lumen solis pertranseat foramen LMN 
30 quod est in hora instrumenti. et foramen lamine quadrate, et perveniat 
ad superficiem aque. quia lumen pertransiens foramen rotundum ampliatur 
semper, per 36 huius. 
Sistatur quoque taliter instrumentum ut lumen cadens super laminam 
secundi foraminis. quod est XYŻ, situm habeat equalem. Et tunc experimen- 
]S tator. reductis manibus ab instrumento secundum omnem situm et modum 
quo prius aspexit numisma. inspiciat ad fundum aque ex parte quarte 
instrumenti cuius hora est abscisa que est AD. Invenietque lumen pertransiens 
ex duobus foraminibus super superficiem hore alterius que est intra aquam, 
et lumen inter duos circulos extremos trium circulorum equedistanter signa- 
40 torum. aut addens super distantiam illorum circulorum modicum, et erit 
additio equalis ex duobus lateribus circulorum. Ex quo patet quod medium 
punctum huius luminis cadit in aliquod punctum circumferentie medij circuli 
illorum trium circulorum, ut in punctum P. Deinde acus ferrea vellignum 
minutum in interiori parte foraminis hore instrumenti applicata pertranseat 
4S medium foraminis dyametraliter, et tunc inspicienti ut prius videbitur umbra 
acus in media lucis opposite, per II huius, dividens eum per equalia. 
Deinde retrahatur acus, donec acumen eius sit in medio foraminis, et erit 
umbra extremitatis acus in media lucis que est in superficie aque et eius 
que est intra aquam. Et universaliter secundum quam proportionem acus 
so periferiam foraminis ut corda absciderit, secundum eandem proportionem 
umbra acus periferiam lucis in superficie aque vel sub aqua existentis abscindet, 
acu vero penitus remota, lumen revertetur. 
Palam ergo ex hijs quod punctus qui est in medio lucis intra aquam 


zz si mg. inser. J.-;,: quod E I equidistans RE I inponatur Cv. ponatur E 23 pinule 
J.-;, 24 superponatur lV M , superponantur mg. co. est ex superponatur in J.-;, I ore RlE 25 
vas rep. et dei. J.-;, 27 ore RlE lobumbrent co. est ex obumbrant in V b I aquam l 30 
ora RlE I et 2 om. l, inser. J.-;,: etiam E 31 pforamen V. I post foramen add. E iIIud 32 
semper om. E 33 Sistatur co. est ex sisquatur in J.-;, 34 habeat equalem tr. J.-;, 35 redutis 
J.-;, 36 aspixit V b I nunmisma E 37 ora Rl E lalescissa R I pertransiens om. 
J.-;, 38 ore RlE I est inser. V. I inter: in J.-;, 39 circulorum: angulorum lJ.-;, I aequidi- 
stanter R 40 aut: ut v., co. est ex ut in J.-;, 42 circumferentie medij tr. lJ.-;, 43 acus: ortus 
V. 44 minutum co. est ex minitum in J.-;, I ore RlE 45 ut prius videbitur: videbitur ut 
prius 1, co. est ex videbitur ut prius in J.-;, 46 oppositam f?] E 48 umbra mg. inser. J.-;, I et 
om. E 
 ascindit l, absciderit co. est ex abscindet in J.-;, 51 abscindet l, abscindet co. est ex 
abscindetin J.-;, I acu mg. inser. J.-;, I post acu ser. et dei. J.-;, acus 53 his R I qui: quael 


,,
>>>
existentis exit a puncto medio lucis in superficie aque existentis et quod 
punctus medius huius luci s exit a luce que est in centro foraminis super- 
ioris. Lux ergo que pervenit ad centrum lucis in superficie aque existentis 
extenditur secundum rectitud-inem linee recte per duo puncta M et Y, 
que sunt centra amborum foraminum. transeuntis. Et hec linea est in superficie 
medij circuli trium circulorum et est pars dyametri illius circuli, que est 
MP, cum sit equedistans dyametro circuli in base instrumenti existentis, 
que est FEG. Punctus ergo qui est medio lucis. in superficie aque existentis, 
est in superficie huius medij circuli. Sed et punctus P in medio lucis intra 
aquam existentis est in circumferentia medij circuli. Hec ergo duo puncta 
erunt in superficie medij circuli. Ergo et tota illa linea erit in superficie 
medij circuli, per lam Xli. 
Quod si lux, que est in superficie aque. non fuerit manifesta. mittatur 
regula minor in aquam et superficies eius in qua signata est linea dividens 
superficiem eius latitudinis per equalia applicetur superficiei aque, ut fiat 
una superficies cum ilIa. et alia eius superficies applicetur superficiei basis 
instrumenti. Pala m ergo ex premissis. in prima huius. quia linea que est 
in superficie regule est in superficie medij circuli. per M et Y centra duorum 
foraminum transeuntis; apparebit que lux que est in superficie aque super 
superficiem regule ei medium illius lucis super lineam que est in medio 
regule. Et si acus fuerit posita super medium superioris foraminis. obumbrabitur 
linea que est in medio regule. Et si acumen acus ponatur super centrum 
foraminis. cadet umnra acuminis _acus in medio lucis que est super regulam 
et ablata acu redibit lumen. .Sic ergo apparebit lumen cadens super super- 
ficiem aque apparitione manifesta. et patebit quod lux incidens centro fora- 
minis superioris ipsa est super lineam transeuntem per centra duorum fora- 
minum. Et quoniam superficies aque transit centrum instrument i, et super- 
ficies regule est una cum superficies aque, superficies itaque regule transibit 
centrum instrumenti. Erit ergo remotio centri luci s a centro instrumenti 
equalis medietati latitudinis regule. que est equalis perpendiculari cadenti 


54 post existentis scr. l et quod punctus medius huius lucis et v" scr. et dei. idem I exit: exijt l erit 
v.. I exit a puncto medie lucis mg. inser. v" 55 exit: erit l 56 que: cum l Ilucis: luos 
f?] V u 58 transeuntes Iv.. transeuntis co. est ex transeuntes in Y& I Et co. est ex per in 
v" I hec: huius l I est inser. v" I superficie: fine v.. 60 cum: tamen l I aequidistans 
R I basi R 61 est 1 0m . v" Ilucis: basis v" I post lucis add. RlY& que est I existentis 
om. R 62 P om. R 62-63 intra aquam existentis: existentes intraquam v.. 63 ergo: 
autem E 64-65 ergo et tota illa...superficie medii circuli om. l 64 et: est v" I linea mg. 
inser. v" 65 post lam add. R p 67 qua: acua l, qua co. est ex aqua in v" I post dividens scr. 
et dei. C ei 68 applicetur superficiei tr. l I superficiei: superficie V& I fiat co. est ex fiet in 
v" 70 in prima rep. E 71 regule est in mg. inser. V& I est om. l I per om. l I centrum 
l I duorum om. E 72 est om. v" I super: in E 73 illius: luminis l, illius co. est ex 
luminis in V& 74 -75 acus fuerit posita super. . . est in medio regule et si om. V u 74 superioris 
foraminis tr. Rl, co. est ex foraminis superioris in v" 75 ponatur rep. et dei. V& 76 mg. inser. 
v" cadet umbra acuminis I acus om. v.. 79 centrum l 79-80 foraminum co. est ex 
foraminim in v" 80 mg. inser. v" superficieis 81 regule: aque v.. I post itaque add. E linee 
vel 82 erit: exit Y& 



 


271 


ss 


60 


6S 


70 


7S 


80
>>>
- 


272 


8
 


a centro foraminis super superficiem basis instrumenti. Erit ergo centrum 
lucis que est in superficie regule vel aque centrum medij circuli. 
Revolvatur ergo regula donec angulus ipsius acutus transeat per centrum 
instrument i et pars inferior linee dividentis angulum eius per equalia sit 
in centro luminis quod est intra aquam. Acuitas ergo superior regule trans- 
ibit centrum circuli medij. Punctus ergo linee superficiei superioris regule. 
qui est in superficie aque. est centrum medij circuli et lucis que est in 
superficie aque, et erit illa linea semidyameter circuli medij. Immitatur ergo 
acus longa in aquam ita ut acumen ipsius sit in puncto anguli regule; 
secabit quoque umbra acus lucern que est intra aquam, eritque umbra 
acuminis acus ad finem regule que est in medio lucis. Et si fixo acumine 
acus. moveatur acus. umbra acus mutabit situm ad diversas partes lucis. 
Umbra tamen acuminis non mutata a medio lucis. ablata vero totaliter. 
acu. redibit lux totalis. Idem quoque accidit in quocunque punc to linee 
que est in superficie regule positum fuerit acumen acus. Ex quo patet 
quod lux existens in aliquo puncto lucis intra aquam procedit a puncto 
sibi sim iii in luce que est in superficie aque. et quod a medio puncto 
lucis que super aquam ad medium punctum lucis intra aquam protenditur 
radius secundum lineam rectam que est medium regule. Ex quo patet quod 
transitus luci s per corpus aque est secundum lineas rectas. per lam xt. 
Et hoc est quod circa propositam propositionem experimentaliter intendimus 
dec1arare. 


90 


9
 


100 


10
 



 


[Propositio]43. In medio secundi dyafani. quod est densius primo dyafano, 
fit refractio radiorum obliquorum ab anteriori superficie dyafani secundi 
ad perpendicularem exeuntem a puncto refractionis super superficiem corporis 
secundi. 
Experimentaliter etiam et hoc propositum theorema potest declarari. Op- 
posito enim foramine superiori ipsius instrument i oblique ipsi corpori solari. 
ita ut radiu s oblique incidat ad horam instrumenti oppositam foramini. 


84 Erit: est E 84-85 centrum lucis tr. v" 87 dividentis co. est ex dividentes in v" 88 
aquam: aliquam v" I post regule add. V. per 89 circuli medij tr. PE 89-90 punctus ergo 
linee...est centrum medij circuli om. l 90 medij circuli tr. Cv.v" I que: qui V. 91 circuli 
medij tr. Iv" I inmittatur P in mittatur V. 92 acus: axis E I longua v" longus E I ante 
in! add. E vel acus longa I mg. inser. acumen v" 93 quoque: que R 94 acuminis co. est ex 
acutiaris f?] in v" I que: qui Cv. I si: sic l 95 mg. inser. ad v" I post ad scr. et dei. v" 
a I diversas: universas l 96 acuminis co. est ex acutioris in v" I mutata co. est ex mutato 
in v" I oblata v" 97 totalis co. est ex totaliter in v" 98 fuerit: super l I quo: ergo 
v" 99 intra: inter v" 100 que est: existenti E 101 post que add. E est lintra: inter l, 
co. est ex inter in v" I post aquam scr. et dei. E illectum voeabulum 103 post transitus scr. et 
dei. O regule f?] I post lucis scr. et dei. E illectum voeabulum I post lam add. R p 104 
propositionem co. est ex proportionem in v", ex illeeto vocabulo in E I experimentaliter: 
proportionem f?] E 
I Propositio [43] C I 43 mg. hab. E I post dyafano scr. et dei. E illectum voeabulum 3 
super om. E 4 post secundi add. R Alha7en 4 n 7. 5 et om. v" I theoreuma E 6 
foramini V. I superioris lEO I ipsius om. R 7 oram RlE I opposita Rl 


........
>>>
273 


et perscrutato per modum quo in premissa centro lucis. que es1 intra 
aquam. signetur illud per puncturam ferri duri in superficie ipsa instrumenti. 
et invenietur illud centrum non in linea GK. perpendiculariter erecta super 10 
G terminum dyametri opposita linee FH [Fig. I], in qua est foramen hore in- 
strumenti. sed decIinabit ab illa linea ad partem in qua est sol. Eritque 
in1er hoc centrum lucis et inter punctum P. quod est communis dJfferentia 
linee G K. perpendicularis super terminum dyametri instrumenti. et circum- 
ferentie circuli medij transeuntis per M et Y centra foraminum. dis1antia 15 
sensibilis. Mittatur itaque regula in aquam et applicetur superficiei lamine. 
ita quod terminus latior regule sit supra centrum lamine; et moveatur 
regula quousque acuitas eius sit perpendicularis super superficiem aque. 
quo ad sensum. Erit itaque centrum lucis que est intra aquam inter acutiem 
regule et lineam GK. perpendicularem super FG. dyametrum basis instrumenti. 20 
Patet ergo ex hoc quod hec refractio est ad partem perpendicularis exeuntis 
a loco refractionis perpendiculariter super superficiem aque. Hoc itaque 
invento. signetur in circumferentia circuli medij trium si!!natorum circulorum. 
super punctum extremum perpendicularis exeuntis a centro eiusdem circuli. 
perpendiculariter super superfis:iem aque. signum fixum per ferri duri puncturam. 25 
Et quia patuit per premissam quod instrumento directe soli opposito et 
radio solis sibi perpendiculariter incidente. lux. que pervenit ad centrum 
lucis que est intra aquam. est lux extensa secundum rectitudinem Iinee 
continuantis duo centra foraminum. que linea pervenit ad centrum medij 
circuli equedistantis superficiei basis instrumenti et est dyameter illius. si 30 
hec linea fuerit ymaginata extendi secundum rectitudinem intra aquam donec 
perveniat ad horam instrumenti. tunc erit totaliter equedistans dyametro 
instrument i et perveniet ad lineam GK perpendicularem super dyametrum 
FG. in interiore parte hore instrumenti ductam. Et quia centrum lucis. 
que nunc est intra aquam, non est super iIIam lineam perpendicularem 35 
in hora instrumenti productam. tunc patet quod lux extensa a medio 
lucis. que est in superficie aque. non extenditur ad medium lucis que est 
intra aquam secundum rectitudinem linee transeuntis per centra duorum 


8 pcrscrulato: pertraclato l perscrutalio v.. , cu. est ex pcrstractato in V b I quo: ut 
v.. I centrum v" 9 puncluram: punctum E 11 opposito l I post opposita scr. et dei. 
C illi I ore Rl E 11-12 hore instrumenti tr. v.. 12 declinabil co. est ex declarabit in 
v" 13 inter 2 om. Rl inser. Vb in v.. 14 linea E I pąSI instrumenti ser. et dei. C quod est 
GK 15 circuli medij tr. v.. I distantium v.. 16 sensibili l sensibilia v" I mutatur Iv" 
E 17 ita quod: itaque PE I post quod add. O huius I centrum: dyametrum v"l 18 
regula co. est ex lamina ;n p 19 senssum v.. I post lucis scr. et dei. V b luminis I que rep. et 
dei. C: quod Rl I post aquam hab. 1 v" et I acutiem: acumen Rł acuciem O 21 quod hec 
om. v" 22 Hoc itaque: Hoc ita R Haec ita l I itaque co. est ex ita in v" 23 signetur rep. 
E I circuli medij tr. v" 25 punturam v.. 28 est 2: et v" 29 medij inser. P 30 
aequidistantis R equedistantes v" 31 hec: huius l I fuerit ymaginata tr. V. I restitudinem 
v" 32 oram Rl E I aequidistans R 34 interiori E I ore R1 E I hore instrumenti tr. 
v.. I quia: quando R1 35 inler V b 36 ora Rl E lextensa: ostensa l 37 extenditur 
co. est ex ostenditur f?] in v" 38 linee: lucis V. I centram f?] v" 


18 - Wilelonis Perspectivac...
>>>
274 


foraminum. sed refrangitur ab illa. Oec1aratum est autem. per primam huius. 
40 quod hec lux extenditur recte a medio lucis que est in superficie aque 
ad medium lucis que est intra aquam. Est ergo huius lucis reflexio apud 
superficiem aque. quod est propositum. 


[Propositio]44. Per medium secundi dyafani rarioris primo radius per- 
pendiculariter incidens a centro corporis luminosi super superficiem corporis 
obiecti penetrat irrefractus. 
Instrumentali similiter experientia propositum theorema potest dec1arari. 
5 Assumatur en im vitri c1ari vel cristalli frusta figure cubice, longitudinis 
dupIe dyametris foraminis hore instrument i, et fiant pIane superficies ipsorum 
equales et equedistantes, et latera ipsorum sint recta et multum poliantur. 
Deinde signetur per sculpturam ferri duri in medio basis instrumenti linea 
recta transiens per centrum ipsius, quod est E [Fig. I], perpendiculariter super 
10 ipsius dyametrum que est FG, super cuius extremitates sunt in hora instru- 
menti producte due perpendiculares FH et GK; et producatur illa lin
a 
in utramque partem superficiei circuli basis. et sit ZEX. Ponatur itaque 
unum vitrorum istorum super superficiem basis instrumenti: et applicetur 
unum suorum laterum perpendiculari ducte. que est ZEX, taliter ut medium 
15 lateris vitri sit vere super punctum E, centrum instrumenti. Et sit totum 
corpus vitri ex parte foraminum, scilicet inter foramina hore et tabule, et 
inter centrum instrumenti. quod est E. Transit ergo dicta dyameter intrumenti. 
que est FG. per medium su pe rficie i vitri superposite basi instrumenti. 
Applicetur itaque vitrum basi instrument i forti applicatione per bytumen 
20 firmum.. taliter tamen quod possit auferri quando placuerit. Deinde ponatur 
secundum vitrum ultra primum, scilicet ex eadem parte foraminum. et appli- 
cetur aliqlla superficierum eills superficiei primi vitri et applicetur basi in- 
strumenti applicatione fixa. Deinde tertium vitrum applicetllr secundo et 
adequetur superficies eius cum duabus superficiebus laterum secundi vitri. 
25 et applicetur basi instrumenti. et sic fiat de pluribus vitris quousque per- 


39 refringetur R I ab: ad V b I iIIo l, iIIa co. est ex iIIo in v" I est autem tr. Y,. I huius om. 
E 40 que: quod l 41 reflexio: refractio RE repletio v" I apud: ad Iv" 
l Propositio [44] C I 44 mg. .inser. EO I secundi inser. v" 3 irrefractus co. est ex 
illecto vocabulo in E I post irrefractus add. RAIhazen 6 n 7. 4 experigentia f?] 
O I theoremma E I post potest ser. et dei. E illectum voeabulum I decłarare v" 5 
assumantur 10m. E I enim vitri om. E lvitri: vinum f?] V b I frusta: frustum Rl, mg. inser. 
alm. v" I frusta figure cubice: figure cubice frustum l I post cubice ser. et dei. v" frustra 6 
dyametris co. est ex illecto vocabulo in E I ore RlE I ipsorum: eorum RlY,., co. est ex eorum 
in v" 7 post equales scr. et dei. E voeabulum illectum I sint: co. est ex sunt in V b , sunt 
O I polliantur v", co. est ex pollantur 10 sint l lora Rl E 13 vitrorum istorum tr. 
C 14 suorum laterum tr. Rlv" I perpendiculariter Rl I ducte inser. v" 15 post vitri 
hab. PO et I sit: sic l 16 scilicet: sit l I ore RIE 17 mg. inser. dicta v" 18 
superposite: subposite P supposite Y,. 19 post basi hab. Y,. 4 f?] I instrumenti co. est ex illecto 
voeabulo in E 20 ante firmum add. Y,. forte I tamen om. E I. auITerri Y,. I post auferri scr. 
et dei. E vocabulum illectum 21 secundum: alterum R super l I scilicet: sed l 21-22 
applicetur: appli E 23 ante fixa add. Y,. forti 24 secundi": inter V w I vitri om. Y,. 25-26 
perveniatur intra: perveniant vitra R 


- 


......
>>>
veniatur intra ad aliam perpendieularem super superficiem basis instrumenti 
aut prope, seilieet versus punetum T. Cum itaque vitra fuerint applieata 
superficiei basis instrumenti seeundum predietum modum. palam quoniam 
premissa dyameter instrumenti. que est FG. transibit per medium omnium 
superficierum vitrorum superpositorum basi instrumenti. Et aItitudo vitrorum 
est dupla dyametro foraminis. dyameter vero foraminis est equalis per- 
pendieulari MF exeuntis a centro IOraminis super superticiem basis instru- 
menti et super dyametrum eius FG. Unaqueque ergo perpen01cularium exeun- 
tium a eentris superficierum vitrorum perpendicularium super dyametrum 
basis instrument i est equalis linee M F, seilieet perpendieulari exeunti a eentro 
foraminis super superfieiem basis instrumenti. Linea ergo que transit centra 
amborum forami num transibit centra superficierum vitrorum perpendicularium 
super superficiem basis instrumenti. 
Aceipiatur ergo regula subtilis, euius formam premisimus, et erigatur 
super horam instrumenti. in superficie basis instrumenti. et ponatur super- 
ficies regule, in qua signata est linea ex parte primi vitri, quod est supra 
E centrum basis instrumenti. Et ponatur regula prope vitrum, et applieetur 
taliter ut linea que est in superficie regule sit in superficie medij circuli. 
Secabit que linea recta transiens per centra amborum foraminum et per 
centra superficierum vitrorum Iineam latitudinis regule perpendiclllariter, et 
transibit ad punetum G. Tunc itaque ponatur instrumentum in vas predictum, 
vaeuum aqua, et ponatur vas in sole directe oppositum centro solis ut 
aecipiat radium perpendicularem. Hoc autem potest fieri si moveatur in- 
strumentum quousque lux solis transeat per ambo foramina et fiat apud 
secundum foramen lux equalis. Et aspieiatur superficies regule opposita 
vitro, et videbitur lux exiens a duobus [oraminibus ipsius instrumenti extensa 
super superficiem ipsius regule. et illud umbrosum quod circumdat lucern 
in superficie regule obumbrabitur per umbram hore instrumenti, eritque 
centrum visus ipsius aspicientis super lineam que est in superficie regule. 
Deinde aeus subtilis ponatur super 'superius foramen, ita quod extremitas 
aeus sit perpendicularis super centrum foraminis. cadetque tunc umbra 
extremitatis acus super centrum lucis in linea que est in superficie regule. 
Tune itaque signetur punetus illius umbre cum ineausto subtiliter, et auferatur 
acus a superior i foramine, et eius extremitas po natur super centrum in- 
ferioris foraminis, cadetque iterum umbra extremitatis acus super punctum 
signatum in superficie regule. ablato quoque acu lux revertitur. Ex quo 


27 scilicet co. est ex si in v" lvitra: intra n;.v" lvitra fuerint tr. E I fuerint: fuerit l 28 
quoniam co. est ex quia in v" 30 superpositoiUm: suppositorum v" 32 MF co. est ex FM in 
v" 33 unaquaque l I ergo: enim l, co. est ex enim in v" I post ergo ser. et dei. E vocabulum 
i/lectum 36 ante centra ser. et dei. C corr 40 oram Rl E 43 ut linea tr. l 47 vas om. 
1v,,0 48 accipiat: excipiat Cv" (in v" co. est ex accidat) accipiatur E I potest: pteost R 
 
secundum om. E I foramen co. est ex foraen in P 53 ore Rl I hore instrumenti om. 
E 55 acus: alicuius v" 58 punctus rep. et dei. P I auITeratur v" alTeratur v" 59 
superiore E 60 iterum: rerum v" 


275 


30 


35 


40 


45 


50 


55 


60
>>>
276 


patet quoniam lux. que est super punctum quod est in superficie regule, 
transit per centra amborum foraminum. Oeinde cum incausto signetur nota 
nigra in puncto in medio superficiei vitri. ex parte regule - potest autem 
6S ille punctus inveniri. per 40 primi huius. quoniam ille punctus est communis 
sectio duarum diametrorum superficiei vitri - et tunc intuens lucern que est 
super regulam. inveniet umbram puncti qui est in medio vi1ri super punctum 
quod est in superficie regule. 
Patet ergo ex hoc quoniam lux que transit per centra duorum foraminum 
70 transit per punctum quod est in medio vitri. Oeinde evella1ur vitrum primum 
quod est super centrum instrumenti. punc1um E. Et in superficie secundi 
vitri signetur punctum medium. ut prius factum est in superficie vitri primi. 
et componatur instrumentum secund() et moveatur qU()usque lux Iranseat 
per duo foramina: perveniet que lu\ transiens per centra duorum foraminum 
7S ad centrum lucis que es1 in superficie regule. Pate1 itaque ex hoc quod 
lux pertransiens centra duorum foraminum transit per punctum quod est 
in medio superficiei secundi vi1ri. et quod lux que transil per centra duorum 
foraminum in prima experimentatione transit etiam per punctum quod est 
in medio secundi vitri. Extrahatur itaque secundum vitrum et opponatur 
80 tertium. et sic de ceteris usque ad ultimum. Et palet universaliter quod 
lux transiens per centra duorum foraminum. perveniens ad superficiem re
!Ule. 
transit etiam per cenIra superficierum vilwrum omnium positorum super 
superficiem lamine. et su nt omnia centra superficierum vitrorum omnium 
in linea una recta con1inuante centra duorum foraminum. 
8S Lux itaque pertransiens centra forami num. tam in corpore vi1ri quam 
extra corpus in aere. extenditur secundum Iineam rectam continuantem cenIra 
duorum foraminum. et est illa linea MP perpendicularis super superficies 
omnium vitrorum oppositas foramini. per 14 am XI'. lila enim linea MP est 
equedistans linee FG. dyametro lamine. que est perpendieularis super super- 
90 ficiem vitrorum. cum sit perpendicularis super differentiam communem super- 
ficei vitri et superficiei lamine. Et si omnibus vitris vel ipsorum aliquo 
premisso modo super fundum instrumenti disposito infundalur aqua vasi 
usque ad concavum superficiei vitri. accidet tamen idem quod prius. quoniam 
radius perpendicularis semper penetrat irrefractus. Item ne putC1 aliquis 


62 in om. V h 64 nigra: magna V. I in 2 um. Rł I parte co. est ex illec/(} vocahulo;/1 E 65 
punctos R post 40 add. R t I quoniam: quias E I pOSI communis ser. et dei. E Uh'lum 
vocahulum 66 post est ser. et dei. E iIIeclUm vocabulum 67 qui: quae IV h I supcr om. 
linser. J-;. 68 est inser. J-;. 71 post est .
cr. et dei. V. in I inser. P secundi 72 medium: in 
medio E 74 transiens per centra: per centra transiens v.. I centrum E I foraminum om. 
E 75 que: quod Rl I pOSI supcrficie ser. el dei. v.. vitri 78 in om. J-;. I ctiam: et 
l 78-79 quod est in medio mg. inser. J-;. 79 apponatur C 84 linea una tr. RłVhE 85 
roramimum co. est ex illecto vocahulu in E 87 iIIa linea tr. V b I supcrlicies co. e.
t ex 
superficiem in J-;. 88 post 14 0m add. R p 89 aequidistans R 91 ipsorum om. E 92 
premisso modo tr. E I aqua: aque f?] O 93 concavum: contactum v" (co. est ex con- 
cavum) I tamen: tum l
>>>
quod rectitudo radiorum perpendicularium adiuvetur per cubicam figuram 
vitri. 
Accipiatur medietas spere vitree c1are vel cristalline. cuius semidyameter 
sit minor distantia. que est inter punctum (' et centrum lamine. quod est 
punctum E. et inveniatur centrum basis ems, super quod signetur linea 
subtilis cum incaus1o. Oeinde ex hac linea. ex parte centri spere, separetur 
linea. equalis linee LN. dyametro foraminis hore ins1rumenti. Erit ergo 
hec linea equalis linee M F que est inter M centrum foraminis. quod est 
in hora instrumen1i. et superficiem lamine. Oeinde super extremitatem huius 
linee separate a dyametro producatur perpendicularis ad utramque partem 
superficiei spence. quod potest fieri per II am 1'. et secetur spera vi1rea 
secundum illam lineam. Planeturque superficies vitri secti. donec sit penitus 
equalis. fiat que perpendiculariter erecta super superficiem planam emisperii. 
quod per angulum rectum cupreum poterit mensurari: erit ergo tunc com- 
mllllis differentia istius superficiei erecte et superficiei basis spere linea recta. 
super quam erit perpendicularis linea prius a centro spere producta: ergo 
et erit perpendicularis super superficiem erectam. Oeinde 111 medio illius 
linee. que est communis sectio, fiat sIgnum cum incausto. deinde vitrum 
istud politum optime super hanc superficiem sectam ponatur super super- 
ficiem lamine instrumenti. ita quod gibositas eius respiciat foramina. et 
medium linee. que est communis sectio duarum superficierum planarum vitri. 
applicetur centro lamine et figatur vitrum super laminam. ne cadat. Oeinde 
ponatur regula subtilis super superficiem lamine instrumenti. sicut in experi- 
mentatione vitrorum cubitorum, ita quod superficies regule 111 qua est 
linea recta latitudinis sit ex parte vitri et prope illud. Deinde ponatur 
instrumentum in vas predictum. et ponatur vas 111 sole vacuum aqua. et 
moveatur instrumentum donec lux solis transeat ambo foramina; cadetque 
lux super supertkiem regule. Deinde ponatur extremitas acus vel stili ferrei 
super centrum superioris foraminis; cadetque umbra extremitatis acus super 
centrum lucis. ablato quoque stilo. revertetur lumen ad locum suum. Idem 
quoque accidit ponenti extremitatem acus super centrum foraminis secundi. 


95 radiorum om. V. I adiavatur E I cubitam PE 96 spere vitree tr. v" I clare inser. 
p I semidyameter: dyameter V. 98 post punctum scr. et dei. v" p 99 sigenetur f?] 
v" 100 separetur: sepetur V. 101 linee om. RPEO I LN: LM l, co. est ex LM in 
v" I ore RlE 102linee om. O 103 ora RlE 105 post 11 0m add. Rp ł06 secti co. est 
ex sicti in v" ł07 post equalis scr. et dei. C erit I ante planam scr. v" 
erectam I hemisphaerij Rl 108 rectum om. E I cupreum co. est ex corporeum in 
v" 109 istius: iIIius v" I recta: erecta v" 111 et: etiam Rl 113 istud: iIIud Rl 116 
vitrum om. l I lamina v" 117 ponatur regula tr. O I instrumenti om. E I sicud 
V.O I mg. inser. sicut in v" 118 cubicorum RC I in om. v" 119 iIIud om. V. in.
er. 
v" I ponatur: imponatur RV. imponitur l 120 et 1 0m . E I ponatur: ponitur Iv" I aque 
v"l I et 2 om. E 122 ponatur: poponatur E. co. est ex ponitur in v" I stili: sili f?] E stilli 
(co. est ex silli) v" 124 silo E 


277 


9S 


100 


105 


110 


lIS 


120 


12S
>>>
278 


Deinde ponatur extremitas acus super centrum spere vitree, cadetque umbra 
extremitatis acus super centrum lucis. Ex quo patet, quia lux transiens 
per centra duorum foraminum transit etiam per centrum spere vitree et 
per medium superficiei lucis que est in convexo vitri. Patet etiam ex hijs 
130 quod lux transiens in corpus vitri extenditur secundum rectitudinem linee 
transeuntis per centra duorum foraminum et est illa linea semidyameter 
spere. Nam perpendicularis exiens a centro basis vitri ad laminam est equalis 
dyametro foraminis et linee exeuntis a centro foraminis perpendiculariter 
ad superfiCiem lamine. Et quoniam hee due perpendiculares cadunt super 
13
 dyametrum lamine, palam quod linea transiens per centra duorum foraminum, 
cum extenditur in rectitudinem. pervenit ad centrum spere vitree. Est ergo 
in illa linea dyameter huius spere vitree; est ergo perpendicularis super 
superficiem huius spere, per 72 primi huius. Quoniam enim transit centrum 
spere, patet quod ipsa est perpendicularis super convexam superficiem spere, 
140 sicut superius patuit in vitris cubicis. 
Auferatur itaque regula SUb1ilis applicata ad superticiem lamine, et ponatur 
instrumentum secundo in vas ut prius, et moveatur quousque transeat per 
duo foramina. Invenieturque lux super horam instrument i, et invenietur 
centrum lucis in puncto P, quod est differentia communis inter circum- 
14
 ferentiam circuli medij et lineam GK perpendicularem in hora instrumenti; 
hoc est in extremitate dyametri circuli medij que est MP transeuntis per 
centra duorum foraminum, M et Y. Ex quo patet quoniam lux transiens 
in corpus vitri et perveniens ad centrum eius, prodiensque in corpus aeris, 
extenditur secundum lineam que extendebatur in corpore vitri. Cum enim 
1
0 linea recta transiens centra amborum foraminum perpendicularis sit super 
superficiem vitri, patet quod ipsa necessario est perpendicularis super super- 
ficiem aeris tangentis vitri superficiem. Itaque si vasi infundatur aqua, 
remanente vitro in sua positione. donec aqua superfluat centro vitri, adhuc 
invenietur centrum lucis super extremitatem dyametri medij circuli; et si 
15
 spera vitrea transvertatur, ita ut convexum eius situetur ad secundum foramen, 
et pIana superficies ad centrum instrumenti. scilicet punctum E, sive aqua 
superfundatur sive non, adhuc omnia alia accident que in priori situ accidebant, 
quoniam semper radius transiens per centra amborum foraminum transibit 


126 proponatur V. 126-127 spere vitree cadetque. . . super centrum om. E 126 umbra om. 
y. 127 -129 Ex quo patet... et per medium superficiei łucis om. V. 127 transiens inser. 
v" post transiens ser. et dei. v" patet I quia: quod EO 128 etiam: et lP 129 hijs: his 
Rl 133 exeunti RE 134 hae Rl 136 in: ad E I rectitudine v" I spere vitree tr. 
v" 138 post 72 add. R t I enim om. E 139 convexam superficiem tr. E 140 cubitis Iv" 
E 141 mg. inser. reguła v" I ponitur v" 142 ante transeat add. Rl łux 143 Invenitur 
que V b I post horam add. E ipsius 145 ora RlE 146 qui EO I MP: in Pl I que est 
MP om. V b 148 rep. et dei. C in corpus I eius om. y. 149 qua C 150 recta inser. 
p I super om. V. 151 est om. E 152 Itaque si: si igitur E I infundantur v" I 153 
remante V. I centra V.Vb 154 medij circuli tr. RIV b 155 vitrea: media Iv" I post 
secundum ser. et dei. v" sum f?] sive: sive f?] E
>>>
etiam per centrum spere. Ex hijs omnibus per vitra cubica et sperica. 
patet quod sive medium secundi dyafani fuerit densius vel rarius. dum 
tamen linea per quam extenditur radius fuerit perpendicularis super super- 
ficiem secundi corporis, quod lux extenditur in secundo corpore secundum 
rectitudinem 'linee per quam extendebatur in corpore primo. Patet ergo 
propositum ; corpus en im vitri est densioris dyafanitatis quam corpus aeris et 
etiam quam corpus aque. 


[propositio]45. In medio. secundi dyafani rarioris primo dyafano fit 
refractio radiorum oblique incidentium a posteriore superficie secundi dyafani 
a perpendiculari exeunte a puncto refractionis super superficiem corporis 
secundi. 
Hoc quod nunc hic proponitur est conformiter prioribus per instrument alem 
experientiam decJarandum. Assumatur enim illud vitrum spericum quo iam 
in precedenti proximo theoremate usi sumus, et ponatur super laminam 
instrumenti. i1a quod superficies piana ipsius respiciat foramina, et quod me- 
dium linee recte que est in ipso sit super centrum lamine, et linea que est 
communis sectio superficierum planarum vitri cadat oblique super dyametrum 
lamine quacunque obliquatione. Palam ergo quoniam linea transiens centra 
duorum foraminum obliqua est super superficiem planam vitri. Coniungatur 
itaque vitrum lamine instrumenti secundum hunc situm firmiter. et ponatur 
instrumentum in vas et vas in sole, moveatur que instrumentum donec 
lux transeat per duo foramina. Cadetque lux in interiori hore instrumenti. 
et centrum lucis erit in circumferentia medij circuli, sed extra illum punctum 
p Wig. I]. qui est communis ditferentia circumferentie medij circuli et linee 
stanti .in hora instrumenti que est GK, et erit declinatio eius ad partem in qua 
est sol; erit ergo ad partem perpendicularis exeuntis a loco refractionis 
super superficiem spericam vitri. Et quoniam hec lux extenditur in aere 
secundum rectitudinem linee transeuntis per centra duorum foraminum. ut 
patet per primam huius. et hec linea. in hoc situ pervenit ad centrum 


159 etiam: et l I his Rl I post hijs ser. et dei. v.. il/ectum voeabulum I omnibus mg. inser. 
v.. 160 sive: su um l 162 secundi corporis tr. v.. 
I Propositio [45] C I 45 mg. hab. EO I sit VuE 4 post secundi add. RAIhazen 
7 n 7. 5 nunc hic tr. v.. inser. v.. I hic om. 1 I post prioribus ser. et dei. V b abus 
f?] I instrumentale C .6 experientia E 7 theoremmate E I post ponatur ser. et dei. v.. 
lineam et il/ectum vocabulum I mg. inser. super laminam V b I laminam: lineam l 8 
respiciat co. est ex respicitur in v" 9 in rep. v.. I ante lamine ser. v.. linee I linea: lamina 
v.. 10 post sectio ser. et dei. E il/eetum vocabulum II post quacunque scr. et dei. E il/ectum 
voeabulum I. obliquatione: obliqua ratione lV u I quoniam: quod R 13 ponatur co. est ex 
ponitur in v... 14 post vas z scr. et dei. v.. vitri f?] coniungatur 15 ora R orae l ore E 18 
stantis E lora Rl I ad: in E 19 ad: in E post perpendicularis add. E que I exeuntis: 
exit E 20 post lux ser. et dei. v.. linea transeuntis [1] 20-21 mg. inser. v.. extenditur in aere 
secundum rectitudinem Iinee 


279 


160 


16S 


s 


10 


lS 


20
>>>
2RO 


spere vitree et es.t ohliqua super superficiem spere planam. palam ergo 
quia terminatio extensionis illius lucis est in centro vitri. 
2S Extenditur ergo lux in corpus vitri secundum lineam rectam exeuntem 
a centro spere ad circumferentiam. que linea, cum sit dyameter, palam. 
per 72 primi huius, quoniam ipsa est perpendicularis super spericam super- 
ficiem vitri; ergo et super concavam superficiem aeris continentis speram 
vitri. Non ergo refringitur in aere secundo sicut neque in primo. sed 
30 neque retlectitur in corpore vitri. neque in convexo ipsius; refringitur ergo 
apud centrum vitri. quia fuit obliqua super superficiem eius planam in 
qua est centrum vi1ri. Palam itaque ex hijs experimentationibus illud quod 
est etiam superius dec1aratum. scilicet quoniam lux. si fuerit extensa in 
corpore suotiliori oblique incidens superficiei corporis grossioris. refringetuI' 
35 ao ipso. el erit eius refrac1io ad partem perpendicularis super superficiem 
spericam corporis grossioris. sieut per 43 huius patuit. Fiat refractio ex 
aere ad aquam. erit illa refractio ad partem perpendicularis exeuntis a loco 
refractionis super superficiem aque. et non pervenit refractio ad perpendicularem. 
Quod si vitrum econtrario situetuI'. scilicet ut superficies eius sperica convexa 
40 respiciat 
Llperius foramen. et punctum medium linee que est communis 
differentia superficierum planarum. quod est linfra] centrum spere vitree. 
sit super centrum inslrumenti. cadatque hec linea oblique super dyametrum 
lamine. ducaturque in ipsa superficie lamine. a centro lamine. linea per- 
pendicularis super lineam que est communis sectio iIIarum planarum super- 
45 ficierum. que necessario erit perpendicularis super superficiem planam vi1ri 
eree' am super superfkiem lam ine. 
I'onatur Itaque instrumentum in va
e sine aqua et moveatur quousque 
lux penranseat duo fmamina; cadetque centrum lucis in circumferenlia 
medij circuli extra punctum P. quod est differentia communis medij circuli 
50 et lince GK. perpendicularis super supertkiem lamine ducie in hora instrumenli. 
quod punctum P est extremitas dyametri medij circuli que est MP. Eritque 
dec1inalio lucis ad partem contrariam illi in qua est perpendicularis educta 
a loen r,,[racl jonis super planam superficiem vitri. Hec autem lux extenditur 


23 my. illser. "i. el es! obliqua I palam um. E 24 terminato V. I post lucis add. l et 26 
que: quia v" I palam om. Iv" 27 post 72 add. R t 29 sicud E 30 ref1ectitur: refringitur 
R I nec RlY" 31 fuit: sunt O 32 his Rl 33 scilicet: sed l I lux inser. P 33-34 
extensa in corpore: in corpore extensa V. 35 post refraclio scr. et dei. V. ex aere ad 
aquam I ad partem perpendicularis om. V. I post partem ser. et dei. C contrariam ei parti in 
qua est linea recta 36 sicut: ut v" post sicut hab. C si ut, ser. et dei. P sint, hab. V. sint, scr. et dei. 
O palet I 43: 93 V. I huius: istius E I post patuit add. R ut si 37 erit iIIa tr. O 38 
pervcnit co. est ex venit in P 39 econverso Rly" I situatur V b I superficies eius sperica: 
spcrica eius superlicies V. I post sperica add. l et 41 diITerentiarum V. 42 centrum om. 
Y" I super: sit f?] E 47 Ponatur itaque: ponaturque R I quousque: donec E 48 
pertranseat: transeat v" I que om. R 50 post et ser. et dei. v" lineas I linee: mg. inser. v", 
linea E I ducta R lora Rl E 51 medij circuli tr. E I Eritque inser. v" erit l 53 Hoc 
CP 


, 
, 
ł. 


.......
>>>
in vitro secundum rectitudinem linee transeuntis per centra duorum foraminum. 
quoniam iIIa linea. cum per centrum spere vitree transeat. est in illa 
dyameter spere vitree. Fit itaque refrac1io lucis apud centrum spere vi1ree. 
quoniam lux transiens centra am
orum foraminum fit o
liqua super super- 
ficiem planam vltri et super superficiem aeris contingentis vitrum. Et si 
aqua infundatur vasi quousque superemineat centro instrumenti. cadet adhuc 
centrum lucis in circumferentia medij circuli. extra extremitatem sui dyame1ri. 
o
lique ad partem contranam illi parti super quam cadit perpendicularis. 
Et quoniam aer es1 subtilior quam aqua. et aqua subtilior vitro. maior 
fiet distantia centri lucis ab extremitate dyametri medij circuli in aere quam 
in aqua. Quod si vitrum ponatur aliter in superficie lamine. scilicet ut 
linea que est communis ditferentia duarum superficierum planarum ipsius 
vitri SIt super Iineam perpendiculariter dyametrum lamine secan1em. non 
tamen sit eius medius punctus. qui est linfra] centrum spere vitree. super 
centrum lamine. et vertatur convexum vitri ad foramma. et figatur regula 
subtilis super superficiem lamine erecta super horam eius. sit que su per- 
ficies eius in quo est linea ex parte vitri: et terminus regule secet dyametrum 
lamine perpendiculariter. palam quia linea transiens per centra fi)raminum 
duorum non transit per centrum spere. sed per aliud punc1um superficiei 
pIane ipsius vitn. et erit obliqua super spericam superficiem. per 72 primi 
huius. 
Ponatur itaque instrumentum in vase. et vas in sole. et moveatur in- 
strumentum quou!\que hl\ transeat per centra duorum foraminum. Et non 
cadet lu.\ directe super superficiem regule. neque centrum lucis cadet in 
linea que est in superficie regule. sed decIinabit oblique extra lineam que 
transit per centra duorum foraminum. ad partem in qua est centrum vitri. 
hnc est ad partem contrariam perpendicularis e'Xeuntis a locn refracti()nis. 
perpendiculariter super superticiem vitri spericam. Erit que linea rertransiens 
centra duorum foraminum perpendicularis super superticiem vi1ri planam, 
per gdm XI'. quoniam iIIa linea est equedistans linee FG. dyametro lamine, 
que. ex ypothesi. est perpendicularis super superficiem planam vitri. Si 
ergo lux transiret per centra duorum foraminum et extenderetur secundum 
rc
:titudinem ad planam vitri superficiem. palam quod tunc extenderetur 


55 in om. R 56 Fit: sit V.Vb 57 fit: sit V. I obliquc l 59 supcremiancnt C I post 
adhuc scr. et dei. E iIIectum voeabulum 61 oblique co. est ex iIIecto voeabulo in E 62 post aer 
scr. et dei. E iIIectum vocabulum I et aqua om. v" I subtilior vitro: quam vitrum E 63 
ccntri: circuli l 64 ponatur aliter tr. V. 66 lineam: laminam l, inser. lineam et ser. et dei. 
laminam v" 67 spere vitree tr. Iv" 68 figatur: figura l I regulae l 69 post superficiem 
scr. et dei. v" spere et O scr. et dei. regule I Ofam Rl E 69-70 sit que superficies eius om. 
l 70 quo: qua RE 71-72 foraminum duorum tr. v"E 72 aliud: iłłud l 73 post 72 
add. R t 78 in inser. v" I obliqua V. I Iineam: regulam V. 79 post foraminum add. V. et 
lux 80 post hoc ser. et dei. v" autem 83 post 8 0m add. R p I illa linea tr. 
v" I aequidistans R 84 post que add. v" est 85 transiet [1] E I centrum 
V. I extenditur v" 85-86 secundum rectitudinem om. V. lvitri: inter l [1] V b I quod: 
quia v" I extenderentur V. 


281 


ss 


60 


6S 


70 


7S 


80 


8S
>>>
282 


secundum rectitudinem in aere. Sed centrum lucis que est in regula cum 
non cadat in rectitudinem huius linee, patet quod lux non extenditur in 
eius rectitudine ad superficiem planam vitri; est ergo lux refracta, sed non 
90 refringitur in aere, neque in corpore vitri. Refringitur itaque ap ud spericam 
superficiem vitri; incidit en im oblique super spericam superficiem. quoniam 
linea transiens centra duorum foraminum non transit per centrum vitri 
et hec lux egrediens a piana superficie vitri, quoniam oblique aeri incidit, 
plus refringitur. 
95 Quod si vitrum econverso disponatur. ut eius superficies piana opponatur 
foramini primo sic quod communis differentia sit super lineam secantem 
dyametrum lamille perpendiculariter. et medius punctus illius linee sit extra 
centrum lamine, tunc ergo linea pertransiens centra duorum foraminum 
non transit per centrum vitri, sed per alium punctum illius plane super- 
100 ficiei, et est perpendicularis super illam superficiem. Moveatur itaque in- 
strumentum in sole, donec lux transeat per ambo foramina. Cadetque 
centrum lucis, que cadit in interiori parte hore ipsius instrument i, in periferia 
medij circuli, extra punctum P, quod est extremitas dyametri medij circuli, 
que est linea MP, sed declinabit ad partem in qua est centrum spere 
105 vitree. Et linea que egreditur a centro huius spere in ymagina1ione ad 
locum refractionis est perpendicularis super superficiem huius spere; est 
ergo perpendicularis super superficiem aeris continentis superficiem spere 
vitree. Hec itaque refractio est ad partem contrariam illi in qua est per- 
pendicularis exiens a loco refractionis super superficiem aeris continentis 
110 speram. Lux vero transiens centra duorum foraminum pertransit corpus 
vitri recte, cum sit perpendicularis super superficiem planam vitri, sed non 
est perpendicularis super superficiem convexam. cum non pertranseat centrum 
spere. Ergo etiam non est hec lux perpendicularis super superficiem aeris 
continentis convexum vitri. Et quia hec lux refracta invenitur, refrangitur 
115 ergo apud convexam superficiem spere vitree. Quod si aqua tunc infundatur 
vasi infra centrum lamine, invenietur etiam lux refracta ad partem in qua 
est centrum vitri. Hoc autem est ad partem contrariam illi, in qua cadit 
perpendicularis exiens a loco refractionis, que extenditur in corpore aeris, 


89 rectitudine: rectitudinem l v.. om. V b I pOSI rectitudine rep. v.. huius Iinee patet quod lux... in 
eius rectitudinem 91 post superliciem 2 add. E vitri 92 duarum v.. 93 hec mg. inser. 
v" I aeeri f?] E 95 econtrario RIVuE I disponitur l I opponatur: apponatur l, co. est 
ex opponitur in V b 96 sic quod: sit que f?] v" 99 post ilJius ser. et dei. V b linee I pIane om. 
E 100 post itaque ser. et dei. P in superficiem 102 interiore R I orae RlE om. 
v.. 104-105 spere vitree tr. RIV b 107 continentis: contingentis R 108 contrariam ilJi tr. 
v.. 109 continentis: contingentis R 112 pertranseat: transit l 114 contingentis 
R I convexxum v.. I refringitur RE 115 infundetur l 116 invenitur v..v"EO 117 ad 
partem: in parte E I qua: quam R 


.....
>>>
perpendicularis super concavam ipsius aeris superficiem convexum vitri con- 
tinentem. Et hoc est propositum. 
[PropositioJ46. Omnem radium incidentem et refractum in eadem pIana 
superficie consistere est necesse. 
Sed et id quod nunc proponitur potest experimentaliter declarari. Quoniam 
enim. omnibus disposi1is ut in 43 huius, lux incidens centro luci s que 
est in superficie aque, et a centro lucis existentis super superficiem aque, 
quod est centrum medij circuli. incidens centro lucis intra aquam existentis, 
quod est in circumferentia circuli medij, transit per centra amborum fo- 
raminum. que similiter sunt in superficie medij circuli, pala m quoniam linea 
secundum quam lumen incidit superficiei aque per medium aerem, et se- 
cundum quam refringitur in aque medio, sunt in eadem superficie, quoniam 
utraque ipsarum est in superficie medij circuli trium assignatorum circulorum. 
Invenitur autem hec refractio in radio solari, quando radius solaris transiens 
per centra foraminum fuerit obliqus super aque superficiem, n
Jil quando 
fuerit perpendicularis. Et propter obliquitatem situs instrument i a centro 
spere aque, nunquam fiet hec linea radialis perpendicularis super superficiem 
aque, nisi sol fuerit perpendiculariter super tenith capitis; sole vero ultra 
vel citra tenith capitum existente, satis. evidens est hec experimentatio omni 
tempore. Patet ergo id quod proponebatur. Et hanc superficiem dicimus 
superficiem refractionis. 
Patet itaque ex hijs omnibus quinque premissis propositionibus, quoniam 
omnis lux pertransit quecunque corpora dyafana secundum lineas rectas. 
Et quam diu linee sunt perpendiculares super superficies corporum, quan- 
tumcunque etiam diverse sint dyafanitatis, semper extenditur secundum re- 
ctitudinem eiusdem linee et non refringitur. In corporibus vero diverse 


119 convexam R convexi l 119-120 contingentum R 120 Et hoc est propositum om. łV M 
inser. v" I post propositum scr. et dei. C ultima pars 46 G . propositionis: Patet itaque ex hijs 
omnibus 5 premissis propositionibus quoniam omnis lux pertransit quecunque corpora.. . refran- 
getur ad partem contrariam perpendicularis predicto modo ducte super superliciem corporis 
secundi scilicet subtilioris. Errore suo observato mg. ser. vacat et Jragmentum in questione de novo 
non rep. in propositione 46 G 
ł Propositio [46] C I 46 mg. hab. EO 2 post ne...esse hab. RAIhazen 5 n 7. 3 post et 
ser. et dei. v" licet [1] I id: hoc E I id quod. nunc: nunc id quod v.. I dec1ararari C 4 
omnibus dispositis tr. v" I post ut add. l est 6 medij circuli tr. Cv.. 10 refrangitur 
C I mg. inser. quoniam PIlipsarum: parum v.., mg. inser. v" 12 radio: medio 
Iv" I solaris transiens tr. 1 13 obliquus RlEO 15 radialis om. E 16 nisi: ubi 
v.. I post sol hab. C sit et inser. fuerit I zenith Rl teniht v.. tenit E 17 citra: contra l circa 
O mg. inser. V b I ante citra ser. et dei. v" contra I zenith Rl teniht V M I existentem 
v" I est inser. P I omni rep. et dei. v" 18 id: illud v.. I proponitur l I post propone- 
batur ser. et dei. v" intra [1] I hanc: han v.. 20 ijs R I quinque premissis tr. E I propo- 
sitionibus om. V b 21 omnis lux tr. v" I quecunque: quacunque l queque v.. 22-23 
quantumcunque: quaecunque l quacunque v..v" quamcunque E 23 etiam diverse tr. 
CPO I sint: sunt O I diafoneitatis 1 dyafonitatis C dyafaneitatis co. est ex diafanitatis in 
v" 24 refrangitur C I corpore RlV b 


283 


120 


5 


10 


15 


20
>>>
284 


H dyafanitatis. omnis lux superficiei secundi corporis ohlique incidens refringitur 
secundum lineas rectas alias ab illis. secundum quas incidebat primo corrori; 
que tamen linee semper erunt in eadem surerficie pIana. ymaginata sccare 
utrumque iłłorum corporum: et hec superficies in inspectione instrlll11cnti 
est medius circulus trium circulorum signatorum in interiore parte hore 
30 instrumenti. cuius dyameter est linea MP. Cum vero lux ohliqlla exiverit 
a corpore suhtiliori ad grossius. rcfringetur ad partem perpendicularis exeuntis 
a loco refractionis. que est perrendicularis super superficiem grossioris se- 
cundi corporis. Et cum lux ohliqua exiverit a corpore grossiori ad suhtilius. 
refringetuI' ad partem contrariam perpendicularis predicto modo ducIe super 
35 surerficiem corporis secundi. scilicet suh1ilioris. 


IProrositioJ47. Radio perpendiculari omne corpus dyafanum penetrante, 
radius ohlique incidens in medio secundi dyafani densioris refringitur ad 
perpendicularem duc1am a puncto im:idcnt ie super secundi dyafani sllpcrficicm: 
et in medio secundi dyafani rarioris refringitur ab eadem. 
5 IIIud quod particularihus exrericn1ijs hactcnus inslrumentaliter prohatuITI 
est. natllrali demonstra1ione intendimus adiuvare. Omnes enim motus na- 
turales. qui fiun1 secundum lineas perpendiculares. sunt fortiores. quoniam 
coadiuvantur virtute universali coelesti secundum lineam rectam brevissimam, 
omni subiecto corpori inftuente. lmpulsiones enim proiectionum factarum 
10 perpendicularitcr slInt fortiorcs eis que fillnt ohliqllc. Et similitcr pcrcllssil\l1eS 
que fiunt perpendiculariter sunt omnibus obliquis percussionibus fortiores, et 
inter omnes obliquas, fortiores sunt i/le que plus accedunt ad perpendiculari- 
tatem. Quia itaque omnis corporis densitas impedit transitum luminis, necesse 
est lumen ymaginari repelli a transitu per resistentiam corporis densi, et plus 
15 per resistentiam corporis densioris. Et per hanc resistentiam qualitatis passive 
que est de nsitas ad activam qualitatem que est lumen intelligimus qllendam 
25 diafoneitatis l dyafonitatis C dyafaneitatis V. I refrangitur C 26 alias om. E 27 in: 
patet V. I imaginate Rl ymaginati V. ymaginata co. est ex ymaginate in v" 28 uterque 
p utrunque RlC I spectione lV.v" 29 est: et v" I parte hore tr. V b I orae RIE 30 
cuius dyameter mg. inser. v" I obliqua: aliqua lV.v" 31 grossius: gressus v" I refrangetur 
C 33 exivit Rl v" 34 refrangetur CO I mg. inser. ad partem P I post partem add. PO 
per I perpendicularis om. Rl 34-35 super superficiem tr. E 
ł Propositio [47] C I 47 mg. hah. EO I dyafanum penetrante tr. E 2 refrangitur 
C 3 incidentem v" I post incidentie add. Y.. et 4 refrangitur C I post eadem add. RAIhazen 
8 n 7. 5 post quod hab. l de et v" scr. et dei. de I hactenus instrumentaliter tr. E 5-6 
probatum est tr. E 7 qui: que Y..v" I post lineas ser. E naturales I posl perpendiculares 
add. Y.. q I post fortiores scr. et dei. P et inter omnes obliquas fortiores et E hab. et 8 
coadunantur l I post virtute mg. inser. alm. v" utili f?] et scr. et dei. universali 9 Inpulsiones 
co. est ex inpresiones in v" I enim om. RE inser. v" I proiectionum: proiectationum Rl 
proiectionem co. est ex proiectationem in v" 10 fortiores co. est ex formaliores in v" 10 
obliquis percussionibus l,., 

 I percutionibus v" co. est ex persecutionibus I ante fortiores scr. 
et dei. Y.. percu I fortiurcs co. est ex formaliores in v" 12 fortiores: formaliores v" I ad: et 
Y.. 14 repelli co. est ex repelle in v" I post transitu add. Y.. repelli I densi co. est ex densiori 
in P et ex densioris in v,,; densioris E 14-15 corporis densi et plus per resistentiam om. 
Y.. I et plus per resistentiam corporis densioris mg. inser. v" I plus per resistentiam corporis 
densioris et om. E 16 activam qualitatem tr. Rlv" I quedam f?] v"
>>>
modum motionis luminis per medium corporum resistentium. que secundum 
plus et minus capacia sunt impressionis luminaris. non quod in transmuta- 
til)ne locali ipsius luminis sil aliquis motus. ut patet per secundam huius. 
sed quia lumen in eodem instant i secundum diversitatem mediorum se plus 
cornprimit vel diffundit. Et hoc vocamus hic motum ipsius lucis. 
Ornnis itaque lux pertransiens corpus dyafanum motu velocissimo et in- 
sensibili pertransit, sic tamen quod per magis dyafana velocir fit motus 
quam per minus dyafana. Omne enim corpus dyafanum plus et minus 
resistit penetrationi lucis secundum quod est participans dyafanitatem plus 
vel minus; grossicies en im corporum resistens est semper luminis penetratiolll. 
Cum ergo lux pertransiverit corpus aliquod dyafanum oblique et occurrcrit 
corpori alij dyafano grossiori. tune corpus grossius resistit luci vehementius 
quam prius corpus rarius resistebat: nccesse est ergo quod propter resistentiam 
illius corporis densioris. motus lucis transmutetur. Et si resistentia fucr;t 
fortis. tunc motus ille ad partem contrariam refringetuI'. quia .,'e:.o non 
resistit fortiter, ideo lumen non redibit in partem ad quam movebatur. 
Si vero resistentia fuerit debilis. propter maiorem raritatem corporis plus 
d
 arani. tune lux incidens non refringetuI' ad contrariam partem. nec poterit 
per ilam lineam procedere per quam inceperat sed mutabitur in situ 
cum vero perpendiculariter inciderit quodcumque corporibus dyafanis et 
quarneunque diverse dyał
mitatis. non mutabitur. sed directe omnia penetrabit, 
q uoniam perpendieularis fortior est omnibus. et obliqui viciniores perpendiculari 
sun1 fortiores omnibus remotioribus obliquis. 
Cum itaquc corpori dyafano grossiori lux incidit oblique, extenditur 
secundum lineam rcctam approximantem ad perpendicularem. exeuntem a 
puneto in quo lux occurrit superficiei corporis dyafani grossi. productam 
super superficiem corporis grossiÓris. ideo quia faeilimus motuum est se- 
cundum Iineam perpendicularem. Si ergo radius lueis inciderit super lineam 
perpendieularem. transibit recte propter fortitudinem motus super perpendi- 
eularem. et si radius inciderit oblique. tune non poterit transire propter 
dcbilitatem motus super lincas obliquas. Accidit ergo ut declinet ad partem 
aJiquam. pcr quam facilior "it transitus quam per illam partem ad quam 


17 l1Iulum Ił:; I mOl1onis um. E 19 aliquis: alius l 20 eadem C I instanti: instrumenti 
1 inslcnti v" 21 vel: illis v.. I hic om. l I motum: motivus C 22 lux: lustrans ['!] vel 
E I ante corpus hab. E lux 23 lit: sit V.E 25 pcnctrationi co. e.
l ex pcnctrationcm in 
v" diaphanitate R 26 vel: quam V
 I grossiores V
 I corporeo f?] v" 27 pertransiret 
l I occurit V. 28 lucis V b 30 ame motus add. E quod I transmittetur PO 31 
rcfrangetur C 31-34 rcfringctur quia vero non... refringetur ad contrariam partem om. 
v.. 32 resistit co. est ex resistat in V
 I movebatur co. esl ex movetur in V b 34 refrangetur 
C 35 posllineam scr. et dei. P lian I inciperat pv.. 36 cum: tamen V. I quodcumque: 
quibuslibet Rl quotcunque EO, co. est ex quodamcunque f?] in v" 37 quamcunque: 
quanlumcunque R quacunque lC 38 fortior co. est ex formalior in V b I oblique 
lV b I perpendiculari: perpendiculares " co. est ex perpendiculares in v" 39 fortiores co. est ex 
formaliores in v" I obliquis om. Rl m
J. inser. V b 41 post lineam add. E obliquam id 
est I adpropximantcm v.. I a: in V. 43 motus E 45-46 transibit rccte... super per- 
pendicularem mq. inser. alm. P 45 P().
I recte aeM. E secundum 


285 


20 


2
 


30 


]S 


40 


4
>>>
286 


per lineam incidentie movebatur. Facilior autem motuum et plus adiutus 
50 celesti influentia est qui super lineam perpendicularem. quod enim vicinius 
est perpendiculari facilioris est transitus quam remotius ab illa. Sit itaque 
ut a puncto A corporis luminosi Wig. 28J incidant radij quam plures per 
medium AB super superfkiem alterius dyafani corporis in qua sit linea 
BCDE, et sit BF linea profunditatis illius corporis. et sit linea AB per- 
55 pendicularis super illam superficiem. Palam itaque secundum rationem pre- 
missam fortitudinis perpendicularium et per experientias instrumentales, per 
42 et 44 huius, quoniam radius incidens secundum lineam AB perpen- 
diculariter penetrat totum corpus BEF. Radius vero incidens secundum lineam 
AC. si directe transeat corpus BEF. tunc non erit diversitas in dyafanitate 
60 corporum ABE et BEF. quod est contra ypothesi. Linea itaque AC propter 
diversitatem resistentie non erit linea continua. Sed si per corpus minus resistens 
movebatur libere per lineam AC. non potest in corpore plus vel minus 
resistente per eandem lineam moveri. 
Si ergo corpus BEF sit densius corpore ABE. patet ex premissis quod 
65 difficilior est transitus per illud. Si itaque linea AC refrangatur a linea 
perpendiculari ducta a puncto C super superficiem corporis BCDE. que 
sit CG, debilitabitur. nec ad aliquid perveniet effectus eius; frustra ergo 
incidebat. Natura autem frustra nichil agit. sicut in principio suppositum 
est. Linea ergo AC. ut etiam ostensum est experimentaliter per 43 huius. 
70 refringitur necessario ad partem perpendicularis CG, ut fortificetur actio 
ipsius. Similiter quoque est de radijs incidentibus secundum lineas AD et 
AE. Quod si corpus in cuius superficie est linea BCDE fuerit dyafanitatis 
rarioris quam sit corp\Js ABE, adhuc propter fortitudinem actionis, radius 
perpendicularis qui est AB penetrat irrefractus, radius vero secundum lineam 
75 AC transiens corpus densius, et in punto C incidens superficiei corporis 
rarioris, non invenit resistentiam quam prius. 
Et quia formarum proprium est semper se diffundere secundum ampli- 
tudinem omnis capacis materie. patet quod radius AC non procedit secundum 
lineam AC. quia sic dispositio dyafanorum corporum secundum resistentiam 


49 incidentie co. est ex incidente in V& I adiutus: adurens V. 50 influentia co. est ex inflentia 
in V& I qui om. RlC I post qui add. E sit I enim co. est ex non in V& .51 post quam hab. 
E quod 52 ut inser. P 53 sit: sint EO, co. est ex sint in P 53-54 linea BCDE: Iinee 
ABDEBC E 54 iJlius: istius V. I post AB add. E linea 54-55 post perpendicularis ser. et 
dei. E voeabulum il/ectum 55 superliciem rep. E 56 post fortitudinis ser. et dei. E vocabulum 
il/eetum I post fortitudinis ser. et dei. E voeabulum il/ectum 57 et om. O I post quoniam ser. 
et dei. E voeabulum il/ectum 57-58 perpendiculariter pcnetrat tr.IV& 59 non: enim l 60 
ABE: ABC E 65 refringitur Rl EO, refrangatur co. est ex refringitur in v" 66 corporis om. 
E 67 aliquid: aliud l I pervenit l 67-68 ergo incidebat tr. v" 68 incidebat co. est ex 
incidebant in v" I Natura: non V M 68-69 suppositum est: dictum est et suppositum E 69 
Linea ergo tr. v" I etiam: enim V. 70 refrangitur C 71 ipsius: eius Rl I post radijs add. 
E ipsis 72 Quod: Et l I BCDE: DE E 73 quam sit rep. E I sit mg. inser. v" 75 
C om. lV.v" 77 proprium mg. inser. v" 78 capacitatis l I post capacis scr. et dei. V& es 


---
>>>
287 


ad receptionem luminis esset uniformis, quod est contra ypothesim. Re- 80 
fringitur ergo radius AC, sed non ad perpendicularem CG, quoniam iIIa 
refractio non fit propter resistentiam materie, sed propter victoriam forme 
agentis super materiam plus dispositam quam prius; unde forma diffundit 
se virtute propria ab incepto progressu secundum Iineam AC et ad partem 
contrariam ipsius perpendicularis CG et eius equedlstantis que BF. Et 8S 
similiter est de omnibus alijs obliquis radijs, ut AD et AE. Motus itaque 
radij incidentis oblique secundum lineam AC in corpore secundi dyafani 
densioris, quod est BEF, componitur ex motu in partem perpendicularis 
AB. transeuntis per corpus BEF, in quo est motu s, et ex motu facto 
super lineam CB. que est perpendicularis super lineam CG. Quoniam enim 90 
transitus perpendicularis est fortissimu s et facillimus motu um. et densitas 
corporis resistit termino motus ad quem intendebat, linea AC necessario 
movebitur ad perpendicularem CG, exeuntem a puncto C, in quo radius 
AC occurrit superficiei corporis densioris. Et quoniam ill i motui resistitur 
propter grossiciem medij, et etiam per naturam alterius motus, qui est 9S 
super lineam CB, qui propter resistentiam medij non omnino dimittitur 
sed tantum impeditur, dec1inabit ergo lumen versus punctum B, semper 
approximans perpendiculari ABF. 
Fit itaque in medio secunde dyafanitatis, grossiore medio primo, refractio 
radij AC secundum lineam CL, propinquiorem perpendiculari CG exeunti 100 
a puncto C, in quo occurrit corpori densiori, quam linea AC, per quam 
incidebat superficiei illius corporis, producta ultra punctum C ad punctum Q; 
propinqua fuerit eidem perpendiculari educte ultra punctum C ad punctum H, 
ita ut angulus ACH sit maior angulo LCG; non concurret tamen cum 
perpendiculari BF versus punctum F, sed versus punctum A. per secundam lOS 
primi huius, quoniam concurrit cum eius equedistante, linea CG in puncto C. 
Cum vero radius AC exiverit a corpore grossiore ad subtilius, mnc, quia 
minus habet resistentie, erit motus eius velocior et magis sui diffusivus. 
Et quoniam resistentia medij densioris impellit semper lucern obliquam. ut 


80 ypothesis f?] P ypothesim co. est ex ypotheticos in J.'b 80-81 Refrangitur CVwJ.'b 81 
quoniam: quia E 82 fit: sit v., mg. inser. J.'b 83 plus: prius V. I prius: plus V. I dilTudit 
p 85 eius om. l mg. inser. J.'b I aequidistantis R 86 radijs mg. inser. alm. in v" I AD co. 
est ex il/ecto vocabulo in V. 87 post dyafani scr. et dei. P corporis 88 quod: que 
V.VJ 88-89 partem perpendicularis AB: perpendicularis AB partem E 90 est om. E 92 
resistit: restit V. I intendebat co. est ex intendenbant in J.'b 94 occurrit: concurrit l 95 
post grossiciem add. E meriti f?] I per: propter Rl 96 super: secundum E 97 ergo lumen 
tr. IV. 98 proximans l, approximans co. est ex proximans in J.'b 99 Fit: sit V. I secunde 
dyafanitatis tr. lJ.'b I medio: secundo J.'b 101 quam 1 : quoniam l 102 Q:H 
RV. 102-103 ad punctum Q...ultra punctum C mg. inser. alm. P I Q propinqua fuerit 
eidem perpendiculari educte... C ad punctum H mg. inser. J.'b 103 propinqua fuerit eidem... C 
ad punctum H om. V. 104 concurrit E I cum inser. J.'b 105 post secundam add. 
R t 106 concurrit cum eius equedistante: cum eius equedistante concurrit E I eius 
equedistante tr. l I eius inser. V. I aequidistante R I post equedistante ser. et dei. J.'b 
eius 107 grossiore om. E 108 velociter R 109 semper: super EO
>>>
288 


110 coadunetur ad perpendicularem lineam a puncto incidentie super super- 
ficiem iIIius corporis productam. que est CG. patet quod in medio rarioris 
dyafani illa resistentia erit minor quam prima. Fit ergo motus lucis ad 
partem a qua per resistentiam repellebatur motus maior. Movetur ergo 
lux in corpore dyafano rariore plus ad partem contrariam parti perpen- 
W! dicularis. ita quod angulus GCK sit maior angulo ACH. Fit tamen semper 
motus lucis AC in reflectione a corpore secundo rarioris dyafani quam 
prim um. inter lineas CG et CE. quoniam cum angulus GCE sit rectus, 
anpulus GCK nunquam potest fieri rectus. Patet ergo propositum. 
[Propositio]48. A superficie pIana corporis diafani omnium radiorum illi 
superficiei incidentium non est possibiIe fieri refractionem ad aIiquod punctum 
unum. 
Quoniam enim. ut patet per premissas. in omni corpore dyafano semper 
Stil refractio vel ad ipsas perpendiculares ductas a punctis incidentie radij 
super superficiem corporis dyafani a qua fit refractio. vel ab illis per- 
pendicularibus (quomodocunque autem hoc contingat), patet, cum iIIe per- 
pendiculares super planam superficiem sint equedistantes, per 6 am XI'. quoniam 
sive ad ipsas perpendiculares, sive ab ipsis, fiat refractio. non est possibile 
10 ut omnium radiorum iIIi pIane superficiei incidentium refractio fiat ad punctum 
unum. Patet ergo propositum. 


IPropositio]49. Nulla refractio transmutat situm partium forme refracte, 
sed solum auget vel minuit figuram. 
Quoniam enim, ut patet per 47 huius, omnis refractio fit in medio 
secundi dyafani. et in rariori a perpendiculari. in densiori vero ad perpendi- 
s cularem. palam quod semper dexter radius remanet dexter et sinister sinister. 
Et similiter de alijs differentijs positionis. Situs ergo partium forme refracte 
non mutantur, sed semper permanent modo suo: cum autem a perpendiculari 
fit fractio. augetur forma secundum dilatationem, et cum ad perpendicularem 
fit refractio, minuitur. quoniam anguli ipsam continentes angustantur. Patet 
10 ergo propositum. 


110 coadunelur: coadiuvetur v.v b (Vb my. inser. et rep. et dei. coadiuvetur) I post lineam ser. et 
dei. C ad ipsam III CG co. est ex G in v.. 111-112 rarioris dyafani tr. v" 112 minor 
rep. E 113 propellebatur v.. 114 in co. est ex a in v" 114-115 perpendiculari l 115 
sit: fit CP 115-116 semper motus tr. V b 116 refractione Rl ref1exione E f1extione 
v" 117 angulus: angulo v.. 118 nunquam: nunquod P I potest: poterit E Patet ergo 
propositum: quod est verum et sic patet propositum E 
l Propositio [48] C I 48 mg. hab. £0 2 aliquem E 4 enim inser. v" Quoniam: 
Queritur f?] E I post patet ser. et dei. v" enim f?] 7 autem om. 1 I cum: quod v" 8 
planam superficiem tr. O I post superficiem scr. et dei. O corporis dyafani I sint: sunt 1, mg. 
inser. v" et ser. et dei. sunt I aequidistantes R I post 6"m add. R p 9 fiat om. v" 10 ut 
om. v.. I incidente l II post unum hab. E num 
I Propositio [49] C I 49 mg. hab. E 3 ut om. O I per: ex E 5 manet 
E I sinister 2 om. E 6 positis l 7 manent CV b I post modo hab. autem suo cum R suo 
autem l I cum inser. v" 8 fit: sit v" I refractio R V b 


'!II.....
>>>
[Propositio]50. In omni sim iii superficie eiusdem dyafani, radij secundum 
equales angulos incidentes, secundum equales angulos refringuntur; et si 
maiores sunt anguli incidentie, maiores sunt anguli refractionum, et si minores, 
mmores. 
Sive enim refractionis modus attendatur a parte superficierum corporum 
in quibus fit refractio. quoniam alia fit refractio a superficie sperica et 
alia a piana, sive a parte dispositionis dyafanorum. quoniam alia fit refractio 
a rariori dyafano alia a densiori. ut patet per plures propositiones libri 
huius, sive attendatur a parte angulorum incidentie, patet semper quod, 
angulis incidentie existentibus equalibus, secundum modum propositum nuHa 
subest causa diversitatis modi refractionis. Fiet ergo semper refractio se- 
cundum angulos equales. Et hoc est propositum primum. Et est huius 
exemplum [Fig. 29], ut si uni corpori sperico dyafano densiori ipso aere, 
in cuius superficie sit circulus ABCDE, cuius centrum sit P, et a puncto 
F corporis luminosi incidant linee radiales que sint AF, BF. CF, DF, 
EF; incidat que radius FC perpendiculariter et alij oblique. Patet quod 
omnes radij incidentes oblique in superficie illius corporis dyafani refrin- 
gentur, per 47 huius. Sit ergo, exempli causa et brevitatis figurationis 
et denominationis linearum, ut omnes illi radij refracti concurrant in puncto 
G, et ducantur perpendiculariter super superficiem corporis linee, que sint 
PDO et PBR et PAX et PEX. Dico quod si angulus incidentie, qui est 
FDQ. sit equalis angulo FBR, quod angulus GDP erit equalis angulo 
GBP. quod patet per premissam, propter uniformitatem omnium premissarum 
conditionum. 
Similiter quoque dico quod si angulus FDQ sit maior angulo FAX, 
quod angulus PDG erit maior angulo P AG. Fiat enim super punctulIi 
A. terminum linee XA. angulus equalis angulo FDQ. per 23 0m r, qui sit 
angulus HAX: refringatur que radiu s HA in puncto A. concurrat que 
cum linea FG in puncto K, eritque per primam partem huius, angulus 
PAK equalis angulo PDG. Est autem angulus PAK maior angulo PAG; 
non enim est equalis. quoniam tunc ex premissis sequeretur angulos incidentie 
esse equales, quod est contra ypothesim. Sunt en im suppositi esse inequales, 


l Propositio [50] C 2 secundum: vel V M I refranguntur Cv" 5 post attendatur ser. et 
dei. v" a f?] sive ex f?] I a: ex l 6 mg. inser. quoniam alia fit refractio v" 6-7 et alia a: 
alia que ex E 7 a 1 0m . PO 8 rarioni P I a 2 om. y. I patebit E 10 propositionum 
l 12 huius om. y. 13 ut sit corpus sphaericum diaphanum densius R I post aere add. 
l medio et v" ser. et dei. medio 14 in cuius inser. V b 15 sint: sunt E 16 EF om. 
y. 17-18 refrangentur Cv" 19 linearum: linee ax f?] C 21 PBR: PBZ l PHT 
Y. I PEX: PEZ R PCX y. I qui est om. C 22 quod: quia y. 23 GBP: GDP 
y. I quod patet om. Rl I per. inser. v" I premissa RCP I premissarum: predictarum 
Rlv" 25 post quoque hab. y.v" ergo 27 post 23 am add. R P 27-30 FDQ per 23 
primi... angulus PAK equalis angulo mg. inser. v" 28 concurrat: concurret l conferat v" 29 
K: B l I eritque om. v" 30 PAK: PAX l 31 sequetur PEO sequeretur co. est ex sequetur 
in v" 32 ypothesim co. est ex ypotheticos in V b I inequales: inquales C, co. est ex equales in E 


19 - Wilelonis Penpeclivae... 


289 


5 


10 


15 


20 


25 


30
>>>
290 


sed neque minor. quoniam sic fieret refractio irregularis, quod est contra 
43 et 45 huius. Est ergo maJor. Ergo et angulus- PDG est maior PAG. 
35 Idem quoque potest demonstrari facilius. ut si angulus FEZ fiat equalis 
angulo FAX. per 8 am III', utpote si arcus AC et CE assumantur equales; 
tunc enim anguli PAG et PEG erunt per premissa equales. Angulus vero 
PDG minor est angulo PEG. quod patet etiam si anguli refractionis ponantur 
esse equales. De hac autem materia hic summarie loquimur, quoniam ipsam 
40 in decimo .huius libro, ubi locum proprium habet, perfectius prosequemur. 
Patet ergo propositum. 


[propositio]51. Datam altitudinem per umbram quanta sit cognoscere 
sole apparente. 
Sit data aItitudo AB. quam proponimus quanta sit cognoscere. sole 
arrarente I Fig. 30]. Et si il1a aItitudo est erecta super surerficiem orizontis. 
5 ducatur in illa superficie linea BD perpendicularis super terminum alti- 
tudinis AB, qui sit B. Et incidat radius solaris per verticem AB, qui sit 
A. ipsi puncto D. et sit AD. Ergo, per ] I huius, erit linea BD umbra 
altitudinis ipsius AB. Erigaturque nota linea EZ, inter umbram BD et 
radium AD. equedistanter altitudini AB,. ut si ZE sit baculus note quanti- 
10 tatis. Erit ergo trigonus DZE, per 2cj'm I', equiangulus trigono ABD; ergo, 
per 4 am VIi vel per 9 huius, erit proportio DZ ad ZE sicut DB ad BA. 
Sed DZ ad ZE proportio est nota, quoniam cum ZE sit assumpta nota, 
potest et linea umbre sue, que est ZD, modica mensuratione fieri nota. 
Ergo OB ad BA proportio est nota; sed DB potest mensurando fieri 
15 nota. Ergo et AB erit nota, quod est propositum, ut si linea AB sit 
aItitudo alicuius turris vel parietis, qui valeat adiri ad mensuranda spatia 
umbrarum. 


33 flet V& I quod: et lV& 3S demonstri 1 36 post 8 am add. R p I AC: AD V. 37 
PEG: PCG V. I premissam lVuv" I equales om. V& 38 patet etiam tr. V. 39 mg. inser. 
hic summarie v" I loquitur [1] v" 40 prosequemur: persequemur Rl prosequamur V& 
I Propositio [SI] C I 51 mg. hab. E 2 post apparente add. R Eucłides 18 theorema 
opt\corum. 3-4 sole apparente tr. E 4 recta O S illam superficiem E I linea om. 
E 6 incidit C 9 aequidistanter R 10 post 29 am add. R P II post 4 am add. R p I DB 
co. est ex BD in v" 13 et: etiam CV& I fieri: scire [1] E I nota: novata [1] V. IS et inser. 
p 16 mensuranda: mensuram PO mensurandum [1] E mensurenda co. est ex mensurendi in 
v" 17 post umbrarum hab. Libri Secundi Finis l Hic finitur liber secundus V& Explicit liber 
secundus O 


- 


. ............
>>>
LlBER TERTIUS 


In premlssls libris, mathematica et naturalia princIpIa premlslmus, per 
que, prout nostra possibilitas fert, nostri propositi consequentia intendimus 
declarare. Volentes autem formarum naturalium actiones sub triplici videndi 
modo prosequi, scilicet illo qui fit per simplicem visionem. et eo qui per 5 
reflexionem, et illo qui per refractionem, in hoc tertio libro, prosequimur 
modum simplicis visionis et dispositionem propriam organ i visivi. Supponimus 
autem hec que secuntur, ut in locis alijs declarata, vel ut per se ipsa 
nota. 


IPETITIONES] 


Visionem non compleri, nisi apud perventum forme visibilis ad animam. 
Item quod per se visibilia sunt tantum duo, scilicet lux et color; quoniam 
lux se ipsa videtur, et ipsa est ypostasis coloris. Alia vero per accidens 
visibilia sunt 20. utpote remotio. magnitudo, situs, corporeitas. _ figura, con- 5 
tinuitas, separatio vel divisio, numerus, motus, quies, asperitas, lenitas, dya- 
fanitas, densitas, umbra, obscuritas. pulcritudo, deformitas, consimilitudo 
et diversitas. Hec en im non solum visu, sed et alijs sensibus comprehenduntur. 
Item petimus lucern fortem ledere visum diutius intuente.m. Item rem maioris 
quantitatis. quam sit oculus, oculo videri. Item rem visam secundum situm, 10 
figuram et ordinem suarum partium videri. Item visum sjmul diversa vi- 
sibilia videre. Item ab ambobus visibus simul unam rem videri. Item quod 


I ante liber tertius hab. R (p. 84) Vitellonis filii Thuringorum et Polonorum opticae. Post Iiber 
tertius hab. C Incipit et post tercius hab Witilionis de modo simplicis visionis prohemium. v"p hab. 
Liber tertius mg. v" tr. Liber tertius. Ante liber hab. EO Incipit et O om. liber 2 mathematicalia 
Rl 3 fert: suITert [1] v" 5 post scilicet ser. et dei. v" vel I post qui 2 add. E fit 6 post et 
add. v" in 7 visivi: visus v" 8 sequuntur Rlv"EO I ut om. Rlv"v" I ut 2 : prout v" 9 
post nota add. R. Petitiones l. et C mg. ser. suppositiones ea que premitimus. 
I [Petitiones] 2 Visionem co. est ex vsionem in v" 3 Ante Item add. R 2 I quod om. 
E I sunt: scilicet v" 4 post lux add. R ex I ypostasis co. est ex ypostatisis in P I post 
ypostasis [1] ser. et dei. v" iIleetum voeabulum 6-7 vel divisio numerus... densitas umbra 
obscuritas om. v" 6 asperitas co. est ex aperitas in V b 8 diversitas inser. P I et 2 om. 
Rlv" 9 ante Item l add. R e et C mg. ser. Petitiones I ante ltem 2 add. R 4 10 ante Itcm 
add. R 5 ł I ante Item add. R 6 12 ante Item l add. R 7 I similis V b I ante Item 2 add. 
R 8
>>>
292 


color non est motivus visus. nisi secundum actum lucidi. Iłem sine contactu 
visionem non fieri. sic ut nec aliquam actionem naturalem. Iłem virtutem 
15 visivam finitam esse et non extendi in infinitum. 


'PRl Ip( ISITU INESJ 


5 


IPropositio]l. Visibili lucern actu non participante. .psum impossibile 
est videri. 
Que enim. ut suppositum est. per se sunt visibilia sunt lux et color. 
Lux autem non est visibilis preter se. Et etiam lux cum sit ypostasis 
colorum. non est possibile colores videri sine luce. Forma en im coloris 
est forma debilior quam sit forma luci s, cum color sit quedam lux in- 
corporata corporibus mixtis. Visus ergo non recipit formam coloris rei 
vise. nisi ex luce admixta cum forma coloris. Et propter hoc alternantur 
colores multarum rerum apud visum, per alterationem lucis orientis super 
ipsas. Et si co lor, qui est per se visibilis. non est motivus ipsius visus, 
nisi secundum actum lucidi. patet quod omni visibili. actu lucern non 
participante, ipsum impossibile est videri. Patet ergo propositum. 


10 


5 


(Propositio]2. Inter quodlibet punctum superficiei rei visibilis et aliquod 
punctum superftciei visus produci posse rectas lineas est necesse. ut res 
actu videatur; ex quo patet. solum in oppositione rei vise ad visum fieri 
visionem. 
Visio enim sive fiat ex eo quod radij egrediuntur a visu super puncta 
rei vise, sive ex hoc quod forme punctorum rei vise per lineas radiales 
perveniant ad superficiem organi visivi. Semper necesse est inter quodlibet 
punctum superficiei rei visibilis et aliquod punctum superficiei visu s produci 
posse lineas rectas ut res videantur actu. Unde. cum hee linee, secundum 
quemcunque propositum modum produci possunt. fit vi sio. nisi forte propter 
alterius impedimenti resistentiam. visus fuerit impeditus. Cum itaque visus 
fuerit oppositus rei vise. videbit ipsam. et cum aufertur ab eius Opposilione. 
non sentiet ipsam. et cum revertetur ad oppositionem. revertetur sensus, 


10 


13 ante Item add. R 9 14 sicud V w I ante Item add. R 10 15 esse: est E 
l Propositio [l] ante Propositio scr. R Theoremata I Propositio C I l mg. hab. EO 3 
post videri add. RAIhazen 39 n l. 5 preter se: praeterquam ex seipsa R I hupostasis Rl 6 
coloris E 8 recipit co. est ex recepit in v" 9 alterantur R 10 alternationem VbEO 
I orientis: exientis E existentis f?] O II est I inser. v" I motivus co. est ex mo'us in P 12 
nisi: non v" 13 ipsum om. E I Patet ergo propositum om. E 
l Propositio [2] C I 2 mg. hab. EO 2 posse: post se Iv" I rectas lineas tr. lE 
6 punctorum mg. inser. V. 7 perveniunt R I necesse est tr. V. 8 aliquod: quodlibet 
E 10 quemcunque: quodcunque l quamcunque V b I nisi: ut v" II inpeditis V. 12 
auferetur C alTertur v" (co. est ex aulTertur) 13 mg. inser. alm. revertetur in P
>>>
quoniam ab alijs partibus quam ab OppOSltlS directe non possunt linee 
produci a punctis visibilium ad puncta superficiei visus. Patet ergo propositum. 


lPropositio]3. Organum virtutis visive necesse est spericum esse. 
Si enim non sit spericum. dico quod impeditur visio. utpote si sit 
superficiei pIane. Tunc enim non videbit uno aspectu. nisi sibi equale. 
Sive enim radij egrediantur a visu super rem visam. sive forme punctorum 
rei vise per lineas radiales perveniant ad superficiem organi visivi. patet 
quod semper perpendiculares sunt breviores. per 21 primi huius; unde res 
ma
!1s approximat visui secundum ilIas. quoniam res visa directe secundum 
ipsas perpendiculares videtur. non per aliquas lineas obliquas que refringantur. 
Quia. ut patet per 48 secundi huius, in corporibus pIani s non potest 
fieri refractio formarum ad aliquod punctum unum. eo quod in talibus 
nullus punctus est omnibus communis. Sola ergo iIIa ab organo visive 
superficiei pIane videri possunt que sine refractione directe perveniunt ad 
ipsum. Hec autem sunt secundum perpendiculares lineas pervenientia ad 
vIsum. 
Sit itaque superficies pIana visus in qua sit linea AB. et sit in super- 
ficie pIana alicuius rei vise equedistantis visui et linee AB linea recta que 
CDE Wig. I]. Et a puncto C ducatur perpendicularis super superficiem 
visus. per I l am Xli. que incidat in punctum A. et sit CA. Et a puncto 
D ducatur similiter super superficiem visus perpendicularis. que sit DB. 
Cum itaque linee AC et BD sint equedistantes et equales. per 23 et 2S 
primi huius. ergo. per 33 primi. linea AB equalis erit linee CD. Et quoniam 
linea AB equalis est linee CD. sed linea CDE est maior quam linea CD, 
ergo non videtur simul to ta linea CDE, quia in hac dispositione non 
potest res visa excedere quantitatem superficiei visus. Et quoniam hoc est 
falsum et contra suppositionem que patet sensui. quoniam possibile est 
rem maiorem ipso oculo videri. palam quia non est possibile ut super- 
ficies organ i visivi sit pIana; sed neque aIterius figure quam sperice. quia 
semper accident impossibilia inequalitatis visionis. Necessario ergo erit sperica 


14 ab 1 0m . V b I ab 2 om. E I possunt linee: potest linea R1 15 post produci add. 
E recte I Patet ergo: quod est E 
l Propositio [3] C I 3 mg. hab. EO I post esse add. RAIhazen 35 n l. 2 post quod 
add. Rl non 3 superficies V. I uno: unico E I sibi equale tr. E 4 a visu super rem 
visam: super rem visam a visu E 5 visivis P 6 post 21 add. R t I primi huius tr. V. 7 
post secundum add. V. lineas I visa: visae lvisas v.v" I directe om. V. 8 ipsas om. 
E I videntur l I refrangantur Cv" res frangantur l 9 post per add. V. 8 1 I post 48 add. 
R t et C scr. et dei. ultimam [1] 10 rep. et dei. refractio v" I formarum mg. hab. v" 11 est 
om. O I visivo R 12 possunt: potest l 13 secundum om. V.v" 16 aequidistantis 
R I linee: linea V.O 18 post lI am add. R p I et l: que E I sit: fit l 20 aequidistantes 
R I 23 et om. R I post 25 add. R t 21 primi huius tr. E I post 33 add. R P et V. 
huius I post primi add. l huius I erit mg. inser. v" I ante linee ser. et dei. v" est 21- 22 
Et quoniam linea. .. est linee CD om. E 22 est l inser. P 25 mg. inser. falsum et ante et scr. et 
dei. falsum v" 


...... 


293 


15 


5 


10 


15 


20 


25
>>>
-- 


294 


30 


superficies organ i VISlVI, In cuius centro fiat eoncursus linearum radialium. 
ex longe maiori magnitudine quam sit ipsum organu m visivum. Palet ergo 
propositum. 


s 


[Propositio]4. Oculus est organum virtutis visive spericum. ex tribus humo- 
ribus et quatuor tunicis, a substantia cerebri prodeuntibus. sperice se inter- 
secantibus, compositum. 
Quomodo sit oculus virtutis visive organum. negotio aIterius partis 
philosophie relinquimus. Quod autem sit spericus. necessarium est per pre- 
cedentem propositionem et etiam ex eo quod est nature aquee, cuius pro- 
prietas est semper rotundari, ut alibi est declaratum. Quod autem sit oculus 
ex tribus humoribus et quatuor tunicis compositus, diligens anathomizantium 
cura edocuit. Primus itaque humorum istorum dicitur cristallinus vel glacialis, 
qui proprie est organum virtutis visive, et est in medio oculi situs; est 
que spera parva, alba. humida, humiditatis retentibilis formarum visibilium 
in qua est dyafanitas non intensa valde, cum sit in ea a1iqua spissitudo, 
unde dyafanitas eius assimilatur dyafanitati cristalli vel glaciei, et ob hoc 
dicitur humor cristallinus. vel glacialis. Quia vero huius dyafanitas mutatur 
in sui parte posteriori versus cerebrum, a qua parte totus oculus recipit 
nutrimentum, quod antequam perfecte uniatur humori cristallino, qui princi- 
paliter intenditur nutriri. nondum plene in formis substantialibus et acci- 
dentalibus eidem assimilatum. necessario est aIterius dyafanitatis ab illo. 
Et ob hoc dicitur aIter humor, et vocatur vitreus. quia similatur vitro 
quasi. frustato. Et quia in.omni quod nutritur semper purum ab impuro 
separatur, iIIud quod ab humore cristallino nutrito, ut sue puritati incon- 
veniens. separatur ad partem oppositam part i nutrimentali. hoc est, ad 
anterius cristallini humoris profluit. Et quia est dyafanum. quoquo modo 
assimilatum humori cristallino. nondum tamen sue perfecte consistentie in 
densitate, eo quod est superfluum nutrimenti corporis densioris, patet quod 
necessario est dyafanum liquidum, unde vocatum est humor albugineus, 
quia simile est albumini ovi in tenuitate et albedine et dyafanitate. Est 
enim humor albus, tenuis, clarus, dyafanus; et habet humorem ad partem 
anteriorem sicut vitreum humorem ad partem posteriorem pro custodia 


10 


lS 


20 


2S 


l Propositio [4] C I 4 mg. hab. E I Occulis v" 2 se om. V b 3 post compositum add. 
RAIhazen 4 n l. I O mg. hab. alm. certas sententias de connexione istae propositionis cum 
Galeno 4 Quomodo: Quoniam V. I post quomodo ser. et dei. V b non 5 Quod: quia 
v" 6 propositionem l 9 humorum istorum tr. v.E I dicitur om. l 11 humiditate 
V. I receptibilis RIO (in O co. est ex recetibilis) 12 intenS8 co. est ex ntensa in V b 14 post 
glacialis add. E et I vero: vera l, inser. V b , om. V. I huius: eius Rlv"E I mutatur: nutritur 
lC (C dei. mutatur et mg. alm. scr. nutritur) 18 ante eidem add. IV. et v" ser. et dei. et 19 
similatur: assimilatur RE 20 frustrato lV b fracto V. I semper mg. inser. V b I impuro co. est 
ex imponito [?] in v" 23 quia om. lV u inser. v" 24 assimilatur v" 26 vocatus 
RlV.Vb 27 est om. v.v" I albumine CV b 28 tenuis clarus tr. Rl I habet: hunc 
RCP 29 sicud V. I post sicut add. l et 


......
>>>
295 


humoris cristallini, ne ab extrinsecis occasionibus vel intrinsecis ci tiu s patiatur, 30 
et cadat ab officio organi visivi, nature sagacitas deputavit. 
Continet autem primos duos humores, scilicet cristallinum et vitreum, 
tela valde tehuis et subtilis separans eos ab albugineo et circumdans ambos 
eos, cuius etiam tele aliqua pars descendens per medium separat cristallinum 
a vitreo; et hec tela, propter sui subtilitatem. tela aranea nominat ur. Cum 3S 
autem humor albugineus sit liquidus per se non consistens. necessarium 
fuit ipsum per aliquod solidum pro oculi custodia retineri; circumdedit 
ergo ipsum natura pelle viscosa solida fort i, non multum dyafana, que 
sui densitate meIius retineat et sui caliditate humorum albugineum temperet, 
ne cristallinus congeletur, et fiat inhabilis receptioni visibilium formarum. 40 
Et quia propter huius tunice densitatern et viscositatem forme visibiles 
ad humorem cristallinum undique taIi tunica circumdatum non pervenissent, 
ideo in anteriori parte oculi. ubi est Ibcus receptionis formarum visibilium, 
natura hanc tunicam intercidit. factum que est foramen rotundum, cuius 
dyameter est quasi equalis lateri cubi descriptibilis intra iIIam speram, vel 4S 
lateri quadrati inscriptibilis circulo magno illius spere. Et est hoc foramen 
ideo rotundum. ut sit magis aptum susceptioni omnium formarum per- 
transiens usque ad eiusdem tunice concavum; et ob hoc hec tunica dicta 
est uvea, quia assimilatur uve in aspectu. Et est hec tunica plurimum 
nigra. sepe tamen viridis et quandoque glauca, et corpus iIIius tunice est so 
tenue densum. non rarum. Ne vero humor albugineus effluat ex foramine 
uvee et ut non impediatur operatio virtutis visive, necessarium fuit nature 
foramini uvee subponere velamen dyafanum solidum ad modum cornu 
albi c1ari. dictaque est hec tunica cornea. Ubi vero coniungitur hec tunica 
alijs partibus corporis circumpositis oculo, ibi cessat dyafinitas sitque alterius ss 
dispositionis tunica solidior quam cornea, non dyafana, cum ipsa tamen 
cornea complens speram unam, que est spera totius oculi, et iIIius spere 
posterior pars non dyafana sed carnosa fit alia tunica. Et hec dicitur 
coniunctiva, vel consolidativa. quoniam coniungit oculum et consolidat ipsum 
cum partibus corporis vicini. Erunt ergo tunica cornea, humor albugineus, 60 
et humor glacialis, et humor vitreus se ad invicem consequentes, et omnia 
ista sunt d yafana propter meliorem formarum visibilium receptionem. 
30 paciatur C patiatur co. est ex patiantur in P 32 primos duos tr. PEO I primum l I post 
cristallinum ser. et dei. v.. i/lectum vocabulum 35 tela 2 om. E 36 humor inser. P 38 ergo: 
autem V b , om. v.. I viscosa: fiscosa C radiosa v.. I forte v.. forti f?] J.-;. 40 inabilis E 41 
huius: eius Rl hoc v..E I tunice om. v.. 42 tali tunica: enim tunice v.. I circundata 
l circumdant PVuJ.-;.O 43 post locus add. E scilicet 44 naturam J.-;. I post natura ser. et dei. 
E hanc 45 descriptibilis: inscriptibilis R I iIIa C 46 inscriptibilis circulo magno iIIius 
spere: circulo magno iIIius spere inscriptibilis E I est hoc tr. J.-;. 47 apta l I susceptioni: 
receptioni 48 hec: hic 10m. J.-;.E 49 assimilabitur l I unae l 50 iIIius: istius 
PEO 51 densum non rarum: rarum densum v.. 52 et: omni f?] J.-;. I inpeditur v.. I fuit: 
fuerit P 53 uve E I post uvee rep. v.. et ut non impediatur operatio virtutis visi- 
ve I supponere RE I solidum: sibi f?] v.. 55 cessant v.. 56 cum om. l 57 post totius 
add. E corporis I et inser. J.-;. 58 post carnosa add. V u et 60 Erit Rl I humor albugineus 
tr. v.. 61 et 1 0m . O 62 receptionem: susceptionem E 


.......
>>>
296 


A substantia cerebri prodeunt humores et tunice oculi. quoniam ex 
anteriori parte cerebri. a duabus partibus ipsius, crescunt duo nervi optici, 
65 id est concavi consimiles habentes duas tunicas ortas a duabus telis cerebri, 
et procedunt hij nervi ad medium anterioris partis cerebri, ubi efficiuntur 
nervus unus obticus. qui in processu iterum dividitur in duos nervos 
obticos consimiles et equales qui, transmutatis suis sitibus, ita ut dexter 
fiat sinister et sinister dexter, sunt procedentes ad convexa duorum ossium 
70 concavorum continentium oculos. quoniam in medijs istonJm duorum ossium 
concavorum sunt duo foramina equaliter perforata que dicuntur foramina 
Pyrationis nervorum concavorum; et quoniam iIIa duo foramina sunt rotunda, 
punctus medius cuiuslibet iIIorum foraminum dicitur centrum illius foraminis. 
IIIi ergo nervi intrant ista duo foramina et exeunt ad concavitatem duorum 
7S ossium predictorum et iIIic dilatantur et ampliantur, et efficitur extremitas 
cuiusque ipsorum quasi instrumentum ponendi vi num in doleis, hoc est 
ad modum pyramidis rotunde concave; et quilibet oculorum componitur 
super unam extremitatem istius nervi et consolidatur cum ipso. Consimiliter 
et a tunicis istorum nervorum oriuntur tunice oculorum. nam tunica cornea 
80 oritur ex tunica extrinseca duarum tunicarum istius nervi et tunica uvea 
oritur ex tu nica intrinseca duarum tunicarum duorum nervorum. Et intra 
istam tunicam uveam ordinatur humor cristallinus, super extremitatem con- 
cavitatis nervi mediante vitreo humore, qui ambo ex medullari substantia 
cerebri oriuntur. Et inter humores istos et tunicam uveam, ex subtilissimis 
85 filis tunice uvee, contexitur tela aranea, quam alij vocant tunicam retinam, 
quia est contexta ad modum retis. 
Sperice se intersecant humores et tunice oculi. Quia enim tunica uvea 
non pervenit intra oculum ad complementum spere, cum, sicut premissum 
est, in anteriori sui parte sit. foramen rotundum, quos tegitur a cornea 
90 tunica, spera ergo tunice cornee necessario intersecat speram uvee et com- 
munis sectio suarum superficierum spericarum est circumferentia iIIius fo- 
raminis. et est linea circularis. per 80 primi huius. In anteriori quoque 


63 prodeunt rep. P I et inser. v" I quoniam co. est ex quia in V b 66 ij Rl I ef1icitur 
Rl 67 nervus unus tr. V. I opticus REO I nervos: terminos f?] V. 69 ossium: 
consimilium V. I post duorum add. V b istorum 71 equaliterr f?] E 72 gyrationis co. est ex 
graitioris in v" I rep. nervorum O I post concavorum rep. et dei. O sint duo foramina equaliter 
perforata que dicuntur foramina gyrationis nervorum 73 ante medius ser. l vero I post 
ilIorum add. E duorum I ilIius roraminis: in ilIa foramine E 74 post IIli add. O videlicet 
f?] lista: ilIa E I antc duorum scr. E ilIorum 75 dilactantur V b 76 cuiuscunque 
EO lvinum: unum v" I dolijs R 77 quilibet oculorum tr. V M I oculorum: ortorum f?] 
C l' quilibet mg. inser. v" 78 Consimiliter co. est ex similiter in v" 80 ex tunica co. est ex 
extunica in v" I' istius: ilIius v" 80-81 istius nervi et tunica... intrinseca duarum tunicarum 
mg. inser. alm. in P 81 intrinsica P I Et om. RlV. 82 uveam ordinatur; uneam ordiatur 
l I super: supra O 84 cerebri: cebri V. I uveam: uneam l I subtillissimus v" 85 
textitur l contexitur co. est ex convexitur in v" I ante tela hab. V. et Itiela V M I tunicam om. 
E I retivam RC retiam l rethinam E 86 rethis E 87 tunice co. est ex tunnce E I post 
enim ser. et dei. est C 88 complenentum E I sicud V. 90 intersecat; secabit l 91 
spericarum: speriorum f?] P superior E 92 80: 79 lCPV.VbEO I ante primi add. R t
>>>
humoris cristallini. propter meliorem formarum receptionem. est compressio 
superficialis parva, minoris curvitatis quam sit superficies cornea continens 
iIIam. spericitas enim superficiei humoris cristallini assimilatur compressioni 
superficiei lenticule. ut patet considerantibus anathomiam oculi. Superficies 
ergo anterior ipsius est portio superficiei maioris spere quam sit spera 
uvea continens ipsam. Et hec compressio equaliter deflectitur ad oppositionem 
foraminis quod est in anteriori parte uvee, quia situs eius ab eo est 
consimilis. Sicut autem foramen rotundum quo;l est in anteriori parte uvee 
est directe oppositum extremitatis concavitatis nervi super quem collocatur 
oculus, sic etiam in parte posteriore concavitatis uvee est foramen rotundum 
quod est super extremitatem concavitatis nervi. et foramen quod est in 
anteriori uvee est oppositum foramini concavitatis nervi. Quoniam nervus 
opticus intersecat tunicam coniunctivam et uveam et penetrat omnes tunicas 
oculi usque ad speram cristallinam, que pyramidem nervi intersecat. sicut 
et humor vitreus. qui in nervi optici pyramidali concavo collocatur. itaque 
communis sectio pyramidis nervi optici et spere cristalline est circulus. 
per 110 primi huius. Spera itaque glacialis est composita in extremitate 
concavitatis nervi obtici et in foramine posteriori uvee rotundo. 
Extremitas ergo nervi continet medium spere glacialis. et est nervus 
iIIe concavus deferens in se spiritum visibilem a cerebro ad oculum, et per 
eius venas parvas pervenit nutrimentum ad oculum, et diffunditur in iIIo 
per vias nutrimenti. Et est in intersectione huius nervi. in anteriori parte 
cerebri. virtus visiva sentiens et diiudicans omne visibile; et consolidatur 
uvea cum glaciali in circulo continente foramen rotundum in posteriori 
uvee. Intersecant quoque se spere iste due, scilicet glacialis et vitrea necessario. 
cum convexum unius obviet convexo alterius sicut enim sunt diverse nature 
et dyafanitatis, sic sunt portiones diversarum sperarum se secantium. Com- 
munis itaque sectio iIIarum sperarum est circulus. per 80 primi huius. Idem 
ergo circulus est basis pyramidis nervi optici, et intersectionis eiusdem py_ 
ramidis et spere cristalline, et consolidationis uvee spere cum spera. cristallina, 
et forte intersectionis earundem sperarum. Corpus vero consolidative continet 
partem pyramidalem nervi, que est intra foramen ossis, per quod transit 


93 meliorem co. est ex maiorem in 
 96 post patet add. n-;. ex I anthonomiam 
CPv..
 97 superficiei maioris spere: maioris spere superficiei E 98 oppositum l 99 
anteriore E 100 Sicut: situs V w I post autem add. v.. formarum I in om. E I uvee co. est 
ex uve in 
 101 est inser. 
 102 sic: si l 104 anteriore E I ante uvee add. 
E parte I uve E I foraminis 
 I Quoniam: quia E I nervuus l 105 obticus 
O 106 sicud v.. 107 obtici O I pyraidali 
 I concavo collocatur tr. v.. 107-108 
itaque communis tr. CPEO 108 obtici CO 109 110: 109 lCPv..
EO I post 110 add. 
R t 110 obtici CEO III ergo: autem E 114 nutrimenti: instrumenti l in. om. 
v... I anteriore O anriori Y" 115 et. inser. 
 I omne visibile: visibile quodlibet E 116 
continente co. est ex continentis in 
 116-117 rotundum in posteriori uvee: in uvee posteriori 
parte rotundum E 118 sicud v.. 12080: 79 lCPv..
EO I post 80 add. R t 121 obtici 
PO I intersectionis co. est ex intersectiones in 
 122 unae l 


297 


9
 


100 


10
 


110 


11
 


120
>>>
298 


12S 


nervus. et intra circumferentiam spere glacialis, et continet speram uveam. 
Ex hiis itaque patet humorem glacialem proprie esse organum virtutis 
visive, nam huius solius dyafanitas est retentibilis formarum visibilium et 
est in medio omnium et humorum et tunicarum collocatus; et si alij 
nllcunque tunice vel humori accidat lesio. salvo glactali humore. semper 
auxilio medicine recipit oculus curationem et sanatur, ac restituitur visus, 
ipsa vero corrupta corrumpitur visus totus sine spe restitutionis per auxilium 
cure medicinalis. Est itaque humor cristallinus vel glaci
lis principaliter 
virtutis visive organum, propter quod est a natura diligentius conservatum. 
Et constituit natura duos oculos propter perfectionem bonitatis visionis 
et complementum eius. Sic ergo patet quoniam humores et tunice oculi 
sperice se iotersecant. et patet declaratio diffinitionis proposite oculi, secundum 
omnium eorum experientiam qui de ipsius anathomia hactenus scripserunt. 
Hec autem omnia que scilicet de compositione oculi, in hac quarta pro- 
positione huius tertii libri nostre perspective, sunt premissa, nunc summatim 
per figuram mathematicam duximus exemplanda, que est talis. 
Sit enim centrum oculi punctum A [Fig. 2], et superficies convexa 
ipsius glacialis arcus BCD. et superficies convexa ipsius vitree arcus BED. 
et tela aranea cooperiens glacialem anterius sit arcus BOD. tela quoque 
aranee inter corp.us glacialis et vitree sit linea recta vel curva que BO, 
teJa quoque cooperiens ipsam vitream posterius sit BQD. exterior quoque 
tunica nervi obtici sit GK dextra et GH sinistra, et interior tunica iIIius 
nervi sit GO dextra et GB sinistra. Superficies quoque uvee sit cuius 
centrum N et in qua sint arcus TMV et BLD, et eius foramen sit cuius 
dyameter est MB. et centrum eius punctum F. Humor quoque albugineus 

11 .:or['lIs HI DU. superficies que intrinsece irsius cornee sit arcus HFK, 
et superficies exterioris cornee sit arcus HXK. Erit ergo medium nervi 
communis punctum G et axis pyramidis totius nervi obtici erit linea GAF 
in qua erunt centra omnium humorum et tunicarum ipsius oc uli. Hec 
itaque est figura totius oculi quam. cum oportunum fuerit. posterius utemur. 


130 


135 


140 


14S 


ISO 


125 speram co. est ex spearam in v" 126 his RI virtuti v" 127 po.
1 huius sa. el dei. 
l e I receptibilis RI 128 pust medio ser. O h 129 cuique V. I glaciali humore tr. 
v" 130 et: ac E I ac et E 13ł spe: sperc v" 133 organu R I per O I a natura: 
ante l I a inser. P I natura mg. inser. v" ,I I post natura add. V. est I conservatum co. est ex 
consevatum in.v" 134 oculos mg. co. est ex angulos in V b I propter perfectionem: per 
resectionem E 135 quoniam: quod Rl I post tunice add. E eius 136 definitionis 
R dispositionis E 137 earum V w I anatomia R I scripserint C 138-139 que scilicet 
de... premissa nunc summatim om. PVwEO 139 nunc: non v" . 140 per figuram mat he- 
maticam... que est talis: in figura mathematica adiecta spectanda propontmus R I exemplan- 
da: explananda EO conplananda V w 141-154 Sit enim centrum... fuerit, posterius utemur om. 
R 141 convexa om. V b 142 vetree PO I BED: BCD l 143 BOD: BED lVbE BCD ['t] 
V. 146 GK: GH l 147 post sit add. V. centrum 148 sint: sit lCv" 149 MB: ML 
CPO I quoque: ergo v" 151 HXK: BEK l HRK [1] v" I nervi: virtutis l 152 optici 
E 154 postea V. I posterius utemur tr. E I post utemur add. O Figura oculi
>>>
299 


IPropositio]5. Impossibile est vlsum rebus visis applicari per radios ab 
oculis egressos. 
Si enim aliqui radij egrediuntur ab oculis. per quos virtus vlslva rebus 
extra coniungitur. aut illi radij su nt corporei vel incorporei. Si corporei. 
tunc, cum visus viderit stellas et coelum, necessarium est ut a visu quid 5 
corporeum exiens impleat totum spatium universi. quod est inter visum 
et partem celi visam. preter diminutionem ipsius oculi, quod et impossibile 
est fieri et etiam tam cito fieri. substantia et quantitate oculi manente 
salva. Si vero detur quod radij sint incorporei, cum sensus non sit nisi 
in re corporali. tunc ipsi radij non sentient rem visam. ergo nec oculus 
corJ1oreus. mediante hic incorroreo non sen1iente. J10terit sentire. Nec enim 
talia incorporea reddunt aliquid visu i, quo visu s posset comprehendere rem 
visam. cum visus non fiat nisi per contactum visus cum forma visa, quia 
sine contactu non fit actio. Radij ergo procedentes ab oculo si nichil 
reddunt vislIi. tllnc non fiet rer irsos visio. Si vero aliquid reddunt visui. 15 
hec erunt luces vel colores. que per se videntur. et que illter radios 
multiplicantur ad visum. Radij ergo non sunt causa applicationis visus 
cum rebus visis. sed aliquid aliud quod se multiplicat ad visum est per 
se causa visionis. Impossibile est ergo radios esse per se causam visionis. 
nisi forte radij dicantur linee descripte per puncta formarum multiplicata 20 
a superficiebus rerum visarum ad visum; quoniam, ut patet per secundam 
huius, inter quodlibet punctum superficiei rei visibilis et aliquod punctum 
superficiei visus necesse est posse produci lineas rectas. ut res actu videatur. 
Tales vero radij ab oculis non egrediuntur. Patet ergo propositum. 


tpropositio]6. Visio fit ex actione forme visibilis in visum et ex passione 
visus ab hac forma. 
Formas visibiles agere in visum ex supposlttone pat et. Leditur enim 
visus ex forti luce, ut in aspectu corporis solaris vel alterius lucis fortis. 
ut lucis reflexe ad oculum a corpore polito. vel ab alio corpore valde 5 
albo. In hiis enim debilitatur visus taliter. ut a sua cad at operatione 
quousque per virtutem intrinsecam naturalem fuerit restitutus. Sed et visus 
patitur a sensibilibus formis. retinet enim quandoque in se fortes earum 


I Propositio [S] C I 5 mg. hab. E 2 post egressos add. RAIhazen 23 n l. item 23 
n 2. 3 egrediantur E S tunc: et patet f?] v.. I quid om. R aliquid l qui Pv"EO quod 
v.. 7 et 1 0m . v" 8 et 1.2 om. E I tancito v" I et 2 om. l 10 corporea E I sentirent 
lRv" I nec inser. P II hic: hoc RlC 12 possit E 13 contactum co. est ex tactum in 
v" 14 nihil Rl 15 fit lCVMv"O 16 hoc CE 17 caussa R I aplicationis Rl 19 
esse per se: per se esse Rl I per se om. v.. I cauS8am R 20 dicantur co. est ex ducantur in 
v" I puncta co. est ex facta mg. in v" 21 a: ad v.. 22 et: est v.. 24 vero: ergo 
E I Patet ergo propositum om. E 
I Propositio [6] C I 6 mg. hab. E I ex: a E 2 post forma add. RAIhazen l. 2. 3. 14 
n l. 3 post ex add. R 2 et 3 4 ut om. l v.. v" inser. P I alterius: alicuius E S ante ut add. 
E alterius I ut lucis om. v" 6 his Rl 7 per inser. P I virtutem: verticem O 8 
quaDdoque in se: in se quandoque V M 


...
>>>
300 


impressiones. Visus en im postquam diu inspexerit fortem lucern vel colorem, 
10 si poste a aspiciat locum obscurum vel locum debilis lu cis. inveniet iłłud 
forte vi si bile, quod prius inspexerat in se ipso cum luce, c6lore. et figura 
sua. Et quandoque color fortis impressus visui permiscebitur coloribus rerum 
visarum in obscuro et videbuntur res ille alio colore mixto colorate. ut 
forte viride visum facit res albas, postea visas in loco obscuriori mixtam 
15 virides apparere; et si cłaudatur oculus. nichilominus occurret visui forma 
prius visa. Forme ergo visibiles agunt in visam et visus patitur ab iłłis. 
Et quia visibilia per se sunt lux et color. et lux est ypostasis colorum. 
lux autem semper sperice ditfunditur ad omnem positionis ditferentiam. palam 
ergo sic etiam colores ditfundi. 
20 Cum itaque visus opponitur alicui rei iłłuminate colorate. tunc muItiplicatur 
lumen vel per se, vel cum iIIo colore rei opposite visui. et perveniens 
ad visus superficiem et agi t in visum. et visus patitur ab illo. Cum itaque 
lux et color veniunt simul ad superficiem visus et agunt in iłłum. et 
visus patitur ab iłłis. et virtus anime. propter unionem formarum visibilium 
25 cum suo organo fit cognoscens. tunc fit visio propter presentiam visibilium 
formarum agentium in visum. Et fit hec actio et passio modo aliarum 
actionum naturalium. quoniam totum agens agit in quodlibet passi. et 
indivisibile. et totum passum patitur a quolibet puncto agentis. Forme ergo 
lucis et coloris. que sunt in aliquo punctorum rei visibilis. perveniunt 
30 ad totam superficiem oculi. et forme omnium punctorum. superficiei rei 
visibilis perveniunt ad unum punctum superficiei oculi. et sic fit actio et 
passio inter ista. Non fit autem actio formarum visibilium in visum. nisi 
forma visibilis sit potens ad agendum et complete ypostasis ex luminis 
presentia. et nisi medium extrinsecum oculo et rei visibili sit lucidum actu. 
35 et nisi organum visus sit receptivum formarum per tunicas medias et humores 
dyafanos sue proprie dyafanitatis. Pars enim tunice cornee superposita fo- 
ramini uvee que primo aeri extrinseco coniungitur et humor albugineus 
implens foramen uvee si a propria ceciderit dyafanitate. utpote mutata 


10 debilcm v.. I iIIud om. R id l 10-11 iIIud forte tr. v.. II post luce add. E et 12 
impressus: conpressus v.. I post rerum scr. et dei. O diversarum 13 obscurro v" I colorato 
v" 14 obscuriori mg. co. est ex posteriori in v" I mixtim RE 15 vinides: vides E I et om. 
Iv" I nihilominus Rl 17 hypostasis Rl 18 autem: enim v.. I sperice mg. co. est ex 
superior in v" 19 sic etiam tr. V v I etiam: et v" 20 iIIuminate colorate tr. E I ante 
colorate ser. Rl vel 21 coloratae l 22 et 1 0m . E iIIo P 24 post illis mg. rep. alm. P Cum 
itaque lux...visus patitur ab iIIis (lineae 22-24 supra) I virtus: visus v.. 25 presentiam: 
sprentiam f?] v.. 26 hec: huius V v 27 ante agens hab. v.. agenti I quemlibet v.. I post 
quodlibet ser. et dei. O passum I post passi add. R punctum etiam et Pv"EA add. etiam I et: in 
Rv"O et in V v usque in E 28 post indivisibile add. ipsius CPv..v"EO 30 totam om. l 31 
unum punctum tr. Rl 32 actio co. est ex fractio in P, mg. inser. v" 33 post complete ser. et 
deI. v" et I hypostasis Rl ypostasis co. est ex ypothosis in V b 34 post et l add. E ctiam 35 
ante visus add. v.. sit I receptivum: receptuum P recept um v.. I post formarum ser. l visibi- 
li um 36 suposita P supposita v.. 37 -38 que primo aeri extrinseco... implens foramcn uvee 
om. v.. 38 inplens C P I unae l
>>>
qualitate sibi propria vel impedimento alio occurrente. vel etiam ipse humor 
glacialis si per nimiam congelationem. vel alio modo a formarum receptione 
fuerit impeditus. non fit visio. quia forma sensibilis organo visivo imprimi 
non potest. 
Forma itaque visibilis, veniens a re visa per medium lucidum usque 
ad superficiem visus, transit per dyafanitatem tunicarum visus. et pervenit 
ad virtutem visivam ex foramine. quod est in anteriori uvee. et pervenit 
ad glacialem, et pertransit in eo secundum modum sue dyafanitatis; et 
ob hoc natura omnes tunicas oculi dyafanas ordinavit, ut a formis sensibilibus, 
actum lucidi habentibus, patiantur. Visus vero licet patiatur a formis vi- 
sibilibus; non tamen tingitur a forma lucis vel coloris post recessum presentie 
corporis lucidi vel colorati, sicut universaliter ostendimus hanc passionem 
convenire omni corpori dyafano. per 4 secundi huius. Et licet quandoque, 
propter fortitudinem lucis et coloris, fiat aliqua impressio in visu m, et 
alteratio secundum illas luces et colores. non tamen ille remanent in visu. 
nisi tempore modico; non est ergo talis alteratio fixa. Visus itaque non 
tingitur et coloribus et formis lucis tinctura fixa. formis sensibilibus agentibus 
in visum. Patet ergo propositum. 


IPropositio]7. Centrum spere totius oculi. et centrum glacialis. et centrum 
superficierum extrinsece et intrinsece cornee. et centrum convexe superficiei 
humoris albuginei necesse est idem esse. Ex quo patet quoniam superficies 
intrinsece cornee superficiei sue ex1rinsece equedistat. 
Resumpta figura oculi quam premisimus in 4 huius I Fig. 2]. dico quod verum 
est quod hic proponitur. quoniam punctum A est commune centrum pro- 
positarum sperarum. Si enim detur quod centrum spere totius oculi. quod 
est punctum A. non sit centrum spere glacialis. palam. per 75 primi 
huius. quoniam linee recte perpendiculares super superficiem spere oculi 
non sunt perpendiculares super superficiem spere glacialis. nisi solum illa 
que transit per ambarum centra; cetere vero omnes que erunt perpendiculares 
super superficiem visus erunt declinantes super superficiem glacialis. Si ergo 
glacialis comprehendat formas rerum visarum secundum mCldentiam Istarum 
linearum que su nt perpendiculares super superficiem oculi et obliquantur 
declinantes super superficiem glacialis. tunc necessario glacialis comprehendit 


39 occurrente: obscurrente v" 40 si co. est ex s in v" I post si scr. et dei. v" est 40 nimiam: 
minimam 40-41 a formarum receptione fuerit: fuerit a formarum receptione E 41 ante 
sensibilis add. E visibilis et I sensibilis: visibilis O I in primi V. inprimi v" 43 lucidi 
E 44 tunicarum om. V. 45 anteriore V.E I uneae l 46 in om. v" I eo om. ł 47 
sensibilibus mg. co. est ex il/ecto voeabulo in v" 48-49 visibilibus mg. co. est ex sensibilibus in 
v" 49 tamen om. v" "I contingitur v" Idecessum PEO 50 colorae V y I sicud 
V. 51 post 4 add. R t 52 et!: vel E 55 et!; ex C a O 56 in visum om. V. 
l Propositio [7] C 2 in trinsece V y 3 Ex: de V. 4 intrinseca E I nequidistat 
R I post aequidistat add. RAIhazen 12 n l. 6 est! om. v" 8 post punctum add. 
E tum I post 75 add. R t 9 oculi: circuli V y 12 ergo: vero V b 14 oblique l 15 
dec1inantur IV. dec1inante V b 


... 
--- 


301 


40 


45 


50 


55 


5 


10 


15
>>>
302 


20 


omnes formas rerum visibilium obliquatas et declinantes a suo situ et figura. 
quam habent extra in superficiebus rerum visibilium. quod est contra sup- 
p0sitionem premissam in principio huius libri. Et quonimn forme incidentes 
medio secundi dyafani densioris secundum lineas non perpendiculares re- 
fringuntur ad perpendicularem. ut patet per 47 secundi huills. sllbstantia 
vero hum0rum et tunicarum oculi densior est aere circumstante. et sub- 
stantie diverse dyafanitatis inter se. ut patet per 4 huius. palam quod 
in ipsa superficie glacialis fiet refractio alia quam in superficie cornee. 
Non distinguet ergo glacialis aliquid in rebus visis propter refractionem 
formarum in sua superficie factarum. 
Manifestum est enim quod linee ohlique incidentes superficiei visus magis 
obliquantur in superficie glacialis. cum glacialis sit aIterius dyafanitatis 
a cornea vel albugineo humore. Est enim in glaciali aliqua dyafanitas 
propter quam recipit form as. et aliqua spissitudo prohibens transitum for- 
marum. et ob hoc figuntur forme in eius superficie et corpore. Nullam 
ergo formarum visibilium comprehendit glacialis secundum eius situm et 
figuram quam habent extra visum. Hoc autem est impossiblle. quoniam patet 
manifeste per suppositionem quod glacialis comprehendit formas rerum visi- 
bilium secundum situm et figuram que habent in rebus extra. Est ergo 
necessarium quod linee que sunt perpendiculares super superficiem oculi 
sint perpendiculares super superficiem glacialis. Erunt ergo superficies oculi 
et glacialis superficies sperarum concentricarum (habentes idem centrum), 
et extremitates omnium linearum ymaginatarum produci a quołibet puncto 
superficiei rei vise perpendiculariter super superficiem oculi concurrunt in 
hoc centro. per 74 primi huius. et sunt perpendiculares super superficiem 
glacialis. per 72 primi huius. Et quoniam superficies cornee anterius compIet 
oculi superficiem spericam et fit cum iIIa una superficies sperica. patet 
quoniam centrum oculi est centrum cornee. per diffinitionem spere. Patet 
itaque quoniam centrum oculi et centrum glacialis et centrum cornee sunt 
idem centrum. 


25 


30 


3S 


45 


16 decIinantur l I figuratur co. est ex figuatur in v" 17 quam habent extra: extra quam 
habcnt v.. I post contra add. R 5 17-18 suppositum v.. 19 secundum: vel v.. I post 
perpendiculares add l huius 19-20 refranguntur Cv..v" 20 ad om. v.. I post 47 add. 
R t I huius om. l mg. inser. v" 21 circunstante l circumstanti P 23 fiat v.. fit E 24 
distinguet co. est ex distnguet in v" I ergo glacialis aliquid: glaciali l 30 finguntur l figantur 
v.. Inulla lCPV b 31 comprehendet Cv.. I glacialem l 32 habuit Rl habeat PO habeat 
[1] E 33 post per add. R 5 34 est: et P 37 superficies om. E I post sperarum scr. dei v.. 
contentarum I concentricarum: contentarum RlC 39 perpendiculariter mg. co. est ex 
comprehenditur in v" I mg. inser. concurrunt v" 40 74: 72 Rlv..v" I ante primi hab. 
R t 40-41 sunt perpendiculares super superficiem glacialis: super superficiem glacialis 
perpendiculares Bunt E I glacialem l v" I sunt perpendiculares super superficiem glacialis om. 
v.. 41 per 72 primi huius Et om. v.. I ante per add. v" tunc I post 72 add. R t I complent 
[1] O 42 una superficies sperica: superficie una sperica superficies v.. 43 per diffinitionem 
spere. Patet: patet per diffinitionem spere v.. 44 post itaque ser. et dei. v" propositum 45 
centrum: centra v.. 


... 
...
>>>
Quia ergo centrum oculi. quod est centrum superficiei exterioris ipsius 
cornee. et centrum spere glacialis sunt unum cum centro totius oculi ex omnibus 
suis humoribus et telis constante. convenientius nature est ut centrum glacialis 
sit ipsum centrum superficiei interioris cornee. ita quod centra. omnium 
superficierum oppositarum forami ni uvee sint unum punctum commune. 
et superficies concava cornee spere fiat equedistans eius superficiei convexe. 
Sic enim. per 72 et 74 primi huius. erunt omnes linee exeuntes a centro 
ad superficiem oculi perpendiculares super omnes superficies oppositas fo- 
ramini. et augebitur bonitas visionis. et erit totus oculus rotundus propter 
unitatem centri cornee cum toto oculo. Et quoniam. per 73 primi huius. 
superficies intrinseca cornee equedistans est superficiei extrinsece ipsius. 
cum ipsarum ambarum sit idem centrum. humor vero albugineus secundum 
eius convexum contingit concavum cornee. ut premissum est per experien- 
tiam anathomizantium in 4 huius tertij. per 79 primi huius. superficies 
convexa humoris albuginei erit pars superficiei sperice. secundum eius convexum 
superficiem concavam spere cornee contingentis. Patet erg(). per 73 prim i 
huius. quoniam convexe superficiei humoris albuginei et concave superficiei 
cornee est idem centrum. Et hoc est propositum. et palet corrolarium. 
[Propositio]8. Speram uveam necesse est toti oculo ecentricam esse. cen- 
trumque eius ad anterius oculi plus accedere, centrum vero oculi amplius 
profundari. Ex quo pate!. centrum uvee cenIris omnium tunicarum et 
humorum anterioris partis oculi amplius elevari. 
Cum enim, ut patet per 4 huius et per precedentem. spera cornea 
secundum eius superficiem manifestam sit continua cum superficie 10tius 
oculi, et pars spere ipsius et totus oculus sit spera maior quam spera 
uvea, quoniam intra se continet maximum circulum spere uvee. patet per 
diffinitionem sperarum se intrinsecus intersecantium quod superficies spere 
cornee est maior superficie spere uvee. Palam itaque, ex diffinitione spere 
maioris, quoniam semidyameter cornee est maior semidyametro uvee. Et 
quia superficies intrinseca cornee superposita foramini uvee est superficies 
concava sperica equedistans superficiei manifeste ipsius cornee. eo quod tota 
cornea est equalis spissitudinis. ut ostensum est in preceden1i. ideo quod 


48 ante nature scr. et dei. C vera f?] 49 centra: centrum lV.v" I post centra scr. E interior et 
exterior ymno 50 foramen l I uneae l I post uvee ser. et dei. v" superficiei I sint: sit Rl 
sunt PEO et sint V. 51 aequidistans R 52 sic: sicud V. I post 74 add. R t , erunt omnes 
tr. v" 54 rotundum V. 55 post per scr. voeabulum illectum P I post 73 add. R t 56 
aequidistans R I superifcie VbO I intrinsece v" 57 post vero scr. voeabulum illeetum 
p 59 ante in scr. P per c f?] I tertij om. R I ante per add. R ergo I post 79 add. 
R t 60 connexi V. I post pars ser. et dei. spere v" I convexam V. 61 Patet om. 
v" I post 73 add. R t 63 idem centrum tr. v" I et patet corrolarium om. l I correlarium 
V.E coroUarium RC 
I Propositio [8] C I 8 mg. hab. O I uneam l I eccentricam R 4 post elevari add. 
RAIhazen 8 n l. 9 definitionem R 10 superficie spere tr. v" t uve E I definitione 
R diffinitionem l II quoniam: que lV.v" 12 quia: quoniam E ł intrinsece 
V. I supposita IV. 12-13 superficies concava sperica: sperica concava superlicies E cornee 
superposita foramini '" manifeste ipsius cornee om. O 13 aequidistans R 14 est equalis rep. E 


303 


SO 


5S 


60 


s 


10
>>>
304 


15 


centrum superficiei intrinsece cornee idem est cum centro superficiei manifeste 
convexe eiusdem cornee. sed superficies concava cornee secat superficiem 
spere uvee super circumferentiam foraminis quod est in anteriori parte 
uvee, ut premissum est in 4 huius et declaratum per 80 primi huius. 
ergo, per 84 primi huius. centrum spere cornee continentis speram uveam 
necesse est remotius esse in profundo quam centrum spere uvee. Patet 
ergo quoniam speram uveam necesse est toti oculo ecentricam esse. centrumque 
eius ad anterius oculi plus accedere. centrum vero oculi amplius profundari. 
quod est principale propositum. 
Et ex hoc etiam patet corollarium. quia cum spera uvee non sit in 
medio consolidative sed antecedens ad partem superficiei manifeste oculi. 
et cum superficies manifesta ipsius oculi sit pars spere maioris. palam. 
ut premissum est. quia centrum eius erit remotius in profundo centro uvee. 
Manitestum vero oculi est superficies ipsius cornee extrinseca convexa, cui 
equedistat eiusdem superficies intrinseca concava. centrum ergo tam super- 
ficiei concave quam superficiei convexe ipsius cornee plus profundatur in 
oculo quam centrum uvee. Et quia superficies concava cornee con1ingit 
superficiem humoris albuginei qui est in anteriori foraminis uvee et super- 
p()nitur ei. patet ex premissa et per 73 primi huius quoniam superficies. 
convexa humoris albuginei est superficies sperica.' cuius centrum est centrum 
superficiei sibi superposite. Superficies ergo convexa cornee et superficies 
concava ipsius. et superficies convexa humoris albuginei attingens concavum 
cornee. cum sint superficies sperice equedistantium sperarum. palam. per 
73 primi huius. quia centrum ipsarum omnium est unus punctus qui amplius 
profundatur centro uvee. 
Et quia superficies an1erioris glacialis est sperica concentrica totali oculo 
per precedentem, et etiam quia superficies spere glacialis convexa secat 
superficiem spere uvee intrinsecus. patet. per 84 primi huius. cum su per- 
ficies e:lacialis sit portio spere maioris quam superficies spere uvee. quod 
amrlius profundatur cenIrum glaciali!'. quam centrum uvee. Centrum itaque 
uvee centris omnium tunicarum et humorum oculi qui sunt anterioris 


20 


25 


30 


35 


40 


4" 


15 est inser. P I cum: omni v.. 16 eiusdem co. est ex ilIecto vocabulo in V. I secat: cecat 
l 17 uvee: unee l 18 post 80 add. R t 19 post 84 add. R t I cornee om. n
 mg. inser. 
v.. I speram uveam om. V. I uneam l 20 - 21 remotius esse in . .. speram uveam necesse est 
om. V. 20 unee l 21 uneam l I eccentricam Rl 22 anterius: terius P 24 etiam 
patet tr. E I correlarium lV.v..O corrolarium E I quia: et V. I unae l 25 consolidantiae 
l I antecedens: anterius R 26 manifesti PV.V..O I pars spere tr. E 27 in rep. 
p I uneae l 28 superficies ipsius cornee: ipsius cornee superficies E 29 aequidistat 
R 30 profunditur l 31 uneae I 3373: 79 R 70 lPV.EO 73 co. est ex 70 in V. I post 73 
add. R t I quoniam: quod CPEO 37 aequidistantium R 38 post 73 add. R t I quia: 
quod EO I est: et v.. 39 profunditur l I centrum Rl 40 est: et V. I concentrica: 
cum centricato l cum centricata v.. 41 spere om. E 42 spere om. E I uneae l I post 
uvee scr. V. quia amplius profundatur centrum glacialis quam centrum uvee centrum itaque uvee 
I patet om. E I post 84 add. R t I cum co. est ex cuius in O 42-43 post superficies add. 
E spere 45 post centris add. E etiam 


..
>>>
partis oculi ad partem aeris extrinsecam respicientes amplius elevatur, quod 
est totum propositum. 


[Propositio]9. Inter centrum oculi et centrum uvee producta linea recta, 
centrum circuli sectionis uvee et medium concavitatis nervi obtici necessario 
penetrabit. 
Ostensum est per 7 huius idem esse centrum totius oculi et centrum 
cornee; sed linea que continuat duo centra cornee et uvee, que in premissa 
figura oculi [Fig. ?], in 4 huius, est linea AF, hec producta pervenit ad centrum 
circuli c;ommunis earum sectionis, per 82 primi huius, ut in punctum F, 
centrum circuli foraminis uvee, secundum cuius periferiam iIIe spere se 
intersecant. Superficies enim concava cornee et superficies convexa uvee sunt 
due superficies sperice secantes se secundum periferiam foraminis uvee, uL 
patet per 4 huius. Palam quoque, per 86 primi huius, quod eadem linea 
producta pervenit ad duo media duarum superficierum cornee inter se 
equedistantium superpositarum iIIi foramini uvee, cuius foraminis periferia 
est circumferentia circuli sectionis. Et quoniam foramen quod est in anteriori 
uvee est directe oppositum foramini quodl'" est in posteriori uvee, quod 
est extremitas concavitatis nervi, palam, per III primi huius, quoniam eadem 
linea producta medium concavitatis nervi optici necessario penetrabit, et 
hoc est centrum circuli basis pyramidis nervi optici concavi. Patet ergo 
propositum. 
[Propositio] I O. [nter centra sperarum g1acialis et uvee linea rt
cta producta 
ad centrum circuli consolidationis sperarum glacialis et vitree cum uvea 
necessario pertinget et super iIIius circuli superficiem erecta erit. 
Patuit ex premissis in 4 huius quoniam spera glacialis intersecat intrin- 
secus speram uveam. Linea ergo per centra istarum sperarum transiens, que 
est linea AN [Fig. 2], per 82 primi huius, erit perpendicularis super centrum 
circuli communis sectionis ipsarum. Iste vero circulus sectionis aut est 
circulus distinguens finem consolidationis harum sperarum ad invicem, aut 
equedistans ei. Superficies enim que est in anteriori parte g1acialis opposita 


l Propositio [9] C I 9 mg. hab. E I uneae l I linea: iam V M 2 uneae l I optici 
R Y" I 3 penetrabunt V. 5 corne Y" I uneae l 6 oculi inser. P I huius inser. P om. 
Y" I AN lCPV.v"EO I producta co. est ex producto in v" I pervenit co. est ex perveniet in 
p 7 circuli co. est ex occuli in v" I post 82 add. R t 8 uneae l I cuitis: eius V M 9 et: 
est V. 10 foraminis om. V. 11 quoque: que l I post 86 add. R t 13 aequidistantium 
R I suppositarum IV. 14 est 1 : et v" 15 oppositum rep. et dei. v" I est 2 om. E I est 
directe oppositum foramini quod est in posteriori uvee mg. inser. alm. in P 16 ante et post palam 
hab. P certa voeabula illecta I III co. est ex 3 in v" et 10 hab. 3 I post 111 add. R t I primi 
huius tr. V. I quoniam: quia f?] E 17 obtici lPV.O 18 nervi om. l ł obtica lPV.O 
l Propositio [10] C I 10 mg. hab. E I uneae l 2 solidationis PEO I unea l 3 
post erit add. RAIhazen 9 n l. 4 intersecat om. E 5 sperarum om. E I qui V. 5-6 que 
est linea AN per om. R 6 post 82 add. R t 7 circuli inser. P 8 post distinguens ser. et dei. 
v" superficiem I finem: superficiem v., inser. V. I harum: istarum E I ad inviemc R 9 
aequidistans R I ei inser. V. I post ei ser. et dei. V. erit 


20 - Witeloni. Penpoct;vlIe... 


305 


5 


10 


15 


5
>>>
306 


10 est foramioi quod est 10 aoteriori parte uvee et situs eius ab eo est situs 
coosimilis, ut patuit io 4 huius. Termious ergo istius superficiei, qui est 
circulus sectioois ioter duas superficies spere glacialis et vitree, aut est 
ipse circulus coosolidatioois istarum sperarum cum uvea, aut equedistaos 
ei. Si ergo circulus sectioois ioter duas superficies, glacialis scilicet spere 
15 et vitree, fuerit ipse circulus coosolidatioois ipsarum cum uvea. iste ergo 
circulus est circulus sectioois ioter superficiem glacialis et uvee, et tuoc. 
ut prius per 82 primi huius, patet propositum. 
Quod si circulus sectioois ioter superficiem spere. glacialis et superficiem 
spere vitree 000 fuerit ipse circulus coosolidatioois sperarum cristallioe et 
20 vitree cum spera uvea, sed fuerit equedistaos circulo coosolidatioois earum 
cum uvea, tuoc superficies spere glacialis si ymagioetur exteodi iotellectu 
mathematico super id quod forma oaturalis sue spere exteoditur, secabit 
speram uvee super circulum equedistaotem isti circulo sectioois spere glacialis 
et vitree. Quooiam iste circulus equalem habet situ m a circumfereotia spere 
25 uvee, et quia iste circulus est equedistaos circulo coosolidatioois, erit oe- 
cessario circulus sectioois ioter superficiem glacialis et speram uveam aut 
ipse circulus coosolidatioois. aut equedistaos ei. 
Quod si circulus iste fuerit ipse circulus coosolidatioois, palam, per 
82 primi huius, quia Iioea traosieos per ceotrum glacialis et per ceotrum 
30 uvee traosibit perpeodiculariter per ceotrum istius circuli. eo quod iste 
circulus est circulus sectioois ioter duas iIIas superficies spericas. Sed si 
iste circulus fuerit equedistaos circulo coosolidatioois et est equedistaos 
circulo sectioois ioter superficiem glacialis et superficiem vitree. est ergo 
cum circulo sectioois ioter superficiem glacialis et vitree io superficie uoa 
35 sperica, que est superficies glacialis. et est equedistaos circulo dicte sectioois. 
Sed si io aliqua spera duo circuli fueriot equedistaotes, lioea traosieos perpeodi- 
culariter ceotrum uoius oecessario traosibit perpeodiculariter ceotrum aIterius, 
ut patet per 68 et per 66 primi huius. Lioea igitur que traosibit per 
ceotrum uvee et per ceotrum glacialis traosit per ceotrum circuli coosoli- 
40 datioois sperarum glacialis et vitree cum uvea secuodum omoes dispositiooes 


10 nateriori R II iIIius v" 12 vitree: uneae l, co. est ex uvee in v" 13 unea 
l I aequidistans R 14 post scilicet add. v" et 15 fuit v.. I iIIe v" I unea l 17 post 
82 add. R t I huius om. lCPv..v"O 20 unea l 21 post tunc hab. O sp I post glacialis 
add. v.. et vitree I ymaginetur: intelligatur E 22 post extenditur scr. et dei. v" spera 23 
aequidistantem R 24 habet situm tr. V. 25 aequidistans R I ante circulo add. v.. a 26 
circulus: circulo v" I speram uveam: superficiem uveae R superficiem uneam l 27 aequidis- 
tans R I ei om. E 28 ilIe V. I fuerit ipse inser. v" I post fuerit ser. et dei. O eq 29 post 
82 add. R t I quia: quod EO I post linea add. C et inser V b recta 30 uneae l I uvee mg. 
inser. v" I post uvee ser. et dei. v" illectum voeabulum 31 post circulus 2 ser. et dei. 
p est I duas iIIas tr. v" 32 aequidistans 1.2 R 34 vitree: uneae l, vitre v" co. est ex vitee in 
E I post glacialis ser. et dei. V b et superficiem I ante vitree scr. v.. superficiem 35 
aequidistans R 36 aequidistantes R I post linea ser. et dei. v" illectum voeabulum I mg. 
inser. v" transiens 38 ut inser. v.. I per 2 om. RIC I post 66 add. R t I ante Linea add. v.. 
Hec I igitur om. V. I transit lEO 39 uneae l 40 unea l
>>>
307 


sperarum et iIlorum circulorum. Est ergo iIIa linea erecta super superficiem 
iIlius circuli. per 66 primi huius, quod est propositum. 
SUnt tamen necessario hij tres circuli circulus unus, quamvis etiam si 
fuerint diversi circuli et equedistantes eadem proposita omnibus occurrunt. 
Secundum eundem en im circulum secant se glacialis et vitrea, et ambe 45 
ille secant uveam et consolidantur secundum eundem circulum cum iIIa, 
et est iIIe circulus basis concavitatis nervi optici; et sic iIIe unus circulus 
optinet otficium quatuor circulorum. 


[Propositio] II. Speram vitream necesse est spere glaciali ecentricam esse, 
centrumque vitree ad anterius oculi plus accedere. 
Quia en im superficies spere glacialis et superficies spere vitree sunt due 
superficies sperice secantes se, centrum ergo superficiei anterioris respectu 
manifesti oculi est remotius in profundo, quam centrum superficiei po- 5 
sterioris, per 84 primi huius, posterior vero harum duarum est superficies 
ipsius vitree, ut preostensum est in 4 huius. Patet ergo propositum. 


[Propositio] 12. Lineam transeuntem centrum g1acialis et uvee, centrum 
quoque vitree et medium concavitatis nervi optici necessarium est transire. 
Quia linea recta transiens centrum spere glacialis et uvee, que in premissa 
figura oculi est linea AN [Fig. 2], producta super centrum circuli consolidationis 
glacialis cum uvea perpendicularis est super superficiem circuli consolidationis 5 
sperarum glacialis et vitree cum uvea. ut patet per 10 huius. Huic autem 
circulo aut idem est circulus intersectionis glacialis cum vitrea, aut equedistans 
ei. Quocunque vero istorum modo rum existente, semper erit predicta linea 
perpendicularis super circulum sectionis spere glacialis cum vitrea; palam 
ergo, per 83 primi huius, quoniam ipsa transit per centrum spere vitree. 10 
Quia ergo linea ista transit per centrum vitree, patet, per 82 primi huius, 
quod ipsa necessario centrum circuli consolidationis perpendiculariter tran- 
sibit. Extenditur ergo in medio concavitatis nervi optici super quem com- 
ponitur oculus, quoniam circulus consolidationis est basis et extremitas 
concavitatis nervi optici, ut patet ex 4 huius. Quia vero ostensum est 15 
supra. per 9 huius. quod inter centrum oculi et centrum uvee producta 


42 post 66 add. R t 43 necessario om. E I hi Rl 44 fuerint: sint lCv..v" I et om. 
E I aequidistantes R I eidem l I occurrant CV. 45 et 1 0m . E I post vitrea add. 
E que 46 uneam l 47 obtici PV..O 48 obtinet Rl 
l Propositio [1 l] C I 11 mg. hab. E I eccentricam Rl 
10 n l. 4 sperice mg. inser. V b I respectu: regulae l 
centrum scr. et dei. C sp sp 6 post 84 add. R t 
l Propositio [12] C I t2 mg. hab. EC I uneae l 2 quoque: que V. I obtici 
PV..O I post transire add. RAIhazen 11 n l. 3 uneae l 3-4 que in premissa figura oculi 
est linea AN om. R 5 perpendiculares l I est om. l I superficiem om. E 6 unea 
l I post uvea add. E superficiem 7 aequidistans RIOpost 83 add. R t Ił Iineaa f?] 
E post 82 add. R t 13 obtici O I quam v..v" 14 quoniam: quam v..v" extremitates 
l 15 optici om. RlC I posc patet ser. et dei. V b per 16 inter: intra V b uneae l 


2 post accedere add. RAIhazen 
5 est remotius om. v.. I post
>>>
308 


20 


linea centrum circuli sectionis uvee et medium concavitatis nervi optici 
necessario penetrat, cum ab eodem puncto. ut a medio nervi optici. super 
eandem superficiem plures perpendiculare non possunt produci. ut patet 
per 20 primi huius, palam quoniam linea eadem per centrum circuli sectionis 
spere uvee et glacialis, et centrum uvee et centrum oculi, et spere glacialis 
et vitree, et per centrum circulis consolidationis est transiens. 
Patet itaque ex premissis quod una et eadem linea. que est QAF, 
transit per medium concavitatis nervi optici et per duo media omnium 
tunicarum oppositarum foramini uvee et est ipsa. per 74 primi huius. 
perpendicularis super superficies omnium tunicarum oppositarum foramini 
uvee; et est perpendicularis super superficiem foraminis uvee. et perpendicularis 
super superficiem circuli consolidationis. et extenditur in medio concavitatis 
nervi optici super quem componitur oculus. Et ipsa est axis totius oculi. 
que in proposita superius figuratione est linea GAF. 
[Propositio]13. Visus non comprehendit res vlsas nISI corpore medio 
dyafano existente. 
Quia enim, ut patet per 6 huius, visio non est nISI ex actione forme 
visibilis venientis a re visa ad visum, forme vero non extenduntur nisi 
in corporibus dyafanis consimilis dyafanitatis, in quibus fit lucis et formarum 
extensio secundum lineas rectas, ut patet per primam secundi huius. Cum 
ergo lineas productas a rebus visibilibus ad visum non abscindit aliquod 
corpus medium non dyafanum, tunc perveniunt forme ad visum et visio 
completur; quod si aliquod corpus non dyafanum intervenerit. impeditur 
multiplicatio forme ad visum. Patet ergo propositum. 
[Propositio] 14. Non fit visio corpore visibili existente similis dyafanitatis 
cum medio. 
Si enim corpus visibile sit dyafanum. tunc non est coloratum. nec 


25 


30 


5 


10 


17 uneae I I obtici y"0 18 cum om. Y" I medio nervi optici: nervi optici puncto medio 
E 20 post 20 add. R t I circuli sectionis tr. Y" 21 spere uvee er. O linter. uvee et spere 
ser. et dei. O et medium concavitatis I post et! ser. Y" medium concavitatis nervi optici super 
eandem superficiem punctis [1] prius [1] perpendiculares erit [1] I et centrum uvee mg. inser. 
alm. P I uneae l 21- 22 oculi et spere.,. et per centrum om. v" 23 et om. Y" I que est 
QAF om. R I que om. l et mg. inser. J-b 24 concavitatis om. Y" I obtici y"0 I et om. 
Iv"V b 25 uneae l I ipsa per 74 primi huius om. Y" I post 74 add. R t 26 post 
perpendicularis add. J-b habet [1] I superificiem Y" 27 uneae l I et est inser. J-b I est om. 
Y" 27-28 foraminis uvee et..,super superficiem om. Y" 27 uneae 2 l I post et 2 add. 
R est 28 circuli: oculi J 29 obtici 10 I quem: quod l 30 qui R I post que add. v"J-b 
est I superius om. l I est: et v", co. est ex et in J-b I post est add. R in rectitudine literarum 
fa, extensa per medium concavitatis nervi optici I linea GAF om. R 
l Propositio [13] C I 13 mg. hab. E 2 exsistente EO I post existente add. RAIhazen 
22. 41 n I. 3 6:9 sexti l 9 6 Y" 5 fit: sit lJ-bEO 7 post 1ineas scr. et dei. J-b 
rectas I rebus co. est ex debus in v" I abscidit J-b 7 -8 non abscindit aliquod... forme ad 
visum om. Y" 10 visus Y" I ergo om. v" 
l Propositio [14] C I 14 mg. hab. E I visibili om. O 2 post medio add. RAIhazen 42 
D I. 3 non est rep. et dei. Y" I coloratum mg. alm. co. est ex colaratatum in C 


.-
>>>
309 


est habens formam luci s, sed solu m lucidi; ergo. non videtur, quoniam, 
ut patet per 4 secundi huius, lux non figitur in corporibus dyafanis taliter, 5 
ut ipsas tingat. vel quod eis prestet actum visibilitatis. Cum ergo dyafanitas 
corporis visibilis fuerit similis dyafanitati aeris, tunc erit eius dispositio 
sicut dispositio aeris et non apprehenditur a visu, sicut nec aer. Et similiter 
est de alio medio quocunque: nullum enim talium videtur, cum dyafanitas 
rei vise non fuerit spissior corporis medij dyafanitate. Si vero corpus visum: 110 
fuerit dyafanum, sed minus quam medium, sicuti cristallus respectu aeris, 
tunc res visa, quoniam habet aliquem colo rem respectu sue spissitudinis, 
videbitur per medium aerem veluti res colorata ; quoniam cum lux oritur 
super ipsum. figetur in ipso aliqua fixione, scilicet secundum illud quod 
est in ipsa de spissitudine. et pertransibit in eo secundum suam dyafanitatem, 15 
et erit in eo forma in aere secundum colorem et lucern, que sunt in 
sua superficie. et illa forma cum pervenerit ad visum, operabitur in visum 
et sentiet visus rem visam. Patet ergo propositum. 


[Propositio] 15. lnter visihile et oculi superficiem distantiam mediam neces- 
sarium est esse. 
Non enim apprehendit visus rem visibilem, nisi quando fuerit in ea 
aliqua lux media. per primam huius; hoc autem non est nisi per mediam 
distantiam. Quando ergo visibile fuerit superpositum visui sine medio, tunc 5 
ipsum non videtur. Res en im per se luminose non possunt immediate 
superficiei visus applicari. Talia enim sunt ut stelle et ignis, que visui 
immediate non possunt applicari. quoniam ex eorum applicatione sequeretur 
corruptio videntis. Reliqua vero corpora non luminosa si visui applicentur, 
iJ!a sine lumine non videbuntur ; requiritur ergo media distantia inter illa 10 
corpora et inter superficiem ipsius visus, in qua se diffundant corporum illorum 
forme mediante luce. Et etiam corporibus visibilibus ipsi visui immediate 
ał'plicatis, tunc corpus oculi secundum situm suum prohibetur a visuali 
operatione. 
Quia enim visio non fit nisi ex parte opposita foramine uvee, ut patet 15 
per 4 huius. si ergo visus comprehendat rem visibilem per immediatam 


S post 4 add. R t I secundi huius tr. V. 6 ipsa RO I contingat v.v" I visibilis l 7 
corpori visibili l I fuerit om. V. 8 sicud 1.2 V. I sicut dispositio rep. O I apprehendetur 
E aprehenditur V. 9 quoque V. lO fuerit co. est ex erit in C 11 minus eo. est ex unus in 
v" I sicut l sicud V. I cristalli E 12 aliquam l 13 aeris lV& 14 post super ser. et dei. 
v" quam f?] I post in ser. et dei. O puncto I secundum om. v.v" I illud: id Rl IS 
transibit lVuv" 16 erit: est l 17 visum l: ussum P 18 sensiet P 
1 Propo!\itio [IS] C I 15 mg. hab. EO 1-2 necesse v.v" 2 post esse add. RAIhazen 
37 n l. 3 in ea inser. v" om. IV. I eo E S suppositum IV. 6 post videtur ser. et dei. 
C corpus I luminose co. est ex luminosa in v" I in mediate V.O 7 stelle: scale V. 8 in 
mediate V.O 9 videntie V. lO requiritur: relinquitur l 11 visus: om. E visua 
v" I dilfundat V. I corpora v" I corporum illorum tr. E 12 in mediante PV. 13 
applicans V u I tunc corpus oculi rep. E I visu ali V. IS post Quia ser. et dei. P om- 
nis I uneae l 16 comprehendit V.v"E I in mediatam Vp
>>>
310 


20 


applicationem, non comprehendet iIIam nisi secundum partem applicatam 
foramini uvee, et non comprehendet residuum rei vise. Et si ymaginetur 
res visa moveri super oculi superticiem quousque visus totam iIIam rem 
contingat. non propter hoc erit judicium per visum, sed potius per tactum ; 
nec enim sic aget in visum forma visibilis, que est forma multiplicata 
extra rem sensibilem, sed res ipsa. Non ergo erit visio nisi inter visibile 
et oculi superticiem sit aliqua media distantia, et hoc proponebatur. 


5 


[propositio] I 6. Visio non tit sine dolore et passione a substantia oc uli 
abiciente. Ex quo patet visum oportere convenientis dispositionis in sanitate 
esse ad hoc ut complete exerceat visionem. 
Quoniam enim glacialis recipit formam lucis et coloris, et lu"X et color 
operantur in glacialem, erit necessario iIla operatio non sine dolore, quamvis 
quandoque non sentiatur ille do lor, ut cum non est valde fortis. Luces 
vero fortes angustiant visum et ledunt ipsum manifeste, ut patet in luce 
solis. vel in luce reflexa a corporibus tersis ad visum. Et quia operatio 
omnis lucis in visum est ex uno genere, non diversiticata secundum magis 
et minus, et maior operatio cuiuslibet lucis in visum est ex genere doloris 
et non diversantur in hoc nisi secundum magis et minus, sic etiam quod 
quandoque latet dolor ipsum sensu m, semper tamen iIla passio quantum- 
cunque insensibilis abicit a substantia oculi. 
Ex hoc ergo patet quod oportet visum convenientis dispositionis in 
sanitate esse ad hoc, ut complete exerceat visionem, quoniam semper com- 
prehensio visibilium a visu est secundum fortitudinem visu s, quia sensus 
visus oculorum diversiticatur secundum vigorem et debilitatem ipsorum; 
humidi enim oculi citius leduntur a lucibus et coloribus et sicci minus. 
Et hoc voluimus declarare. 


10 


15 


[Propositio] I 7. Visio distincta tit solum secundum perpendiculares lineas 
a punctis rei vise ad oculi superticiem productas. Ex quo patet omnem 
formam visam sic ordinari in oculi superticie sicut est ordinata in super- 
ticie rei vise. 


17 comprehendant Y" apprenendet O 18 uneae l 22 nisi om. V.y" 23 proponebatur co. 
est ex proponitur in Y" 
I Propositio [16] C I 16 mg. hab. O 2 abijiciente Rl lvisum alm. inser. 
p I dispositione Y" 2-3 in sanitate esse: esse in sanitate E 3 esse ad mg. inser. 
Y" I excerceat P I post visionem add. RAIhazen 26. item l. 2 n l. 6 Luces: lu [1] E 8 
tersis: politis Rl 9 visium f?] V b I est om. Y" I ex co. est ex in in Y" I ante secundum 
add. R nisi 10 est: et Y" II diversificatur Rl diversificantur V. I in: secundum V. I nisi 
om. l I quod om. Y" 12 semper tamen tr. Y" 12-13 quantumque CV. quacunque f?] 
Y" post quantumcunque scr. et dei. Y" visibilis f?] 13 insensibilis co. est ex invisibilem in 
Y" abicit a substantia oculi: a substantia oculi abicit V. I post oculi ser. et dei. Y" 
et 14-15 dispositionis in sanitate esse: esse dispositionis in sanitate E 15 excerceat 
p 16 a: ab l I quia: et V. 18 sicci: situ V. 19 hec RlY" I volumus PV. 
I Propositio [17] C 3 sic ordinari tr. V. I est: esti V. 4 post vise add. RAIhazen 15. 
18 n l.
>>>
Licet emm. ut ostensum est In 6 huius, tota forma rei visibilis agat 
in visum et in quodlibet punctum superficiei visus. Quia tamen, per 20 
primi huius, forma tantum unius puncti totius su perficie i rei vise opposite 
visui perpendiculariter incidit uni puncto superficiei visus, et fonne omnium 
punctorum residuorum superficiei rei vise veniunt ad illud idem punctum 
superficiei visus super lineas declinantes, per 13 undecimi. et in quolibet 
puncto superficiei visus transeunt in eodem tempore forme omnium punctorum 
que sunt in superficiebus omnium visibilium oppositorum visui in iIIo 
tempore, quoniam suppositum est in principio huius, visum simul diversa 
visibilia videre, sola vero forma puncti que perpendiculariter incidit iII i 
puncto superficiei visu s, per 47 secundi huius, transit recte per dyafanitatem 
omnium tunicarum oculi; forme vero omnium aliorum punctorum refringuntur 
et transeunt per dyafanitatem tunicarum visus secundum lineas declinantes 
super superficiem visus, et etiam ex quolibet puncto superficiei glacialis 
exit una tantum perpendicularis super superficiem visus, quoniam cum spere 
glacialis et totius oculi sit idem centrum, ut patet per 7 huius, quecunque 
linea fuerit perpendicularis super superficiem unius, et super alterius super- 
ficiem perpendicularis erit, per 74 primi huius, sicut autem ex eodem 
puncto superficiei spere glacialis secundum ponentes radios egredi a visu 
exeunt linee infinite ad superficiem visus que sunt declinantes super super- 
ficiem visus, sic a puncto aliquo superficiei glacialis ex quo exit perpendicularis 
super superficiem visus et pertransit foramen uvee exeunt linee alie infinite 
transeuntes in foramen uvee et pervenientes ad superficiem visus declinantes. 
Et sic ut radij ymaginati egredi a visibus quando fuerint imaginati ref ringi 
secundum modum differentie dyafanitatis cornee a dyafanitate aeris, per 
47 secundi !luius. perveniunt ad diversa loca et ad puncta diversa in superficiebus 
rerum visibilium oppositarum visui in uno tempore, et nulla istarum linearum 
occurrit puncto quod est apud extremitatem perpendicularis. sic etiam secundum 
nos ponentes radios non egredi, sed formas diffundi ad visum, forme punctorum 
visibilium que sunt apud extremitates !larum linearum, extenduntur secundum 
rectitudinem !larum linearum et perveniunt ad superficiem visus et, per eandem 


5 agit v.. 6 quolibet pv.. I Quia: quod O I tamen: tantum v.. I post 20 add. R t 7 
vise om. v.. 8 incidet l 10 post 13 add. R p II post eodem add. v.. spere I tempore 
forme fr. v..
 l2 que sunt inser. P 13 post huius add. R 6 suppositione I simuł co. est ex 
substantias f?] in 
 14 incidet RlCP 15 post 47 add. R t 16-17 omnium tunicarum 
oculi.. . et transeunt per dyafanitatem om. v.. 16 ałiorum punctorum tr. E I refranguntur 
C
 17 transiunt E I secundum rep. E 18 quołibet: quo v.. 19 exit: erit Rl, mg. inser. 

 I post cum ser. et dei. v.. cum visus et etiam ex quołibet puncto superficiei glaciałis 21 
linea mg. inser. V b I superficiem unius om. E 22 post 74 add. R t I sicud v.. 23 ante 
glacialis mg. hab. E et małe f?] 24 visus: rei v.. 25 exit: erit l 26 foramen: foramini 
l I uvee: uneae l 27 post et add. l quod et 
 ser. et dei. quod 28 sicud v.. I fuerunt 
l 29 secundum rep. et dei. 
 I a om. l 30 post 47 add. R t I pervenerint l I puncta: 
predicta f?] v.. 31 istarum linearum tr. E 32 occurrunt l I ante perpendicułaris mg. hab. 
et bene E 35 eandem om. R 


311 


5 


10 


15 


20 


25 


30 


35
>>>
312 


47 secundi huius, refringuntur ad idem punctum superficiei glacialis. Solus 
autem punctus qui est apud extremitatem perpendicularis non refringitur, 
sed semper extenditur secundum rectitudinem perpendicularis et pertransit 
ad iłłum punctum glacialis. Si itaque glacialis secundum lineas non per- 
40 pendiculares sentiat, tunc puncta que sunt in superficiebus visibilium nunquam 
ordinabuntur in sensu secundum modum ordinis sui in superficie rei vise, 
quoniam in eodem punto occurrunt forme admixte ex multis formis diversis 
et ex coloribus diversis et non distinguetur aliquis in iłłis. Sed si glacialis 
secundum lineas perpendicularis tantum sentiet, tunc distinguentur in ea puncta 
45 que sunt in superficiebus visibilium, nec erit differentia situs et ordinationis 
formarum visibilium in superficie glacialis et in rebus visibilibus que sunt 
extra. Quoniam autem secundum suppositionem nostram, forme visibilium 
perveniunt ad visum sub figuris quas habent in rebus extra, patet quod 
secundum solas perpendiculares lineas fit visio; tunc enim solum forma 
50 visa sic ordinatur in oculi superficie sicut est ordinata in superficie rei 
vise. Patet ergo propositum. 
Omnes itaque linee diffusionis quarumcunque visarum formarum que sunt 
perpendiculares super superficies tunicarum visus continentur in pyramide 
cuius vertex est centrum visus et cuius basis est circulus foraminis uvee, 
55 vel pars superficiei iłłius circuli. Et quanto magis extenditur hec pyramis 
et removetur a visu, tanto magis amplifiatur, et omnes forme rerum cadentium 
intra iłłam pyramidem extenduntur in rectitudinem linearum radialium et 
pertranseunt tunicas oculorum irrefracte. et hanc pyramidem dicimus pyramidem 
radialem. Forme vero rerum visibilium que sunt extra hanc pyramidem 
60 nunquam incidunt per aliquam iłłorum linearum perpendicularium, sed forte 
accidit ipsas extendi per lineas rectas, que sunt inter ipsas et superficiem 
visus oppositam foramini uvee, et iłłe forme refringuntur a dyafanitate 
tunicarum visus, et non perveniunt ordinate ad virtutem visivam; unde non 
fit distincta visio secundum illas, verumtamen illas formas refractas aliqualiter 
65 accidit videri, sed indistincte, in concursu scilicet ipsarum cum lineis per- 
pendicularibus a centro oculi extra pyramidem radialem productis. Dicimus 
autem nunc superficiem visus iłłam partem superficiei oculi que est opposita 
superficiei foraminis uvee. 


I 
. 


'\ 


36 post 47 add. R t I refranguntur C I post superficiei ser. et dei. Y" visus 37 refrangitur 
C 39 ilIum: ilIam V. ilIud EO 40 sentist: sentiant P sensiat E I puncti R I que: qui 
RlC 41 sensum E 43 ex om. E I post ex scr. et dei. P mult I aliquid O aliquis f?] 
E 44 sentient l I eo puncti R ea puncti J 45 qui Rl 47 post extra add. E vel 
[?] I Quoniam: quam f?] Y" I post secundum add. R 5 49 sola V. 50 sicud 
V. lordinata: ordinatum l ordinati V. 52 Omnis O 54 cuius: centrum V. I post 
circulus ser. et dei. Y" sectionis 55 quanta V. 56 amplifiatur co. est ex amplicatur in 
Y" I rerum cadentium tr. V. 57 rectitudine PV.EO 58 pertransiunt E I irrefracte: 
refractse RlV., co. est ex refracte in Y" I rep. dei. et inser. dicimus Y" 58-59 pyramidem 
radialem om. l 59 visibilum V. 62 uneae l I refranguntur C 63 viso V. 64 
verumtamen: veruntamen Rl verumptamen V.Y"O 65 instincte V. 66 post extra ser. et dei. ) 
 
Y" hanc [7] 68 uneae l , \ 


..... .........
>>>
Quod autem visus comprehendat quandoque iIla que sunt extra pyramidem 
radialem. patet experimentaliter. Extremitas enim acus vel stipule subtilis 
posite in postremo oculi. ut inter palpebras vel in parte lacrimali. quiescente 
visu. videbitur. cum tamen illa extremitas sit extra pyramidem radialem. 
Similiter quoque in eisdem locis circa oculum erecto indice vel alio digito. 
extra pyramidem radialem, que valde subtilis est. quoniam pyramidalitas 
eius non est ampla. unde nichil sui pervenit ad loca que circumdant 
oculum, videbitur tamen superficies ipsius indicis vel alterius digiti. Forma 
itaque istorum visibilium pervenit ad superficiem visus per lineas obliquas 
que sunt extra pyramidem radialem. Patet ergo quod forme rerum taliter 
situatarum respectu pyramides radialis perveniunt ad superficiem visus per 
refractionem factam in superficie visus ab aere, qui est rarioris dyafani 
quam sint tunice ipsius visus. 
Quod autem refractio fiat a superficie ipsius visus formarum oblique 
visui incidentium. patet etiam in iIlis quorum forme, nisi prohiberentur. 
caderent intra pyramidem radialem. Si enim acus, vel alia res subtilis 
minuta, directe opposita foramini uvee, interponatur visui et p ariet i albo, 
videbitur tamen forma totius parietis. cum secundum veritatem forme partis 
parietis directe opposite acui et visui directe non perveniat ad super- 
ficiem ipsius visus, pervenit autem. ut patet, quoniam videtur. Palam ergo 
quoniam pervenit "per refractionem factam in superficie ipsius visus. Omnia 
autem hec videntur indistincte, unde reductis ipsis intra pyramidem radialem, 
et ablato quolibet corpore interposito, videbuntur ilIarum forme distincte 
et perfectius quam prius. Fit ergo visio distincta solum secundum lineas 
perpendiculares a punctis rei vise ad oculi superficiem productas. indistincta 
vero visio fit per Iineas non perpendiculares. et ita vi sio indistincta coadiuvat 
distinctam. 


[Propositio]18. Omnium formarum visibilium distincta visio fit secundum 
pyramidem cuius vertex est in centro oculi. basis vero in superficie rei 
vise; ex quo patet omne quod videtur sub angulo videri. 
Cum. per 6 huius, omnis ,visio fiat ex actione forme visibilis in visum, 
et quelibet pars forme visibilis et punctus se multiplicat per medium extrinsecum 


)\ 


69 autem: antem l I pyramides l 70 radiales l 71 lachrimali R 74 quoniam; quando 
O 75 eius om. E I non om. lY.v" I ante ampla add. E magna nec I nihil 
RlC I provenit lCP 76 oculum: circulum v" 77-79 Iineas obliquas que sunt...ad 
superficiem visus per om. y. 82 ipsius om. E mg. inser. v" I ipsius visus tr. y. 83 visui om. 
y. 84 Si enim acus om. y. I alia: aliqua l I res subtilis tr. V& 85 uneae l 86 tamen: 
cum V&, om. E I cum: tamen EO I secumdum l 87 parietis om. v" I perveniat: 
perveniunt E 88 ipsius visus tr. V w I autem: tunc y. 91 distinctincte P 92 ante visio 
ser. v" Iineas [1] 92-93 Iineas perpendiculares tr. Rl 
ł Propositio [18] C I 18 mg. hab. E 3 post videri add. R Euclides 2 hypothesis opticae. 
Alhazen 19 n l. 4 fiet y. 5 multiplicet RIC 


313 


70 


75 


80 


8S 


90 


9S 


s
>>>
314 


ad oculi superfkiem totam. et tota superficies rei vise ad unum punctum 
oculi, quia tamen oculorum tunice sunt alterius dyafanitatis quam aer 
extrinsecus, sole ille linee formarum a superficie rei visibilis ad super- 
ficiem oculi producte, que protracte centrum oculi penetrant, cum sint 
10 perpendiculares super superficiem oculi. non refringuntur in medio dyafani 
ipsius cornee, ut patet per 72 primi huius et per 47 secundi huius et 
per premissam; alie vero linee omnes refringuntur quia incidunt oblique, 
unde non fit visio secundum illas. Quoniam autem sola glacialis proprie 
est organu m visus, et non superficies oculi, que est pars spere cornee, 
H oportet necessario ut linee per quas debet fieri visio perveniant ad glacialem. 
Et quia non est possibile ut visus comprehendat rem visam secundum suum 
esse nisi quando apprehendit formam unius puncti rei vise ex uno tantum 
puncto sue superficiei, quoniam, ut in premissa ostensum est, omnis forma 
rei vise sic ordinatur in oculi superficie sicut est ordinata in superficie 
20 rei vise, non est ergo possibile ut glacialis comprehendat rem visam secundum 
suum esse nisi quando comprehendit colorem vel formam unius puncti 
rei vise ex uno tantum puncto superficiei visus venientem ad se. 
Et cum centrum oculi et centrum spere glacialis. sicut patet per 7 huius, 
sit idem punctum. necesse est quod omnes linee perpendiculariter producte 
25 a punctis visibilium super superficiem oculi dyafanam concurrant in centro 
glacialis; erunt que quidem dyametri in superficiebus tunicarum oculi per- 
pendiculares super ipsas tunicas oculi. Eritque quelibet perpendicularis occur- 
rens superficiei cornee in puncto uno et occurrens superficiei glacialis in 
puncto uno. Et una tantum perpendicularis transit per punctum aliquod 
30 glacialis a centro cornee per ipsam superficiem cornee superpositam illi 
puncto glacialis que sit perpendicularis super superficiem rei vise, quoniam, 
per 20 primi huius, ab aliquo puncto super superficiem unam una tantum 
perpendicularis duci potesL Unde cum superficies rei vise fuerit equedistans 
superficiei ipsius visus, erit. per 23 primi huius, illa linea perpendicularis 
35 super superficiem visus et super superficiem rei vise, alie vero linee omnes 
sunt oblique super superficiem rei vise, quamvis producte ad centrum visus 
fiant perpendiculares super superficiem visus et super superficiem ipsius gla- 


6 oculi: oculum E I post oculi add. E totum vel ipsius I totami tota V. I superficiem totam: 
superlicies totam E I et tota superficies rei vise: et superficies tota rei vise P et tota rei superficies 
vise V. et superficies tota rei vise EO 7 post quam add. V. ad I aer: aliter f?] V. 8 linee 
om. V b 9 sint: sunt C 10 refranguntur C II cornee inser. P I 72: 73 l I post 72 add. 
R t I per 2 om. RE I post 47 add. R t 12 refranguntur C 13 visio secundum illas: 
secundum iIIas visio E I solus R 15 post linee scr. et dei. v" illectum voeabulum 16 visus: 
visu um V b I rem visam: revisam v" 17 post ex scr. et dei. P quo 18 sue superliciei tr. 
v" 19 oculi superlicie sicut: superlicie oculi sicud V. 20 est ergo tr. PEO I comprehendat 
rem tr. v" 22 uno tantum puncto superficiei visus: tantum uno superficiei visus puncto E 23 
spere glacialis tr. Y u I sicud V. 24 sit: sic P 25 diafoni l I concurrunt PEO I centro: 
centra f?] O 26 glaciali l I quidam CP 28 glaciali l 29 puncto: punctum l 31 
glaciali lV b 32 post 20 add. R t I superficiem: sphaeram l I una mg. inser. v" 33 
aequidistans R 34 post 23 add. R t 35 linee alm. inser. in P 


--
>>>
cialis. Forma ergo cuiuslibet puncti superficiei rei visibilis, mota ad visum 
secundum lineam unam perpendicularem productam ab eo ad superficiem 
visus, occurrit superficiei visus super unum punctum. super quem non occurrit 
ei aliqua fonnarum aliorum punctorum rei visibilis. 
Productis ergo a quol.ibet puncto superficiei rei visibilis ad centrum 
oculi lineis, palam quoniam iste linee producte in diversis punctis oculi 
superficiem spericam oculi secabunt et omnes in centro oculi concurrent, 
quia omnes linee iste continentur quasi in uno corpore continuo, quia 
a punctis quasi continuis unius superficiei rei vise ad unum punctum qui 
est centrum oculi terminantur. Palam ergo, quoniam omnes iste linee ymagi- 
nande sunt in quadam pyramide verticem habente in centro oculi et basem 
in superficie rei vise; erit enim forma cuiusque puncti superficiei rei vise 
extensa secundum rectitudinem linee que est inter iIIud punctum et verticem 
pyramidis qui est centrum visus. Et omnes tunicarum oculi et humorum 
superficies secant hanc pyramidem, quoniam forme penetrant per iIlas; et 
ob hoc quia superficies glacialis convexa secat hanc pyramidem quasi eque- 
distanter basi, figuratur in iIIa superficie glacialis quasi nova pyrami s, cuius 
basis est in ipsa superficie glacialis, et vertex ubi prius, et bases iłłarum 
pyramidum fiunt quasi similes, ut patet per 99 et per 100 primi huius. 
Et ex hoc patet omne quod videtur sub angulo videri, quem continent 
Iinee radiales concurrentes in centro visus. Patet ergo propositum. Linea 
itaque recta transiens per omnia centra tunicarum visus ad locum gyrationis 
concavi nervi super quem componitur oculus, quia illa, ut patet ex premissis 
et 12 huius, transit per centrum visus, et per centrum foraminis, quod 
est in anteriori uvee, et per centrum ipsius uvee extenditur in medio 
pyramidis radialis, dicatur axis pyramidis radialis ; alie vero linee huius 
pyramidis dicantur linee radiales. 


[propositio] 19. Corpus visibile oportet ut sit alicuius quantitates respectu 
superficiei visus ad hoc ut actu videatur. 
lam enim ostensum est. quoniam visio semper fit per pyramidem cuius 
conus est in centro oculi et basis in superficie rei vise, per premissam, 


38 superliciei inser. 
 I post visibilis ser. et dei. vel [1] 
 41 aliqua co. est ex obliqua in 
C I ei aliqua tr. E I aliorum punctorum tr. Rly'
 42 superifcie l 43 quoniam: quod 

 I ante in scr. et dei. V b a I in inser. 
 I post oculi add. E super 44 superficiem 
spericam tr. y. I post oculi add. se PVu
EO I centrum Rl I oculi 2 om. PY.EO 45 post 
quasi ser. et dei. O al [1] 48 pyramidem l I basim R 49 cuiuscunque RIO I post 
puncti ser. et dei. 
 voeabulum illectum 50 illud: idem 
 53 secant O 53-54 
aequidistanter R 54 basis 
 I quasi: quia l 55 glaciali l I post glacia!is add. y. 
quia lubi: ibi y. 56 per 2 om. R I post 100 add. R t 57 post patet add. E quod I post 
videri add. E necesse est 59 recta inser. P lvisus: visuum lvisum Y.V b 61 ante 12 add. Y. 
ex I centra l 62 uneae 1.2 l I post uvee 2 add. E et 64 dicantur !inee radiales om. 
y. I radiales co. est ex radi in 
 
l Propositio [19] C I 19 mg. hab. EO 2 post videatur add. RAIhazen 40 n l. 3 visio 
semper tr. V b I semper om. y. 


-- 


315 


40 


4S 


SO 


ss 


60
>>>
316 


5 


et quod ista pyramis distinguit ex superficie membri sentientis parvam 
partem in qua ordinatur forma rei vise, ut patet per 17 huius. In rebus 
ergo valde parvis erit pyramis parva, et pars distincta per ipsam ex super- 
ficie convexa glacialis, que est primum membrum sentiens. erit quasi punctus, 
vel valde parva. Sed membrum sentiens non sentit formam nisi quando 
pars sue su perficiei ad quam pervenit forma fuerit quantitatis sensibilis, 
respectll totills oClili. qlloniam virtlltes sensus sllnt finite et non extenduntur 
in infinitum; unde sunt secundum unum aliquem terminum ad quem per- 
venire potest virtus sensitiva. Cum ergo pars membri sentientis ad quam 
pervenit forma non est quantitatis sensibilis apud totum membrum sentiens, 
tunc non sentit membrum actionem quam agit forma rei visibilis in ilIa 
parte, propter parvitatem ipsius, quare non comprehendit formam rei tam 
parve. Sole itaque res sunt sensibiles actu, quarum pyramides inter visum 
et centrum visus distinguunt ex superficie glacialis partem aliquam sensibilis 
quantitatis respectu totius superficiei glacialis; iIIe ergo res oportet ut sint 
alicuius quantitatis respectu superficiei visus. Et hoc est propositum. 


10 


15 


20 


5 


(Propositio]20. Visio non completur nisi cum ordinatio forme recepta 
In superficie glacialis ad nervum pervenerit communem. 
Quoniam enim, ut patet in 4 huius. in concursu amborum nervorum opti- 
corum in anteriori parte cerebri constituta est virtus visiva sentiens et diiudicans 
omne visibile, propter quod in uno vidente est unitas sensus visus, ob 
cuius unitatern ambobus visibus unam et eandem rem simul accidit videri, 
patet quod visio non complebitur nisi cum forma visibilis unietur virtuti 
sentienti. que est in concavo communis nervi; oportet enim cognoscibile 
semper uniri ipsi cognoscenti. Quia vero, per 17 huius, formarum visibilium 
fit ordinatio in ipsius oculi superficie sicut ordinate in superficie rei vise, 
et ex suppositione huius res visa secundum situ m, figuram et ordinem suarum 
partium videtur, necesse est ergo fieri ordinationem forme in ipso nervo 
communi secundum modum ordinationis qua est recepta in superficie gla- 
cialis, et aliter non complebitur visio. Patet ergo propositum. 


10 


5 quod: quia v", om. E I distinguitur l I ex: cum v" I ante membri add. v" ex 7 
distincta: districta l 8 que: quod V y 9 Vel: et l I post vel add. E pars superficiei 
I valde parva tr. v" I menbrum v" I formam : foramen l 12 sunt om. E I post 
terminum add. E sunt 12-13 rep. pervenire v" 13 membri sentientis: menbri sentiens v" 
menbri sensientis v" 14-15 membrum 1.1: menbrum v" 17 sensibiles actu tr. CPEO 18 
viso v" I glaciali l I sensibilem l 19 iIIe ergo res oportet ut: oportet ergo ut iIIe res 
E 20 post quantitatis scr. E que debet videri que sit quantitas sensibilis 
l Propositio [20] C I 20 mg. hab. EO I receptu E 2 pervenit O pervenerit v" co. est ex 
pervenete I post communem add. RAIhazen 25 n l. 3-40bticorum CPEO 4 est: sit 
E 7 patet om. E I quod: quia v"v" I visibili uniretur l 8 sentienti: sensitivi v" 
sensientis v" I in: iu l 9 semper inser. P I forma v" 10 ocu\i: circuli v" I ante in add. 
R sunt 11 post ex scr. R 5 12 fieri om. v" I ipso om. E 13 communi: quoniam 
Iv" 13-14 glaciali l 



 
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>>>
[Propositio]21. Humorem vitreum aIterius dyafanitatis a gl acial i necessarium 
est esse. 
Si en im dyafanitas istorum duorum corporum, glacialis scilicet humoris 
et vitrei. sit consimilis. tunc (ut patet per primam secundi huius. et per 
17 huius. et per 72 primi huius). quoniam forme visibiles recepte in super- 
ticie glacialis non reflexe secundum lineas radiales concurrent in centro 
oculi. propter consimilitudinem dyafanitatis. et ibi se intersecantes uIterius 
se ditfundanł. Quia vero ut patet per premissam. visio non completur nisi 
postquam ordinatio forme que recipitur in superticie glacialis pervenit ad 
nervum communem, situs autem partium forme secundum suum esse in 
superticie g1acialis non potest pervenire ad nervum communem nisi per 
ex ten sio nem eius in concavo nervi super quem componitur spera glacialis. 
quia aliter est ipsam impossibile pervenire; forma vero non potest extendi 
a superticie glacialis ad concavum nervi communis secundum extensionem 
linearum rectarum et conservare situs suarum partium secundum suum esse. 
nisi natura aIterius dyafani clarioris sibi occurrat antequam perveniat ad 
centrum oculi. quoniam si non sit medium alterius dyafani, omnes iste 
linee concurrent apud centrum oculi et efficietur quasi unum punctum. 
Et quia hoc centrum oculi est ante locum unionis nervorum opticorum. 
patet, per 91 primi huius. quod si iIIe linee uItra centrum oculi debeant 
extendi. necessario erit linearum iIIarum intersectio in centro. et post centrum 
creahitur nova ryramis. cuius linee longitudinis secundum positionem et situm 
priori pyramidi modo contrario se habebunł. Convertetur ergo totus situs 
figure rei vise quoniam habet in superficie rei vise et in superficie gla- 
cialis, taliter ut iIIud quod est in superficie glaciali dextrum fiat sinistrum 
apud sensum et econtrario. et surerius fiat inferius et econtrario: nec 
perveniet aliquid forme directe ad nervum communem. nisi solum unum 
punctum. quod est in extremitate axis pyramidis. Omnis ergo res. secundum 
modum suo naturali situi contrarium videtur. quod est contra suppositionem 
et manifeste contra id quod accidit in sensu. Patet ergo quod necessarium 
est quod isti humores sint diverse dyafanitatis. quod est propositum. 


I Propositio [21] C I 21 mg. hab. EO I ulterius v.. I necesse Y,. 2 post esse add. 
RAIhazen 2 n 2. 3-4 humOfis et tr. E 4 consimilis: communis v.. I post primam add. 
R t 4-5 per 17 huius et om. r.. 5 post 72 add. R t I primi om. lPy"O I quoniam om. 
R I visibilis Rl I receptis v.. 6 glaciali l I reflexe: refractae l 9 glaciali l 10 post 
esse add. EO est et P ser. et dei. est II glaciali l I non: nec E 12 quem: quam 
ly"łb 13 est ipsam tr. C lipsum R I extendi: ostendi v.. 14 glaciali 1 16 naturao 
łb I post dyafani scr. et dei. P communis 16-18 c1arioris sibi occurrat... si non sit medium 
alterius dyafani om. Y,. 17 omnes: communis RlPy"EO liste: ille Y,.łb 19 oculi om. 
PY,.EO I locum om. E I unionem E I obticorum CPY,.O 20 post 91 add. 
R t I debeant co. est ex debebant in C 21 linearum illarum tr. łb I post illarum add. 
E communis 22 nova: nove v., inser. łb I post pyramis add. E quedam I positionem et 
situm: situm et positionem E 23 prioris łb I pyramidis Iv..Vb I totus mg. inser. łb 24 
quoniam: quem R quam E 25 illud: idem łb I glacialis EO 26-27 necperveniet aliquid 
forme: aliquid forme nec perveniet aliquid forme v.. 27 communem: concavum PVuEO 28 
post extremitate ser. et dei. Y,. qui contra id quod accidet I omnis: Omnes Rlłb, om. v.. 29 
modum om. v.. lvidentur Rl I post contra add. R 5 30 id: illud Y,. 


- 


317 



 


10 


B 


20 


2
 


30
>>>
318 


[Propositio]22. Superficiem communis sectionis spere glacialis et vitree 
ad anterius centro oculi sitam esse, humoremque vitreum et spiritum visibilem 
eiusdem quasi dyafanitatis, et utraque plus dyafana humore gl acial i necesse 
est esse. 
5 Quoniam, ut patet per 20 huius, omnis forma rei vise secundum situm, 
figuram et ordinem suarum partium pervenit ad nervum communem. palam, 
sicut in premissa ostensum est, quod necessarium est quod fiat aliqua 
refractio ante perventum forme ad centrum oculi, quia etiam si fiat re- 
fractio post centri transitum, erunt necessario forme converse, quoniam et 
10 tunc, per 91 primi huius, erit mutatus situs partium forme. Refractio 
vero cum solum fiat ad perpendicularem vel a perpendiculari, ut patet 
per 47 secundi huius, palam, quia non transmutat situm partium, sed solum 
auget vel minuit figuram. per 49 secundi huius. Quia vero glacialis ad 
quam perveniunt forme secundum rectitudinem tota est unius dyafani, 
15 refractio vero non fit nisi medio alterius dyafani, palam, quia non potest 
fieri refractio formarum nisi apud humorem vitreum. cuius corpus, ut in 
precedenti ostensum est, diverse est dyafanitatis a corpore glacialis. 
Hic ergo humor necessario antecedit centrum oculi; ideo ut refringantur 
forme apud ipsum priusquam perveniant ad ipsum centrum oculi, quod 
20 idem est centrum humoris glacialis, per 7 huius; alias enim in centro 
illo fieret concursus omnium linearum radialium. per 72 primi huius. Quia 
ille linee sunt omnes perpendiculares super superficiem glacialis, accideret 
quoque illis formis ulterius progredientibus transmutatio secundum situm, 
per 91 primi huius, ut premissum est. et quia hoc est impossibile, patet 
25 ergo quod humor vitreus antecedit centrum glacialis. Quamvis itaque glacialis, 
in qua est principium sensu s, indigeat lineis radialibus extensis secundum 
rectitudinem. eo quod impossibile est ut forma rei vise sit ordinata in 
superficie visu s, propter magnitudinem rei vise et per unitatem superficiei 
corporis visus, nisi per istas lineas per quas completur comprehensio rei 
30 vise secundum suum esse, perventus tamen formarum ad ultimum sentiens 
non indiget tantum extensione. formarum secundum rectitudinem istarum 
linearum. quoniam receptio formarum in membro sentiente non est omnino 
similis receptioni formarum in corpore dyafano: membrum en im sentiens 
recipit istas formas propter suam dyafanitatem et sentit eas propter eius 
35 virtutem sensibilem. Et sic recipit formas secundum receptionem sensus 
cum alia corpora dyafana recipiant formas tantum ad representandum ipsas 
visui. non autem ad sentiendum. 
I Propositio [22] C I 22 mg. hab. EO 3 plus dyafana tr. O (et inser. alm.) EO 4 post 
esse add. RAIhazen 30 n l. Item 4. 5. 6 n 2. 6 pervenerit v" 7 sicud V u 8 quia: patet 
v.. 9 centrum Y b E 10 post 91 add. R t 12 post 47 add. R t I secundi huius tr. E 13 
post 49 add. R t 14 quem R I totus R IS post nisi add. E in 17 glaciali l 18 
refrangantur C 19 perveniat V b 20 idem est tr. RlV b I est rep. V u I ante alias add. Rl 
q\Jia et V b ser. et dei. quia 21 omnium om. E inser. alm. P I post 72 add. R t 24 post 91 
add. R t I impossibile co. est ex possibile in v" 26 quo R I radiabus f?] O 30 
formarum: forrum V b 33 receptionem v" 34 sentit: senat f?] V b I eas: ens V u I post 
propter scr. et dei. P suam 36 recipiunt O 37 sentiendum: sentiandum v" sensiendum E 


':....
>>>
Qualitas ergo receptionis formarum in humore vitreo secundum lineas 
refractas est propter diversitatem sue dyafanitatis a corpore glacialis et 
propter qualitatem receptionis sensibilis que non est complete in humore 
glaciali. Sed et corpus subtile, quod est in concavitate nervi inter humorem 
vitreum et nervum communem, quod corpus nominatur spiritus visibilis 
quoniam in ipso primo discurrunt spiritus visibiles, necesse est dyafanum 
esse, quoniam forme rerum visibilium quando perveniunt in corpus humoris 
vitrei extenditur sensus ab iłło in corpus sentiens extensum in concavo 
nervi continuati inter visum et anterius cerebri, et secundum extensionem 
sensus extenduntur forme ordinate secundum suam dispositionem. Patet 
ergo quod ordinatio partium corporis sentientis formas et ordinatio virtutis 
sentientis equaliter est necessario in corpore vitreo et in omni corpore 
subtili extenso in concavo nervi. Cum enim forma pervenit ad aliquod 
punctum superficiei vitree, extenditur directe et non alteratur eius situs 
in concavitate nervi in quo extenditur corpus sentiens, et erunt forme omnium 
punctorum consimilis ordinationis adinvicem. Corpus itaque sentiens quod 
est in concavo nervi erit necessario dyafanum propter receptionem formarum 
visibilium, eritque dyafanitas eius quasi eadem cum dyafanitate humoris 
vitrei, ut non obliquant vel fiant monstruose forme apud perventum earum 
ad ultimam superficiem vitrei vicinantem corpori quod est in concavo 
nervi. Pertranseunt ergo forme in isto corpore subtili ratione dyafanitatis 
et apparent virtuti sensitive ratione spissitudinis eiusdem corporis. Sentiens 
itaque ultimum, quod est in nervo communi, comprehendit lucern ex ilIumi- 
natione corporis huius et colorem ex eius coloratione, quoniam horum 
forme transeunt et figuntur in ipso. Fit autem refractio formarum apud 
humorem vitreum tam propter diversitatem qualitatis receptionis sensu s quam 
propter diversitatem dyafanitatis humoris glacialis et vitrei. Et si dyafanitas 
suorum corporum esset consimilis. esset forma extensa in corpore vitreo 
secundum rectitudinem linearum radialium propter consimilitudinem dyafani- 
tatis, et esset refracta propter diversitatem qualitatis sensus inter hec duo 
corpora; et sic fieret forma aut monstruosa aut essent due forme. 
Quando vero propter dyafanitatis diversitatem fit refractio, et diversitas 
qualitatis sensus affirmat illam refractionem aut obliquationem, tunc erit 
forma post obliquationem refractionis forma una ordinata secundum suarum 


39 est inser. v.. I glaciali l 40 equalitatem v" I compIeta RlE 43 visibiles: visibile v" 
visibilis o 43-44 dyafanum esse tr. E 44 quando: quoniam v" 45 extenditur: conten- 
ditur v" I sensiens v" I extensus V. 46 extentionem co. est ex contentionem in v" 48 
corporis: corpus v" I sensientis P 49 corpore: parte E 50 concava: continuo 
v.. I Cum: dum l 51 superficiei vitree tr. E I vitree om. v.. 53 sensiens v" 54-58 
erit necessario dyafanum... corpori quod ist in concavo nervi om. O 56 obliquentur R I post 
vel add. v" non I post earum ser. et dei. v.. isto corpore subtili 57 corpori quod tr. l 58 
ergo: autem E 60 est in nervo communi: in nervo communi est E I communi: quod 
l communis C 61 ex om. v" 62 figuntur: figurae l 64 diversitatem co. est ex receptionem 
mg. in v" 65 consimilium v.. I post consimilis scr. et dei. nec v" 66 radialium om. V. 68 
fieret: fient l I forma: formae l I menstruosa v.. 69 dyafanitatis diversitatem tr. 
C I post diversitatem scr. E et 


319 


40 


4.5 


.50 


.5.5 


60 


6.5 


70
>>>
320 


partium sit um. figuram et ordinem quam habet forma in re extra, et virtus 
sensitiva sentit formam rei vise ex toto corpore sentiente extenso a super- 
ficie visus primo'sentientis et sensibiles formas recipientis usque ad concavum 
75 nervi communis. quod est ultimum corpus sentiens. quoniam in ipso consti- 
tuta est virtus sensitiva. Sunt itaque humor vitreus et corpus quod est 
in concavitate nervi eiusdem quasi dyafanitatis, quia inter ipsa non fit 
refractio aliqua sensibilis diversa. sed regulatur per unitatem virtutis sensitive 
ad unitatem simplicis extensionis forme post refractionem in superficie 
80 vitree. Et quoniam in hijs ambobus corporibus fit progressio formarum 
ultra centrum oculi. patet quod illa refractio facta est a perpendiculari 
erecta a puncto refractionis super superficiem glacialis; utrumque ergo 
illorum corporum est plus dyafanum corpore ipsius glacialis, per 45 vel 
47 secundi huius. Patet ergo propositum. 


(Propositio]23. Superficiem communis sectionis spere glacialis et vitree 
necesse est planam esse, aut partem spere maioris quam sit spera glacialis 
et ecentricam superficiei oculi. 
Istarum sperarum, glacialis scilicet et vitree, communis sectionis super- 
5 ficies est necessario pIana. aut talis qualis proponitur, quoniam oportet 
superficiem huius sectionis esse similis ordinationis ita quod eius extremi- 
tates ordinentur in consimili et eadem distantia a centro oculi; ut non 
appareant forme monstruose post refractionem. Superficies autem consimilis 
ordinationis aut est pIana. aut est sperica. Hec autem superficies non potest 
10 esse ex spera concentrica oculo, tunc enim essent linee radiales que sunt 
perpendiculares super superficiem glacialis perpendiculares etiam super ipsam, 
ex 74 primi huius, et non fieret refractio formarum, sed concurrerent 
in centro. et fierent forme monstruose sicut per prt::missam ostensum esL 
Est ergo illa superficies, si fuerit pars spere, necessario ecentrica oculo. 
15 Ergo non potest esse ex spera minore quam sit spera concentrica oculo, 
quoniam ratione diversitatis centri forme concurrerent ante perventum suum 
ad centrum oculi; minoris enim spere minor est dyameter quantum est 
de natura spericitatis; et propter maiorem dyafanitatem spere vitree super 


73 sentiva P I sentit mg. co. est ex sensieit [1] in v" I extense [1] v" 74 sensientis 
p I sensibibiles E I usque mg. co. est ex visus in V 6 I concavum co. est ex concvum in 
v" 75 sensiens V 6 77 ipsas V 6 78 regulariter Rl I unitatem: virtutem V M 79 
unitatem co. est ex virtutem in v" I extensionis forme tr. E 80 ijs Rl I post hijs add. v" 
omnibus I formarum co. est ex forma in v" 83 iIIarum l I ipsius om. E I vel: et 
v..E 84 post 47 add. R t I post propositum add. E demonstrandum 
I Propositio [23] C I 23 mg. hab. E I communis: eon munis v.. 2 plenam PO planam 
co. est ex palanam in v" 3 eceentricam Rl I post oculi add. RAIhazen 3 n 2. 5 necessario 
om. E I plena v" 6 ita quod eius: itaque eius lPv..V 6 0 eius itaque E 8 menstruose 
v.. VbEC I post: per l I autem om. Rl 9 pIana aut est om. E I est 2 co. est ex in in 
v" 10 essent: erunt lPv..O 12 post 74 add. R t I concurcent v" 13 menstruose 
v..VbE I sicud v.. 14 ilIi v.. I eccentrica R ecentricum v.. I oculo: circulo v.. 15 Ergo 
om. v.. I concentrica: eccentrica R ecentrica l 16 concurrent 10
>>>
'T 


glacialem. que ostensa in premissa. refringerentur forme ab ipsa perpendiculari. 
per 47 secundi huius. ratione rarioris dyafani, cui incidunt; ratione vero 
spere minoris in superficie communis sectionis frangerentur ad perpendicularem. 
Sic ergo efficerentur forme monstruose, quoniam procederent ad per- 
pendicularem. ratione sue perpendicularis super superficiem spericam. que 
perpendiculares semper transeunt etiam centrum, per 72 huius. et reflecte- 
rentur a perpendiculari. Ista ergo superficies est aut pIana aut sperica, 
utpote pars spere alicuius bone quantitatis. ita quo d spericitas eius conveniat 
ordinationi secundum proportionem refractionis a perpendiculari; que fit 
propter naturam alterius dyafanitatis. Omnes ergo forme pervenientes in 
superficiem glacialis extenduntur per corpus glacialis secundum rectitudinem 
linearum radialium quousque perveniant ad istam superficiem; tunc reflec- 
tuntur apud ipsam secundum lineas consimilis ordinationis secantes lineas 
radiales. Forma itaque perveniens in aliquod punctum superficiei glacialis 
semper extenditur super eandem incidentiam Iinee ad idem punctum superficiei 
visus et ad idem punctum loci nervi communis. A quibuslibet ergo duobus 
punctis consimilis situs in respectu duorum nervorum extenduntur due forme 
ad idem punctum in nervo communi, donec fiat perfecta unitas formarum. 


[Propositio]24. Inter omnes lineas pyramidis radialis necesse est solam 
axem transeuntem per centrum foraminis uvee super superficiem communem 
glacialis et vitree et super posteriorem superficiem vitree perpendicularem 
esse. 
Axis enim hic, si non fuerit perpendicularis sed declinans super aliquam 
istarum superficierum, accidet diversificatio ordinationis formarum pervenien- 
tium ad iIIam superficiem. et mutabuntur dispositiones iIIarum formarum 
propter declivitatem axis. Solum enim tunc, cum axis fuerit perpendicularis 
super superficiem glacialis, perveniet forma rei vise in superficiem g1acialis 
ordinata secundum ordinem partium superficiei rei vise, et perveniet forma 
puncti quod est apud extremitatem axis in superficie rei vise ad punctum 
quod est super axem in superficie glacialis, ut patet per 17 huius. Et 
quia axis. radiałis est perpendicularis super superficiem glacialem, pala m, 
ex 18& VI', quoniam omnes superficies pIane exeuntes ab axe et secantes 
superficiem glacialis erunt perpendiculares super istam superficiem. Et quia 


19 post ostensa add. RE est I refrangentur C 20 post 47 add. R t I rarioris dyafani co. 
st 
ex dyafani narioris in 
 I diafani: dyafanitatis E 22 efficerent v.. I menstruose 
v..
E 24 pertranseunt v.. I etiam: per Rl in E I post 72 add. R t 24-25 refringerentur 
R 2S est om. v.. I plena 
 270rdinationem J.-;, 28 propter: per Iv.. 30 pervenerint 
Rl perveniunt J.-;,EO 30-31 refringuntur R 33 super: secundum J.-;, 34 quibus licet 
PO 35 post consimi\is scr. et dei. P ordinationis secantes 36 unitas: utrum f?] vitrum f?] v.. 
I Propositio [24] C I 24 mg. hab. EO I solum R 2 con munem v.. 4 post esse add. 
RAIhazen 7 n 2. 8 declivitatem: declinationem Rl I tunc om. l 9 forma om. v.. 10 
ordinata secundum rep. E 12 glaciali l 13 glacialis RC 14 post 18 add. R p I post 
superficies inser. mg. alm. P linee 15 glacialem 1 


21 - Witelonis PenpectivllC... 



 


321 


20 


25 


30 


35 


5 


10 


15
>>>
322 


20 


superficies humoris vitrei respiciens ipsam speram glacialem, que est communis 
sectio spere glacialis et vitree, ut patet per premissam. aut est superficies 
piana aut sperica, et centrum eius non est centrum visus. si ergo axis 
radialis est declinans super istam superficiem et non est perpendicularis 
super ipsam, non exibit ab axe superficies piana perpendicularis super 
istam superficiem, nisi una tantum superficies, illa scilicet que transit per 
inequalitafem maximam angulorum, que patet per 39 primi huius; et omnes 
superficies residue exeuntes ab axe erunt declinantes super ipsam superficiem 
vitree. 
Si enim due superficies vel plures exeuntes ab axe sunt perpendiculares 
super dictam superficiem, cum ilIe superficies de necessitate se intersecent 
et sua communis differentia sit axis pyramidis radialis, erit, per 19 8m XI'. 
axis perpendicularis super eandem superficiem; datum autem ruit quod esset 
declinans. Sit itaque centrum oculi punctum C [Fig. 3]; in superficie quo- 
que oculi, sive in ipsa superficie glacialis, que per 7 huius et per 73 primi 
huius equedistat superficiei ipsius oculi, sit linea BAD, et in superficie 
humoris vitrei respiciente humorem glacialem sit linea EGF. sitque axis 
pyramidis radialis linea AC. Ymaginemur ergo superficiem ABCD exeuntem 
ab axe et erectam super superficiem glacialis transeuntem per centrum oculi, 
quod est C. et hec superficies sit erecta etiam super superficiem humoris 
vitrei. que est CGF. Sitque communis sectio huius superficiei erecte ABCD 
cum ipsa superficie glacialis linea BAD, et sint puncta B et D equaliter 
distantia a puncto A, quod sit terminus axis pyramidis visualis. Et sit com- 
munis sectio eius cum superficie humoris vitrei linea EF. Exeant quoque 
due linee a centro C. que sint CB et CD; erunt ergo iste due linee CB 
et CD cum axe CA in superficie communi perpendiculari super super- 
ficiem EGF. per 1 0m Xli. quoniam omnia puncta CBD sunt in illa super 
ficie. Eruntque per ypothesi duo anguli ACB et ACD equales. quod patet 
per 8 8m Ii, si illis arcubus BA et AD subtendantur corde BA et DA. 
Sint quoque linee CB et CD secantes lineam EF, que est communis sectio 
dicte superficiei erecte et superficiei vitree super duo puncta E et F, 
secetque axis CA eandem lineam EF super punctum G. 


25 


30 


35 


40 


45 


16 humoris om. v.. I s
ram: superficiem Rl sperram mg. inser. alm P I glacialis R 20 axe 
co. est ex aere mg. in V b 22 angulorum rep. et dei. C I 39: 29 l 30 R I post 39 add. 
R t 23 superficies om. v.. I axe mg. co. est ex eodem aere in v" 26 se intersecent co. est ex 
intersecent se in v" 27 post 19 am add. R P 28 esse v.. 29 post itaque add. E sig- 
natum 30 glaciali Rl I et om. v.. I post 73 add. R t 31 huius om. v.. co. est ex huis in 
v" I aequidistat R I ipsi oculo v.. 32 recipiente RlCP 33 pyramis V u I ymagiemur 
v.. ymaginenus f?] O I post ymaginemur ser. et dei. v" videtur 34 glacialem l glacia 
E I centra Y u 35 sit erecta tr. Rl 36 EGF RW b 37 glaciali lPEO I BAD co. est ex 
BCD in v" I sint: sicut V b 39 cum: in v" I superficie: sit l I exeuot lV b 40 sint: sit 
C v.. v" I CD: CB l 40-41 ergo iste due linee...cum axe CA mg. inser. V b 41 post CD 
add. Iv.. et 42 post lam add. R P 43 Eruntque: erunt P que v.. om. E I per: ex 
Rl I hypothesi Rl ypothesim O 44 post 8 am add. R P 45 quoque: que E 46 E et F: 
F et E R 47 post EF ser. et dei. P que est communis sectio
>>>
Si ergo superficies que est communis sectio spere glacialis et vitree 
est piana, erit differentia communis, que est EGF, linea reeta. Et si axis 
AC fuerit declinans super superficiem vitree et ipsa est in superficie ABCD 
erecta super superficiem EGF, tunc necessario erit axis CA declinans super 
lineam EF. Erunt ergo anguli EGC et FGC inequales, quoniam linea a puncto 
G perpendieulariter produeta super lineam EGF, ex lI a ( faciet angulos 
equales cum linea EF. Cum itaque anguli EGC et FGC sint inequales, 
angulus quoque CGF sit, exempli causa, minor angulo CGE, et duo anguli 
ACB et ACD sint equales, erunt, per 24 am Ii, due linee EC et CF 
inequales; est .enim linea CF brevior quam linea EC. Si enim ille linee 
sint equales, cum anguli ECG et FCG sint equales et linea GC communis 
ambobus triangulis, erunt, per 4 am t, anguli EGC et FGC. equales, quod 
est contra dat um, cum axis AC sit declinans super lineam EF. Sit ergo 
linea CH equalis linee CE, et ducatur linea HG que, per 4 am Ii et ex 
premissis, erit equalis linee EG; et a puncto G dueatur perpendicularis 
GL super lineam CH, per 12 8m ( Ex penultima ergo prim i, latus GH 
oppositum angulo recto in triangulo HLG est maius latere GL; ergo, 
per 19"m eiusdem Ii, erit linea d'H maior quam linea GF. Cum enim 
angulus GFH sit extrinsecus angulo GLF recto, palam quod angulus GFH 
est obtusus; est ergo maior angulorum trigoni FGH. Ergo linea EG, que 
est equalis linee GH, maior est quam linea GF. Erunt ergo duo puncta 
E et F diverse distantie a puneto G, et ista duo puncta E et F sunt 
iIla ad que perveniunt forme duorum punctorum superficiei glacialis, scilicet 
B et D, que sunt equaliter distantia ab axe. 
Puneta itaque equaliter distantia ab axe in superficie glacialis inequaliter 
distant a puneto axis incidentis superficiei vitree, quod, cum ita sit, palam, 
quia cum forma pervenerit a superficie glacialis ad superficiem humoris 
vitrei, erit ordinatio forme non secundum esse quod habet in superficie 
glacialis neque seeundum suum esse in superficie rei vise. Quando ergo 
axis fuerit declinans superfieiem planam, que est communis sectio super- 
ficiei glacialis et vitree, et erit linea, que est differentia communis cuiuslibet 
superficiei exeuntis ab axe erecte super superfieiem vitree, et superficiei 
ipsius vitree, continens cum axe duos angulos inequales, preter quam in una 
tantum superficie, que secat secundum angulos reetos superficiem transeuntem 


49 plena v.. I erit: et l I difTerentiam C I recta: erecta l 51 post CA ser. E dedi 
ca 53 post 11. add. R P 54 anguli: angulo v.. 55 CGF: IiGF E I caussa R 56 24: 
28 v.. I post 24. m add. R p I CF: EF RlPv..v..EO 57 post inequales ser. et dei. v.. 
sunt I CF: EF RlPv..EO I CF co. est ex EF v.. I EC alteravit alm. ad FC in C I post si 
ser. et dei. P n 58 et 1 0m . E I equales: inequales E 59 triangulis: trigonis v.. I post 4. m 
add. R P 61 et 1 0m . l inser. v.. I post 4. m add. R p 63 GL: Gl R GB v.. I post 12. m add. 
R P I GH: GHA V M 64 in om. V M I HLG: HlG R I GL: Gl R 65 post 1
 add. 
R P I eiusdem om. R 66 post angulus scr. et dei. O scilicet [1] I GLF: GIF R ł GFH: 
FGH v" 67 FGH: FHH [1] v.. I que om. O 68 est 2 om. Iv.. mg. inser. v" 70 forma 
v.. 71 B: H v.. 72 Puncta itaque equaliter distantia ab axe mg. inser. v" I glaciali l 73 
ita om. v.. 74 glaciali l 76 glaciali l I neque: nec RC non l necque v" I suum om. v.. 
SUum esse tr. O 78 et 2 om. R 81 tantum superficie tr. v.. 


323 


50 


55 


60 


65 


70 


75 


80
>>>
324 


. per dec1ivitatem axis, quoniam huius tantum superficiei communis differentia 
continebit cum axe angulos rectos. Et cum duo anguli predicti fuerint 
inequales, et anguli apud centrum glacialis equales, erunt due partes dif- 
8.5 ferentie communis, que est in superficie vitrei, inequales. Forme ergo 
secundum ista puncta, que su nt in extremitatibus istarum differentiarum 
pervenientes ad superficiem vitree, erunt diverse distantie a puncto axis 
quod est in ista superficie. Sed. quia puncta istarum linearum in super- 
ficie g1acialis equaliter distant a puncto axis, in eadem superficie videbuntur 
90 forme non secundum suam ordinationem in superficie glacialis et in rei 
vise superficie. 
Similiter quoque demonstrandum si superficies vitree fuerit sperica, et 
fuerit axis declinans super ipsam: tunc enim axis non transibit per centrum 
vitree et tamen transibit per centrum glacialis. Linee ergo que exeunt a centro 
9.5 glacialis ad puncta, quorum distantia a puncto axis in superficie, glacialis 
est equalis, continent cum axe apud centrum g1acialis angulos equales. 
Et quia centrum glacialis non est centrum vitree, ut patet per II huius, 
distinguent iste linee ex superficie vitree arcus inequales. Cum .enim linea 
EC, ut predictum est, sit maior quam linea CF [Fig. 3A], sit linea CH 
100 equalis linee CE, et protrahatur linea GH, super quam descripta portio 
circuli EGF, que sit GH. erit equalis portioni EG, per 23 8m III', ideo 
quia corda EG est equalis corde GH, per 4 8m Ii. Producta ergo per- 
pendiculari GL, erit, ut prius, corda GH maior quam corda GF. Ergo 
arcus GH erit maior arcu GF. per 23 8m III'. Ergo et linea recta, que 
10.5 est EG, equalis linee GH. erit maior quam linea GF recta. Arcus ergo 
EG est inequalis arcui GF, per 28 8m lIIi. Nulle ergo linee continentes 
cum axe angulos rectos et exeuntes cum linea AC in eadem superficie 
distingunt ex superficie vitree duo s arcus equales, nisi due tantum linee 
que sunt in superficie secante orthogonaliter superficiem erectam super 
110 superficiem vitree. . 
Cum ergo axis fuerit declinans super superficiem vitree, forme pervenientes 
ad superficiem vitree erunt diverse ordinationis, sive sit superficies vitree 
piana sive sperica. Cum vero axis fuerit perpendicularis super superficiem 
vitrei, erit perpendicularis super omnes differentias quarumcunque super- 
83 axe angulos tr. y. I post angulos ser. et dei. P ani [1] 84 inequales: equales E . 84-85 
differentie om. y. 85 que om. E 86 ista: illa y. 87 a puncto mg. inser. v" 89 glaciali 
l I distat O I eadem: aliquid [1] v" 90 suam: sui y. I glaciali lY.v" 93 per inser. 
v" 94 tamen: cum Iv" non O 95 glaciali l I quorum: quoque V b I glacialis 2 : glaciali 
l 96 continent: concurrent y. 98 distingent y. 99 CF: EF lCPY.VbEO I sit om. 
y. I post CH ser. et dei. v" equalis linee CH 100 supra E 101 circuli: oculi l I 23: 24 
R I ante IIII hab. R p 102 chorda R I chordae R I post 4 am add. R p 103 GL: Gl 
R Ichorda 1.2 R 104 post erit add. O ergo I ante per add. v" et I 23: 28 R 27 E I post 
23 am add. R p et V b et I ergo et tr. E 105 post est add. V b nulle [1] superficie 106 28: 27 
lCPY.v"EO I post 27 am add. R p 107 exeuntes: existentes R I post linea ser. et dei. v" 
axe 108 distinguunt RlC 109 ortogonaliter Cy' 111 Cum ergo tr. v" 112 post superfies 
add. E spere 113 pIana sive sperica: sperica sive piana E I vero: ergo E 114 perpen- 
diculares V b I omnesque Y. 


t.....
>>>
ficierum planarum ductarum per lineam AC et superficiei ipsius vitree. 
Et erunt quelibet due linee exeuntes a centro glacialis, quod est unus 
punctus axis, continentes cum axe angulos equales, et distinguentes ex 
differentia communi, que .est in superficie vitree, duas partes equales, sive 
sit superficies illa pIana, sive sperica. Et comprehenduntur forme a sensu 
secundum suam ordinationem in superficie glacialis et in superficie rei vise. 
Et quia talis est comprehensio formarum, ut patet ex suppositione, palam, 
quia semper axis pyramidis visualis est perpendicularis super superficiem 
humoris vitrei anteriorem et posteriorem, quoniam eadem est causa et eodem 
modo demonstrandum. Omnes vero alie linee erunt dec1inantes super has 
superficies, quoniam procedunt, ac si secare possint axem super centrum 
glacialis, et nulla ipsarum transit per centrum vitree, si fuerit sperica, 
nisi axis tantum, per 72 primi huius, quoniam sola illa est perpendicularis 
super ipsam. Patet ergo propositum. 


[Propositio]25. Motu oculi secundum se totum existente possibili; non 
est possibile situm suarum partium mutari. 
Ostensum est in 4 huius foramen esse in concavo ossis per quod 
transit nervus opticus. Sed inter hoc foramen ossis et inter circumferentiam 
glacialis coniunctam cum uvea est spatium aliquantulum, et nervus opticus 
extenditur in isto spatio ex fine foraminis usque ad circumferentiam glacialis 
secundum pyramidalitatem, et amplificatur quousque perveniat ad circumferen- 
tiam spere glacialis cum qua consolidatur. Cum ergo iste nervus dec1inatur, 
erit eius dec1inatio apud foramen concavitatis ipsius ossis. Et quoniam 
concavitas ossis continet totum oculum, dec1inato sic nervo, et oculus 
movebitur secundum totum in ista concavitate; consolidativa en im, que 
consolidatur cum eo quod est in anteriori oculi ex nervo et ex tunicis 
residuis, semper est custodiens situm eius. Dec1inatio ergo nervi apud motum 
oculi non est nisi a posteriore totius oculi. Non est ergo possibile situm 
partium oculi mutari, quoniam, ut per 7 huius patuit, centrum superficierum 
tunicarum visus oppositarum foramini uvee ut cornee est idem cum centro 
oculi. Sicut ergo cum movebitur oculus, non mutabitur centrum oculi, 


115 ipsius vitree co. est ex vitree ipsius in 
 116 centro rep. C I glaciali l 117 post axe 
scr. E equa I angulos equales tr. PEO I distingentes v" 118 communi co. est ex communis 
in 
 119 superficies iIIa tr. v" I senssu v" 120 ordinationem co. est ex ordinem in 
p I glaciali l 121 patet om. E I post ex add. R 5 123 quoniam om. v" I caussa 
R 125 post possint add. C secant 127 post 72 add. R t I solus iIIe R 
l Propositio [25] C 2 post mutari add. RAIhazen 5. 13 n l. 3 foramen: formam 
v" I quod mg. inser. 
 4 obticus v" I hec v,,
 I circumferentia 
 E 5 glacialis: 
licet [1] 
 I unea l I nervio v" I obticus v,,0 6 iIIo Rl I glacialis om. v" 7 post 
ad add. v" speram ad 8 post nervus ser. et dei. O consolidatur 9 eius om. E I ossis: 
communis v" 10 et: etiam R 11 secundum rep. C I post secundum add. R se 13 
residuus v" 14 posteriori E 15 ut rep. E I post ut ser. et dei 
 patet 16 vissus 
O I uneae l I ut: et R 17 sicud v" sic 
 


325 


115 


120 


125 


5 


10 


15
>>>
326 


quoniam, spera aliqua aliqualiter mota, non propter hoc mutatur situs 
centri. Sic nec centrum superficierum tunicarum oppositarum foramini uvee 
20 mutatur. Ergo neque situs tunicarum oculi mutatur. 
Quia enim linea transiens per centra omnium tunicarum et humorum 
oculi transit per medium concavitatis nervi orthogonaliter erecta super basem 
pyramidis nervi, ut patet per 9 huius, et linea que transit orthogonaliter 
per centrum circuli basis alicuius pyramidis necessario attingit verticem 
25 pyramidis, per 89 primi huius, in pyramide vero concava nervi optici 
vertex pyramidis, moto oculo, non mutatur, necesse est moto ocuło secundum 
se totum, partes eius nullo modo mutari, quoniam linea que transit centra 
illarum partium transit per medium concavitatis nervi optici, per 9 huius. 
Ex quo patet quod partes oculi nullo modo mutant ur. Dec1inatio enim 
30 partes pyramidalis nervi super superficiem circułi consołidationis est semper 
declinatio consimilis. Partes ergo oculi secundum suum situm non mutantur. 
Et hoc est propositum. 
Et quoniam oculi ambo sunt consimilis dispositionis in suis tunicis et 
partibus et in figuris suarum tunicarum, et in situ cuiusłibet tunicarum 
35 respectu totius ocułi, patet quod non est diversitas inter illos, quo ad 
hoc quod proponitur de suarum partium situs mutatione, ipsis ocułis motis. 
Situs enim linearum ambarum transeuntium per centra tunicarum visus 
in utroque oculorum est semper situs consimilis in omnibus dispositionibus 
ocułorum. Patet itaque illud quod proponebatur. 


(propositio]26. Uno ocuło moto, necesse est ałium eidem conformiter 
moven. 
Quoniam enim situs partium ocułi non mutatur in utroque oculorum 
et motus unius oculi fit per motu m nervi optici in centro foraminis ossis, 
5 motum vero nervi partialis procedit a puncto nervi cornmunis, quoniam 
semper illud quod movetur in partibus aliarum movetur circa aliquod fixum, 
motus itaque nervi partialis incipit in puncto nervi communis ambobus 
nervis opticis amborum ocułorum, in quo est virtus anime sentientis et 
moventis. Et quoniam illa virtus est indivisibilis et uniformis et principium, 
10 quo primo movet, est corpus naturale secundum sui formam naturalem 
indivisibiłem, pałam quod movendo unum oculum movet et alterum. Nec 


18-20 situs centri. Sic nec.. .oppositorum foramini uvee mutatur mg. inser. alm. in P 20 
necque 
 nec E 21 et: ex 
 22 per: er l I basim R 23 patet per 9 huius: per 9 huius 
patet E I ortogonaliter Cv. 24 centra V v I post circuli scr. et dei. V.luminis 2S post 89 
add. R t I obtici PO 27 post se scr. et dei. quoniam 
 I post transit add. RlE per 28 
illarum co. est ex illorum in 
 I obtici PV.O 29 Declinatio: Declaratio v.
 31 similis 
E" 34 ante partibus ser. et dei. V. dispositionibus 36 de: ex E I mutatio E 37 situs: 
sicud V. I transiuntium E 39 ante oculorum add. C o 
1 Propositio [26] C I 26 mg. hab. E 4 fit: sit 
 I obtici PV.O S motum: motus 
RlC I vcro: enim E 6 post aliquod ser. et dei. 
 formarum 8 obticis O 9 moventis 
mg. co. est ex m8nentis in 
 I uniforformis E 11 indivisibile lCP
>>>
327 


enim est maior ratio qua unum oculum moveat quam qua alterum; uno 
itaque oculo moto, ambo oculi moventur, et unus conformiter alteri movetur, 
ut sicut ab eodem principio motus amborum incipit, sic ad eundem tenninum 
terminentur ambo motu s, et sicut ab uno indivisibili incipiunt, sic ad unum 15 
divisibile tenninentur. Palam est ergo iIIud quod proponebatur. 


[Propositio]27. Duobus visibus uno visibili directe oppositis, necesse est 
duas figurari pyramides quarum communis basis est superfieies rei vise 
et axis cuiuslibet transit per centrum foraminis uvee et per centrum sui 
VISUS. 
Quoniam enim, ut patet per 17 huius, situs partium superficiei rei vise 5 
pervenit ad superficiem utriusque visus, et in iIIa figuratur secundum lineas 
perpendiculares ab omnibus punctis superficiei rei vise ad oculi illius super- 
fieiem productas, quarum omnium concursus secundum puncta suarum inei- 
dentiarum respicit centrum oculi, cuius superficiei incidit, et demum post 
refractionem quelibet ilIarum figurarum pervenit ad medium punctuin nervi 10 
communis, ambarum itaque iIIarum formarum concursus fit in puncto medio 
nervi communis cui ineidunt. Quia itaque centra duorum visuum sunt duo, 
palam quia in visione eiusdem rei a duobus oculis due pyramides visuales 
modo proposito figurant ur. Superficies enim rei vise semper erit basis 
utriusque pyramidis ab utroque oculorum prodeuntis, propter muItiplicationem 15 
forme cuiuslibet puncti superfieiei rei vise equaliter ad visum, et axis 
cuiuslibet earum transit per centrum foraminis uvee ad centrum sui visus. 
Sicut enim visibile directe opponitur uni visui, sic directe opponitur 
et alteri, ex ypothesi. Et quoniam ambo visus equaliter moventur ad aIiquid 
videndum, per premissam, patet quod semper in visione unius rei, medium 20 
punctum superficiei visus oculi opponitur medio puncto superfieiei rei vise, 
vel propinquo illi; medium autem punctum superfieiei visus vel oculi est 
centrum foraminis uvee, per 4 huius. Forma ergo illius puncti medij super- 
ficiei rei vise vel puncti propinqui i11i per centrum foraminiS" uvee per- 
venit ad centrum sui visus. Et hoc est propositum. 25 


12 enim: eius v.. I ante qua ser. et dei. v.. a I unum oculum tr. v.. I quam qua mg. co. est ex 
quamquam [1] in v.. 13 itaque: ita PC I itaque oculo tr. V b 14 sicud v.. I principio: 
puncto l I eandem v.. I!'! sicud v.. I uno om. E sic: se v.. 16 divisibile: divisibilem 
l indivisibile E I terminentur co. est ex termientur in v.. I Palam: patet E I est rep. V M om. 
E I ergo om. V. I iIIud quod om. E I proponebatur: responsio [1] E 
1 Propositio [27] C I 27 mg. hab. EO 3 foramis V. 7 iIIius: ipsius V. 8 productas 
om. V. I omnia [1] sive obiectiva [1] V. 8-9 incidentiarum: incidentium l' 9 cuius: eius 
V. II ambarum co. est ex ambabus in v.. lilIarum: iIIa v.. I puncto om. v.. I!'! 
utriusque mg. co. est ex cuius li bet in Y b I oculo prodeuntis E 17 centra Rl I uneae 
l 18 sic directe opponitur rep. E 19 alteri co. est ex astri in v.. I ypothesio v.. 20 quod 
mg. inser. v.. 23 uneae l I Formarum PyMv..EO 24 post puncti ser. et dei. v.. pingui 
I propinquo V. I uneae J
>>>
328 


5 


[Propositio]28. Duobus existentibus oculis unius rei unam tantum formam 
accidit videri. 
Quoniam enim, ut prius pluries dictum est, forma recepta in superficie 
g1acialis pertransit corpus glacialis, deinde extenditur per corpus subtile 
quod est in nervo optico, et venit ad anterius cerebri in quo est sentiens 
ultimum, quod est virtus sensitiva, comprehendens sensibilia, cuius virtutis 
oculus est instrumentum, recipiens formas rerum et reddens eas ultimo 
sentienti, sic quod apud nervum communem ambobus 
culis, cuius nervi 
situs a duo bu s oculis est situs consimilis, demum completur vi sio, licet 
ergo due forme perveniant in duobus oculis ab una re visa. IIIe tamen 
forme ambe, quando perveniunt ad nervum communem, concurrunt et 
fiunt una forma, et per unionem harum formarum comprehendit ultimum 
sentiens formam rei vise, et sic unius rei tantum unam formam accidit 
videri; nisi forte, per aliquam occasionem intervenientem, accidat formas 
duo bu s oculis acceptas non uniri, eo quod non concurrunt in unionem 
amborum nervorum opticorum. Tunc en im duas formas accidit videri; ut 
cum aspiciens mutaverit situm unius oculi ad anterius, et aJius oculus 
fuerit immotus. 
Quando vero situ s duorum oculorum fuerit naturalis, tunc, qui a situs 
ipsorum ab una re visa est situs consimilis, pervenit forma ab una re visa 
in duo loca consimilis situ s ; et cum situs unius oculorum fuerit declinans, 
tunc diversatur situs oculorum ab iIIa re visa, et sic perveniunt due forme 
iIIius vise rei diversi situs; sed hoc non inest visui naturaliter, sed solum 
per violentiam, quam facit voluntas, vel naturalis debilitas consuetudini 
nature. Quando itaque situs oculorum fuerit naturalis, tunc semper ambobus 
visibus unius rei unam formam accidit videri, quod est propositum. Duo ergo 
forme visi puncti infiguntur in doubus medijs duarum superficierum amborum 
visu um, et quilibet punctus alius forme vise infigetur in duo bu s locis consi- 
milis positionis in duobus visibus. Deinde, due forme vise perveniunt ad 
concavitatem communis nervi, et perveniunt due forme que sunt in puncto 
quod est in duo bu s axibus iIIarum duarum pyramidum radialium, secundum 
quas fit visio, ad punctum quod est in communi axe, et efficiuntur una 
forma. Et quelibet due forme que sunt in duobus punctis consimilis po- 
sitionis a duobus visibus pervenient ad idem punctum punctorum circum- 
stantium, punctum qui est in axe communi. Sic ergo due forme totius 
rei vise superponuntur sibi et efficiuntur una forma, et sic visum comprehen- 
ditur unum. 


10 


15 


20 


25 


30 


35 


I Propositio [28] C I 28 mg. hab. O 2 post videri add. RAIhazen 27 n l. Item 
9 n 3. 3 Quia E 4 glacialis l: glaciali O 5 obtico O 6 sensibilia: visibilia v" 7 post 
instrumentum ser. et dei. v" respiciens 9 post situs add. V. est 12 fiunt: sunt 
V. I conprehendit V. 14 accidit l 15 unione P 16 obticorum PO 18 in motus 
V. 19 post vero add. l nullus 20 est situs consimilis... una re visa om. V. I pervenerit 
l 21 consimilis: nsimilis C I et cum situs mg. inser. V b 23 vise rei tr. RIE I visui: visu 
v" 24 vel mg. inser. V b 25 Quando: cuivardo f?] O 27 infinguntur V. 29 due om. 
R I vise: viso V. 36 visus V.
>>>
329 


[Propositio]29. Omnem punctum forme incidentem superficiebus visuum 
per axes radiales ad centrum foraminis gyrationis nervi concavi pertingere 
est necesse. 
Quoniam enim quilibet axium transit per centrum foraminis uvee ad 
centrum visus, ut patet per 27 huius, ergo et pertransit centrum ipsius s 
spere uvee, per 8 huius. Omnis vero linea recta producta inter centrum 
oculi et uvee, centrum circuli sectionis uvee et medium punctum concavitatis 
nervi necessario penetrabit, per 9 huius. Palam ergo, cum perpendicularis 
semper maneat inconfracta per 47 secundi huius, quod omnem punctum 
forme incidentem superficiebus visuum per axes radiales ad centrum gyrationis 10 
nervi communis pertingere est necesse. Ab hoc autem puncto diffunditur 
forma ad medium punctum nervi communis, et quoniam medius punctus 
nervi communis est tantum unus, palam quod axes amborum visuum in 
uno puncto nervi communis semper concurrunt. Patet ergo propositum. 


[propositio]30. Si a terminis linee inter duo centra foraminum gyrationis 
nervorum concavorum producte due recte linee ad medium communis nervi 
producantur, necesse est in constituto triangulo angulos ad basem equales esse; 
ex quo patet quod linee iIle producte sunt equales. 
Sint duo centra foraminum gyrationis nervorum concavorum R et T s 
[Fig. 4], inter que producatur linea RT, sitque medius punctus nervi communis 
A, et constituatur triangulus RAT. Dico quod angulus ART est equalis 
angulo A TR. Cum enim positio duorum nervorum in respectu concavitatis 
nervi communis sit positio consimilis, quia concavitas nervi unius est omnino 
similis concavitati alterius per 4 huius, ergo et medium concavitatis unius 10 
est simile medio concavitatis alterius, unde axis nervi unius equalis est 
axi nervi alterius. Sed, per eandem 4 huius, positio duorum nervorum 
in respectu duorum foraminum est positio consimilis, in quorum nervorum 
medio feruntur linee RA et T A, ut axes. 
Palam ergo quoniam positio duarum linearum RA et T A apud lineam 1 s 
RT est positio consimilis. Hoc autem est impossibile, nisi anguli ART 
et A TR sint equales, quoniam ad inequalitatem istorum angulorum sequitur 
inequalitas positionis medij axis ipsorum nervorum concavorum et, ex conse- 


I Propositio [29] C I in .cidentem v.. 2 gyrationis inser. I) I pertingere: contingere 
l 4 quaelibet l I uneae l !Ii ipsius mg. inser. I) 6 uneae l 7 uneae 1.2 l 9 
inconfracta: irrefracta R in confracta PVMI)O I post 47 add. R t II Ab: ob l 12 medium 
v.. 13 quod: quia Rlv.. 
l Propositio [30] C I 30 mg. hab. E 2 recte linee tr. l 3 basim R I equales esse tr. 
E 4 post equales add. RAIhazen 6 n 3. !Ii post duo ser. et dei. V b puncta I R co. est ex T in 
O 6 inter que: uterque v.. I linea co. est ex linee in I) I nervi co. est ex nervis in I) 7 
cunstituatur I) 8 ATR: RTA E I positio co. est ex puncto in O I duorum nervorum tr. 
I) 9 concavitas: concavitatis l 11 post medio scr. et dei. I) a I nervi unius equalis est: 
unius nervi est equalis E 12 nervi alterius tr. E 13 in 1 0m . v.. 14 feruntur: fuerint lCv.. 
ferentur P IRA: RQ l 16 positio consimilis tr. E I inpossibile v.. I angulus E 17 
istorum: horum E 



-
>>>
330 


20 


quenti, ipsorum nervorum. Su nt ergo illi anguli ad basem equales. Ergo, 
per 6 am Ii, linee ille producte sunt equales, scilicet linea AR linee AT. 
Patet ergo propositum. 


5 


[propositio]31. Uno puncto rei vise superficiebus amborum visuum per- 
pendiculariter incidente, necesse est axes radiales in centris foraminum gy- 
rationis nervorum concavorum angulariter ref ringi. 
Quoniam enim. ut patet per 27 huius, quilibet illorum axium pertransit 
centrum foraminis uvee et centrum oculi, motu s autem cuiuslibet oculorum 
fit in centro foraminis gyrationis nervi optici, patet quoniam secundum 
motum oculorum variantur axes illi radiales in quibus sunt semper idem 
semidyametri oculorum, que scilicet ab ipsorum centris ad centra foraminum 
uvee protenduntur. Partes autem superiores illorum axium, quibus a centris 
foraminum gyrationis nervorum concavorum forme perveniunt ad punctum 
medium nervi communi, semper manent secundum modum unum. 
Cum itaque alie partes illorum axium semper sint immobiles et alie 
semper mobiles, cum per ipsas unus punctus videtur, patet per I lUD XI', 
quoniam ille linee non sunt linea una. Utpote, si forma puncti B [Fig. S) 
videatur secundum ambos axes BR et BT et, sicut factum est in premissa, 
ducantur linee RA et T A ad medium punctum nervi communis qui sit Ą 
patet, per lam XI:, quoniam linee BR et RA non sunt linea una: eius 
enim partem in sublimi partem in plano accideret esse, quod est impossibile. 
Patet ergo quoniam angulariter coniunguntur, quod est propositum. Et 
licet axes premisso modo refringantur, formatio tamen pyramidum visualium 
fit, ac si axes integri ad verticem pervenirent, neque accidit visui aliqua 
diversitas ex illo. 


10 


1S 


20 


5 


lPropositio]32. Necesse est axes pyramidum visualium amborum vlsuum 
transeuntes per centra foraminum uvee semper coniungi in uno puncto 
superficiei rei vise, etiam motis visibus per superficiem rei vise. 
Cum enim videns intuebitur aliquam rem visam, tunc uterque visus erit 
in oppositione illius rei vise, per 2 huius, et utraque pupillarum dirigetur 
ad iIlum visum directione equali, propter visuum equalitAtem, per 4 huius. 


19 rep. et dei. anguli C I basim RE 19-20 Ergo per 6 am : per 6 am ergo E 20 post 6 am add. 
R p linea: linia l 21 Patet ergo: quod est E 
l Propositio [31] C I 31 mg. hab. O 3 refrangi lCv" 4 enim om. E I quodłibet 
l I post il10rum ser. et dei. O per 5 uneae l I aut C 6 foraminis gyrationis tr. V b 8 
semidiameter E 9 uneae l I protendenduntur E I Partes autem tr. v" 12 alio partes tr. 
V. I in mobiles V. lalie l : alij l 13 post semper ser. et dei. v" sint I per inser. P I post 
primam add. R p 15 BR: KR v" I BT: TR lPV.v"O, TB co. est ex TR in C I sicud 
V. 16 RA [1] O 17 post lam add. R p 18 sublimi partem in plano: plano partem in 
sublimi E 19 Patet om. V. 20 refrangantur Cv" I pyramidis V. 22 ex rep. E I post 
il10 add. V. patet ergo propositum 
l Propositio [32] C I 32 mg. hab. O I pyramidum om. V. I visualium amborum tr. 
V. 2 transeuntes co. est ex transteunstes in v" I uneae l 3 per: super E I post vise add. 
RAIhazen 2 n 3. 6 inequalitatem V b 


..
>>>
Sint ergo duo centra duorum visuum E et G [Fig. 6]. et sit medius 
punctus nervi communis punctus A; et sit superticies rei vise BCDF que 
sit. exempli causa, equedistans linee centra visuum connectenti. que sit 
EG. Palam ergo quoniam a centris visuum perpendiculares super ipsam 
superficiem BCDF producte sunt equedistantes, per 6 am Xli. que sint EQ 
et Gx. In hac itaque superficie BCDF signentur punctus qui sit V. Dico 
quod propter equalitatem amborum oculorum in omnibus suis dispositio- 
nibus, si aIter visus tilerit motus ad videndum punctum V. statim etiam 
reliquus movebitur ad videndum idem punctum V, ita quod axes amborum 
pyramidum visualium transeuntes per centra foraminum uvee coniungentur 
in puncto V, una ipsarum ibi pertingente. 
Si enim una illarum axium incidente in punctum V, alia incidat in 
alio puncto, sit illud punctum Z, eruntque duo axes EV et GZ, inter 
quorum terminos linea ZV producatur. Et quoniam axes sic protensi a duobus 
visibus non concurrunt in aliquo punctorum linee ZV, sicut neque coc.:urrunt 
si solum secundum perpendiculares lineas, que sunt EQ et GX fiat visio, 
palam quod nuIlum punctorum linee ZV videbitur ambobus visibus sed 
tantum uno. Alter ergo oculorum videtur superfluere, cum unus oculorum 
secundum sui axem omnia puncta linee ZV possit inperceptibiliter transcur- 
rere. Constituit autem natura duos oculos propter perfectionem bonitatis 
visionis et complementum eius. ut ipsorum virtus unita sit fortior, ut patet 
per 4 huius. 
Si ergo axes visuales non concurrant in aliquod punctum unum linee 
ZV, sequitur vel naturam superfluere, vel ipsam modo debiliori quo potest 
operari, quorum utrumque est impossibile. Natura enim nichil agit frustra, 
nec deficit in necessarijs. ut patet per suppositionem. Accidit autem hoc 
impossibile si axes solum incidant diversis punctis superficiei visibilis, im- 
possibile autem nunquam accidere si incidańt in idem punctum. Palam 
itaque quoniam in idem punctum incidere axes pyramidum amborum visuum 
semper est necesse. quoniam operatio amborum visuum est uniformis. Cum 


8 sit om. l I BCDF: BCDE E 9 caussa R I aequidistans R I connectenti: convertenti 
l conectenti V.V& 10 ipsam om. E II aequidistantes R I post 6 0m add. R P I sunt 
E 12 GX: GQ V. 14 ad: propter E lvidendum punctum V: punctum V videndum 
E 15 reliquus: reliquuus V. lita quod: itaque l 16 uneae l I coniungentur: coniun- 
guntur l coniungantur Y" 17 una ipsarum: uno ipsorum Runa ipsorum Y" 18 una iłłarum: 
uno iłłorum RE I incidet l I puncto Rl I alius R I incidat: incidit Rl 19 GZ co. est 
ex GT in Y" 20 ZV co. est ex TV in Y" 21 ZV co. est ex TV in Y" I sicud 
V. I concurrant V. 22 solum: super l I secundum om. l I EQ: GQ co. est ex EQ in 
V. I GX: EX co. est ex GX in V. 23 ZV co. est ex XV in Y" 24 videtur: movetur 
l I superllue IV. 25 ZV co. est ex TV in Y" I imperceptibiliter R interceptiliter l in 
perccptibilis O 26 Constituit mg. co. est ex consistant Y" 27 post virtus add. E u I unica 
Iy"EO 29 aliquod: aliud l 30 modo om. E 31 utrumque: utcrque lV& I nihil Rl nichil mg. 
co. est ex naturalis in Y" 32 necessaris Y" I post per add. R 8 I post suppositionem add. 
R 2 huius 34 nunquam: umquam PV. unquam EO rep. et dei. nunquam Y" nunquam co. est ex 
umquam in C I accideret RlC (co. est ex accidere in C) 34-35 idem J ,2: iłłud l 35 
quoniam in idem: in idem quoniam V. I axcs: axis l axas f?] V. I pyramidis y"EO 


".-.. 


331 


10 


15 


20 


25 


30 


35
>>>
332 


40 


etiam VISUS fuerit motus super rem visam, tunc uterque visus movebitur 
super illud, et axes congregate in uno puncto superficiei rei vise, moto 
visu ambo movebuntur simul ad aliud unum punctum super superficiem 
iIIius rei vise. Ambo enim oculi sunt equales in omnibus suis dispositionibus 
et est ambobus oculis unus nervus communis. 
Et quoniam motus oculorum procedit ab una virtute, necesse est virtutem 
motam per unitatem nervi procedere. Hec ergo mot o uno oculo, ambos 
oculos movebit. ut patet per 26 huius. Actio itaque et passio oculorum 
semper est equalis et consimilis; et si alter visu um motus fuerit ad aliquid 
videndum, statim alter movebitur ad hoc idem videndum iIIo eodem motu; 
et si alter visuum quiescat, reliquus quiescet. Impossibile est enim aIterum 
visuum moveri et alterum quiescere, nisi alter fuerit impeditus, ut patet 
per 26 huius. Et sicut etiam dec1aratum est per 18 huius, superficies 
rei vise semper erit basis utriusque pyramidis ab utroque oculorum pro d- 
euntis, quoniam tunc positio puncti in quo ambo axes su nt coniuncti 
est positio consimilis, quia est oppositus duobus medijs amborum visuum. 
Pala m ergo propositum, dicemus que punctum concursus amborum axium 
in superficie rei vise punctum coniunctionis. 


45 


50 


5 


[propositio]33. Si a puncto medio nervi communis ad medium linee 
connectentis centra foraminum gyrationis nervorum concavorum linea recta 
producatur, necesse est productam super divisam perpendicularem esse, et 
eam puncto viso cum axibus incidente trigonum ab wdbus et divisa linea 
contentum per equalia dividere. 
Quod hic proponitur, patet per premissam et per 31 primi huius. Ut 
autem particularius demonstretur, sint omnia disposita ut in 30 huius, et 
sit linea RT [Fig. 7] divisa per equalia in puncto S. Sitque visibile aliquod 
oppositum ambobus visibus, quo d sit BC, in cuius puncto medio. quod 
sit B. concurrant per precedentem ipsi axes radiales, qui sint RB et TB; 
et producatur a puncto A, quod est medius punctus concavitatis nervi, 
ad punctum S linea AS. Dico quod linea AS est perpendicularis super 
lineam RT. Quoniam enim angulus ART est equalis angulo ATR, per 


10 


37 etiam: igitur Rl 38 illud: iIIam RP 39 visu: uno l I ad inser. v" laliud: aliquod R, 
om. V. I super mg. inser. V b 42 quoniam: quia E 43 motam: motivam V. motivam [1] 
C I unitatam v" I Hec: hoc l, om. E 44 Actio co. est ex ncti in v" 45 visuum: visus 
E 46 hoc: iIIud E I videndum 2 : modum v" 47 quiescat mg. inser. v" I reliqus 
v.v" I reliqus quiescet. Impossibile est enim alterum visuum mg. inser. v"v. 49 superficies: 
superficiem l superficiei v" 51 post quo add. E scilicet 52 ante oppositus add. E punc- 
tum I oppositus: oppositio R, co. est ex positus in v" 54 post coniunctionis add. V. utrum [1] 
sive ut videtur [1] 
l Propositio [33] C 2 conectentis v" S post dividere add. RAIhazen 7 n 3. 7 
particularius: perpendicularius V. I omnia: unam V. 8 RT: SR V., co. est ex RS in v" 9 
ambobus visibus tr. E I BC: OC R BT l BE E I puncto medio tr. E 10 quae l I post et 
ser. et dei. O b I TB co. est ex IB [1] in V b 11 et om. V. 12 post punctum add. l scilicet et 
om. S 


'.-.
>>>
I - 


333 


30 huius, et linea AR est equalis Iinee AT; sed linea AS est equalis 
sibi ipsi, ergo, per gam 1', trigona ARS et ATS sunt equiangula. Angulus u 
ergo AST est equalis angulo ASR. Ergo, per diffinitionem perpendicularis, 
linea AS est perpendicularis super lineam RT.