Optics and the Rise of Perspective. A Study in Network Knowledge Diffusion 9781905622313

156 12 11MB

English Pages [262] Year 2014

Report DMCA / Copyright


Polecaj historie

Optics and the Rise of Perspective. A Study in Network Knowledge Diffusion

Citation preview

Optics and the Rise of Perspective

GEMAS Studies in Social Analysis

Series Editors: Mohamed Cherkaoui, Peter Hamilton, Bryan S. Turner This series makes available in English the best work of a leading French research group, GEMASS (Groupe d’Etude des Méthodes de l’Analyse Sociologique de la Sorbonne). The group’s aim is to contribute to the production of empirical sociological knowledge and to the renewal of sociological theory. Their work is as much concerned with the relevance of classical sociology as the contemporary issues on which it can be focused, and has three interconnected elements: general sociological theory; epistemology and methodology of the social sciences; and the application of theories and methods to areas such as collective action, norms and values, culture and cultural practices, morality, debate, knowledge, and education and social stratification.

Series titles include: AGENT-BASED MODELLING AND SIMULATION IN THE HUMAN AND SOCIAL SCIENCES Edited by Denis Phan and Frédéric Amblard BABY BOOMERS Catherine Bonvalet and Jim Ogg BOUNDLESS YOUTH Olivier Galland DARWIN IN FRANCE Dominique Guillo DURKHEIM AND THE PUZZLE OF SOCIAL COMPLEXITY Mohamed Cherkaoui FAMILY AND HOUSING Edited by Catherine Bonvalet, Valérie Laflamme, Denise Arbonville THE FUTURE OF COLLECTIVE BELIEFS Gérald Bronner GOOD INTENTIONS Mohamed Cherkaoui


INVISIBLE CODES Mohamed Cherkaoui




MODELLING EDUCATIONAL CHOICE Nathalie Bulle NEW EUROPE Edited by Sven Eliaeson and Nadezhda Georgieva


Optics and the Rise of Perspective A Study in Network Knowledge Diffusion

Dominique Raynaud

The Bardwell Press, Oxford Groupe d’Etude des Méthodes de l’Analyse Sociologique de la Sorbonne Centre National de la Recherche Scientifique Université de Paris-Sorbonne (Paris IV)

First published 2014 by The Bardwell Press © 2014 The Bardwell Press

All rights reserved. Apart from any fair dealing for the purposes of research or private study, criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, no part of this publication may be reproduced, stored or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission in writing of the publishers. Published by: The Bardwell Press Tithe Barn House, 11 High Street Cumnor, Oxford, OX2 9PE, UK www.bardwell-press.co.uk Cover image: Portrait of Luca Pacioli, attributed to Jacopo de’ Barbari, c. 1495–1500

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

ISBN: 978-1-905622-31-3

Typeset by The Bardwell Press, Oxford, UK Printed in Great Britain


List of Tables and Illustrations Notations Acknowledgements Introduction CHAPTER 1 Perspective and its Optical Backing

vii xi xv 1 11





CHAPTER 4 The Studia Generalia Network


CHAPTER 5 Knowledge Diffusion Simulation



APPENDIX 1 List of OFM University Lectors


APPENDIX 2 List of OFM Universities


APPENDIX 3 List of OFM Provinces


Bibliography Index of Names Subject Index

209 229 237



TABLES Table 1.1: Ghiberti’s references to classical authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 Table 1.2: Ghiberti’s references to Arabic authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 Table 3.1: Distribution of optical manuscripts, by treatise and author. . . . . . . . . . . . . .64 Table 3.2: Distribution of optical manuscripts, by author and obedience . . . . . . . . . . .65 Table 3.3: Docking’s borrowings from Grosseteste’s De iride . . . . . . . . . . . . . . . . . . . . . . . .73 Table 3.4: University cosmopolitanism index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79 Table 3.5: University density by province . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83 Table 3.6: University recruitment pools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85 Table 4.1: University network distance matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98 Table 4.2: University network adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 Table 4.3: University centrality indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 Table 4.4: University network equivalence partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 Table 4.5: University network dynamic partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115

FIGURES Figure 2.1: Figure 2.2: Figure 2.3: Figure 2.4:

The perspective of the circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 Intersecting the visual pyramid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 The route to vanishing point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 The rectilinear propagation of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44


OPTICS AND THE RISE OF PERSPECTIVE Figure 3.1: Manuscript power law fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 Figure 4.1: Distribution and correlation of degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97 Figure 4.2: Studia generalia in-degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 Figure 4.3: Studia generalia betweenness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 Figure 4.4: Studia generalia closeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 Figure 4.5: Studia generalia constraint (A) and transitivity (B) . . . . . . . . . . . . . . . . . .109 Figure 4.6: The law of cosmopolitanism/closeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 Figure 4.7: University network dendrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112 Figure 4.8: University network reduced graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 Figure 4.9: Reticular roles typology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116 Figure 4.10: Directed filters typology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 Figure 5.1: The logistic function and its derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140 Figure 5.2: Diffusion curve irregularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142 Figure 5.3: Curves of diffusion (any receptor) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 Figure 5.4: Curves of diffusion (receptor within subset P 7) . . . . . . . . . . . . . . . . . . . . . . .144

MAPS Map 2.1: Distribution of the madha¯hib of Isla¯m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 Map 3.1: Optical MSS geographical distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 Map 3.2: OFM provinces in the mid-14th century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Map 3.3: OFM studia generalia at the end of the 14th century. . . . . . . . . . . . . . . . . . . . . .82 Map 3.4: Oxford’s studium recruitment pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86 Map 3.5: Paris’s studium recruitment pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 Map 3.6: Bologna’s studium recruitment pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 Map 4.1: University network directed graph at the end of the 14th century. . . . . . . . .101 Map 5.1: Network diffusion from Oxford. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Map 5.2: Network diffusion from Seville . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Map 5.3: Network diffusion from Vienna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130 Map 5.4: Network diffusion from Palermo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 Map 5.5: Network diffusion from Florence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132 Map 5.6: Susceptible adopter pool (Strasbourg). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146 Map 5.7: Susceptible adopter pool (Florence) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147 Map 5.8: Isotropic diffusion in the studia generalia network from Dijon . . . . . . . . . . .148 Map 5.9: Anisotropic diffusion in the studia generalia network from Esztergom . . .149

PLATES Plate 2.1: Uncoordinated frontal projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 “Prince and Attendants”, Dı¯va¯n by Ja¯mı¯ (Safawids) S 1986.49 ˙ Institution, Washington D.C. Courtesy of Freer and Sackler Galleries, © Smithsonian Plate 2.2: Coordinated frontal projection “Bahra¯m Gu ¯ r in the Turquoise-Blue Pavilion”, Khamsa by Niza¯mı¯ (Safawids) F 1908.275a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 ˙ Courtesy of Freer and Sackler Galleries, © Smithsonian Institution, Washington D.C.


L I S T O F TA B L E S A N D I L LU S T R AT I O N S Plate 2.3: Uncoordinated oblique projection “Ulugh Beg with Ladies of his Harem”, in a book from Samarqand (Timurids) F 1946.26  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47 Courtesy of Freer and Sackler Galleries, © Smithsonian Institution, Washington D.C.

Plate 2.4: Coordinated oblique projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47 “The Feast of ‘I¯d” from a Dı¯va¯ n by Ha¯fiz (Uzbeks) F 1932.51 Courtesy of Freer and Sackler Galleries, © Smithsonian Institution, Washington D.C.



a a aij A = (aij) bjk(i) C Ci CO CBi , ClBi ClB CCi , ClCi CC*i , C°Ci CC* CDi , ClDi ClD | drR dij dij D = (dij)

Diffusion curve parameter (early adopters) Cutting threshold of the dendrogram Element of the adjacency matrix A = (aij) Adjacency matrix Probability for i to be on the geodesic between j and k. Equivalence class (cluster analysis) Burt constraint Cosmopolitanism index (Oxford) Betweenness centrality (absolute, normalized) Betweenness centralization Closeness centrality in an undirected graph (absolute, normalized) Lin closeness centrality (in a directed graph) Lin closeness centralization (in a directed graph) Degree centrality (absolute, normalized) Degree centralization Network cohesivity Lector recruitment mean distance Geographic distance between vi and vj. Euclidean distance between i and j Geographical distance matrix xi


D = (dij) d d dL ei ejk {ek} E = (e1, e2, . . . e,) E Eit e = | E{ek} | el zG = max dij z(n) g G = (V, E) c H(x) i, j (i, j) Ji k , li Li m Ml0 Mll1, Mll2 MGt Mit M *it ni ni = [vi, . . . , z] nt N Nl

Euclidean distances matrix University density Network density Lector density Edge in the graph G Probability to be connected for i and k excess degree vertices Vertices emitting a message to k. Edge set Set of vertices {ek} Emitter status of vertex i at time t Emission surface of vertex k (absolute) Emission surface of vertex k (normalized) Diameter of the graph G Delaying function (logistic) Number of vertices in the graph G Graph of vertices V and edges E Diffusion curve parameter (slope) Heaviside function of x Graph vertices, licence for vi, vj Edge between vi and vj, licence for (vi, vj) Number of vertices reachable by i Vertex degree (random graph) Number of edges in the graph G Number of foreign lectors at studium i Number of lectors at studium i Network parameter in the law cosmopolitanismcentrality Maximum of the diffusion curve first derivative Extrema of the diffusion curve second derivative Number of messages in graph G at time t Cumulative number of messages at vertex i at time t Proportion of messages owned by vertex i at time t Limit of the series Mit Path from vertices vi and z Number of individuals having adopted at time t Population Number of susceptible adopters in N xii


Nit pk P qk r {rk} R t = | R{rk} | tl S SR v vq2 t T vi, vj (vi, vj) V = [v1, v2, . . . vg] xik z |X| k nn . k i n b l k /

Instant number of messages received by i at time t Degree distribution Equivalence class (dynamic) Excess degree distribution Degree correlation index Vertices receiving a message from k Set of vertices {rk} Vertex k reception surface (absolute) Vertex k reception surface (normalized) Province surface Studium recruitment pool surface Studium native recruitment pool surface Variance of degree distribution qk Time, step Transitivity (clustering) Vertices of graph G Edge between vertices vi and vj. Vertices set Value of the edge between i and k Vertex mean degree in a random graph Cardinal number of X Mean degree of the neighbours of vertex i Binomial, combination of n objects taken k to k Structural equivalence



UNIVERSITY CODING 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22


Røskilde Magdeburg Erfurt Cologne Prague Regensburg Vienna Esztergom Dublin Cambridge Oxford Strasbourg Dijon Paris Angers Toulouse Montpellier Avignon Valencia Salamanca Lisbon Seville

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44



Asti Milan Genoa Pisa Bologna Padua Venice Rimini Florence Siena Perugia Assisi Todi Ascoli Chieti Djurazci Cività Barletta Napoli Reggio Palermo Rome


This study, which was originally a part of my habilitation à diriger les recherches, obtained at the University of La Sorbonne, Paris, has benefited from much advice since I first planned to supplement The Oxford Hypothesis. I was given a warm welcome at a seminar of the Centre for the History of Arabic and Medieval Science and Philosophy in Villejuif. I then gave a conference on this topic at the Académie de France in Rome, developed the point at an International Conference on Perspective held in Rome, and afterwards unveiled some undetected consequences on the occasion of a Collège de France International Symposium and the writing of two book chapters. I am grateful to Reda Alhajj, Jean-Michel Berthelot, Alain Berthoz, François Chazel, Marisa Dalai Emiliani, Pascal Dubourg Glatigny, Pascal Engel, Michel Forsé, Jean Gayon, Marianne Le Blanc, Tony Lévy, Nasrullah Memon, Régis Morelon, Roshdi Rashed and Roland Recht for giving me the opportunity to make my ideas clearer. I would like to thank Mohamed Cherkaoui warmly for having accepted, with complete confidence, the manuscript in the GEMAS Studies in Social Analysis series. I am also greatly indebted to Peter Hamilton and Toby Matthews for revising my text. This book would have been of another tonneau without their valuable suggestions. St. Pierre, Winter 2014



According to the standard view, linear perspective is an invention of the Italian architect Filippo Brunelleschi (1377–1446). Shortly before 1413, he performed a public experiment using a device for superimposing two images, one painted and the other reflected, of the Baptistery San Giovanni in Florence. Through this experiment, Brunelleschi demonstrated the exact match between the painted and the reflected image of the Baptistery, and thus, the accuracy of the rules of linear perspective he had just devised. This is, in a nutshell, the thesis that is commonly drawn from reading the Vita di Filippo di Ser Brunellesco written by Antonio di Tuccio Manetti (1970). In a book published in 1998, outlining the results of the experiment I reproduced in Florence in April 1995, I explained in detail why it was still possible to doubt the truthfulness of this story. Almost fifteen years after the publication of The Oxford Hypothesis, this record remains strongly supported by many scholars. We must, therefore, begin by recalling the “compelling evidence” from which this thesis is derived: 1) Domenico da Prato declared Brunelleschi prospettiuo ingegnoso in August 6, 1413, thus Antonio di Tuccio Manetti (1423–1497) was yet to be born when Brunelleschi performed his experiment in Florence. At best, he saw the picture, not the experiment. 1


2) It is accepted that Manetti wrote his Vita di Ser Brunellesco around 1480, that is more than sixty years after Brunelleschi’s alleged experiment. 3) In Manetti’s report there is not a single occurrence of the term experiment. He only tells us that Brunelleschi mostro, fecie una pittura. 4) Manetti sometimes recounts the experiment in a very problematic conditional tense: e pare che sia stato a ritrarlo … se l’auesse ritratto. 5) In Manetti’s report, the experimental conditions (viewpoint, visual field and size of the pinhole) are mutually exclusive, thereby prohibiting any safe reproduction of Brunelleschi’s experiment (Raynaud, 1998a: 132–150). 6) The tavoletta is lost. Nobody can judge its correctness in perspective. 7) There is not a single perspective view signed by Brunelleschi. 8) The only perspective view that has been attributed to him (Longhi, 1940) is the silver plate entitled The Healing of the Possessed.1 Inspection of the geometric layout shows that it does not obey the rules of linear perspective (Raynaud, 1998a: 74). Therefore, the standard opinion is based on very fragile foundations. The literature on the origins of linear perspective only rarely mentions these contradictions and historical gaps. Since 1998, I have not found them mentioned, except in the work of Hersey (2000: 162) and Lubbock (2006: 322), whose cautious attitudes contrast with the considerable number of reports again crediting the myth of the Florentine invention. Recent books and articles have repeated that Filippo Brunelleschi was indeed the “inventor of linear perspective” (Capretti, 2003: 28), that linear perspective was duly “demonstrated in two experiments” (Belting, 2011: 164). According to the historian of art, Brunelleschi’s tavoletta was the experiment “by which the Florentine architect demonstrated the laws of perspective” by means of “mathematical construction” (Wirth 2006: 274). The historian of science agrees: his tavoletta was a “simple and rather straightforward scientific experiment” tout court (Feyerabend, 2001: 95). Elaborating on these fragile foundations, others logically come to the conclusion that Brunelleschi’s act was a “founding experiment” (PérezGómez, 2000: 25) if not “a revolutionary achievement” (Brown, 2013: 55). Where do such important qualities—invention, laws, mathematical demonstration, scientific experiment or revolution—come from, in the absence of any document? 2


Let us be clear. The doubts surrounding Brunelleschi’s contribution do not mean that he played no role in the development of linear perspective. These doubts simply mean that his precise contribution will remain unknown until new evidence is added to the file. In a sense those material gaps are an opportunity for the historian because they help us in changing the scale, departing from narrow matters of date and attribution in quattrocento Florence, to question the sociohistorical conditions that fostered the rise of linear perspective. This is the framework in which attention must be paid to the nexus between optics and perspective. In The Oxford Hypothesis, and some further papers, I have shown that the development of Renaissance linear perspective was coextensive to the diffusion of medieval optics. This is not just the case because all perspective problems can be solved by geometry and because medieval optics provided geometrical devices to painters. The main reason is that optics increased artists’ awareness of some principles of visual perception—within the Euclidean tradition (see Euclid’s Optica, props. 6, 11, and 42) or not (see for instance Ptolemy’s and Alhacen’s contributions to binocular vision and aerial perspective). Another reason is that optics—especially from the Arabic–Latin translations made at the end of the twelfth century—provided a powerful Weltanschauung, drawing attention on a broad range of phenomena (from burning mirrors to visual illusions, from rainbows to ocular anatomy), thus paving the way for a naturalist–realist ethics of experiencing the world. This keen interest in optical topics resulted in various phenomena, which are mainly documented by three series of facts. First, the study of sources establishes the dependence of the perspective treatises written by Alberti, Ghiberti, Piero della Francesca and Leonardo da Vinci on medieval optical literature. Early Renaissance treatises are rich with about forty textual borrowings to Grosseteste, Bacon and Pecham. Those scholars attended, as master or as student, the Franciscan studium of Oxford University. Second, a comparative analysis of Italian libraries’ inventories dating from the very beginning of the quattrocento, shows that Franciscan libraries possessed more optical literature than others. On this basis, we have concluded that a dominant—though not exclusive—flow of diffusion linked Oxford’s Franciscan studium to Rome, Assisi and Florence, where early perspective techniques were awakening. Third, a survey of optical literature written during the thirteenth and fourteenth centuries show that Grey Friars contributed more than any other secular or religious organization to the diffusion 3


of optics, in number of treatises composed and copied (see Raynaud, 1998a, 2001ac). The present book aims to address more specific questions that remained unanswered in The Oxford Hypothesis. (1) Why did perspective not arise in medieval Islam when the major improvement of geometrical optics was by scholars such as al-Kindıˉ, Qustˉa b. Luˉqaˉ, ˙ Ibn ‘I¯saˉ, Ibn Sahl or Ibn al-Haytham? (2) Why did perspective arise in central Italy rather than in any other European region when Franciscan studia and libraries were disseminated throughout Europe? 1. FIRST STEPS TOWARDS LINEAR PERSPECTIVE If we set aside Roman Antiquity, the earliest trials in perspective occurred first in Italian painting at the end of the duecento. In this respect, the pictorial composition of some outstanding frescoes is remarkable in that it foreshadows Renaissance perspective. Let us cite, for example, The Confirmation of the Rule, St. Francis Preaching before Honorius III and The Recovery of the Wounded Man of Lerida in San Francesco’s Upper Church of Assisi, The Presentation of Christ in the Temple, Christ Among the Doctors in the Lower Church, or St. Francis before the Sultan in Santa Croce of Florence, Cappella Bardi. Assisi’s Frescoes Let us take a closer look at the frescoes of San Francesco at Assisi. According to Vasari (1967, II: 100), the inferior register of the upper church could have been executed during the years 1296–1305. The terminus ante quem is deducible from the fact that the tower of the Palazzo del Capitano, seen unfinished on the frescoes, was completed at that date. On the other hand, the terminus post quem has been recently predated because we now know, thanks to an early fourteenthcentury manuscript, Religiosi viri, that pope Nicholas IV was the commissioner of the fresco campaign: nec vidimus in ecclesiis fratrum sumptuositatem magnam picturarum nisi in ecclesia Assisii, quas picturas dominus Nicolaus IV fieri precepit (MS. Rome, Archivio del Collegio di S. Isidoro, 1–146, fol. 263r; Cooper and Robson, 2003: 32–33). Since Nicholas IV reigned from 1288 to 1292, the frescoes terminus post quem must predate 1288. Certain art historians have attempted to restrict the interval to 1288–1305 on the basis of stylistic criteria, but there is endless debate on the criteria that would enable us to do so (Bellosi and Ragionieri, 2002; Wesselow, 2005). Among those frescoes, The Recovery of the Wounded Man of Lerida presents a noticeable feature: 4


the chequered pattern of the flat coffered ceiling is drawn so that the coffers’ orthgonal lines converge on a central point and coffers’ diagonals on two lateral points. Since the central and lateral points are located on the same horizontal, the fresco is an example of correct foreshortening in perspective (Raynaud, 2008).2 There is a fresco of the same type on the lower church north transept. Christ Among the Doctors exhibits a flat coffered ceiling, whose lines regularly converge on one central and two lateral points located on the same horizontal—as required in linear perspective. The two frescoes should thus be regarded as early steps towards a faithful representation of space. Practical and Theoretical Perspective These trials cannot be referred to as perspective as such, because we know almost nothing of the meaning Medieval and Renaissance men attributed to the geometrical loci involved in pictorial composition. The central and lateral points of convergence are not “vanishing points,” since the mathematical properties we now currently attribute to vanishing points were only highlighted during the sixteenth and seventeenth centuries. These contributions are due to leading mathematicians like Egnatio Danti (1583) or Guidobaldo del Monte (1600)—both spurred on by reading Vignola’s Due Regole della Prospettiva pratica (Thoenes and Roccasecca, 2002). Before this time, there was no consensus on perspective’s mathematical meaning. Despite a common opinion, linear perspective was not an established, secure system during the Renaissance: (1) the idea of Albertian perspective as a construzione legitima is a nineteenth-century German invention;3 (2) there was no agreement on the fundamental concepts to be used;4 (3) basic geometrical properties of perspective, such as the fact that the vanishing point must be located at the eye’s height above the ground, or that two parallels always converge on a point at infinity, were unknown until the end of the sixteenth century. Therefore, we can hardly admit a revolution between the Middle Ages and Renaissance on this point. In the Middle Ages too, some concepts of perspective were available through native optical literature. Aside from Qustaˉ b. Luˉ qaˉ’s specific study of the law of diminution (see below Chapter 2, Figure 2.4), let us review other possible sources. In the De Aspectibus, II.3, Alhacen explains how sight operates in the perception of size. When an object is excessively remote it has no (apparent) size and ceases to be visible: “The small portions that lie at the outer limit of the space (ultimum spatium) are not perceived or distinguished by sight, for 5


a small magnitude disappears from sight at an extreme distance” (Smith, 2001, I: 179, II: 485 [158]).5 Alhacen resumes his analysis in Chapter III.7, devoted to the study of visual illusions. He declares more specifically that, when the object is receding, it is seen as a point before ceasing to be visible: “And as the distance increases, the error is reinforced until the distance can become so great that a body will be judged to be the size of a point (quasi punctualis), and if the distance is further increased, that body will disappear from view” (Smith, 2001, I: 305, II: 603 [16]). Both passages appear to be milestones in the conceptualization of the vanishing point.6 It should be noted that, following Alhacen, medieval scholars had at their disposal two terms with a semantic nuance: ultimum is used to describe the endpoint of a real interval; punctum is used to refer to the tiny image painted onto the eye’s surface. 2. FRESCOES’ COMMISSIONERS Whatever the meaning and terminology, the correct foreshortening of end-duecento frescoes at Assisi appear to be a direct consequence of the sciences—mainly geometrical optics, as we will see—that were cultivated in Italian Franciscan circles. The easiest way to demonstrate this fact is to focus on the commissioners of the works. The Minister General Mendicant church building and decoration depended mainly on two authorities: first, the Pope since the church was a papal property—in 1253, Innocent IV asked for the work to be completed in those terms: nobili compleri structura et insignis praeminentia operis decorari; second, the minister general, approved as ministratore ecclesiae Sancti Francisci. In 1292, Assisi’s Statuta generalia stressed his duties in these words: Sed neque testudinatae ecclesiae amodo fiant, nisi super maius altare, absque licentia generalis Ministri (Bihl, 1941: 51). The minister’s artistic involvement is well known as regards architecture. Renato Bonelli (1985) has shown that the architectural options for Assisi’s basilica were changed under different ministries: the Angevine gothic transept under Aymon of Faversham (1240–44); bare walls and visible framework in the days of Giovanni da Parma (1247– 57); gothic ribbed vaults under Bonaventure (1258–73). Who were elected ministers general of the Franciscans? Aymon of Faversham, an Englishman, was serving as magister regens at the universities of Tours, Bologna and Padua. Giovanni da Parma, educated at Paris, became 6


master at Bologna and Naples. Bonaventure is famous for his regency at the studium parisiense. So all ministers had served as faculty members before they took administrative positions within the order and, in this capacity, commissioned Assisi’s artworks. Now, if we ponder on the ministers posted in the days of Assisi’s perspective first trials (after 1288), members of the faculty may have been directly involved in the pictorial revolution that occurred then. In this respect, it is noteworthy that the commissioning of Assisi’s frescoes coincides with the reign of the first Franciscan Pope, Nicholas IV (1288–92). The minister general of the Franciscans was then Matteo di Acquasparta (1287–89). The Minister’s Academic Training Matteo studied at Paris sub magistris regentibus Iohannes Peckam et Guillelmo de Mara, in the days when the Franciscan convent hosted another renowned perspectivist: Roger Bacon. After he took his degrees in Paris (ie baccalarius biblicus ca. 1269–70, baccalarius sententiarus 1272), Matteo di Acquasparta served as magister regens at Bologna and Paris, and became lector sacri palatii to the papal court at Viterbo in 1279, where he directly succeeded John Pecham. He was appointed minister general of the Friars Minor in 1287. Among the 76 MSS the doctor mellifluus left in his will to the friars of Assisi and Todi, we find: Arismetrica (then Todi, n. 278), Geometria cum commento et cum pluribus aliis (then Assisi, Biblioteca segreta, n. 269), Alpharganus’ astronomical compendium, Libri naturales, some Scripta super libros naturales and Questiones super omnes libros naturales (then Todi, n. 263). These books attest to Matteo’s interests in the quadrivium (Menestò, 1993: 285–89). Comparatively, his writings seem of little interest at first sight, because he apparently contents himself with light metaphors—theological: lux aeterna—metaphysical: lux spiritualis/ corporalis—psychological: lux/intellectus.7 Therefore, his topic is mainly “metaphysics of light” (Crombie, 1953; ten Doesschate, 1962; Lindberg, 1986).8 But this harsh judgement must be balanced, for being biased by a cultural distance: medieval optics is not always in line with modern optics. In my view, Smith (1981) and Tachau (1988) have convincingly argued that perspectivists’ main concern was not with light but with vision, and more specifically, that they searched for a theory of visual cognition, with a full range of epistemological consequences. Matters that we no longer consider as optical actually were so in the Middle Ages. Matteo’s optical education is salient, at least occasionally. Consider, for 7


instance, his Questiones de anima separata, q. 6: Album imprimit speciem albedinis in visum et quamvis oculus per speciem albedinis vere sentiat albedinem, numquam tamen fit vere albus, sed ut albus. In this question, he supports both the thesis of species’ reality and the intromission theory, which was quite unusual in those days (ca. 1278). Nevertheless, Matteo’s choice is understandable when we refer to his sources or to the concession he makes quod lumen naturale sufficiat ad certam cognitionem sine influentia illarum rationum (Acquasparta, 1957: 222 sq.). Finally, his Questiones disputate de productione rerum are interesting from the viewpoint of the history of art, because they contain an account of aesthetics that may have spurred early trials of spatial coordination. Matteo’s definition of beauty insists on binding and proportioning the work’s parts with each other: Pulcritudo autem et decorum consistit in distinctione, in gradatione, in connexione, in proportione (1956: 243). Since I have hypothesized Acquasparta’s role in the development of end-duecento perspective techniques (Raynaud, 1998a: 314–15, 321), art historians have independently argued that he was, in fact, responsible for the design of the iconographic programme of the nave and supervision of the work (Zanardi, 2002: 228–29; Cooper and Robson, 2003: 32). But, as he was familiar of optics since his Parisian stage, his involvement may have consisted of introducing optical matters in painting, by means of commissioning art. The end-duecento pictorial revolution—as a striking encounter between art and science—is perhaps due to such fortuitous circumstances. In a nutshell, I hypothesize early perspective to have resulted mainly from the transmission of optical theories to craftsmen by the commissioners who were given training in optics in the course of their former academic lives. This is a fresh hypothesis from which to embark on the study of linear perspective’s beginnings. OUTLINE OF THIS BOOK Chapter 1 sets out the problem to be solved: “Why did linear perspective arise in central Italy and not elsewhere?” by classifying standard explications of the rise of perspective in three categories: the firstrank factor (optics), second rank factors (realism, iconophilia, science as a backing for upward mobility), third-rank factors (the role of the Florentine bourgeoisie, artistic patronage, Renaissance humanism, urban development, training at a faculty of arts, training at abacus schools, disciplines outside optics, etc.). 8


Chapter 2 clarifies by a kind of reductio ad absurdum, how much this singularity has been dependent on aesthetical realism. Since linear perspective proceeds from Latin optics, which is mostly in continuity with Arabic optics, the beginnings of perspective should have occurred in the Arabic world. And yet no paintings and miniatures were drawn in linear perspective in any region in which Arabic was spoken. Tackling this paradox, I will show that the disregard of perspective must be explained by Islamic reluctance towards realism. Chapter 3 turns to the Latin Middle Ages, with the study of optical literature dissemination within the medieval university network. I first consider the distribution of optical treatises according to authors and obedience. Then I pass to the lector recruitment procedures in medieval universities, in order to specify university cosmopolitanism—the very condition for the diffusion of optical literature in European universities. The conclusion is that the cosmopolitanism was higher at Bologna than Paris, and higher at Paris than Oxford, thus accounting for a more intense lector turnover in Italy. Chapter 4 applies a full network analysis to the problem of the diffusion of medieval optics. The major finding is that university cosmopolitanism varies in accordance with the closeness centrality in the directed graph associated to the network of medieval universities. This confirms the dependence of cosmopolitanism on recruitment procedures. Moreover, a study of reticular roles shows that central Italy is skirted with “directed filters” favouring in- over out-connections, hence determining central Italy as a knowledge confinement area. Chapter 5 presents a simulation of diffusion within the medieval university network. The simulation obtains a final distribution of manuscripts matching the actual European distribution of optical treatises quite well. The chapter also provides theoretical remarks on diffusion dynamics. The main difficulties of diffusion analysis are reviewed. Standard diffusion modelling can only be applied with difficulty in sociology because social groups are basically finite, heterogeneous and anisotropic. Appendix 1 gives a detailed analysis of lists of lectors who attended the universities of Oxford, Paris and Bologna—roughly from 1230 to 1350. Appendix 2 provides a list of studia generalia, and pays special attention to the rules that ensured academic exchanges throughout Europe. Appendix 3 consists of the transcription and translation of the Provinciale ordinis fratrum minorum that gives a census of all Franciscan 9


settlements ca. 1340. This document provides a firm basis for testing the hypothesis that territorial organisation determined diffusion flow. The reader will find a complete list of references in the bibliography, along with two indices (an index of names and a subject index). NOTES 1. Healing of the Possessed, ca. 1425, silver plate (7 × 11 cm), Paris, Musée du Louvre, Département des Objets d’Art. 2. Fresco’s correctness is limited to the coffered ceiling. Firstly, side ceilings are drawn in oblique perspective when the central ceiling is in perspective. Secondly, the horizontal line is situated 722 mm above the eye line, with which it should coincide according to linear perspective. 3. This expression is an Italian translation of the German words “legitime Verfarhen”, first used by Heinrich Ludwig in 1888, and then popularized by Panofsky from 1924 onwards (Roccasecca, 1999). 4. Renaissance painters, architects and mathematicians have used a broad spectrum of terms to refer to the vanishing point, like “punctum centricum” (Alberti, 1435), “punto visibile” (Piero della Francesca, 1470), “punctus principalis” (Viator, 1521), “poinct de la veue” (Cousin, 1560), “punto principale” and “punto orizontale” (Danti, 1583), “punto dell’orizonte” (Serlio, 1584), “punctum perspectivae” (Benedetti, 1585), “punctum concursus” (del Monte, 1600), “poincte oculaire” (Vries, 1605), “point declinateur” (Caus, 1612), “punctum primarium” (Aguilon, 1613), “punto perspettivo” and “punto di concorso” (Accolti, 1625), etc. 5. The passage also appears in the Italian version De li aspeti, as quoted by Ghiberti: “Ed ancora quando la remozione fosse massima, le parti piccole dello spazio fossero per modo non [che] si comprendessino, le quali sono nell’ultimo dello spazio non si comprendono dal viso e non si distinguono per la grande remozione” (1947: 128). 6. Ibn al-Haytham, Kita¯b al-mana¯z·ir, III.7 [16], speaks in terms of “an imaginary point” (nuqt·a mutawahham) that “is not perceptible, the sense-faculty being able to perceive only that which has magnitude” (Sabra, 1989, I: 285). The passage remained untranslated into Latin. 7. For example: «Sed lux non potest esse sine lucere, nec lucere potest separari a luce; ergo nec intellectus sine intelligere, nec intelligere separari ab intellectu» (Gondras, 1957: 268). Nevertheless, it should be note that I have not looked into his Commentarius in I, II et III Sententiarum (ca. 1271–72). This analysis should be undertaken to reach definitive findings. 8. “Metaphysics of light” is, in fact, a misleading expression because it implicitly admits the subordination of optics to theology—a thesis that was rarely championed as such. We can simply disambiguate this expression by speaking in terms of “metaphysical derivations of light” as Eastwood (1970) or “invasion of theology by mathematics” as Asztalos (1992).



Perspective and its Optical Backing

As a starting point, the present inquiry requires agreement upon key conditions regarding what is significant vs. insignificant in the problem at hand. Sociology and the history of science often provide complementary analyses of the same empirical data. Science historians select and order the historical works from their intrinsic meaning: the criterion is mainly progress or failure. Sociology of science is driven by other criteria: it focuses on the collective aspect of a scientific practice (exchanges, social interests, controversies, etc.) regardless of the work’s scientific ranking. A good example is the contrast between Freiberg’s correct—but practically unknown—qualitative explanation of the rainbow (4 MSS) and Roger Bacon’s widespread—but not so good—explanation of the rainbow (39 MSS) (see Chapter 3.1 for further details). Science historians will look at the first work, sociologists at the latter. A second typical feature of sociology is that it frequently endorses Weber’s “principle of accentuation” that consists of considering a phenomenon explained as far as its main component is. I have chosen to focus the present study on optics–perspective ties. This drastic choice proceeds from a critical examination of the factors that presumably exerted an influence on early Renaissance perspective development. 11


1. INSIGNIFICANT FACTORS Albeit we all have reasons to endorse the assumption that any historical event is the product of an “infinite number of causal factors” (Weber, 1922), we do not need to pile up causes that have little to do with the facts. Therefore, for the sake of clarity, I will attempt to sort all causes in order to reject all second- and third-ranking factors and concentrate on the first one. Third-ranking Factors I call third-rank factors those that played little influence on the growth of perspective and are consequently of negligible significance. A short review is needed to understand this low ranking. 1. The role of the Florentine bourgeoisie is an old explanation of the development of perspective in Italy. It is quite simple to disprove it by a geo-historical Gedankenexperiment. Not every bourgeois context necessarily paid attention to perspective or visual matters. For instance, there is no record of this type in the case of the Hanseatic League that flourished between twelfth and fifteenth centuries in northern Europe. 2. Artistic patronage. With this explanation, we come closer to the practise of visual arts, but it is hard to see why painters would have drawn in perspective rather than using other techniques of representation. All things considered, Picasso too benefited from the patronage of Maurice Raynal, Daniel-Henry Kahnweiler or Eugenia Errázuriz. Artistic patronage can but explain art blooming, not the use of perspective. As regards the Renaissance period, the commissioners were much more concerned with price, deadline, warranty, theme and size of the work, than with the perspective techniques to be used by the painter (Baxandall, 1985: 18). 3. Renaissance humanism. Elsewhere, I have provided a full critique of this thesis about the development of perspective in quattrocento Italy (Raynaud, 1998a: 213–258). Philosophical speculation was a second-rate concern behind the training of future princes or merchants. Humanism was much more fond of moral philosophy than of mathematics, often limited to the very rudiments. For instance, in his De nobilitate, Coluccio Salutati denounces the “uselessness of mathematics” (mathematicorum vanitatem) and, even though he possessed some scientific manuscripts, he annotated none. 4. Urban development. This connection is assumed rather than demonstrated. For instance, we could mention the many urban civilizations 12


(from Babylon to China) that did not contribute at all to the history of perspective, theoretically or practically. 5. Training at the faculty of arts. At risk of anticipating a little the demonstration, I shall say that the European network of faculties of arts was as dense in southern France as in Italy. So, there is no distinctive reason why perspective developed in central Italy: it should also have appeared in Provence or Languedoc. 6. Training at abacus schools. The Italian scuole d’abaco may have increased the awareness of craftsmen of the question of proportionality that is the basis of practical geometry, with application to height, length and depth measurement, but this thesis is faced with two objections: it does not account for the many borrowings from medieval optics that characterize perspective treatises of the Renaissance; abacus notes state that teaching was mostly devoted to arithmetics, rather than to optics. 7. Role of the other mendicant orders, such as the preacher friars. Aside from the rarity of optical works by Dominicans, and the rare references to them in quattrocento treatises, another objection is the structural properties of their studia generalia network. The Dominican network was less dense in Italy than the Franciscan one. At the end of the fifteenth century, there were only 0.2 Italian Dominican provinces (4 in 20) vs. 0.44 Franciscan ones (14 in 32), and 0.33 Italian Dominican studia generalia (6 in 18) vs. 0.49 Franciscan ones (21 in 43). Preachers seem to have been concerned with homogeneity in territorial organization. 8. Disciplines outside optics. Several studies have uncovered other possible sources, such as cartographic projection (Edgerton, 1975), astrolabe use (Aiken, 1995), practical geometry (Camerota, 1998, 2006), or even the association of all sources (Bøggild-Johansen and Marcussen, 1981). None of these sources supplant optics, for the aforementioned reasons. For example, Sinisgalli (1980) has convincingly argued that, by associating Ptolemy’s planisphere with perspective, Commandino broke away from the Renaissance perspective tradition. Second-ranking Factors Other contributing factors played more salient roles. Accordingly, they must be acknowledged just behind the main explanation, which is based on optics–perspective relationships. Among second-rank factors, I shall consider the following: 1. Realism. Theologians’ position towards representation seems to have included didactic functions, vis-à-vis the illiterate, and 13


mnemotechnical functions, towards clerics (Simi Varanelli 1996). It is quite easy to see that didactics and mnemotechnics both urge the drawing of realistic images. Should it be otherwise, the illiterate/cleric would not have seen/recognized what paintings represented. This conclusion is evidently reinforced by the explanation of the disregard for practical perspective in the Islamic world (see Chapter 2). 2. Iconophilia. By allowing all types of representation—including God—, medieval taste for images (Besançon, 1994) spurred the representation of architectural structures, whose edges converge on the vanishing point. This was probably the primitive practical situation in which the problem of how to draw in perspective was laid down. 3. Science as instrument of mobility. Another factor that contributed to the success of perspective is the artists’ desire for upward social mobility in Renaissance times (Wolf, 1972). Architects and artists based their claims on the fact they too used science—mainly geometry and optics—to reduce the gap between the mechanical and liberal arts. And they carried this out successfully. 2. THE MAIN FACTOR: AVAILABILITY OF OPTICS As regards the early stages of linear perspective, optical knowledge is unquestionably the main factor that contributed to its development. This statement is supported by several facts. The very name of “perspective” comes from perspectiva (the Latin name for geometrical optics). Had the abacus been the main source for painters and architects, we would have talked about some “artificial abacus”—not “perspective”. Many textual parallels, entangling Renaissance perspective treatises with medieval optics, provide a second proof. Let us take a look at an example to understand how Renaissance artists worked. Although Da Vinci considers himself an “uomo senza lettere”, we know that he owned or read books by Euclid: “euclide in geometria” (Madrid II, 2va), Aristotle: “aristotile de cielo e mondo” (Cod. Atl. 97va), Ptolemy: “cosmografia di tolomeo”, (Madrid II, 3rb), Archimedes: “archimenide” (L. 94v) and Vitruvius: “vitruvio” (F 1v). He had some knowledge of medieval authors such as Tha¯bit ibn Qurra: “tebit” (M 11r), Jordanus: “gordano de ponderibus” (Quaderni II, 22v), John of Sacrobosco: “spera mvndi” (Madrid II, 3rb), Albert of Saxony: “alberto decelo e mundo” (F 1v), Bacon: “rugero bachone stampato” (Br. M. 71v), Pecham: “prospettiua comvne” (Madrid II, 2vb) and Witelo: “vitolone in san marco” (Br. M. 79v). Finally, Da Vinci read the latest literature: Luca Pacioli: 14


“arimetrica di maestro luca” (Madrid II, 3rb), Alberti: “vnlibro di ba alberti” (Madrid II, 3ra) as well as Giorgio Valla: “libro di gorgo valla” (Madrid II, 2va). All promoters of perspective of the Renaissance were probably working in similar conditions. They were using multiple sources, and, as the case may be, they chose or merged them with one another. Therefore, it is difficult to report the birth of perspective as a linear history, with its own stages and heroes. However, it may be asked which, among these sources, were the most important. Only a thorough comparison of texts can answer this question. I have used a refinement of the standard search for text parallelism: the “tracers method”, which consists of comparing a whole set of texts on the basis of the less frequently used words that have a high discriminating power (Raynaud 1998a, 2011b). Some of those silent borrowings are as follows (tracers are denoted by matching numbers: ①, ② etc.): Vinci’s Codex Atlanticus, fol. 594r “La luce operando① nel vedere le chose contra se converse② alquanto le spezie di quelli ritiene. Questo si pruova per li effetti③, perche la vista in vedere luce alquanto teme. Ancora dopo lo sguardo rimangano④ nel locchio similitudine della chosa intensa, e fanno parere tenebroso il luogo di minor⑤ luce, per insino che dallocchio sia spartito il vestigio⑥ de la impression de la magiore⑦ luce” (Vinci 1970: 24)

Pecham’s Perspectiva communis, I, 1 “Lucem operari① in uisum supra se conuersum② aliquid impressive. Hoc probatur per effectum③ quoniam uisus in uiuendo luces fortes dolet et patitur. Lucis etiam intense simulachra in oculo remanent④ post aspectum et locum minoris⑤ luminis faciunt apparere tenebrosum donec ab oculo euanuerit uestigium⑥ luci maioris⑦” (Pecham 1970: 62)

Ghiberti’s Commentarii, III “Ma restano le venti① altre cose sensibili② cioè il sito la corporità la figura la grandeza la continuatione la diuisione la separatione③ el numero et mouimento o riposo l’aspreza et la delicateza la diafanità la spesseza e ll’ombra la belleza et la pulcritudine turpitudine e lla similitudine et la diuersità, tutte le cose

Bacon’s Opus maius V, 1, 3 “Sunt autem uiginti① alia sensibilia② scilicet remotio situs corporeitas figura magnitudo continuatio discretio vel separatio③, numerus, motus, quies, asperitas, laevitas, diaphaneitas, spissitudo, umbra, obscuritas, pulchritudo, turpitudo, item similitudo et diuersitas in omnibus his, et in omnibus



sono composte④ di queste⑤ venti" (Bergdolt 1988: 50)

compositis④ ex his⑤” (Bacon 1964: 6)

Piero della Francesca’s De prospectiva pingendi, I, 30 “Da la intersegatione de doi nervicini che se incrociano vene la virtù visiva al cintro de l’umore cristallino, et da quello se partano i raggi et stendonse derictamente①, devidendo la quarta parte del circulo de l’occhio, sicommo o posto fanno nel cintro angolo recto② … aciò che l’occhio receva più facilemente le cose a lui opposte bisogna che se rapresentino socto minore③ angolo che il recto④” (ed. Nicco-Fasola, 1984: 98)

Pecham’s Perspectiva communis, I, 32, 39 “Ab anteriori parte cerebri oriuntur duo nervi concavi directi① ad anteriorem partem faciei, qui primo coniunguntur, qui inde ramificatur in duos ad dua foramina concava sub fronte | Ergo si ab extremis huius foraminis linee in centrum ducantur, constituent super eum angulum rectum② … nunc … maximus angulus sub quo est visio radiosa est minor③ recto④” (Pecham 1970: 116, 122)

Alberti’s De pictura, I, 5 “Centricum radium dicimus eum qui solus① ita quantitatem feriat ut utrinque anguli angulis sibi cohaerentibus respondeant … uerissimum est hunc esse omnium radiorum acerrimum② et uiuacissimum③ | Ex quo illud dici solitum④ est visum per triangulum⑤ fieri” (Alberti 1992: 90)

Pecham’s Perspectiva communis, I, 38 “Unde sola① perpendicularis illa que axis dicitur, que non frangitur, rem efficaciter representat, et alii etiam radii quo ei sunt propinquiores eo fortiores② et potentiores③ in representando | Dicunt communiter④ loquentes quia omne quod videtur sub angulo vel forma triangulari⑤” (Pecham 1970: 120)

Application of this methodology to the literature on perspective uncovers some forty silent borrowings from medieval optics (Raynaud 1998a: 163–209). Therefore, even though optics is not an absolute necessity to draw in linear perspective—since we can reconstruct most perspective theorems by means of similar triangles only1—the fact remains that Renaissance scholars did actually use optics for that purpose. Had it been otherwise we could not understand why Renaissance perspectivists assumed the principles of the visual pyramid and the rectilinear propagation of light, borrowed ideas concerning camera obscura and aerial perspective, wondered about extramission vs. intromission, 16


and at times tackled the problem of binocular vision. All issues indeed belong to optics. Many authors have investigated the links between optics and perspective (Parronchi 1958; Ten Doesschate, 1964; Federici Vescovini, 1964; Bergdolt, 1988, 1989; Simi Varanelli, 1989; Baggio, 1994; Field, 1997; Cecchini, 1998, 2006). This nexus is even incidentally acknowledged by those who are working on post-Renaissance periods (e.g. Andersen, 2007: 10–11). Classical, Arabic and Latin Optics Which kind of optical literature most influenced the development of linear perspective: is it classical, Arabic or Latin optics? This is a difficult question that cannot be answered from crude indices, because conceptual similarities can proceed from either direct or indirect quotation. Let us take the example of Ghiberti’s Commentario terzo. It is a case study, because Ghiberti’s authorship is now known to be around a hundred lines—about 2% of the text (Ten Doesschate, 1940; Bergdolt, 1988). As regards ancient sources, Ghiberti’s Commentario contains many references to Euclid, both to the Elements and to the Optica. But amazingly, even when Euclid is clearly mentioned in the text, the wording corresponds to posterior treatises: Lorenzo Ghiberti’s Commentarii, III “Le piramidi piu brieui essere piu lontane non procedenti da quella medesima base piu forti et parte piu deboli. Essere piu brevi impero che eglè necessita le ottuse piu, si come apparisce per lo primo de Euclide.” “Addunque AB, CD siano pari momenti distanti, questo si propongi impero che di poi niuno inconueniente seguita, sara la linea AC perpendicularmente tratte per ypotesim dalla linea pari BD et per consequente la linea BA pari al DC, si come apparisce per la 33 et 34 de Euclide” (ed. Bergdolt 1988: 204, 208).

Pecham’s Perspectiva communis, I,19 and 29 “Pyramides breviores, quia breviores, partim esse longioribus ad eadem basi procedentibus fortiores, partim debiliores. Breviores siquidem quia breviores sunt obtusiores esse necesse est, sicut patet ex primo Euclidis.” “Cum AB et CD sint equidistantes (hoc supponatur quia inde inconveniens non sequitur), erit linea AC perpendiculariter extracta per hypothesim equalis linee BD, et per consequens linea BA equalis DC, sicut patet ex 33a et 34a primi Euclidis” (Pecham 1970: 94, 110).



Here Ghiberti does not cite Euclid—not even in the medieval version of De visu (Theisen, 1979: 81). He, in fact, quotes John Pecham’s Perspectiva communis to the letter. But there is another interesting point. Since Pecham’s passage “sicut patet …” is to be found only in MSS BAWF, since W omits “ex,” since A gives “23a et 34a,” and F adds “elementorum” before Euclidis, we know that Ghiberti used a manuscript closely related to B that is: Oxford, Bodleian Library, MS Ashmolean 1522 (fourteenth century). Some quotes found in Ghiberti’s treatise are in fact “quotes of quotes” and they must be taken as such to avoid a biased viewpoint on Ghiberti’s aims, scientific education, and even perhaps the very way in which we should read his treatise. There is a quite simple device to estimate the bias. We just need to distinguish between (direct) quote vs. (indirect) reference, and class them throughout the Commentario terzo (Table 1.1). Euclid is a paradigmatic case: he is referred to twelve times by Ghiberti, and yet he is not quoted once. All Ghiberti’s Euclidean references are borrowed from Bacon’s Perspectiva (8 times in 14) and Pecham’s Perspectiva communis (6 times in 14). Now, as regards Arabic sources, it has been shown that some manuscripts on Arabic optics were available to Renaissance painters and architects. Almost half of Ghiberti’s Commentario terzo is based on the Italian version of Alhacen’s Kita¯b al-Mana¯z.ir. The De li aspecti was first discovered by Narducci (1871) and later thoroughly

Table 1.1: Ghiberti’s references to classical authors Explicit references to2

are quotes of


23 Aristotle

15 Bacon

49B:2, 60B:5, 60B:5, 60B:5, 60B:5, 60B:5, 72B:27, 79B:30, 80B:30, 82B:34, 84B:36, 87B:40, 88B:40, 98B:44, 163B:141 50P:96, 105P:126, 105P:128, 105P:128, 110P:128, 110P:128 50A, 59A 156B:131, 156B:131, 156B:131, 157B: 132, 159B:135, 159B:135, 162B:140, 162B:141 50P:96, 103P:122, 103P:122, 106P:94, 106P:96, 107P:110 49B:2, 99B:45, 158B:134

6 Pecham

14 Euclid

2 Add. 8 Bacon

6 Pecham 3 Ptolemy

3 Bacon



studied by Federici Vescovini (1964, 1998). The remaining parts of the Commentario terzo are based on materials received from a variety of sources (Ten Doesschate, 1940; Bergdolt, 1988). And again, some quotes are in fact “quotes of quotes:” Ghiberti’s Commentarii, III “Io tratterò la compositione dell’occhio spetialmente secondo tre oppinioni d’auctori cioè Auicenna ne’ libri suoi et Alfacen pel primo libro della sua prospetiua, Constantino nel primo libro dell’occhio, però che questi auctori bastano e più certamente tractano quelle cose no uogliamo. Non dimeno noi possiamo seguitare le parole di ciascuno però che alcuna uolta si contradicono per la cattiua traslatione” (ed. Bergdolt 1988: 42).

Bacon’s Perspectiva, I, dist. II, cap. 1 “Recitabo compositionem oculi praecipue secundum tres auctores, scilicet Alhazen in primo Perspectiuae, Constantinum in libro de oculo et Auicennam in libris suis, nam hi sufficiunt et certius pertractant quae uolumus. Non tamen possum sequi uerba singula cuiuslibet, quia aliquando contrariantur propter malam translationem” (Bacon 1964: 13).

Here Ghiberti does not quote the Arabic scholars—which would indicate that he made a “faithful copy” or an “intelligent critical choice” of sources. He presents, in fact, a copy of a copy, following Bacon’s Perspectiva. Therefore, we may question whether Ghiberti had, or not, a knowledge of (translated) Arabic sources other than the De li aspecti. Let us proceed as previously (Table 1.2). More often than not, Ghiberti’s classical sources are indirect references. Hence, we should be in total agreement with Federici Vescovini, when she writes: “Thus, Ghiberti quotes Avicenna, Galen, Averroes, Alhacen (he calls him Alfacen), Constantine the African” (1998: 69) except for one word: “quotes.” Ghiberti’s explicit references to Avicenna are, in fact, mostly given through Bacon’s Perspectiva (13 times in 16). Averroes too is referred to through Bacon (2 times in 3). The other borrowings from Avicenna and Averroes come from a Latin (unknown) medical treatise, for Ghiberti ends his quotations with the following words: “[…] Et haec de ossibus secundum Avicennam” and “[…] Et de ossibus Averoijs haec dicta sufficient,” which are not Avicenna’s and Averroes’s own wording. The single mention of Galen is not significant since Galen is quoted in many different sources (59). In his knowledge of Constantine the African—the 19


Table 1.2: Ghiberti’s references to Arabic authors Explicit references to

are quotes of


38 Alhacen

32 Bacon

16 Avicenna

3 Pecham 3 Add. 13 Bacon

49B:2, 59B:13, 62B:13, 63B:15, 64B:16, 65B:17, 65B:17, 66B:18, 66B:19, 67B:19, 67B:19, 67B:20, 68B:20, 69B:21, 69B:21, 69B:22, 70B:26, 72B:27, 74B:28, 74B:29, 82B:34, 83B:34, 83B:34, 99B:45, 99B:45, 100B:45, 100B:45, 157B:132, 158B:134, 160B:137, 161B:139, 164B:143 50B:62, 104P:126, 109P:114 173A, 193A, 194A 59B:12, 60B:5, 60B:5, 62B:13, 62B:13, 63B:15, 63B:15, 63B:15, 64B:16, 65B:17, 65B:17, 65B:18, 72B:27 205M, 206M 59A 59B:13, 61B:13, 61B:13, 62B:13, 63B:15, 65B:17, 65B:17 88B:40, 98B:44 209M 157B:132 105P:126

7 Constantine 3 Averroes 2 Alkindi

2 Med. 1 Add. 7 Bacon 2 1 1 1

Bacon Med. Bacon Pecham

translator of H. unayn b. Ish. a¯q’s Ten Treatises on the Eye and ‘Alı¯ b. ‘Abba¯s’s Complete Book of the Medical Art—Ghiberti is exclusively dependent on Bacon’s Perspectiva (7 times in 7). The same feature applies to Alkindi, who is referred to only through the writings on optics by Roger Bacon (1 time in 2) and John Pecham (1 time in 2). Ghiberti quotes Alhacen extensively. Nonetheless, we can check in the reference editions of Ibn al-Haytham’s Kita¯b al-mana¯z.ir/De aspectibus (Sabra, 1989; Smith, 2001–2010) that he does not quote him in the passages he refers to. There is no exception to the rule. In fact, when he explicitly mentions Alhacen or the “autore di prospettiva”3 he almost always quotes Bacon’s Perspectiva (32 times in 38) or Pecham’s Perspectiva communis (3 times in 38). For that reason, the rise of linear perspective cannot be explained through a direct influence of Arabic optics. Only Alhacen’s De aspectibus and De li aspecti—the Latin and Italian versions of Ibn al-Haytham’s Kita¯b al-mana¯z.ir—and medieval commentaries were known to perspectivists. All these Latin texts notably differ from the Arabic model.4 20


Neither can the birth of perspective be explained by Witelo’s influence, though he sojourned at the papal court in Viterbo. The references to Witelo in Renaissance treatises of perspective are more unusual than those to Pecham and Bacon, as evidenced by the study of textual parallelisms (Raynaud, 1998a: 170–194). Ghiberti’s Commentario terzo again provides a good example, for it mostly consists of 95 pages copied from Bacon, 31 pages from Pecham, and only 4 pages from Witelo (after the ‘Synopsis des dritten Kommentars’ by Bergdolt, 1988: 570–573). There is a very simple reason for this imbalance. With only 25 MSS, Witelo’s Perspectiva received a rather confidential diffusion compared to that of Bacon’s and Pecham’s (Federici Vescovini, 2007: 109). For a complete analysis of the distribution of manuscripts, see Chapter 3, Table 3.1. Another medieval scholar who is often said to have influenced the development of linear perspective is Biagio Pelacani of Parma (d. 1416). The reader may wonder why I do not mention the relationships between Biagio Pelacani, the author of a series of scholastic questions on Pecham’s Perspectiva communis, and Filippo Brunelleschi, the Florentine inventor of linear perspective. It is not for lack of trying, notably after the impressive attempts by Graziella Federici Vescovini (from 1961 to 2009). This opinion is based on the assumption that Pelacani can solve both the problems discussed in The Oxford Hypothesis, which also underlie this book: (1) the conversion of perspectiva naturalis (optics) into perspectiva artificialis (perspective), since Pelacani’s Quaestiones perspectivae are supposed to include significant improvements in this sense; (2) the birth of perspective in quattrocento Florence, as Pelacani briefly sojourned in that city in 1388. However, this assumption that some scholars take as proven fact, is still without any evidence. 1. It has been claimed that Biagio Pelacani was “the first to have taught and demonstrated that the decrease in apparent magnitudes is proportional to distance.” Regrettably, most medieval scientists precisely discarded the “angle axiom” that Panofsky saw in the proposition 8 of Euclid’s Optics (e.g. Pecham, Perspectiva communis, I, 73–74), and Euclid himself used the “distance axiom” later on in propositions 19–20. Consequently, the Renaissance artists may have found this idea in any optical treatise that was circulating in Italy at the end of the trecento. This odd slip comes from Panofsky’s misunderstanding of proposition 8 of Euclid’s Optics, along with a biased reading of an article by Graziella Federici Vescovini (1961: 201), where the idea of a “ratio of the distances” is a gloss on the Quaestiones perspectivae, I, 16. The Latin 21


text appears at the septima conclusio (see 1961: 226 sq.). The reader will verify that the interpretation is made from only two words, virtus illativa, which, in turn, are extracted from an extremely broad argument by Pelacani that visual faculty is a cognitive faculty (incidentally, this was already Alhacen’s opinion). 2. So far, the scholarly literature has never reported any single textual parallel between the Renaissance treatises on perspective and the Quaestiones perspectivae by Biagio Pelacani. I have unsuccessfully sought such parallels (Raynaud, 1998a: 163–194). While Renaissance artists were used to plagiarising whole passages of medieval optics—as evidenced by forty borrowings ad litteram from the optics of Alhacen, Bacon and Pecham—one may well wonder why they never copied nor quoted the treatise that is thought to have exerted the strongest influence on the birth of perspective in quattrocento Italy. 3. The search for textual parallels establishes that some Renaissance writers do not follow Pelacani when they address the key issue whether the knowledge of size of a visible object is based on the angle or on the distance: Ghiberti’s Commentarii, III “La quantitade dello angholo sotto el quale si uede la cosa non bastare alla quantità della cosa uisibile […] La comprensione della quantitade per la comprensione della piramide raggiosa procedere per comperatione della basa alla quantità dello angulo et la lungheça della distantia. Addunque la cognitione sola della quantitade dell’angulo non basta alla quantità di scientia pertanto comporta a

Pecham’s Perspectiva communis I, 73–74 “Quantitatem anguli sub quo res uidetur minime sufficere quantitati rei uisibilis capiende […] Comprehensionem quantitatis ex comprehensione procedere pyramidis radiose et basis comparatione ad quantitatem anguli et longitudinem distantie. Sola igitur cognitio quantitatis anguli ad quantitatem non sufficit discernendam, confert tamen adhoc, sicut


Pelacani’s Quaestiones perspectivae, I, 16 “Idem uoluit Magister in hac prima parte Perspective in hac forma ‘angulum sub quo res uidetur, non sufficere pro quantitate rei uisibilis capienda’. Et postea subiungebat alibi ‘quantitatem rei apprehendi per quantitatem anguli et proportionem distantie’ et cetera” (ed. Federici 2009: 223).


questo, si come apparisce sopra alla 40 propositione” (ed. Bergoldt 1988: 260).

patet supra ex 41 propositione” (Pecham 1970: 144).

Moreover, according to Pelacani, the size of objects is known through the similar triangles whose common base is the axis of the visual pyramid. This axis is called “radius centricus” by Leon Battista Alberti. The search for textual parallels in classical and medieval optics establishes that there is only one source that refers to the centrum of the pyramid. This is al-Kindı¯’s De aspectibus, prop. 12, thus matching Alberti’s paraphrase. Like any other perspectivist, Pelacani attributes the quality of being fortior to this radius, however he does not call it “radius centricus” but “radius perpendicularis,” thus following the standard terminology fixed by Ptolemy (Optica II, 18), Alhacen (De aspectibus I, 6.24 and II, 2.30), Bacon (Opus maius IV, 3), Witelo (Perspectiva III, 17) and Pecham (Perspectiva communis I, 38): Leon Batista Alberti’s De pictura I, 8 “Centricum radium dicimus eum qui solus ita quantitatem feriat ut utrinque anguli angulis sibi cohaerentibus respondeant. Equidem et quod ad hunc centricum radium attinet uerissimum est hunc esse omnium radiorum acerrimum et uiuacissimum” (Alberti 1972: 44).

Al-Kindı-’s De aspectibus 12 “Dixerunt enim quod radius supra centrum circuli pinealis basim continentis erectus perpendicularis super centrum existit … Potentior ergo uisus conuersionem efficit perfectam. Ipse igitur efficit radium perfectum, scilicet fortem … Dico ergo quod super centrum uisus cadit radius fortior” (ed. Rashed 1997: 471).

Pelacani’s Quaestiones perspectivae I, 9 “Radius perpendicularis est fortissimus radiorum et incessus perpendicularis est fortior et efficacior omni alio. Ad confirmationem cum dicitur ‘si fieret fractio, hoc esset propter resistentiam et cetera’, dico quod hoc non est propter resistentiam, sed hoc ideo magis est quia per hanc fractionem radium melius conservatur in esse” (ed. Federici 2009: 151).

4. Pelacani wrote his Quaestiones perspectivae between 1379 and 1390/9 (quaestio 3A) and 1416 (quaestio 3B). Two manuscripts are 23


preserved in Florence: Biblioteca Laurenziana, Pluteo XXIX, 18, fol. 1r–83rb [F1] and Ashburnham 1042, fol. 1–144 [F2]. The first was copied in 1428. The second, most corrupt, is later (Raynaud, 1998a: 229–231). Therefore, neither copy can have fostered Brunelleschi’s experiment, which we know to be prior to 6 August 1413 (see the introduction). To support the thesis of the influence of Pelacani on Brunelleschi at all costs, some scholars have distorted a passage by Graziella Federici Vescovini, holding that the text “è verosimilmente databile intorno al 1403 per un esplicito riferimento a certi fenomeni luminosi che avevano dal miracoloso, accaduti appunto nel 1403” (1965: 242). In so doing, they confuse the Pluteo XXIX, 18 manuscript with Pelacani’s autograph, and the terminus a quo with the terminus ante quem of the autograph. Firstly, Federici Vescovini has always explicitly dated the Pluteo XXIX, 18 to 1428: Expliciunt quaestiones perspectivae magistri Blaxii de Parma expletae per me Bernardum Andreae de Florentia die undecima mensis Martii anni 1428 ad honorem et laudem Dei. Secondly, in the passage quoted above, the year 1403 is established as the terminus a quo of the treatise. Moreover, the text provides evidence that the optical illusions Pelacani saw in Lombardy were described at a later date. In the passage: Unde recordor in domini anno 1403 quod in Lombardia iuxta quoddam castrum quod dicitur Busetum, apparuerunt per tres dies, omni die, ante horam tertiarum, turba magna […] (Pluteo XXIX, 18, fol. 82 rb), the words unde recordor and the perfect apparuerunt indicate that Biagio was stating an ancient memory, which is only mentioned by the year. Finally, even when the erudite Italian philologist approaches the optics–perspective nexus, she does so with caution, either by focusing on pure gnoseological questions (1965, 2003, 2007), or by considering all late-medieval optical works as possible sources of perspective. In a little-known article aimed at pointing out the contribution made by fourteenth-century perspectivists, Federici Vescovini concludes that “the technical-mathematicalgeometrical model of perspectiva artificialis surely is original and peculiar to Alberti, but in the above-mentioned works on perspective circulating in Florence in his time [from Alhacen to Pelacani], he could find the idea of a geometrical-perceptive representation” (2000: 81). By concealing those deliberate omissions and biased reading, it was assumed that the influence of Pelacani on Brunelleschi was established once and for all. I am obliged to repeat the conclusion drawn in The Oxford Hypothesis (1998a: 231). For the time being, the assertion that Pelacani’s Quaestiones perspectivae were known to Florentine artists before 1413 must be considered as a hypothesis awaiting evidence. 24


In the current state of knowledge, the question of how medieval optics was transformed into linear perspective cannot be solved by any of the oversimplified sequences: Alhacen–Witelo–Brunelleschi or Alhacen–Pecham–Pelacani–Brunelleschi. However, in my view, the question of whether perspective comes from Classical vs. Arabic vs. Latin optics is not the upmost controversial matter, because there is an unquestionable continuity—apart from the language used—in optical research from Euclid to Kepler. Bacon, Pecham, Witelo re-elaborate or criticize Euclidean material; they strongly depend on Alkindi’s and Alhacen’s optical conceptions too. In sum, until the classical period, scholars posed problems and provided solutions in the same manner. Even when they were innovative, as was Ibn al-Haytham, they belonged to the same tradition. As is well known, this optical tradition came to an end with the understanding of the retina as the seat of vision (Vésale 1543, Platter 1583), and the subsequent discovery of the retina’s function by Kepler (1604).5 From then on, geometrical optics reached its own limits, because the propagation of visual percepts through the optic nerves became a mere question of physiological optics. Optics and the Translatio Studiorum Scholars have asserted that some medieval schools reached higher levels in optics than others, thus contributing predominantly to the progress of that science and allied techniques. Which kind of optical literature most influenced the development of linear perspective: is it English, Parisian or Italian optics? At risk of disappointing my audience, I will not engage further in this debate, because I think the correct position is, in fact, to accept all three theses at the same time—as long as each is restricted to strict limits. Grosseteste’s disciples were active at Oxford from 1230 to 1258 (Adam Marsh’s death). Paris’ studium then gained leadership from roughly 1257 to 1267 (Bacon’s stay). Finally, the studium Curiae near the papal court of Viterbo took over from 1267 to 1277 (John XXI’s death). This is just an illustration of the thirteenth century translatio studiorum from England to central Italy. But this gradual transition corresponds to the peregrinatio academica of many northern scholars southwards. None of these schools is independent from each other. As a matter of fact, Bacon and Pecham frequented Oxford’s studium in the days Adam Marsh, a disciple of Robert Grosseteste, taught there. A decade later, both were to be found at Paris. At the end of his Parisian sojourn, Bacon sent his Perspectiva and De multiplicatione specierum to Pope Clement IV (1267); 25


interest, a sort of passion in which some friars engaged in dilettantism, but rather a matter of institutional interest contracted within the orders’ educational structures. Optics Outside the Quadrivium MS. Assisi 196 is a central piece of the puzzle. This manuscript, described by Pouillon (1940), is an academic collection of questions disputed in the faculty of theology at Oxford. Entirely written in the same hand, this manuscript was compiled by John Basset, regent master at Oxford from 1295–96. The scattered references made to the authors of such recorded questions show that they mostly come from the Oxonian Minorites of the late thirteenth century: Thomas de Bungey (1270–72), Nicolas Ocham (1285–86), Hugh de Hertilpole (1287–88), John of Pershore (1288–89), John de Berwick (1289–90), etc. This compilation contains a broad range of questions centring on the arts of the quadrivium, mainly twenty-eight questions about place, void, time and motion (fols. 5r–15r) and fifteen questions on motion and infinity (fols. 26v–30v). But the MS. Assisi 196 compiler especially recorded a series of questions concerning the multiplication of species, refraction and sunlight: q. 176. Utrum lux multiplicet se subito uel successiue; q. 177. Sed qualiter est quod oculus uidet rem quasi maiorem si fuerit in aere et res in aqua; q. 178. Tunc queritur qualiter producitur lumen a sole ex quo; q. 179. Sed estne lumen in medio. The manuscript also contains three disputed questions in which optical statements are used for solving theological problems (qq. 43, 99, 150). Other academic texts show the same enthusiasm for optics. MS. Assisi 158 (Henquinet, 1931; Glorieux, 1936) is an English manuscript, which was for the personal use of friar Nicolas Comparini of Assisi. It outlines a series of questions ca. 1280–90, combining optics with psychology and metaphysics: q. 46. Utrum anima formet ymagines in semetipsa; q. 118. De luce; q. M, which is devoted to a review of visual illusions (deceptiones in uisu). The MS. 174 Assisi is another collection of disputed questions, whose composition has been described by Pelster (1931).8 The manuscript, in particular, reports a question by a certain Guillaume—probably Guillaume de la Mare, lector at Paris (1268–69) or Guillaume de Falegar (1271–75)—entitled: q. 4. Queritur utrum lumen in medio sit uera res ut forma substancialis uel accidentalis, uel sola intencio (fols. 27v–29r). Similarly, we find in the questions disputed by Roger de Marston, lector at Oxford (1282–84), a series of Quaestiones disputatae in lux naturalis. 27


Another way to appreciate the interest the friars had in optics within the educational structures of the order is to investigate the optical passages recorded in lecture notes (reportationes) whose titles do not necessarily refer to the problem of vision. The haul is no less fruitful. The fondness for optics is attested by many Franciscan Sentences commentaries that refer to this science (Tachau, 1988). Bonaventure [Giovanni da Fidanza]’s Commentaria in librum II Sententiarum, dist. XIII, contain a series of questions on light, in particular: art. 2 on the nature of lux, i.e. light itself, q. 1: Utrum lux sit corpus vel forma corporis, q. 2: Utrum lux sit forma substantialis vel accidentalis; art. 3 on lumen, i.e. the radiating light, q. 1: Utrum lumen, quod exit a corpore luminoso sit corpus, q. 2: Utrum lumen sit forma substantialis an accidentalis (Bonaventure, 1885: 317–333; Lindberg, 1986: 17–18). Pierre de Jean Olieu—called Pietro Giovanni Olivi, in the Italian context—discusses in his Quaestiones in II librum Sententiarum several questions related to time vs. instantaneous propagation of light (q. 26), to the sense of visual rays spreading, in which he concludes against the extramission arguments of Platonists (qq. 58, 73). He consequently embarked on a revision of the concept of visual species (q. 74) (Jean Olieu, 1922). In the first decade of the fourteenth century, optical matters were also discussed by Alessandro [Bonini] d’Alessandria, lector at Bologna and Paris, in his Commentarius in IIII libros Sententiarum: “Utrum in luce proprii generis possit aliquid sciri sine speciali influentia alterius luminis.” Among his successors—both at Bologna and Paris—Pierre Auriol, devoted large parts of his Scriptum in I librum Sententiarum to explaining certain optical phenomena such as the persistence of visual impressions in the eye, diplopia, or the breakup of the image by refraction (Auriol, 1953). Auriol’s Reportatio in II librum Sententiarum also discusses the question of time vs. instantaneous propagation of light (fol. 67 ra–rb). William of Ockham, graduate baccalarius sententiarum of Oxford in 1317–19, is now well known for his conceptions in logic and philosophy of language. But his work also contains many references to optics (Tachau, 1982). We find evidence of that in the quotes of Alhacen that appear in Ockham’s III Libros Sententiarum and Reportatio. In particular, he attempted to devise a deflationist theory of perception, hence magnifying the role of sensation and breaking from the tradition of species (Reportatio, II.15). Ockham’s theory was subjected to John of Reading’s and William Chatton’s further criticism. In his commentary on the Sentences, Chatton comments on a full range of visual topics 28


related to the concept of species up to the transparency of coloured medium—a theme that is not without some Alhacenian flavour: rubedo diffusa per radium transeuntem per vitrum est vere color (Florence, BNCF, Conv. Soppr. C.5.357, fols. 194–95; Tachau, 1988: 409).9 Geraldus de Odo, minister general of the Friars Minor between 1329 and 1342, also commented on optical matters in his Quaestio de lumine, including in his Commentarium in I librum Sententiarum (Lindberg, 1975). Another comment appears in the writings of Adam de Woodham, magister regens at Oxford ca. 1338. His Lectura secunda discusses at length Pierre Auriol’s accounts of visual illusions, like after-images: Species potest manere, or refracted images: Tertia experientia de apparentiis fracture baculi cuius pars est in aqua (Tachau, 1988: 430–31). Other optical questions are discussed in a work formerly attributed to Duns Scot, and now restored to Simon Tunsted, master regent at Oxford in 1351. In his Commentarius super libros Metheororum Aristotelis, Tunsted describes the primary and secondary rainbows (Duns Scotus, 1639; Boyer, 1958). The primary rainbow is given the exact value of 42° following Roger Bacon.10 Although he admits the existence of the secondary rainbow (which is a dim and reversed rainbow), Tunsted draws the conclusion that the third rainbow is impossible, thus confusing existence and conditions of visibility: the tertiary rainbow actually exists, though invisible because of its dimness and position (it makes a circular halo around the sun, absorbed by its luminosity).11 During the fourteenth century, friar Egidius de Baisiu tackled another optical question that scholars came up against before Kepler and Maurolico: “Why the sun, passing through a triangular-shaped opening, produces a round image onto a distant screen?” Egidius’ demonstration, which is essentially a disproof of Pecham and other unnamed presumptores, is better than those Henry of Langenstein and Biaggio Pelacani composed at the end of the fourteenth century. Egidius maintains the rectilinear propagation of light, while his rivals admitted an exception in this case (Mancha, 1989: 19–20). He also abandons the concepts of primaria and secundaria lux to explain the formation of pinhole images and correctly states—from his own experiments—that the source’s shape is recovered in proportion of the distance (redditur distantia proportionaliter) lying between screen and aperture (1989: 4, 19–20). All these texts have the fact that they were not written outside the academic framework in common. All quaestiones belong to the exercise of disputatio in which a trained bachelor (baccalarius formatus), sometimes assisted by the master himself, presented the question 29


and responded to objections from the audience. The content of some reportationes, as well as the authorship of certain tracts, establishes the interweaving of theological and natural questions (Pouillon, 1940). Optics was discussed not only in the faculty of arts—its natural place as a geometric science—but also by masters and trained graduates within the faculty of theology. This deduction is even easier in the case of the Franciscan Commentarii in libros Sententiarum frequently sprinkled with optical themes. After having joined the faculty, a bachelor became baccalarius biblicus for two years. Before he could participate in disputes, he spent another two years in explaining the book of Sentences as baccalarius sententiarum. Finally, at the age of twenty-five or so, he redacted his own commentary. Accordingly, the optical topics that appear in disputed questions and Sentences commentaries do not evade the regular academic curriculum, even in the case of the mendicant orders.12 Two Testimonies Roger Bacon offers the first evidence for collective interest in optics, in the course of the apology he presented to the Pope for the delay he took before sending his works to the Roman Curia. He explains: I had as yet composed nothing, except that I had sometimes compiled in cursory fashion certain chapters now on one science, and now on another at the request of friends (ad instanciam amicorum).13 John Pecham offers further proof of the collective drive to write on optical matters in the prologue of the Perspectiva communis. While declaring himself unsatisfied with the use his alumni made of a first draft, he fully recognizes the passion they put into that science: A while ago, at the request of my associates (rogatus a sociis), I wrote certain unpolished mathematical notes (mathematicae rudimenta), which, since I was occupied by other matters, I left uncorrected. These appeared in public against my intention. Consequently, I will endeavour to correct them slightly in order that they may be of benefit to young students (ut prosint iuvenibus studiosis). Now, among the investigations of physics, light is most pleasing to students. Among the glories of mathematics it is the certitude of demonstration that most highly exalts the investigators. Therefore, 30


perspective is properly preferred to all the traditional teachings of mankind (Proemium).14 Lindberg estimates Pecham’s notes (mathematicae rudimenta) were composed between 1269 and 1279, and more precisely: either in 1269–75 during his Paris/Oxford professorship, or in 1277–79 while he stayed at the papal court of Viterbo (Lindberg, 1970: 17). It is widely agreed that the second period is more probable. First, there is almost exactly the same wording in Pecham’s and Witelo’s treatises, which makes it almost certain that one borrowed from the other (Lindberg, 1971: 77). Second, Perspectiva communis is of a higher standard than Pecham’s Tractatus: after 1277, he could have benefited from Wilhelm de Moerbeke’s and Witelo’s help in optics. However, this opinion is based on conjecture. As a matter of fact, there is no proof that Witelo’s Perspectiva was composed prior to Pecham’s. I feel particularly uneasy about two details. First, Pecham evokes his socii, a term of legal origin, that has a precise meaning in medieval academic life (Courtenay, 1999: 85–86). Socius meant: A) either a “fellow” in case of persons standing at the same level, as students or masters; or B) an “assistant” that is, a junior colleague having a status inferior to the master’s—for instance fr. Iohannes Scotus et fr. Thomas eius socius; or C) a “fellow” in the case of a member holding a fellowship in a college—as in socius collegii de Merton or socius domus de Sorbonna parisiis. Now, if Pecham composed his Perspectiva communis when officiating as lector sacri palatii, his audience was the Cardinalice College itself. As Pecham was not a cardinal, he could not have treated cardinals as socii—both for reasons A and C that he was not one himself, and for reason B that lectors had no precedence over cardinals. Next, our eyes are drawn to the expression “young students” (iuvenes studiosi) that cannot have applied to the Sacred College’s members. Why would Pecham have taken advantage of his papal sojourn to fulfil the needs of young students he was now freed from? This is quite anomalous. Thus, we must return to Lindberg’s first statement that Perspectiva communis was composed in the 1269–79 period, and that “borrowing [between Pecham and Witelo] could have occurred in either direction” (1971: 81). If Pecham ever composed his second optical treatise before he joined the papal court, it is likely that his socii were among the students that attended Paris’ studium in the days of his regency. Among them were Matteo di Acquasparta and Pietro Giovanni Olivi—both in Italy during the next decade. 31


Whatever the exact date of composition of Pecham’s Perspectiva communis, the passage “At the request of my assistants” (Scripsi … rogatus a sociis) is reminiscent of Bacon’s lines, when he declares he composed much of the Opus maius “at the request of friends” (Composui … ad instanciam amicorum) (Lindberg, 1971: 71–72). These two quotes establish that optics, considered as a geometrical science, and taught in the faculty of art in this respect, was far from being excluded from other faculties. Such accounts attest to the fact that optical research undertaken by friars was not a private matter, but was rather subjected to collective interests. The passion for optics was encouraged within institutions, and was even closely relayed by academic activities in which bachelors and masters took part. I guess these remarks will be sufficient to redirect the analysis of the rise of linear perspective in central Italy through (1) the availability of optics as a condition for emergence, (2) academic education as a condition for the transmission of optics, (3) the collective dimension of the diffusion process. Indeed, if sociology is concerned with private scientific interests, it must focus on individual actions, beliefs and values. Now, if institutional interests support knowledge, the sociologist may change scales. In accordance with the postulates of methodological individualism, the units of observation can be macro-individuals (Boudon, 1992). In a study of the diffusion of knowledge, they consist of academic structures in which the texts were written and circulated. Since universities disseminated throughout Europe during the fourteenth century, we can reflect on whether academic contacts and exchanges were responsible for a non-uniform distribution of optical knowledge. Should the case arise, we can search for whether the high level perspective reached in Italy depended on the differential diffusion of optics. Now, the problem is entirely rewritten, and it becomes possible to consider the question: “Why did linear perspective arise in central Italy?” as a genuine question for the sociology of science. NOTES 1. Applications for perspective are countless. I can but give some examples. Similar triangles are underlying the demonstration of interval diminution by Piero della Francesca (1984, I: xiii), the proof of perspective construction of a rectangle by Benedetti (1585: cap. I), the proof of the existence of a point of convergence for any set of orthogonal lines by Danti (2003: prop. xxxiii), or the same proof,






5. 6.



extended to any set of parallels in any plane, by Guidobaldo del Monte (1600, I: props. xxix–xxxii). A reference such as [50P: 96] has to be read as “Ghiberti, p. 50, copies Pecham, p. 96.” 50 refers to the target page in the passable edition of the Commentarii by Morisani (Ghiberti 1947) that I have preferred to the solid, but hardly available edition by Klaus Bergdolt (1988); the letter refers to the source: A (Additions), B (Bacon 1964), P (Pecham 1970), V (Vitruvius); 96 refers to the source page. I have not considered the low frequency sources: Agatharcos 215V, 215V, Anaxagoras 204V, 215V, Apollonius 49B, Archimedes 49B, Boethius 98B, Democrite 215V, 215V, Empedocles 205V, Euripides 204V, Galen 59A, Heraclites 204V, Hippocrates 59A, Plato 48A, 105P, Pythagoras 205V, Seneca 163B, 163B, Socrates 210V, Thales 204V, and Theodosius 67B. Alhacen 164B has also been credited as the author of the Tractatus de crepusculis et de ascensionibus nubis, which is, in fact, a work by Abu¯ ‘Abd Allah Muh. ammad b. Mu’¯adh al-Jayyanı¯—that is to say: from Jaen (Spain). The text existed in various Arabic, Latin, Hebrew and vernacular Italian translations (see Smith, 1992; Smith and Goldstein, 1993). In his recent book, Hans Belting (2011) admits that linear perspective is dependent on Arabic optics. However, he entirely accounts for Alhacen’s optical theory from the Kita¯b al-mana¯z.ir (ed. Sabra, 1989) without referring to the authoritative 8-volume edition of Alhacen’s De aspectibus (ed. Smith, 2001–2010). Furthermore, his book makes a single mention of Bacon, Pecham and Witelo’s works, a sign that the Arabic and Latin versions are muddled up. The Latin filter presented in these lines is further confirmed by the comprehensive study of the Arabic legacy to perspective. For instance, it has been shown that no propositions of al-Kindı¯’s De aspectibus were accessed by artists. Da Vinci’s awareness of al-Kindı¯’s optics was in fact limited to the medieval echoes he could grasp in Bacon, Witelo and Pecham (Raynaud, 2011b). For the significance of Kepler’s Ad Vitellionem paralipomena, see Field (1987). See, in particular, the following passages: “E però che li raggi non sono altro che uno lume que viene dal principio de la luce per l’aero infino a la cosa illuminata …” (II, 9, 5); “Queste cose visibili, sì le proprie come le communi in quanto sono visibili, vengono dentro a l’occhio—non dico le cose, ma le forme loro—per lo mezzo diafano, non realmente ma intenzionalmente, sì quasi como in vetro trasparente …” (III, 9, 6); “Veramente Plato e altri filosofi dissero che’l nostro vedere non era perchè lo visibile venisse a l’occhio, ma perchè la virtù visiva andava fuori al visibile: e questa oppinione è ripprovata per falsa dal Filosofo …” (III, 9, 11), etc. See, for example: “Among all these onliche on is the instrument of sight; that is the humour cristallinus and had that name of cristal, for he is iliche cristal in colour …” (V, 5); “also lux and lumen is light in englische, but in latyn is difference bitwene thilke tweyne nounes, for lumen is a stremynge out of light, byt lux is the substancial welle vppon the whiche lumen is i-oned …” (VIII, 28); “As he auctor of Perspectif seid, in thre maner byschinynge is ifounde, for som bischinynge is iclepid lumen reflexum, and som fractum, and som directum …” (VIII, 29), etc. Against the common thesis of Franciscan reluctance to science, we must note that those quaestiones disputatae are part of a compendium of scientific texts: Tractatus communis de spera (fol. 59r–68v), Compotus communis solaris et lunaris fratris Iohannis de Sacro Bosco (fol. 80r–95v), Compotus fratris P. (fol. 95v–101r),



9. 10.



13. 14.

Tractatus quadrantis. Geometrie due sunt species theorica et practica (fol. 101r– 104r), Tractatus astrolabii Messehallarum (fol. 105r–113r), Tabula stellarum fixarum (fol. 113r–113v), Algorismus uerificatus (fol. 113v–115v). This MS formerly belonged to the Franciscan convent of S. Croce. The angle formed in the observer’s eye by the anti-solar line and the rainbow depends, in fact, on the radiation: it ranges from 40.5° for blue light to 42.4° for red light. In this respect, Tunsted is far behind Theodoric of Freiberg. However, medieval works must be compared to the state of the art. We should remind ourselves that Freiberg is correct only about the qualitative explanation of rainbows, and that he gives quite strange quantitative values such as 22° for the primary rainbow, and 33° for the secondary rainbow. On this point we can question whether Weishepl (1965, 1968) and Federici Vescovini (1969, 1973) are entirely right when they exclusively place perspectiva in the quadrivium. Such a positioning in the classification of sciences does not account for all existing texts. We must refer here to recent works, such as those by Livesey (1985, 1898, 1990) on the subalternatio scientiae, by North (1992) on the quadrivium teaching, by Asztalos (1992) about the place of theology and by Hugues (1983) and Roest (2000: 140–146) on the Grey Friars’ interest in the sciences. They all admit, though with different focuses, a less neat division between the faculty of arts and the faculty of theology curricula. The picture should also be amended resulting from the contribution of science historians. As regards the role played by the faculty of arts, Guy Beaujouan writes: “At Paris, mathematics was hardly the subject of a regular teaching […] It seems that there was no actual chair of mathematics in Paris like in Bologna, Krakow, or Salamanca, for example” (1997: 193–94). From there, Guy Beaujouan concludes that sciences were “treats” (gourmandises) kept for festival days (1997: 240). Sed proculdubio nichil composui nisi quod aliqua capitula nunc de una scientia nunc de alia ad instanciam amicorum (Lindberg, 1971: 71–72). Scripsi dudum rogatus a sociis quedam mathematice ruditer rudimenta, que tamen aliis occupatus incorrecta reliqui, que etiam contra intentionem meam in publicum prodierunt, que idcirco intendo ut potero perfunctorie corrigere ut prosint iuuenibus studiosis. Igitur inter physice considerationis studia lux iocundius afficit meditantes. Inter magnalia mathematicorum certitudo demonstrationis extollit preclarius inuestigantes. Perspectiua igitur humanis traditionibus recte prefertur (Lindberg, 1970: 60).


Part I

Why Did Optics Not Lead to Perspective in Medieval Islam?


The Axiological Foundations of Perspective1

In the foreword to his Studies in the Sociology of Religion, Max Weber questions the conditions under which many cultural phenomena appeared as Western particularities, but nonetheless acquired universal value (Weber, 1922). He wonders in particular about the rise of linear perspective in the West. His general solution to the problem is based on the idea of a “rationalization process” that develops in specific societies, and gradually pervades all human activities according to the particular conditions men meet or create. Taking the perspective problem anew, it has been shown that the favourable conditions reputed to be specific to the Renaissance go back in fact to the Medieval optical literature that played a prominent role in the rationalization of sight (Raynaud, 1998). There are many clues to this contribution, such as the borrowings and explicit references that are to be found in the treatises by Alberti, Piero della Francesca, Ghiberti, or da Vinci, at the least. But, on the other hand, the growing number of critical editions of Arabic optical literature now enables us to see that the Latin optical treatises the Renaissance scholars relied on contain thorough commentaries on Arabic optical studies. As a result, we must recognize a second-order continuity between Renaissance and Medieval Arabic achievements, that pose new and specific questions. The problem I want to raise is the following: if Renaissance perspective proceeds from Medieval Latin optics, whose contents go back 37


to Arabic optics, we might then expect the first stages of perspective advancement to be found in Arab civilization. But there is nothing to confirm this deduction, because in all the areas where Arabic was spoken—including Persia, Central Asia and al-‘Andalu ¯ s—painting, miniatures and technical diagrams were drawn according to other types of representation: frontal and oblique projections, for the most part. So how to explain that first rank optical researches developed in Islam without any application to representation? Is it the consequence of a decisive improvement of optical knowledge after it was transplanted into Latin World? Is it due to the limited diffusion of optical treatises in the Orient? Is it a consequence of the well known Islamic aversion for images? The facts are unclear, and require a new approach. After having detailed the nature and soundness of optical knowledge tied to perspective questions (Part I), I will scrutinize three sets of reasons: sociability, religious beliefs, and axiological orientations of those who were the potential users of perspective in Islam (Part II). I. ARABIC THEORETICAL KNOWLEDGE ON PERSPECTIVE This section is devoted to the argument that all optical sources that were in fact combined to give birth to linear perspective were already available in Medieval Islam. Before establishing the point, I will say a few words about the four main texts that will be referred to in the course of the demonstration: (1) Euclid’s Optics was translated into Arabic by Hiliya¯ b. Sarju¯n—a scholar of Greek origin as his name shows: “Elias filius Sergii”. His Kita¯b Uqlı¯dis fı¯ ikhtila¯f al-mana¯zir (Book of Euclid on the Difference of ˙ Perspectives) dates from ca. 212 H. / 827–28 (Kheirandish, 1999); (2) Among other optical works, al-Kindı¯ wrote a Book on the Rectification of the Errors and Difficulties due to Euclid in his Book of Optics (Kita¯b Abı¯ Yu¯suf Ya‘qu¯b b. Isha¯q al-Kindı¯ ila¯ ba‘d ikwa¯nihi fı¯ taqwı¯m al-khata’ ˙ ˙ ˙ wa al-mushkila¯t allatı¯ li-Uqlı¯dis fi kita¯bihi al-mawsu¯m bi-al-mana¯zir). ˙ Composed before 866, this treatise is important for apprehending the shift between al-Kindı¯’s optics and the Euclidean tradition, to which he nevertheless pertains (Rashed, 1997); (3) Qust¯a b. Lu¯qa¯ was a friend ˙ and a collaborator of al-Kindı¯. He also undertook studies in optics, and wrote a treatise entitled The Book on the Causes of the Difference of Perspectives that appear in Mirrors (Kita¯b fı¯ ‘ilal ma¯ ya‘ridu fı¯ al-mara¯ya¯ ˙ min ikhtila¯f al-mana¯zir). This treatise moves in the direction of a geo˙ metrical-physiological synthesis. Being dedicated to al-Muwaffaq 38


(229–78 H. / 843–91), it is later than Al-Kindı¯’s works, possibly ca. 870 (Rashed, 1997); (4) Finally, no study can ignore Ibn al-Haytham’s Kita¯b al-mana¯zir. In his masterpiece of optics, Ibn al-Haytham champions ˙ intromissionism and sensorial image (su¯ra) against Euclidean visual ˙ ray theory and Ptolemy’s emissionism. Being included in the List III transmitted by Abı¯ Usaybi‘a in his Tabaqa¯t, the treatise was composed ˙ ˙ between 419–29 H. / 1028–38 (Sabra, 1989). The Perspective of the Circle Ibn Sarju ¯ n: “Wheels of chariots are seen at times as circular and at other times as distorted (*). Let then be a wheel, ABGD, and we draw in it two diameters, AB, GD, intersecting at right angles at point E; and let the eye be on a plane parallel to the plane of the circle. Then if the line drawn from the eye’s position to the circle’s centre is perpendicular to the circle’s surface, or else equal to half its diameter but not perpendicular, then the diameters are seen equal; therefore it is demonstrated that the wheel is seen as circular; but if the line drawn from the eye’s position to the circle’s centre is not perpendicular to the circle’s surface and not equal to half its diameter, then the diameters are seen as unequal; therefore it is demonstrated that the wheel is seen in this case as distorted (*). And that is what we wished to demonstrate” (Figure 2.1A).2 Al-Kindı¯: “The wheels of a chariot when moving (**) are seen at times as circular, at other times as non circular […] Demonstration: if the line drawn from the eye to point E, that is the centre of the wheel, is perpendicular in E to lines AB and GD, then all diameters belonging to it are seen equal, and the wheel AGBD necessarily appears as circular […] If the line drawn from the eye to point E is neither perpendicular to the circle’s plane AGBD, nor equal to its semi-diameter, then the diameters cutting the circle are seen as different. It is then clear, for that reason, that wheel AGBD will be seen as non circular. And this happens for all circle and all circular figures: if the line drawn from the eye to its centre falls either perpendicularly to the circle’s plane, or non perpendicularly but being the line equal to



f A




D 2.1A Figure 2.1: The perspective of the circle 39

D 2.1B

a c

b d

O P T I C S D I D N O T L E A D T O P E R S P E C T I V E I N M E D I E VA L I S L A M the semi-diameter of the circle, the circle is seen, for that reason, circular, and if this line is unequal, it will be seen deviated and non circular (***). And that is what we intended to demonstrate” (Figure 2.1A). 3

Commentaries. * The Arabic version of Euclid’s Optics deals here with the problem of how a circle is seen in perspective. According to the eye’s position, a wheel, as any other circular figure, will appear circular when the visual ray falling on the centre of the circle is perpendicular to the circle’s plane; elliptical otherwise. Ibn Sarju¯n and al-Kindı¯ use the words * mu‘awwaj “distorted” and *** manhir “deviated”, instead of ˙ nuqsa¯n “ellipse, lack”, to refer to the appearance of the circle. So they ˙ describe the ellipse by the angle formed by the visual ray falling on the centre of the circle and the line perpendicular to the circle’s plane, which presupposes an affine transformation K = fa / fb = gc / gd (Figure 2.1B). This geometrical property was known from Archimedes’ Conoids and Spheroids, prop. 4 (1970: 166–170) and Ibra¯hı¯m b. Sina¯n expounded it in his Maqa¯la fı¯ rasm al-qutu¯‘ al-thala¯tha (Epistle on the Drawing of ˙ the Three Sections) (Rashed and Bellosta, 2000: 263–89). ** Al-Kindı¯’s Rectification usefully complements the proposition, insisting on continuity in experiencing the various perspectives of a circular figure. The point is made clear, in considering the rectilinear movement of a chariot passing by the spectator. Renaissance scholars would gradually make clear that a circle is always seen as a conic section—generally an ellipse—and that it can be obtained by an affine projection (Vinci, Codex Atlanticus, fol. 115rb, ca. 1510). This fundamental law of perspective was already a source of geometrical interest before the 10th century. Intersecting the Visual Pyramid

Ibn Sarju ¯ n: “[Of] planes below the eye, the most distant one is seen at the highest. For let the eye’s position be point A, which is higher than plane BDEF, and let the ray[s] AB, AD, AE, AF, fall. Then I say that FE is seen higher than ED, and ED [is seen as] higher than DC. So let us cut line BD in half at point C, and draw GC at a right angle. Now because the visual ray falls upon GC (*) before falling upon CF, then ray AF falls upon line GC at point G, and ray AE [falls] at point H, and ray AD [falls] at point I; then all the rays fall upon line GC; but because point G is higher than point H, and point H is higher than point I, and the ray that has point G on it [also] has point F on it, and the ray that has point H on it [also] has point E on it, and the ray that has point I on it [also] has point D on it, and since FE is seen because of GH, and ED is seen because of HI, then FE is seen higher than ED, and in the same way, ED is also seen higher than DC. And that is what we wished to demonstrate” (Figure 2.2).4




F E D Figure 2.2: Intersecting the visual pyramid



Al-Kindı¯: “Among surfaces disposed below the eye in the same plane, the most distant from the eye are seen higher than the nearer from eye […] Demonstration: Draw a line from A to B, and lines AF, AE, AD, AC, that we suppose to be the [visual] rays. The size they enclose is what the ends of the ray encompass. The ends of the ray under which we see EF are the two lines AF and AE, the ends of the ray under which we see DE are the two lines AD and AC and the ends of the ray under which we see CD are the two lines AC and AB. Draw a perpendicular at point C towards point G on AF, and mark point H where line GC cut line AE, and mark point I where line GC cut line AD (*). Then point C is seen by line AC and point D is seen by line AID; from A we see point H higher than point I according to the size of line IH. But point E is seen from A by line AHE, therefore point E is seen more deviated upwards, depending on the size of line IH (**). But point F is seen at the same time that point G, therefore ray AGF is higher than H depending on the size of line GH; therefore point F is seen higher, i.e. more deviated in the direction of A than point H, depending on the size of line HG (**). Therefore it is seen higher than E by line HG. As a result F is seen more deviated upwards than E, and E more deviated than D, and D more than C, and C more than B, by the same way. Therefore most distant places and surfaces, having all their width on line BF, are seen higher, i.e. more deviated in the direction of A, than the points of line BF nearest to A. And that is what we intended to demonstrate” (Figure 2.2). 5

Commentaries. * Ibn Sarju ¯ n and al-Kindı¯ explicitly use the system known as the “intersection of the pyramid of visual rays” in which picture plane GC intersects visual rays AF, AE, AD, AC … This is nothing but the definition of linear perspective given by Alberti (1992: 103), Vinci (1970: 34), or Danti (2003: 119, def. i). **, *** Al-Kindı¯ goes beyond the Arabic version of the Optics for where Ibn Sarju ¯ n considers only the position of segments GH, HI, IC (higher or lower), al-Kindı¯ estimates the length of segments (“point H is seen higher … depending on the 41


size of line IH”). So al-Kindı¯ was perfectly aware that GH