Measuring Shadows: Kepler’s Optics of Invisibility 9780271077338

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MEASURING SHADOWS

M EASU R I N G SH AD O W S kepler’s optics of invisibility RAZ CHEN-MORRIS

The Pennsylvania State University Press  |  University Park, Pennsylvania

Library of Congress Cataloging-inPublication Data Chen-Morris, Raz, author. Measuring shadows : Kepler’s optics of invisibility / Raz Chen-Morris. pages cm Summary: “Focusing on the astronomer Johannes Kepler’s 1604 treatise on optics, explores Kepler’s radical break from scientific and epistemological traditions and shows how he posited new ways to view scientific truth and knowledge in the early modern period”—Provided by publisher. Includes bibliographical references and index. isbn 978-0-271-07098-8 (cloth : alk. paper) 1. Kepler, Johannes, 1571–1630. Ad Vitellionem paralipomena quibus astronomiae pars optica traditur. 2. Geometrical optics—History. 3. Science, Renaissance. 4. Science—Philosophy—History—17th century. I. Title. QC380.K4633 2016 535’.32—dc23 2015030525 Copyright © 2016 The Pennsylvania State University All rights reserved Printed in the United States of America Published by The Pennsylvania State University Press, University Park, PA 16802-1003 The Pennsylvania State University Press is a member of the Association of American University Presses. It is the policy of The Pennsylvania State University Press to use acid-free paper. Publications on uncoated stock satisfy the minimum requirements of American National Standard for Information Sciences— Permanence of Paper for Printed Library Material, ansi z39.48–1992. This book is printed on paper that contains 30% post-consumer waste.

TO JOANNA

He brushed away the thunder, then the clouds,

Then the colossal illusion of heaven. Yet still

The sky was blue. He wanted imperceptible air.

He wanted to see. He wanted the eye to see

And not be touched by blue. He wanted to know.

—WALLACE STEVENS, “LANDSCAPE WITH BOAT”

CONTENTS

viii List of Illustrations ix Acknowledgments

1 introduction 24 one. The New Optical Narrative: Light, Camera Obscura, and the Astronomer’s Wings 48 two. “Seeing with My Own Eyes”: Introducing the New Foundations of Scientific Knowledge 63 three. The Content of Kepler’s Visual Language: Abstraction, Representation, and Recognition 97 four.“Non tanquam Pictor, sed tanquam Mathematicus”: Kepler’s Pictures and the Art of Painting 123 five. Reading the Book of Nature: Allegories, Emblems, and Geometrical Diagrams 159 six. Nothing and the Ends of Renaissance Science 186 postscript

191 Notes 216 Bibliography 235 Index

ILLUSTRATIONS



18 1. Kepler’s musing on the falling Icarus



26 2. Frontispiece of Kepler’s Ad Vitellionem



65 3. Kepler’s system of conics



68 4. Kepler’s geometric scheme of reflection in a plane mirror



124 5. Giuseppe Arcimboldo, The Librarian



129 6. Giuseppe Arcimboldo, Vertumnus (Emperor Rudolf II)

132 7. Alchemist’s Laboratory, in Heinrich Khunrath, Amphitheatrum sapientiae aeternae

132 8. The gate to the amphitheater of eternal wisdom, in Khunrath, Amphitheatrum sapientiae aeternae



134 9. Emblem viii, in Michael Maier, Atalanta fugiens



135 10. Emblem xxi, in Maier, Atalanta fugiens



142 11. Robert Fludd’s weather glass



153 12. Kepler’s copulating squares

ACKNOWLEDGMENTS

Through the long process of researching and writing this book it was the love and support I received from my lovely wife Joanna that kept me going. Her intellectual input as a reader and commentator was essential in bringing this book to completion. I thank my children, Jasmine (and her husband, Itzik Sade), Emily, and Daniel, who accepted Kepler into our household and who suffered my many moments of absent-mindedness while pondering some of the grave historiographical questions involved in this research. I have a special intellectual debt to Sabetai Unguru and Rivka Feldhay. While I was a student at the Cohn Institute for the History and Philosophy of Science and ideas at Tel Aviv University, they first introduced me to the intricacies of the history of the mathematical sciences. During those early years I had the privilege of studying with Amos Funkenstein, Yehuda Elkana, Michael Heyd, and David C. Lindberg, who have all since passed away. My overall historiographical approach has been molded by an inner dialogue with their​ lingering influence. I began working on this book during my stay as a research fellow at the Unit for the History and Philosophy of Science at the University of Sydney in 2006–2009 as part of an Australian Research Council–supported project, The Imperfection of the Universe (DP0664046). Further research was done during my stay as a Rosenblum short-term fellow at the Folger Shakespeare Library in Washington, D.C., during the summer of 2009. I am deeply grateful to the library staff, especially to Carol Brobeck, for all their assistance. I would also like to thank Gail Kern Paster and David Schalkwyk for their support and interest in my project. A much shorter version of chapter 4 was published as “Optics, Imagination, and the Construction of Scientific Observation in Kepler’s New Science,” Monist 84 (2001): 453–86, published by Oxford University Press; versions of certain sections of chapters 4, 6, and 7 were published in “Shadows of Instruction:

Optics and Classical Authorities in Kepler’s Somnium,” Journal for the History of Ideas 66 (2005): 223–43, published by the University of Pennsylvania Press, and “From Emblems to Diagrams: Kepler’s New Pictorial Language of Scientific Representation,” Renaissance Quarterly 62, no. 1 (2009): 134–70, © 2009 University of Chicago Press. I did much of the fine-tuning for this book while participating in the “Before Copernicus” research group directed by Rivka Feldhay and Jamil Ragep at the Max Planck Institute for the History of Science in Berlin. The lively scholarly exchanges that took place were crucial in refining some of my bolder assertions, and I thank the prominent scholars who constituted this group: Nancy Bisaha, Christopher S. Celenza, Ihsan Fazlioglu, Maria Mavroudi, Robert Morrison, Sally P. Ragep, Michael Shank, and Edith Sylla. It was a pleasure during these years to be part of the faculty of the Science, Technology, and Society Graduate Program at Bar Ilan University. I owe so much to my friends Noah Ephron, Oren Harman, Nurit Kirsh, and Anat Leibler for all their encouragement. I am indebted to them and to the students of the program for providing me with a stimulating intellectual environment. Today I am honored to be part of the History Department at the Hebrew University of Jerusalem and am proud to present this book as my debut contribution to its intellectual reputation. It was fortunate that, throughout the years I spent working on this book, there were many colleagues who were willing to comment and read drafts of my research. Special thanks to Anthony Grafton for his exceptional generosity and inspiring suggestions. Lorraine Daston, Sven Dupré, J. V. Field, Paula Findlen, Menachem Fish, Stephen Gaukroger, Giora Hon, Antoni Malet, Sheila Rabin, John Schuster, Bruce Smith, and A. Mark Smith all read and commented on different sections of this book on sundry occasions, and I am grateful for their input, though the final product is solely my responsibility. While preparing the final draft of my manuscript, I spent the winter of 2014 as a short-term fellow at the Folger Shakespeare Library (again) and had the opportunity to discuss some of these issues with Ronda Arab, Kenneth Gouwens, Gerard Kilroy, Pamela Long, Julie Park, Goran Proot, and Michael Witmore. Many of the good friends I have made throughout my intellectual career share with me an interest in early modern culture and science, and I have certainly benefited from conversing with them on many of the topics treated in this book: Gadi Algazi, Patrick Boner, Miriam Eliav-Feldon, Ayelet Even-Ezra, Snait Gissis, Yakir Levine, Yaakov Mascetti, Lyle Massey, Ohad Nachtomy, Jane

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ACKNOWLEDGMENTS

Newman, Andreas Niederberger, Gerard Passanante, Oded Rabinovitz, Eileen Reeves, Elchanan Reiner, Rami Reiner, Yossef Schwartz, Zur Shalev, J. B. Shank, Noa Shein, Moshe Sluhovsky, John Sutton, Dorit Tanay, Dror Wahrman, Daniel Weidner, Hanan Yoran, and Gur Zak. I would like to acknowledge Ofer Gal, whose companionship with me has survived great distances over the years. The many discussions we shared concerning early modern science and philosophy resonate through the pages of this book. I thank the two anonymous referees for their enlightening comments and suggestions; my editor, Eleanor Goodman, for believing in this book and seeing it through to publication, and Charlee Redman, John Morris, and the other staff members at Penn State Press for all their assistance. I know that my father, Yeshayahu Chen, and my mother-in-law, Diana Morris, would have been so proud of this book. I take some consolation in the fact that I am able to present it to my mother, Esther, my sister Edit, and my father-in-law, Gerald, and I thank them for their love and support. My thoughts took shape while walking in the forest surrounding my village, nestled on the slopes of the Judean Hills. My final thank you goes to the forest and to my lively companions—Oscar, Dustin, Pluto, Angie, and Pudding.

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INTRODUCTION

toward copernican optics In January 1604 Johannes Kepler, the young imperial mathematician at the court of Rudolf II, dedicated a treatise on optics to the emperor.1 The treatise presents some novel ideas on the formation of images on the retina, and an elegant geometrical explanation of the operation of the camera obscura, together with a manual of measurements of astronomical observations.2 Throughout these topics a startling notion is threaded: that in order to account for real physical motions, one has to investigate artificially produced shadows and reflections. This book sets out to explore this paradoxical notion, its epistemological function in shaping early modern practices of observation, and especially the role it played in Kepler’s own scientific endeavor.3 Recent accounts of Copernicus and of Copernican astronomy have concentrated on the way astronomy changed its disciplinary character and its social role following the publication of De revolutionibus orbium coelestium in 1543 or on Copernicus’s philosophical underpinnings and indebtedness to Aristotelian logic and principles of natural philosophy. These accounts, while widening the scope of scholarly understanding of the evolution of Copernicanism, fail to pay due attention to the changing notions of visibility crucial to the new astronomy.4 One of the core aspects of Copernican astronomy is the replacement of daily sense-experience with the invisible motion of the earth, thus rejecting perception as the basis of human knowledge. This book concentrates on the way in which Kepler tackled this question of visibility, setting his paradoxical optics as the cornerstone to his

radical Copernican astronomy. The new optics enables him to get “beyond the threshold of perception,” thus merging physical considerations with the vast invisible space of his new astronomy.5 It was less than three years after Kepler arrived in Prague to assist the great exiled Danish astronomer Tycho Brahe, then an imperial mathematician. Both Kepler, the young provincial mathematics teacher, and Tycho, prince of astronomers, were refugees from the crises of late sixteenth-century European culture and politics: the first a victim of the Counter-Reformation and the growing religious hostilities in the lands of the Holy Roman Empire; the other exiled due to courtly intrigues symptomatic of the burgeoning early modern centralized state. The two advocated divergent solutions to the problems that haunted Renaissance astronomy in establishing its practices and theories on firmer philosophical foundations, motivated by a desire for complete and pure transparency. Overcoming the opacity of the celestial realm, which hid its secret causes from the frustrated gaze of human observers, Tycho devoted his efforts to improving astronomical instruments, practices of observation, and methods of calculating the results, aspiring to present his European audience with the most exact astronomical data possible. Kepler, on the other hand, set out early in his intellectual career to construe, using mathematical speculations, an extraterrestrial point of view from which the celestial mechanism could be revealed. Their meeting in Prague, the glorious capital of Mannerist culture under the irenic rule of Emperor Rudolf II, could not have been more emblematic. Prague was the nerve center of various strands of late Renaissance philosophy and a lodestone attracting philosophers, spiritualists, alchemists, and mathematicians, most notably (besides Kepler and Tycho) John Dee and Giordano Bruno. Prague was also the appropriate geographical and cultural location for an intellectual shift from Brahe’s Renaissance-style astronomy to Kepler’s new mathematical mode of practicing a philosophical inquiry into the celestial realm. Kepler’s assignment was to help Brahe and his few assistants in completing his astronomical tables. The idea was to supply Europe with the most complete astronomical data ever, which would replace older tables and compendiums of observations. Under Tycho, Kepler worked on a theory for the planet Mars, a task that had eluded Tycho’s other assistants. For Kepler, it was also an exceptional opportunity to use Tycho’s most accurate astronomical data to corroborate his own speculation on the structure of the heavens, which he published in 1596 as Mysterium cosmographicum.6 In this treatise Kepler claimed that the gaps between the planets exactly fit a structure composed of the five Platonic

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MEASURING SHADOWS Introduction

solids nested one within the other. This was intended to prove Copernicus’s heliocentric theory not only astronomically but also physically. Some discrepancies, however, existed between Kepler’s speculations and his numerical data. For this reason, Tycho’s treasure of observations was attractive to him, and he hoped to find the data needed to establish that God’s harmonic plan of the universe is Copernican.7 Kepler’s turbulent years from 1598 to 1601 and his pugnacious relationship with Tycho Brahe are well documented, yet Kepler did not sway from his aim of establishing astronomy on physical grounds. He stayed with Tycho, investing his time in complicated calculations of Mars’s orbit. In 1601 Tycho died suddenly, and Kepler found himself the new imperial mathematician. His job included responsibility for Tycho’s unfinished publications, in particular the completion of astronomical tables now officially dedicated to the emperor and proclaiming his name. Kepler had to tackle some of the basic problems of early modern astronomy, and at the beginning of 1604 was on the verge of major discoveries that would radically change the nature of astronomical knowledge.8 Yet, instead of rapidly declaring his achievements and showing his patron his progress in accomplishing a theoretical basis for the new astronomical tables, Kepler preferred to dedicate to Rudolf II a book on optics as his first major treatise. This was a strategic move on his part. In a letter to Herwart von Hohenburg on November 12, 1602, Kepler outlined his plan of forthcoming publications: first Optics, and only then his “Commentaries on the Theory of Mars . . . or the Key to Universal Astronomy.”9 Kepler, then, kept at bay his difficult research into a comprehensive theory of the planetary orbits, supposed to supply Europe with a “key to universal astronomy.” Instead he embarked on a time-consuming work, according special urgency to the publication of his optical treatise. While some historians of science assume that Kepler’s scientific curiosity is sufficient to explain such a diversion in his work,10 most of the historiography of early modern optics concentrates on whether Kepler’s optical investigations truly constituted a revolutionary break with the medieval tradition.11 However, only the short second chapter of the work is dedicated to this problem. The reader may ask why a whole treatise was written around what seems to occupy only a small part of his optical theory. Furthermore, is the understanding of the camera obscura’s operation on par with the first two laws of planetary motion? The title of the optical treatise increases the reader’s puzzlement— Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur (Paralipomena to Witelo, which is the optical part of astronomy). It states at the

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outset the question of tradition and innovation, and Kepler’s relationship to his medieval perspectivist precursors. The name of Witelo, a medieval philosopher and mathematician, appears here as a metonym of an entire discipline, and as that of the writer whose work embodies the great synthesis of medieval perspective. Witelo’s treatise on perspective from 1270 was a thorough interpretation and adaptation of the work by the eleventh-century Arab mathematician Ibn al-Haytham (Alhacen). The two works were published together in 1570 by a follower of Petrus Ramus, the mathematician Fredericus Risner.12 Kepler rightly took them as representatives of the apogee of medieval geometrical optics and acknowledges his debt to this tradition. This indebtedness is turned awry by the adjunct tag paralipomena. Kepler does not proclaim it to be a “new optics,” since Witelo’s name appears as the framework for his own ideas. Paralipomena, however, is no mere supplement, as it is so often translated, but stresses the act of omission, dealing with those things that Witelo disregarded, left unnoticed to the side. From the outset, Kepler portrays the perspectivist tradition as defective, incomplete, and in need of a thorough reform, insinuating that those things neglected and refused by the perspectivist tradition should become the cornerstone of the new mode of artificial observation.

a historiographical remark The prevailing historiographical attitude toward Kepler’s optical investigations defines their import by placing them in a progressive historical narrative. In an endeavor to define Kepler’s positive achievement, historians seek to determine whether his optics is a revolutionary gambit leading to the great scientific triumphs of Descartes, Newton, and Huygens, or whether his optical theories are the culmination of medieval perspectiva tradition. This attitude concentrates on immediate problem solving, such as Kepler’s analysis of the process of image formation through a pinhole, and thus on the physical aspects of optics. On one side of this historiographical debate one finds Alistair Crombie and Richard Straker, who underline how Kepler, in following Renaissance “rational artists,” formulated a new model of the eye as a camera obscura and thus initiated the mechanization of sight.13 On the other side, David C. Lindberg points out the continuity of the geometrical procedures Kepler applied with the medieval traditions of perspectiva communis, and the marginal role the identification of the eye with the camera obscura played in Kepler’s general scheme.14 Lindberg further stresses that portraying Kepler as a problem solver overlooks the great

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MEASURING SHADOWS Introduction

mathematician’s indebtedness to Neoplatonic speculations on the metaphysics of light and that this consideration is of foremost importance to an understanding of his optics.15 In suggesting a Neoplatonic frame for Kepler’s optics, Lindberg reveals the problems involved in philosophically characterizing Kepler’s work in general. Lindberg is in accord with the prevailing view, which stresses Kepler’s Neoplatonic and somewhat mystical disposition.16 J. V. Field, however, rejects this interpretation, claiming instead that Kepler was a “radical Platonist” who renounced Renaissance mathematical mysticism as irrelevant to his scientific endeavor.17 Contrariwise, William Donahue, among others, emphasizes Kepler’s indebtedness to the Aristotelian tradition in establishing the logical and philosophical foundations of both his optics and his astronomy.18 As noted above, others see Kepler as a forerunner of the mechanistic worldview of the seventeenth century.19 This interpretation is poignantly criticized by Patrick Boner, who stresses instead Kepler’s conception of the universe as an organic living body and not a world operating like clockwork.20 For my part, I embrace Sheila Rabin’s observation that in employing a variety of philosophical views and concepts in his investigations, Kepler tends to turn them on their head (a most suitable idiom for Kepler’s inverted retinal pictures).21 In order to understand Kepler’s project, one has to follow the way he mobilizes certain intellectual resources, thus seeking novel modes of knowing that, while providing groundbreaking solutions, reframe the old questions he pursues in a manner irreducible to any preconceived philosophical system or worldview. This approach is in accord with developments in the historiography of medieval optics that embed the scholastic discourse of optics within a more general theory of human cognition relating to certain epistemological problems that vexed theologians and philosophers through the thirteenth and fourteenth centuries.22 While in medieval studies this attitude embedded the geometrical discussions of optics in a rich tapestry of theological and philosophical debates,23 the discussion of Kepler’s optics remained chiefly concerned with its historical significance in the emergence of the New Science of the seventeenth century. Mark Smith synthesizes Lindberg’s magisterial account of the development of medieval theories of vision with insights taken from Alistair Crombie’s comparative analysis of Alhacen’s and Kepler’s models of sight, and suggests that the import of Kepler’s optics was in severing the bond assumed by medieval perspectiva between “the perceiving subject and perceived object.”24 Kepler thus followed the medieval tradition in his application of geometry to solve and explain optical problems and issues, but his conclusions diverged from the medieval path to open a new understanding of visual experience.

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Accepting this assertion raises a crucial difficulty: if the geometrical techniques at Kepler’s disposal were not different from those of his predecessors, and if his metaphysical convictions descended from medieval light metaphysics, then what motivated Kepler to initiate such a bold move that turned medieval perspectiva upside down? In tackling such a historiographical difficulty, this book places Kepler in a wider context, examining his work on optics both in relation to his own intellectual endeavor to establish astronomy on a more secure epistemological foundation and in relation to general cultural concerns at the turn of the seventeenth century. It thus provides an additional comment on the place of sight in early modern culture. One should mention in this context a few historians who have disputed the role vision played in early modern culture. In The Art of Describing, Svetlana Alpers locates a common ground for Keplerian analysis of the formation of pictures over the retinal screen with a northern European (or Netherlandish) mode of pictorial representation as an art of describing.25 According to her view, Kepler’s theory of optics takes its cue from the concerns of northern European artists, thus giving Dutch aesthetic sensibilities scientific form: [I]t was established concerns of artists in the north that were in effect taken up by Kepler. He turned his attention not only to the camera obscura, but also to lenses, mirrors, and even to glass urinary bottles filled with clear liquid, all of which served him as models of refracted light. Contemplating these models, we cannot help but be struck that these are the phenomena that had traditionally fascinated northern artists. . . . We have a case of traditional crafts and skills sustaining or keeping alive certain interests that eventually become the subject of natural knowledge. Northern art came of age, came into a new age, by staying close to its roots. In Kepler’s study of the eye, natural knowledge caught up with the art.26 Jonathan Crary, instead of focusing on the convergence of optics and painterly style, suggests analyzing the role of the camera obscura as an organizing metaphor for early modern visual experience.27 Both of these suggestive interpretations, however, assume the camera obscura was simply given and not a matter of epistemological contentions and concerns, thus neglecting a certain cultural dynamic that guaranteed the camera obscura its significant role in shaping early modern modes of perception. More recently, Stuart Clark, in his authoritative Vanities of the Eye, criticizes the prevailing assumption that

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MEASURING SHADOWS Introduction

visual experience, and especially visual metaphors of knowledge, hold hegemonic position in Western culture, and that their assent to this position began in the seventeenth century.28 Instead Clark presents the early modern field of vision as haunted by epistemological anxieties and doubts. His portrayal of the early modern apprehension of sight is static, however, as if a component of an early modern mentalité. Furthermore, while he uses and analyzes a variety of theological, literary, and medical sources in his vast historical synthesis, he relegates geometrical optics to the background, taking for granted the explanation of vision prevalent in that period. In concentrating on Kepler’s optical investigations, this book suggests a more dynamic approach where various practices and theories are formulated in order to secure the epistemological foundation of observation and knowledge through rethinking the relationship between mathematical procedures, instruments of observation, and the human mind. In following Kepler’s struggle with the insecurity of visual knowledge, one can come to understand the role of optics in his philosophical and astronomical endeavor, and its bearing on the early phases of the New Science of the seventeenth century.

kepler and the question of reading the optical tradition Stepping beyond this old and deficient science of optics and adjusting it to a new physics of the heavens, Kepler breaks away from the medieval and Renaissance modes of reading the ancient authorities. Embracing a critical stand toward his traditional sources situated Kepler in the midst of an emerging new conception of reading that treated the text not “as a universal and permanent memory,” but as “a heuristic exercise, proffering topoi that the reader may appropriate and reorganize in relation to an unforeseen mental horizon.”29 What effects did these changes in the modes of reading, in the status of the auctoritas and the figure of the author, have on scientific discourse? To the medieval practices of commentary and interpretation Renaissance humanists added the critical editing, emending, and reconstructing of ancient scientific texts.30 In accordance with Ciceronian ideals, Renaissance humanists hailed the imitation of ancient texts, ascribing intellectual and stylistic authority to these ancient textual patterns and models.31 Kepler’s position in regard to the Hellenistic and medieval traditions was different. The classical sources were no longer the origins to be restored into their correct form, nor were they the authoritative suppliers of textual patterns. They were to be historicized as steps

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in a long tradition, but with no exceptional claim to truth.32 Kepler had to introduce, instead, new grounds for his own claims to truth. This was why he had to formulate anew the principles of optics as the obvious foundation of astronomy. The classical texts thus become mere triggers, offering Kepler an opportunity to initiate his own analysis. As the new discourse on optics unfolds, it leaves the classical authorities behind, presenting a new mode of observation with new entities to see and a new language to assess them. How, then, did Kepler reread his precursors? Which new entities were neglected by the medieval perspectivist tradition that Kepler now suggested positing as the foundation of his new edifice of scientific observation? Placing Witelo in the title of his treatise shows Kepler’s historical awareness, and his appreciation of the great advancement made in optics during the Middle Ages. It also points, however, to the stability of this tradition. The thirteenth-century synthesis was still at the core of the late Renaissance understanding of vision and other optical phenomena. This synthesis consisted of a geometric analysis of visual phenomena together with a physical explanation of sight. Medieval optics thus supplied a coherent and attractive theory of human visual experience.33 In the words of John Pecham, one of its foremost exponents, “[A]mong the investigations of physics, light is most pleasing to students of the subject. Among the glories of mathematics it is the certitude of demonstration that most highly exalts the investigators. Therefore, perspective, in which demonstrations are devised through the use of radiant lines and in which glory is found physically as well as mathematically so that perspective is adorned by the flowers of both, is properly preferred to [all the traditional] teachings of mankind.”34 The main physical principle of medieval optics was that there is no action at a distance; for an object to be seen, it somehow has to come in contact with the eye. For that to take place, a stream of immaterial species must flow from every point in the visual field into the eye. These species communicate the essence as well as the accidents of the visible object and imprint them on the crystalline humor. This physical description was corroborated by a geometric analysis. The species flow from each point to the eye in straight lines analogous to rays of light. Only those rays that hit the eye at a straight angle participate in the act of sight. All the other rays are either reflected or refracted. They are thus weakened and cannot communicate their information to the eye.35 Preference was thus given to what was directly evident to the eyes. Only those objects that were clear and present in front of the eyes could testify to their visual truth. Whatever was perceived obliquely, indirectly through whatever mediation, or

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MEASURING SHADOWS Introduction

was too far away for the sense of sight to discern clearly was degraded, given secondary epistemological value. Such perception could not serve as a basis for a sound philosophical account of the natural world. The emphasis on direct visual perception was in accord with the place assumed by the eye in the Aristotelian tradition. The eye, according to Aristotle’s De anima, is precisely suited to perform its task of visual discernment, and whatever falls out of its field of sight cannot be perceived in its reality. This state of affairs did not change through the fifteenth and sixteenth centuries, although some scientific and philosophical problems continued to haunt the optical tradition (the formation of images through a pinhole, the ontological status of the visual rays, and whether the eye is an active or a passive organ of perception). In contrast to astronomy, optics seemed to have a solid and coherent theoretical exposition of the act of sight. Rather than an abundance of competing hypothetical, fictitious constructions, there was an accepted geometrical description of the motion of visual rays with successes such as the law of reflection, the manipulation of burning mirrors, and an ability to explain away errors of vision and illusionary images. Treatises on optics (in contrast to artificial perspective) all through the fifteenth and sixteenth centuries did not win much public attention.36 Problems such as the formation of pinhole images were not considered disturbing, as they may appear to a modern reader of histories of optics. Maurolyco’s solution to the problem, though written in the 1520s, remained unpublished until after Galileo emerged with his telescope in 1610, when Clavius understood its importance and saw to its publication.37 Kepler, however, thought that the optical tradition urgently needed a supplement. He even neglected his work on astronomy and diverted his efforts to rewriting medieval perspective, an established, and relatively stable, scientific system. A closer examination of the nature of the supplement to Witelo suggested by Kepler will disclose that its aim was not the fortification of traditional optical theories, but a radical inversion of the perspectivist system.

supplementing witelo The second part of the title—astronomiae pars optica—declares that the supplement to Witelo and the medieval tradition of perspective is the optical part of astronomy. This immediately raises a question: adding optics to astronomy is a supplement to the great arch-astronomers, such as Ptolemy or Copernicus, not

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to Witelo. Why did Kepler think that the optical part of astronomy signified a change in optics? Further examination of this phrase against the backdrop of classical and medieval philosophy detects a surprising oxymoron:38 can there be optics, that is, an account of visual experience, of the heavens? Since classical theories of visual perception emphasized direct contact with the perceived object, the remoteness of the stars leaves the observer with only indirect and uncertain appearances. Furthermore, one of the basic propositions of classical optics asserted that the distance at which an object is seen is inversely proportionate to the size of the angle of vision, and that beyond a given distance it is no longer visible on account of the decrease in the size of the angle.39 The heavenly visual signs are only hints of a reality one cannot perceive directly. The observer has to go beyond appearances and cannot trust the eyes as to the actual meaning of the given visual experience. Plato asserted such a separation most clearly. The heavenly phenomena are not given to human vision. Gazing at the stars does not produce knowledge but only void sense perception. True knowledge “deals with being and the invisible,” and thus the senses cannot provide it. Heavenly phenomena serve only as hints for the philosopher to contemplate the essence of motion according to number.40 The Platonic astronomical research program was thus a priori speculation based on geometry but without the need for empirical support. The aim of Platonic speculation was to go beyond visual appearances, and, by formulating a probable tale, to elevate the soul to contemplate the moral significance of the world.41 Denouncing visual experience altogether in favor of a supreme hidden truth meant that there was no need for optics to account for such experiences. This emphasis on an epistemological barrier between what is immediately given and can be fully known and what is distant and beyond human perception is succinctly formulated in Aristotle’s De partibus animalium. Aristotle sets the power of human vision as the limits of human knowledge. Certain things, he says, are too distant and alien for humans to know. “Of things constituted by nature some are ungenerated, imperishable, and eternal, while others are subject to generation and decay. The former are excellent beyond compare and divine, but less accessible to knowledge. The evidence that might throw light on them, and on the problems which we long to solve respecting them, is furnished but scantily by sensation.” This is in contrast to animals and plants, “living as we do in their midst,” for which ample data are available. An example of such distant and eternal entities is celestial bodies. Of these, although they give so much pleasure in their excellence, one has only “scanty conceptions.”

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MEASURING SHADOWS Introduction

Still, humans are excited and curious about them, “just as a half glimpse of persons that we love is more delightful than a leisurely view of other things, whatever their number and dimensions.” Terrestrial things, because of their affinity and nearness, are known with certitude and “in their completeness.” Human knowledge of celestial things is lofty; they are beyond one’s grasp and are known only as far as “our conjectures [can] reach.”42 The outcome of this classical epistemological position is a divorce between observation and the actuality of the heavens. Yet Aristotle’s critique of astronomical observation focuses on those a priori geometrical elements that Plato and the mathematicians cherished so much. Aristotle asserts that “[t]he minute accuracy of mathematics is not to be demanded in all cases, but only in the case of things which have no matter. Therefore its method is not that of natural science; for presumably all nature has matter.”43 Aristotle further claims that this assertion concerning the limits on the applicability of mathematics concerns not only “what nature is,” but the way humans experience and perceive natural events. The end result of these Aristotelian notions was an utter rejection of the aspiration to mathematically based observation: And at the same time not even this is true, that mensuration deals with perceptible and perishable magnitudes; for then it would have perished, when they perished. And astronomy also cannot be dealing with perceptible magnitudes nor with this heaven above us. For neither are perceptible lines such lines as the geometer speaks of (for no perceptible thing is straight or curved in this way . . . ), nor are the movements and complex orbits in the heavens like those which astronomy treats, nor have geometrical points the same nature as the actual stars.44 This Aristotelian dichotomy between mathematics as the realm of pure intelligibility and perception pulled the rug out from under the epistemological claims of astronomy. As a discipline based on a combined contribution of observers of heavenly phenomena and mathematicians’ explanations of such occurrences, astronomy could not come to terms with these Aristotelian assertions. In practice this meant a separation between observation with a dubious status, mathematical descriptions as fictive constructions, and physical explanations based on metaphysical principles with very little attention to empirical verification. Scant empirical data (for instance, the apparent circular motions of the planets) were used as mere starting points for such speculations; the mathematical construction was only a theoretical tool for predictions of planetary positions. Yet

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none of them was expected to support a physical worldview of the heavens, nor was the physical speculation expected to support the mathematical theory.45

curiosity and the threshold of perception The epistemological reservations concerning the possibility and value of formulating a physical theory of the heavens founded on observation and mathematical calculation had an intellectual affinity with religious prohibitions and warnings. In the Jewish tradition, Ecclesiastes celebrates direct experience at the expense of inquiring into and speculating on the nature of things not given to human perception. The author of Ecclesiastes emphatically declares, “Better [is] the sight of the eyes than the going of the soul” (Eccl. 6:9). The biblical author denies the human soul’s desires for what is not directly given. This point was emphasized in ancient Christian commentaries; Saint John Chrysostom, for instance, wrote, “Better to enjoy with the eyes . . . than things that are stored away which the eye does not see.”46 The moral dimension, then, aiming to quiet desires and passions, has immediate epistemological implications. One should embrace appearances that are immediately present before one’s eyes and not inquire into and speculate about hidden meanings and causes. Later generations emphasized this epistemological warning against human desire for knowledge and vain curiosity. In the apocryphal book of Sirach, one is warned, Seek not (to understand) what is too wonderful for thee, And search not out that which is hid from thee. Meditate upon that which thou must grasp, And be not occupied with that which is hid. Have naught to do with that which is beyond thee, For many are the conceits of the sons of men, And evil imaginations lead astray.47 A succinct formulation of this prohibition on inquiring what is beyond human direct perception is found in the Mishnah in a passage dealing with mystical experience and esoteric knowledge: “Whosoever gives his mind to four things it were better for him if he had not come into the world—What is above? What is beneath? What was before time? and what will be hereafter?”48 A similar and parallel negative attitude concerning curiosity is found in Hellenistic literature at the beginning of the Christian era, from Plutarch to

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MEASURING SHADOWS Introduction

Apuleius.49 Lucian, for instance, in several treatises mocks the intellectual vanity of philosophers, historians, and scientists. His main target is what he takes to be their ludicrous claim to know. In his treatise on Icaromenippus, Lucian expresses his disappointment with the different astronomical schools and their vain claims about the structure and nature of heavens. He aims his satirical arrows at their boastful theories about things they cannot assess directly with their eyes: You will laugh, my friend, when I shall tell you of their pride and impudence in the relation of extraordinary events; to think that men, who creep upon this earth, and are not a whit wiser, or can see farther than ourselves, some of them old, blind, and lazy, should pretend to know the limits and extent of heaven, measure the sun’s circuit, and walk above the moon; that they should tell us the size and form of the stars, as if they were just come down from them; that those who scarcely know how many furlongs it is from Athens to Megara, should inform you exactly how many cubits distance the sun is from the moon, should mark out the height of the air, and the depth of the sea, describe circles, from squares upon triangles, make spheres, and determine the length and breadth of heaven itself: is it not to the last degree impudent and audacious? Lucian criticizes any sort of astronomical knowledge, be it mathematical calculations and measurements, or speculations concerning the physics of the heavens. “When they talk of things thus obscure and unintelligible, not merely to offer their opinions as conjectures, but boldly to urge and insist upon them: to do everything but swear, that the sun is a mass of liquid fire, that the moon is inhabited, that the stars drink water, and that the sun draws up the moisture from the sea, as with a well-rope, and distributes his draught over the whole creation?”50 The insistence on direct perception as the only solid foundation of knowledge persisted throughout the Middle Ages. The dichotomy between direct “facie ad faciem” perception and mediated perception “per speculum” was an organizing principle in theological writings, in mystical experiences, and in philosophical treatises.51 Such direct experience, however, was beyond the reach of human senses as well as beyond human reason. The fourteenthcentury scholastic philosopher and mathematician Nicole Oresme asserts that exact knowledge of the heavens is impossible for a human observer, as the motions of the heavens are “innumerable for men” and “because of

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defects of the senses.” Oresme concludes that “all things lie hidden behind him who numbers the multitude of the stars, and who governs the world by reason everlasting. No one, therefore, should presume to judge so facilely about the cause of uncertain things.”52 One is urged to pursue the study of astronomy so to “repudiate errors scientifically by solid demonstrations and not by empty babbling as do those who are ignorant.” Furthermore, through the observation of “visible things the perfect works of God might magnify the invisible creator.” Oresme ends his treatise with a clear specification of the role of the “good astronomer” and the kind of knowledge such an astronomer may legitimately aspire to: “Thus it is sufficient for a good astronomer to judge motions and aspects near a point, and that his senses do not observe and judge the opposite.” Whatever is beyond these limited and approximated mathematical measurements, Oresme admonishes, is vain “and impairs the spirit and is foolishly presumptuous. . . . Except in a very general and doubtful way, no one ought to speak but rather restrain the tongue about things which are in the hands of God, for only he knows,” to whose eyes “all things are naked and open.”53 Renaissance humanism revived classical satirical sentiments. Erasmus echoes Lucian’s sarcasm and mocks the epistemological presumptions of the astronomers: Theirs is certainly a pleasant form of madness, which sets them building countless universes and measuring the sun, moon, stars and planets by rule of thumb or a bit of string, and producing reasons for thunderbolts, winds, eclipses and other inexplicable phenomena. They never pause for a moment, as if they’d access to the secrets of Nature, architect of the universe, or had come to us straight from the council of the gods. Meanwhile Nature has a fine laugh at them and their conjectures, for their total lack of certainty is obvious enough from the endless contention amongst themselves on every single point.54 Erasmus’s epistemological mocking of astronomical presumptions resonated well with sixteenth-century understanding of Ecclesiastes and its confinement of human curiosity. In Luther’s commentary on this book of the Old Testament, the point is emphasized over and over again. Fallen humanity can know and handle only those things that are sub sole. Humanity must be satisfied with the ability to exercise its power over those things placed under human dominion in Genesis 3. Any attempt to go beyond humanity’s daily tasks and vocation is

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MEASURING SHADOWS Introduction

vain, resulting in despair and doubt: “Thus the subject matter of this book is simply the human race, which is so foolish that it seeks and strives for many things by its efforts which it cannot attain or which, even if it does attain them, it does not enjoy but possesses to its sorrow and harm.”55 Towards the end of the century, this understanding of Ecclesiastes won even greater popularity. In 1586 Antonio de Corro, a Spanish-born reader in divinity at Oxford, published a commentary on Ecclesiastes, Solomons sermon of mans chief felicitie. In it he emphasizes that the postlapsarian state of humankind prevents it from deciphering the secrets of the universe. Commenting on Ecclesiastes 7:13 (“Consider the work of God: who can make straight what he has made crooked?”), he writes, “Either for that [man] cannot attain to the ful & absolute knowledge of things, because they are lapped & inwrapped in so manifold knots & marueilous difficulties, & beside the things themselues be so infinite in nūber: or for that there happen so many peruerse, crooked, and ouerthwart chāces in the life & doings of men, which by no reason can be ordered or amended.”56 Sixteenth-century astronomers had very little to say in their defense. The nobility of their scientific endeavor depended on the fact that the objects of their research are located in the highest realm in the universe, on the verge of the invisible realm of Platonic ideas. In fact, it is exactly because of this that the objects of astronomical research are not given to direct perception. This difficulty resulted in a severe methodological problem. The ideal scientific reasoning rested on the ability to proceed and induce from visible effects the true causes. In astronomical speculations, however, this did not work, as Peter Barker and Bernard Goldstein succinctly summarize this sixteenth-century view: “in the case of astronomy, there is no valid way of choosing among the possible causes, limiting ourselves to the one actual cause and then explaining the motions of the planets a priori or propter quid.”57 The fact is that the heavens are remote and the details of celestial events are basically invisible and cannot supply any means of verification beyond immediate effects. “God the Creator placed these bodies so far away from our senses that we are unable to produce principles of demonstration for them (as we can in the sciences of other things) or to discover what is natural and familiar, by means of which we may afterwards set out the causes of particular appearances.”58 Such an epistemological state of affairs brought to a dead end any aspirations to reach a realistic theory of astronomical knowledge. Instead, in the second half of the sixteenth century the growing attitude was that astronomy should be satisfied with fictitious constructs and should give up on any attempt to reach any casual explanation. In

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his attack on Tycho Brahe’s world system, the former imperial mathematician Ursus declares, A hypothesis or fictitious supposition is a portrayal contrived out of certain imaginary circles of an imaginary form of the world-system. . . . I say a contrived portrayal of an imaginary form of the world-system . . . , not the system itself, but a form of it of the kind which we think up by imagining and proclaiming as a conception of the mind. These contrived hypotheses are nothing but certain fabrications which we imagine and use to portray the world-system. So it is not in the least necessary . . . that those hypotheses correspond altogether . . . to the world-system itself . . . , provided only that they agree with and correspond to a method of calculation of the celestial motions, even if not to the motions themselves.59 In 1617 Mulerius, who was in charge of the third edition of Copernicus’s De revolutionibus, could not decide between the heliocentric and the geocentric models of the universe. His position was one of resignation and irony: “But what good is it to linger on hypotheses, which are nothing more than fictions with which men try in vain to discover the world system. . . . We must recognize the supreme wisdom of God the Creator and the weakness of our intelligence, which must regard with awe more than comprehension of the world machine.”60 During the last decades of the sixteenth century, the problem of astronomical observation became even more complicated. The results of Tycho Brahe’s observations of the comet of 1577 suggested that there are no solid orbs in the heavens and that there is nothing corporeal that the astronomer can measure other than moving, shining dots. Accordingly, rather than merely being concerned with the validity of observing remote and inaccessible objects, what does an astronomer observe and measure if there are no solid wheels in the sky? Giordano Bruno, for instance, rejected optics and geometry as legitimate venues for capturing heavenly phenomena. I would like to know how it is possible to deduce the reckoning of the propinquity and remoteness of luminous bodies from their size and, conversely, how it is possible to deduce the proportional variation of size of like bodies from their distance and propinquity. I would like to know by what principle of perspective or of optics we can definitely establish the correct distance, or the greatest and smallest difference, from any

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MEASURING SHADOWS Introduction

variation of the diameter. . . . We cannot establish the true size or distance of a luminous body from its apparent size.61 Instead, Bruno suggested that only an extraordinary direct experience of the heavens can provide a new foundation for astronomical knowledge. The Nolan (the persona of the true philosopher in his dialogues) has no recourse to either authority or any mediated experience: “But in truth it signifies nothing for the Nolan that the aforesaid [motion] had been stated, taught, and confirmed before him. . . . For he [the Nolan] holds [the mobility of the earth] on other, more solid grounds of his own. On this basis, not by authority but through keen perception and reason, he holds it just as certain as anything else of which he can have certainty.”62 Traditional astronomical knowledge accepts the limits of the human sensory apparatus and has to be satisfied with circuitous and flawed perception of celestial phenomena. This state of affairs is the cause for the folly of astronomers and their monstrous inventions and theories: The Nolan . . . has freed the human mind and the knowledge which were shut up in strait prison of the turbulent air. Hardly could the mind gaze at the most distant stars as if through some few peepholes, and its wings were clipped so that it could not soar and pierce the veil of the clouds to see what was actually there. It could not free itself from the chimeras of those who, coming forth with manifold imposture from the mire and pits of the earth (as if they were Mercuries and Apollos descended from the skies), have filled the whole world with infinite folly, nonsense, and vice, disguised as so much virtue, divinity, and discipline. What can the Nolan suggest instead of optics, geometry, and peephole cameras? Bruno’s radical suggestion is to accept the testimony of the one who has been to the sky and had direct experience of the true nature of the universe: How shall we honor this man [the Nolan] who has found the way to ascend to the sky, compass the circumference of the stars, and leave at his back the convex surface of the firmament? . . . Now behold, the man [the Nolan] who has surmounted the air, penetrated the sky, wandered among the stars, passed beyond the borders of the world, [who has] effaced the imaginary walls of the first, eighth, ninth, tenth spheres, and the many more you could add according to the tattlings of empty mathematicians and the blind vision of vulgar philosophers. Thus, by the light

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1 Kepler’s musing on the falling Icarus, in Franciscus Sanctius, Commentaria in Alciati Emblemata (1573), fols. 242v–243. © The British Library Board, Egerton MS 1234.

of his senses and reason, he opened those cloisters of truth which it is possible to us to open with the key of diligent inquiry; he laid bare covered and veiled nature, gave eyes to the moles and light to the blind, who could not fix their gaze and see their image reflected in the many mirrors which surround them on every side.63 Bruno’s wondrous flight of philosophical imagination stresses the crisis felt by many astronomers at the turn of the seventeenth century: they asked how astronomical knowledge was possible if the two pillars of astronomy—observation and geometrical measurements—were crumbling and demanded radical reformation. It was the figure of the falling Icarus that embodied the combination of moral admonition and the sense of epistemological folly involved in the astronomical aspiration to know the heavens. The recognition that no one can inform astronomers of any direct experience of the heavenly host and that the powers of human senses are limited drew strict boundaries to the astronomical knowledge that humans can attain.

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MEASURING SHADOWS Introduction

epistemological transgressions Challenging commonsense epistemological boundaries was no easy matter. An indication of the difficulties can be detected in an epigram Kepler wrote in a small commentary on Alciati’s book of emblems, which served as an album amicorum owned by Nicolaus Olaus of Skara in Sweden. Facing an emblem scolding astrology with a falling Icarus, the mythical epitomization of the vanity and presumptuousness of those seeking after knowledge, Kepler wrote, “Nemo cadit, recubans, terrae de cespite planae; O curans hominum, o quanta est in rebus inane” (No one falls off the grassy plain of the earth lying on the back; oh, the toils of men, oh, so much emptiness in their affairs!) (figure 1).64 This is a puzzling response from one of the leading astrologers of the first decades of the seventeenth century. One would have expected Kepler to defend Icarus’s endeavor and not urge him to linger on earthly plains, nor indict Icarus’s aspiration to reach the heavens. The second part of Kepler’s epigram is a direct quote of the opening sentence of Persius’s Satires, lamenting the decaying morals and especially the deterioration of the arts in Rome of his time. This, however, does not dispel the reader’s puzzlement as to the context of the falling Icarus within a denunciation of astrological pursuits. A more immediate source for Kepler’s complaint over the inanity of humans, and closer to the context of Kepler’s musing, is Martin Luther’s commentary on Ecclesiastes discussed above. In discussing verse 1:8 (“All things are full of weariness”), Luther adjoins Solomon’s intentions with Persius: “The greatness of the vanity of men, he says, is unspeakable. Thus, Persius also exclaimed: ‘Oh, what vanity there is in things.’”65 Luther’s commentary is a forceful indictment of vain human pursuit after ephemeral possessions, yet Luther excludes scientific inquiry from this admonition. On the contrary, he defends the study of astronomy as useful and religiously acceptable: “Therefore we should not follow the imaginations of the interpreters who suppose that the knowledge of nature, the study of astronomy or of all philosophy, is being condemned here and who teach that such things are to be despised as vain and useless speculations. For the benefits of these arts are many and great, as is plain to see every day. In addition, there is not only utility, but also great pleasure in investigating the nature of things.”66 Luther, however, admits astronomical knowledge only in accordance with the place God has allocated to humans “under the sun.” The audacity and vanity of mankind consist in its incessant attempts to transgress these topographical limitations: as noted above, Luther saw mankind as “so foolish that it seeks and strives for many

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things which it cannot attain or which [it] . . . possesses to its sorrow and harm.”67 Kepler’s assertion that the astronomer can attain knowledge while lying on the grassy plains of the earth, together with the allusion to Luther’s vindication of certain types of astronomical observation, divulges the outlines of a radical epistemological project. Investigating the position of the observer will refigure the visual regime involved in the production of scientific knowledge. It will further assign the eye’s different capabilities, changing its role and function and establishing a new mode for capturing celestial truth from our lowly and fallen human state. Kepler’s new optics intended to do just this: it would allow observation of distant and almost invisible objects to grant knowledge as valid as the observation of things at hand. Invisible lines and circles could be measured, and yet these measurements would substantiate physical truths far more certain than the measurements of static solid and corporeal objects. Paradoxically, Keplerian optics would prefer indirect visual experience of shadows and virtual images over direct experience of tangible entities. What is the mechanism that allows such apparently absurd optics to work? What are the epistemological foundations of such a scientific oxymoron: optics—or a science of visual experience of things that are beyond the power of one’s eyesight? Kepler aimed at an a priori astronomy based on physical causes. In order to present his new conception of astronomical knowledge, he first had to tackle ancient and Renaissance critiques of astronomy, which was not an epistemological instrumentalism, but a frustrated acceptance of the limits of human experience. Because of their remoteness, one cannot ascertain the reality of geometrical theories of the heavens, and therefore the only criterion for a successful astronomical theory is its predictive ability. Kepler, however, aimed much higher (to “direct the eyes to the central mystery of the cosmic machine”).68 The only way to achieve this task was by reformulating the nature of visual experience in general and of scientific observation in particular. Such a new construal of visual experience aspired to connect observations with the invisible structure of the heavens. In order to accomplish this, Kepler had to devise an optical theory that concerned itself with invisible elements, an optics that would supply astronomers with the means to measure an empty space. Kepler posits these questions in an intricate manner in the letter of dedication and introduction to his treatise. The analysis of these prefatory texts in chapter 1 reveals the rich humanist and artistic context of Kepler’s optics together with his epistemological stakes. These readings are corroborated by

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MEASURING SHADOWS Introduction

diverse paratextual elements, including the treatise’s frontispiece and introductory poems and Kepler’s letters from the 1590s. These elements locate Kepler’s interest in light as connected to his aspiration to an all-inclusive physical theory of the heavens. In these texts Kepler establishes new relationships between the visible and the invisible. At the core of this new foundation one finds, paradoxically, not direct observation of scientific objects, but the measurements of shadows and their analysis. In Kepler’s view, these measurements are not secondary means for perceiving astronomical truth. Shadows and other artificial images are, in fact, the sole venue for grounding science. In order to allow shadows and artificially produced images to fulfill such an epistemological task, Kepler presents two bold moves: one on the level of practice and instruments of observation, to reallocate their position as mediators between the eye and physical reality; the other on the level of mathematical language and its relation to motion. Chapter 2 presents the way Kepler bypasses the prevalent notions regarding vision and light and suggests a new point of departure for his optics. This new turn allows Kepler to introduce light as the agency of sight and the camera obscura as a model of the visual process. Chapter 3 follows Kepler in critically rewriting the medieval perspectivist tradition. Kepler rejects the ancient and medieval prevailing attitude toward mediated vision as an epistemological excess. Instead he posits the camera obscura as a paradigmatic model for the way instruments of observation correct visual errors and supply visual data not available to the naked eye.69 However, to decipher the formation of shadows and pinhole images, Kepler has to think anew the way mathematical entities relate to the physical world. In reviewing classical and medieval catoptrics, Kepler discards the Aristotelian concept of mathematical entities as abstractions from solid physical bodies. Instead Kepler sets the outlines of a theory of cognition based on the understanding of mathematical entities as representations of physical motion itself. Having analyzed the way mathematics and instruments are combined in the process of producing visual data, the fourth chapter discusses Keplerian retinal pictures as an end product. Kepler’s understanding of pictorial representation is set against the backdrop of Renaissance humanists’ and painters’ treatment of the epistemological status of the art of painting in general, and of the value of artificial perspective in particular. Renaissance humanism, with its rhetorical bias and the medieval inheritance of Aristotelian psychology, frames painting as imitatio of nature. Kepler dissociates his retinal pictures from this dependence on physical bodies. On the contrary, he emphasizes the artificial-mechanical way in which visual images are produced. Such a move brought about a divorce

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between painters’ use of geometry for the description of natural objects and the aspiration of early modern scientists to mathematize nature. The fifth chapter, through an extensive analysis of Kepler’s polemics with various occult philosophers, describes the new mode of interpretation and reading that Kepler’s new concept of mathematical paintings demanded. In several places (his polemics with Robert Fludd, his famous letter to Tanckius of 1608, and his introduction to the Astronomia nova) Kepler rejects traditional qualitative modes of reading nature as a text and suggests instead a new quantitative description of natural motions. Visual experience of natural processes does not constitute an obscure emblem, or a deficient shadowy picture of hidden and secret essences, waiting to be interpreted. Instead, Kepler suggests that the entire range of visual entities—reflections, refractions, shadows, and virtual images—is the vehicle for an exact scientific depiction of the harmonic order of the universe. Thus, the celebrated occult and kabbalistic conception of De umbris idearum is shifted from the symbolic to the realm of concrete physical and measurable representations. Through measuring these shadows, Kepler aims at exposing the meaning of visible phenomena as an exact representation of the hidden Godhead. Instituting shadows as a basic entity for any astronomical research shifts the borders of the domain of what there is in the physical world. Throughout medieval and Renaissance Neoplatonic speculation, the infinite and nothing were the end points of both physical being and human knowledge. Kepler’s novel treatment of the relationship of mathematics and physical motion and his investigations of shadows and other refractions provide him new paths to account for nothingness. In several short texts, such as De nive Sexangula, Kepler tackled the paradox involved in the concept of nothing, turning it from an object of mystical contemplation into an object of manipulation. The sixth chapter will follow the epistemological implications of Kepler’s treatment of nothing in the context of early modern science. This unique optical research program is an important key to Kepler’s scientific project. Optics was the foundation for Kepler’s claim that the distant reality of the heavens can be physically known and that science can discern the invisible structure of the universe on the basis of visible phenomena. In his optics Kepler supplies the vehicles for such a heroic undertaking. Paradoxically, these are shadows and artificially constructed images, entities that in medieval and Renaissance epistemology were considered secondary and deficient. Yet these optical phenomena can be measured and are the products of mathematically constructed instruments. In fact, mathematics is the essence of such images

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MEASURING SHADOWS Introduction

and shadows. Thus, mathematics can vouch for the ability of optical constructs to represent physical reality accurately. It was with good reason that the first major treatise Kepler dedicated to Emperor Rudolf II was on optics as the foundation for his new astronomical understanding. It is for this same reason that Kepler depicts himself in the following words in his epitaph: I used to measure the heavens, Now I shall measure the shadow of the earth. Although my soul was from heaven, The shadow of my body lies here.

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1.

THE NEW OPTICAL NARRATIVE Light, Camera Obscura, and the Astronomer’s Wings

the peculiarities of a frontispiece Let’s look again at the frontispiece of Ad Vitellionem1 (figure 2). As noted above, coupling astronomy and optics is not obvious and smacks of oxymoron. Furthermore, the notion of paralipomena is ambiguous, leaving the reader with a riddle: what exactly is missing in Witelo, and is it something that Witelo and the medieval tradition as a whole overlooked? What is this crucial element that Kepler assumes to be a blind spot in Witelo’s field of vision? As the reader’s eye goes down the list of titles and subtitles, the tension gradually begins to resolve: the topic missed by medieval perspectiva is concerned with artificial observation and its performances, and the crucial step for Kepler is turning this topic into the essence of optics. The eye and the process of human visual experience are relegated to the end of the list, where in small italics, as if just an afterthought, the reader is invited to the multa noua which are contained in a brilliant treatise about the mode of vision and the use of the ocular humors and (still in small letters) opposing the opticians and the anatomists. This is the dramatic inversion that takes place through this frontispiece—the traditional topics of classical and medieval optics. The eye and the theory of vision are now subsumed under artificial observation, which aims at going far beyond “the threshold of perception.” Kepler, in a sense, adopts and explicates Dürer’s famous and erroneous translation of perspectiva as “seeing through”—optics,

according to Kepler, is not necessarily about seeing but about the instrument through which optical data are transmitted and received.2 This practice of inverted representation is summarized in an alchemical novel of 1610, Le Voyage des Princes Fortunez, by François Béroalde de Verville, where steganography is defined as the art of representing plainly that which is easily conceived but which under the coarsened features of its appearance hides subjects quite other than that which seems to be represented; this is practiced in painting when some landscape or harbour scene or portrait is shown which conceals within itself some other figure which can be discerned by looking from a certain viewpoint determined by the artist. This is done also in writing, when an author discourses at large on plausible subjects that unfold some other excellencies which are known only when read from a secret angle which uncovers splendours concealed from common appearance.3 In one of the opening epigrams to the Optics, Kepler makes the eye proclaim, “I lessen life for fame, sensation for the name: Know this [my] soul, it is more useful to die and not to decay.”4 Beyond visual reality lies divine meaning. It is the duty of the mind to work this meaning out.

reclassifying the sciences The reshuffling of optics into astronomy is not concerned, from Kepler’s point of view, solely with a redefinition of optics. From the opening paragraphs of the treatise’s preface, Kepler makes it clear that the concern of these optical studies is to assist in his endeavor to reestablish astronomy as a physical science. The first sentence is a radical and challenging definition of astronomy: “Astronomy, which deals with the motions of the heavenly bodies, principally has two parts. One consists of the investigation and comprehension of the forms of the motions, and is mainly subservient to philosophical contemplation. The other, arising from it, investigates the positions of the heavenly bodies at any given moment, and has a practical orientation, laying the foundations for prognosis.”5 Astronomy deals with motions of real physical bodies, and its investigations receive their principles from philosophical contemplation, that is, physics. These

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2 Frontispiece of Kepler’s Ad Vitellionem (1604). Reproduced by permission of The Huntington Library, San Marino, California, RB708372.

contemplations constitute the theoretical part of astronomy, whereas its praxis is the mapping of the heavens, at each and every moment, as the basis of astrological prognostication. Kepler stresses that astronomy is the investigation of the heavenly bodies’ motions themselves. These words explicitly defy the traditional view that astronomy is primarily a geometrical science. Kepler’s following assertion that astronomy is subservient to philosophy puts him in clear opposition to the hegemonic Aristotelian tradition’s treatment of the scientia media. Aristotle discusses the mathematical sciences such as astronomy and optics as exceptions to the general methodological rule that scientific demonstrations must be from proper principles and cannot pass from one subject matter to another. In the Posterior Analytics Aristotle asserts, “One cannot, therefore, prove anything by crossing from another genus [metabasis eis allo genos]— e.g. something geometrical by arithmetic.”6 Astronomy and optics seem to violate this rule because in them one can use mathematical principles and demonstration in order to arrive at conclusions about physical phenomena. Aristotle allowed this breach because such sciences are related to pure mathematics as one under the other (altera sub altera, hence subalternate or subservient). In the lower science one knows the fact (oti, quia), while in the higher one knows the reason why (dioti, propter quid). Aristotle thus called such sciences the “more natural of the branches of mathematics” (τα φυσικοτερα των μαθηματων).7 Commentators on the Posterior Analytics in the Latin West, from Robert Grosseteste in the early thirteenth century onwards, argued that despite the usefulness of mathematical proofs in the subalternate sciences, such proofs demonstrate only quantitative accidents of the physical subject, and are restricted to formal causes only. Grosseteste arrives at the conclusion that in optics, for instance, geometry changes its demonstrative status from propter quid to quia. Geometry can prove that the visual angles, in the case of reflection, are equal, but not why they are so. In contrast, Thomas Aquinas, who coined the term “middle sciences” (scientiae mediae), asserts that the subjects of higher and lower sciences are related not as genus to species or universal to particular, but as form and matter, so that the middle sciences are formally mathematical; consequently, only mathematical demonstrations can be propter quid in them. Such demonstrations are available only to the mathematician and are limited to formal causes, while the practitioner of the lower science is restricted to quia demonstrations and sensible experiences.8 The decline of medieval universities in the late fourteenth century brought an end to these discussions. The interest in these methodological

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questions resurfaced in the sixteenth century with the efforts of religious and humanist reformers to establish education systems on new principles. Together with the thorough humanist interpretation of the Aristotelian corpus, they brought further clarifications and emphases of the question of metabasis. The prominent Paduan commentator on Aristotle’s opus Jacopo Zabarella, for instance, argued that both the higher and the lower sciences consider their subject matter mathematically. Where they differ is in what they consider (in res considerate), in that the lower adds a sensible quality to the subject of the higher. By virtue of this sensible quality one can acquire knowledge quia through the senses in the lower sciences. The mixed sciences are formally and essentially mathematical in that in them only mathematical demonstrations can be propter quid, but, unlike pure mathematics, they can also avail themselves of sensible knowledge.9 In the German context, these discussions held a prominent place in Kepler’s alma mater, the University of Tübingen. Jacob Schegk, a professor of logic and philosophy at Tübingen, argued in 1564 that “[t]here is no place for geometrical reason [ratio] in music. For the interval of 9/8 can be divided geometrically in two: in arithmetic and music it cannot, since obviously a tone is divided unequally into a major and a minor semitone. Parmenides and Melissus committed similar errors, transferring to physics not physical reasons but mathematical ones, confusing therefore the limits of the disciplines.”10 Schegk warned his readers that “one should diligently be cautious not to confuse the limits and terms of the sciences; this is plainly the first step in any method.” The example for such a transgression is taken first from the optical sciences, and then from astronomy: When demonstrating something in optical or mathematical reasoning, in whatever way the lines are described, it makes no difference, but physically [Physice] it makes a great difference, because there are principles of other [types of] demonstrations. We have seen in our time an astronomer who placed in heaven a mobile earth outside the centre of the world, and a Sun that stands still, and using mathematical principles, had nevertheless most accurately given an account of the phenomena. Therefore, it is necessary for a demonstrator to have known the difference amongst principles, and by having correctly established [those principles] everything which agrees with truth is concluded by everyone, but by [setting up] the principles falsely, things which not only disagree with truth, but also those false things which disagree among themselves are gathered

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MEASURING SHADOWS The New Optical Narrative

together, and then the consequent false things also show up the falsity of the principles.11 In the midst of these debates over methods of the different sciences, it was Kepler’s mentor at Tübingen, Michael Maestlin, who attempted to legitimize the application of mathematics to the study of the heavens. His move consisted in qualifying astronomical research, contending that this concerns only apparent motions and not the real motions of the heavenly bodies. In 1582 Maestlin affirmed that “[p]rincipally, astronomy deals with the heavenly bodies.” Yet he emphasized that astronomy’s business was not with the real physical motions of these bodies but “to catalog, measure and investigate the causes of their apparent motions.” This emphasis on apparent motions allows Maestlin to turn astronomy into a mathematical science: “It is plainly a composite [science] in itself. For it is properly subordinated to the object or matter of physics: mathematics, truly, is a different [part of astronomy], as Geometry and Arithmetic, evidently supply it [astronomy], with demonstration as if with a form. Indeed, apparent motions are not demonstrated physically, but are demonstrated according to mathematical reasons.”12 In trying to keep within the set boundaries of the sciences, Maestlin ends up with a very epistemologically narrow definition of astronomy. Mathematics, as in Thomas Aquinas, supplies merely the formal cause. Following a notion similar to Zabarella’s, there is a material part based on sense-experience. Such sense-experience in astronomy, however, yields only apparent motions of the heavenly bodies, and therefore the mathematical formal cause is not applied to real motions, as the astronomical observer has no sensory access to a complete and direct experience of the natural phenomenon and its causes. The astronomer has to do only with indirect experience of the causes of apparent motions. A few years later, in 1599, another Italian commentator, Lodovico Carbone, would bring these arguments to their extreme form, arguing that in astronomy one demonstrates either through appearances without determining whether they are causes or not, or through false premises (such as eccentrics and epicycles) assumed merely to save appearances. In any case, astronomy cannot give true and valid knowledge of the real heavens and the motions of the heavenly bodies. With such sentiments abounding, and in an ambience that restricted and limited the possibilities of astronomical knowledge, Kepler felt it was his duty “to uphold astronomy’s dignity and to subdue the hostile fortress of doubt.”13 The first step in this campaign was his treatise on optics and his succinct

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opening definition of astronomy, which ignores traditional warnings and road signs to stop at metabasis. Astronomy, according to Kepler, deals with real motions of real physical bodies, and is not subservient to geometry, but to philosophy (i.e., causal knowledge, or physics). Kepler does not stop at that, but shifts the meaning of “theory” as an astronomical term. Since Hellenistic times theory had been accepted as the part of astronomy that deals with calculating the different positions of the planets. Kepler assigned this job to astronomical practice, associating it with astrological prognostications. The theoretical, that is, the real science of astronomy, is no longer concerned with appearances, but with physical causes.

optics and the wings of the astronomer How can one seek after a causal physical account when the heavenly bodies are so distant, that is, without direct sensory knowledge of the quia, the facts? In his preface Kepler provides a bold and radical solution to this problem. Adopting the Platonic allegory of the charioted soul, he turns it against its common sixteenth-century meaning: “These two [parts of astronomy, the philosophical and the practical,] fly up into the heavens, supported, as Plato used to say, by (as it were) a pair of wings, geometry and arithmetic.”14 The way to approach the heavens in order to capture basic factual knowledge is through the mathematical sciences. In defining astronomy as a physical science, Kepler does not give up mathematics. On the contrary, the Keplerian definition of astronomical science flouts the Aristotelian dictum against metabasis. The new Keplerian science is a mathematical-physical research into the heavens. The mathematical sciences are supposed not only to supply it with explanations propter quid, but, in a way, to transport the observer to the heavens themselves in order to grasp the experiential facts themselves together with their physical causes. This dual role of geometry is made clearer in the following section. Kepler suggests that geometry grants the infrastructure for two kinds of astronomical activity: the search after physical or metaphysical axioms, and the molding of astronomical observations into astronomical facts. In his later treatise “The Key to a Deeper Astronomy” (which would eventually be published as the Astronomia nova), Kepler promises to deal with the manner in which geometry can be a part of the physical explanation of the motions of the heavenly bodies and their causes. Kepler divides the second kind of activity into three parts. The first two, which are concerned with the instruments of observation and

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the records of observational data, were covered by the work of Tycho Brahe, but Kepler hints that without a theoretical framework they will be rendered worthless. The palace of astronomy that Tycho built is only an empty shell, nicely decorated and the foundation laid correctly, but with no solid content. The Optics will supply the palace with windows that will enable the light of the stars and the luminaries to enter the palace and reveal their truth, enabling the astronomer to turn the appearances of the heavenly bodies into physical facts. Optics, Kepler asserts, deals primarily with light and shadow and how they mediate celestial phenomena to the observer. Optical analysis treats the ways different factors affect one’s observation (the modifications of the media, the position of the observer, etc.). It is thus able to correct visual errors originating from the great distance between the observer and the celestial bodies.15 The reader can now perceive that this is not an ordinary treatise on optics. The function and place of optics within Kepler’s classification of the sciences have radically changed. In order for astronomy to become an integral and legitimate part of natural philosophy, it needs optics not as a technical assistant merely correcting and fine-tuning astronomical instruments of observation, but as a vehicle for the transformation of distant appearances into solid facts. This transformation is initially made possible through the ability of optical analysis to do away with the separating media and the other distortions between the celestial bodies and the human observer. This ability is due to optics being part of geometry. Yet geometry in its turn is not an abstract mathematical science anymore: “And I have not satisfied my soul upon speculations of abstract geometry, namely upon pictures ‘of what there is and what is not,’ to which the most famous geometers today devote almost their entire time. But I have investigated the geometry that, by itself, expresses the body of the world following the traces of the Creator with sweat and puffing.”16 Kepler is turning the tables, and what appears to be a technical treatise on astronomical observations is now a radical program for the integration of natural philosophy, geometry, and astronomy.

the glory of shadows To understand the meaning of Kepler’s synthetic program, the reader has to continue and excavate the complex and layered textual structure he presents. Kepler cannot reiterate traditional optical theory; he has to redefine the tenets of optics in order for it to accomplish these epistemological expectations. The main revision is ontological. The basic objects of optical research are new and

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different from those of classical and medieval optics. The alteration takes place almost unnoticed: “The things to be considered optically in astronomy are . . . the things themselves set before the sense of sight, by way of considering the species of things, that is, light and shadow.”17 The identification of species with light and shadow turns medieval optics and theories of cognition on their head. Robert Grosseteste, for instance, defines species as a radiation of likenesses or power from the visual body: “A natural agent continuously multiplies its power from itself to the recipient, whether it acts on sense or on matter. This power is sometimes called species, sometimes a likeness, and it is the same thing whatever it may be called.”18 Roger Bacon follows and asserts that “species is similar in essence and definition to the agent and the things generating it.” Light for medieval perspectivists was only a more visible example of such an emanation or multiplication: “And to explain the meaning of species with an example, we say that the lumen of the sun in the air is the species of the solar lux in the body of the sun.” Whatever the debates concerning the status of species in the thirteenth and fourteenth centuries, the basic insight of scholastic philosophy was that species is a physical expression or communication of the visible body itself. The multiplication of species was molded into a visual ray in the case of sight and into physical radiation in other cases of action at a distance (such as in the case of a magnet), light being just a paradigmatic case. Kepler, in his almost unnoticeable move, abandons visual rays and physical emanations and reduces them all to a radiation of light. The objects of sight are not emanations of physical things, which can protect the cognitive process and keep direct contact with external reality, but merely light, or its absence, reflected back from the visual field. This maneuver releases optical research from psychological considerations of intention and cognition, allowing Kepler to take optics as the physical study of light. It allows him, further, to reduce these studies to the question of the transition of light through media: “The medium through which light the carrier of species travels, by the cause of it, one way or another, light reaches us refracted.”19 The last thing optics deals with is the instruments of observation and the eye as receptors of these refractions. In this way Kepler reduces all our visual experience to refractions of light. Therefore there is no difference if one observes distant heavenly bodies or an object in the immediate visual field; the images produced are, in the final account, effects of refractions of light, and are similarly treated and analyzed. This becomes clear when Kepler moves on and narrows the general definition of optics and discusses the celestial bodies specifically. He emphasizes that

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the astronomer considers only the species of these bodies transmitted as light, and that these species are nothing but their mathematical properties, namely, figure and size. Therefore, in order to capture visual phenomena, there is no need for full imagery, with all its sensual details of colors and texture. Since light transmits quantifiable properties only, mere shadow also suffices as the foundation for true analysis of physical phenomena. Kepler thus reduces the entire visual world to a theater of reflections and refractions of light, turning the absence of material bodies into a foundation of true knowledge of creation. His intuition is based on the role of eclipses (in other words, the absence of the luminaries) in instituting astronomical science: “For the most noble and ancient part of astronomy is the eclipse of the sun and the moon, a subject that, as Pliny says, is in the entire study of nature the most wondrous, and most like a portent.”20 The disappearance of the luminaries is the basis of any inquiry and is the invitation God left the human race to study his creation. This invitation is ordered like a theater with signs and allusions to be deciphered by the human mind, which is the likeness of God himself: “Anyone who ponders this [the eclipsed luminaries] carefully will find . . . both that there is God, founder of all nature, and that in the very mechanics of it he had care for the humans that were to come. For this theater of the world is so ordered that there exist in it suitable signs by which the human minds, likenesses of God, are not only invited to study the divine works . . . but also are assisted in inquiring more deeply.”21 Emphasizing the theatrical metaphor, Kepler proceeds to compare the eclipsed luminaries to playful jests nature performs to lure the human mind to contemplate through them the true order of the world. Thus a triple relationship is constituted in this theater between the divine Godhead, the human mind, and the body of the world. This relationship is geometrically based. As Kepler writes to Fabricius, “God has ordained certain animal faculties in this earth, participating somehow in the mind in order to perceive in a certain way geometrical beauty, or even discrete quantities, which are perceived as being active themselves in exuding vapors. This is certainly that peculiar ordination of God: these faculties are images of God, sensing the geometrical beauty, as God.”22 There is almost an identity between human knowledge of God, human perception of geometrical beauty, and the geometrical structure of the physical world. This identity enables the senses, with the help of the mind, to extract geometrical knowledge from the vapors of the natural world. This process is aptly described in the introductory epigrams to Ad Vitellionem. One epigram, which presents a dialogue between the eyes and

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the mind, ends with the latter guaranteeing to restore the damage suffered by the eyes on its behalf and to radiate starry brilliance even if it is stained with blemishes.23 These lines retrace the tension in Horace’s Satires between scattered moles and the well-formed body: “If my nature, otherwise virtuous, be found faulty for small, ordinary things, as if you were censuring the scattered moles on a well-formed body.”24 Kepler, however, undermines this Horatian dichotomy between essential perfection and the insignificant moles that tarnish it; the blemishes do not hide the brilliance but become an integral part of this process of enlightenment.25 What are these imperfections and blemishes radiant with knowledge, and what is the knowledge Kepler hoped to gain from such a defect? Kepler explicates these words in the preface: Now, one may consider that all the rest of astronomy is closely associated with the motion of the sun and the important assistance given us by the moon, participating in the days just as in the nights, when all other means failed us: it is believed rightly that universal astronomy is born from this obscurity of the luminaries. Just as these darknesses may be the eyes of the astronomers, these defects may be a rich source for doctrines, and these “stains” may illustrate the most precise pictures on the mortal mind. O most excellent and commendable argument for all the nations about the glory of the shadow.26 Kepler, in a surprising turn of phrase, set the Platonic cave on its head. It is the shadows, the obscurity, and the material effects of light that become the source of knowledge. The astronomer is not supposed to observe the sun itself but to examine its shadow in order to portray the true picture of the universe and reveal its secrets. The quantification of shadows (i.e., the absence of the material object) is the key to nature’s powers. In sharp contrast to both Platonic and Aristotelian philosophy, Kepler asserts that a true investigation of the shadows, one that will define them correctly, then measure and analyze them accurately, will finally expose the true structure of the universe: “And thus the quantity of the image which the moon or the sun, whether whole or eclipsed, shows us, and of the shadow which the earth stretches out to the moon, must be carefully investigated by the astronomer.”27 Kepler points to the difficulties involved in such research: “Even though these images are obvious to everyone’s eyes, all practicing astronomers complain that it is with difficulty that they are measured. This is partly because the bodies have a very narrow apparent size, and partly because they constrict the

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eye with their exceptional light, so as to prevent their fulfilling their function in seeing.”28 It is here that Kepler introduces instruments of observation as a corrective to, yet actually a replacement of, the feeble human eyes. The prototype of such instruments is the camera obscura: “Nature has not forsaken those desirous of learning, for she has shown us a procedure by which we may accomplish in darkness, without detriment to the eyes, what is completely impossible in clear light, with the eyesight directed towards the sun.”29 Kepler constructs a new place for astronomical observation: instead of the human eye as the natural and obvious instrument of sight, it is the camera obscura as a theater of light and shadows. Within this theater the main characters in nature’s play are eclipses or blemishes, which baffle the human senses and intrigue the human mind. The camera obscura is a place of inversion and deceit, where the world outside appears hanging upside down on a screen and insubstantial apparitions seem like real things. Kepler converts this place into a location where a hidden aspect of external reality can be disclosed to the careful observer, where the shadowy and artificial conceits of the camera are transformed into a new genus of facts.

a theater of shadows In order to understand the logic guiding Kepler’s peculiar utilization of shadows as means to solid knowledge, it is beneficial to look to another contemporary theatrical vision that investigated the blurred lines between fantastical conceit and truth, between shadows and substance. In Shakespeare’s A Midsummer Night’s Dream, several metamorphoses take place, all concerned with the power of the imagination over sight. The play celebrates the power of lunar light and of darkness to “turn melancholy” into “the spirit of mirth” and to change the discordant key of injuries and violence into a harmony of love and reveling. As a play within a play, a group of simple artisans are preparing to put on a show of the tragedy of Pyramus and Thisbe, an endeavor that will in the end become “tragical mirth,” a concord of discord. This transformation, however, does not take place in an unidentified location, and as the artisans ponder the role of the moon in their play, they describe the stage as a camera obscura: “Why, then may you leave a casement of the great chamber window, where we play, open, and the moon shine in at the casement.”30 A shaft of light entering through an opening, flooding the room, transforms the actors to shadows and visions, as Puck confesses in his famous epilogue:

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If we shadows have offended, Think but this and all is mended: That you have but slumbered here While these visions did appear.31 The Shakespearean theater is a place of conversion where real people are transformed into shadows and apparitions and visions appear as corporeal entities. It presents, however, not some idle fantasy of a “seething brain,” but a “great constancy” and a new manner of observing and grasping a hidden truth.32 A similar logic operates in Kepler’s notion of the camera obscura as a location for revealing truth through the observation of shadows. The shadows appearing on the Shakespearean stage blur the lines between the artificial and the authentic, the imaginary and the real, indicating a new scheme for producing knowledge not available to human rational and direct sense-experience. Kepler takes this new theatrical mode of investigation a step further. Not satisfied with rejecting direct sense-experience in favor of the new constancy of the imagination, he proposes a new procedure to establish scientific truth. Adding geometry as a necessary element in the operation of his theater of knowledge establishes observation on a new foundation: instead of verification through the certainty of direct experience, observation is valid only due to the mathematical reasoning that governs the mediation of the new instrument. In the theatrical setting of the camera obscura, the observer is able to artificially produce shadows and stains of light of the heavenly bodies, and to measure them. An important part of the optics will have to account for the possibility of observation through artificial instruments as a substitution for the natural naked-eye gaze at the heavens. The astronomer should follow a manual that combines the ingenuity of artificial instruments with geometrical demonstrations, embedded in specific examples. Kepler provides one such example and testifies that he applied this method during the 1600 eclipse. This happened around the time he applied for a position as mathematician at the court of Archduke Ferdinand. Kepler wrote an open letter in which he pointed out the possible benefits of his being hired.33 Kepler used July 10, the date of the impending solar eclipse, as his case study. The letter’s argument focused on calculating the eclipse and rejecting Tycho Brahe’s lunar theory. Tycho tried to save the motion of the moon while preserving a uniform circular motion for the moon and the planets. Thus, he introduced ever-new epicycles in his theory of the moon, whose motion exhibited especially complicated inequalities.34 Kepler, who had not yet read Brahe’s

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calculations but had heard of the general outlines of the theory, rejected and opposed it, since “simplicity is more in agreement with nature.” Consequently, he accepted a nonuniform motion of the moon in its orbit. He further postulated a virtue in the earth that causes the moon to move (“In Terra inest virtus, quae Lunam ciet”). However, the virtue being in inverse proportion to the distance between the earth and the moon, the moon will therefore move more slowly at a greater distance. After formulating the geometrical deduction, Kepler could now conceive and construct an apparatus that could help obtain exact numerical values of the progress and size of the eclipse. Furnished thus with the observation of the eclipse at hand, he hoped “to remedy any imperfections in Brahe’s lunar theory and to be able to test his allegedly certain conclusions by the all-revealing experience.”35 This attempt at a large-scale (for a mere mathematics teacher in a provincial town) scientific endeavor was to prove to any future patron his capability of managing such operations. A few months before, Kepler had written to his patron and friend the courtier and Bavarian chancellor Herwart von Hohenburg that he would like to dedicate a few more years to the study of astronomy, or else go study medicine. However, the only position that would have enabled him to practice fulltime astronomical research was as a mathematician at one of the courts. Kepler fancied that such a position would enable him “to pay readers in order to take care of my eyes which already grow feeble and to save time. I would send messengers here and there to obtain books and gather the advice of learned men. I would build instruments. I would appoint others for observing because I am less suited thereto.”36 In order to prove his ability to execute such a scheme, Kepler erected his new instruments in the marketplace in Graz and with them, on July 10, 1600, observed the eclipse that he had previously forecasted. While evaluating the results of this observation on July 22, he clearly saw the optical reasons for the apparent diminution of the lunar disk at the time of the solar eclipse.37 He formulated the geometrical deductions that explain image formation through small apertures and thus overcame one of the difficulties that had caused Tycho Brahe so much trouble.

the nature of light In introducing the study of the refraction of light as the kernel of his new optics, relegating the study of vision to a secondary place,38 and thus emphasizing the measurement of shadows and artificial instruments of observation, Kepler

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discloses that the stakes of his treatise are higher than only treatment of technical issues. For once, Kepler’s objective is the rewriting of the whole of optical theory by giving “pride of place to a priori reason.”39 These a priori expositions will have to proceed from an account of “the whole nature of light.” Such an account enfolds a complementary subject matter, which Kepler is well aware is somewhat off the traditional track and the usual topics of the optical sciences. Yet Kepler asserts that this addition is necessary to the study of astronomy: “Although this [the nature of light] is not completely beyond the accepted principles [of optics]: indeed not a little is to be gained by the astronomer by means of all types of rays.”40 These further implications of the nature of light are barely alluded to throughout the preface, though a couple of times Kepler does mention, in a sleight of hand, the subject of planetary motion. Turning to Kepler’s correspondence for the period before the publication of his optical treatise may add further weight to the supposition that his secret gambit in his optical researches is the role of light as a vehicle of motion in the universe, a theory dear to him as a pillar of his celestial physics. On September 14, 1595, in a letter to his teacher, Maestlin, Kepler explicated for the first time his new theory of the specific order of the planets as based on the proportions of the five regular bodies. By then, he had already formulated a connection between the moving soul (which he later referred to as “virtue” and even “force”) and light: “Next, there is a moving soul and infinite motion in the Sun, the resulting motion in the mobiles decreasing unequally because of two [reasons]: initially, it is caused by the unequal size of the orbs, even if the moving force were the same in all the orbs; but, second, indeed that strength of motion (as is lux in optics) is the weaker the farther it is from the source.”41 The analogy between the moving force and light helps Kepler explain the fact that the speed of the planets is not the same but changes according to their distance from the sun. This analogy between the “forces” that shape the solar system and light was of crucial importance to Kepler, since he was not satisfied with a purely geometrical description of the universe but aimed at a complete physical explanation.42 In order to causally explain the changing speed of the planets, Kepler had to come up with an outline of a physical theory. This was formulated around the intuition that the sun, as the source of all motion, is at the center of the system. From this premise, Kepler sought a pattern to connect the (linear) speeds of the planets with their distances from the sun. The key for deciphering these ratios is to be found in light.43 Kepler develops this notion further in another letter to Maestlin on October 3, 1595:

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“The moving soul [is], as I said, in the sun.” Yet although the force spreads uniformly, it takes one planet longer than another to complete its revolution, and the primary reason is that the sizes of their orbits are different. “If equality of motion and the same moving power were to come from the sun to all the orbits, one would still probably go around more slowly than the other because of the inequality of their orbits.” However, one can try to establish a connection between the slower speed of the more distant planets and their distance from the sun: “the periods of time would be [measured] according to [the size of] the circles. For it [the size] measures the quantity of motion. However, circles are [to one another] as their semi-diameter, that is, their distances. Thus, we may easily derive from the well-known mean motions also the mean distances.” Thus, the orbital length increases directly with distance, while the planets’ speed varies inversely with the distance. Yet the periodic times increase, not simply in proportion to the size of the orbit, as they would if all the planets moved at the same speed, but more, indicating that the distant planets are slower in absolute speed. Thus, Kepler was looking for a mathematical relationship between distance and periodic time, attempting to work out how the moving soul located in the sun operates at different distances. He supplies an analogy in order to explain it: But what makes the remote [planets] slower is due to another cause. We comprehend this from our experience with light. For light and motion are connected by origin [i.e., the sun] as well as by action, and probably light itself is the vehicle of motion. Therefore, in a small orb, and so also in a small circle [of light] near the sun, there is as great [a quantity of moving force or light] as in the greater and more distant [orbit]. The light in the larger [orbit] is certainly more rarefied, whereas in the narrower it is denser and stronger. And this power again is in a proportion of circles, or of distances. The ratio Kepler has in mind is that the speed of the planet decreases by twice the difference between the radii (i.e., the distances) of the two planets’ orbits. “Whereof half the difference of two motions joined to the minor motion will give the distance of the further: the smaller motion itself will be the distance of the nearer.”44 Kepler was looking for a physical analogy for his concept of a moving soul to be measured quantitatively. In his Mysterium cosmographicum he speculates that light issuing from the sun embodies this moving soul:

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A single moving soul is in the center of all the spheres, that is, in the Sun, and it impels each body more strongly in proportion to how near it is. In the more distant ones, on account of their remoteness and the weakening of its power, it becomes faint, so to speak. Thus, just as the source of light is in the Sun, and the origin of the circle is at the position of the Sun, which is at the center, so in this case life, the motion, and the soul of the universe are assigned to the same Sun.45 He needed this analogy in order to explain not only the order of the universe but also its special arrangement. The order is given by the formation of the five regular bodies enclosed within spheres, which are measured by the diameter of the sphere of the earth. However, the special physical arrangement, the distances between them and their speeds and motions, in relation to each other is to be given by a proportion—better still by harmonic proportions. Kepler was not satisfied with static symmetry, with the reduction of the universe to mere Pythagorean numbers, or even to Platonic geometrical bodies. He attempted to reduce physical phenomena to something more dynamic—namely, to mathematical regularity based on proportions. Proportions can accommodate different sizes and the progression from one quantity to another, as long as this transformation keeps the same proportion. These proportions are not speculative. There is no need for special visual abilities to perceive them, since they are embedded in the visible world, in its shadows, as well as in the human mind.46 Light is the natural vehicle for such visibility and also one of the few natural phenomena to deserve geometrical treatment throughout antiquity and the Middle Ages. Kepler’s attempt to associate a geometrical explanation of planetary motion with the geometry of light, therefore, seems obvious enough. This attempt to formulate a comprehensive analogy between planetary motion and its relation to periods of time and the diffusion of light encountered certain difficulties, however. The decrease in planetary speed did not exactly follow the increase in distance, and was not always in the same proportion for all the planets. In the final published version of his Mysterium cosmographicum (1596), Kepler left these ideas in an incomplete state. In the following years he returned to these issues in an attempt to improve and reformulate this scheme until it received its final structure in his Astronomia nova (1609) and Harmonices mundi (1619) as what is known as the second and third laws of planetary motion. Yet what is important here is that he never dropped the basic analogy between the solar moving force and light.

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In a letter to Maestlin on August 19/29, 1599, Kepler raises his latest ideas on the role of proportion in structuring the universe. There are two main problems in his initial description of the universe in Mysterium cosmographicum. The first is that there is no simple relationship between the weakening of the moving virtue that emanates from the sun and the length of the solar ray. If that had been the case, then that proportion would have been not only between two proximate planetary orbits and speeds but also between remote ones, which it was not. The second, and for Kepler the more basic problem, was the appropriateness of only selected proportions, the harmonious ones, for the job at hand and their correspondence to the regular bodies that define the geometrical structure of the universe. “There are infinite ratios [logoi]. . . . However, it is still to be demonstrated that only a few [ratios] are distinctly beautiful, and are [necessary] to the similitude of regular solids.” This selected and complete set of logoi is to be found naturally as “the sense reveals them in music.”47 How can one limit these logoi, since these are not only numerical values in proportion to other numerical values but specific and special logoi between commensurables? In searching through different geometrical options (the ratio between single bodies, or between plane angle and solid angle, etc.) for the origin of such ratios, Kepler could not find any geometrical relation between celestial bodies that would give the complete set of eight musical proportions. The movement from geometry to harmony is based on the equivalence between geometric knowables (scibilitas),48 that is, those that can be constructed with the straightedge and compass. Musical consonances are determined by the arcs subtended by the knowable polygons. However, in order to limit the number of possible subtended arcs to correspond to the number of harmonic intervals, Kepler adds a supplementary principle: the two parts of the circle, subtended by one side of a polygon, must be in consonance with each other and with the total circumference. Thus, Kepler succeeds in setting up a genealogy of basic harmonies which correspond to the complete set of musical harmonies. All the basic consonances are produced by the subdivision of the circle, by its diameter (taken as a degenerated polygon), a triangle, a square, a pentagon, a hexagon, and an octagon. Therefore, in dividing the circle in two or in four a regular ratio is constituted, corresponding to the perfection of regular bodies, and the parts often have a ratio between themselves and the whole, [as the ratio] of some demonstrated regular figure [to the whole circle]. This is the definition. For just as the regularity of bodies is defined through the regularity

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of planes, so the perfect ratio, and as I say, the solid ratio is considered through simpler ratios. . . . A solid ratio exists, and often the subtended arc is related to the diameter, and the [ratio of] the arc to the rest [is] expressible by the same mode and number, as the other arc [is] related to the subtended [arc] and [as it is related] to the whole circle. Kepler takes for granted the establishment of relationships between the regular solids in simpler proportions deduced from the division of the circle—which for Kepler is the basic geometrical form!49 However, these proportions are not merely abstract geometrical calculations, but pervade and constitute the entire universe (xosmopoihticoi) since “nature loves these relationships in everything that is capable of thus being related. They are also loved by human reason, which is the image of the Creator.”50 From these eight ratios one can produce a complete musical philosophy, as well as the rules that govern musical instrumentation. However, not only sounds receive these logoi but also colors. “As in tones from G to g [there] are infinite voices, but nature makes [only] few of these discriminate according to geometrical proportions, thus between light and darkness there are infinite variations, but nature teaches to the eyes [only] few names, which, no doubt are constituted in the terms of the said ratios.”51 The spectrum of colors is a result of refraction (as it happens in twilight, in solar eclipses, and in the rainbow). “Thus frequently I have reflected, whether the proportion of the angles of refraction constitutes the terms in which it is said that a color is green, blue, etc. So, if refraction is direct, [that is] the angle is zero, it creates yellow, or rather this is what is most splendid in yellow, light itself. . . . Thus by the reason of the divisions of the right angle colors are constituted.”52 Kepler continues his speculation and suggests that these geometrical considerations may be found in all other sensual phenomena, such as smell and touch. Further, these ratios are also found in the relationship between the different members of the human body. Kepler concludes that all these ratios are originally the product of geometrical figures: “Figurae sunt causae λογων.”53 These proportions and harmonies are especially relevant in astronomy; they could be traced in astrological aspects as well as in the motions and speeds of the planets. Kepler says, “Therefore, what is the proportion of motion? It is the same as musical harmonies. Give the heaven air, and the truest harmonious tune will prevail [and be heard].”54 The geometrical proportions that govern the motions of light, its refractions and reflections, supply Kepler with a powerful key to physical phenomena, and

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even more so “to disclose the very Pythagorean harmonies of the universe.”55 Qualities, such as colors, and physical motion can be directly related to quantitative values, and thus reduced to full-fledged geometrical regularity. In his 1602 De fundamentis astrologiae certioribus, Kepler applies this insight to the physics of the heavens, and especially to deciphering the modes of planetary influences of earthly occurrences. In this short treatise Kepler attempts to glide from his role of a mathematician to physical conclusions, declaring in the title that in his role as mathematician he proposes a “physical prognosis” intended for the philosophers (i.e., philosophers of nature). In the opening sentence of the first thesis he merges, in a sleight of hand, the duty of the mathematician and the duty of the philosopher: “It is commonly considered the duty of the mathematician to write annual prognostications. With that in mind, I have decided, therefore, to fulfill this obligation for the coming year 1602 from the birth of Christ Savior by confining them [the prognostications] not so much to the curiosity of the public as to the duty of the philosopher.” The only way mathematicians (that is, astrologers) can fulfill their professional task is by discovering the physical causes underlying their practices. Applying this geometric-physical analysis to “other specific fields” would enable the astrologer to “exceed the bounds of his own profession” and to “become a farmer, doctor, chemist, etc.” In order to obfuscate the disciplinary borderlines, Kepler posits two principal tenets. The first is that one can learn of celestial objects through comparing them with earthly phenomena: Philosophers will forgive me as I draw conclusions from bodies that we investigate close at hand to heavenly bodies. For clearly philosophers themselves disclose no differences at all among those bodies. Why, as they probably think, should they transfer base and terrestrial differences with equanimity to heavenly bodies? But it is preferable to say something that falls into no palpable absurdity than to say nothing at all. And clearly I am not sure whether we ought not to call divine and heavenly rather that quality in the carbuncle gem that produces its little light, than that elementary quality of the sun’s for heating.56 The second principle is that light is the chief communicator of heavenly influences to the earth and through its agency one can gain knowledge of the distant heavenly bodies: “The physical means by which the planets influence the earth is the light that comes down.” Since matter is inert and cold,57 the rays of light flowing from the sun as well as from the other celestial bodies are assigned with the function of carrying forth motion and life to the sublunary region.

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For cold and dryness are not positive dispositions, but ones [that arise when] deprived of light and related life. For cold and dryness are greatest where there is an absence of all light, all life, and thus all heat. Since, therefore, nothing descends to us from heaven but the light of the stars, certainly cold and dryness will not come down by themselves.58 Kepler borrows from traditional optics the differentiation between direct and reflected radiation, and applies it to the physical influence the planets exert upon the earth. In contrast to Aristotle’s linguistic scheme of four elements based on simple pairs of oppositions (dry/moist; cold/hot), Kepler supplements his distinction (direct and hot / reflected and moist) with a geometric formula consisting of three elements of mathematical proportion: two contraries and a mean. The result is a more complex grammar of possible physical effects produced by light’s radiation: “The physical means by which the planets influence the earth is the light that comes down. Insofar as it is reflected light it humidifies (as with the moon), and insofar as it is radiated light it warms (as with the sun). Further, each type of light can exist in three degrees: excess, moderation, or deficiency. These six types differ from the four tactile sense contraries of Aristotle (hot, cold, wet, and dry) by the inclusion of the moderate term between the extremes, which is suggested by the example of geometry.”59 The main element that determines the degrees of light’s power to influence the earth is the reflecting surface of the planets and its color and shape. Since we have deduced their light as borrowed from reflection, we must consider the varying kinds of reflection on the basis of the difference that exists on varying surfaces. I am not speaking now about that reflection that issues from a mirror and is reflected from any point of it to only one other point; but rather [I speak] about the reflection that we see from some wall, that is of uneven and rough surface, which reflects the light with which it is illuminated, tinged with the color that it [the wall] possesses, from any point of it onto the whole hemisphere.60 All this pertains to a radiation from one single celestial body to the earth, and it depends on a visible contact between them, that is, the celestial body must be above the horizon in order to affect the earth. Another and more sublime mode for the heavens to produce effects on the earth consists of the angle the light rays of two stars or planets create on the earth. In this case the heavenly bodies do not have to be visible over the horizon, and one may as

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MEASURING SHADOWS The New Optical Narrative

well imagine these rays and their earthly meeting point. “For [this cause] does not draw its power along straight lines from single stars, but it takes into account the rays of pairs of stars as they meet on earth and form a geometric or non-geometric [αλογως] angle.61 And this [cause] is not extinguished along with the very face of the silent moon, when no rays descend to earth, but the descending ray is then imagined. It is not impeded when the earth is placed between us and the stars; but it makes the stars hiding below [the horizon] also efficacious above.”62 Geometric and harmonic angles are active, and their effect is immediate and not dependent on the amount of light emanating from the star or reflecting from its surface. The agency of light thus facilitates the intimate relationship between geometry, motions, and physical qualities: “The motions of bodies, which constitute the life of the world, are all perfectly in tune, or act strongly in unison, when they take their proportion from the regular planes [of the solid figures]. For as the plane is the representation of a solid, so the motion is the representation of a body.”63 Light is not corporeal, and it flows instantaneously in all directions. Its immaterial emanation, however, is subject to the laws of optics only: “For their method of operating consists in a certain flowing out of light continued as far as these sublunary bodies; although this flowing out is without matter and time, nevertheless it is not without quantitative dimensions. For it is made in a straight line; it is attenuated with distance from a star; it increases or decreases with the face of the shining planet; it is impeded by the interference of an opaque body; and, on the other hand, given the visible presence of the star, it flows continuously.”64 Investigating the geometrical operation of light, and especially the angles traced by its radiation, provides Kepler with a key to the physical world. The motions of light are the source of life in the universe and an embodiment of God’s creative power. “A man is like God in the universe, and Man’s dwelling place is the earth, just as God’s, if he has any material dwelling, is certainly the inaccessible light of the Sun.”65 In the Mysterium cosmographicum, light and visual experience are vehicles for contemplating divine order. Observations are the material substrate of a pensive mind in its abstract meditations, which aspire to disclose the higher realm beyond one’s daily corporeal experiences. In the dedicatory letter of his treatise, Kepler reiterates Aristotle’s celebration of human desire for knowledge for its own sake. Kepler, however, detaches mental meditation from sensory experience, emphasizing its transcendent and divine origin.

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We accept painters, who delight our eyes, musicians, who delight our ears, though they bring no profit to our business. And the pleasure which is drawn from the work of each of these is considered not only civilized, but even honorable. Then how uncivilized, how foolish, to grudge the mind its own honorable pleasure, and not the eyes and the ears. . . . For would that excellent Creator, who has introduced nothing into Nature without thoroughly foreseeing not only its necessity but its beauty and power to delight, have left only the mind of Man, the lord of all Nature, made in his own image, without any delight?66 While the eyes find pleasure in visual experience for its own sake, the mind is not satisfied with such worldly delights, and is searching for the hidden causes to which such experience can only allude. “There is no need to ask why the human mind undertakes such toil in seeking out these secrets of the heavens. For the reason why the mind was joined to the senses by our Maker is not only so that Man should maintain himself . . . but also so that from those things which we perceive with our eyes to exist we should strive towards the causes of their being and becoming, although we should get nothing else useful from them.”67 Kepler’s interest in light and the practice of astronomical observation has much higher stakes than solving the technical problems Tycho Brahe had encountered in his observations and measurements of solar eclipse. From his early work Kepler regarded light as the divine agent operating and ordering the celestial machine. What was an interesting speculation in the Mysterium cosmographicum of 1596, however, turned into the foundation for a new celestial physics in the astrological treatise of 1602. It is clear from this treatise that before Kepler could propose his new astronomy, he had another crucial task. Kepler’s research into the foundation of optics in 1604 was not concerned only with technical details of observation and instruments. In deciphering light and its different modes of radiation and reflection, Kepler aspired to establish the principles for the understanding of the working of the heavens. These investigations, however, also transformed Kepler’s ideal of knowledge: instead of meditative contemplation of abstract geometrical bodies, Ad Vitellionem celebrates carefully executed manipulation of rays of light within the camera obscura. The astronomer is no longer brooding over drawings of different figures, speculating on their arrangement, but is now applying an instrument to produce and measure light and shadows in order to disclose the hidden causes over and beyond appearances.

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MEASURING SHADOWS The New Optical Narrative

Kepler’s plan for his treatise on optics was to set a theater with a play within a play. Staging the camera obscura allows Kepler to present light in its multifunctional capacity. Light initially appears in its epistemological functioning as a carrier of information. Light, and not the “visual rays,” is the mediator between the external world and the human sense of sight. Kepler expects light, as the main epistemological agent, to mediate celestial information to the astronomical observer. Simultaneously, light appears in a different role: it is expected to fulfill a physical task, connecting and mediating solar moving virtue to the planets, or at least to supply Kepler with the rules for such a propagation. Lastly, optics as the science of light (à la Kepler) is a geometrical science, and thus Kepler expects light to mediate between mathematics and physics. Geometry will be, so Kepler assumes, the common ground between his new optics and the celestial physics he is hoping to execute in the near future. Ad Vitellionem is Kepler’s initial attempt at a full-fledged account of physical phenomena in geometrical terms. It is not a hastily written treatise meant only to satisfy the demands on Kepler as an imperial mathematician. It is a well-calculated treatise intended to provide Kepler with both an epistemological foundation and justification of his astronomical research and a winning example of a fully worked out geometrical physics.

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2.

“SEEING WITH MY OWN EYES” Introducing the New Foundations of Scientific Knowledge

geometrical images for common notions “Albeit that since, for the time being, we here verge away from Geometry to a physical consideration, our discussion will accordingly be somewhat freer, and not everywhere assisted by diagrams and letters or bound by the chains of proofs, but looser in its conjunctures, will pursue a certain freedom in philosophizing—despite this, I shall exert myself, if it can be done, to see that even this part be divided into propositions.”1 With these contorted sentences Kepler begins the first chapter of Ad Vitellionem, “On the Nature of Light.” The awkward style shows that Kepler is well aware of the thorny challenges he is facing in introducing his new science of the optics of the heavens. This new science, according to Kepler, will have to transgress and erase the boundaries between mathematics and physics, two domains that Kepler’s predecessors, the medieval perspectivists, invested so much of their intellectual energy in keeping apart. John Pecham, for instance, in prefacing his Perspectiva communis of ca. 1280, asserts that while optics benefits from both mathematics and physics, his aim is to set clear demarcating lines between these separated parts of optics, “according to the type of the subject matter,” since existing treatises “are presented with great obscurity.” Pecham’s aim is to avoid the confusion that especially haunts the students of optics: “As the master [Aristotle]—the light of all men—deems the investigator of light worthy of illumination.”2

Such worries do not bother Kepler, who on the contrary understands that his endeavor necessitates such an act of transgression and confusion. The new science transgresses the demarcating line between the heavens and the earth; blurs the dividing line between mathematics and physics; is supposed to be founded a priori; and finally integrates instruments and artificial construction into the knowledge of natural phenomena. In order to confront all these challenges, Kepler has to provide a new starting point that will simultaneously allow the intertwining of mathematics and physics and be able to produce a coherent body of knowledge. This move means formulating new premises that will allow such a priori reasoning together with a way to reposition instruments of observation as vehicles of epistemological certitude. Furthermore, Kepler has to formulate geometry anew so as to allow him to capture and analyze physical motions. This last move involves a radical confrontation with deeply entrenched notions of geometrical optics of the perspectivist tradition since Euclid. The crucial step is taken at the outset of the treatise through Kepler’s bold move of treating the nature of light in the framework of a treatise on perspective. This in itself already assumes that physical matters are integral to mathematical sciences,3 an assumption that requires a new set of “common notions,” that is, a novel axiomatic structure. For such a structure, Kepler cannot turn to traditional treatment of optics. From Euclid onwards, treatises on optics had begun either with a set of optical phenomena or with mathematical definitions. Both options assumed a separation between the mathematical realm and physical phenomena: the first could not have followed the a priori path Kepler wishes to take, while the second assumed mathematics as a mental abstraction with an uncertain relationship with the physical world itself. Where is Kepler to find definitions and axioms that will enable him to establish his science of optics on them? “In place of Common Notions I am giving a preliminary admonition on vocabulary.”4 Kepler, in humanist fashion, first turns to etymological analysis in order to detect the origins of the words applicable to the behavior of light’s rays: that is, to find “in which [words] of the common folk, the use of the thing and its nature is most properly experienced.”5 Kepler points out that linguistic use is based on analogies and metaphors that relate different things with the same words. For instance, the Latin flectere is most accurately used for something that is curved under the influence of some force. However, as the forces cease it is supposed to straighten out. Thus, the application of this word to the behavior of light is through analogy, and “for the sake of avoiding the undesirable implication, the word flectere is clearly to be eschewed”6 (my emphasis). Kepler uses

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the same arguments in the case of other Greek and Latin words for refraction, concluding that “because rays are affected in one way by mirrors and in another by water (for the one rebounds from the mirror towards those parts whence it came, while the other bends down from the surface of the water into the depth, and towards those parts opposite those whence it came), let us therefore follow here the convention of the optical writers.” It should be noted that in the final account the words that Kepler suggests the optical writers use are either taken from poetic sources (Virgil’s Aeneid) or linguistic novelties made up by Kepler himself.7 Kepler’s distrust of language as a conveyer of truths about the nature of things reflects humanist theories of the role of metaphor in language in general and in poetic language in particular. Various sixteenth-century philosophers and theoreticians of rhetoric were concerned that the relationship of words to things is liable to become merely metaphorical. Metaphor occupies an ambiguous status in the late Renaissance worldview. While it was acknowledged as a sign of the divine—“For just as God brings forth that which is, out of that which is not, so wit makes something out of nothing: it makes a lion become a man, and the eagle a city”8—in the final account it was recognized as a fallacy. Aristotle himself defined metaphor as “the application of a name by transference.”9 In terms of these Aristotelian poetic notions, Kepler’s aim is to show that even the word “in general use among the people” is, in the case of light, “the application of name by transference either from genus to species, or from species to genus, or from species to species, or by analogy.” This Aristotelian attitude survived into the Renaissance and courtly culture, as Castiglione warns the courtier: “Do you not realize that these figures of speech which give such grace and clarity to what we say, are all abuses of grammatical rules, but are accepted and established by usage because . . . they are pleasing and insinuate their charm through our sense of hearing?”10 Puttenham asserts the same in his Arte of English Poesie (1589). He rejects metaphor as a “Figure of Transport”; metaphors “deceive the eare and also the minde.”11 While Kepler followed the precepts of humanistic philology in adopting simple, normal terminology, he rejected language as a solid basis for his research in natural philosophy. If language, at least in the case of optics and light, is nothing more than metaphors, then it cannot serve as a basis for scientific investigation into the true nature of light. The only thing that is accepted by “all philosophers and optical writers,” and thus can serve in a way as “common notion,” is that there is a certain relationship between physical bodies and their motion and light. What, then, can serve as a basic definition that will capture

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MEASURING SHADOWS “Seeing with My Own Eyes”

both this relationship between light and physical motion and the mathematical aspects of optics as a science of light? In groping with this problem, Kepler takes a bold leap: the relationship between light and physical motion is founded not on a verbal definition of the nature of things but on a certain geometrical imaging of the Godhead. Kepler abandons the verbal dimension, with its ambiguities, and follows instead the path of figures, shapes, and visual representations. His point of departure is not a qualitative definition of light but divine geometrical archetypes: At first the nature of all things created by God, as much as their own essence is able to, must represent [Him]. . . . [The Creator] found nothing more beautiful or more excellent than Himself. That is why, when he conceived the corporeal world, he gave it a form as like Himself as possible. Thus originated the whole genus of quantities, and in it the distinction between the curved and the straight and the most excellent of all figures, the spherical surface. For in forming the sphere, the all-wise Creator produced for his pleasure the image of his holy Trinity.12 The reader moves from the careful linguistic analysis to the simple, and for Kepler atomic, geometric components—the point, the straight line, the curve, and the sphere. These are the clearest representation of the Trinity, which is the quantity that expresses the Godhead. The relationships of the Trinity to the sphere are for Kepler beyond mere analogy: “Therefore, the central point may be regarded as the origin of the sphere, the surface as the image of the innermost point, and each path coming to the surfaces an infinite movement of the point outside itself, so as to produce a certain equality of all movements.”13 Similar to the Trinity, the sphere is a shape made up of three components held together. So the sphere becomes a unique body, which is made up of three (i.e., the point, the surface, and the radius). Moreover, the sphere is transformed into a physical body: it is a geometrical figure constructed through constant motion from the center toward the surface. This motion does not exist in time, as the undetectable movement of the perceiving eye as it gazes at the figure of a sphere. That is why one perceives perfect unity and harmony when perceiving a spherical figure: “Hence, everywhere between the point and the surface, there is absolute equality, closest unity, loveliest harmony, connection, relation, proportion, and symmetry. And although there are plainly these three things—center, surface, and space between—nevertheless they possess such unity that none of them could be removed, even in thought, without destroying the whole.”14

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This image of the expanding sphere in the context of both theological speculations and metaphysics of light is not new with Kepler.15 An ongoing tradition from Plotinus, through al-Kindi and Grosseteste to Nicolaus Cusanus, employed this image in similar contexts. Yet these philosophers applied this image with a view to metaphysical ascension. Grosseteste, for instance, conceived his physico-mathematical speculation on light as an exercise in meditation, supposed to transform the reader and to allow a deeper speculation on perfect numbers and harmonic, Pythagorean ratios. His short text De luce begins with a physico-theological account of creation of the corporeal world that culminates with a picture of the entire world in front of the reader. It then reverses, abstracting from this image of the world the numerical ratios that lead the mind to the uppermost limit of intellectual speculation. Grosseteste ends his exposition with a celebration of the number ten. From this the road is open to a new kind of superrational understanding beyond scholastic dialectic. Cusanus, who in all probability is the immediate source of Kepler’s metaphysical engagement with the sphere,16 presents a playful absorption with circles and spheres that involves motion. However, he ends his speculations with a paradoxical climax, mystically asserting that “God is a circle whose center is everywhere.” Kepler’s aim is different. His image of a sphere is thoroughly constructed through motion, and Kepler takes it not as a step toward a mystical understanding of the divine, but as a tool toward a scientific description of physical nature. The sphere is not only a representation of God but an image of the corporeal world, and spherical motion is something assumed by all physical bodies: “This, then, is the authentic, this is the most fitting, image of the corporeal world, which anything that aspires to the highest perfection among corporeal created things takes on, either simply or in some respect.” All things aspire to that perfection by radiating certain virtues into geometric dimensions, which in their turn aspire to sphericity. Magnetic virtue is one such emanation. Another is the virtue that moves the planets. However, light is the most beautiful and important of these radiations: How marvelous, then, if that source of everything beautiful in the world, which the divine Moses introduces into the newly created matter on the first day [of creation] for the shaping and animating of all things—how marvelous, I say, if this source and most excellent thing in the whole corporeal world [i.e., light], origin of the animal faculties and link between the corporeal and spiritual worlds, should proceed according to

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MEASURING SHADOWS “Seeing with My Own Eyes”

the same laws by which the world was to be ordered. Thus, the sun is a particular body, in which resides that faculty of communicating itself to all things that we call light. On this account, it requires the middle place in the entire world, the center, so that it can uniformly and perpetually diffuse itself into the whole sphere. All other things that participate in light imitate the sun.17 Light is the most refined expression of the basic spherical representation of the divine. Its nature is not any specific quality of the things that radiate it, but a geometrical figure, a visible image of a spherical propagation, or in other words of geometry in motion. If light represents most accurately the spherical image of the Godhead, then visibility and visual perception (the result of the activity of light) are the surest path to recognizing the true nature of things as well as of the soul and its faculties. Whereas the verbal logos is conventional, arbitrary, and merely human, the true and primary language of creation is visual and geometrical; it is a language of logos, as beautiful and harmonious geometrical proportions formed through the geometrical division of the circle. This image of spherical propagation is turned into Kepler’s fundamental premise for his treatment of light and of his science of optics. Kepler believes that the principles of traditional perspective can actually be derived from these speculations: “From this consideration there arise, in a way, certain propositions, which are among the principles in Euclid, Witelo, and others.”18

light and motion Kepler’s point of departure is that light is an embodiment of physical motion, and that motion is a constitutive aspect of the fundamental geometrical shape of a sphere. The “certain propositions” contain Kepler’s explication of the manner in which a geometrical sphere is created by the motion of the outflowing of light: “We have said that light strives to attain the configuration of the spherical. However, its true geometrical genesis lies in the equality of the intervening spaces, through which the middle point is spread out into a surface.” This is Kepler’s solution to the Aristotelian problem of metabasis: light is corpus geometricum and at the same time a body in constant physical motion (or “motion is a property of light”).19 What makes this conflation of mathematics and physical motion possible is an act of separation: light’s essence is physical motion in itself without material substance—“light has no matter, weight, or resistance.” Furthermore,

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light has no corporeality or solidity, for “to light . . . there belong only two dimensions.” Light is made of two-dimensional minima flowing out of a central point (“it always goes from a single source to all regions”) to create a sphere; it is truly geometry in motion. Kepler, however, is careful to define light’s motion as flowing out and not as some Neoplatonic emanation: “To light belongs not uniting, but something similar to separating, and a certain most violent projection or flowing out.” This concept of light allows Kepler to analyze motion per se in geometrical terms with no apparent contradictions. Kepler’s initial premise is that light is “the chain linking the corporeal and spiritual world.” As such, light is the carrier of the mathematical laws of creation and a central agent in communicating it to the world (by being an agent of motion) and to the human mind (through the sense of sight). This premise, originating from positing the visual image of a sphere as an axiom, enables Kepler to envisage a mathematical physics that will overcome Aristotle’s admonition against metabasis.

observing through instruments Laying the foundation for a mathematical physics of light, Kepler’s new optics had to confront two further urgent problems: it had to justify the astronomer’s presumption to measure the heavens, and to introduce artificial instruments of observation as a primary and legitimate vehicle of knowledge of the natural world. Both problems relate to the epistemological status of the camera obscura, especially in the context of Tycho Brahe’s observations of solar eclipses. The measurements Tycho carried out following these observations resulted in a shrinking lunar diameter compared with that on a regular night. These results cast a shadow on Tycho’s entire research program and its emphasis on accurate instruments of astronomical observation. Tycho invested his effort in improving his instruments to their utmost. The confirmation of his instrumental abilities was crucial to fulfilling his ambition to reestablish the empirical foundations of the science of astronomy. For this reason he published the Mechanica, a list of instruments for the restoration of astronomy with large detailed pictures and extensive descriptions of the construction and use of each of his twenty-two most important instruments. Through these discussions one can get a fair picture of the limits set on Tycho’s application of his instruments by the Aristotelian paradigm. In the case of astronomical aiming, for instance, until Tycho this was done through pinhole sights attached to the ends of the alidade. Tycho found this method to locate and sight on stars extremely difficult. His

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MEASURING SHADOWS “Seeing with My Own Eyes”

first attempt to achieve a better sighting was an adjustable slit sight, consisting of two posts, at the front and the back of the alidade. As the observer aligned the pointer on a star, the width of the slit diminished, until perfect alignment was achieved at the vanishing point of the star. This instrument had one shortcoming, however. The posts could give only one accurate alignment, and any attempt at a second alignment introduced an eccentricity into the reading. Tycho approached this problem first by calculating away the distortion, and then by ingeniously improving the sighting. However, he never questioned the instrument itself as a distorting mediation between the eye and visual reality, but only searched for its technical deficiencies, either solving them or calculating them away, in the hope that a technical solution would be found in the future. On April 21, 1598, Tycho reported to Maestlin on some observations he had taken of the solar eclipse in February that year and the two lunar eclipses which preceded it. His conclusion about these observations was quite disturbing. “It must be recognized that the moon when it is on the ecliptic and when it is new does not appear to be the size which it is at other times at full moon even though it is then removed the same distance from the earth; but it is, as it were, constricted by about one part in five.”20 Tycho, who could not find any technical fault with his instruments, concluded that the paradoxical effect must represent some physical reality, and that the diminution in the lunar diameter must be due to the disappearing lunar atmosphere under the illumination of the sun. Even if the strange phenomenon was due to some optical distortion, this effect was located in the relationship between the observed objects (the sun and the moon) and not in the optical instrument. This technical problem of observation and mismeasurements loomed over the Tychonic research program, which prided itself on accurate measurement and sophisticated instruments. The insecurity of astronomical knowledge in the late sixteenth century in general, and especially in regard to human ability to measure the heavens, contributed to this atmosphere of crisis that Kepler perceived. One should also note Kepler’s involvement in the polemic between Tycho and Ursus in those same years on similar topics concerned with the epistemological status of astronomical knowledge.21 In this context, Kepler came to realize that technically improved instruments in themselves would not suffice and that a new theory of artificial observation was needed. The problem was not in any shortcoming of one of Tycho’s instruments; rather, a general reformulation of the position of the instrument as a mediator between visible reality and the human sense of sight was called for.

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The epistemological status of instruments of observation as distorting mediation between the human eye and the visible object had been a commonplace since antiquity. Direct vision was privileged over any sort of mediation. This dichotomy between direct and mediated vision underlay the various medieval discussions and treatises on optics. The common ground was the Aristotelian assumption that human perception in general and visual perception in particular forms a continuous flow from the object to the eye. The object, according to this assumption, impresses its form on the transparent medium that surrounds it, and the medium transmits this form and impresses it on the eye. In such a scheme, any instrumental intervention would only obstruct the flow of forms through the medium to the eye and consequently distort the true picture of reality received within the eye. The result was a clear-cut differentiation between the flow of species directly into the eye and the mediated way they arrived in the case of optical instruments such as mirrors. This was not only a technical difference but combined epistemological as well as methodological borders. In Pecham’s Perspectiva communis we find the following definition of the image in a mirror: “What then is an image? I say that it is merely the appearance of an object outside its place. For example, sometimes the eye judges one thing to be two . . . because the object appears not only in its true place but also outside it. So in the present case, it is the object that is really seen in a mirror, although it is misapprehended in position and sometimes in number.”22 The resulting mirror image does represent the object, but this representation adds information beyond necessity and thus contains false elements that disrupt perception. This superfluous addition is the result of the unnecessary stages of reflection, which disrupt the straight flow of species from the object to the eye. The basic assumption is that “[s]traightness is naturally associated with the propagation of light (as well as with any other action of nature), it arranges and orders nature, for every action is strong in proportion to its straightness.”23 In the mirror the species hit the reflecting surface and only then reach the eye. This breaks the straight ray into a tortuous path: “A species produced by a visible object has the essential property of manifesting the object of which it is the likeness. . . . Therefore, even though it is reflected, it maintains its essence and thereby reveals the object—albeit in another position.”24 The image is not only in a wrong position but has no material-physical reality: “It is evident that nothing is impressed there.”25 The conclusions are clear: an instrument as a mediation of vision disrupts the natural straight flow and adds unnecessary visual information. Furthermore, since the natural straight flow of species has no geometrical variation, there is no need for elaborate geometric diagrams

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and explanation. In contrast, any explanation of reflection and refraction that disrupts the flow of information is thoroughly mathematical. In this way, to Pecham’s great satisfaction, the separation between the physical aspect of vision, which needs only a slight modicum of geometry, and the explanation of virtual mirror images with their thorough geometrical elaboration is preserved.

the mathematics of pinhole images To this critical attitude toward instrumental mediation was added a specific problem of pinhole image formation. This problem was involved in the camera obscura as the main astronomical instrument for observing and measuring eclipses in the sixteenth century. It was a locus classicus of ancient and medieval optics, and various theoreticians had offered different solutions to several of its aspects since its earliest appearance in Pseudo-Aristotle’s Problemata. The main problem was “Why does the sun penetrating through quadrilaterals form not rectilinear shapes but circles, as for instance when it passes through wicker-work?”26 In the Middle Ages, the different aspects of this problem were explored. Emphasis was given to the relationship between the distance of the screen on which the image is formed from the pinhole and the distance of the pinhole from the light source. The result of this relationship is that light rays passing through any small aperture would initially produce the shape of the aperture and, as they receded further beyond the aperture, would take the shape of their source. The different solutions formulated attempted to accommodate these aspects into a coherent whole. However, toward the end of the thirteenth century, it seemed that these attempts had reached a dead end. John Pecham’s Perspectiva communis presented the various options, rejected them, and formulated an obscure solution. Pecham presented three options, the first of which took the roundness of the sun (the source of light) as the cause for the roundness of the image. He rejected this, since the image would have been rounded immediately after passing through the aperture, which obviously was not the case. The second explanation sought a geometrical cause dependent on the intersection of pyramids of light in the aperture. Pecham did not elaborate on this explanation much further than his predecessors and asserted that the geometrically based explanation “non totam causam ministrare.” Instead, Pecham preferred the explanatory power of “the nature of light.” “The spherical shape is associated with light and is in harmony with all the bodies of the world as being to the highest degree conservative of nature, all parts of which join

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together most perfectly within itself. This is why a raindrop assumes roundness. Therefore, light is naturally moved toward this shape and gradually assumes it when propagated some distance.27 The peculiar problematic of optics as a scientia media in the medieval sense comes to the fore. Mathematics can be applied to describe the propagation of light, and it can even explain (i.e., supply a causal account of) the formation of mirror images and other illusions (resulting from reflected and refracted rays of light), but it cannot be the sole supplier of the causes of real physical effects. When, in the early fourteenth century, Levi ben Gerson tackled this problem, he did this not as part of a systematic causal optical account, but as a heuristic device for the manipulation of pinhole camera images in the service of astronomical theory. Gerson explained the rounding of the corners of the image as a result of the intersections of sun rays at the aperture. This solution implies the later Keplerian solution, but, as Lindberg points out, this understanding is retrospective.28 There is no reason to assume that Gerson had thought about the convergence of the different solar images into a unified round image. Through the 1490s, Leonardo picked up this trail of reasoning from Levi ben Gerson’s work, probably in an attempt to follow his astronomical research program. A careful reading and analysis of his notes and diagrams exposes the peculiar character of Leonardo’s work. Despite his vast collection of case studies, Leonardo made almost no attempt to achieve a definitive diagram that would exemplify the geometrical essence of pinhole images. Even his seemingly general descriptions are ultimately particular in nature.29 Inspecting Leonardo’s diagrams, one gets the impression of a collection of instances of different shapes of the aperture, different arrangements of the pinholes, or a screen being positioned at different distances. Each of these depictions is accurate, but none of them is generalized, and no sequence of these diagrams can be regarded as an inductive argument. Leonardo, absorbed with an almost infinite multiplicity of cases, never attempted to present a definitive diagram.

illuminating the dark room It is at this point that Kepler picked up the line of argumentation and, through his solution to the camera’s image formation, aimed to solve simultaneously the problem of the diminution of lunar diameter in times of eclipse and that of the epistemological status of instruments of observation in general. As a premise, just as at the beginning of his treatment of the nature of light, Kepler posits a

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MEASURING SHADOWS “Seeing with My Own Eyes”

visual image, this time not an image of contemplation but a concrete material representation. After discussing the dead end reached by the ancient and medieval investigations of pinhole image formation, Kepler feels confused and the solution evades him. The solutions offered by the perspectivist tradition are inept and mix together elements of truth and error: Witelo “in dealing in these ambiguities . . . had not understood the true cause”;30 Ioannes Pisanus (Kepler’s mistaken identification of John Pecham) “rejected . . . the actual true cause, and withdrew with Witelo into the hidden recesses of the arcane nature of light.”31 All these authors were trapped in the confusion of Aristotle, who, in introducing “irrelevant cause,” brought “darkness upon those things,” so that all his successors could not understand what he meant. The ancient books and authorities were of no avail; a reinterpretation of the traditional solution just brought more confusion and vagueness to the whole matter. Again, just as in his maneuvers concerning the nature of light, Kepler contrasts the inept linguistic method to a visual-geometrical intuition. In order to break through this deadlock, he resorts to a visual demonstration: “Several years ago, some light shone forth upon me out of the darkness of Pisanus. Since I was unable to understand the very obscure sense of the words from a diagram drawn in a plane, I had recourse to seeing with my own eyes in space.”32 The emphasis is on “seeing with my own eyes.” The original expression in Greek is αυτοψια, an expression used in Neoplatonic literature for intuitive and direct perception of the matter at hand. Kepler posits his visual demonstration as a shortcut bypassing elaborate verbal description and allowing immediate comprehension of the behavior of light. This is not an experiment, nor exactly a model, but a visual representation (in the same way that a sphere is a representation of God and simultaneously of light’s radiation), in a sense theatrical, with a book in the role of the light source and threads for the light rays: I set a book in a high place, which was to stand for a luminous body. Between this and the pavement a tablet with a polygonal hole was set up. Next, a thread was sent down from one corner of the book through the hole to the pavement, falling upon the pavement in such a way as to graze the edges of the hole, the image of which I traced with chalk. In this way a figure was created upon the pavement similar to the hole. The same thing occurred when an additional thread was added from the second, third, and fourth corner of the book, as well as from the infinite points of the edges. In this way, a narrow row of infinite figures of the hole outlined the large quadrangular figure of the book on the pavement.

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It was thus obvious that this was in agreement with the demonstration of the problem, that the round shape is not that of the visual ray but of the sun itself, not because this is the most perfect shape, but because this is generally the shape of a luminous body.33 The material three-dimensional construct is in a way an immediate demonstration of the theory of pinhole image formation. The threads stretched from the corners of the book through a hole appear with no reference to a human eye, and the image Kepler traced on the floor was formed mechanically with no intentional significance. In the first chapter, Kepler defined the nature of light as motion and communication; now he can examine how to manipulate this motion through material apparatus. The different figurations light assumes are effects of material constraints such as distance, the size and shape of an aperture through which light flows, and the position of a screen. The human eye is only the last element in this process of the figuration of light, and its role is determined only because of “the limits of sense perception.”34 These limits construe the final image the observer perceives, but this image is a result of miscalculation and not of the actuality of the light falling on the screen. “It follows hence that from any point of a luminous surface a pyramidal ray is cast upon an interposed wall, the base of which ray is similar to the shape of the window. Thus the ray descending from the whole luminous surface to the illuminated wall consists of shapes that are potentially infinite, similar to the window, mutually overlapping, and falling approximately on the same region of the wall. These shapes individually would nonetheless have their own boundaries, if they were separated.”35 The eye perceives this multitude of individual shapes, each in the shape of the aperture, as part of one continuous shape not of the aperture (or window) but of the source of illumination. The image supplied by the instrument of observation is not a superfluous, additional, and unnecessary image of the same object, like Pecham’s mirror image, but its formation contains an element the human eye cannot grasp. This shortcoming of the human eye is supplemented, in an act of gestalt, by the mind that takes the blurred borders of the separate images as forming one line, tracing their layout on the screen (a layout that represents the shape of the source of light). For Pecham the rays of light constitute an image of their source—a triangular source will produce a triangular image, and so on. In the final analysis, the instrument (the aperture) can hardly affect this process, and the eye cannot intervene in the production of the image. That is why he rejects the intersection

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of the rays as a cause for the ultimate shape of the image, and looks for this cause in the ray itself. Kepler, in contrast, takes the eye as an active participant in the process, as an additional material constraint on the propagation of light. Therefore, in order to understand the formation of images through a pinhole, one has to remove the eye from the process. The course of the propagation of light through the aperture unfolds independently of the human observer.36 In that exact moment when the eye is understood as a factor in forming the image of the source of illumination on the screen of the camera obscura, Kepler creates the possibility of observing the act of observation. Through this process rays of light pass through an aperture, and because light is mathematical it is easy to analyze the process and its outcome. One can observe how the image is formed without the intervention of the human sense of sight, and assign the incompetence of that sense its due role in the final resulting image. This procedure allows Kepler to examine the role of the instrument itself as a geometrically shaped material constraint on the propagation of light. Tycho’s anxiety that if instruments of observation distort the resulted image, the whole project to reform astronomy will collapse, was unfounded. The mathematical nature of light as a transmitter of quantifiable shadows and radiations guarantees that an instrument with a known geometrical shape will not distort the image arbitrarily. The instrument shapes and configures the radiations according to set rules that make it easy to calculate away its effect on the resulting image. In this way Kepler dismisses Tycho’s fears and presents his solution to the diminution of the lunar diameter in the time of solar eclipse. Kepler’s analysis asserts that the “pointed shape of the luminous surface [i.e., the eclipsed sun] is necessarily mixed with the . . . shape of the window.” Yet that does not “detract from the likeness of the shape.” It only affects the size of the image as it “is augmented by the semi-diameter of the window.” And the perceived shapes of the horns of the eclipsed luminary “sent in through a round window . . . appear, not sharp (as they are in the sky), but drawn back and blunted by the little circle of the window.”37 With these distortions taken into account, one can calculate them away and find out that the lunar diameter in the time of a solar eclipse is the same as that of a usually observed full moon. This solution allows Kepler to celebrate the camera obscura and any instrumental mediation between the eye and the sky as a solid basis for exact measurements of the heavenly bodies. This instrument afforded previous astronomers, such as Reinhold, Gemma, and Maestlin, “the opportunity to apply” it to “measure the magnitudes of solar eclipses, the

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ratios of the diameters of the sun and the moon, and the inclinations to the vertical of the circle drawn through the centers of the luminaries.”38 The instrument enabled astronomers to avoid “the inadequacy [damnum] of the eyes, and . . . the error which generally occurs in a bare estimation.” However, these astronomers, and especially Tycho Brahe, did not take into account the limitations on the instrument itself: “believing in the theorem [that the ray of the eclipsed sun being similarly eclipsed when it is taken through a small hole] without restrictions they fell into a large error.”39 Kepler, unlike his predecessors, suggests that the solution of the pinhole image formation cannot be an ad hoc solution, but has to be taken as an integral part of a systematic explanation of light and vision. The astronomers’ mistake was in believing that the instrument cannot be allowed to intervene and distort the received image; therefore, their explanations were in the tradition of the scientia media, merely explaining away the phenomena. In Kepler’s explanation, light as a carrier of quantitative information is affected by the geometrical shape of the aperture, and therefore the process and the resulting image can be analyzed completely in a geometrical (i.e., quantitative) manner. The distortion introduced by the instrument does not affect its credibility; on the contrary, it only emphasizes the astronomer’s ability to manipulate light and to use its effects to measure the distant sky: “For aside from this no other sure procedure can be established for measuring something that happens in the sky.” This is Kepler’s straightforward answer to the age-old ridicule and criticism of astronomical knowledge. Lucian, Erasmus, and sixteenth-century astronomers all assumed the heavens to be so far off that they cannot really be measured and known. Turning the presence of light or its absence into the only communicative agents of visual reality makes discrimination between visual perception of distant objects and of objects right in front of one’s eyes redundant. Keplerian astronomy manages to bridge the gap between the observer and heavenly appearances in three moves. The first is the definition of light as a mathematical body. The second is by showing how to manipulate instruments of observation mathematically in order to achieve exact representation of these distant and almost invisible heavenly occurrences. The third is outlining a new language for scientific observation by reformulating the status of mathematics and its relationship to phenomena.

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3.

THE CONTENT OF KEPLER’S VISUAL LANGUAGE Abstraction, Representation, and Recognition

the shape of light and kepler’s analysis of conic sections In the first two chapters of Ad Vitellionem, Kepler establishes light and artificial instruments as the core of any mode of scientific observation. This move denotes a new understanding of optical phenomena: that is, a new conception of light’s behavior, a new theory of the eye, and finally a new mode for the application of vision to the study of physical reality. By implication, geometry, as the most significant visual language, acquires a completely new basis and origin. Instead of locating geometrical entities in abstract intellectual speculations divorced of matter, Kepler searches for geometrical regularities in the motions of the physical world itself. This has far-reaching consequences, leading to a new theory of cognition that emphasizes a different set of faculties within the human mind, altering its relationship to external reality. Kepler defines light as motion and communication and suggests that artificial instruments (i.e., any physically induced limitation on the motion of light) are what confer on light the particular shapes it acquires through its dispersion. Any crack or hole that stands in the way of the flow of light can shape a beam into a ray: “From now on, the word ‘ray’ is almost always taken to mean that portion or shape of light that has fallen upon a surface after passing through, and being configured by, some window or hole.”1 The instrument contracts the flow of light, forcing it to present the visual data it carries. The outcome is a complicated puzzle made of infinite stains of light configured by the artificial

hole or window. There is no occult power that flows from the visible object to the eye, but simple physical constraints on the flow of light: “It follows hence that through the individual points of any window, which are infinite, individual (and thus infinite) inverted images of the luminous surface are transmitted to the illuminated surface, following each other in the same order which the points of the window themselves have.”2 The passage of light through a pinhole is epistemologically perplexing. The projected stains of light are a mixture of the shapes of the luminous body flowing through the mathematical points of “any window” and the shapes of the aperture projected by the flow of light from any single mathematical point on the luminous surface. The only thing that allows the spectator knowledge of the source of the projected image is the order of the stains of light and their proportions. The meaning of the image is not in its essential resemblance to an external object but only as part of a patterned series of dots. Only the complete set of the infinite number of dots gives the shape of the external source of illumination, whereas each dot in itself is a mixture of the shape of the pinhole and of that of the luminous source: “The shape of a ray on the wall is a mixture of the inverted shape of the luminous surface and the upright shape of the window.”3 The final figure projected on the screen corresponds to the luminous surface only on account of the geometrical motion of the flow of light. In order to decipher the data treasured in such a serial assembly of dots of light, the mind has to apply a certain mode of mathematical operation. Kepler’s notion of the significance of series and structural sets is best exemplified in his mathematical analysis of conic sections. In Apollonius’s classical treatment of conic sections (ca. 200 b.c.e.), each curve is constructed separately from a visible concrete cone.4 Kepler’s treatment is the first to depart from such a cone and present a unified system of the sections and their property.5 Kepler does not consider the resulting curves as abstractions of specific sectioning of a specific cone, but rather emphasizes the analogical relation between them. Instead of geometrical constructions of specific cones, he begins his exposition with a quick inventory of the different possible cones. Then he specifies the five species of sections of the cones, emphasizing that the characteristics of these sections are independent of the specific kind of cone they bisect.6 This enables Kepler to present the reader with an a priori system of conic sections, where the value of each of its different elements is internally determined and interrelated to the other elements.7 In this system the age-old ontological dichotomy between the basic species of lines, the straight and the curved, collapses.8 Through systematic and self-regulating transformations, Kepler unifies these

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MEASURING SHADOWS The Content of Kepler’s Visual Language

3 Kepler’s system of conics, in Ad Vitellionem, chap. 4, 94. Reproduced by permission of The Huntington Library, San Marino, California, RB708372.

contradictory elements into one geometrical continuum: “For the line on the surface of a cone established by a section is either straight, or a circle, or a parabola or hyperbola or ellipse. . . . [T]here exists among these lines the following order, by reason of their properties: it passes from the straight line through an infinity of hyperbolas to the parabola, and thence through an infinity of ellipses to the circle.”9 The straight line is nothing more than an infinitely stretched hyperbola, and a circle is but an obtuse ellipse: “For the most obtuse of all hyperbolas is a straight line; the most acute a parabola. Likewise, the most acute of all ellipses is a parabola; the most obtuse, a circle. . . . Therefore, the opposite limits are the circle and the straight line: the former is pure curvedness, the latter pure straightness. The hyperbola, parabola, and ellipse are placed in between, and participate in the straight and the curved, the parabola equally, the hyperbola in more of the straightness, and the ellipse in more of the curvedness.”10 Kepler, however, is unable to establish this system of interrelated elements on firm geometrical grounds.11 Discarding Apollonius’s “cone-based” generation of the different kinds of curves, Kepler is unable to suggest the quantitative reasons for his system, and all he has to fall back on are qualitative analogies. Kepler celebrates these analogies as “the most faithful teachers,” which assist the scientist in exposing the “hidden secrets of nature.”12 Yet he knows that these analogies cannot deliver a geometrical proof for his system. Kepler puts forward instead a physical analogy, emphasizing the role of the conic sections in the analysis of curved mirrors that focus the rays of the sun. His description

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of the foci (probably a linguistic invention of Kepler’s) transforms the physical points into geometrical terms. Kepler explains how the single focus of the circle and the two foci of the ellipse are related, and continues: “In a parabola one focus D is within the section, the other is to be supposed on the axis, either outside or inside the section, removed at an infinite distance from the former one, so that a line HG or IG drawn from that hidden focus to any point of the section G is parallel to the axis DK.”13 Kepler can now complete his system of analogies: “It follows by analogy that, for the straight line, the focus (as we will use the term, in relation to the straight line, without precedent, to complete the analogy) in both cases coincides with the line itself, so that there is only one focus, as for the circle. Then in the circle the focus is at the center itself—the point furthest from the circumference—in the ellipse it is less far from the circumference, and in the parabola much less, in the end being the minimum distance away: that is, it falls on the line itself.”14 Kepler’s method for the construction of conic sections, and the a priori reasoning that brought this system into being, are not enough to make it meaningful.15 This system becomes meaningful only if the mind is able to recognize these curves in the movements of the physical world, that is, as a surface that would turn into parallels all the rays of light falling upon it and would describe a pattern of the angles of refraction. Kepler is looking for a geometrical surface that will have an exact physical meaning. Yet he has to admit that mechanics shows that a physical surface of this kind only resembles a hyperbola, and that “it is a little more acute than the hyperbola itself near the vertex.” Dissatisfied, he concludes his attempt to measure refractions by setting a challenge to his reader: “When you shall have acquired perfect knowledge of this surface by some procedure, know that you have achieved something great in mechanics.”16 This failure accentuates the two crucial elements that, for Kepler, make visual representation physically meaningful. The first is that particular elements of a series (geometrical curves or stains of light) acquire their meaning only through their interrelatedness with other elements of the same series. Deciphering the meaning of the system thus created does not depend on external bodies (the cone or the visible body), but only on its inner mathematical coherence. The second is the way motion is involved in the production of the visual representation. Motion unveils the geometrical proportions that constitute a visual representation as a coherent system. Philosophical knowledge is possible only on account of a certain correspondence between these two elements. This amalgamation of self-regulated structures, artificial order and physical motion, meant a reconceptualization of mathematics as a representational language. In

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MEASURING SHADOWS The Content of Kepler’s Visual Language

order to establish such an oxymoronic language (mixing structures and motions), Kepler had to reject deeply entrenched classical and medieval conceptions of cognition that had dominated scientific thought since antiquity. According to the traditional scheme of scientific cognition, “concepts are formed in continual dependence on ‘natural,’ prescientific experience, from which the scientific concept is ‘abstracted.’ The meaning of this ‘abstraction,’ through which the conceptual character of any concept is determined, is the pressing ontological problem of antiquity; it becomes schematized in the medieval problem of universals, and, in time, fades away completely.”17 Kepler’s move toward a representational language of mathematics, at the expense of abstracting mathematical entities from natural concrete experiences, redefines the relationship of mathematics to the natural world. Accordingly, as with Kepler’s treatment of conic sections, this new mode of mathematics implied that “[n]o longer is the thing intended by the concept an object of immediate insight. Nothing but the internal connection of all the concepts, their mutual relatedness, their subordination to the total edifice of science, determines for each of them a univocal sense and makes accessible to the understanding their only relevant, specifically scientific, content.”18 Kepler’s rejection of traditional notions of abstraction turns mathematics into a representational language and reallocates mathematical entities from some ideal world into the physical world, as well as into the divine and the human mind.19 This shift does not take place unknowingly, or as an unconscious epistemic rupture from the past. In formulating his new visual language, Kepler takes full cognizance of the optical tradition, rereading Aristotle, Alhacen, and Witelo and rewriting perspective on a different metaphysical basis. Thus, a new historical reading of Kepler must also account for his new mode of reading the optical tradition and his novel adaptation of Hellenistic and medieval visual metaphors to his new science.

kepler’s critique of traditional catoptrics Kepler’s recourse to traditional geometrical tools suggests the historiographical image of “the last of the medieval perspectivists.” The novel manner, however, in which Kepler redefines and applies these, as tools of new mathematicalphysical knowledge, places him at the forefront of early modern science.20 This ambiguous place is most clearly delineated in the way Kepler reinterprets the role of the perpendicular in the formation of reflected and refracted images. In

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4 Kepler’s geometric scheme of reflection in a plane mirror, in Ad Vitellionem, chap. 3, 56. Reproduced by permission of The Huntington Library, San Marino, California, RB708372.

these cases, his intricate relationship to ancient textual authorities as sources of knowledge is fully revealed. He rejects them as erroneous, thus discarding them as invalid for the mathematical sciences. Instead, he attempts to establish a priori knowledge of optical truth. However, while this leads him to the verge of new discoveries, he is unable to cross the line and make full use of his novel approach. Kepler is unable to accept the validity of a law of refraction based on complex proportions that lack any visual equivalent. The third chapter of Kepler’s Ad Vitellionem deals with the foundations of catoptrics and the determination of the place of images in mirrors. Its first part aims to refute the fallacies in the tradition of classical and medieval authorities. Kepler states, “At the very foundation of catoptrics, the demonstrations of the optical writers are still clouded, in that they require from sense perception the very thing that was to be demonstrated.”21 Kepler discards the humanist program (Maurolyco’s, for instance) for the reform of classical mathematical knowledge. It is not enough to correct the ancient texts and to supply the scientific community with better and revised versions of the sources, since the sources in themselves are erroneous. This point is further explicated when Kepler presents one of Euclid’s errors in his Catoptrics and shows that the error is in Euclid’s axioms (i.e., the whole

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MEASURING SHADOWS The Content of Kepler’s Visual Language

Euclidean system is to be rejected). Euclid asserts that whatever “falls under vision is seen according to its perpendicular to the surface of the mirror.”22 Euclid proves this by relying on an axiom that states that if point C on the mirror CF, where the perpendicular AC from the visual object A hits the mirror, is covered, the object will not be seen, even though the rays from the eye B to the point of reflection D are unobscured. Kepler does not interpret away this mistake in a futile attempt to save the classical foundations of catoptrics, but shows that in following Euclid’s supposition, the perspectivist tradition preserved his errors (figure 4). Kepler develops his argument and claims that Euclid has no other proof for the superiority of the perpendicular than his mistaken axiom. Euclid’s only resort is to talk, in an artificial fashion, of the visible rays, thus ascribing some occult “skill” to the perpendicular line that produces an image when it hits the mirror’s surface.23 Alhacen and Witelo,24 claims Kepler, understood the absurdity of the Euclidean axiom, since they omitted it from their treatises (although they followed Euclid in other respects). Witelo may have tried to suggest the cause for the role of the perpendicular ray in producing an image in a mirror, but his failure made things worse since “the obscurity of the subject produced hallucinations in him.”25 Against Witelo’s explanation,26 Kepler argues that one cannot deduce the place of the image only from the place of the object, since there are “many differences between the object and its image.” In other words, Kepler argues that since the place of the image does not necessarily follow exclusively from the place of the object, the latter cannot be the direct cause for the existence or disappearance of the image.27 Witelo assumes the image to be a direct physical derivative of the visible object, thus a veridical representation of its source. Kepler, in contrast, defines the image as a result of the flow of light, which only indirectly relates to the visible object. Whereas Witelo turns to Euclid’s postulates for help (which Kepler has already rejected as false), Kepler tries to see whether Alhacen can save the argument. Alhacen derives from empirical experience the fact that the image is always seen along the perpendicular falling from the visual object to the surface of the mirror. This maneuver fails, and Alhacen ends up with an obscure sentence: “the true state of natural things depends on the position of their principles, and the principles of natural things are obscure.”28 Kepler can only comment that “hoc non est demonstrare.” A different way of interpreting Alhacen is to assume some preconceived natural order of creation, or, following Witelo, some preestablished order within the soul.29 For Kepler these attempts fail because “all these visual affections are

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drawn from material necessity, where there is no place for consideration of final [causes] or of beauty.”30 The argument from considerations of beauty assumes the perpendicular to have some special quality about it, some essential power to unite the image and the visible object and thus to communicate the visual object’s true size. The emphasis of the perspectivist tradition on the role of the perpendicular line of sight assumes a necessary connection between the ideal and static geometrical order (the perpendicular to a diameter divides a circle into four equal parts) and optical phenomena. Kepler rejects this necessary connection and the imposition of ideal geometrical beauty and order on physical phenomena. The perspectivist tradition’s celebration of the power of the perpendicular fails to explain why in curved mirrors, under the same perpendicular line of sight, the wrong magnitude of the visible object appears, whereas in plane mirrors the correct magnitude of the reflected object is perceived. The challenge Kepler poses to the perspectivist tradition relates to the way geometrical shapes of different mirrors determine different sizes of images. In contrast to the emphasis laid by Alhacen and Witelo on the power and strength of the perpendicular, Kepler asks for a mutual causal connection between the geometrical shape of the physical surface of the mirrors and of any differently rarefied media and the geometrical description of the passage of light.

kepler’s new mathematics: from abstraction to representation The traditional relationship between the mathematical domain and physical reality was set by the Aristotelian formula that “[w]hile geometry investigates natural lines but not qua natural, optics investigates mathematical lines, but qua natural, not qua mathematical.”31 The science of optics is truly secondary and dependent on entities whose origin lies in another discipline. Any observer of nature creates, through abstraction, geometrical figures that are then artificially applied according to accepted rules of the intermediate disciplines such as astronomy, optics, and music. In optics the scientist constructs artificial lines and angles, although, according to Aristotle’s analysis in De anima, in reality there are no rays, nor reflected rays or angles of incidence. The scientist, by assuming these imaginary constructs to be real, can explain away optical puzzles in order to save optical phenomena. On the other hand, medieval perspectivists asserted that the scientist abstracts his lines and visual angles directly from observed optical phenomena.

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MEASURING SHADOWS The Content of Kepler’s Visual Language

In other words, they somewhat turned the Aristotelian formula on its head: “Every line along which light . . . reaches . . . is a natural sensible line . . . within which a mathematical line is to be assumed imaginarily.”32 Witelo proceeds by saying that “[The] minimal light [does not] fall on a mathematical point . . . but on a sensible point.”33 Aristotle emphasizes that the optician talks about mathematical lines as if they were concrete physical entities having physical effects. In contrast, Witelo brings the opticians’ practice closer to that of the mathematicians, who directly abstract geometrical forms from material forms. However, the principle remains the same—the two domains are separate and the geometric shapes originate by abstraction from concrete bodies located in the physical-material realm.34 Furthermore, the adverb Witelo uses for allowing the use of geometrical entities into the science of optics is “imaginarily.” This term refers to Aristotle’s account of the way the mind separates different aspects of the external and sensual world.35 These terms of mental separation are embedded in Aristotle’s theory of cognition, explicated in his De anima. Aristotle assigns the faculty of imagination the role of producing images of external objects, transmitting them to the intellectual faculties for contemplation.36 Aristotle asserts that “[w]hen the mind is actively aware of any thing it is necessarily aware of it along with an image; for images are like sensuous contents except in that they contain no matter.”37 The imagination is the location where a synthesis of sensual data is given a coherent shape. However, since the imagination does not rely on special sensibles (like each of the senses) but attempts to combine them and determine what the perceived object is, the imagination is liable to errors and to the production of false images. “Perception of the special objects of sense is never in error. . . . Next comes perception that what is incidental to the objects of perception is incidental to them: in this case certainly we may be deceived. . . . Third comes the perception of the common attributes which accompany the incidental objects to which the special sensibles attach (I mean, e.g., of movement and magnitude); it is in respect of these that the greatest amount of sense-illusion is possible.”38 This propensity of the imagination to distort sense-data requires the control of the soul’s faculty of rational judgment to correct its errors.39 At least for some of the scholastic philosophers, this critique of imagination and its place within a cognitive hierarchy meant that those mental concepts, such as mathematical entities, are far removed from the reality of the physical world. Nicole Oresme, for instance, in his commentary on Aristotle’s Politics, remarks concerning the possibility of an empire:

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If everyone wished to avoid war and to obey one sovereign who could always understand every matter, and judge it well and properly order everything . . . it would be a splendid thing, as it seems. But in fact this is like the fiction of the poet, or the speculation of the mathematician [ymaginacion mathematique]. For as I have said earlier the world is not run by hypothesis. It must be taken as it is . . . and taking it as it is by nature, it scarcely seems possible that anyone could be a sovereign [universal] monarch and last for very long.40 Mathematical hypotheses are ideal cases, but their application toward an understanding of the world (be it the natural or the political world) is dubious. Witelo’s words, that “a mathematical line is to be assumed imaginarily” within the sensible light ray, invite the Oresmian critique. The mental production of mathematical images, abstracted from matter by the imagination, is liable to errors and thus is to be considered true only under the strict rules of definition and deduction. Mathematical sciences can discover universal principles of beauty and order, but their relevance for the knowledge of concrete objects and events is controversial. However, I may infer that like comedy, which applies general understanding concerning the nature of man to hypothetical events, mathematics is less convincing than physics, which takes as its subject matter concrete events and objects (as in tragedy). Mathematics can thus supply ideal cases, but when compared to “nature as it is,” it is like poetic fiction or political ideals, never to be truly fulfilled or implemented in reality. Kepler reads Witelo’s application of abstraction to the science of optics as creating an unnecessary separation between the mathematical aspect of reality and its material necessities and effects. In the first chapter of Ad Vitellionem, Kepler emphasizes that the lines considered by the opticians are not abstractions of material phenomena but representations of the motion of a twodimensional (i.e., mathematical) surface: “a ray is nothing else but the motion of light itself.”41 Thus, the treatment of light under the laws of geometry is not an artificial discourse ideally imagined, but must include physical necessities and effects as well. Kepler’s is not a mechanical-materialistic science, where objects merely affect one another according to their material mass in movement. The basis of Keplerian science is in his declaration that material objects tend to assume geometrical shape, and it is these geometrical shapes that are the main factor in Keplerian optics.42 Light, as a geometrical surface, interacts with physical objects not through their material qualities, but only through their geometrical surfaces. “Light is not impeded by the solidity of the bodies,

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inasmuch as they are solids, because it can least pass through them. . . . Light is affected by the surfaces of any opposed bodies.”43 Summarizing this point, Kepler argues that the definition of light has two aspects: one relates to what is the essence of light, the other relates to its quantity. The first is light’s ability to illuminate, and the other is local motion. The motion of light (i.e., the expression of the quantitative aspect) has two objects: the medium through which it is transmitted, and the physical object that it strikes. These two objects affect light by refracting it (the various interfaces of adjacent media) or by reflecting it (the polished surface). However, these effects concern the motion of light, that is, they are quantitative effects and are created by the interaction of geometrical surfaces.44 It is Kepler’s challenge to prove, according to these principles, basic optical theorems, such as that the angle of reflection is equal to the angle of incidence. Kepler wants to show that what affects the motion of light is not any occult quality of light, or of “nature” in general, but the geometrical shapes assumed by material bodies. Kepler’s proof is divided into two steps. First he establishes that light, flowing to a surface, is reflected to the opposite side. Then he demonstrates that the angle of incidence is equal to the angle of reflection. In both cases Kepler uses elements that had already been applied by Alhacen and Witelo, that is, the analogy to a solid body that hits a solid surface, and the composition of oblique motion from perpendicular and parallel surfaces.45 However, Kepler reinterprets these elements. For Alhacen the comparison of the reflection of light to the rebound of an arrow from a solid surface is limited and serves as a simile, not an identity. In Lindberg’s words, “these comparisons [were] . . . meant to elucidate the geometry and causes of reflection and refraction but not the nature of the reflected or refracted entity.”46 Kepler, on the other hand, isolates the element of motion and states the case as a rule of anything that violently strikes an opposing object.47 The second element (i.e., the decomposition of oblique motion) serves Alhacen and Witelo as an artificial and heuristic construct; the true cause for reflection according to equal angles is that nature tends to operate according to the shortest path.48 Kepler rejects this line of reasoning.49 The geometrical model is constructed as a representation of possible paths for the motion of light, and their interaction with other physical surfaces creates the true and real linear path of the motion of light. Kepler applies to refraction the same analysis and separation of the element of motion from the other qualities of light. “For as in the case of physical motion, a javelin is sometimes made to collide with a thing toward which we aim it and, adhering together, they both proceed on the same path with one

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motion, so the same thing happens in the case of light and the more dense surface which light penetrates, yet without matter or the dimension of solidity [sine tamen materia aut soliditatis dimensione].”50 Toward the end of the same proposition, Kepler clarifies this point even more emphatically: “This happens at all inclinations. . . . And so this disposition of physical violent motion to flow back is however of the same kind as that of light.”51 Kepler’s analysis of refraction compares the motion of light to oars penetrating the surface of the water and to theories regarding a balance. He can thus separate the element of motion from the mobile and turn geometrical lines into representations of possible paths within a geometrical model (either a circle in the case of a balance or a table in the case of the oars). The perpendicular, drawn within that model, is a limiting case: in the case of the balance, the perpendicular is drawn to the center of the balance and to the center of the circle it describes. This perpendicular is the limit for the ascent or descent of the balance; in the case of the oars, it is that line to which the oars deflect in case of an inclined violent motion of the river. In the case of light, the density of the medium limits the dispersion of light. The oblique ray cannot proceed in the same direction, nor can it be deflected in a direction further away from a perpendicular ray emanating from the same source. Thus, an oblique ray must be refracted from its point of incidence somewhere between the path of a ray perpendicularly passing the denser medium and the continuation of its original oblique path (i.e., it will be refracted toward the perpendicular). Again, although the outlines of Kepler’s explanation are not different from the medieval perspectivists’ theory of refraction, the ontological significance of Kepler’s scheme is radically different. The perpendicular line, in Kepler’s explanation, does not stand for any concrete ray of light but represents a possible path that may be described by the passage of light (that is, by the motion of twodimensional surfaces). Geometrical lines can represent all physical motions. The peculiar aspect of light is that its motion can be described with no reference to matter, since light is an incorporeal being. The tendency of the refracted ray of light toward the perpendicular is neither because of an innate virtue of light or of perpendicular lines nor because of any diminution in the quantity of light in the oblique ray. The only part played by the perpendicular is its being a limiting case of direct radiation:52 that is, the perpendicular line must be taken into account as a representation of one of the possible paths for a ray of light that falls on the surface of a denser (or rarer) medium. The perpendicular is neither an abstraction of some concrete entity, nor a sign, nor a conventional symbol of something that was there before (the way

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written letters are signs for the sound once uttered), nor an ideal entity existing in some noncorporeal realm. The perpendicular (or any line, for that matter) is a representation of real motion. The line is described and created by the possible movements of any body in the world (including the two-dimensional surface of light). Kepler’s conception of geometrical physics was more than mere Renaissance Platonism. Kepler’s geometry did not constitute a separate realm, but was created by the actions and passions of the physical world itself. Furthermore, while most of the Keplerian moves are not original with him, and can be found in his medieval and classical sources, he not only reinterprets these moves but purges the language of perspective from any recourse to the occult and inherent qualities of light and geometrical lines. The natural world does not aspire to imitate and imperfectly fulfill an ideal divine plan. On the contrary, despite its blind and material motions, the natural world somehow directly and exactly represents the divine archetypes to the human eyes and intellect. In the third chapter of Ad Vitellionem, Kepler applies this conception of geometry to explain the role of the perpendicular in determining the position of the image in the case of reflection. The classical and medieval traditions had to fall back on some inherent virtue of the perpendicular in order to explain why the image in the mirror lies along the perpendicular from the visible point to the mirror’s surface. Kepler’s point of departure is the eye’s visual distortion. The eye imagines the object to be where its refracted or reflected image is. Then it imagines its line of sight to be in a straight line to that image, and does not perceive the point of its refraction or reflection. The place of the image, then, is that point where the produced lines of sight from each of the eyes meet after they have passed through the point of refraction or reflection. Each eye and its line of sight is on a surface of reflection (or refraction). The place of the image is where these two surfaces meet, and one point along their common section is the visible point. Now these surfaces are set perpendicularly above the mirror, and therefore their common section is a perpendicular line and one of its points is the visible object. The object and its image are on that common section between the two surfaces on which each of the eyes is situated, and this common section is the perpendicular from the visible point to the surface of the mirror. Kepler can now claim that he managed to causally explain why the image is placed along the perpendicular without resorting to the “occult nature of light” or to “the ingenuity of a universal nature.”53 The laws of geometry win the day, the boundaries between natural causes and geometrical ones having been breached. Instead of the classical and

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medieval landscape of concrete physical objects and their direct geometrical abstractions, Kepler presents a geometrical landscape produced out of possible lines and shapes that materialize by the real motions of physical bodies. Onto this model created according to geometrical necessity, Kepler molds ancient experimental results. He aims to formulate a geometrical model that will accommodate the different angles of refraction supplied by Witelo (after Alhacen). Although this failed, Kepler still believed that only geometry could give meaning to experimental results. The main aim of Kepler’s new reading of the optical tradition was not only to come up with a true theory of optics, but also to propose a new scientific language. Medieval Aristotelians and Renaissance humanists relied on linguistic analysis to attain the true nature of things. Instead, Kepler turns to geometrical figures as the sole vehicles of natural truth. The “geometrical turn” obliged Kepler to reconceive the ontological status of geometrical entities. Keplerian optics transformed the perpendicular from being either an artificial line or an abstraction of a concrete line, with some occult quality to produce images; instead, for Kepler, the perpendicular becomes an exact representation of real motions of physical bodies. Kepler attempted to disentangle geometry from the Aristotelian grip that assumed its secondary role in any explanation of natural processes. In order to accomplish this philosophical endeavor, the Keplerian effort had to confront a deeper layer of traditional theories of cognition, however. Kepler realized that solving problems in traditional optics and reinterpreting its traditional concepts (such as the role of the perpendicular to the surface of a mirror) was not enough. The application of his new mathematical tools required discarding some of the most deeply entrenched cultural as well as philosophical tenets concerning visual cognition. It was the apprehension that traditional uses of mathematics are dependent on a certain understanding of visual cognition that led Kepler to add a lengthy appendix to the first chapter of Ad Vitellionem. This appendix presents a thorough critical attack on Aristotle’s theory of vision as the fundamental scheme for all future generations of optical discussion until his own times: “The inept human studies have become so vain that no work is celebrated unless it either erects or destroys the temple of Diana: that I say either to support the authority of Aristotle, or, striving after fame, to hold a battle against him.”54 In diagnosing late Renaissance optics, Kepler points out that it is still under the sway of Aristotelian convictions concerning vision, and he sets out to bring Aristotle into “the school of optics” and to shatter Aristotelian illusions that have led the whole subject of vision into error.55 In what follows I will trace

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the intricacies of the Aristotelian tradition and its impact on medieval and Renaissance basic beliefs concerning the way vision takes place and its relationship with the scholastic philosophy of mathematics and with the physical world.

the aristotelian framework of sight The medieval paradigm of perception and cognition assumed a direct link between the senses and the physical objects they confront.56 The senses and particularly the eye assimilate or capture the sensual object by either emitting some power or passively receiving the power transmitted by the objects. This means that the image produced in the eye has a direct link with the external object and thus somehow resembles it. In this sense the image is almost like its origin, always depending on it but never completely identical with it. Kepler identified Aristotle’s exposition of sense perception (especially visual) in De anima as the classical formulation of this cognitive paradigm. The most obvious point of contention was Aristotle’s denial of the existence of light as a separate substance. Aristotle asserted that light is but the actual transparency of the potentially transparent media. Moreover, Aristotle denied any ontological status to light rays, or any ability of light to “travel.”57 Instead of light being a distinct agent communicating between eye and visual object, Aristotle suggested this role was reserved to the transparent medium. He assumed that the transparency of the medium serves to offset the distance that separates the eye from the perceived object. The role of the transparent medium is to enable the eye to touch the visual object and so to render it directly to the sense of sight. These principles of visual perception point to a basic ambiguity in Aristotle’s treatment of sight. He accepted the Platonic visual metaphors of true cognition, thus admitting that sight is the most discriminating sense—probably because it is the one sense that can supply the imagination with an immediate and a completely delineated image. This means that sight can deliver the basic image to the higher intellectual faculties. To this image, all other sensorial details (such as sound and texture) could be applied. Moreover, since the mind demands images for its intellectual processes, vision, as the image-producing faculty, becomes the central metaphor for intellection in general: “The process [of intellection] is like that in which the air modifies the pupil in this or that way and the pupil transmits the modification to some third thing”58 (my emphasis). This close affinity between visual images and the substrate of intellectual

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processes allowed Aristotle to assert that “[a]ll men by nature desire to know. An indication of this is the delight we take in our senses . . . and above all others the sense of sight. . . . [W]e prefer sight to almost everything else. The reason is that this, most of all the senses, makes us know and brings to light many differences between things.”59 Aristotle felt uncomfortable with this Platonic delight with vision, however. For the Platonists, vision represented the possibility of grasping the shape of physical objects, seemingly without the mediation of time, as an example of the separation of forms from the flux of being and sensation. Aristotle contends that this quality of sight has an affinity not only with intellectual processes but with the imagination, thus placing the outcome of the visual process on less solid ground: “As sight is the most highly developed sense, the name fantasia (imagination) has been formed from φαος (light) because it is not possible to see without light.”60 In most cases our imagination is wrong and leads (if it escapes the control of discursive thought) to errors and false assumptions: “imaginations are for the most part false.”61 Aristotle rejects the assumption that through the sense of sight form is separated from matter. Therefore, he prefers, as the basic metaphor for sensation, the more empirical sense of touch.62 The language of touch and physical impression dominates the general description of sensual perception: “Generally, about all perception, we can say that a sense is what has the power of receiving into itself the sensible forms of things without the matter, in the way in which a piece of wax takes on the impress of a signet-ring without the iron or gold.”63 Touch is the indispensable sense. It is the universal sense that each animal qua animal has. All the other senses are secondary for the survival of the animal: “It is clear that without touch it is impossible for an animal to be.”64 This perception is not an action at a distance but takes place through the medium that is affected by the physical object and serves to transmit this affection to the organ of sense. Thus, in the case of sight, Aristotle asserts, “That is why in the case of reflection it is better, instead of saying that the sight issues from the eye and is reflected, to say that the air, so long as it remains one, is affected by the shape and color. On a smooth surface the air possesses unity; hence it is that it in turn sets the sight in motion, just as if the impressions on the wax were transmitted as far as the wax extends.”65 Aristotle compares the thinking process itself to the sense of touch, suggesting the soul, similarly to a hand, touches the different forms produced for it by the imagination.66 Just as the hand adjusts itself to receive the form of the object it holds, so the soul receives and becomes the intellectual objects it

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grasps. This correspondence produces in the soul, not the external objects in themselves, but their forms: “It is not the stone which is present in the soul but its form.” Thus the senses receive the sensual content as if it were impressed on them. These sensual forms are further impressed on the intellectual faculty to produce the different abstractions: “The objects of thought are in the sensible forms, viz. both the abstract objects and all the states and affections of sensible things.”67 Aristotle assumes a cognitive sequence leading from a concrete spatial and material entity, situated outside the soul, to its realization as an abstract and universal form within the intellect. This chain is created by the direct and tangible impression of the object on the medium (what Aristotle calls the activation of the medium), and through the medium, it is impressed on the organ of sense. Thence the image received within the soul is impressed on the intellect. This Aristotelian analysis is based on two major assumptions. The first assumption is the adequacy of the organ of sense to receive its appropriate sensual object and become whatever it receives: color in the case of sight, sound in the case of hearing, and so on. The second assumption posits the medium as an indispensable requisite for the sensual action to take place, since it creates the direct link between the external object and the organ of sense.68 The two axes along which the Aristotelian theory of vision was constructed were the rejection of light as a separate substance (thus the rejection of geometrical ray analysis as having any physical reality) and the centrality of the medium as the transmitter (of touch) from the visual object to the eye. The second element avoids any hypothetical and mysterious powers that cross the gap between the observer and the observed in favor of a physical impingement and direct contact between the eye and the visual object. The special object of sight (that is, color) is imprinted on the surface of the eye, and an accurate cognition of that special object is guaranteed. Thus, Aristotle could assert that “perception of the special objects of sense is always free from error” and that “sensations are always true”!69

scholastic vision The Aristotelian visual paradigm was accepted in the medieval scholastic centers with mixed reactions. While Aristotle’s assertion concerning the need to assume a direct contact between the eye and the visual object was generally accepted, his other assertion, denying light any separate and independent existence, was a point of philosophical contention. The Aristotelian “party,” in

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rejecting the substantiality of light, had to come to terms with the crucial place occupied by the metaphor of light in the New Testament and in the church fathers. Furthermore, the Aristotelian commentators had to face the successful solutions to optical problems suggested by Alhacen and his followers in the West. On the other hand, the perspectivist “party,” though supporting ray analysis of the visual process, accepted the hegemony of the Aristotelian discursive framework. Thus, such questions as the role of the transparent medium and the status of the geometrical analysis of the propagation of light were of critical importance to medieval perspectivists. The discussion of light, upheld by Albertus Magnus and Thomas Aquinas, can serve as a good illustration for the way Aristotelian commentators confronted the problem of light. Aquinas denies light a separate existence as a substantial form and reinterprets the appearances of the word “light” in the New Testament and in Augustine as a metaphor and allegory. In the solutio of his commentary on distinction 13 of the “Sentences,” Aquinas asserts that lux is a sensible quality per se, which has no separate existence from its place in the sensual process, that is, it is not a substance but a quality of the visual process. Furthermore, this formulation of light as a quality assists Aquinas in his reinterpretation of the spiritual dimension of light in a metaphorical sense. The spiritual meaning of lux is that quality that produces an intellectual manifestation to the mind, just as corporeal and sensible light is a quality that produces illumination among sensibles. Therefore, when John the Evangelist and Augustine speak of lux, they should be understood, according to Aquinas, as speaking in this spiritual sense and not as if light is some divine substance. Light should not be interpreted as a physical entity even in considering biblical verses or Augustinian speculations. When lux is discussed, it refers to a spiritual manifestation and to an integral quality of intellectual enlightenment. After suggesting the way of interpreting lux in a religious and spiritual context, Aquinas turns to natural light. He attempts to accommodate ray analysis into his Aristotelian ontology. In order to achieve this, he differentiates between four instances of light: 1. Lux is that quality by which any luminous body illuminates another, as in the case of the sun. 2. Lumen is that which the illuminated transparent body (i.e., the medium) receives. 3. Radius is the illumination according to a direct line from a luminous body; wherever the ray is, there also is lumen, but not vice versa. Aquinas preserves

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the Aristotelian dictum that rays are fictionally applied to the analysis of vision and concludes that in drawing a ray from the visible object to the luminous body, one assumes the existence of a transparent illuminated medium, but there are no rays within that medium. 4. Splendor is the reflection of lumen (conceived as a ray) by any clean and polished body. In this way, geometrical ray analysis is reduced to no more than a heuristic procedure with a definite dictionary for its translation into the real physical state of affairs. Each time one of the perspectivists uses the term “ray,” Aquinas can translate it into Aristotelian vocabulary without harming the coherence of either geometrical optics or Aristotelian physics. In general, Aquinas disregards the theory of perspective. He does not mention the basic optical law of reflection or the phenomenon of refraction, and only incidentally speaks of species and their activity.70 Albertus Magnus’s attack on the claims of medieval perspective was even more vigorous. The only way to utilize geometrical analysis in optics is by adopting Aristotelian principles. The Albertian conviction is that it is possible and necessary to formulate an Aristotelian perspective.71 In his commentary on the Sentences, Albert asserts that whatever there is in the perspectivists’ treatises concerning nature cannot be accepted when it is in contradiction with authors of natural philosophy (i.e., Aristotelian natural philosophy). The question of whether lux is a body is not within the competence of the authors of perspective, as Anaritius (al-Nairizi) says in his Commentary on Euclid’s Geometry. Hence the “author of Perspective acted unwisely in interposing this discussion.” Moreover, says Albert, we do not lack books on perspective by other philosophers, such as Aristotle, who posit principles of natural science different from the perspectivists’ texts. At the end of his discussion of mirrors, Albert summarizes the relationship between the geometrical explanation and the physical meaning: “However, all these [questions of mirror images] are to be treated according to perspective, because these and their causes are impossible to be known without geometry; and therefore what these assertions are, they are assertions according to an ideal [exemplarem] explanation, and not because they are [really] so intended. But it establishes out of all these that vision needs a medium, that is, an actually transparent [body], through which such radiations could be produced, and that colors are not seen unless in such a medium.”72 Like Aquinas, Albert differentiates between the need to apply geometrical analysis in specific cases (such as the formation of images in mirrors) and their physical

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truth. He restates the Aristotelian case concerning the ontological status of light and rejects any claim of the perspectivists to contribute to natural philosophy by their mathematical tools. It was against this Aristotelian background that Roger Bacon asserted the perspectivist case. In his De multiplicatione specierum, Bacon attempted to reformulate the geometrical analysis of visual phenomena as the paradigm for any natural process. This he performed by considering the nature of species and their multiplication according to the Aristotelian scheme of the four causes. Bacon preserved the geometrical analysis by asserting that species multiply according to straight lines: “Quod autem multiplicatio ex natura sua appetat fieri secundum rectas lineas.”73 These lines, however, are not geometrical lines, as they possess “profunditatem et latitudinem,” that is, physical reality and presence. Rejecting the Aristotelian denial of the existence of “traveling” rays of light, the perspectivist paradigm still preserved the medium as a necessary condition for a physical contact between the eye and the object of sight. Roger Bacon reformulated the role of the medium not as what touches the eye and the object simultaneously, but as the material substrate that enables the preservation of the species in their multiplication, providing the material cause for the multiplication of species.74 The medium functions as a place in which a species can exist. It is through the medium that a species comes to resemble the agent, preserving the material categories of place and dimension. It is the body that provides the substratum that enables a species to be “a corporeal thing actually existing in body.”75 Two further conclusions follow from Bacon’s argument that the species exist in the medium and that through the medium they acquire corporeal dimensions. First, species are corporeal effects of corporeal bodies; and secondly, it is the medium that enables the species to mix and to produce one coherent effect in the recipient. It should be stressed that Bacon understands the role of species and the medium as part of an Aristotelian natural philosophy. A species is an actor with an ability to produce effects that resemble its source, seemingly enabling an action at a distance (such as the sun’s ability to kindle fire on earth). This process must be explained away, since there can be no action at a distance in the Aristotelian worldview. For that reason, Bacon differentiates between the causes of the generation of a species and the causes of its multiplication, which fills the gap between the agent and the recipient. This process of multiplication can explain the transmission of certain influences through a plenum (i.e., the medium) without permanently transforming the medium and without being an occult phenomenon.

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The complex situation and the apparent contradictions between the rigorous Aristotelian line and the perspectivist interpretation forced later commentators to perform intellectual gymnastics in an attempt to grasp the stick at both ends. Nicole Oresme, for instance, in his commentary on De anima, rejects Aristotle as the supreme authority on optical questions and declares that he is irrelevant to a theory of light and vision.76 Yet he still conforms to the Aristotelian requirements of De anima. The result is that Oresme represses the mathematical aspects of Perspectiva and adopts mainly causal and qualitative arguments. Following the interpretations supplied by Thomas Aquinas and Albertus Magnus, Oresme distinguishes between two sorts of light. Lux is that quality inherent in the luminous body, while lumen is its manifestation in the transparent medium: “It should be known that the quality of a luminous body—that is, one generating lumen as does the sun—is commonly called lux. . . . [L]umen is said to be . . . the quality of the diaphanous medium through which illumination is made—[a medium like air, heaven, or water]. And thus lux is in the sun and lumen is in the air.”77 Oresme fully adopts the Aristotelian stance that light has no independent and separate being. The qualitative differentiation between the two aspects of light enables Oresme to apply perspectivist terms and principles. Thus, for instance, Oresme states that lumen is the reflection of lux. Since every medium has some density, it follows that in every point some reflection occurs in the medium. Therefore, Oresme asserts that lumen is reflected everywhere in the medium given the presence of lux. Through this analysis, Oresme is able to explain away action at a distance: lux acts through its reflection, while lumen exists continuously throughout the medium and fills the gap between the luminous body, the visual object, and the eye. Therefore, when the eye perceives the light of day, it actually perceives the brightness of the medium. This ambiguity in the Oresmian stance surfaces most conspicuously in his commentary on Aristotle’s Meteorology. Aristotle’s explanation of meteorological phenomena, such as the rainbow, applies geometrical methods utilizing perspectivist terminology of rays and power of sight, concepts rejected in De anima. Oresme asks whether “visus frangatur in occursu medii densioris et rarioris puta aqua vel aeris.”78 In his explication, Oresme rejects the possibility that what is refracted is a power of sight, since that assumes sight to be an active element, whereas Aristotle asserted in De anima that it is passive. Instead Oresme adopts the concept of a visual ray. The problem, then, is that a visual ray may imply an extramission theory of vision (i.e., that the eye sends out a ray in order to capture the externally visible). In dealing with this difficulty,

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Oresme resorts to the Aristotelian assertion concerning the fictional status of geometrical entities in natural philosophy in general and in optics in particular. Initially, Oresme contends that one cannot take Aristotle’s assertions concerning vision in different places in the Aristotelian corpus as constituting a coherent whole. According to Oresme, Aristotle applied various concepts at different places. It is the interpreter’s role to notice that and judge accordingly: “And then it would be said in reply to Aristotle in the book On the Animals that he spoke according to the manner of the ancient mathematicians; in the second book of Aristotle’s On the soul, he spoke according to the manner of speaking of the moderns and according to truth [secundum modum loquendi modernorum et veritatem].”79 As to the application of rays, the reader should remember that these geometrical concepts are products of the imagination and thus cannot determine the relevant physical aspects of the phenomenon: It is to be noted that insofar as concerns perspective and the proposition, it does not matter whether vision occurs by extramission, as was imagined by the first opinion, or by internal reception, as was imagined by the second opinion; since according to both opinions the lines and angles and refractions are imagined in the same way—just as for astrology, it does not matter whether the motion of the heavens is natural or violent, since in each way the circles, spheres, and aspects of the planets are similarly imagined, and similarly concerning other things.80 (my emphases) The diverse medieval attitudes toward perspective, while disagreeing on the ontological status of rays, still accepted the central role played by the transparent medium in communicating and sustaining the species of the visible object. The medium supplied the material framework for avoiding any occult and mysterious communication between the eye and the visible object. Through the transparent medium, the external body is able to imprint its image on the eye. According to Bacon’s perspectivist account, the transparent medium supplies the substrate for a physical three-dimensional ray created by the process of multiplication of species. Oresme ascribes to the medium the same role assigned to it in De anima, with some qualifications: the transformation of the medium to transparency occurs gradually, and the geometry of rays can be applied to it, but not as a physical causal account. Thus, according to both options, primacy is given to the physical impression of the visible body on the eye. The geometrical analysis of sight is derived from, and dependent on, the physical

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impression—either as an abstraction, in Bacon’s case, or as a hypothetical supposition and a heuristic instrument, according to Oresme. This debate was assimilated, in a typically eclectic manner, into the humanistic discourse of the sixteenth century. A typical example, and one relevant to Kepler’s own educational and intellectual background, is Philipp Melanchthon’s commentary on De anima. Melanchthon holds that the principle of investigation is a dialectic one, in accordance with the Ciceronian method. “However, although many questions are argued about the manner of vision, yet we . . . shall put forward, without much disputation, the opinions [which are] not monstrous, that the scholars accept, those [opinions] they have considered more diligently.”81 Following this precept, he accepts the Aristotelian definition of light: “However, it is difficult to give the definition of light. We say it is the visible quality in heavenly bodies, and fire, air and water, and other such bodies.”82 In order to elaborate on this definition, Melanchthon turns to Genesis: “However, the prophetic narrative, in calling the heavenly bodies themselves light, they being the origin of the lumen of the bodies, is much revealing. For since it is said: ‘Let there be light’—undoubtedly, the heavenly bodies are signified, in which God by his singular wisdom has distributed this goodness, that we call light.”83 Now Melanchthon attempts to synthesize the perspectivist and the Aristotelian approaches to light. Although rays are no mere abstract geometrical notions, still the true cause for vision is the transparent medium: However, the transparent medium, what they call διαφανες, between the object and the eye is of necessity for vision. For without the object being carried through the transparent medium to the eye, it cannot be perceived. Certainly, the transparent medium is the space where the lumen of the colors is able to spread, and this space is transparent, evidently air or water, because without external light, the lumen of colors is so much weakened, as not being able to be perceived, nor to be discerned. It is commonly said about vision, that the medium is nominated a causa sine qua non, or as it is said in Greek, ωυ ουκ ανεν.84 Melanchthon situates in a specific context the meaning of the role of the medium as well as of the nature of the image and its mode of communicating the object to the eye: “Ut in oculo fiunt imagines, velut in speculo, sic ipse nostris mentibus imprimit imagines de sese, velut speculo.”85 Two terms should be noticed: first, the use of the word imprimit (impressing upon), and secondly,

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the comparison with the mirror. The act of impressing discloses that the element of touch, so important in the Aristotelian analysis in De anima, did not disappear. The impression on the eye, together with the allusion to the image in the mirror, means that the image is accepted as a complete whole on the surface of the eye. These two components, the element of impression and the mirror metaphor, capture the essence of the Aristotelian and medieval perspectivist discourse of vision I have reviewed to this point.86 Although each point on the object sends forth its form (or species) in all directions, the eye receives a complete and unobstructed picture corresponding to the object in all details. The species is not only a mathematically constructed representation of the object, but mainly its physical semblance—in this manner, its epistemological accuracy is guaranteed. Moreover, there is no need for interpretation of sense-data in this cognitive model, only the accurate identification of the object whose species was received in the eye. The imagination does not construct the representation of the object for the mind, but only abstracts a more rarefied image, to be transmitted and impressed on the intellect, from the initial semblance (i.e., the species) received complete on the crystalline humor.87

kepler and the demise of the aristotelian theory of vision In disentangling himself from these hegemonic Aristotelian assumptions and the complex of debated issues they defined, Kepler sights Aristotle’s De anima, book 2, chapter 7 as his main target. Kepler sees this chapter as that temple of Diana to be demolished to release optics from the shackles of scholastic intricacies. Kepler’s thorough reading of the chapter is a radical overthrow of the Aristotelian worldview, not presenting another commentary on a master text, but discarding Aristotle’s text altogether, making it obsolete and irrelevant to any future natural philosophy. Kepler’s interest in this appendix does not rest in the position of optics or of mathematical analysis within a system of knowledge, but in the cognitive domain. Kepler identifies the Aristotelian paradigm of perception as the main obstruction to his new language of vision. The list starts with the ontological status of light and color and continues to the question of the way the medium allows the visible object to move the eye. Kepler’s initial step is to reject Aristotle’s assumption that vision is produced by the alteration of the medium into actual transparency by a luminous body, and by the color of the visual object imbuing this transparent medium.

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Kepler rejects the need for these two sources of alteration, contending that color as such has no potency to move the instrument of vision. Furthermore, in the final account, color is of the same nature as light. “Therefore, color, since [it comes] from light, has the power to move the instrument of vision. Hence, it primarily [owes its nature] to light, and is capable by itself to alter the screens (and hence the eye).”88 Kepler emphasizes this point further: “It is not enough for colors to be related to light, it is necessary that they be actually illuminated by light and so they absorb a certain light, which in this chapter is called transmitted light.”89 The state of things is not that vision is dependent on color as the first cause, but that color is dependent on light for its communication to the eye, and for the following alteration that takes place within the eye. If light is the cause for alteration, the Aristotelian definition of light as a state of transparency cannot hold. Kepler concludes that Aristotle’s definition avoids a deeper examination of the very nature of light and is related only to its accessibility to the human sense of sight. Kepler suggests instead an indirect method of investigation: since light in itself is invisible, one must examine its action and function in order to learn of light’s nature. This sort of examination will make clear that light is not another name for a state of transparency.90 The Aristotelian attempt to reduce light to the transparent medium fails because it assumes light to perform two contradictory actions at the same time. Light cannot be the color of actual transparency (i.e., a transparent color, or no color at all) and at the same time be an activation of the color of the visible object. If light is not an activation of the transparent medium, then it has to be substance in its own right, an entity that in flowing luminous bodies illuminates the entire medium: “If the sun is present in the air, and yet it is fixed in the sky, it will therefore be present in the air through flowing out.”91 Further, this flowing is not a concrete and present ray but something that “was or almost was there.” Kepler recapitulates in these obscure words his eighth proposition: “The ray of light is never about the flowing out of light itself. For the ray is nothing else but the motion of light itself. . . . [Just as in physical motion] so in light the motion is in the same manner, a straight line, [while] the moving thing is a certain surface. And as that straight motion does not pertain to the [physical] body, so this straight motion does not pertain to the surface [of light].”92 While Kepler accepts the medieval notion of a continuous flow of species of light from the luminous source, there is no corresponding flow of species from the visible object. Further, the continuous flow of two-dimensional surfaces from the luminous source only describes a ray, but does not create a material three-dimensional line (i.e., there are no Baconian three-dimensional rays

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possessing profunditatem et latitudinem). The ray can only be delineated after the motion of light has described it in the transparent medium. The ray is but an exact representation of patterns of the motion of two-dimensional surfaces of light from the colored surfaces of the illuminated objects in the medium.93 Keplerian light cannot be reduced to the physical media through which it is extended. Moreover, its ability to “travel” implies a denial of any sort of contact between the perceiving eye and the visible object. The role of the medium in the Aristotelian scheme was exactly to create the possibility of the organ of sense touching its sensible object. In denying the need for such direct contact between the eye and the visible object, Kepler instantiates light as a mediator transmitting color and geometrical shapes onto the retina. Colors are small enclosures of luminosity within materiality, however, and not any Aristotelian material qualities: “Color is light in potentia, light that slumbers within transparent matter.”94 The grades of colors are determined by the grades of density, or transparency of matter. Light excites those enclosures of luminosity within the material body and captures its color, which is determined by the density of the body. Kepler preserves the Aristotelian distinction between potentiality and actuality but transforms its meaning. It points now not to a change within the material domain, whereby one quality becomes another, but to a different process by which a hidden element is coming into view. “In the same way that the potential heat of ginger is different from the actual heat of fire, this light in the colored material seems to be different from the light in the sun.”95 The potential of color is excited by the light of the sun, or any external source of active light that brings out the light contained and enclosed in the limits of the colored body. This is reminiscent of the Platonic theory about the two species of fire—the one generated from the eye, the other from the sun— whose unification creates vision. Kepler’s theory, however, makes one crucial change: the light from the sun does not meet an active and spiritual light from the eye, but passive light embedded in the perceived body! Kepler mobilizes his conception of potentiality to a full-fledged rejection of the Aristotelian theory of color transmission in the medium. The principal element of the Aristotelian theory is that the medium, which in potentia is transparent, can take on colors because it has no color of its own. Kepler attacks this notion of potentiality, which assumes the same body to become something else: that is, a colorless body is transformed into a transparent one that can take on any color whatsoever. If, he contends, the transparent body has no color, the moment it receives the visible body’s colors, it is no longer transparent: “the more the transparent touches the colored (since nothing is

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absolutely transparent) the less transparent it will be.” These ambiguities in the Aristotelian scheme concerning the relationship of light to the transparent medium and the concept of color lead Kepler to a broader critique of the Aristotelian conception of change and motion. The underlying assumption of the Aristotelian conviction that the transparent air becomes a direct extension of the visible body, thus facilitating the transmission of a qualitative change into the eye, is that colors are perceived only through the illumination of the medium, and not by the illumination of the colored surface. Against this view, Kepler brings the evidence of experience: “Where there is no illumination of colors, they are not seen, whatever becomes the transparent [medium].”96 Aristotle bases the necessity of the medium for any visual experience on the everyday experience that when a colored object is put on the eye, the object is not perceived by the eye. Kepler rejects this line of reasoning, contending that there are other reasons why in this situation the eye cannot see: for instance medical reasons (the pain the object causes the eye by touching it) or mathematical (the eye perceives only what falls within the circle of the pupil, and as the object is brought closer to the eye less and less of it can be perceived). The main reason is physical—when the object is put on the eye, no light can illuminate its colors, and no perception of the visible object can take place. Thus, it does not follow necessarily that the medium is required to inform the eye or to put it in motion. Furthermore, the medium cannot supply a full physical account of the perception of distinct colors. Aristotle assumes that the action of change is not taking place through local motion but as an alteration. Accordingly, “the transparent is changed from being with no color to becoming colored, and anew from green to red and to black.”97 The medium, then, has to be imbued with one color, and still each person perceives different colors according to what stands in his line of sight. Kepler supplies a different causal account that leaves the medium as an unnecessary vehicle: Accordingly, since it [the medium] is a fictitious and an inadequate cause of the effect [non sit adaequata haec commentitia causa effectui], and it does not accept at one and the same time various modes corresponding to the variety of vision, for it will receive none, therefore, . . . from the Sun, and from the colors illuminated by the Sun, species exactly alike are flowing, diminished by the flow itself, until for whatever reason, they fall on an opaque medium, where they paint their source: and vision is

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produced, when the opaque screen of the eye is painted this way . . . and it is confused when the pictures of the different colors are confused, and distinct when they are not confused.98 Thus, Kepler shifts the metaphors that describe the act of sight from the Aristotelian act of impression, which involves direct contact, to the act of painting a picture, which emphasizes the process whereby the object is represented on a screen. Kepler emphasizes the indirect relationship between the eye and the visible bodies, or the sources of illumination. “For it is not the body of the sun or of colors, nor the medium [that] moves the eye, but species that is lights, that is the rays of the sun and of colors descending through the medium and glittering circularly to the [entire] hemisphere [of the visual field].”99 The rays of light, with their instantaneous local motion, are what cause motion and alteration on the screens of the eye. Thus, it is not the species of the visible body but a species of light that is “descending from the illuminating bodies, qua light,” and is imprinted on the opaque screen of the eye.100 In contrast to the Aristotelian conviction that when the eye receives the visible qualities it becomes what it senses, the Keplerian mediation of light, which transmits colors to the eye, maintains the sense organ and the visible body as separate and distinct entities. There is no need for direct touch, and there is no impression of the visible body’s simulacrum on the eye. Furthermore, no chain of abstractions from the corporeal image on the eye, through inner mental impressions of the perceived image on higher faculties, to a refined form to be contemplated intellectually is possible. The origin of abstract images and ideas (such as geometrical figures) is not in any material object from which they are extracted by a mental process, but must be sought somewhere else. In what follows, I will attest that Kepler sought the sources for the truth of geometry in other mental processes that identify the geometrical patterns created by natural motions. One such natural motion is exemplified by the nature and activity of light that brings into visibility the geometrically ordered motions of the physical world as the image of the Creator.101 The other source is the innate content of the human mind informed at the time of its creation by the divine mind. The conflation of these two elements supplies Kepler with a metaphysical scheme for the divine origin of geometrical figures as well for certitude in applying geometrical causes to physical events. Light moves in an instant and brings out those qualities that are susceptible, and in a certain sense identical, to it (such as color, which is light in potentia). Through the activity of light these qualities (like color) are depicted on

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the opaque screen in the eye just as a painter draws a figure on his canvas: the canvas is not transformed into the depicted object, and the painter, by his activity, represents the perceived colors on it. However, since light is a mathematical being (a surface that moves), the picture received by the eye is amenable to mathematical analysis and especially to proportional analysis (as Kepler, following the perspectivist tradition, performed in the catoptrical part). Through mathematical analyses one can determine the place and actual figure of the source of the optical picture, whether it is the real picture within the eye or a reflected image in a mirror.

from representation to recognition The question still remains: if there is no process of abstraction, how can the mind identify the natural motions in the external world, manifested to it by light, as exhibiting geometrical figures and proportions? How can the mind supply those geometrical figures and proportions that structure meaningfully the colored stains transmitted into the eye? It seems that, although Kepler attempted to embed geometry within the optical reality, he finally had to assume a different origin for geometrical shapes than physical and sensual objects. Kepler admits that “that construction is never drawn from sensible things in a diagram, though it is assisted by them; and it does not arise from the assembling of many individual sensible things into one axiom, but it is obtained a priori.”102 What is then the source of this ability for a priori reasoning, and what are the consequences for the constitution of geometrical figures? As I pointed out in the case of the perpendicular, mathematical entities are not abstracted from the material world, nor are they mere hypothetical and aesthetic constructions. Lines and angles are not abstracted from concrete bodies but represent possible paths of the motion of physical bodies. Thus, the question of their identification by the human mind is crucial. The role of the imagination, as the locus for the creation of specific pictures and for the identification of geometrical elements represented by the physical world, has to be redefined and radically changed. A literal abstracting of a geometrical shape from some particular and concrete object is impossible, since it is not the likeness of the object that touches the eye but only radiation from its colored surface. Therefore, there must be some innate ability that produces within the mind the recognition that a certain geometrical line or figure has been described by

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a specific movement of some external body. The relationship between the motion and the line is one of representation—there is no line outside, and there is no motion in the line conceived within the mind, but they share an analogous relationship. The mind recognizes that a particular motion of a physical body has exactly described a geometrical figure: “Adde quod secundum demonstrata hoc capite radius in perspicuo (quatenus perspicuum) non est, sed fuit, vel quasi fuit.”103 Thus, no concrete material line exists in the material realm, but the mind recognizes that the motion of a body is represented by a geometrical line. The process of representation is fully revealed in the case of sounds: “A string [chord] is here taken to mean not the line subtended by an arc of a circle, as in geometry, but any length which is capable of emitting a sound; and as a sound is elicited by motion, ‘string’ is to be understood in the abstract in reference to the length of any motion whatever, or to any other length whatever, even if it is conceived in the mind.”104 The human soul is always in search of harmonic proportions. These proportions are detected in the motions of bodies. However, just as the mind recognizes the straight line in the motion of light, so it recognizes in other physical motions geometrical proportions. “[T]he operations and motions of bodies, which imitate the harmonic proportions, are on the side of the soul and the mind, assigning them a cause for their delight in consonance. . . . [P]roportions are entities of Reason, perceptible by reason alone, not by sense, and . . . to distinguish proportions, as form, from that which is proportioned, as matter, is the work of Mind.”105 The harmonies the mind identifies in the motions of the air that produce different sounds are the same harmonies that the mind identifies in other natural motions such as the movements of the planets or the motion of light. “Throughout we shall indeed speak of melody, that is harmonious intervals which are not abstract but realized in sound; yet to the educated ears of the mind the underlying reference throughout will be to the intervals abstracted from the sounds. For it is not only in sounds and in human melody that they yield their charm, but also in other things which are soundless.”106 Kepler turned on its head the process of abstraction and with it the process by which mathematical entities originate. Not only can the mind conceive mathematical figures and entities from within itself, it can form an image of any mathematical entity only through an image of motion. “Thus the mind, without imagining certain motion, does not discern harmonic proportions from the confused infinite [proportions] surrounding them within a given quantity” (my emphasis).107 Only through motion can the mind differentiate between harmonic ratios and unmelodic ones: “In general in everything in which quantity,

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and harmonies in accordance with it, can be sought, their presence is most evident in motion rather than without motion. For although in any given straight line there are its half, third, quarter, fifth, sixth, and their multiples, yet they are lurking among other parts which are incommensurable with the whole.”108 The mental act of contemplating harmonic proportions is accompanied always by an image of a hand drawing a line: “[The mind] evinces by contemplation what the hand evinces by drawing a line.” The mind, “through the eyes,” picks out from an infinite confusion of things “which are present without motion . . . the more important” only through an image of the motions of the hands, which sketch out, for instance, the chord of a circle subtended by a third of it.109 What identifies the external motion as represented exactly by a ray or any other geometrical figure or proportion is an innate and divine element within the human psyche. The content of the divine element in the human mind is constituted from geometrical archetypes. These archetypes function like a twosided mirror that on the one hand represents the mind of God and on the other hand identifies the divine plan in the physical realm. It is a triple reflection where both physical creation and the human mind reflect the mathematical archetypes coeternal with the divine essence: “that the mathematical reasons for the creation of bodies were coeternal with God, and that God is pre-eminently soul and mind, whereas human souls are images of God the Creator, even in essentials in their own way, is known to Christians.”110 The balance is set between the three components: God the Creator, the human mind, and the structure of the physical world. The fundamental component of the Keplerian system is God, whose essence is identified with mathematical forms: “Geometry, which before the origin of things was coeternal with the divine mind and is God himself (for what could there be in God which would not be God himself?).”111 Then those geometrical forms were embedded in the souls that govern the physical world as “patterns” that inform the motions of physical bodies. Lastly, geometrical forms were implanted in the human mind as the image of God. For geometry . . . is coeternal with God, and by shining forth in the divine mind supplied patterns to God . . . for the furnishing of the world, so that it should become best and most beautiful and above all most like to the Creator. Indeed all spirits, souls, and minds are images of God the Creator. . . . Then since they have embraced a certain pattern of the creation in their functions, they also observe the same laws along with the Creator

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in their operations, having derived them from geometry. Also they rejoice in the same proportions which God used, wherever they have found them, whether by bare contemplation, whether by the interposition of the senses, in things which are subject to sensation, whether even without reflection by the mind, by an instinct which is concealed and was created with them, or whether God Himself has expressed these proportions in bodies and in motions invariably, or whether by some geometrical necessity of infinitely divisible material, and of motions through a quantity of material, among an infinity of proportions which are not harmonic, those harmonic proportions also occur at their own time, and thus subsist not in being but in becoming.112 The human mind knows geometry not through any sensual experience but from within, as if by an instinct. In sharp contrast to the Aristotelian theory of the cognitive origin of mathematical entities, Kepler asserts, “Geometry . . . passed over to Man along with the image of God; and was not in fact taken in through the eyes. . . . [Q]uantities possess constructibility not by virtue of the figures’ passing before the eyes, but in virtue of being clear to the eyes of the mind, in virtue not so much of having been abstracted from sensible things but of never having been associated with them.”113 The human mind orders and shapes the sensual faculties, and especially those of the eye. The aim of the senses is to supply the human mind with the reflection of its own content in the material world. An evidence for that is the joy the mind feels when it detects the divine harmonies in the world. “For they [human souls] take joy in the harmonic proportions in musical notes which they perceive, and grieve at those which are not harmonic. . . . But if we also take into account another harmonic proportion, that of notes and sounds which are long and short, in respect of time, then they move their bodies in dancing, their tongues in speaking, in accordance with the same laws. Workmen adjust the blows of their hammers to it, soldiers their pace. Everything is lively while the harmonies persist, and drowsy when they are disrupted.”114 The soul identifies its own content in the world outside: Indeed to the human mind . . . quantity is known by instinct, even if for this purpose it is deprived of all sensation. Of itself it understands a straight line, of itself an equal distance from a given point, of itself it forms for itself from these an image of a circle. . . . If so, it can . . . perform the function of the eye in seeing the diagram. . . . Certainly the mind

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itself, if it never had the use of an eye at all, would demand an eye for itself for the comprehension of things which are placed outside it, and would lay down laws for its structure which were drawn from itself. . . . For the recognition of quantities, which is innate in the mind, dictates what the nature of the eye must be; and therefore, the eye has been made as it is because the mind is as it is, and not the other way round.115 In these passages, Kepler breaks away from the Aristotelian tradition and the dichotomy between imagination and ideas and the material world. The mind is an active force that shapes our sensual experience of the world. The passivity of the eyes and of the intellect in confronting the world is rejected. Instead of conceiving the imagination as an obstructive force in coming to grasp the world as it is, Kepler posits the imagination as the place where mathematical understanding of the world is made possible. These passages make the human imagination relevant to physical understanding. Following the Hellenistic tradition, the medieval commentators conceived the imagination either as a passive receptor of replicas of external bodies or as an active servant assisting the intellect in creating hypothetical and fictitious constructions in order to explain natural phenomena. Their conception of the relationship between imagination, mathematics, and the world rested on the assumption that the external world impresses its essences on the eyes. Thence the imagination passively synthesizes, from these impressions, images of external reality that are but secondary entities in relation to the real physical objects. Finally, geometrical figures are abstracted from these images, and by now they are twice removed from natural phenomena. The Platonic worldview, on the other hand, assumed that physical objects are mere shadows of some real ideas in a different world. Therefore, whether according to Plato or Aristotle, the gap between mathematical figures and material reality is preserved. The significance of this gap between mathematical entities conceived in the mind and physical bodies is that any attempt to describe the natural world mathematically is doomed to the status of a mere hypothesis. In order to avoid this gap, Kepler proposes two assumptions: (A) The physical world follows a geometrical pattern, not in the abstract but in concrete material motions, and (B) these motions represent themselves to the human mind through the senses. However, this representation does not take place through a physical impression of the external body on the eye but through the mediation of light. Light, as a two-dimensional surface (i.e., as a mathematical entity), represents through its motion the geometrical structure of reality on

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the retina. The mind can now recognize its own geometrical archetypes in the picture of physical reality depicted on the retina. Kepler’s attack on the role of the Aristotelian medium released the eyes from their subjection to physical impressions and transformed them into sensitive filters for geometrical proportions embedded in the created world and transmitted by the action of light. Kepler rewrites medieval optical theory and posits an active light as a mediator between physical reality and the human sense organ. Medieval perspectivists discussed optical phenomena in the framework of the Aristotelian cognitive chain. One of the consequences of this chain is that mathematicals are but secondary abstractions from those sense impressions. Kepler, on the other hand, conceives mathematicals to be embedded in the motion of light and in the visible motions of physical bodies. The human mind recognizes its own constitutive archetypes in the mathematicals represented by light falling on the opaque screen within the eye. The activity of light is complemented by the activity of the mind that recognizes itself in the realm of appearances. The Keplerian mind does not dissolve into the cognitive continuum of abstractions proceeding from natural experience to universal concepts. Nor is it a constructive modern mind that subjects natural phenomena to its own inner logic. Instead it is “a recognizing mind”—a separated subject that recognizes the reflection of its own mental contents (i.e., the divine geometrical archetypes) in the autonomous realm of nature. The complementary activities of light and the human mind, while supplying a solid foundation (from Kepler’s point of view) for the certitude of geometrical descriptions and explanations of the natural world, suggest a psychological problem. How can one differentiate between true pictures of external reality and other fantastic pictures conceived by the mind? In other words, what is the relationship between the formation of the Keplerian optical picture and the creative process behind artistic and imaginary paintings?

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4.

“NON TANQUAM PICTOR, SED TANQUAM MATHEMATICUS” Kepler’s Pictures and the Art of Painting

the “illiberal” drawings of kepler Early in the Thirty Years’ War Sir Henry Wotton, the English ambassador to the Holy Roman Empire, was traveling through Lower Austria. In Linz he met Kepler and saw him perform a “Philosophical Experiment,” as he tells Lord Francis Bacon in a letter on October 20, 1620. Kepler presented him with a “masterly done draught of a Landskip on a piece of Paper.” After a short inquiry, Kepler admitted that he had drawn the picture, quickly adding that “he had done it, Non tanquam Pictor, sed tanquam Mathematicus.” Kepler explained the way he used a tent as a portable camera obscura, projecting images of the world outside through “a long perspective Trunk, with a Convex glass fitted to the said hole, and the concave taken out at the other hand,” onto a piece of paper and tracing over the images to produce his drawings. The resulting paintings provide “the whole Aspect of the Field,” and thus Wotton recommends this instrument to “Chorography.” Yet his final remark is that “to make Landskips by it were illiberal; though surely no Painter can do them so precisely.”1 Wotton explicitly asserts that optical accuracy, though valuable, is not the only or even the most important criterion by which to judge the aesthetic value of an artistic drawing. Wotton’s phrase contrasts a passive, physical-mathematical representation of nature with an artistic imitatio of natural objects; between the natural necessity expressed through mechanical devices and the expression of human talents and ingenuity through artistic means.

Kepler’s laconic dictum and Wotton’s critical remark focus on the picture as the end product of the optical process. Although the innovative instrument is the object of Wotton’s wonder, it is the aesthetic status and value of the picture that Kepler produced that preoccupies Wotton’s speculative reflections. This anecdotal short exchange circumscribes “What is a picture?” as a crucial question for the entire edifice of Kepler’s new optics, not only by demarcating the picture produced optically by instruments from an artistically painted picture, but by pointing to the epistemological difficulties involved in Kepler’s endeavor: an ontology of light and shadow based on a new nonmimetic mathematical language of representation (as discussed in the above chapters) necessarily denied any direct grasp of the visual object, turning visual experience into a mediated process. Kepler attempted to overcome this epistemological impediment by suggesting that the mind recognizes its own geometrical content in the infrastructure of the natural world. This act of mental recognition needs an epistemological device to guard its veracity, saving the scientific cogency of observational practices, and differentiating authentic human visual experience from apparitions and illusions. In meeting the challenge of physically bridging external stimuli and internal perception, Kepler selects the term pictura to designate the necessary boundary object. This multivalent term, however, situates Keplerian optics squarely within the semantic field of Renaissance theory of painting. This association of scientific inquiry with visual arts is not unique to Kepler. Historians have noted the intimate association of art and science in early modern culture, including the intricate personal relationships between Galileo and several early Baroque artists such as Rubens and Cigoli and their intellectual exchange concerning sunspots, lunar secondary light, and telescopic observations of the heavens;2 or the manner in which, for instance, Galileo, Scheiner, and Descartes, when reporting observations and theorizations, fancied themselves as the famous painter Apelles. Others have stressed Galileo’s training in perspective and drawing as essential to his ability to decipher the lunar landscape observed through the telescope; or the application of aesthetic values and standards of symmetry and proportions to the advent of Copernicanism as an argument against the monstrosities of the old Ptolemaic system of the world. In several cases these conflicting cosmological systems were even identified with specific artistic styles.3 Modern historiography has singled out, however, the (re)discovery of the geometrical technique of artificial perspective as the principal “trading zone” between artistic creativity and scientific methodology. Erwin Panofsky

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identified it as a unique symbolic form that set the framework of the modern mathematically informed scientific attitude. Others perceived Renaissance perspective as the missing link between medieval scientific achievements and the scientific revolution of the seventeenth century, or as a meeting point where artistic practices are converted into intellectual pursuits. Special attention was given to the interface between the medieval mathematical science of perspectiva and the new mode of integrating mathematics and sight in the practical context of Renaissance painting. Historians of science, such as Stephen M. Straker, have suggested the artistic context as specifically constitutive of Kepler’s new revolutionary optics. Albrecht Dürer’s application of strings to capture the operation of visual rays, or illustrations of Platonic solids and polyhedral bodies by Leonardo da Vinci and Wentzel Jamnitzer, indicated the relevance of artistic sensitivities to Kepler’s new science: his use of strings to represent the motions of light rays, or his fascination with Platonic solids as the geometrical archetypes of the Copernican universe. Svetlana Alpers suggests that Kepler shared the same cultural grounds with Dutch painters who evaluated description and representation over allegories and narrative. How can one accommodate these assumed cultural undercurrents and connotations with Kepler’s clear differentiation between mathematical draftsmanship and painterly creativity in Wotton’s succinct anecdote? As noted above, in designating the image on the retinal screen as pictura, Kepler was fully aware of these connotations, yet a closer examination of his analysis of pictorial formation on the retina reveals a poignant and thorough critique of the epistemological claims of Renaissance painting tradition. Kepler’s pictura undermines Renaissance concepts of visual depiction and representation, suggesting instead a new economy of artificiality and naturalness, of imagination and knowledge.

alberti’s humanist geometry Reviewing the indebtedness of Renaissance artificial perspective to medieval optics sheds new light on Kepler’s sharp distinction between painterly depictions and mathematical representation. Instead of the historiographical question of continuity, the emphasis shifts to the way various cultural strands and interests transmuted medieval mathematical pursuits into the humanistic dilemma over the depiction and value of visual reality.4 Modern historiography has attempted to embed Leon Battista Alberti’s groundbreaking De pictura (ca.

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1435), in which he laid the foundations for the theory of artificial perspective, within a progressive historical process narrating how “out of these medieval components [Alhacen and Pelacanni (i.e., Blasius of Parma)] emerged a new alloy that showed artists the relevance of science as a way of making sense out of the experience of nature.”5 The historical figure of a rational artist is thus presented as a necessary step on the way toward the seventeenth-century revolutionary mathematization of nature.6 My detailed examination of the way Alberti applied certain geometrical elements to his explanation of sight will discard the historiographical attempt to portray the Renaissance technique of perspective painting as the missing link between the achievements of medieval speculation about nature and the New Science of the seventeenth century. Alberti’s account of the pyramid of vision is not a simple modification of its medieval equivalent, but a conscious effort to come to terms with humanistic concerns and anxieties. The accepted medieval exposé of the visual cone was an elaboration of the model suggested by Ibn Al-Haytham (Alhacen) in the eleventh century.7 Disregarding some modifications that Alhacen utilized in order to solve some problems of the visual power of oblique rays,8 it is possible to summarize the structure of Alhacen’s pyramid of vision in the following way. The pyramid of vision is constituted only from the collection of the rays flowing from the field of vision perpendicularly to the surface of the eye. All other rays are either reflected or refracted, but in any case do not participate in the process of clear visual perception. The rays originate from each and every point on the surface of the visual object,9 that is, through the decomposition and reconstruction of the visual object in the process of vision. This flow of visual rays creates an infinity of pyramids in the medium around the object. In each of those pyramids only one ray is perpendicular to the surface of the eye and can transmit the visual power.10 Except for the size of the ray’s angle of incidence on the surface of the eye (i.e., a quantitative difference), there is no qualitative difference between the rays; each transmits with it the form of the point from which it was initially originated. Alhacen’s basic assumption was that every point on the surface of the visual object radiates its form in all directions. Turning to Alberti’s highly poetic and metaphorical description of the manner in which the visual rays form the visual pyramid, one can detect the radical divergence of his analysis from that of his medieval Arab predecessor: “[The philosophers] say that surfaces are measured by certain . . . visual rays, since by their agency the images of things are impressed upon the senses. These rays, stretching between the eye and the surface seen, move rapidly with great

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power and remarkable subtlety, penetrating the air and rare and transparent bodies until they encounter something dense or opaque where their points strike and they instantly stick.”11 Although Alberti does not commit himself to the issue of extramission or intromission of the rays of vision, he suggests a clear image of the origin and direction of the visual rays: in describing the motion of the rays as aiming, encountering (offendant), and striking at the visible surfaces, it is obvious that Alberti assumes the origin of these rays to be in the eye. This impression is strengthened when reading the following paragraph: “Let us imagine the rays, like extended very fine threads gathered tightly in a bunch at one end, going back [recipi] together inside the eye where lies the sense of sight. There they are like a trunk of rays from which, like straight shoots, the rays are released and go out towards the surfaces in front of them” (my emphasis).12 Clearly, Alberti believed that visual rays do not originate in visible objects. These objects are not subjected to punctual analysis that decomposes and then recovers their form. On the contrary, the rays originate from the eye and capture the whole form of the object. They then turn back into the sense of vision according to the visual angle that is located (consistat) in the eye. In contrast to the medieval geometrical model, the difference between the rays composing Alberti’s pyramid is not quantitative, that is, by reason of differences in angles of refraction and reflection, but qualitative “according to [their] force and function.”13 There are three kinds of rays, so Alberti explicates: the extrinsic rays (extremi), the middle rays (medii), and the central ray (centrici). The extrinsic rays transmit the dimensions of the body by touching and holding its outlines; the middle rays transmit color and light; and the central ray constitutes the perpendicular between the eye and the perceived object and ensures keen sight.14 Alberti diverges from the medieval model on all the crucial points: the visual rays flow from the eye to capture the image of the visual object; the visual rays differ qualitatively and not according to their angle of refraction; and there is no punctual analysis of the visible object and its image is transmitted as a whole into the eye. Instead of the medieval perspectivist tradition, one should follow some obvious clues that point unwaveringly to Euclid’s Liber de visu as the main source for Alberti’s model of the visual pyramid. Since its reception in the Latin West during the twelfth and thirteenth centuries, Euclid’s treatise was widely read and discussed, and thus was surely available to Alberti.15 Euclid’s treatise is concerned almost exclusively with problems of perspective, and Alberti adopts the Euclidean assumptions without hesitation. Most importantly, Alberti’s fundamental tenets are already included in Euclid’s

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opening proposition: “It is posited that straight lines drawn from the eye sweep away an immense space. And that the figure contained by the rays of sight is a cone, having its vertex in the eye while the base [lies] at the end of the visible objects.”16 Similarly to Alberti’s description, the Euclidean visual rays are drawn from the eye, creating a cone (or pyramid) that falls on the visible object, capturing it completely and transmitting its image back into the eye. Furthermore, Euclid claims that “visible [objects] are seen more keenly when seen under more angles”17 (Sub pluribus autem angulis visa perspicacius videri), because objects that encounter a larger number of visual rays are captured under more angles that are created between neighboring visual rays, and are therefore seen more clearly. This assumption is similar to the advice Alberti offered: “I usually give my friends the following rule: the more rays are employed in seeing, the greater the quantity seen will appear, and the fewer the rays, the smaller the quantity.”18 Although Alberti must have encountered and studied the medieval treatises on optics during his studies in Bologna, in his De pictura he ignores the obvious theoretical advancements that they had to offer, and, in a typically humanistic move, embraces the classical theory of Euclid instead. In adopting Euclidean optics as the geometrical infrastructure of his treatise on painting, Alberti indicates that perspectival painting is to be one more tool in the arsenal of the humanist program of the revival of antiquity and its virtues, preferring this cultural goal over mathematical sophistication.19 Alberti’s humanistic adaptation of optical topics does not consist only in promoting ancient authority; as has often been noted, he dressed up Euclid’s optics in rhetorical garments.20 In his careful and insightful analysis of the structure of Alberti’s treatise, D. R. Edward Wright suggests that it was constructed after Quintilian’s pedagogical Institutio oratoria. Alberti thus assumed the role of the painter to be near-identical to that of the orator; both have to utilize their tools to represent absent sublime things in a persuasive manner that will induce their audience to aspire for moral improvement. Furthermore, as Michael Baxandall points out, Alberti’s adaptation of rhetorical models to his treatise on painting does not end with the fashioning of the painters’ persona and their recommended course of studies, but involves elements concerned with the painterly practice per se. In specific cases Alberti molds meanings borrowed from rhetoric into visual and geometrical vessels. Terms, such as proportio or compositio, that have strong allusions to the rhetorical world are associated in De pictura with modes of extracting the right measures in vague cases, and with assembling different parts of the picture into a coherent and

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beautiful whole.21 Alberti applies these modes in an effort to fashion an ideal representation counter to the phenomenal world. Either through positing the ideal measure of a human braccia22 or by assembling the different segments of the painting as if they were parts of a well-formed flowery sentence, Alberti aspires to transform a discordant phenomenal world into a visual experience of beauty and harmony.23 Applying rhetorical models to geometrical techniques of perspective and to painting in general suggests that these are forms of human expression governed by the same rules of eloquent speech. Just as rhetoricians expect to reform their audience through eloquent speech, so Alberti aspires, through perspective, to educate the spectator ad bene spectandum. This significant move elevates visual experience by casting it into rhetorical molds. It is the outcome of Alberti’s diagnosis of the failure of the tools of humanists’ eloquence to convey virtue into the social world. Cultivating the orator’s skills is not the effective remedy to a corrupted human world made of deceptive disguises and deceits.24 A stronger potion, Alberti suggests, is the reform of visual and aesthetic experience, that is, the fashioning of proportionate and harmonious images produced through new mathematical procedures and devices. Perspective, for Alberti, is a vehicle to infuse beauty into a corrupt social world that rejects virtue in the name of ephemeral riches and fame. Casting his theory of perspective in a rhetorical mold, Alberti employs it to confront certain contradictions he perceives within the humanist worldview. Ever since Eugenio Garin countered Jacob Burckhardt’s image of Alberti as the universal man of the Renaissance, bringing to light those darker and pessimistic aspects of the great humanist’s persona, historians have replaced the heroic image of Alberti with the figure of a scholar ridden with contradictions, anxieties, and cultural discontent.25 Alberti critically follows the logic of the humanist program to its radical conclusion, the immediate implication of the humanists’ pursuit of virtue and eternal glory. At the same time, there is a renunciation of temporal fame and material riches that undermines their aspiration to become civic educators and reformers. Alberti describes vividly the social frustration of the scholar, who strives after “fame and honor,” which he believes will come to him “through erudition.” Alberti realizes that this mode of thinking and behaving is “not very realistic” and does not tally with “the ways of the world.” He confesses that “by repeated experience and by having to deal with society” he has to comply with “those arguments concerning the disadvantages of a scholar’s life.” Scholars cannot achieve those things valued by the social world, such as wealth or social recognition; instead their life is

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“necessarily hard and harsh.”26 Scholars are excluded from the prizes granted by fortune, such as financial wealth or political power. What they gain, however, is knowledge of the loftiest matters, which far surpasses all other worldly successes. Scholars know the true value of ephemeral things, such as public flattery and popularity. These, he says, are “far less important than knowledge and wisdom.” Public affairs are “transitory, unstable, fragile, full of pointless effort, rife with fears and suspicions, fraught with mishaps and downfalls.” True tranquility and peace of mind can be attained only through virtue and the “full remembrance of forgotten antiquity and its wisdom.” Only through an exacting kind of virtue can one “rise above and endure any attack or accident of fortune.”27 Virtue thus exists only outside and beyond the social realm of civic activity. A critical gap is formed between the humanist ideals of vita activa and Alberti’s aspiration to true virtue. This gap between the realm of virtue and the wretched social world is portrayed on the famous medallion of Alberti made by Matteo de’ Pasti. The front of the medallion displays the profile of the youthful Alberti; the back shows a winged eye, and below the eye is the inscription “Quid tum?” (What next?). As Jarzombek brilliantly shows, this question appears for the first time in the Albertian corpus in a dialogue titled “Somnium” in the Intercoenales.28 Written between the late twenties and mid-thirties of the fifteenth century, the Intercoenales are a collection of forty-three pieces of varying lengths, comprising dialogues, dreams, fables, and allegories.29 Lepidus (Pleasant and Witty) meets Libripeta (the Book Fanatic) emerging from the sewer. Libripeta tells him of what he saw in the underworld in the valley of lost things: “enormous bladders filled with license, lies, the sounds of flutes and trumpets . . . favors, which consist of silver and gold hooks; and next to these are leaden wings, which they say are men’s high offices.”30 Lepidus, horrified by this tale, asks, “Quid tum?” The question represents the moment of bitter revelation, the transition from innocence to skepticism, from naïveté to an understanding that wisdom and the arts have been lost in the sewer. Above this expression of cynical sobriety fly the winged eyes that Alberti associates in another piece with Pegasus: “The Pegasean horse represents the course of life and the fleeting age by which we are rushed along. As we hasten towards the havens of a better life, we must use wings to avoid sinking into the waves. These wings are the powers of our human intellect and the gifts of our minds which help us attain even the heavens in our study of nature, and which join us to the gods in piety and virtue.”31 Alberti’s humanistic program in fashioning a novel mode of ocular experience aims to give wings to the human eye, enabling it to gaze beyond immediate

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appearances and to capture the realm of virtue and truth. Alberti’s theory of perspective redefines the fundamental concepts and procedure of geometry in visual terms, lifting the human eye toward a revelation of beauty embedded in the physical world. In the opening paragraphs of his De pictura, Alberti boldly demands that abstract entities should become visible: “Mathematicians measure the shapes and forms of things in the mind alone and divorced entirely from matter. We, on the other hand, wish to set this thing up as visible, and will therefore use, as is said, a cruder Minerva for writing.”32 Contrasting the mathematicians’ abstract mode with the painters’ need to embed mathematics in a concrete visible form challenges the dichotomy between the divine and spiritual realm and the material world. Alberti insists upon blurring this dichotomy between the visible and material signs and the abstract supersensorial mathematical entities. In order to accomplish his goal, he adopts several measures. Initially he implements what seem at first glance merely crude definitions for geometrical entities: “A point is a sign which one might say is not divisible into parts.” This is not mere reiteration of the Euclidean definition, as Alberti stresses immediately that a sign is “anything which exists on a surface so that it is visible to the eye.”33 A “point” turns out to be any very small mark on any surface, because “things which are not visible do not concern the painter, for he strives to represent only the things that are seen.” This does not mean the painter is limited to a description of present phenomena. On the contrary, Alberti celebrates painting’s “truly divine power” to literally make the “absent” and invisible “present” and so, for instance, “represent the dead to the living many centuries later.” The visible mark on a surface is therefore not a crude and dull material stain but, as part of the art of painting, has the magical ability to materialize invisible elements. This quality of the painted visible marks is embedded in the nature of the picture itself. Pictures are the material representation of the invisible surface that bisects the pyramid of vision. Just as though this surface which they colour were so transparent and like glass, that the visual pyramid passes right through it. . . . But as it is only a single surface of a panel or a wall, on which the painter strives to represent many surfaces contained within a single pyramid, it will be necessary for his visual pyramid to be cut at some point, so that the painter by drawing and colouring can express whatever outlines and colours that intersection presents. . . . Therefore, a painting will be the intersection of a visual pyramid . . . represented artificially with lines and colours on a given surface.34

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In order for the picture to correspond exactly to this particular and invisible intersection of a pyramid, Alberti devises his famous veil, disclosing that “among my friends I call [the veil] the intersection.” By stretching threads that crisscross each other over a wooden frame, Alberti does not merely fix the painter’s eye on the painted object. His initial aim is to stabilize the visual pyramid and to materialize the otherwise transparent intersection: “I set this [veil] up between the eye and the object to be represented, so that the visual pyramid passes through the loose weave of the veil. This intersection of the veil has many advantages, first of all because it always presents the same surfaces unchanged, for once you have fixed the position of the outlines, you can immediately find the apex of the pyramid you have started with, which is extremely difficult to do without the intersection.”35 In this way, the veil divulges the pyramid of vision together with its special points, such as the apex, and its base with its outlines. This technical procedure for the materialization of abstract concepts and ideas (in this case of invisible mathematical entities) was also used for even more intangible and illusive entities such as beauty. According to Alberti, the artist does not imitate and represent nature itself, but aims at the forms of beauty that are recondite within natural phenomena: “But, considering all these parts [of the body the painter wishes to represent], he should be attentive not only to the similitude of the things, but primarily to their true beauty.”36 Indeed, Alberti forbids the painter to follow only his own mind, but urges him to observe nature. By “nature,” Alberti means especially the form of beauty that is concealed within it. The forms of beauty are not given, but are concealed behind the phenomenal realm. One has to collect and reconstruct them from the different bodies in order to create the form of beauty that is beyond the appearances of real nature: “Therefore from the beautiful bodies all the esteemed parts are to be selected. Accordingly, no small effort is demanded in order to grasp, and indeed to have and experience by industry and study beauty. Even if that thing is the most difficult of all, because all the beautiful qualities are not discovered from one place but are scattered or dispersed, yet in order to investigate or to learn this thoroughly all labour is to be displayed.”37 The main intention of Albertian perspective was to enable the artist to reveal the “beauty” that exists within and beyond appearances. Painting over a two-dimensional surface makes salient those values crucial to Alberti’s ideal of beauty, such as “symmetry” and “harmony” between the different parts of the painting. Albertian perspective constitutes the perfect relationship between the bodies that appear in an artificially organized space. Hence, the main goal

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of perspectival painting, according to Alberti, is not to represent faithfully space as a means to imitate visible nature, but to represent “beauty.” Painterly representation is an instrument for the improvement of nature and especially for self-improvement: “The gifts of nature are to be cultivated and enriched by industry, study and especially practice, and hereafter nothing that pertains to glory should be seen to be negligently overlooked by us.”38 Remodeling painting as a liberal art, Alberti renounces the mechanical context of this artistic practice dealing with corporeal bodies. Instead, he stresses that the goal of the art of painting is the cultivation of the human spirit in contributing to the realization of the rational soul. The view seen from Alberti’s window does not merely encompass material and corporeal appearances, but calls for an active act of reading aesthetic and moral values into them. The technique of perspective was a means toward turning abstract and invisible notions into a concrete visual experience, and thus affecting the spectator’s moral attitude. Painting, as a visual embodiment of geometrical concepts, supplies the wings that lift the humanist scholar to the heavens on the way to acquire virtue. It makes it possible to concentrate and disclose the “beauty” hidden and dispersed throughout the natural world. Improving one’s intellectual and especially moral capacities demands a recognition that the true nature of the world is more beautiful and varied than what the human eye is physically exposed to. The picture is transformed into a mathematical intersection between an invisible and abstract realm of beauty and the dejected materiality of the human world, that is, abstract values and virtues are transformed into concrete visual experience. The exhausted scholar of De commodis litterarum atque incommodis and the ailing sufferer from the nightmarish social experience of “Somnium” find solace in the aesthetic experience provided by pictures drawn in perspective. In rearranging geometrical procedures according to rhetorical precepts, and in embedding mathematical entities in visual experience, Alberti aspires to bridge the realm of virtue and concrete human existence, salvaging the humanist social-reform program.

leonardo’s disillusioned eye The anxieties and inner tensions that shaped the humanistic role Alberti assigned to the technique of artificial perspective did not affect its initial enthusiastic reception in the fifteenth century by the likes of Uccello and Piero della

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Francesca. Perspective implied a magical power of transformation. Just as a two-dimensional surface metamorphosed into three-dimensional space, so the emotions of those viewing the painting were deeply affected. It was this potential of painting to affect and transform emotions and to conjure absences that attracted Leonardo da Vinci in the last quarter of the fifteenth century. If indeed these powers could truly create the images of absent persons and objects, then the painter shared certain attributions of the divine Creator: “If the painter wants to see beauties with which he will fall in love, he is a lord who can generate them, and if he wants to see monstrous things which may terrify or be buffoonish, and be laughable or truly arouse compassion, he is their lord and God.”39 Leonardo disparaged human language, highlighting instead the ability of painting to convey the sense of things directly: “Painting represents the works of nature to the [common] sense with more truth and certitude than words or letters do, but letters present words to the [common] sense with more truth than painting does.” Language, as a human construct, could only communicate human intentions, whereas painting implied the creative power embedded in nature itself. “We will declare that the more admirable science is the one that represents the works of nature rather than the works of the worker, that is, the works of man, which are his words such as poetry and similar things which pass [to the common sense] through human language.”40 In order to avoid the futility of language, Leonardo suggested a different subject matter, different tools, and a different sense experience in order to acquire true and useful knowledge. “Painting acts through a more noble sense than poetry, and renders the figures of the works of nature with more truth than the poet [does]. And the works of nature are far more worthy than words, which are the works of man, because there is the same proportion between the works of man and the works of nature as between man and God. Therefore, it is a more worthy thing to imitate things in nature, which are actual similitudes in fact, than to imitate facts and the words of man in words.”41 The superiority of painting over poetry, according to Leonardo, was established in its ability to produce the simulacra of things, enabling invisible and sublime objects to appear in tangible forms: “thereby lovers are moved by the simulacra of their beloved to speak with painted imitations. Thereby, with fervent vows, people are moved to seek out the simulacra of gods, and not the sight of the works of poets, which figure the same gods in words. Thereby, animals are deceived.”42 Leonardo was captivated by the narrow gap between the painted image and the object it represents; indeed, for Leonardo this gap had a magical quality that could be manipulated, inducing people’s (and animals’) behavior:

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The end result of painting is communicable to all generations in the universe because its final effect is dependent upon the virtu visiva. . . . So [painting] needs no interpreters of different languages as letters do. [Painting] satisfies the human species immediately, not differently from things produced by nature, and it satisfies not only the human species but also other animals, as was shown by painting which portrayed the father of the family, whom little children tried to caress when they were still in swaddling clothes, and also the dog and cat of the same household. What a marvelous thing it was to observe this scene!43 In summoning these powers of painting, Leonardo gave much importance to the mathematical procedures as guarantors of the certitude and truth of the pictorial representation: “The elements of mathematics, that is to say number and measure, termed arithmetic and geometry, discourse with supreme truth on discontinuous and continuous quantities. Here no one argues that twice three makes more or less than six, nor that a triangle has angles smaller than two right angles, but with eternal silence every dissension is destroyed, and in tranquility these sciences are relished by their devotees.”44 Applying mathematical certitude to the depiction of the natural world, Leonardo confronted nature’s capacity to playfully generate variety that undermines any sense of order.45 This endless metamorphosis, Leonardo perceived, was not initiated by some rational plan but according to blind drives—“Lust is the cause of generation.” In his investigation into the motion of water, all that Leonardo could have detected was a constant process of change that turned order into disastrous chaos. Leonardo found himself immersed in a Heraclitean,46 or better still, a Lucretian view of the natural world; a world that was in a constant flux that made any human experience epistemologically futile: “Nature is full of infinite causes which were never set forth in experience.”47 In his efforts to decipher these powers of mutability, Leonardo eventually had to concede that the distance between the realm of mathematics and the physical world is unbridgeable: “There is an infinite difference between the mechanical point and the mathematical point, because the mechanical is visible and consequently has continuous magnitude and everything continuous is infinitely divisible. The mathematical point, on the other hand, is invisible and without magnitude and where there is no magnitude there is no division.”48 This ontological gap between the realm of mathematics and natural phenomena undermines Alberti’s program to endow visual experience with proportionate order and harmony. As Leonardo’s investigations into the theory of perspective progressed, he arrived at the frustrating conclusion that

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the physical world does not correspond to the painter’s geometrical conception of the visual process. Through the early 1500s, Leonardo became aware of the abstraction involved in the painter’s perspective. The painter’s pyramid of vision was deprived of physical reality, and the image within the eye was further complicated by various distortions in the external world. In addition, “perspective made by art” in two dimensions was itself subject to the distortions of “perspective made by nature.”49 Leonardo examined several problems, such as angular problems of apparent size,50 lateral recession, the fact that “the eye does not know the edge of any body,”51 and binocular vision. In depicting the elusive appearance of leaves on trees, he noted that each leaf is affected by four tonal factors, “that is to say, shadow, brightness, luminous highlight and transparency.” Their specific combination transforms each leaf ’s apparent color at any temporal instance. Furthermore, the apparent leaf is subject to more variables: “Remember, O painter, that the varieties of light and shade in the same species of tree will be relative to the rarity and density of the branching.” All these minute changes are blurred at a distance, and the celebrated sense of vision finds it “difficult to recognize the parts, in that they make a confused mixture, which partakes more of that which predominates.” The attempt to capture nature as it is seems doomed to failure. In addition to the various optical transformations, nature exhibits an astonishing variety of forms: “You imitator of nature, be careful to attend to the variety of configurations of things.”52

rhetoric and perspective in the sixteenth century Leonardo’s program suppressed the rhetorical edge of Alberti’s humanist artificial perspective in order to secure sight as a source of knowledge. Its failure nevertheless heralded one of the dominant features of sixteenth-century criticism of the power of sight and of the deceptions and illusions involved in artificial perspective.53 The critical attitude toward the epistemological shortcomings of visual images and pictorial representations gained further dramatic impetus with the advent of the Reformation. The iconoclastic eruptions through the sixteenth century challenged Renaissance pictorial aspirations to convey spiritual truth. Whereas the radical reformers claimed, in the words of Zwingli, that “die Wänd sind hüpsch wyss” (the walls are beautifully white), Luther and Calvin allowed pictorial representation only of “those things which can be beheld with the eyes,” neutralizing any allegorical interpretation of the visible signs.54 Catholic theorists of art facing this demand had to redefine the representational

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function of visual images and their relationship to what is imagined. In this context, the technique itself of “perspective”’ became a complicated, somewhat obscure, and highly mathematized domain eventually turned into an arcane and secret subject (though its secretive aspect was “only a matter of acquiring some mathematical knowledge”).55 Some of the associations concerned with perspective turned out to be negative, and Pontormo, for instance, argued that painting “disturbs his [the painter’s] mind rather than enriching his life” (piotosto fastidi di mente che aumento di vita), implying that the painter’s eagerness to exceed nature by extravagant inventions may result in melancholic symptoms. Dürer put much perspective into his Melancholia I, while Hans Holbein applied anamorphosis in order to disguise the skull in his Ambassadors, emphasizing that death has a special perspective—“Where Death is present, the figures are not and vice versa.”56 Perspective, notwithstanding the religious complexities involved in its being a tool of visual representation, became associated with saturnine humors. The religious notions of visual representation and the melancholic flavor associated with perspective pervaded the imperial court in Prague in the late sixteenth century. In this context, artists and scholars were seeking after new modes to define the nature and effect of pictorial representation. The tension between painting from life and painting from the imagination, and between the mathematical order of the painting (its proportions and harmonious appearance) and its ability to baffle the senses, was central to this late Renaissance art scene. The hopes Alberti and Leonardo pinned on artificial perspective and their eventual frustrations gained new coloring in Counter-Reformation Prague. The mathematical aspects of perspective had turned into deceptive elements, not a vehicle of epistemological certitude. Giovanni Paolo Lomazzo, one of the more important theoreticians of painting in the late sixteenth century, who was connected to the Hapsburg court through his association with Giuseppe Arcimboldo, opens his remarks on perspective with the following statement: “Such is the vertue of Perspectiue, that whiles it imitateth the life, it causeth a man to oversee and bee deceaved, by showing a small quantity in steed of a great.”57 Whereas for Leonardo it was still important to overcome visual errors and to save the cognitive value of sight, Lomazzo accepts these at face value and suggests that the main characteristic of perspective is deception. Many sixteenth-century writers on art shared Lomazzo’s emphasis on the illusionary nature of painting. Daniele Barbaro, for instance, asserts that painting deceives the eye to see “what it does not see.”58 The most obvious characteristic of painting is not the depiction of truth but

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the way the painter transforms the figures painted on a two-dimensional panel into an illusion of three-dimensional scenes. The ability to control these illusions and to amaze the spectators with imaginary scenes through the manipulation of basic perspectivist techniques became one of the most estimable gifts of the sixteenth-century painter. This power endowed the painter with godlike creative capabilities defined in relation to his ability to apply perspective as a universal geometrical procedure of his art. It also emphasized the subjective imagination as the active force that transforms perspective into art. The realization that individual ingenuity is responsible for great works of art caused the elaboration of idiosyncrasy and thus the fragmentation of the universal rule into different maniere. Instead of a unified sense of space and geometrical order, nature and perception were differentiated into individual instances and subjective acts of creative vision. Although painters apply the same measures and techniques, they still produce “almost miraculously” very different paintings, “each in his own manner and fashion.”59 Perspective becomes the chief instrument in the hand of painters to prove their ability in producing illusions and deceptions: “Moreouer Paintinge hathe this of the Arte Perspectiue, that it deceiueth the sighte, and in an Image diuersely placed, both caste many fourmes ouer the eies of the beholders . . . and with counterfaited measures, maketh the thinges seene whiche are not, as those whiche are, and maketh the thinges that are not so, to appeare in an other manner.”60 Denigrating the techniques of perspective led late Renaissance art theorists to stress the rhetorical dimension of painting at the expense of its mathematical accuracy. Instead of visual order as a means to convey pictorial meaning, they revived the notion of a picture as a mnemonic device, stressing the allegorical nature of visual signs; stains of color or a drawn line were regarded as mere mediators between depicted objects and a sublime and imperceptible realm. In creating forms that “do not exist outside the mind” (Cose che non hanno lessere fuor della mente),61 the painters around Rudolf ’s court employed detailed empirical observation and a careful study of real objects, while simultaneously fragmenting and reassembling them in order to create their new inventions. Throughout the sixteenth century, perspective was questioned by placing optical reality against the painted surface, stressing that artistic representation is merely a human-made image that cannot be reduced to the works of nature it aspires to depict.62 By bending the rules using optical instruments such as convex mirrors or the camera obscura, or by the excessive aggregation of realistically depicted details, the painter put into question the lines dividing the real from the fantastic.63

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kepler’s pictures and the limits of human imagination In claiming that his pictures were drawn in the mathematicians’ way and not the painters’ way, Kepler did not profess self-depreciation and humility (as Sir Henry Wotton might have perceived it) but boldly asserted a demarcating line separating his mode of observation from artistic experience. The transformation of geometry in the hands of late Renaissance painters into a tool of distortion and illusion undermined the cognitive value of visual experience and of optical analysis altogether. In redeeming observation’s claim to knowledge, Kepler had to redefine the notion of pictorial representation as the interface between the human mind and its physical environment. Kepler turned the picture into an ocular site where the human mind can recognize its own archetypal geometrical proportions embedded in sensory experience of physical reality. Kepler posited the picture at this interface as both a concrete material locus and a mathematical construct that facilitates such an act of recognition.64 In purging the act of sight of any rhetorical and allegorical signification, Kepler molded the picture into the crucial element in the causal chain forming visual experience: “vision is brought about by a picture [picturam] of the thing seen being formed on the concave and white wall of the retina.”65 What the mind sees is not an immediate expression of the visual object itself but its processed and somewhat distorted picture (formed on a concave surface), which furthermore is inverted as “those things that are on the right outside, are depicted on the left side of the wall, the left at the right, the top at the bottom, the bottom at the top.”66 How can such a picture serve as the foundation to veridical visual knowledge? In answering this question, Kepler diverges from the painter’s conception of the picture as a tablet intersecting the pyramid of vision. Instead, he redefines the picture as the result of a process of multiple refractions and intersections of light rays within the eye. The new significance of pictura demands a reassessment of the epistemological value of refractions and optical mediation, a reassessment that is encapsulated in Kepler’s definition in chapter 5 of Ad Vitellionem: “Since hitherto an image has been a rational being, now figures of objects that really exist on paper, or upon other screens, are called pictures.”67 Kepler poignantly contrasts the physical presence of the picture and the unreal image that is formed by an act of reason calculating ratios. This differentiation between virtual and real images, as modern commentators have noted,68 was not new with Kepler. Pecham’s Perspectiva communis, although not as explicitly as Kepler, supplies a clear demarcation between species (as the physical effect of direct vision) and ydolum (the optical result of reflected

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and refracted rays). The focal place for this difference is in Pecham’s discussion of mirrors, where reflected “light and color . . . manifest to the eye the objects of which they are the species.” The physical effect of the visible object on the mirror is through the species, whereas the reflected light and rays “reveal the object—albeit in another position.” The appearance of the visible object in a different place is what Pecham defines as ydolum (in Lindberg’s translation, “an image”): “What then is an image? I say that it is merely the appearance of an object outside its place. For example, sometimes the eye judges one thing to be two . . . because the object appears not only in its true place but also outside it. So in the present case, it is the object that is really seen in a mirror, although it is misapprehended in position and sometimes in number.”69 The resulting mirror image does represent the object, but this representation adds redundant and superfluous information disrupting the straight flow of species from the object to the eye, distorting one’s perception. Pecham’s basic assumption is that “[s]traightness is naturally associated with the propagation of light (as well as with any other action of nature), it arranges and orders nature, for every action is strong in proportion to its straightness.”70 In the mirror the species hit the reflecting surface and only then reach the eye. This interrupts the straight ray, forming a tortuous path where the final image is not a direct emanation from the body but an artificial and mediated product of the mirror: “A species produced by a visible object has the essential property of manifesting the object of which it is the likeness. . . . Therefore, even though it is reflected, it maintains its essence and thereby reveals the object—albeit in another position.”71 The image is not only in a wrong position but has no material-physical reality. “It is evident that nothing is impressed there.”72 This differentiation involves a more fundamental demarcation between the world of physical causes and effects and the realm of mathematical calculations and reasoning. Carefully reading Pecham’s Perspectiva communis, one may note that the sections discussing species are almost completely devoid of geometrical diagrams. The term “species” appears for the first time in part I, proposition 26, and is repeatedly discussed until part II, proposition 5, as this textual space is dedicated in its entirety to the physical, anatomical, and psychological aspects of vision. Outside this space the term hardly appears in the treatise, and Pecham is careful to use the term “rays” when he discusses the propagation of light, and “image” for the optical effects of reflection and refraction. Whereas the identification of visual rays with species does not present any special problem, the second identification of species with reflected rays and color forces Pecham to clarify the different epistemological status of these two

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entities. While the species supplies direct physical contact with the visible body, images are but mediated and nonphysical entities. Therefore, the analysis of species concentrates on their physical essence, whereas the analysis of reflected images finds its truth in geometrical diagrams and illustrations. The conclusions are clear: an instrument, as a mediation of vision, averts the natural flow of species from their straight course and adds unnecessary visual information. Furthermore, since the natural flow of species has no geometrical variation (a simple straight line), there is no need for elaborate geometric diagrams for the description of direct vision. In contrast, any explanation of reflection and refraction that disrupts the flow of information is thoroughly mathematical. In this way, a sharp distinction is preserved between the physical aspect of vision, which needs only a slight modicum of geometry, and the explanation of virtual reflected images, with their thorough geometrical elaboration. This distinction is well in accord with Pecham’s contention in the preface to his treatise that the main error of former optical treatises is the obscure way of combining mathematical and physical demonstrations. While preserving this difference between real and virtual optical effects, Kepler gives it a new epistemological twist by redefining the relationship between geometrical analysis and the physics of sight. Kepler seems to echo Pecham’s wording as he defines the image as “when an object is seen, with its own colors and parts, but not in its own place, not showing the proper quantities, and its parts holding wrong proportions. In short, the image is the vision of a certain object linked to an error of the faculties concurrent with vision. The image itself, therefore, is hardly anything and it would have been better called an imaginary fabrication. It is something made up from the species of real color and light and [from] intentional quantities.”73 In emphasizing the intentional character of the image and its contrived quality, Kepler transports it into a new and peculiar domain of associations between mathematical analysis and physical phenomena, however. In this domain, Kepler could reformulate and cast new meaning into the medieval differentiation between species and images. In what follows, I will describe how Kepler manages to rearrange the geometrical considerations and to recombine them with physical and psychological variables to produce a new entity (i.e., a picture) that is both a true physical effect and yet the result of a process of several reflections and refractions within the eye. He formulates this new term by redefining the place and function of the imagination in the visual process. Thus, Kepler’s theory does not eliminate psychological issues from optics, but some of his crucial moves assume a specific psychological theory in order to explain the formation of images and the completion of visual perception.

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At the beginning of chapter 5 of Ad Vitellionem, Kepler rehearses the epistemological context of his optical discourse. The astronomers place at the foundation [of their science] the diameters of the luminaries and the quantities of solar eclipses; now, some visual deception is produced, partly from the conduct of the observation, which we discussed in chapter 2, above [where Kepler discusses the formation of images in a camera obscura], partly from simple vision itself. . . . [A]s long as the latter is not dissipated, it will create considerable difficulties for the astronomers and diminish their judgment capacities. Therefore, the occasion of such errors in vision has to be investigated, and that according to the shape and functions of the eye itself.74 The main part of the fifth chapter, dealing with human eyes and the sense of sight, is devoted to the eye’s failings and limitations, treading the fine line between deception and illusion and solid observational knowledge. Kepler draws this line with the help of a particular set of experiments that follow his summary of the anatomical description of the eye’s structure. The anatomical details serve as a backdrop to his optical theorems, and are utilized to isolate the two main components that produce the sensation of vision: the crystalline humor and the retina. In his experiments, Kepler forms an analogical and idealized structure where a screen stands for the retina and a crystal ball or a glass urinary flask filled with water, set against a window in a room, stands for the crystalline humor. In this setting, Kepler discusses two fundamental phenomena: the appearance of a picture on paper, or any other sort of screen, and the disappearance of this picture when the eye is positioned where the paper was before. In other words, when a screen is placed in a closed room behind a refracting object in front of a window, the light beaming through will project a picture of the outside on that same screen. When, however, the screen is removed and a human spectator—that is, a human eye—is placed where the screen had been standing, the picture will disappear and the spectator will see nothing. Kepler asserts that the main factors responsible for these phenomena are the refractions of rays of light and the convex shape of the glass globe filled with water.75 In the following propositions, he attempts to explicate these phenomena, and especially to delineate the psychological factors that determine the disappearance of pictures on paper and the appearance of images on the surface of the ocular lens—namely, the crystalline globe. The presence of the eye transforms one’s

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optical experience and turns a physical picture into a mere image. The first step is to explain how the formation of the image is dependent on the activity of the eye and is not the effect of the refracting or reflecting surface alone. Following his catoptrical propositions, Kepler proves that the geometrical locus of the image is between the eye and the globe.76 If that is the case, why does the eye see the image on the surface of the globe, and not at that point where the picture had appeared before on the sheet of paper, that is, at the locus of the image? The answer rests on the natural disposition of the eye, which cannot capture a place for “the image of a thing hidden behind the ball” in front of the ball or globe of water.77 Kepler defines three factors that affect the eye in this context. The first factor is physiological: since the eyes tend to preserve their parallel position, any attempt to perceive a nearby object strains the eyes to turn and contract toward it. The result is that “the sense of vision perceives with more difficulty the nearer [objects] than the remote ones.”78 The second factor is the natural disposition and function of the eye to perceive light: “Vision is attracted by vividness, but it is hardly drawn to the fleeting and the feeble.”79 Kepler establishes this disposition not only on common experience, but on the essential property that sight is moved by light: the stronger the light, the more agitated the eye is. Therefore, the eye follows the stronger illumination at the expense of the lesserilluminated areas. The last factor is the ability of vision to actively create images. This ability is destroyed by strong light, which diverts the beholder from the supposed location of the image. Kepler proves thus that the image cannot be seen between the eye and the globe because the eye is attracted to the illumination coming from the globe. Furthermore, this illumination will overpower and disperse the image. This analysis leads Kepler to define the image as an entity that the sense of sight creates as an active agent. The human mind is forced to supplement the missing data not because of the external environment, but because of its own disposition. In these cases, the faculty of vision naturally constructs the mathematical ratios that inform visual reality and present it coherently. Kepler provides an explicit case of such in his catoptrical discussion. Since the eyes cannot perceive the point of reflection (or refraction), the faculty of vision has to fill in the sense data and structure it geometrically. Kepler contends that this is the reason for one’s perception of the image over a perpendicular line drawn from the object to the mirror. As noted above, in chapter 3 of Ad Vitellionem Kepler contends that the perpendicular line is produced neither by the object nor by the form of the mirror’s surface, but is in fact a mental construct:

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For it makes no difference to the place of the image, what sort of mirror surface is placed opposite the object, since the proportions of the image being formed are all taken from that part of the mirror upon which are the two points of reflection of light to the two eyes. So it is at this part of the mirror, not at the actual perpendicular from the object, that the cause of the image’s place being on that perpendicular lies. And so one should understand mentally the continuation of the pattern of curvature that had created the reflection on the whole circumference, and above this imaginary sphere one should also draw a perpendicular from the object for defining the place of the image.80 (my emphases) For Kepler, illusions are not games played by nature in order to delude the human mind, but result from the rational operation of the mind itself. In contrast to artistic musing over the deception inherent to the sense of sight, Kepler stresses the coherence of human perception. The limited nature of the human senses forces the mind to play mathematically and to invent those geometrical constructions in order to produce an intelligible visual reality. While an image is determined by the natural disposition of the human mind and of the human sense of sight to produce a coherent perception of reality, it depends for its particular appearance on a concrete set of physical circumstances.81 A paper or screen does not respond to these effects of strong light or color and does not accommodate itself actively to such changes in the environment. Therefore, the image is stabilized and appears over the paper or screen when it is positioned in the correct geometrical locus where the rays intersect. In this sense, the paper is a passive agent, and the appearance of the image is wholly determined by external causation and its mathematical considerations. This does not hold for human perception. The convergence between the mind’s natural inclination to mathematical regularity and the particular set of physical conditions causes the image to appear “confused and doubled.” Hence the significance of Kepler’s terminology: Ens rationale and Ens intentionale emphasize the active and inventive role the faculty of vision plays in rationally producing an optical image. In contrast to the medieval ontological distinction between species and ydolum, Kepler’s differentiation between an image and a picture does not concern the process through which they are produced mathematically or physically, but only the psychological aspects that force the mind to actively compensate for missing visual data. Kepler desperately attempts to preserve a distinction between true perceptions that correspond to real extra-mental state of affairs and erroneous

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perceptions fabricated by flaws in the limited human sensory processes. In either case, picture formation on the retina or virtual images in mirrors or in midair, a mental activity is needed in order to decipher the inflowing visual data; Kepler emphasizes that in the case of retinal pictures, the human mind has no need to actively conjure the missing visual information. As the picture is symmetrically inverted, the mind only has to turn the picture around its symmetrical center to get the exact upright depiction of visual reality. The picture is a passive product of the visual process, and this passivity secures its epistemological status as a true depiction of visual reality. Kepler writes, Just as vision is not an action because of illumination’s being an action, but is an effect contrary to an action, so also, in order that the places correspond, the recipients of the action must be directly opposite the acting things. Further, places are perfectly opposite when the same center forms the midpoint in all the lines of the oppositions, which was not going to occur if the picture had been erect. And so, in the inverted picture, even if from a universal perspective and with respect to some common line the right parts are transformed into the left, nevertheless the right parts of the object are perfectly opposed to the right parts of the picture, and the upper parts of the object to the upper parts of the picture (each in relation to itself), as well as concave to concave. Nor is there any fear that the sense of vision might err about the region. . . . Rather, it would have been in error had the picture been erect. . . . Therefore, no absurdity is committed by the inversion of the picture.82 In the case of virtual images, sensuous data is lacking or obstructed: the eye cannot perceive the point of reflection, or strong illumination erases other optical phenomena. However, in the formation of an optical picture, all data external to the eye arrive at the retina. The picture preserves the mathematical arrangement of the visible object, and there is no need for the mind to supplement it with imaginary geometrical constructs in order to produce a coherent perception of visual reality. The mathematical regularity of the inverted picture ensures there will be no mistaken identification of place or area. While Kepler’s images are still products of the human imagination, they do not differ from pictures in their formative process, both being the result of refracted and reflected rays of light. The difference rests in the entirety of the visual data supplied by the process. In picture formation, all the reflections and refractions take place within the eye, thus preserving their geometrical arrangement, though still

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finally producing an inverted picture. In image formation, some data are missing and the intervention of the human imagination causes the final result to be a certain artificial entity, where things are not what they appear to be. Kepler’s retinal picture is a mediated product of a series of refractions and not the result of an emanation from the visible object itself; it needs an active mental intervention to set it right. This Keplerian notion runs counter to deeply entrenched medieval and Renaissance beliefs as to what counts as correct visual perception. The medieval theorists defined species as the direct physical impression on the eye that travels along perpendicular lines to the crystalline humor. They differentiated between species as physical effects and species as images that are mediated imaginary constructs governed by geometrical laws of reflection and refraction. The main concern of medieval optical theoreticians was to safeguard the differential criteria between direct visual perception and other types of optical experiences that gave way to delusions and inner visions. The latter were concerned mainly with mediated vision, involving reflection and refraction of the rays of light arriving at the eye. Any mediation between the visible object and the mind reduced the epistemological certainty presumed by the Aristotelian view as belonging to raw sense experience. Intellectual efforts to create such an epistemological bridge between the external world and the eye collapsed with the nominalist critique of Roger Bacon’s theory of species in the fourteenth century.83 In the final phase, Nicole Oresme contends that there is no vision without mediation. Not only should mirrors and other reflective or refractive visual instruments be considered unreliable producers of images, but the various media, and the eye itself, are obstructing mediators between the intellect and visual reality.84 Alberti abandoned the fragmented medieval visual field with its many species and cones of visual rays, adopting instead a simple Euclidean pyramid of vision. The picture was but a materialization of an abstract intersection of such a pyramid, attaining its coherence by embedding its geometrical elements in structures of rhetorical signification. The ensuing cultural review of this new painterly concept of pictorial representation subjugated the act of sight to various poetical ends. Yet both Alberti’s humanist program to reform vision and its critical reviews through the sixteenth century were still embedded in the medieval dichotomy between direct rays falling perpendicularly on the eye and mediated images formed through reflected or refracted visual rays—the painting, even as an intersection of the pyramid of vision, remains a mediated and fabricated re-presentation of an authentic object of sight.85 Kepler’s new definition of pictura sought to release the discourse of vision from this epistemological

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straitjacket by radically relinquishing the differentiation between direct and mediated vision while establishing certitude on new grounds.86 Instead of a species as an emanation from the visible body itself expressing not only its visible qualities but its essence as well,87 Kepler’s pictures are the products of refractions of light as an external and mediating agent. The result of the visual process is, according to Kepler, an inverted and reversed picture of the external object; the parts on the right are now on the left, and those at the top appear at the bottom. For medieval and Renaissance sensibilities, such retinal pictures are distortions verging on presenting a chimera, or a monstrous depiction of reality. As noted above, what saves the retinal picture from absurdity is, for Kepler, the geometrical regularity of the inversion of the retinal picture. The stable center, through which pass all the lines from the corresponding points on the object and on its inverted picture, limits the mind’s ability to play with the sensual data and thus guarantees one’s visual knowledge. In a sense, Kepler’s retinal picture is a sort of a “serious joke”: though it represents the world upside down, it allows one to perceive the truth. However, artistic jokes in the manner of Arcimboldo are human creations that suggest playful leaps between multiple levels of meaning and signification. Thus, their truth is hidden and evasive. In contrast, Kepler’s retinal pictures are natural creations and thus limit human creative playfulness; by their geometrical regularity, they allow only one true depiction of visual reality. Kepler asserts that in contrast to the workings of the human mind, “[n]othing designated by nature is wasted.”88 He differentiates between the realms of human creativity and natural truth. While the first produces illusions that are the result of epistemological uncertainty, the latter, when carefully analyzed, exhibits geometrical regularity and certitude. Kepler reformed optics in a bold attempt to save it from the late Renaissance denigration of sight as a locus of deceiving phantasms. His optics redefined ocular experience as a basis for an emerging mode of scientific observation that shared several elements with Renaissance artistic practices: in both cases artificial instruments were merged with a daring application of traditional mathematical tools, creating new forms of knowledge of the natural world. Kepler, however, clearly distinguished his mathematical view of nature from the painters’ practice. Rejecting the tradition from Alberti to Arcimboldo that imbued vision with rhetorical values, and thus apprehended ideal and perfect measure as artificial and external to mundane experience, Kepler found God’s eternal geometrical ideas embedded in the motions of the phenomenal world. Geometrical ratios govern the very process of change in nature, and Kepler was willing to accept even second-best figures (the ellipse instead of the circle

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for the motion of the planets) and inverted pictures on the retina, as long as their regularity was preserved. Kepler’s illiberal drawings, purged of rhetorical signification, were causally formed according to the strict geometrical rules of refraction and the material surfaces that determined their appearance. Such a strictly geometrical setting insulated the knowledge extracted from visual experience from the ambiguities of rhetorical interpretation and from the threat of illusions and deception. Yet two questions are still to be noted: what kind of knowledge can be obtained from such optically formed pictures, and how can that knowledge be read from these pictures with no rhetorical tools?

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5.

READING THE BOOK OF NATURE Allegories, Emblems, and Geometrical Diagrams

of making many books there is no end A lifted curtain; a pyramidal pile of leather-bound books emerges. The books are not ordered properly: some of them are laid one on top of the other, while others are standing, leaning on an almost monochromatic pile; a big tome is lying over a smaller one, and a heavy red volume towering over it bends it slightly; at the top of this strange pile rests an open book. Its pages are fanned as if a wind is blowing through them. The perspective is awkward, and the books do not cohere into a single vanishing point; their arrangement gives the impression of a late medieval naïve manipulation of the depicted objects to create a sensation of pictorial depth (figure 5). The peculiar bulkiness of the pile of books invites the viewer to discern quite easily the facial features of literally a bookish man. Sven Alfons identified the painting as a composite portrait of Wolfgang Lazius, the court historiographer to Rudolf II, interpreting it as satire at the scholar’s expense. Others interpreted the painting as a criticism of book collectors who are captivated by the materiality of books, amassing them indiscriminately without ever reading their contents. Arcimboldo’s composite portraits are more than just crude satires, however, and are invested with serious allegories. Two elements draw the viewer’s attention to this allegorical dimension. First, the curtain is not pulled to reveal a hidden scene, but the bookish persona is bursting out, clumsily pushing the curtain, entangled in its folds, as if stating that occult knowledge can be

5 Giuseppe Arcimboldo, (copy after?) The Librarian, ca. 1566? Skokloster Castle, Sweden.

divulged only through richly covered printed books. The book becomes an allegory of knowledge in the exact sense of Benjamin’s assertion that “[a]llegories are in the realm of thought, what ruins are in the realm of things.” The book is the remains of a once existing fullness of knowledge that now is concealed behind a curtain, beyond the reach of the seeker for wisdom. This allegorical significance is amplified by the second element—the open book at the top of the bookish head, the pinnacle of this pyramid of scholarship. Inspecting its open leaves, one notices that this is not printed, but handwritten. Together with the many bookmarks, it gives the impression of a commonplace book; a book in the middle of metamorphosing into a printed book that will eventually become part of the heap below. In visually depicting this vicious circle of reading, the painting is more than a mere satire of humanist erudition. It presents a poignant question: can pictorial representation break through the deadlock of reading? Is a different mode of reading that can peer at the veiled knowledge concealed by piles of books possible?

allegories and jokes The late sixteenth century witnessed the subjection of the different elements of painting—from the geometry invested in it to its actual content—to rhetorical intentions. As noted in the previous chapter, the technique of perspective, evolving as a geometrical antidote to humanistic discontent, became an allegorical symbol of painters’ self-conscious awareness of the illusionistic nature of their trade. Geometry was applied in a playful manner to create intriguing distortions and awkward viewpoints. These were combined with an ability to paint lifelike details in order to create bizarre and grotesque images, wavering between artificiality and mimesis. Late Renaissance painters and theoreticians of art embraced such elaborated techniques of anamorphosis and optical distortions as means to induce spiritual contemplation of abstract rules through concrete appearances.1 As geometry was relegated to another poetical tool in the hands of the painters, blurring the borderline between painting and the written text, so these curious images became an elaborated allegory burdened with hidden and sophisticated meaning. While Kepler’s retinal pictures exhibited similar curious characteristics of playful inversion and distortion, he resolutely aimed to disentangle these optically formed experiences from any allegorical interpretation and its ambiguities. By doing so, he established observation on more solid ground. Refashioning mathematical analogies, Kepler

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aspired to capture the meaning of mathematical figures and proportions without resorting to allegorical or emblematic possibilities, suggesting a new mode of reading visual signs and the world of appearances in general. This Keplerian endeavor was carried out against the backdrop of the Rudolfine court in Prague and its taste for such pictorial devices and literary and visual paradoxes. Set against this literature, certain aspects of Kepler’s optical analysis of pictures and diagrams emerge as direct responses to the epistemological problems addressed by such pictorial experimentation. Under the patronage of the emperor, different tendencies flocked together, so one could find side by side the taste for the imaginary together with a disposition in favor of “painting from life,” and all implied an attempt to invest visual signs with verbal meaning. As Kaufmann has pointed out, these tendencies found expression in the manner in which the artists of Prague embraced the mythological figure of Hermathena.2 The hybrid of Minerva and Mercury, this figure symbolized the academic ideal and the unity of the arts and sciences.3 As part of the cycle of arts, painting must be linked with the other liberal arts and share their principles as well, since, as Cicero asserted, “The whole of the content of the liberal and humane sciences is comprised within a single bond of union; since when we grasp the meaning of the theory that explains the causes and issues of things, we discover that a marvelous agreement and harmony underlies all the branches of knowledge.”4 The aim of Rudolfine painters was also to save and regain this unity of the arts and sciences, which had been lost through the deviant and devious nature of the fallen human languages. Words have deviated from their “origin” in reality, and their concealed duplicity undermines the assurance with which they are commonly used.5 The symbolical marriage of Mercury with Minerva was supposed to recreate the lost unity between eloquence and true science, and the place for such an occurrence to take place was the painter’s easel. This concept of the art of painting implied that visual art cannot be satisfied with visual reality, but must present ironic depictions of reality that are always conducive to a higher level of meaning. Consequently, Prague painters tended to transform their pictures into imaginary emblems, always looking for an ideal viewer to decipher and apprehend the mysterious meanings. Artistic pictures were by no means passive reflections of external reality but acts of the painter’s imagination that appropriated fragments of visual reality and recomposed them into novel and unified meanings that transcended immediate experience. The only way for the eloquent artist to achieve this and to overcome the gap between the realm of meaningful ideas and ordinary and ever-changing appearances

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was to construct ever more complicated visual puzzles and allegories. Artistic visual language had been wholly transformed into rhetorical invention, and the unified truth of the nature of things that Leonardo had aspired to disclose still evaded the painterly gaze. The mode of painterly allegories was best exemplified in the work of Arcimboldo, who blurred the dividing line between fantasy and empirical facts: “The fantastic virtue . . . is highly developed in Arcimboldo, who, by combining images of things seen by him, forms strange caprices that are invented by the power of the imagination. That which seems impossible to combine, he unites with great dexterity, making all that he wants.’”6 In combining idealized artificiality together with a meticulous attentiveness to natural objects, Arcimboldo’s style and, following him, that of other Central European artists highlighted the paradox involved in the late sixteenth-century conception of painting. In order to accommodate detailed descriptions of particular natural objects together with an abstract and universal concept, paradox had to step in and take over the spectator’s attention. Just as paradox sets language against itself in order to convey meaning, Arcimboldo’s pictures test one’s sight in order to evoke a sense of surprise and knowledge. The significance of these grotesque and excessive figures was in their combination of humor and serious contemplation, and in their playing with and questioning the value of optical illusion; they questioned the reality of visual appearances and of human perception in general.7 Arcimboldo’s paintings made simple observation an impossible act and forced the spectator to view ordinary objects and animals in a new and fantastic light. While it did aim at pleasure,8 it was not mere comic pleasure and relief. It was meant to turn the spectator’s gaze from the visible to the invisible, from mundane and ephemeral objects to spiritual and eternal ideas. In order to bridge these two realms, the painter presented the world either as through a convex mirror, distorting all in sight, or as a visual pun or a joke, embracing the literary topic of the “serious joke” or “serious play” and giving it a visual translation.9 The aim of the serious joke was to grapple with inherent mysteries that lay beyond one’s usual perceptual abilities. By mocking and challenging ordinary experience, the serious joke exposed the inexpressible mysteries that influence one’s reality. In this sense, fictitious constructs shifted from mere playfulness and fancy inventions to positions of meaning. Arcimboldo’s paintings followed this late Renaissance literary sensibility, mobilizing humor in order to subvert ordinary visual experience and to posit the invisible and universal as the true aims of visual pleasure: “Arcimboldo’s

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pictures of the seasons, elements, and Vertumnus may have a joking exterior, but within lies what may be called a ‘long, mysterious message,’ related to imperial majesty.”10 Employing fantastic imaginations, humorous distortions, and visual puns, Arcimboldo and other artists at the Rudolfine court enticed the spectator to search beyond mere appearances. The spectator was forced to capture the hidden meaning concealed under ingenious forms of expression that were beyond surprise and comic delight, and beyond all physical and concrete experience. Fantastic imagination was conceived not merely as a distortion of reality but as a serious way to influence it and force it to disclose its secrets. Comanini, commenting on Arcimboldo’s painting of Vertumnus as Rudolf II (figure 6), underlines this obscure border between the joke, the serious and the knowledgeable: On the outside, I appear to be a monster, But a beautiful resemblance And a royal image I hide inside. . . . . . . . . . . . . . .  Now go, spectator, Since I have said in few lines What I am, and what I adumbrate.11 By doubling the identity of the depicted figure and adding to it the dimension of a pile of vegetables, Arcimboldo multiplies the painting’s levels of signification. It is not only that the deity disfigures the prince and that both are part of a harvest of different types of vegetables, but that they are also transformed into a higher level of significance. Nature is history, and both act as the container of the mythical figure of Vertumnus. The lines dividing these forms of human experience are blurred, and a transformation is hinted at but never takes place. Art builds static compounds that ultimately reject metamorphosis. Such a playful approach surfaced in Filippo Gherardino’s “Flora,” a poem based on another painting by Arcimboldo: Neither Flora was changed into Flowers, Nor were Flowers changed into Flora By the good painter, who painted Flora As she is, Flora of Flowers.12 Thus, a paradoxical depiction emerges, depicting one entity by means of another, blurring the dividing lines between the formerly distinct entities. Yet

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6 Giuseppe Arcimboldo, Vertumnus (Emperor Rudolf II), 1590. Skokloster Castle, Sweden.

the painter preserves each entity as it is: each flower remains a distinct flower without being transformed into any bodily member. Thus, by concentrating on shifting appearances, the painter could point toward the ideal order beyond phenomenal reality.

mathematical symbols and the emblems of alchemical knowledge Arcimboldo’s celebration of the fragmentation of visual experience resonated with a new sense of allegory that, in piling together different items, formed a succession of incongruous and ludicrous comparisons and mixed metaphors in a search after deep spiritual meaning. Such allegorical musing was also embedded in the growing vogue in the second half of the sixteenth century for the literature of emblems. The emblems played with disjointed images and words that were often staged on the backdrop of ruins, and their objects always presented what they were not in a desperate attempt “to rescue from historical oblivion that which threatens to disappear.”13 Thus challenging the gap between human mental construction, which aspired to the realm of eternal forms, and ever-changing physical reality, the emblem applied paradoxical games that conflated different levels of interpretation with different systems of sensory experience.14 “The emblematic worldview” had emerged from the Renaissance interest in hieroglyphs, Roman coins and the symbols impressed on them, and fables and epigrams collected from classical sources.15 The original aim of the “inventor” of emblematic literature, Andrea Alciati (1492–1550), was to compose epigrams that were especially enigmatic, so the readers would feel a pleasant and surprising enlightenment when they succeeded in solving them with the help of the attached interpretation. As Alciati notes, emblems aim “to pierce the soul, to nourish the eye, to fill what is empty with meaning, and to give the power of speech to what is mute.”16 Alciati’s original plan did not include an picture to accompany the epigram, but when Emblemata was first published in 1531, engravings were added, and in midcentury the visual picture was an integral part of the emblem. Composed of a picture, a short motto, and a longer epigram, the emblem was a focal point for different kinds of knowledge from different sources. As a verbal expression, a puzzle, and a picture, it enabled the reader to bridge gaps and inconsistencies in different textual traditions. The objects and persons appearing in the emblem suggest a multitude of perspectives

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for interpretation, and thus both the emblem itself and its meaning are fragmented into conflicting contexts of signification. The emblematic tradition flourished all over Europe. Alciati’s book came out in dozens of editions and extensions, and these led to dozens of other books of emblems. Around 1600, hundreds of emblem books were in print, and this flow kept increasing until, around midcentury, it started to recede. Emblems enjoyed an exalted intellectual status among certain circles of alchemists, natural historians, and other scientifically minded persons in central Europe during the sixteenth and seventeenth centuries. Emblems usually elucidate and clarify the written epigram; the alchemical emblem emphasizes the function of the emblem as a means to decipher reality. The alchemical emblem’s main purpose is to indicate how to unveil corporeal reality and unearth its significance.17 The most notable treatise of alchemical emblems associated with Rudolf ’s court was Heinrich Khunrath’s Amphitheatrum sapientiae aeternae (1609).18 Khunrath’s pictorial program was endorsed by Michael Maier, who served as Rudolf II’s personal physician and published most of his alchemical treatises in the years immediately following Rudolf ’s death. Yet, as Evans comments, “the labour of inventing and preparing [these books] must have occupied him through his Prague years.”19 The fundamental tenet of this program was that each natural object contains an allegorical element that directs the contemplative mind to recognize it as a symbol for spiritual and supersensual ideas that constitute the divine realms of the universe. In this framework, mathematical entities and the technique of perspective lose their promise of certitude and coherence and are metamorphosed into one more allegorical element in the metasensory realm of fossilized nature and ruins. The alchemical emblematists followed in the footsteps of John Dee, who, in his Monas Hieroglyphica, created according to careful mathematical calculations a potent amulet that was an allegory both of the planets and of the “wonderfull sciences, greatly ayding our dymme sightes to the better vew of his [God’s] power and goodness.”20 Playing on the astrological symbols of the planets (especially the combination of Mercury and Venus), Dee suggested a visual device that would be both a powerful mnemonic image and a path to transform the viewer’s mind. While alchemical emblems did not preserve Dee’s rigorous mathematics, those mathematical elements (such as perspectival constructions) included in them were transformed into allegories. For instance, in an engraving of the magus room in Khunrath’s Amphitheatrum sapientiae, a hall is depicted in strict one-point perspective (figure 7). This method of depiction creates the illusion

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7 Alchemist’s Laboratory, in Heinrich Khunrath, Amphitheatrum sapientiae aeternae . . . (Hanau, 1609). Courtesy of the Gershom Scholem Library, Jerusalem National University Library. 8 The gate to the amphitheater of eternal wisdom, in Khunrath, Amphitheatrum sapientiae aeternae. . . . Courtesy of the Gershom Scholem Library, Jerusalem National University Library.

of progressing from the musical instruments and measuring devices in the foreground to the alchemical furnace, and then to the magus, who kneels in front of an altar of enlightenment within a majestic tent, where, with the help of geometrical diagrams and combinations of Hebrew letters, he aspires to illumination. Finally, the spectator’s gaze is directed toward a window open to an invisible world. The same process occurs in the depiction of the gate leading to the doors of the amphitheater of eternal wisdom (figure 8). A tunnel is depicted under a one-point perspectival grid, directing the spectator’s gaze from a world of visible light to an invisible world, yet illuminated by a mysterious source. In both examples, the perspectival tunnel both embodies an allegory of the sublimation of sight and invites the actual transformational process to take place within the viewer’s mind.21 This allegorization of mathematical procedures receives a clear exposition in Michael Maier’s Atalanta fugiens of 1617.22 Maier characterizes his treatise as “[p]artly adapted to the eyes and the intellect, with copper etchings, and added sentences, epigrams, and notes, partly [adapted] to the ears and to the recreation of the soul with less than fifty musical fugues in three voices, . . . to be seen, read, meditated, understood, judged, sung, and listened to with particular pleasure.”23 Maier’s intention in this combination of solitary contemplation and application of sensual pleasure becomes clearer when one contemplates the emblems themselves. In emblem VIII, Maier presents the reader/viewer/ singer with the peculiar structure of a perspectival tunnel (figure 9). In the background of a highly theatrical scene, in the back wall of the theater’s closed space, a vestibule opens up, depicted in one-point perspective, leading to an invisible area. The motto to this engraving reads, “Take the egg and pierce it with a fiery sword.”24 The epigram develops the allegory: the egg is the source from which, through the combined help of Vulcan and Mars, a bird will arise that will conquer the realm of fire and metals.25 As with other emblems, Maier knits a complex web of associations and allegories leading from Greek mythology to Christian allegory and alchemical symbols dealing mainly with the transmutation of metals. The alchemical process, according to the emblem and the discourse appended to it, is portrayed in analogy to other processes of transformation: from an egg to a mature bird, from the womb to an adult human, and especially from Mary’s womb to divine perfection. The alchemist, with the help of his furnace and the process of distillation, can decipher the hidden meaning of resurrection and transmutation. The alchemist’s task is to follow these processes in his attempt to realize his aspiration to perfect control over nature. The perspectivist vestibule guides and directs the

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9

Emblem VIII, in Michael Maier, Atalanta fugiens, hoc est, Emblemata nova de secretis naturae chymica (Oppenheim, 1618). Reproduced by permission of The Huntington Library, San Marino, California, RB 600059.

reader from the material and sensual gaze, and from the alchemical processes per se to a different kind of gaze, a perfected one, structured according to perfect mathematical proportions. This gaze will be made possible when the egg is cracked open in the material theater from which the magus embarks on his way to supreme wisdom. Hence, the meaning of the vestibule drawn in perspective is the transformation of human cognitive processes and the shift to another form of consciousness and awareness, which exists on a separate level of reality. However, mathematical symbols acquire additional meanings in the emblematic world. For instance, in emblem XXI in Atalanta fugiens, the motto commands, “Make a circle out of a man and a woman, out of this a square, out of this a triangle, make a circle and you will have the philosopher’s stone” (figure 10).26 The epigram further develops the symbolic nature of the geometrical shapes:

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10 Emblem XXI, in Maier, Atalanta fugiens. Reproduced by permission of The Huntington Library, San Marino, California, RB 600059.

Make a circle out of a man and a woman, From which a quadrangular body arises with equal sides, Derive from it a triangle, which is in contact on all sides with a round sphere: Then the stone will have come into existence. If such a great thing is not immediately clear in your mind, Then know that you will understand everything if you understand the theory of geometry. Maier then asserts that geometrical shapes and their manipulation acquire new meaning—the squaring of the circle is now a symbol of the unification of contraries. The shapes themselves acquire a dimension of sexual identity and

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difference that the magus aspires to transgress. One of the means the magus can apply is the activation and combination of images, a process that will bring about unification and harmony in the universe—and vice versa, processes in the universe (like the creation of a supersexual being) will cause the squaring of a circle. Maier is following here the tradition that Ficino and Dee propagated in the Renaissance, which employed geometry to mobilize external forces, especially astral influences. The association between the geometrical shape and the natural object is not a quantitative representation but a qualitative allegory. In other words, the geometrical figure itself becomes an emblem that concentrates within itself a web of verbal, that is, textual, associations. Following the above examples, one can point to two main uses of mathematics within the emblematic world: initially, mathematical entities are conceived as both causing and symbolizing the metamorphoses of the so-called magical gaze. The magus’s vision moves from the perception of the material world to the conception of a different realm arranged according to pure proportions. These proportions guide the contemplative eye to a higher sphere of being.28 The other use of mathematical entities fixes them as symbols of magical processes that cause the unification of contraries, or as talismans that concentrate within them the powers of the universe—this is usually the case in Renaissance theories of astrology. This utilization of mathematical symbols aims at influencing the material world itself. It is the use of the knowledge acquired through the first process for the transformation and salvation of the visible realm. Maier laments that human memory is but some feeble sparks of knowledge hidden in the human mind that must be “protected, fanned, and strengthened” and then put into operation by exercise, education, and the textual tradition. Only then can these sparks become a true recollection of true knowledge: recollecting the essence of geometric shapes will lead to the understanding that geometrical figures are not mere abstractions but potent means to capture allegorically the essence of alchemical qualities. The process described in the emblem is not just the squaring of the circle (which Maier claims the natural philosophers knew) but an alchemical process in which the four elements are transformed into three: body, spirit, and soul. These in turn correspond to the three primary colors: the earth or material body is black (the color of Saturn); the spirit is water and appears in the whiteness of the moon; and the soul, as the air, is the yellow color of the sun. The triangle must be transformed into a circle, which is unity, and its color is red. This is the process through which a woman turns into a man to become unity. In such a way, the number one subsumes

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and perfects all the other numbers into “rest and eternal peace.”29 Through the contemplation of these analogous processes, the initiate will advance toward true enlightenment. The alchemist perceives the natural world to have deteriorated from its initial divine grace and perfection. Nature has been corrupted and disintegrated as a result of human original sin. This lapsed natural state is not limited only to the sublunar realm: many philosophers during the sixteenth century watched with anxiety how decay and corruption spread to the eternal heavenly bodies. Maier and his circle saw in this state of affairs an urgent call for an extreme effort to save the universe. Following Paracelsus and other alchemical theoreticians, Maier conceived the transmutation of metals as a salvation of their primary, exalted, and divine nature. The task of the alchemist is to retrieve and recollect these memories of the divine Creator’s act that are enclosed within fallen matter. The process Maier describes takes place on two analogous levels in the human mind and in physical nature. Contemplating emblematic scenes initiates a mnemonic process whereby the human mind dissociates itself from the material world to gaze instead on spiritual and divine essences. The human contemplator aims for the retrieval of divine sparks imprisoned in the material realm of bodily passions, and for their reunion with their divine origin. This process is complemented by the alchemical transmutation of metals, whereby debased metals are transformed back to their noble and golden essence. These two processes are connected, each reflecting the advent of the other. The alchemist experiments with fire in the material realm to inspire and induce the mental processes and vice versa. Thus, the salvation of nature will take place simultaneously with the eschatological salvation of the human race. The alchemical emblems addressed the Aristotelian gap between the realm of concrete visual signs and the realm of universal concepts by turning this gap into a paradox, into serious play.30 The visual image is set in order to overcome the loss of Adamic language, together with the deep wisdom associated with it that allowed words to penetrate the hidden corners of the universe. The visual image enables putting the different texts side by side: the Bible together with the Platonic dialogues, the Hermetic texts together with Aristotelian speculations or Paracelsian treatises. It combines the meaning of the various textual traditions and produces a coherent and unified truth. The emblem thereby disconnects memory from normal sensual experience and the external world, and enables it to begin its travel toward the resurrection of lost knowledge.31 This becomes possible through the fantastic nature of the visual image. Although it is a sensuous image, it depicts unusual occurrences that transform natural

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phenomena into playful puzzles. Yet in this play the serious conclusions are always ephemeral, always hinting that things are not what they seem and that any serious truth arrived at will evaporate immediately in front of the frustrated gaze of the sinful human mind.

emblems and intellectual vision in robert fludd In order to understand this ambiguity of the emblem as a mnemonic device, on the one hand, and as pointing beyond memory to the source of divine wisdom, on the other, one has to look into the psychological system of vision as invested in the alchemical emblems, for instance the one suggested by Robert Fludd (1574– 1637). Though Fludd was not in direct contact with the court in Prague, he shared the spiritual aspirations of this intellectual circle. In his short psychological explication of sight, he supplies a possible key for the theory of visual cognition that underlies these emblems and their epistemological agenda.32 Overlooking it may foster a misunderstanding of Fluddian rhetoric, embracing the false assumption that Fluddian science envisions a protoempiricist scientific method. In the second volume of his monumental work dealing with the history of the microcosm, Fludd gives an idiosyncratic account of the triple division of the soul’s vision. This account is a strange mixture of Neoplatonic, Hermetic, and Aristotelian psychology. The first stage in the soul’s visual perception is the corporeal vision, which perceives the colors and dimensions of external bodies.33 The soul needs this corporeal dimension, since otherwise it is unable to perceive external physical reality due to its being itself an invisible spiritual substance. The second type of vision discerns the spiritual images of corporeal bodies. Although these images originate from material bodies, Fludd names them spiritual because they do not possess a corporeal body. Furthermore, these images are created by an occult power, which enables the external bodies to express themselves in the same manner as images are produced in a mirror.34 The third type of vision is the intellectual vision, which aims at a realm beyond the physical world and perceives the truth itself. Fludd’s next step is to put these three types in an ordered hierarchy according to their cognitive value. In contrast to Aristotle in the De anima, who assumes the perception of the special objects of each of the senses to be infallible, Fludd declares that the material sense of vision itself is responsible for errors of perception in the soul.35 He gives different topical examples of visual errors, from sailors who imagine the shore to withdraw, to a stick that appears

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to be broken in the water. Fludd concludes this section by denigrating vision as a prime deceiver of the soul: “On the contrary, inasmuch as vision is the most excellent among all the other senses, it is mostly deceived.”36 All these matters are treated in the science of optics; however, for Fludd, optics does not seem to solve the problems concerned with visual deception but only to trace the problems and the extent of visual illusions in daily human experience. Fludd then rejects the Aristotelian definition of special objects of sense, and locates the visual distortions in the eye itself and not in the judgment of the soul. Moreover, the soul can adjust (as well as produce) these distortions by using the science of optics and perspective. According to Fludd, the power residing in the more spiritual elements of the soul is greater than in the senses concerned with material reality as such. Therefore, the distortions produced in the spiritual parts are more potent and have more lasting effects. These distortions not only cause the soul to fail, but also bring suffering and vexation.37 Guiding and shaping these effects is the principle that the second type of vision is concerned not with the things in themselves but with their similitude, and the mind perceives these simulacra as in a polished mirror. However, these images are not only the product of things in themselves but are their reflection, the image of these things exerted from the unreliable external senses.38 It is through this type of vision that one discerns the three-dimensional outlines of the corporeal world, which is the place where the images of the celestial bodies and of the zodiac are constructed. Fludd, however, is careful to emphasize the difference between this spiritual vision and the intellectual vision. The latter can never err and is always contemplating higher realities such as “God, the rational mind and intellectual reason, the cardinal virtues, chastity, piety, and whatever else of this kind”; whereas through the former type of vision one sometimes acquires correct perceptions and sometimes wrong ones. Thus, the spiritual vision sometimes agitates the soul and sometimes brings it to tranquility. The main cause for this problematic state of affairs is that the spirit is not a body, but participates in bodily qualities either by being the form and similitude of the body, or by participating in the nature of the lower and material kind of light. While one is sleeping, the more agitating effects of the spiritual imagination are revealed in sexual dreams and nocturnal emissions of semen.39 Although it may have some cognitive value, the imagination is more dangerous and false than the external senses because it can move the flesh while the normal inhibitions are weakened in sleep. Thus, even chaste and religiously minded people are not protected against its evil temptations.40 For if the soul

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adheres to the superior mind, then its vision is turned to more mental and intellectual pursuits. However, when the soul is subjected more to the sensual elements of cognition, it employs more of its inferior vision. If one attempts to gaze at the truth, one must first release the middle spirit from its corporeal temptations. Fludd implies the means for liberating the soul from the grip of material passions in his discussion of dreams. While for the most part dreams, the hallucinations of the madman, and daydreams are just reenactments of daily material desires, there are certain kinds of dreams in which one is not viewing external and material bodies but the spiritual content of the soul itself. In this kind of internal vision, the soul perceives not those simulacra it received from external bodies, but its own inner forms. In order to achieve this kind of vision, Fludd testifies that he contemplates remarkable emblems (admiratio insigni) that bring him to amazement and wonder. This vision is disconnected from the material corporeal vision: “Yet, whatever the nature of that vision is, it is for certain not corporeal. For no [corporeal] body produces those images in the spirit. Nor has it this power to shape something spiritual; but the spirit by itself presents it in its own wonderful speed, as one might expect of [something] spiritual, intellectual, or rational.”41 According to Fludd, the material eye can only produce meaningless visual impressions; it gazes at the passing ephemeral reality and receives its shadowy impressions, which become pictures in the internal eye of the spirit, where fantasies retain the content of one’s memory. However, as long as this pictorial content is associated with the external world, it has dangerous implications. It excites the human imagination to false dreams and moves the human psyche mainly in the direction of sexual stimulation. Thus, it moves one’s mind away from its aspiration to wisdom and truth. The only way to turn the internal mind toward the superior domains of true wisdom is to arouse the spirit’s inner content and kindle the intellectual eye. This can be performed by refurnishing the theater of memory with pictures that have no origin in the external and sensual world. Although it has a sensory aspect, the emblem is a fantastic picture divorced from ordinary human experience that allows the spiritual eye to overcome sensual temptations and turn toward intellectual contemplations of God. Yet although the stakes were high and the promise of the alchemical emblem was tempting, at the heart of this tradition was an epistemological obstacle. The gap between the concrete phenomenon and the realm of universal concepts was preserved. In bringing the particular and the universal ever so close, the serious play still left the spectator-reader of the emblem in uncertainty: had one glimpsed truth or just another level of allegory?

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fludd’s experiments and his mode of reading the book of nature A radical attempt to establish such an allegorical mode of reading visual signs was suggested in Fludd’s experiment with the “Weather-glasse.” In this experiment Fludd goes beyond the reading of images, emblems, and fantastic pictures and suggests that the physical world itself can become an emblem, not through its normal and daily behavior, but through humanly fabricated wonders. As the fallen human senses cannot supply access to the divine message veiled in worldly appearances, examining one’s own experiences is worthless. Quotidian physical events present only their fallen appearances to the distorted human sense of vision. The regular course of nature remains opaque, leaving the human mind enmeshed in intellectual doubt that generates moral corruption. Only a physical wonder produced by some natural magic can break the material bondage that prevents the human mind from regaining lost wisdom. The power of physical wonders is such that they disperse the material fog that covers the true message of natural events. Furthermore, by their irregular appearance, they awaken the intellectual vision within the mind to reflect, contemplate, and search after the divine meaning embedded in nature. Fludd asserts that physical and visual reality in itself is meaningless. Therefore, the only way to extract meaning from a physical event is by transforming it into a text, searching for the verbal meanings contained within it. Fludd transforms the physical wonder into a text liable to one of the traditional ways of textual interpretation: according to the letter, allegorical, moral, and anagogical. In several places Fludd suggests how to transform his experimental model into an elaborate proof for the existence of God and for the truth contained in the Scriptures.42 The degenerated state of the human race, “erected and soared up, even unto the highest pitch of infidelity,”43 prevents it from grasping the clear message of the Scriptures. Instead, Fludd’s “ocular demonstration” provides a new circuitous road that leads toward the spiritual realm by initially delving into material processes. Although experience is the “mistresse of fools,” yet even “the brute beast, who warned by experience . . . doth make his choice of that, which it hath proved good, and escheweth that, which it hath found naught and dissonant to his nature.”44 The ordinary course of nature cannot change the minds of those who deny the plain truth of Scripture. To uproot incredulity, a special sign has to be fabricated as an “ocular demonstration” that combines natural reason together with scriptural authority to ultimately confute the “Ethnick philosophers.” The

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11 Robert Fludd’s weather glass, in Robert Fludd, Mosaicall Philosophy: Grounded upon the Essentiall Truth or Eternal Sapience (London: Humphrey Moseley, 1659), 9. By permission of The Folger Shakespeare Library, Washington, D.C.

device chosen for this purpose is “commonly styled by some, the CalenderGlasse, and by others, the Weather-glasse” (figure 11).45 Fludd’s “ocular demonstration” proceeds from constructing the mechanical wonder of a primitive barometer whose principles he found “Graphically specified, and Geometrically delineated, in a Manuscript of above five hundred years antiquity at the least.”46 Following these ancient instructions, he takes a “Matras [i.e., matrass], or Bolts-head, and the small vessell of water, into which the nose or orifice of the Matras . . . ought to enter.”47 Placing the matrass’s pipe into the vessel perpendicularly, Fludd divides its neck into equal parts, assigning each a number. Next, he heats the matrass’s head “against the fire, till it be very hot; for the heat of the fire will rarifie and dilate the Ayre in the glasse, and cause by that means a good portion of it to flye out of the glasse’s orifice, and so it will remaine in that estate, so long as the glasse is in the degree of heat.”48 At a certain moment the nose or the pipe is inserted into the water. Thus, “as the bolts head doth

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keel or waxe cold, so also will the water by little and little mount upwards into the neck of the glasse.”49 Having presented this peculiar instrument, Fludd is now able to reveal his interpretation. He begins with an interpretation according to the letter, which is in this case “the usage of this demonstrative Instrument.”50 Through the weather glass Fludd can observe “the temper of the externall aire, or catholick element, in heat and cold; for the higher that the water doth climbe in the neck or pipe of the Matras, it argueth, that the firmer & stronger is the dominion of cold in the aire; so that by this means we may daily judge, of the increase or decrease of cold in the aire; and by consequence, we may guesse at the proportion of heat.”51 Fludd’s second and higher grade of interpretation is allegorical. The device is a reflection of the macrocosm and the activity of the spirit that enlivens it: “not only this experimentall Organ hath a relation unto the great world, but also the spirit included in this little modell doth resemble and imitate the action of that which is included in the great or macrocosmicall Machin.”52 The instrument not only signifies specific elements and processes in the physical world, but it resembles and imitates the physical world as a complete whole. “I would in this regard have each discreet Reader to understand, that, when he beholdeth this Instrument’s nature, he contemplateth the action (as it were) of a little world.”53 This little world resembles in its structure the “great world,” as both have poles and an “Aequinotiall line” and two “Tropicks.” Thus, the instrument imitates the phenomena of the weather on a smaller scale. Going beyond human philosophical reasoning, Fludd’s next step is to apply his “Instrument” as a vehicle for spiritual and mystical understanding of invisible divine acts that took place at the time of the creation. The “aire included in the glass of the Instrument” is the likeness of the “Catholick Aire or Sublunary Heaven,” which is the likeness of the divine spirit and its operation at the time of creation. The air’s activity is expressed in the winds that move “water, that is the Catholick passive,” and is analogous to the creative action of the divine spirit on inanimate matter.54 The process that took place at the time of creation through contraction and expansion is the same that takes place regularly, whereby the air moves the water, and particularly within the weather glass: “It must needs follow, that he [the spirit of the Lord] is the agent, as well in the contraction and dilatation generally, without the Glasse, as particularly within the Glasse.”55 The weather glass is a resemblance of the local weather conditions, of the processes of entire natural world, and of the mysterious activity of the divine spirit regulating the world in the seven days of creation and ever since through the processes of “Rarefaction and Condensation” (heat and cold). These processes are enacted

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most clearly in Fludd’s instrument: “Thus you see it evidently confirmed by an ocular demonstration, that cold doth contract, inspissate, and make gross the included aire [in the glass]. . . . And again, that heat doth dilate and dissipate, by the enlargement of the aire in hot weather, or by laying of the hand on the bolts head, which is made evident by the beating down of the waters.”56 “The universal mystery of Rarefaction and Condensation”57 is the underlying process whereby the divine spirit (Ruach Elohim) applies “from the four corners of the heavens” heat and coldness to one “catholick sublunary Element” in order to create the various effects in the material realm.58 Fludd claims to have derived this insight from the weather glass, which presents to the spectators’ eyes how the expansion of air and water is transformed by the impact of the four winds. Elaborating on this process of rarefaction and condensation, Fludd proceeds to a mystical speculation on the inner structure of the Godhead. The two principles of contraction and expansion (or darkness and light) represent God’s will and action in all the spheres of the universe. Understanding these principles will eventually lead into the hidden mechanism of the act of creation itself, associating different biblical passages together with several observations of natural processes, kabbalistic speculations, and Platonic ideas into one coherent whole.59 In considering the primal unity, Fludd cannot discern any diversity, any duality: “it existeth for evermore, but onely one and the same Identity.”60 The primal unity gives rise to two principal motions—contraction (darkness) and expansion (light), or privation and the creative will of God. Therefore, Fludd concludes “that a two-fold aeviall effect or principle, clean opposite unto one another in condition and disposition, must needs spring and arise from these two severall properties, in that one entire essence; namely, Nolunty and Volunty, whereof the first was expressed by darknesse, and the other by light.”61 God can be revealed either as pure nothingness and privation (which is the darkness) or as pure will, that is, an enlightened act of creation. From this speculation Fludd asserts the unity of God as a unification of these opposites whereby “that Nolunty in it, is nothing else but its Volunty.”62 Light and darkness are unified as different expressions of a single and eternal unity, and their interaction explains the creation of the entire universe from that primal unity according to numerical values, that is, from the number 1. “Out of obscurity” light emanates and “reduceth the universal Nothing into an universall Something.”63 This mystery is revealed in the divine name “Mitattron or Donum Dei catholicum,” and can be philosophically demonstrated by a Platonic “Heptachord,” which consists of seven proportions arranged in a pyramid.64 This geometrical construction “most lively expresse[s] the generall kinds of all creatures, with their

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harmony.”65 However, one should give careful attention to the word “expresse” in the Fluddian terminology: the numbers, geometrical shapes, and (both arithmetical and geometrical) proportions are transformed into emblems of variously different levels of meaning. The role of mathematical shapes in the Fluddian description is not to outline concrete bodies or to convey quantities but to symbolically clarify through visual means mysterious processes: 1 is primal unity, 2 is the material number, and 3 is the formal number. The number 3 is perfection, since all perfect things contain three elements (“all perfection consisteth of 3. tearmes, namely a beginning, a middle, and an end”).66 Thus, if unity had not been vivified by its emanation (the material number 2), “the 2. or deformed matter of the Chaos had stood in his duall, confused, or imperfect estate.”67 Thus, the unity of light informs the primal dual matter to produce greater perfection in the number 3. For Fludd this is a fundamental insight: “This mystery (I say) being rightly understood, all science even in the abstrusest Philosophy, may easily be decyphered.”68 The expression 1 + 2 = 3 is not only an arithmetical calculation, but a symbol of the processes of emanation whereby the higher and more spiritual numbers and what they symbolize create the Anima mundi and the lower material numbers and their symbolic equivalents: “The root of matter therefore which is 2. imports the dark Chaos, the root of form which is 3. imports that the root 2. or the dark waters, is animated by the formall or bright emanation of Unity or 1. and so the first 2. was accomplished, and the soul of the world created, namely by the Angelicall emanation. And thus was the kabbalists Mitattron or Anima mundi first produced.”69 Fludd proceeds in this fashion to interpret the other numbers that constitute his pyramid. He combines numerological elements together with kabbalistic gematria and quotations from different and sundry philosophers. All this is symbolically comprised “by two equall Piramidicall shapes, whereof the one is formall, the other material.” The formal pyramid originates in the “infinite and onely bright Unity” that penetrates the material cone “unto the very center of the dark earth or abysse.” The material cone, on the other hand, ascends “unto the center of the basis of the formall Pyramis.”70 At the point where the two cones penetrate each other, Fludd artificially circumscribes a circle: “we therefore with the Platonists . . . tearm it, the orbe or sphere of the soul of the world.” This sphere is in the middle in order to symbolize that it takes equal parts “of the spirit of the waters, and of the formall fire descending from God or Unity.”71 The circle is not an abstraction of the form of the soul or a representation of the soul, but rather an allegorical emblem that captures a trace of the essential quality of the soul of the world. By contemplating the circle, one can acquire

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insight into the true meaning of the world soul as layer after layer of different allegories is peeled off from the world soul’s geometrical symbol. The contemplative itinerary is now coming to an end. Starting with a physical experiment, Fludd seeks deeper insights into the meaning of the divine creative act. He finally returns to “mysticall Irradiations which spring occultly from the two foresaid opposite Principles, Light and Darknesse, with their Sympatheticall and Antipatheticall effects.”72 Starting with quotations from various religious authorities (especially “Hebrew Rabbies, and profoundest Cabalists”73), then proceeding to explain astrological principles, Fludd ends his treatise with experiments concerning magnetic forces. The spectator can now grasp the significance of these experiments as a final emblem reconstituting and reestablishing a complete mystico-philosophical system. Fludd expects a radical reeducation of the reader’s mind. Moving from the wonder of the weather glass and proceeding through its fourfold interpretation, the reader can then discern in the more common wonders the act of the creative God. The Fluddian interpretation produced an emblem out of natural phenomena. In contemplating this emblem, the mind moves from understanding the material function of the instrument to understanding the principal powers behind it (extension and contraction). This understanding leads the mind to penetrate into the most spiritual realm, contemplating the principles (light and darkness, will and matter, the numerical symbols 1, 2, 3) that emanated from the primal unity on the first day of creation. Fludd’s reading of visual experiences attempts to associate allegorically biblical passages with artificially produced physical phenomena, thereby retrieving the act of creation and through it revealing a true and coherent knowledge of God. Fludd rejects any novel scientific methods and theories since all knowledge is traditionally hidden in the Scriptures. The fallen and degenerated human mind can no longer understand these passages or grasp their truth by simply contemplating the created world. It therefore, so Fludd contends, needs an artificially produced wonder that will force it to turn its gaze from the material domain to its spiritual meaning. Fludd’s cosmic philosophy is derived from “authority, as Wel sacred as Philosophicall” and is founded on the doctrine of “the most high Prophet and prime Theosopher Moyses.” Yet to overcome human ignorance, its proof is in the “easy familiar and ocular demonstrations, the very pointing finger of experience, which is able to instruct the rude and rusticall, yea and very fooles them selues.”74 The artificially produced experiment diverts the human mind from contemplating “carnal senses” and “outward sence, Which are only compositions of

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antipathy and discorde . . . to behould with spiritual eyes the central and hidden truth.”75 As reading the allegorical meaning embedded in the world of creation and ascending to higher spiritual truth is no longer possible, only humanly made wonders can “convert euery outward affection into the inward self.” And only through contemplating the “inner self ” can one “clime up unto God.”76

kepler’s nature: computable proportions instead of textual interpretation Fludd’s allegorical reading of experimentation is an effort to retrieve the lost spiritual competence to decipher the divine message embedded in natural appearances. In order to accomplish this, Fludd transforms experimentation into an emblem that demands a specific mode of reading in order to decrypt its hidden significance. The experimental apparatus is both part of one’s visual experience and at the same time a wonder that baffles human understanding and thus invites an allegorical interpretation. Kepler’s diatribe against Fludd is targeted against this process of allegorization of nature, rejecting the emblematic worldview it implies in favor of mathematical representation. In promoting his new mathematization of nature, Kepler suggests also a new and different mode of reading the book of nature, translating it into geometrical figures, although not transforming it into an imperfect allegory of the Godhead.77 The point of departure of this new mode of reading is that the observed natural phenomena we experience in the present are not defective signs of the sublime realm that the human race lost touch with after the Fall. The physical world is illuminated by the same light as the light of the first day of creation, and this light, according to Kepler, is a perfect mathematical entity that makes the motions and passions of physical objects mathematically intelligible. There is no need, therefore, to turn the mind’s eye from the material world in order to grasp other or higher realms of being. On the contrary, the mind’s own spiritual content evades one’s internal inspection, and worse still, an internal inspection leads quite often to fantastic chimeras. By recognizing the mathematical structure that the Creator has installed in the matter of the physical world, one can discern the geometrical archetypes that furnish the human mind, which is a perfect image of the divine mind. Therefore, geometrical knowledge of the physical world is the foundation for any theological inquiry. Mathematical diagrams are not mere theoretical pictures, nor, by turning the human gaze to a realm beyond, are they a mnemonic means of recalling a lost perfection;

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rather, they are representations of real physical motions. Mathematical diagrams constitute a real means of bridging optical phenomena, the human mind, and the divine geometrical shape. Kepler encountered emblems on various occasions throughout his scientific career. He read books of emblems and gave serious consideration to the role of visual representations as a means for acquiring knowledge.78 In his Mysterium cosmographicum of 1596, Kepler had already supplied the reader with an impressive image of the heavenly globe constructed from the Platonic solids. The intention of this globe was to “direct the eyes to the central mystery of the cosmic machine.”79 This interest in visual representations persisted throughout his scientific career, always in an attempt to demarcate his own method for the utilization of images from artistic practices from the alchemists’ application of visual imagery in their quest for knowledge.80 In 1617 Kepler found himself in direct opposition to the emblematic worldview of Robert Fludd and his alchemical-spiritual milieu. Kepler succinctly summarizes the differences: In [Fludd’s] work there are many pictures; in mine, mathematical diagrams keyed with letters. You may also note that he takes delight in the shadowy mysteries of things, while I strive to bring these same things, wrapped in obscurity, into the light of understanding. . . . The harmonies he aims to teach are mere symbols . . . poetic or rhetorical rather than philosophical or mathematical. . . . Following the celebrated axiom of Hermes, he makes things above similar or analogous to things below. But in order for this analogy to apply everywhere he must often drag things in by the hair so that they will apply on both sides. My view on analogies . . . is clear; they are apt to run into infinity.81 What is new in Kepler’s diagrams? What exactly is the difference, for instance, between Kepler’s symbolically potent heavenly globe composed of Platonic solids and Fludd’s emblematic depiction of microcosms and macrocosms? What different ontological status did Kepler ascribe to his pictures, in contrast to those of Fludd and his milieu? Both Fludd’s emblems and Kepler’s diagrams addressed the gap between knowable universals and concrete physical events. In order to surmount this gap and to disentangle his science from the multileveled interpretations of emblematic representations of the secrets of nature, Kepler had to redefine the epistemological status of pictures. Kepler is careful to distinguish true diagrams, which

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are supposed to comprise an integral part of scientific reasoning, from “pseudodiagrams,” which serve a didactic or rhetorical purpose. In defending Brahe’s diagram of parallax calculations for the position of comets from Galileo’s violent criticism, Kepler stresses that it was nothing but a “pseudodiagram.”82 Brahe’s diagram was meant only as an explanatory illustration, and one cannot draw physical or mathematical conclusions from it. True diagrams are not simply visual illustrations artificially appended to corporeal reality, and they are not abstracted from it. On the contrary, Kepler turned the process of abstraction on its head, and with it the process by which mathematical entities originate.83 Instead of the scholastic dictum that “nothing is in the intellect that was not first in the senses,” Kepler declares that “[i]ndeed, to the human mind . . . quantity is known by instinct, even if for this purpose it is deprived of all sensation. Of itself it understands a straight line, of itself an equal distance from a given point, of itself it forms for itself from these an image of a circle. . . . For the recognition of quantities, which is innate in the mind, dictates what the nature of the eye must be; and therefore, the eye has been made as it is because the mind is as it is, and not the other way around.”84 Kepler further asserts that a diagram is an expression of a priori principles, and that its role is to assist the mind in performing geometrical constructions: “Yet that construction is never drawn from sensible things in a diagram, though it is assisted by them; and it does not arise from the assembling of many individual sensible things into one axiom, but it is obtained a priori.”85 In order to decipher natural phenomena, Kepler has no need to rely on ancient knowledge or to find how to allegorically relate them to specific passages from Scripture. Kepler ironically discards any attempts to interpret the encounters of the gods in Homer as planetary conjunctions with astrological value or the battles between the gods in the Iliad 20 and 21 as an expression of the stellar conjunction that will take place on the last day of the world.86 When coming to interpret an astronomical description in a literary text or a book of history, like Lucan’s De bello civili, there is no need to impose on it a mathematical-astrological analysis; one should instead examine its poetical function.87 Classical pagan literary texts are not the only kind of texts excluded from the realm of knowledge of natural events. In Kepler’s view, the Scriptures themselves should be seen as a rhetorical device for the correction of human morals rather than the fountain of all knowledge. He deals explicitly with the role of textual interpretation in his introduction to the Astronomia nova, where he warns against dragging Holy Writ “into physics class.”88 The aim of Scripture is moral, teaching humans of “God’s greatness and potency in a creation of such magnitude, so solid and stable.”89

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There is no need to physically allegorize on biblical verses. They are not obscure fables and enigmas containing some hidden truth about nature to be excavated by the diligent and contemplative reader. Kepler contrasts ancient authorities with the role of the ever-present reason: “while in theology it is authority that carries the most weight, in philosophy it is reason,”90 he writes, differentiating between the world of texts and the realm of nature. The first is a world that refers to the domain of memory and authority; it is specifically created by humans and pertains to reforming human forgetfulness and corrupted moral attitudes. The latter is a domain of creation itself, of the divine revelation and playfulness. Kepler asserts that play is an essential part of the divine creative act, and thus emphasizes the affinity between play and knowledge. The central position of the concept of playfulness in the acquisition of knowledge is elucidated in Kepler’s answer to Feselius’s attack on judicial astrology and the theory of signatures it implies. Feselius contends that “[t]hese imagined signatures are nothing but a jolly fantasy of idle heads that cannot remain without occupation and like to make up tales.”91 Kepler’s answer asserts the positive aspect of play, as well as interpreting signatures as geometrical structures: “God Himself, since because of His supreme goodness He cannot remain without occupation, has therefore played with the signatures of things, and has represented Himself in the world; and so I sometimes wonder whether the whole of Nature and all the beauty of the Heavens is not symbolized in Geometry. . . . Just as God the Creator has played, so he has taught Nature, His image, to play, and indeed to play the same game that He has played before her.”92 Kepler ends by asserting that human reason, in contemplating the world, imitates the divine game. However, this is a special kind of game that can produce true knowledge, and not “a silly child’s game, but a natural instinct implanted by God.”93 In a letter to Joachim Tanckius, an alchemist from Leipzig, Kepler traces the rules of this play, demarcating the lines between the proper way to play and the erroneous and absurd game. Tanckius asks for Kepler’s help in deciphering Reinhardus’s arguments and contentions against established harmonic theories, particularly against Clavius’s. Tanckius declares that he is not looking for disputes or paradoxes but seeks a solid and mathematical opinion on Reinhardus’s claim that he can deductively derive musical harmonies from the heavenly harmony. Kepler’s answer is an excellent example of his artful irony. He is wholly aware of the intellectual paradigm from which he attempts to disentangle himself, and his criticism is subtle and playful. Kepler begins with compliments to

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Reinhardus’s erudition and knowledge of the ancient writers. He even admits his defeat; he cannot remember reading in ancient sources how to compose the human stature out of musical notes: “Reinhardus wins by reading.”94 This confession of defeat soon turns into a full-fledged offensive not only against Reinhardus’s reading of the ancients, but against the ancient authorities themselves. After a short explication of the Platonic theory of harmony and its symbolic origin, Kepler rejects it as a whole. His premises for rejecting Plato’s system of harmonies are grounded in the causes ascribed by Plato to the existence of harmony as well as to its effects. Plato ascribes the origin of harmonies to arithmetical ratios, and Kepler contends that their origin is geometrical. Secondly, Plato errs in dividing the pleasant tones, and Kepler adds, following Ptolemy, three supplemental tones on account of the judgment of the ears.95 Kepler explains this point more fully later in his career. In the Harmonices mundi, he repeats his attack on the Pythagoreans and Plato for not following their ears and for establishing a tyranny of abstract and numerical harmonies.96 The harmonic proportions the Pythagoreans accepted were limited to those ratios made out of the tetractys (1, 2, 3, 4). Plato followed them and made things worse (according to Kepler) by accepting the Pythagorean semitone, or Platonic limma 256:243, which is the difference between two major tones and a fourth.97 Ptolemy was the first, according to Kepler, to rebel against this tyranny and to “uphold the sense of hearing against the Pythagorean philosophy.” He not only accepted their proportions but “also admitted the proportion of one and a ninth to one (10/9) as equivalent to a minor tone and that of one and a fifteenth to one (16/15) as equivalent to a semitone.” However, Ptolemy later in his work, after he “restored the judgment of the ears to its rightful place, deserted it again, as even he adhered to the contemplation of abstract numbers.”98 Kepler establishes the criterion of differentiating between senseless analogies and meaningful ones. The criterion is the judgment of the sense, or in other words, physical effects, and not abstract mathematical numerical ratios. Ignoring these principles and following abstract analogies leads to ludicrous conclusions, as in Reinhardus’s exposition of the numerical ratios that generate male and feminine principles in nature. Kepler takes this opportunity to come to terms with a well-established tradition that goes back to Pythagoras, Plato, and Hermes Trismegistus, who in the harmonic division of minor and major find the way “nature divides . . . the human race through their generative organs to femininity and masculinity, of which to that something is wanting, to this something is overflowing.”99

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These titillations (titillatione) stimulate Kepler to attempt a symbolic play with the feminine and masculine principles of nature. His point of departure is a special kind of proportion: the divine proportion, which is a special case of the golden section (a : b :: b : c). The main quality of this proportion is “when out of the three terms the two minor ones equal the larger one, or when the whole is divided in this fashion in two, so that between the parts and the whole there is a continuous proportion.” A further property of this proportion is that “a similar proportion is composed again from the larger term and the whole. For what initially was the larger [term], now becomes the lesser, what [was] initially the whole now becomes the larger term, and it calculates the composition from both sums [that is, when a > b > c, b + c = a, a : b :: b : c :: (a + b) : a :: a : b].” These combinations and recombination can go on to infinity, always maintaining the same divine proportion. Kepler suggests that this kind of divine proportion governs the generation of “similis ex simili.”100 Kepler finds evidence for this speculation in the existence of regular pentagonal figures, which result geometrically from similar divine proportion in botanical generative processes and, more importantly, in the planetary motions.101 Kepler brilliantly shows how this perfect divine proportion produces images of male and female. As integers cannot form a divine proportion since they satisfy only one condition, that the greatest term be the sum of the two lesser, but cannot produce geometric proportion (that is, that as the greatest is to the middle term, so the middle term is to the least or in algebraic formulation when a > b > c then a/b = b/c or b2 = ac). If one calculates this sequence with integers (Fibonacci numbers), ac either exceeds or falls short of b2 by unity. In other words, as the square of the middle term progresses, it comes infinitely nearer to the rectangle of the two other terms. Kepler speculates that the relationship between the insemination of plants (where, according to his botanical understanding, generation is ex similis) to the generation of animals is that between “geometrical proportion, or the expression of lines, [and] an arithmetical proportion, or the expression of numbers.” Kepler’s only way to make this speculation clear is by pointing to some small diagrams showing some squares either missing a piece or with an additional piece in position that can be interpreted as copulation (figure 12). Finally he comments, “I do not assume to be able to explicate the thing more clearly and palpably, than if I tell you to see the images, this of a male organ [mentulae], that of a womb [vulvae].”102 Thus, “we never articulate the divine proportion by integers [numeris], and yet we always fall nearer to it.”103 To emphasize the somewhat humorous character of his elaboration, Kepler summarizes his point

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12 Kepler’s copulating squares, in Kepler, letter to Joachim Tanckius, May 12, 1608. Courtesy of St. Petersburg Branch of the Archive of RAS. SPb ARAS, deposit 285, inv. 1, unit 8, fol. 39v.

by answering a rustic satyr’s inquiry as to why there is a difference between plants and human generation. Kepler’s answer is that plants have semen in them, therefore one is enough in order to generate a similar one. But in humans it is necessary to fit the male organ into the womb, since the latter is wanting and it desires what completes it (like the missing unit to complete a square in the diagrams), whereas the other is overflowing with its excess of a unit and is inclined to supply that desired unit. “The union of two humans and the insertion of the one’s excess into the deficiency of the other give birth to a similar third.”104 Kepler amusingly concludes that these speculations were forced out of him by “the lecherous feelings roused by Reinhardus’s speculations.”105 This joking mode has, however, a serious dimension as well. It is through such humorous games that he draws and entices the reader to separate false and empty analogies from the true depiction of reality. By suggesting the joke, Kepler is able to draw more clearly the dividing lines and classifications of the different procedures, with no need for elaboration or presenting obscure definitions. As Kepler argues that a geometrical aspect has real meaning only when it is considered as an objective representation of physical events, the reader can now understand why it is so in comparison to abstract verbal symbolization:

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As in meteors, it is some active cause, capable of reason and geometry, that according to the rising aspect [a geometrical matter] in the heavens accommodates the modes and paroxysms of its operations, which is to excite the vapors of the subterranean humors as they evaporate. The erroneous others support this by symbolization [Symbolisationibus], expecting out of Saturn snow, out of Mars thunder, from Jupiter rain, and from Venus dew, out of Mercury wind, etc. But the geometry of an aspect is an objective cause [causa obiectiua], moving the subterranean arche to some impulse on account of which all the things mentioned, with no distinction, result—now this, now that, according to the circumstances.106 Similar arbitrary symbolization is the Pythagorean analogy between odd and even numbers and the nature of male and female. Whereas Pythagoras and Plato had an idea similar to Kepler’s deficient and excessive squares, their numerical speculations resulted in a grotesque joke: “Because an excess appears also in an odd number, by comparison with an even number. Therefore the Pythagorean males have their organ, but the Pythagorean females do without wombs.”107 As in his analysis of the conic sections discussed above, Kepler’s geometrical entities acquire their value and meaning in relation to other figures and refer to other geometrical quantities only. A geometrical figure does not serve as a mnemonic for some other concept or entity; when one is contemplating such a figure, all that will come to mind should be whether there are other geometrical quantities that are in a certain proportion to it. The only analogy possible is between a geometric proportion or arrangement and exactly the same proportion or arrangement between physical objects, preferably their motions (as in music, or in the case of light). The observer of physical events can recognize in them nothing but the geometrical archetypes that were implanted in it at the particular moment of creation by the divine Creator. No correlated divine text appears on the horizons of Kepler’s reading of the “Book of Nature,” but mute physical objects, their motions and geometrical shapes, and their proportionate arrangement. The rigor of geometrical deduction at the core of the Keplerian reading of nature assists in distinguishing it from the multiple levels of interpretation ascribed to natural events and their symbolization by Fludd and his milieu. Kepler suggests only one possible meaning for a physical phenomenon, and there is no method for transforming one thing into another through allegorical reasoning. In the final account, the meaning of each physical phenomenon is its geometrical representation and not some elusive verbal (i.e., qualitative) concept.

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This line of reasoning comes up most clearly in the second part of Kepler’s letters to Tanckius. Rejecting scholars’ attempts to deduce the proportions of the chord from the heavenly spheres, Kepler begins his research with sounds and musical harmonies, inferring from the human ability to discern between pleasant and unpleasant musical tones. Instead of Platonic abstract and arbitrary arithmetical reasoning, Kepler suggests that the ears can recognize only rational geometrical differences as harmonious proportions.108 Kepler admits that the ears present a certain freedom and uncertainty of judgment, especially when two voices are “to be optimally harmonious [concordauerint], one by keeping the tone, the other gradually intensified [paulatim intensa].”109 Critically abjuring Ptolemy’s futile attempt to enforce arbitrary reasoning on this process, Kepler follows the judgment of the ears in determining the division of the chord into the different quantitative relations that constitute the tones and voices. Going beyond the empirical assessment of human capacities of auditory discernment, Kepler endeavors to provide its underlying geometrical causes: “What is the cause that the ears approve as consonant [approbent consonantiam] first [the proportion] between parts 1, 1, and the whole 2, then [that] between parts 1, 2, and the total 3, between parts 1, 3, and the total 4, parts 1, 4, and the total 5, parts 1, 5, and the total 6, parts 2, 3, and the total 5, parts 3, 5, and the total 8, and yet between the total 7, 9, 11, 13, and whatever parts, [the ears] find no harmony [nullam inueniant concordantiam].”110 The reason is not that these proportions have any arithmetical significance concerned with these numbers being odd or prime, as both odd and prime numbers express harmonious or disharmonious proportions.111 The cause of harmonious or disharmonious proportions is the geometry of the circle. If a chord is stretched to a circular shape, one may discover that it cannot be divided geometrically into exactly 7, 9, 11, or 13 equal parts. “Therefore I say the cause to be this, because a circular line (that signifies the straight chord, and sounds alike if it is able to stretch circularly) can be divided geometrically in 2, 3, 4, 5, 6, 8, and cannot be divided geometrically in 7, 9, 11, 13, not because of a defect [uitio] of our intellect, not because of the imperfection of the geometrical science, but because of nature [sed natura]. Of course it implies a contradiction, to divide a circle geometrically into 7 equal parts.”112 Kepler’s far-reaching conclusion is that the human sense of hearing responds only to geometrical quantities and can differentiate intuitively between those and irrational sizes: “Accordingly, what presides [praeest] over the faculty of hearing is capable of the highest reason [summae rationis capax], rejecting those sounds that flow according to nongeometrical [αγεωμετρητoις] lengths,

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and approving those that originate [pendent] from geometrical lengths.”113 This sensory aptitude to recognize specific geometrical ratios is a divine endowment implanted in humans at the time of creation. The created world was established on certain geometrical ideas, and whatever can be inferred from these ideas could have been created. This is the only limit to which Kepler subjects the divine act of creation, eliminating any arbitrariness in the government of the world or any speculation of imperfection in the divine creation. As the divine mind is beyond imperfect human intellectual capacities, a true theology is possible only through gauging, through observation of the material world, the geometrical ideas that “existed in the divine essence itself ” (in ipsa essentia diuina) previous to the time of creation.114 Moreover, the shape of the created world cannot be other than a perfect imitation of God, whose essence is imaged in the sphere, because “[t]he will and authority of God rules according to the laws of its own essence; he does not wish to deceive [mentiri], because he cannot, he does not wish contradictions, because they fight with his own essence [pugnant cum ipsius essentia].”115 Thus, God can only mold his creation according to those ideas that are to be viewed and contemplated in his own essence. These ideas, then, “do not admit a seven-angled regular shape, a nine-angled regular shape: certainly it is never expressed in such a way, neither in bodies, nor in motion, nor in the presiding faculties of the senses.116 Following, thus, the “ratiocination and the judgment of the ears,” Kepler can conclude that in contrast to Reinhardus, he has deduced the divisions of the monochord wholly from geometrical considerations without playing around with ideas about the structure of the heavens.117 Kepler gives two reasons why one cannot move from musical tones directly to the contemplation of heavenly harmonies. One reason derives from the different physical objects considered by astronomy and music. The other reason is derived from a formal difference between three-dimensional bodies and two-dimensional geometrical shapes. The physical difference is constituted by what the geometrical shapes represent: in planets, their journey in the sky, in music the pulsed rhythm of the chord (celeritas chordae pulsae). This implies that the rules that govern these motions are the result of different causes: “For the planet is not [in the same relation to] its circuit as the quill is to the chord” (Non enim id est planeta suo circuituj, quod est plectrum suae chordae). In contrast to physical chords, the speed of the planets is not ruled only by the length of their orbits, but also by their distance from a center.118 Even if one may find a similarity (for instance that planets and chords both move through air—if there is air in the heavens) and thus ascribe sounds to the

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places where a planet moves more slowly or quickly, this comparison will face two problems: the cause for the change of the planet’s speed is the distance from the center, and the proportion between the diameters of two planets and their motion is not only due to intension and remission (“intentionis inquam et remissionis mensura,” i.e., their changing distance from the center).119 The second reason why Reinhardus and his ancient predecessors are wrong in attempting to deduce the heavenly spheres from musical harmonies is located in pure geometrical considerations: Since diameters are understood as diameters of spheres, truly a sphere is either a body, in the opinions of the physicists [secundum physicos], or the three-dimensional region of space [locus solidus] in which the planet may be found in the course of time. Therefore, the diameters of the spheres produce proportions from solid figures, as has been pointed out in Mysterium cosmographicum. On the other hand, sound is motion, truly all motion [is along] lines, therefore the proportion of motion is more fitly sought from the geometry of plane figures, by dividing the circle (the line of celestial motion). For as a line has itself [related] to a solid body, so motion has itself in a certain way [related] to a body.120 In these sentences, Kepler releases the physical world from its dependence on textual interpretation for acquiring meaning. For Fludd, the natural world is a text, and its understanding requires all the intricacies of mystical reading, as each phenomenon is an emblematic sign with several levels of possible interpretation, from the literal to the anagogic. Fludd considers verbal analogies and metaphors as legitimate tools for understanding the interconnectedness of the whole cosmos. For Kepler such a reading of the book of nature runs the risk of falling into an interpretative abyss of infinitely delusive polysemic symbols and emblems. Kepler’s new geometric mode of reading the book of nature sets a rigorous limit on the application of analogical reasoning, rejecting the application of language, and especially of metaphors, for the understanding of the natural world. Instead of employing linguistic signs, Kepler represents each physical phenomenon by a geometrical figure, locating its cause in geometrical proportions. Not every geometrical proportion has physical meaning, however; only those that can physically affect the senses have physical reality. Abstract geometrical archetypes are embedded in material natural phenomena, and this materiality protects the contemplating mind from arbitrary geometrical reductions. On the other hand, only through recognizing these geometrical

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archetypes embedded in corporeal motions can the mind understand the meaning and coherence of the physical world. Fludd’s milieu of Paracelsian alchemists and spiritualists emphasized the process of retrieving lost knowledge through the reading of physical signs. Moreover, in order to reconstruct that lost knowledge, one had to repress sensual perceptions and transform them into a higher and more spiritual instrument. In contrast, Kepler does not propose any esoteric method of reading nature’s text, since physical objects are signs of a single unified meaning, that of the divine act of creation. This meaning can be understood directly through pure sense-experience of the world. The human senses can perceive and be affected by the geometrical proportions embedded in physical reality from the time of creation. Thus, the scientist has no need for special mnemonic capabilities or for special textual knowledge as a storehouse of allegories and metaphors. All one needs in order to decipher nature are sense-data (especially of observation) and the properly applied language of geometry. While the judgment of the senses can guarantee that no arbitrary humanly conceived mathematical deductions are applied to physical reality, geometrical knowledge and analysis can protect from imaginary inventions and delusions. Since the human senses are sensitive to geometrical proportions, the human imagination tends to produce artificial geometrical quantities when physical reality does not supply them directly; for instance, in the case of reflecting or refracting surfaces, the eye produces imaginary straight lines in order to perceive the visual object not in its true place but as a reflection or refraction in an imaginary place. It is only through the application of a rigorous geometrical analysis to these phenomena that one can differentiate between illusions and the physical truth. Fludd and the occult philosophers blur exactly these dividing lines by subjecting natural phenomena to human verbal analysis. They produce imaginary speculation either by neglecting the reality of sense-data or by overlooking the geometrical causes of physical phenomena. Thus, in order to escape infinite symbolization and allegories, Kepler rejects memory and textual authorities and establishes observation and geometrical understanding as the foundation of true knowledge. The sequence that the Christian tradition aspired to establish from the book of nature to the book of Revelation is severed with no regrets. Kepler celebrates the geometrically ordered autonomy of natural phenomena, relegating human imagination to an auxiliary role in his search for divine truth.

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6.

NOTHING AND THE ENDS OF RENAISSANCE SCIENCE the quality of nothing The waning King Lear sets the scene for a dramatic investigation of invisible nonentities such as “love” and “nothing.”1 Lear begins his inquiry with the assumption that man is the measure of all things, and thus human linguistic means are adequate to capture the unknown and imperceptible realms. When Cordelia admits that her only answer to her father’s demand to measure her love is nothing, all that Lear can suggest to her is to “mend your speech a little.” The correct use of the right words is the principal tool to disclose the truth of things, whether corporeal or spiritual. The Earl of Gloucester is the one who indicates how it is possible to observe “nothing.” His bastard son Edmund pretends to hide a letter incriminating his brother, the legitimate heir. Gloucester inquires, “What paper were you reading?” and Edmund replies, “Nothing, my lord.” The earl marvels, “No? What needed then that terrible dispatch of it into your pocket? The quality of nothing hath not such need to hide itself. Let’s see. Come, if it be nothing, I shall not need spectacles” (my emphasis).2 This is an obvious play on various late sixteenth-century proverbs as “A man needeth not spectacles to see the Sunne shine.”3 Gloucester, with this double negation, discloses the truth. In order to observe what is unobservable— that is, nothing—one actually needs lenses, as artificial means that go beyond the capability of human senses.4 Toward the end of the play, as Lear carries the

body of Cordelia in his arms, the power of artificial lenses and mirrors is fully revealed. Lear cries out, “Howl, howl, howl, howl!” and protests against the inadequacy of human language to reflect his feeling and of the human eye to see reality, stressing the first vowel in the shape of a zero: O, you are men of stones: Had I your tongues and eyes, I’d use them so That heaven’s vault should crack. However, neither the human eye nor human language can penetrate the heavens to come face to face with the mystical nothing (the kabbalistic Ayin) that governs life and death. In order to penetrate into these depths of nothingness, Lear needs artificial means of observation: I know when one is dead and when one lives; She’s dead as earth. Lend me a looking-glass; If that her breath will mist or stain the stone, Why, then she lives.5 The only way to observe breath, that is, an almost invisible nothingness and simultaneously the essential sign of life, is to manipulate it through the application of a mirror, the device that produces false images and distorted reflections. King Lear confronts the emerging paradoxical awareness at the turn of the seventeenth century that “nothing” is no mere negation. At the beginning of the play, “nothing” still presents the limits of speech and of human understanding. As Lear’s tragic journey evolves, he discovers that though “nothing” is beyond human measure and the representative abilities of human linguistic skills, it is still a powerful force that can be reckoned with artificially. Lear’s fascination with the representation of nonhuman measures was shared by early seventeenth-century painters, mathematicians, and natural philosophers, who sought to transgress the bounds of human senses and to examine and calculate “nothing.”

beyond the defects of sight Toward the end of his Ad Vitellionem, Kepler turns back to one of the main issues that prompted his optical investigation: the diminution of the lunar disk

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during solar eclipse. Against Tycho Brahe’s assertion that the moon’s diameter diminishes due to solar illumination, which reveals the true dimensions of the lunar disk, Kepler contends that the causes of these appearances are to be sought in the “structure of the sense of sight” and not in the physical interaction between the luminaries themselves. The astronomer’s task is therefore “to distinguish carefully here between those things that happen to the sense of sight and those that happen when the consideration of the sense of sight is removed.”6 Kepler severs visual sensations from “those things that in actual fact occur” and forewarns astronomers not to “trust the sense of sight.” Having rejected Brahe’s calculations of the true size of the lunar diameter and his conclusions concerning the impossibility of a complete solar eclipse, Kepler suggests his own table of parallaxes to determine stellar positions. As the eye is beyond remedy, all the astronomers can do in order to represent only “what happens outside of consideration of the sense of sight” is to recalculate the parallax involved in their observations. In the following chapters, Kepler sets out to avoid the sense of sight by subjugating it to geometrical procedures and instruments of observation.7 In this methodological maneuver, Kepler sets to undermine the Renaissance dream of pure visibility, rebuking both Brahe’s astronomical practice and the relentless effort by various Renaissance men of letters to establish vision as a secure foundation of knowledge. The Aristotelian primacy of perception as the basis of valid and certain knowledge resonates in Alberti’s ardent demand (discussed above) to transform abstract geometrical entities into visible signs, or in Pomponazzi’s sneer that “it is ridiculous and foolish to forsake what is observable, and what can be proved by natural reason, to seek what is unobservable, and cannot be proved with any verisimilitude.”8 It was exactly this trend that was the immediate target of Kepler’s onslaught in declaring that “there exists no way to grasp motion visually,” and more firmly that “whatever is in our senses concerning the motion of the heavens, we have absorbed thanks to the intervention of reasoning.” In inverting the scholastic dictum “Nihil in intellectu quod prius non fuerit in sensu,” Kepler has to answer how one’s reason can ponder over things that were not sensed or perceived previously. In discussing parallaxes, Kepler suggests that a different faculty is involved in gauging imperceptible entities: Now in geometry, when we don’t know how to describe a spiral or a conic section in a single process, we imagine some points which the lines passes through, from which the whole trace of the line may be

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distinguished. So likewise, in astronomy . . . we by no means perceive the motions of the heavenly bodies with the eyes, but we compare various positions with each other, and we thence seek the form of the motion, by which all those places previously noted, and similar ones that are going to be, may be set in order.9

new sources of knowledge Kepler is well aware of the problems involved in this epistemological attitude that relegates to the mathematically informed imagination the task of grasping and analyzing imperceptible entities such as planetary motions. Already in his 1596 Mysterium cosmographicum, the young Kepler had predicted the critics’ reaction to his speculations on how Platonic solids determine planetary orbits: “I shall have the physicists against me because I have deduced the natural properties of the planets from immaterial things and mathematical figures, and furthermore because I dare to investigate the origin of the orbits out of bare [nuda] imaginary cross sections.”10 Kepler describes his critics sitting pensively, reminiscent of Dürer’s melancholic angel, pondering at a row of Platonic solids lying in front of them and wondering how such imaginary constructs can supply scientific explanation to real physical bodies. At first glance this assumed criticism is an extrapolation of the traditional Aristotelian principle of metabasis, admonishing scientists against confusing the categorical boundaries between different disciplines. Reading this paragraph again, however, divulges Kepler’s self-reflecting perplexity: can one move confidently from imaginary mental constructs to physical realities? It is this problem of accommodating imaginary constructs with reality, to find a common measure between inner artificial pictures and external sensory phenomena, that haunts Kepler. The old Aristotelian metabasis demarcating mathematics and physics acquires new significance. Late Renaissance science perceives this Aristotelian principle as a dichotomy between artificial constructs and natural phenomena, between human inner imaginations and external corporeal motions. The traditional distrust of human sensory knowledge was complemented by apprehension over the creative power of human imagination. In this way, interpretations of sensory data are likely to be further distorted by one’s inner fantasies. How can one guarantee that inner speculations are true reflections of the truth? No matter how long one observes the world (and speculates over it), the true causes that govern it and its motions will remain beyond human reach.

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Kepler’s bold intuition that the space between the planetary orbits can be filled with Platonic solids originated from his playful drawing of triangles connecting the points of the great conjunction of Jupiter and Saturn along the zodiac. It was a leap of his imagination to notice within the complexity of lines and circles that “the sides of the triangles . . . sketched out a smaller circle,” and what facilitated this leap was divine intervention answering Kepler’s prayers and presenting him “fortuitously” with what he was “never able to obtain by any amount of work.”11 How can one verify such an inspired cognitive leap? This question is double-edged: it is concerned not only with Kepler’s sources of knowledge in intuiting a certain order in the universe, but also with the manner in which he established this intuition, extrapolating physical motions out of imperceptible geometrical proportions and figures. In this new system of the universe, geometrical order had physical meaning. At the turn of the seventeenth century, Kepler thus evinced a bold blueprint of physical astronomy. At its kernel he aimed to measure the force that moves the heavens as the principal explanation of their shape and their diverse appearances. It was a radical attempt to capture what traditionally was considered “that lofty fantasy. . . . [t]he Love which moves the sun and the other stars.”12 In order to implement his ambitious plan, Kepler had first to find a way to convert “love” as a motivating force into an entity that can be measured and manipulated. Kepler’s bold program aspired to fuse fantasy with meticulous observations, intuition of abstract mathematical entities with careful measurements of physical phenomena, into a new mode of doing natural philosophy. The difficulties and dissonances involved in Kepler’s attempt to establish knowledge on inspired intuition and to transform geometry into physics came to the fore a few years later in his De fundamentis astrologiae certioribus (1602). The treatise opens with an admonition against enthusiastic fantasies as a source of knowledge: Truly astrologers consider as causes of their predictions, to be sure, some that are physical, and some that are political, but when they yield the pen to enthusiasm, [they consider causes that are] in greater part inadequate, for the most part imaginary, foolish, and false, and finally altogether worthless. If they sometimes speak the truth when they are carried away by their enthusiasm, it ought to be marked up to good luck, and it must not be thought that it issues, more often and for the most part, from some higher and occult instinct.13

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One cannot, therefore, trust intuition alone, and even if intuition proves correct, it does not testify to its divine origin. There is a need to outline the different causes according to the ability of the human mind to know and perceive them, from the more obvious to the mysterious. “Some physical causes are known by everyone, others, by very few; moreover, many things indeed exist naturally, but from causes comprehended by no one to this day. And of the causes that we consider to be known, there are some whose kind and explanation we all understand in common; there are others whose kind, or related causes, either very few understand, or no one at all.”14 Responding to Renaissance critics of astrology, who rejected any celestial forces communicating specific qualities to earthly matter,15 Kepler posits the heating power of the sun as the most obvious and basic physical heavenly influence. Investigating this power supplies Kepler with a more solid point of departure for his causal astronomy. The “approach and withdrawal of the sun,” its height above the horizon, and the angle at which its rays fall onto the earth present a clear correlation between geometrical values and physical phenomena. The geometrical determination of the sun’s physical effect is most evident when “the low sun strikes our horizon obliquely and weakly, but the high sun [strikes] more at a right angle, and more strongly.” In this case the “immaterial ray of the sun” behaves like colliding physical bodies, providing the natural philosopher with a mysterious link between the realm of abstract and invisible entities and the corporeal world of mechanical interactions. Kepler embraces his opponents’ main critical point that “the heavens have no particular force beyond the universal influence of motion and light.”16 Furthermore, he assumes these two elements as fundamental to astrology’s aspirations to physical knowledge. As the motion of light is geometrically considered, Kepler can reduce astrology’s physical conclusions to geometrical speculations. After allocating the strength of the light and heat coming from the sun and the planets to quantitative values, Kepler can proceed to a more abstract level of analysis of celestial influences upon the earth. In the hierarchy of knowable causes, there is one cause further removed from this visible correlation of a ray of light from the sun and the planets (though geometrical and noncorporeal) and physical event, as it consists of the physical effects of an invisible and purely geometrical relation. These effects are the result of the meeting of rays coming from two stars on the earth. Some of these rays, however, are to be imagined and are not part of the observational data available to the astronomers: “And this [cause] is not extinguished along with the very face of the silent moon, when no rays descend to earth, but the

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descending ray is then imagined. It is not impeded when the earth is placed between us and the stars; but it makes the stars hiding below [the horizon] also efficacious above.”17 What began as an attempt to purge astrology of “imaginary” causes suggested by astrologers “carried away by their enthusiasm” results in imaginary lines and invisible and wondrous geometrical relations. These imaginary constructs turn out to be the most significant operating causes in the universe, and as “nobler . . . possessing far more wonder” than the physically direct influence of light. These incorporeal causes affect the earth not through any direct contact (such as the warming effect of the sun’s rays or the humidifying effect of reflected lunar radiation) but through the inner cogitations of the earthly animated faculty. “For it is not a material cause but rather a formal one, and the form which it has is not simple form, but form in animate faculty, in intellect, in geometric cognition.”18 The convergence of rays in itself does not embody any power to excite effects. It is only the special godlike disposition of the animal faculty or soul of the earth to perceive, recognize, and respond to special harmonic angles that can produce a physical event. Just as in the act of sight, it is not the visible object that incites bodily reactions but the soul’s act of perception. Now this faculty, which adds force to aspects, is not located in the stars themselves. For these aspects we are discussing touch upon the earth and are a pure condition flowing not from the formal motion of the stars, but from the accidental arrangement of pairs of stars with the earth. In the same way, then, that the spirit moving the body is located not in the object, but in the place where the idea of the object is displayed, just so is it necessary for this force, which makes the aspects powerful, to be located within that great globe earth, as it is in all the sublunary bodies. To be sure, every animate faculty is the image of God the Geometer in creation, and He is inspired to His task by this celestial geometry of aspects or harmony.19 The ability of celestial aspects to affect earthly creatures is dependent on the soul’s inner disposition to recognize these special angles and ratios. The animal faculty acts as a mediator that can both discern abstract geometrical ratios and control the concrete reactions of the physical body. The external event is accidental, “For this [influence] does not come about because rays join together to form any angle.”20 There is no special value to the angle created by the

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convergence of the planetary rays, as this happens constantly and an angle exists “both the day before and the day after an Aspect, and two rays always form some kind of angle.”21 There is an infinite number of angles, and only those special angles that correspond to a harmonic ratio are effective. The convergence of the rays in a harmonic angle, however, does not in itself produce physical effects, unless its image is perceived by the animal faculty. In other words, there is a need for a soul to perceive and recognize the specific angle as a harmonic ratio. “But there is no efficient power per se in ratios or schemata. The same thing obviously happens here that usually happens in the locomotion of animals. If one should say that the objects of vision were able to move the animal as they enter the eye, and that therefore there was no need of an animate faculty in the animal itself that was moved, one would be philosophizing strangely.”22 Thus, Kepler’s geometrical account of celestial physics turns the inner cogitations of the earth into external physical reality. The earthly soul must actively fathom these invisible geometrical proportions in order for physical phenomena to take place. The universe is full of infinite rays and their geometrical relations and angles, and only the active imagining of the earthly soul can differentiate the significant ones from the rest. Whereas in the Mysterium cosmographicum it was an animated virtue flowing from the sun that activated and moved the planets, in Kepler’s 1602 treatise the earthly animated faculty residing in the body of the planet (in this case the earth itself) has to recognize the particular constellation as harmonic before it produces a corporeal motion. The earthly soul identifies these constellations and proportion intuitively, but how can the conscious human observer detect these invisible and incorporeal angles, how can invisible entities be measured, and what criteria can be applied to differentiate the physically real from fictitious imaginations?

the new visual economy and the perils of melancholy Two years later, writing his Ad Vitellionem, Kepler sought to solve these knotted epistemological problems with a radical rewriting of the optical tradition. As noted in the preceding chapters, in the preface to Ad Vitellionem Kepler declares that in observing the celestial bodies “we consider nothing but their species,” and this species is reduced to “light and shadow” on the one hand and to “shape and quantity” on the other. These quantifiable illuminations are the mise en scène of “this theater of the world” and are signs suitable to “human minds, likenesses of God,” assisting them in their investigations after deeper meaning.

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The way, however, in which humans can perceive these signs is principally as shadows and deficiencies, that is, as no-things that capture true knowledge of the world.23 The only way to capture these images and shadows is by applying the camera obscura as a microtheater of light and shadow. The camera assists the astronomer in avoiding “the inadequacy of the eyes” and is the only “sure procedure . . . for measuring something that happens in the sky.”24 Within the darkness of the camera the astronomer can “accomplish . . . what is completely impossible in clear light.” The eye cannot measure shadows, as it is attracted to the strong light of the luminaries and to their visible effects, whereas within the camera one can observe and measure the shapes and figures created by a ray of light coming through a window onto a wall. The stains of light on the wall, just like clouds, were considered in Renaissance culture as an opportunity and stimulation for artistic invention, as Leonardo da Vinci suggested: “I remind you that it is worth your while to stop sometimes in order to look at the stains on walls, or . . . clouds . . . or similar things, in which, if you consider them well, you will find marvelous inventions . . . because in confused things the ingegno is stimulated to new things.”25 Yet for Erasmus these stains on the wall are no mere playful invention of the human imagination but are representations of nothing, as he explains in his Adagia: “Clouds upon a Wall. In a letter to his son, Gregorius, Ausonius used the phrase ‘clouds upon a wall’ for something most similar to nothing or a dream; ‘have you ever seen a cloud painted upon a wall?’ he says. [By this] he indicates that the subject [lemma] of the poem subjoined to this letter is trifling and empty; for, a cloud is too unsubstantial to be expressed by colors.”26 In reproducing the visible and corporeal world as a projection of an insubstantial shape on a wall, Kepler can apply to it mathematical measurements, circumventing the Aristotelian admonition against metabasis. Furthermore, Kepler not only reduces corporeal objects to shadows and images on the wall but defines the light as the agent that produces these stains on the wall as a two-dimensional, noncorporeal entity, for “light has no matter, weight, or resistance.”27 The ray of light is but a representation of its motion and thus has no material reality: “the ray is not in the transparent [medium] . . . but there was [a ray], or almost was.”28 Kepler asserts that the noncorporeal geometrical line can express and represent motion, “since motion cannot keep the one thing that is complete about it, its past.”29 Light is the purest embodiment of motion as it penetrates the transparent medium and collides with dense surfaces “without matter or the dimension of solidity.”30

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One can sketch Kepler’s main principles of his new visual economy:

A. In order for a truly mathematical account of physical phenomena, one is to apply instruments of observation to reduce these phenomena to insubstantial, yet perfectly measurable, shadows and stains of light. B. Motion is not a qualitative process of change defined by its beginning and end points, but as a vanishing continuum produced by the mobile. C. Geometrical lines and points must be considered not as hypothetical devices or aesthetic factors. They are embedded in the physical material realm determining its possible motions. A point at infinity is not merely a geometrical playful speculation but can be considered a real cause in determining the path taken by light in a parabolic mirror, for instance. This visual economy is fully played out within the camera obscura, where instead of focusing on the contact between the corporeal visible object and the eye, the observer concentrates on shadows as the embodiment of imaginary mathematical entities artificially produced by the instrument. Enwrapped in darkness, the astronomer does not consider light directly but only concentrates on its shadowy effects. The operation of light as a two-dimensional incorporeal entity can be known only through the examination of other insubstantial appearances on the camera obscura’s screen. Kepler molds the astronomer, inside the camera obscura, hiding from the blinding light of the sun, concentrating on fleeting shadows and incorporeal apparitions, as the epitome of the melancholic persona. Kepler’s astronomer is like “Melancholike men,” who are “enemies to the Sunne, and shunne the light,” and whose spirits, “bending themselves to the eye, doe set before the imagination all manner of darke and obscure things.”31 The melancholic embraces “solitarynes,” contemplating inner states of mind, as the “spirits and blacke vapours continually passe by the sinews, veines and arteries, from the braine unto the eye, which causes it to see many shadowes and untrue apparitions in the aire.”32 Adopting the melancholic disposition to favor shadows and artificiosa observatione means that the astronomer too is in danger of treating these apparitions as real. Observing the stains of light and shadows on the screen of the camera obscura can lead astronomers to “devise with themselves a thousand fantasticall inventions and objects, which in deede are not at all.”33 Kepler’s mode of astronomical observation structured around shadows, incorporeal forces, and invisible angles and aspects resonates significantly with the melancholic vision. In order to demarcate his search after scientific

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knowledge from mere fanciful inventions and rid his new natural philosophy of melancholic doubts, Kepler has to find a way to guarantee that nature is geometrical through and through, and that these procedures are not mere artificial hallucinations within the human mind. Confronting these difficulties, Kepler has to assume that nature herself is governed by a formative faculty corresponding to the human mind. Understanding these animal faculties that reside within the earthly matter supplies a powerful key to the mysteries of the universe. In fact, it provides Kepler with the true rules of the divine game.34 As the divine game is incorporeal, Kepler’s pawns evaporate into nothing while in play and reveal the bare grid of rules to his investigative mind.

investigating nothing This pattern of investigation into “nothing’s” positive aspects occupied Kepler’s mathematical researches into natural philosophy. Kepler applied infinitesimally small figures and values, both in his astronomical calculations of the areas traced by the planetary motions as well as in calculating the volumes of wine casks. In these Kepler ignored the Archimedean formal limitations on such calculating techniques and applied them freely, taking such “nothings” as legitimate mathematical entities in considering physical problems.35 Kepler, however, did not adopt the infinitesimally small only as aids in his calculations, but in several instances attempted to ground such “nothing” ontologically. In 1611, just a few years after King Lear was first performed in London, Kepler attempted his own play with nothing.36 In a small treatise entitled Strena, seu De nive sexangula (A New Year’s gift, or On the six-cornered snowflake), Kepler seeks an object that will be small enough to be considered nothing, yet will “give promise of a geometrical speculation” and “excite the desire for invisible things.” He hopes to give such an object as a New Year’s gift to his benefactor Johannes Wacker, who is most appreciative of such trifles. The first move is to “review the elements, that is, whatever is smallest in each kind.” As Daniel Tiffany notes, this alludes to Lucretius’s famous analogy of letters as elements of human verse to the atoms as elements of natural bodies. Kepler, however, in ending the sentence with a quantitative categorization of the thing he seeks, transforms the analogy. Instead of written letters, the elements Kepler alludes to are the elements of Euclidean geometry. Nature is no text to be deciphered but a geometrical shape to be calculated and constructed from ephemeral geometrical shapes and proportions.37

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After considering and rejecting different minute things and animals, while crossing over Karlsbrücke in Prague Kepler found that a snowflake is “something smaller than any drop, yet with a pattern.”38 Snowflakes are even better than clouds as representations of nothing:39 they evaporate almost immediately and “melt into nothing,” or “they are entangled in larger plumes.” Better still, while snow is an earthly meteorological phenomenon, it is associated with astronomical research, as “it comes down from heaven and looks like a star.” Contemplating the flakes with their six corners and feathered radii triggers Kepler’s inquiry into nature’s formative faculty: “There must be some definite cause why, whenever snow begins to fall, its initial formations invariably display the shape of a six-cornered starlet.”40 In order to answer this question, Kepler has first to isolate the snowflake as a unique case. Other six-cornered shapes in nature are formed with a view to their utility. An external factor, such as cold, cannot be the cause of the vapor’s particles assuming the six-cornered shape of snow. Neither can material necessity resulting from the clash of the inner heat with external cold be the cause. Kepler concludes that a solution will be formulated only if he manages to “bring to light a way for the internal heat to fix the drop of vapor on three diameters, in the shape of an octahedron, or at any rate in a six-sided shape, on which matter may accumulate by condensation.”41 Kepler infers that the snowflake is formed over a “skeleton (so to say) of the octahedron with its three feathered diameters that intersect at right angles.”42 This skeleton is the bare mathematical form that operates from within the plumed particles as “the formative power” that resides in the center “disseminates itself equally in all dimensions.” This formation is not random but is part of the “creator’s design . . . preserved in the wonderful nature of animal faculties.” In the case of shaping snowflakes, this formative faculty operates with no obvious purpose and thus reveals itself in its pure form: “Formative reason does not act only for a purpose, but also to adorn. It does not strive to fashion only natural bodies, but is in the habit also of playing with the passing moment.”43 In detecting the way nature follows the mathematical rules of play, one can gather the causes of physical processes. The formative principle in nature, or its soul (anima), is geometrical and seeks to fulfill itself in the orderly shape of a regular body, imitating the Creator playing with geometrical forms. These are not merely imaginary constructs but incorporeal causes that operate from within the material realm as active agents.44 Underneath the multicolored physical reality are hidden skeletal mathematical figures and shapes. These are not part of the realm of Platonic ideas but are the constitutive elements of material shapes and their motion. The natural

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philosopher should not be deluded by external sensible qualities but strive to explore and reveal the “nothingness,” those bare mathematical figures and proportions, that govern and form material reality. The human mind can know the world not as it appears to the senses but as it can be reduced to invisible geometrical components. Kepler’s stress on the active role of mental ability to recognize abstract, supersensory mathematical proportions turns the Renaissance sense of melancholic contemplation on its head. According to the Renaissance medical understanding, melancholic imaginations spring from an ever-widening gap between human inner abstract intellections and external corporeal phenomena. Kepler, in contrast to Dürer’s melancholic angel studying artificial regular solids and pondering over their physical reality, reduces the physical world to bare geometrical forms and their motions.45 In Kepler’s analysis, basic corporeal qualities, such as cold and heat, are not sensations but expansion and contraction, that is, motions and their direction. To know a snowflake is not to marvel like the psalmist at its sensual similarity to wool, but to expose its skeleton of abstract geometrical structure.

shadows and the dream of knowledge In concluding his “New Year’s gift” with the ironic “Nothing to follow,” Kepler sets the outline of the answer of his new science to late Renaissance melancholic distress: scientific endeavor has to discard sensorial qualities and instead set its sights on “nothing.” No sensory criteria can establish a clear differentiation between wakeful states of mind and dreamy states and delusions. The natural philosopher has to apply her imagination in order to relate the bare mathematical figures to physical reality. This calls not for a skeptical retreat but for a more nuanced control over one’s imagination. Directing his telescope at the moon, Kepler reported to Matthias Bernegger, An experiment with the telescope that I carried out recently, produced a marvelous sight, altogether remarkable: cities and walls, which were circular because of the shape of the umbra. What more should I say? Campanella wrote his City of the Sun. And if we were to write a City of the Moon? Wouldn’t it be excellent to paint the cyclopean mores of our times in lively colors, but leave the earth behind and go to the moon, for the sake of prudence? But what is the good of such evasive action,

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since neither More in his Utopia nor Erasmus in his Praise of Folly were so well protected that they didn’t have to defend themselves? We must forsake the political tar pit and stay within the green and pleasant plains of philosophy.46 The images refracted by the telescopic lenses, like the shadows of the luminaries within the camera obscura, are confused, insubstantial stains or clouds upon a wall. These apparitions can lead the mind toward poetic, ingenious inventions that can entertain but cannot lead to true philosophy. In order to scientifically investigate celestial phenomena, the astronomer has to discipline his imagination, expurgate its playful disposition to form novelties, and direct it, through careful analysis and comparison, at the true structure of physical reality. One has to mobilize the imagination, without which no scientific play can ensue, yet one should be careful not to be carried away and search beyond the imaginary symbols and differentiate between false constructions and true causes. In analyzing celestial phenomena, one should ignore the metaphorical and mythological significance of the heavenly bodies and their sensual qualities. All there really are in the sky are geometrical proportions and relations. In order to discover these geometrical aspects, one has to play, but this play has to be censured and meticulously adjusted in order to decipher the bare truth hidden underneath what appears to the senses in general and the bodily eye in particular.47 Instead of the humanistic serious play with its melancholic undertones, Kepler seeks to fashion a new theater of knowledge, where shadows can lead the investigation after scientific truth. In this novel setting, apparitions and illusions will be transformed into accurate depictions of the physical world, and the melancholic imagination of the spectators will be converted into a vehicle of scientific observation. This fine balance between observation, imagination, mathematical measurements, and the act of reading receives its final and comprehensive expression in Kepler’s posthumous treatise Somnium. In this unique mixture of literary prose and scientific discourse, Kepler relates a strange dream that culminates in a detailed picture of the way the heavens appear to a lunar observer. This knowledge is revealed to the protagonist of Somnium through a daemon that teaches and explains the main tenets of the lunar astronomy. This daemonic instructor, however, presents itself not as a source of illumination and clarity, but as a shadowy creature: “For as a group we [the daemons] inhabit the earth’s shadows. . . . Up there [on the moon] we quickly withdraw into caves and dark places. . . . As soon as a spot begins to be free from the sun, we close

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ranks and move out into the shadows.”48 Instead of the enlightened Platonic teacher, the reader encounters shady daemons, who live in dark caves where they are granted leisure “to exercise our minds in accordance with our inclination.” The reader might associate this curious characteristic with the general occult ambience of the dream, and might relate it to the protagonist’s motherwitch. Kepler, however, explains that this fascination with shadows and darkness is directly related to scientific observation and knowledge: “The allegory compares the journey through the shadow to the observation of eclipse . . . the time spent in the caves, to continuous speculation based on observation of the eclipse.” Kepler proceeds to his own personal experience: “In Prague I had a residence in which no spot was more suitable for observing the sun’s diameter than the underground beer cellar. From the floor of the cellar I used to aim an astronomical tube, described in my Optics, through an opening at the top toward the noonday sun on the days of the solstices.”49 The reader is transported into the dark room, with its technologies of observation as a place of true and clear knowledge. Shadows and darkness are transformed from obstacles into vehicles of knowledge; the daemon and its shadowy world are the true teachers of astronomy: “However, to the extent that . . . the daemon stands for the science of astronomy, there is seriousness in the assertion that for the mind there is no passage to the moon except through the earth’s shadow and the other things which depend on it.”50 Recapitulating his theory of optics, Kepler reasserts that scientific knowledge is acquired not through direct experience of the phenomena in full daylight, but through the use of instrumental manipulation of shadow, projections, or any other sort of images and reflections. The shadowy instructor of Somnium is an embodiment of Kepler’s conception of scientific observation. Its central place in Somnium suggests that this treatise is more than mere entertaining pastime with Copernican undertones. The central place of shadows suggests it is a bold attempt to convert the reader’s actual mode of vision. Thus, Kepler expects his reader not only to embrace Copernican theory but to adopt a new ontology of vision comprising instruments and shadows as positive elements.51 In advancing his arguments for a new mode of vision, Kepler tackles the well-entrenched dichotomy between the empiricist preference of direct vision and the Neoplatonic embrace of a supersensual mode of perception. Kepler detects the general outlines of these two modes of vision in the text’s two main classical sources:52 Lucian’s A True Story and Plutarch’s Concerning the Face Which Appears in the Orb of the Moon.53 In Lucian, only direct experience can

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be counted as a source of knowledge, whereas in Plutarch, truth can be partially glimpsed through speculative myths that point beyond rational scientific procedures. The conclusion is the same: the epistemological foundation of astronomy (which deals with objects not given to direct experience) is doubtful, and therefore certainty of knowledge is unobtainable. To confront these criticisms of astronomical knowledge, Kepler’s Somnium mobilizes both rhetorical tools and his new optics, and thus, while disentangling the reader’s point of view from the clutches of Lucianic cynicism, converts the relationship between myth and knowledge in Plutarch’s story. Kepler constructs a narrative within a narrative, unfolding a hectic political situation in contemporary Prague.54 The narrator follows the people of Prague in searching for the meaning of this situation in ancient Bohemian legends.55 The search turns into a dream of a book of memoirs belonging to an Icelandic sage named Duracotus. In a moment of rage, his witch-mother, Fiolxhilde, gives him away to a passing ship. The boy arrives in Hven, at Tycho Brahe’s astronomical observatory. He stays there for a few years learning astronomy, alchemy, and other sciences. Missing his mother, Duracotus then returns to Iceland. His mother is impressed by his newly acquired knowledge and introduces her son to a daemon, her secret mentor in occult knowledge. The narrative now turns into a lecture delivered by the daemon on the way astronomical phenomena are observed from the moon. The daemon adds details of lunar geography and weather conditions. The daemon’s lecture ends abruptly as Kepler wakes up from his dream, leaving the reader puzzled as to its meaning. This complex narrative within a narrative has an extensive apparatus of footnotes and comments. In this apparatus, Kepler presents himself as astronomer and mathematician in order to explain the other elements of the treatise. He also states explicitly the reason for writing this strange and intricate story: “The purpose of my Dream is to use the example of the moon to build up an argument in favor of the motions of the earth.”56 Kepler follows here his own rhetorical method, wrapping an uncomfortable truth in brilliant and attractive camouflage. In his astrological exposition of 1601, Kepler states these rhetorical precepts explicitly: “We may observe (in order to cure the crowd’s craving for marvels) what physicians observe in the sick, that we may make use of the unnatural and pernicious appetites of the crowd to get them to swallow (as medicine) such advice (disguised as prognostication) as may serve to remove this disease of the mind, and which otherwise we could scarcely have persuaded them to take.”57 The physician-philosopher-astronomer humors his patients and lulls them into believing that his moral advice is a marvelous prognostication of their

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future. In many cases this disguise has a jocular character, and in his letter to Bernegger (mentioned above) where he discusses extensively his work on the dream, Kepler complains that “[t]here are just as many problems as lines in my writing, which can only be solved astronomically, physically or historically. But what can one do about this? How few people will attempt to solve them? The people wish that this kind of fun, as they say, would throw itself around their necks with cozy arms; in playing they do not want to wrinkle their foreheads. Therefore I decided to solve the problems myself in notes, ordered and numbered.”58 Kepler’s aim appears to be to introduce an argument in favor of Copernicanism disguised as a fabulous dream that will attract various readers in search of amusement. Somnium, however, is more than just magnificent wrapping paper to be discarded as the reader approaches the scientific core of the treatise. Kepler points out in his notes that the aim of his dream is a total conversion of the meaning and place of observation in the hierarchy of knowledge. In the note quoted above, Kepler explains the allegorical significance of the mother-son relationship in the dream: “However, I wanted to make this further suggestion: untutored experience, or, to use medical terminology, empirical practice is the mother who gives birth to Science as her offspring.” Kepler, then, identifies this empirical practice as ignorance. Such empiricism ends up in witchcraft, since it cannot supply sound scientific reasoning for its results. Furthermore, in the final account, what crude empiricism concludes through magical rites, the scientist achieves through reason. When Duracotus comes back to his witch-mother in Iceland, she is “deliriously happy that I had become acquainted with that science [astronomy]. Comparing what she had learned with my remarks, she exclaimed that now she was ready to die, since she was leaving behind a son who would inherit her knowledge, the only thing she possessed.” Later, the mother asserts that most of the things Duracotus learned through science she had learned from the magical daemon. Kepler claims that the main advantage the scientific method has over empirico-magical practices is that through theory one can reach conclusions not available to direct sensory experience alone. In the case of astronomy, such radical empiricism is most harmful. Empirical experience on its own forms “the universal opposition of mankind” to the notion of a moving earth. Later on, in note 96, Kepler restates the thesis of Somnium as “an argument in favor of the motion of the earth or rather a refutation of the argument, based on sense perception, against the motion of the earth” (my emphasis).59

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Kepler confronts crude empiricism that celebrates sensual perception as the only viable source of knowledge. This sort of empiricist thought rejects any theoretical considerations as fictitious human inventions and reasserts only what is immediately given to the senses. Such an empiricist attitude cannot accept the Copernican worldview. Copernicanism mathematically deduces that what we see is not the truth of the matter, and that in contrast to our visual perception, the earth revolves around a stationary sun as well as around its own axis. Kepler’s epistemology, however, is more complicated than just reasserting the Platonic dislike for sense-experience as a source of knowledge about the truth. Kepler seeks an integrated method where theoretical deductions find their affirmation in naïve observation and where sensual wonders find their explanation in a geometric theory of the heavens. In one of many digressions in his notes, Kepler relates several stories of ancient sages vainly sacrificing themselves in search of the key to natural phenomena. Kepler mentions Empedocles, who “with blind audacity” hurled himself into the flames of Mount Etna “searching for the causes of the everlasting fire”;60 and Pliny the Elder, who was “suffocated by stinking sulfur and cinders” in his vain attempt to investigate the nature of Vesuvius’s eruption.61 Kepler suggests a different method for the acquisition of knowledge. When Duracotus meets Tycho Brahe, Tycho is “quite delighted and [begins] to ask me many questions.” Duracotus adds that “it was the habit of that true philosopher never to stop asking questions, acquiring information, valuing such reports highly, thinking about them repeatedly, and applying them to the laws of nature.”62 Kepler envisions a division of labor between those who observe nature, naïvely reporting what they see in their travels, and those who use these reports to depict the true picture of reality. A case in hand is the Dutch mission that searched for the northern passage to the Far East, wintering in icy Novaya Zemlya. Their accounts, Kepler asserts, “supplied a large number of astronomical exercises incorporated by me in my Optical Part of Astronomy in 1604.”63 Later Kepler remarks that “the Dutchmen in the Arctic Ocean . . . found everything occurring just as we astronomers here at a distance have known and taught.”64 This is the main tenet of Somnium’s epistemological program: direct experience by itself is limited and cannot give the causes of phenomena. An emphasis on direct experience severs the relationship between the senses and their guiding mind, creating a rupture between experience and theory. This rupture leads scientists to assume magical and occult causes for natural phenomena. The task of true science is to regain the supremacy of theoretical reasoning over empirical observation. In such a way the vague occult causes are transformed

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into manifest geometric proofs.65 In his Somnium, Kepler suggests the antidote for the naïve belief in direct experience as the source of knowledge. Instead of direct experience, the astronomer can ascertain the true causes of celestial phenomena through the application of geometrically based instruments: “There is a popular joke: ‘I’ll believe in it rather than go into the matter personally.’ And many people ask whether we astronomers have just fallen down from heaven. They were answered by Galileo’s Sidereal message [i.e., the application of optical instruments, in this case the telescope]. . . . But even stronger is the judgment of reason [i.e., theory], being testimony which prevails over every objection, as the Dutchmen found by experience in their winter sojourn.”66 In the final account, experience will always corroborate what reason has already proved a priori. The framework of the dream allows Kepler to follow his theoretical principles of astronomy and to depict what is invisible: “In a dream it is necessary to have the freedom sometimes to invent that which was never perceived.”67 In his Somnium, Kepler does not mention his immediate adversaries, but identifies this skeptical attitude in the two classical texts he mentions as the framework of his own treatise. In Lucian’s A True Story, he finds an example of biting ironic criticism of human curiosity. In Plutarch’s Concerning the Face Which Appears in the Orb of the Moon, he finds a mixture of academic skepticism together with an invaluable compendium of classical physical astronomy. In Somnium, Kepler develops a dialogic relationship with these treatises, borrowing ideas and examples from them, while discarding their underlying epistemological claim. In doing so, he aims to release his readers from the sway of such skeptical attitudes. Lucian’s A True Story poignantly mocks travelers’ stories from Ulysses and Herodotus to his own contemporaries.68 He relates his own fantastic voyages across the Strait of Gibraltar; his flight to the moon; his being trapped in the belly of a giant whale; his sailing to the Isles of the Blessed; and his further travels to other islands and strange lands. Lucian ends his story abruptly, promising that “what happened in the other world I shall tell you in succeeding books.”69 This meandering narrative, Lucian promises, will supply scholars with “the sort of reading that, instead of affording just pure amusement based on wit and humour, also boasts a little food for thought that the Muses would not altogether spurn.”70 Parodying various poets, historians, and philosophical schools, Lucian admits that the only epistemological merit to his work is that his lying “is far more honest than theirs, for though I tell the truth in nothing else, I shall at least be truthful in saying that I am a liar.”71

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Lucian’s goal, however, is not only to expose false reports on distant lands but to admonish vain curiosity that leads the mind to seek what is beyond one’s reach: neither forward in space (it serves no rational purpose to venture crossing the Pillars of Hercules), nor into the future (Lucian’s greatest lie, as one of the earliest commentators adds in the margin, was his promise to tell more stories in his future work), nor into what is above (as the result is the grotesque voyage to the moon), nor into what is below us (into the belly of the whale that inhabits the depths of the ocean). Finally, any attempt to truly know the past is futile and will result in absurd answers, such as Homer’s disclosure that he is actually a Babylonian.72 In Lucian, curiosity and disorientation are equated with futile speculation that distances one from the good life.73 The indecisiveness of the different philosophical schools and their inane contemplations lead Lucian to reject philosophy in favor of the here and now.74 In contrast to this commonsense empiricism stands Plutarch’s Concerning the Face Which Appears in the Orb of the Moon, which is Kepler’s other source for lunar voyaging. Written in the spirit of Platonic academic philosophy, this treatise rejects mere sense perception as a source of knowledge and sets reason and philosophical myths as the true guides to moral life. The treatise opens in the middle of a discussion of the reason for the face that appears in the orb of the moon. The main problem is how one can perceive what is not available to the senses directly. The main thrust of the discussion is that when rational philosophizing and ordinary scientific procedures fall short, unconventional solutions come into play. Dreams and ancient myths come in handy when philosophy goes awry; they can bring a discussion closer to the truth than reason and science, as Lamprias, the brother of Plutarch, explains: “What else would you expect us to have done,” I said, “since it was the difficulty in these opinions that drove us from our course upon those others? As people with chronic diseases when they have despaired of ordinary remedies and customary regimens turn to expiations and amulets and dreams, just so in obscure and perplexing speculations, when the ordinary and customary accounts are not persuasive, it is necessary to try those that are more out of the way and not scorn them but literally to chant over ourselves the charms of the ancients and use whatever means to bring the truth to test.”75 The treatise thus addresses an impasse resulting from conflicting philosophical accounts concerning the nature of the heavens. Following the presentation of

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the different scientific arguments concerning the physics of the heavens, the main debater claims that the moon, a heavenly body, is no different than the earth, and the face that appears in its orb is nothing but mountains and oceans.76 If indeed the moon is no different than our human world, the question arises concerning its inhabitants. At this point in the discussion, Sulla, the guest, interrupts and tells a fabulous myth he heard from a strange traveler who used to serve as a priest to Cronus. This story turns to the moral meaning of the moon as a celestial earth, a body at the border between the material realm and the heavenly spiritual domain. The academic stand portrayed in Plutarch’s treatise suggests that there are different ways to save the phenomena, some even absurd and sacrilegious (assuming the earth to be mobile like a planet, or alternatively that the moon is a celestial earth). The more significant speculation, however, leads one to understand the moral meaning of truth through interpreting and contemplating ancient myths and traditions. Sulla brings into the scientific discussion an entirely different source of knowledge, an ancient myth, related by word of mouth from the priests who serve Cronus on a northern island. The emphasis in this story is on the blessed form of life that exists around Cronus, and on the passage of the soul to the moon after the death of the corporeal body and its expectation for a second death, leading to a mystical union of the intellectual soul with the sun. Plutarch asserts a triple partition of the human being: the material body dies on earth and a second death occurs on the moon, symbolizing the separation of mind from soul. This is a process where souls are purged of earthly pollution, and only those souls that had practiced reason manage to hold fast to the moon and gain initiation into her secrets. The moon is proportionally constituted of earthly matter and ether, and thus it is both a star and an earth. It is larger than the earth, and its appearance as smaller than the earth’s shadow is due to her hastening her “motion in order that she may quickly pass through the gloomy place bearing away [the souls] of the good which cry out and urge her on because when they are in the shadow they no longer catch the sound of the harmony of heaven.”77 Sulla ends his tale of the stranger’s story with a peculiar comment: “This, said Sulla, I heard the stranger relate; and he had the account, as he said himself, from the chamberlains and servitors of Cronus. You and your companions, Lamprias, may make what you will of the tale.”78 This is not the kind of ironic remark to be found in Lucian’s True History. The aim of Sulla’s comment is to urge Lamprias and his companions to do something with the story. From the opening sentence of the text to this last comment, Plutarch contemplates the

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moral meaning of myth as the object of true philosophical discourse.79 One cannot inquire if the myth is true or false, but can gather that only by allowing the intellect to rule one’s passions can true happiness be acquired. Only the intellect can gain immortality and return to its solar origin, while bodily desire and emotional passion either remain buried in the material earth or hang desperately to the moon. These two treatises gave Kepler the general framework of his own narrative on traveling to the moon, and assisted him in shaping his student dissertation on lunar astronomy into literary form. Plutarch, specifically, supplied him with additional details of ancient theories of physical astronomy, critically portraying the different alternatives.80 Kepler saw this discussion as a solid basis for a beginner’s guide to astronomy, and even suggested this treatise as a textbook to his friend Bernegger. He cites extensively from Plutarch in the relevant chapters of his Optica pars astronomiae, on the moon and how it reflects the sun’s light. In Somnium, Kepler alludes to these texts in several ways. Lucian’s treatise, because of its satiric nature, plays only a minor role in Kepler’s text. It supplies, however, the formula for the combination of satire and philosophical argumentation. Lucian’s celebration of commonsense empiricism is criticized throughout the Somnium, and Kepler emphasizes, in turn, the role of rational-​ geometrical procedures in the acquisition of knowledge. At the end of Somnium, Lucian’s satire is completely inverted. Leaving the Isle of Punishment, Lucian and his companions land on the Isle of Dreams, where for thirty days and thirty nights they meet all sorts of dreams and fantasies, having a “fine time—sleeping! Then of a sudden a great thunder-clap came; we woke up, sprang out of bed.”81 A similar scene appears in Somnium: “When I had reached this point in my dream, a wind arose with the rattle of rain, disturbing my sleep. . . . I returned to myself and found my head really covered with the pillow and my body with the blankets.”82 In adding this comic detail, Kepler gives new meaning to Lucian’s sudden awakening. As Rosen remarks, the covered head signifies the observers of solar eclipses. Instead of waking up from vain dreams and empty phantasms, Kepler is waking up from scientific observation. Despite the fact it took place in a dream, this observation still vouches for the truth contained within the dream. Lucian’s conception of dreams as a source of deception is rejected by Kepler in favor of the covered head of the astronomer, who measures the motions of distant planets in the process of disclosing their causes. Kepler’s reading of Plutarch’s treatise proves to be more complex. Plutarch posited myth and divinely sent dreams as higher sources of knowledge than scientific reasoning in his epistemological hierarchy. Furthermore, Plutarch

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rejected mathematics as an instrument for the discovery of the laws of nature.83 In order to reverse this epistemological path, Kepler restructures Plutarch’s narrative. His own treatise begins with a dream and a superrational revelation, only to disclose that the daemon relates a scientific picture of the universe. As Kepler’s apparatus of footnotes reveals, the daemon’s knowledge is powered by rational scientific reasoning. The notes are a safeguard for speculation gone astray, an attempt to control the seventeenth-century reader’s tendency to look for the occult and fantastical. Kepler’s aim is to fortify the authorial position and fix the interpretational context, thus undermining any allegorical reading. He explains the names he chose for the different characters and places populating his story, describing how he came across these names in contingent contexts.84 In explaining the name “Levania,” ascribed to the land on the moon, Kepler notes that “[t]he moon in Hebrew is Lebana, or Levana. I could have called it Selenitis. But Hebrew words, which are less often heard by us, are recommended for occult arts by the greater aura of superstition attached to them.”85 Instead of a kabbalistic interpretation, searching for the original meaning of a Hebrew word and its poetic power, Kepler emphasizes the arbitrary nature of human language and its consensual nature. Furthermore, human languages express only human sensory experience of physical reality, not its essence.86 The other task of the apparatus of notes is to provide an astronomical validation of the daemon’s description of the moon. Just as Duracotus and his mother have to perform a magical rite in order to summon their daemon, so Kepler has to perform his magic in order to call forth the mathematical interpretation. As soon as the moon becomes crescent and Saturn enters the sign of the bull, Duracotus and his mother stand at a crossroads and evoke the daemon.87 In a parallel manner, Kepler describes a magical game he used to perform in Prague that evoked mathematics and astronomical observational techniques. This also is a magical ceremony. The corresponding feature in the teaching of astronomy is that the method is not in the least voluble or spontaneous. On the contrary, every prompt action requires repose, recollection of ideas, and set words. During those years in Prague I often carried out a special procedure in connection with a certain observation. Whenever men or women came together to watch me, first, while they were engaged in conversation, I used to hide myself from them in a nearby corner of the house, which had been chosen for this demonstration. I cut out the daylight, constructed a tiny window out of a very

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small opening, and hung a white sheet on the wall. Having finished these preparations, I called in the spectators. These were my ceremonies, these my rites. . . . In capital letters I wrote with chalk on a black board what I thought suited the spectators. The shape of the letters was backwards (behold the magical rite), as Hebrew is written. I hung this board with the letters upside down in the open air outside in the sunshine. As a result, what I had written was projected right side up on the white wall within. If a breeze disturbed the board outside, the letters inside wiggled to and fro on the wall in an irregular motion.88 Kepler identifies the enchanted crossroad in Duracotus’s tale with a specific location in his house, as well as with a specific position of the sun and the moon. This position “invites the astronomers to observe the latitude of its [the moon’s] limits.”89 The fun of the game increases when the optical principles underlying the magical appearances are revealed. “I used to tack on these very games, which the spectators enjoyed all the more for realizing that they were games.” This courtly amusement turns out to be the paradigm for the acquisition of knowledge. The visual pun, with its magical undertones, supplies Kepler with the method of applying optical instruments to astronomical observation of the world: “With this very rite (ha, how magically magical!), . . . we had observed a solar eclipse on October 2/12, 1605. You remember, O envoys from the Count Palatine of Neuburg, because you were present. For on the balcony of the pavilion in the emperor’s gardens we lacked a dark room. Therefore we covered our heads with our coats and kept out the daylight in that way.”90 In this anecdote, the magic of Duracotus’s mother is converted into science, and the significance of human letters is inverted. Attention is turned not to their apparent meaning but to the technique of observation. Kepler suggests optical instruments like the camera obscura as reliable vehicles of knowledge. From this point onwards, the irony and playfulness of the notes gradually disappear, and Kepler points to the “physical considerations” that underlie “the jocular explanation.”91 The daemon’s tale of lunar astronomy is told through mathematical analysis. The different features of the terrestrial-based universe are relative to the position of the spectator and “exist only in the imagination of the earth-dwellers. Hence, if we transfer the imagination to another sphere, everything must be understood in an altered form.”92 The motions of the heavens create diverse appearances for differently positioned spectators: “These appearances of their [the planets’ motion] own are mingled with the motions in which the earth and moon travel, as things look

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to the earth-dwellers and the moon-dwellers.”93 The analysis of these appearances serves Kepler in introducing the moon as an inverted, mirror-like image of the earth: “To those who are on the moon it does not seem to revolve, but they regard it as stationary, just as our earth seems stationary to us.” This leads the moon-dwellers to transfer their path in the heavens to earth: “However, this lunar path bending around the earth as though the latter were stationary in the center of the universe is transferred by the moon-dwellers, who imagine their own home to be stationary, to this earth, that is, their Volva.”94 These differences of perspective redefine all the basic concepts of astronomy. The poles, as the apparent stationary points around which the heavens seem to rotate, are different for the moon-dwellers. This is due to the fact that “the axis of the lunar globe is not parallel to the earth’s axis, but is always intersected at right angles by the line from the center of the earth. Consequently the moon’s axis is not directed toward those points toward which the earth’s axis is directed.”95 Since the altitude of the poles helps in determining the latitude of places on all the meridians, this difference in the position of the celestial poles means an alternative map of the universe. The astronomer takes into account these differences, together with the moon-dwellers’ belief that their “lunar plane and the ball of Volva hanging up high over it remain in one place,”96 in order to depict the way the universe is for the lunarians. However, the ability of the mathematician to disclose indirect experiences is not limited to this. The mathematician can further disclose the topography of the moon through the decipherment of obscure spots observed on the body of the moon: “For by absolutely certain principles of optics we prove that that variety of spots and light is connected with the roughness and smoothness of the surface: what is bright, is both high and hilly; what is dark, is both flat and low.”97 These tools supply Kepler with the length of the lunar day and night, with the possible physical influences the sun, the earth, and the other planets have on the moon, and with weather conditions on the different parts of the moon. Furthermore, knowledge of the topography of the moon enables Kepler to speculate as to other living conditions of the lunarians. These speculations come in the last notes, and Kepler emphasizes that they are not scientifically based—“[t]herefore, in this place, too, my prophesying breaks down”—and that these speculations are based on “pure reasoning, divorced from any visual evidence.”98 The evidence is taken partially from literary sources concerning different tribes in remote corners of the world, albeit it is embedded in the wellestablished astronomical and optical framework. This is in accordance with Kepler’s astrological project, which emphasized physical determination against

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mental dispositions.99 The analysis of the lunarians’ way of life is limited to “their bodies . . . their functions, breathing, hunger, thirst, waking, sleep, work, and rest.”100 Thus, there are hardly any speculations on their social organization and mores. Instead, the reader gets a grotesque and inverted picture of human life on earth: oversized creatures with rapid growth and short life who prefer to dive under water than to boil in the sun and are of a serpentine nature, growing out of pine-shaped objects. These images emphasize the treatise’s main point: the moon is an inverted picture of the earth. If for us the earth stands still and the universe revolves around it, for the moon-dwellers the moon stands still and the entire universe, together with the earth, revolves about it. These beliefs are a result of our immediate visual experience and its distortions: “Everyone screams that the motion of the heavenly bodies around the earth and the motionlessness of the earth are manifest to the eyes. To the eyes of the lunarians, I reply, it is manifest that our earth, their Volva, rotates, but their moon is motionless. If it be argued that the lunatic senses of my lunarian people are deceived, with equal right I answer that the terrestrial senses of the earth-dwellers are devoid of reason.”101 The false beliefs of the lunarians are a mirror image of the earth-dwellers’ false beliefs. In transferring the reader to the moon, Kepler forces the reader to invert these beliefs and to perceive the correct picture of the universe: the earth is not static in the center of the universe but revolves around the sun and around its own axis. This principle of inversion is strongly associated with the camera obscura. Kepler opens this section of notes with this instrument, and he ends his comments with it. The context of his last note is the problem of the diminution of the lunar disk during solar eclipses. As I noted at the beginning of this chapter, in his Ad Vitellionem of 1604 Kepler associates this astronomical problem of correct measurement with the shortcomings of the human sense of sight and the drawbacks of Tycho Brahe’s observational methodology. In 1606, however, this debate resurfaced in a disputation presided over by Kepler’s teacher Maestlin. In his Somnium, Kepler confutes these arguments. He points out that in measuring the resultant shadows and images, the observer should take into account the effects of the instrument of observation and the process of vision as essential factors in shaping them. The observed “extension beyond the body of the moon” is not an effect of the real physical shape of the moon, but a product of visual distortion, which spoils “the image of the visible object on the retina.” At different times of the day, “bright parts expand and encroach on the bordering dark region,” distorting the moon’s image. This, however, does not detract from the epistemological value of observation. The mathematician

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can calculate this distortion away, and in the end “this picture on the concave retina within the eye corresponds exactly and invertedly to the vision of the external object.”102 The stable geometric center, through which all lines pass from the corresponding points on the object and on its inverted picture, limits the mind’s ability to play with the sense-data and thus guarantees one’s visual knowledge.103 In a sense, Kepler’s retinal picture is a sort of a “serious joke”: though it represents the world upside down, it allows one to perceive the physical truth hidden in the world of appearances.104 Reading Kepler’s Somnium in light of the two notes on the camera obscura discloses the treatise itself as a camera obscura, where the daemon’s fantastic lunar astronomy is an inverted picture of the terrestrial worldview. The role of the notes is to supply the reader with the inversion principles, thus revealing the real structure of physical reality of the heavens. Furthermore, Kepler’s dream inverts Plutarch’s line of literary exposition. Plutarch’s treatise begins with a scientific debate that ends with a narration of a myth; Kepler’s Somnium begins with a fantastic tale that ends with a scientific description. This methodological inversion allows Kepler to rebuff Lucian’s skepticism. The optics of the camera obscura as well as of other instruments of observation supply accurate knowledge even of objects and phenomena not available to direct experience. The Somnium is a Copernican treatise that aims to convert the reader’s modes of vision and modes for the acquisition of knowledge. Observation of shadows through instruments is superior to direct and unaided sight, and scientific observation is superior to textual tradition. Somnium begins with the narrator as an avid humanist in search of a historical and textual meaning. It ends with the conversion of the narrator, his head covered with a pillow, into a scientific observer of the heavens. In order to overcome one’s immediate sense perception, the astronomer has to muster the power of the imagination, and so to prove Copernicanism; Kepler suggests an imaginary point of view on the moon that would upset our normal perception and allow the invisible motion of the earth to be observed.

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POSTSCRIPT “Making Nothing All”

Kepler’s Somnium is a dream that shatters the Renaissance celebration of perspective as a tool to transform opaque surfaces into transparent media in order to allow absolute and unobstructed visibility. This follows Alberti’s demand “to set [mathematical entities] up as visible” (sub aspectu rem positam esse volumus). In meeting this challenge, Alberti introduced two further requirements: initially, the visual field is made according to human measure so that “however small you paint the objects in a painting, they will seem large or small according to the size of any man in the picture.”1 Furthermore, the painter’s task is intimately connected with humanistic modes of reading and interpreting the historical textual traditions. It is a dream of perspicuity, of turning spiritual realities into an artificial extension of the immediate visual field, of celebrating what can be observed as a supreme criterion of knowledge. Expanding the field of sight was the hallmark of the Renaissance aesthetic endeavor, starting with Alberti and continuing through Leonardo da Vinci to Nicolas Cusanus. Notwithstanding their different occupations, aspirations, and philosophical sensitivities, they all sought to put objects of inquiry in front of one’s eyes as a condition for their knowability. The core of this Renaissance dream of complete transparency consists in locating an exact point of view or in construing a subtle contraption to view directly the ideal infrastructure of the world or to glimpse the Godhead itself. Alberti’s theoretical outline of artificial perspective seeks to determine the position of the viewer in order to grasp the painting as the geometrical section of the pyramid of sight; Cusanus’s beryl lens provides the observer with a concrete

position from which to grasp God visually as both the absolute infinite and the absolute minimum; anamorphic painting construes a secret angle that reveals the hidden meaning that lies underneath worldly appearances (“secret endroit qui descouvre les magnificences occultes à l’apparence commune”).2 An anxiety accompanied this aspiration for transparency, however. In disrobing the opaque material layers that veil the world, nothing will be revealed but more appearances, brute force, and chance occurrences. On the horizons of such Renaissance aspirations for transparency appears a Machiavellian world of masques and deceptive façades, and no anamorphic playful skill can locate the position from which its true meaning can be unraveled. Instead of clarity of sight, Leonardo da Vinci, for instance, portrays a nightmarish world of betrayal, plots, and conspiracies that entraps the observer in an endless labyrinth: The lord of the manor must be able to go through the entire fortress, including the upper, the middle and the lower parts, using tunnels and underground passages, which shall be disposed in such a way that none of them could be used to reach the dwelling of the lord, without his agreement. And through these ways using portcullis and other devices he must be able to imprison in their residence all those of his retinue, who may plot against him, and may close or open the door of the main entrance and the relief route. And this danger is even greater than the enemy itself because those on the inside have greater opportunity to do harm than the enemy who is shut out.3 Renaissance ideals of visibility, embedded in the techniques of perspective in painting, turned out to be ambiguous, exposing undercurrents of illusion and visual distortions that misled human ocular experience to believe what is not. In following the ideals of visibility, artificial perspective blurred the threshold between the visible world and the world of spiritual and abstract entities. Its success in creating optically convincing images further blurred the demarcation line between perceptions imprinted directly on the sense organs and mediated images associated with artificiality and fabricated imaginations. This success contributed significantly to the growing doubt in the sixteenth-century European world of letters and the arts concerning the ability of the human eye to observe accurately the hidden recesses of the physical world. In Leonardo da Vinci’s claustrophobic manor of machinations and counterschemes, there is no way for the innocent observer to disentangle the complex system of suspicions and deceptions. In a world made of mere

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ephemeral appearances, there is also no way to advance beyond them to an open vista of reasoned causality and eternal truth. It is in this context that one may understand Copernicus’s phrasing: “And in fact, [the evidence] by which natural philosophers attempt so very hard to confirm the immobility of the earth depends for the most part upon appearances. All [their evidence] falls apart here in the first place since we overthrow the immobility of the earth also by means of an appearance.”4 If there are only appearances in front of the observer, then Copernicus seems to suggest that there is no use in contriving a position within the system from which to decipher it, but instead a new and fictive viewpoint outside the world must be created in order to discern its true motions. This suggests a new poetics, not merely of new modes of literary inventiveness but, as Sir Philip Sidney claims, the poet’s craft of making “things either better than Nature bringeth forth, or, quite anew, forms such as never were in nature. . . . So as he goeth hand in hand with Nature, not enclosed within the narrow warrant of her gifts, but freely ranging only within the zodiac of his own wit.”5 Such a notion of poetic novelty that constructs a wholly new platform for observing the universe raises an immediate question concerning the epistemological status of observations and their function in corroborating or refuting a theoretical statement. If mathematical reasoning can construe an imaginary point of view for observing cosmic order, a question begs to be asked: how is this viewpoint related to observational data acquired from a terrestrial position? In abandoning the Renaissance ideal of perspicuity, Kepler mobilizes his optics to establish observation on radically different grounds, thus answering this question by accommodating visual experience to a world where sight and perception have lost the obvious primacy assumed by the Aristotelian tradition. In rejecting the medieval ideal of direct visual perception in broad daylight through a transparent medium, as well as Alberti’s framed and transparent window looking to “a putative substitute world,”6 Kepler’s observer deciphers and molds a picture of reality out of a play of flickering light and shadows on the screen within a dark room, a play of refractions, inversions, and displacements. And instead of the humanist aspiration to read through visual experience, Kepler’s optics disentangles his observation from its textual inheritance. Reducing visual experience to artificial specters enables Kepler to relegate visual qualities to a secondary role; it also enables him to investigate physical reality as fundamentally composed of insubstantial skeletal geometrical figures and their motions. In construing physical phenomena as a theater of shadows and light, Kepler measures the incorporeal and invisible force that governs the

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motions and order of the universe. The camera obscura’s filtering and molding of visual experience is crucial to Kepler’s attempt to capture the invisible causes that govern the universe. Its ephemeral imaging points beyond human language and human senses, posited in the domain of the imagination. Kepler’s optical dream is to tame and harness the imagination through artificial means. He does this in order to direct the mind’s eye to know those bare and imaginary mathematical characters and how they form the natural properties and paths of the corporeal world. The Keplerian observer, like the tears of the lover in John Donne’s “A Valediction: of Weeping,” makes the world out of nothing: On a round ball A workman that hath copies by, can lay An Europe, Afrique, and an Asia, And quickly make that, which was nothing, All, So doth each teare, Which thee doth weare, A globe, yea world by that impression grow[.]

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NOTES

In the notes, KGW refers to Kepler, Gesammelte Werke.

introduction





1. Kepler, Ad Vitellionem. 2. For the historical development of the camera obscura, see Gernsheim and Gernsheim, History of Photography; Hammond, Camera Obscura. 3. Gianna Pomata suggests that observation as a novel epistemic genre emerged in the early sixteenth century, replacing experience with an “emphasis on seriality, mathematization . . . exact and calibrated measurement . . . clear-cut distinctions between direct and indirect experience, and the separation of observation from theory” (Pomata, “Observation Rising,” 69). My book goes beyond this precise and perceptive historical analysis to point to Kepler’s reformulation of optics, which radically transformed the practice of observation into artificiosa observatione, thus exposing the inner epistemic tensions that shaped this new practice and establishing a new visual economy as a necessary prerequisite for his new astronomy. 4. Westman, Copernican Question; Guddo, Copernicus and the











Aristotelian Tradition. The question of visibility did not arise in the debate that ensued in these major historical analyses: Barker, “Goddu’s Copernicus”; Heilbron, “Robert Westman on Galileo”; Swerdlow, “Copernicus and Astrology”; Westman, “Copernican Question Revisited.” 5. “Getting beyond the threshold of perception” is Gerard Passanante’s phrase in Passanante, Lucretian Renaissance, 90, where Lucretian poetics enables the reader “to move from the figure of dust colliding in a ray of sunlight to an unseen world of atoms.” 6. Kepler, Mysterium cosmographicum. 7. See, for instance, Caspar, Kepler, 116–21, and Voelkel, Composition of Kepler’s “Astronomia Nova,” 82–129. 8. Voelkel, Composition of Kepler’s “Astronomia Nova,” 137. 9. Kepler to Herwart von Hohenburg, [November 12, 1602], KGW, vol. 15, no. 232:108–11. 10. For instance Caspar, Kepler, 142–43, or Voelkel, Johannes Kepler, 60, who ascribes it to “Kepler’s legendary inability to focus on one problem at a time [getting] the better of him.” 11. Stephen M. Straker asserts that “[s]timulated by certain optical problems raised in the pursuit of accurate astronomical data—especially those





















concerned with the quantitative observation of solar eclipses and with the refraction of light in the atmosphere—Kepler undertook a systematic study of the optics known in his day.” “Kepler’s Optics,” ii. See also Straker, “Kepler, Tycho”; and Lindberg, “Laying the Foundations.” 12. Risner, Opticae thesaurus. 13. Crombie, “Mechanistic Hypothesis,” “Science and the Arts in the Renaissance,” and “Expectation, Modelling and Assent” (parts 1 and 2); Straker, “Eye Made ‘Other.’” 14. Lindberg, “Alhazen’s Theory of Vision,” “Theory of Pinhole Images,” Theories of Vision, and “Laying the Foundations.” 15. Lindberg, “Genesis of Kepler’s Theory of Light.” 16. Martens, Kepler’s Philosophy, 34; Voelkel, Composition of Kepler’s “Astronomia Nova,” 74. 17. Field, Kepler’s Geometrical Cosmology, 188; Field, “Kepler’s Harmony of the World.” 18. Donahue, “Kepler as a Reader of Aristotle”; Jardine and Segonds, “Kepler as Reader and Translator of Aristotle.” 19. To Crombie and Straker, mentioned above, one should add Dijksterhuis, Mechanization of the World Picture, 310. 20. Boner, Kepler’s Cosmological Synthesis. 21. Rabin, “Was Kepler’s Species Immateriata Substantial?,” 54. 22. Smith, “Getting the Big Picture”; Tachau, “The Problem of the Species in Medio”; Tachau, Vision and Certitude. 23. Phillips, “John Wyclif and the Optics of the Eucharist”; Denery, Seeing and Being Seen. 24. Smith, “What Is the History of Medieval Optics Really About?,” 194. See also Smith’s comprehensive account in From Sight to Light. 25. Alpers, Art of Describing, 33–71.

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MEASURING SHADOWS Notes to Pages 4–20















26. Ibid., 71. 27. Crary, Techniques of the Observer. 28. Clark, Vanities of the Eye; for the cultural critique of the primacy of sight and of visual metaphors in Western culture, see Jay, Downcast Eyes; Levin, Modernity and the Hegemony of Vision; Levin, Sites of Vision. 29. Cave, “Mimesis of Reading,” 163. 30. Rose, Italian Renaissance of Mathematics, 178. 31. For Kepler’s own estimation of Copernicus’s indebtedness to Ptolemy, cf. Kepler, KGW, 1:50 and 3:141. For a modern reassessment of this estimation, cf. Price, “Contra— Copernicus,” 215. 32. See, e.g., Kepler’s emblematic depiction of the history of astronomy on the frontispiece of the Rudolfine Tables and its interpretation in Jardine, Birth of History and Philosophy of Science, 287ff. 33. Smith, “What Is the History of Medieval Optics Really About?” 34. Pecham, John Pecham and the Science of Optics, 61. 35. The best summary of medieval theories of vision is still Lindberg, Theories of Vision; see also Smith, “Getting the Big Picture.” 36. See, e.g., Jean Pena’s complaint about the meager contemporary interest in optics in his preface, “De usu optics,” to the 1557 edition of Euclid’s Optica et catoptrica. Pena, however, considered optics the science which “like the sun, throws its light on all the other sciences” (Quae solis instar, reliquis lucem impertiat). Ramus, in Ramus and Taleus, Collectanae, 140. 37. Frangenberg, “Perspectivist Aristotelianism”; Lindberg, “Optics in Sixteenth-Century Italy.” 38. Kepler’s “audacious association of terms” in his titles is also noted in his Epitome astronomiae Copernicanae; see Pantin, “Kepler’s Epitome.”













39. Euclid, L’optique et la catoptrique, propositions III and V, 3–4. 40. Plato, Republic 7.529c–d, in Collected Dialogues, 761. 41. See also Lloyd, “Plato as Natural Scientist” and “Plato on Mathematics.” 42. Aristotle, On the Parts of Animals (De partibus animalium), 1.5.645a21– 645b6, in Complete Works, 1:1003–4. 43. Aristotle, Metaphysics 3.2.995a15–17, in Complete Works, 2:1572. 44. Aristotle, Metaphysics 3.2.997b33– 998a6, in Complete Works, 2:1576. 45. For the different interpretations of the epistemological status of astronomy, see, e.g., Mueller, “Physics and Astronomy”; Lloyd, “Saving the Appearances”; Jardine, Birth of History and Philosophy of Science, 225–57; Jardine, “Epistemology of the Sciences”; Barker and Goldstein, “Realism and Instrumentalism.” 46. John Chrysostom, St. John Chrysostom: Commentaries on the Sages, 182. 47. Sirach, 3:21–25, The Apocrypha and Pseudepigrapha, 1:324. These phrases are quoted all through the Middle Ages and the Renaissance in setting the limits of human understanding to the earthly realm. See, e.g., Petrarch, “On His Own Ignorance and That of Many Others,” in Cassirer, Kristeller, and Randall, Renaissance Philosophy of Man, 76. 48. Mishnah Hagigah 2.1, in The Mishnah, 213. 49. Plutarch, Περι Πολυπραγμοσυνης (On minding your own business), in Moralia 515b–523b; Apuleius, Metamorphoses. See also Walsh, “Rights and Wrongs of Curiosity.” 50. Lucian, Icaromenippus, or the SkyMan. In Lucian, 2:277–79. 51. For further discussion of this attitude, see Gal and Chen-Morris, Baroque Science, chap. 2.













52. Oresme, De proportionibus proportionum, pt. II, proposition XIX, 427. 53. Ibid., proposition XX, 427–29. 54. Erasmus, Praise of Folly, 151. 55. “Simpliciter itaque libri seve subiectum sive materia est humanum genus, quod adeo stultum est, ut multa quaerat et conetur suis studiis, quae tamen assequi non potest aut si assequitur, non tamen fruitur sed cum dolore et camno possidet culpa non rerum sed stultissimorum affectum.” Luther, Annotationes in Ecclesiasten, 12; 10. 56. Quoted in Hattaway, “Paradoxes of Solomon,” 521. 57. Barker and Goldstein, “Realism and Instrumentalism,” 248. 58. Ibid., quoting Nicodemus Frischlin, De astronomicae artis (1586), 41, translated in Jardine, “Epistemology of the Sciences,” 700. 59. Quoted and translated in Jardine, Birth of History and Philosophy of Science, 41. 60. Mulerius, Institutionum astronomicarum libri duo, 14–15, quoted in Hallyn, The Poetic Structure of the World, 153. 61. Bruno, Ash Wednesday Supper, 139–40. 62. Ibid. 63. Ibid., 88–90. 64. Sanchez of Salamanca, Commentaria in Alciati Emblemata, 242v–243. For the general theme of Icarus, see Ginzburg, “High and Low.” 65. Luther, Annotationes in Ecclesiasten, 18. 66. Ibid., 9. 67. Ibid., 10. 68. Kepler to Maestlin, June 1/11, 1598, in KGW, 13:225, and see also Epitome astronomiae Copernicanae for his definition of the astronomer’s role of placing before the eyes the form of the universe: KGW, 7:25.

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69. For the camera obscura’s cultural role as an organizing metaphor in the seventeenth and eighteenth centuries, see Crary, Techniques of the Observer, 25–66.







1. Kepler took the graphic format of his publications most seriously, as he commented in a letter to Maestlin on the publication of his Mysterium cosmographicum: the titles of a treatise are not “to deceive the community of mathematicians.” Quoted in Voelkel, Composition of Kepler’s “Astronomia Nova,” 217. On the evolution of layouts of title pages, see Proot, “Converging Design Paradigms.” 2. One may compare this inversion to a prevalent pictorial style in early modern Europe from Pieter Aertsen’s Butcher’s Stall with the Flight into Egypt (1551) and Peter Bruegel the Elder’s Landscape with the Fall of Icarus (1567) to Diego Velázquez’s Christ in the House of Martha and Mary (1618), where through the inversion of background and the painting’s subject the viewer is invited to probe the borderlines between artificial and authentic, between daily tangible experience and the spiritual and incorporeal significance, presenting the painting itself as a transitional ground between these two realms. See the extensive and brilliant analysis of such bifurcated pictures in Stoichita, Self-Aware Image. 3. Verville, L’Histoire veritable, fol. 2, translated in Klossowski de Rola, Golden Game, 28. 4. Vitam pro fama minuo, pro Nomine sensum: Disce anima vtilius, ne moriare, (my translation)

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chapter 1



MEASURING SHADOWS Notes to Pages 21–36











5. Kepler, KGW, 2:14, translated in Kepler, Optics, 13. 6. Aristotle, Posterior Analytics 1.7.75a38, in Complete Works, 1:122. 7. Aristotle, Physics 2.2.194a7, in Complete Works, 1:331. 8. For extensive discussions of this theme, see Laird, “Robert Grosseteste on the Intermediate Sciences” and “Galileo and the Mixed Sciences.” See also Ribeiro do Nascimento, “Le statut epistemologique”; McKirahan, “Aristotle’s Subordinate Sciences”; Livesey, “Metabasis”; Livesey, “Oxford Calculatores”; Livesey, Theology and Science in the Fourteenth Century. 9. For Zabarella, see especially Mikkeli, “Foundation of an Autonomous Natural Philosophy” and Aristotelian Response to Renaissance Humanism. 10. Schegk, De demonstratione, fol. 147, quoted in Kusukawa, “Lutheran Uses of Aristotle,” 177. 11. Schegk, De demonstratione, fol. 451, quoted in Kusukawa, “Lutheran Uses of Aristotle,” 178–79. 12. Maestlin, De astronomiae principalibus, fol. A2v: “Ipsam ergo compositam esse patet. Physicae enim propter obiectum seu materiam subiacet: Mathematica vero secretior, videlicet Geometria & Arithmetica suppeditant ei demonstrationes velut formam. Apparentia enim motuum non physicis, sed mathematicis rationibus demonstrantur.” Quoted in Methuen, Kepler’s Tübingen, 192. 13. Kepler, KGW, 2:8 (my translation). 14. Kepler borrows this identification of arithmetic and geometry with the Platonic wings of the soul from Melanchthon, “Praefatio in arithmeticen Joachimi Rhetici (1536),” in Opera quae supersunt omnia, 11:288: “Sunt igitur alae mentis humanae, Arithmetica et Geometria.” Another possible source is Digges’s preface to his treatise on the new star of 1572,









where he states, “whoever wears these Platonic or—to use a more accurate expression—Mathematical Wings and heads upwards into the ethereal realm, leaving behind entirely the elemental regions, will see that [the star] is much further away than the place of the comets.” Digges, Alae, sigs. A1r–B1r and B1v–B3r, quoted and discussed in Goulding, “Wings (or Stairs) to the Heavens,” 48. 15. See Hon, “On Kepler’s Awareness of the Problem of Experimental Error.” 16. “Neque animum explevi speculationibus Geometriae abstractae, picturis scilicet Kai twn ontwn kai mh ovtwn, in quibus pene solis hodie celeberrimi Geometrarum aetatem transigunt: sed Geometriam per ipsa expressa Mundi corpora, Creatoris vestigia cum sudore et anhelitu secutus, indagaui” (Kepler, KGW, 2:10) (my translation). 17. Kepler, KGW, 2:15 (my translation); for Kepler’s use of the term “species” and the question of how to translate this term, see Rabin, “Was Kepler’s Species Immateriata Substantial?,” 49–50. 18. Grosseteste, “De lineis angulis et figuris,” in Die Philosophischen Werke des Robert Grosseteste, 60. 19. Kepler, KGW, 2:15 (my translation). 20. Ibid., 2:16, translated in Kepler, Optics, 15. 21. Kepler, KGW, 2:16, translated in Kepler, Optics, 15. 22. “Deum ordinasse facultates quasdam animales in his Terris, mentis quodammodo participes ad percipiendas geometricas pulchritudines, seu etiam quantitates discretas, quibus perceptis ipsae essent operosae in exsudandis vaporibus. Haec est igitur illa peculiaris Dei ordinatio, facultates illae sunt imagines Dei, percipientes geometricam pulchritudinem, ut Deum” (Kepler to Fabricius, November 10,











1608, in Ioannis Kepleri Astronomi Opera Omnia, 2:357). 23. Mens: Qua licet expediam; faueat modo fama loquenti; Mortales chartis perpetuabo meis. Hic etiam referam, quae pro me damna tulistis: Hicque suum et naeuis irradiabo iubar. (Kepler, Ad Vitellionem, Epigramma Avthoris) 24. [A]tqui si vitiis mediocribus ac mea paucis mendosa est natura, alioqui recta, velut si egregio inspersos reprendas corpore naevos. (Horace, Satyrarum libri 1.6.65–67) 25. After 1613 and the debate over sun spots between Galileo and Scheiner, Kepler’s reinterpretation of Horace’s lines acquired fresh astronomical poignancy. Emanuele Tesauro used this allusion in the frontispiece of the 1670 edition of Il Cannocchiale Aristotelico, where poesis in the guise of a seated woman observes the spotted sun through the telescope following Aristotle’s instructions. See also Johnson, Hyperboles, 98. 26. Kepler, KGW, 2:16 (my translation). 27. Ibid., translated in Kepler, Optics, 16. 28. Kepler, KGW, 2:17, translated in Kepler, Optics, 16. 29. Kepler, KGW, 2:17, translated in Kepler, Optics, 16. 30. Shakespeare, Midsummer Night’s Dream, 3.1.56–58. 31. Ibid., 5.1.425–28. 32. For a different interpretation of Shakespearean shadows as creatures from purgatory and the Shakespearean stage as a cultural compensation for and replacement of the afterlife abolished in Protestant theology, see Greenblatt, Hamlet in Purgatory, 161–64. 33. I follow here Caspar, Kepler, 109–11. 34. For a more detailed account of Tycho’s lunar theory, see Thoren,

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Lord of Uraniborg, 317–33, and app. 3, “The Tychonic Lunar Theory,” 486–99. As for Kepler’s final formulation of the lunar theory in his Epitome of Copernican Astronomy, see Stephenson, Kepler’s Physical Astronomy, 175–201. 35. Kepler to Archduke Ferdinand, July 1600, KGW, 14, no. 166, 22, quoted in Caspar, Kepler, 111. 36. Kepler to Herwart von Hohenburg, September 14, 1599, KGW, 14, no. 134, 19–23, quoted in Caspar, Kepler, 109. 37. On Kepler’s pinhole camera and its use, see Hon and Zik, “Kepler’s Optical Part of Astronomy (1604).” 38. For the repercussions of this radical move, see Gal and Chen-Morris, “Baroque Optics.” 39. Kepler, KGW, 2:17. See also the discussion in Dupré, “Optics Without Hypotheses.” 40. Kepler, KGW, 2:17. Another apologetic presentation of this subject about the “Nature of Light” is in the “Dedicatory Letter”: “Cumque fuerit tota visus ratio ex integro explicanda, quae refractionibus perficitur, et simulachris rerum visarum, et coloribus: non debet cuiquam videri mirum si . . . capite I. in naturam lucis et colorum, et alibi in alia paulo longius sum digressus. Nam etiamsi ad Astronomiam nihil conducerent hae materia, per se tamen dignae cognitu sunt” (Kepler, KGW, 2:9, my emphasis). 41. Kepler, KGW, vol. 13, letter no. 23, 38. 42. In the preface to his first book, the Mysterium cosmographicum of 1596, Kepler summarizes the motivating questions for his investigations: “Above all there were three things of which I diligently sought the reasons why they were so, and not otherwise: the number, size, and motion of the spheres.” Kepler, KGW, 1:9; quoted in Stephenson, Kepler’s Physical Astronomy, 8.

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43. Stephenson argues that “[s]ome sort of moving virtue varied in a mathematical defined way, but Kepler had neither explanation nor analogy for how or why it varied in this way. He was not trying to understand the cause responsible for the different speeds, nor to relate them to any physical process, but only to find a pattern in them” (Stephenson, Kepler’s Physical Astronomy, 9). Overlooking Kepler’s earlier letters, Stephenson misses Kepler’s clear analogy between this virtue and the movement of light. 44. Kepler, KGW, 13, no. 23, 38; see also Stephenson, Kepler’s Physical Astronomy, 12–18. 45. Kepler, Mysterium cosmographicum, 199. 46. See, for instance, his letter to David Fabricius, November 11, 1608, quoted above, note 22. 47. Kepler, KGW, 14, letter no. 132, 46. 48. That is, the possibility of constructing these polygons with rule and compass: the three-, four-, five-sided are “knowable,” but the seven-, nine-, eleven-sided are not. This corresponds to Kepler’s later definition in the Harmonices mundi: “To know in geometry is to measure in terms of some known measure. In this matter of inscribing figures in circles the known quantity is the diameter of the circle.” 49. Kepler, KGW, 14:47. 50. Ibid., 49. 51. Ibid. 52. Ibid., 50–51. 53. Ibid., 51. 54. Ibid., 52. 55. “Et jam tempus est ipsissimam Pythagoricam mundi harmoniam patefacere” (ibid., 51). For the way Kepler adapted Pythagorean philosophy to his version of Copernican astronomy, see Joost-Gaugier, Pythagoras and Renaissance Europe, 133–41.











56. Kepler, “Johannes Kepler’s On the More Certain Fundamentals of Astrology,” 95. 57. “For whatever partakes of matter, insofar as it partakes of it, is by nature cold” (ibid., 92). 58. Ibid., 93. 59. Ibid. 60. Ibid., 94. 61. As Field comments, the meaning of geometric and nongeometric angles is dependent on whether or not “the angle between them can be constructed ‘geometrically,’ i.e. using only a straight edge and compasses” (Field, “Lutheran Astrologer,” 250n47). 62. Kepler, “Johannes Kepler’s On the More Certain Fundamentals of Astrology,” 96–97. 63. Ibid., 97. 64. Ibid., 96. 65. Kepler, Mysterium cosmographicum, 167. 66. Ibid., 55. 67. Ibid.

chapter 2 1. Kepler, KGW, 2:18, translated in Kepler, Optics, 17. 2. Pecham, John Pecham and the Science of Optics, 62. 3. Optics was part of the mathematical sciences in the medieval university. In Oxford, according to the statutes of 1431, candidates for the degree of Bachelor of Arts had to study “Geometriam per duos anni terminus, videlicet librum Geometrie Euclidis, seu Alicen Vitulonemve in perspectivam.” Statuta antiqua Universitatis Oxoniensis, 234–35, cited in Beaujouan, “Motives and Opportunities,” 220–21. 4. Kepler, KGW, 2:18, translated in Kepler, Optics, 17. 5. Kepler, KGW, 2:18 (my translation). 6. Ibid., translated in Kepler, Optics, 18.



7. See Donahue’s comment on this passage in Kepler, Optics, 18n14. 8. Tesauro, Il Cannocchiale Aristotelico, 82–83, quoted in Gilman, Curious Perspective, 74. 9. Aristotle, Poetics 21.1457b, translated in Aristotle’s Theory of Poetry, 77. 10. Castiglione, Book of the Courtier, 80. 11. Puttenham, Arte of English Poesie, 128, 148–49. 12. KGW, 2:19. I have followed here Lindberg’s translation in “Genesis of Kepler’s Theory of Light,” 29–30. 13. KGW, 2:19 (my translation). 14. Ibid. 15. For the implications of this view, see Gal and Chen-Morris, “Archaeology of the Inverse Square Law” (parts 1 and 2). 16. See, e.g., Kepler’s comment on Nicholas of Cusa’s comparison between God and the curved line in Mysterium cosmographicum, 93. 17. KGW, 2:19–20 (my translation). 18. Kepler, Ad Vitellionem, 7, translated in Kepler, Optics, 20. 19. For the way Kepler grapples with the geometrical nature of light and later in his work reconsiders it as a material substance, see Rabin, “Was Kepler’s Species Immateriata Substantial?” 20. Brahe, Opera Omnia, 8:55, quoted and translated in Straker, “Kepler’s Optics,” 1:5. 21. See Jardine, Birth of History and Philosophy of Science, 9. 22. Pecham, John Pecham and the Science of Optics, 171. 23. Ibid., 159. 24. Ibid., 161. 25. Ibid., 171. 26. Aristotle, Problems (Problemata) 15.6.911b3–4, in Complete Works, 2:1417. See also Lindberg, “Theory of Pinhole Images.” As E. Broydrick Thro notes, Leonardo lists Aristotle’s Problemata in the record of his books (Madrid MS. II (fols. 2b–3a, ca. 1504). Thro, “Leonardo’s Early Work,” 23n12.

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27. Pecham, John Pecham and the Science of Optics, 70–71. See also Lindberg, “Laying the Foundations,” esp. 26–29. 28. Lindberg, “Theory of Pinhole Images,” 303. 29. For Leonardo’s investigations of pinhole images, see Thro, “Leonardo’s Early Work” and “Leonardo da Vinci and the Problem of the Pinhole Camera”; see also Veltman, “Leonardo and the Camera Obscura.” 30. Kepler, Ad Vitellionem, 37–38, translated in Kepler, Optics, 55. 31. Kepler, Ad Vitellionem, 37–38, translated in Kepler, Optics, 55. 32. Kepler, Ad Vitellionem, 38, translated in Kepler, Optics, 56. 33. Kepler, Ad Vitellionem, 38, translated in Kepler, Optics, 56. 34. The case discussed here is that lines of illumination, although arriving in the eye from a single luminous point (lucente puncto), appear to be parallel when the proportion to the base of origin is large enough. Donahue interprets “tum aequidistantibus ad sensum aequiparari” (Kepler, Ad Vitellionem, 40) as “to be regarded as equivalent to equidistant lines within the limits of sense perception” (Kepler, Optics, 57). 35. Kepler, Ad Vitellionem, 42, translated in Kepler, Optics, 59. 36. See also the discussion in Gal and Chen-Morris, “Baroque Optics,” esp. 192–94. 37. Kepler, Ad Vitellionem, 54–55, translated in Kepler, Optics, 70. 38. Kepler, Ad Vitellionem, 39, translated in Kepler, Optics, 56–57. 39. Kepler, Ad Vitellionem, 39, translated in Kepler, Optics, 57.

1. Kepler, Ad Vitellionem, 436. A similar operational definition of a ray reappears at the opening of Newton’s

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Optics: “The least Light or part of Light, which may be stopp’d alone without the rest of the Light, or propagated alone, or do or suffer any thing alone, which the rest of Light doth not or suffers not, I call a ray of light” (Opticks, bk. 1, def. 1, p. 2). 2. Kepler, Ad Vitellionem, 44, KGW, 52, translated in Optics, 61. 3. Kepler, Ad Vitellionem, 45, KGW, 53, translated in Optics, 62. 4. See Fried and Unguru, Apollonius of Perga’s “Conica.” 5. See Davis, “Systems of Conics in Kepler’s Work”; Field, “Two Mathematical Inventions.” 6. The sections of all these, regardless of kind, fall into five species. 7. Here I follow Klein, Greek Mathematical Thought, esp. 121–23. 8. See Plato, Republic 10.602c, in Collected Dialogues, 827; Aristotle, Metaphysics 1.5.986a22–26, in Complete Works, 2:1559 (Aristotle’s mention of the Pythagorean dichotomy of the straight and the curved). See also Aristotle, Posterior Analytics 1.4.73b19–24, in Complete Works, 1:119; Proclus, Commentary, 84–92. 9. Kepler, KGW, 2:90, translated in Kepler, Optics, 107. 10. Kepler, Optics, 107. 11. Ibid., 106, where Kepler presents his own analysis of the conics in the following way: “However, because the consideration of the section is difficult, because it is too little pursued, it would be good to say a few words about it in a mechanical, analogical, and popular vein. Geometers, be indulgent!” 12. Kepler, KGW, 2:92, translated in Kepler, Optics, 109. 13. Kepler, KGW, 2:92, translated in Kepler, Optics, 108. 14. Kepler, KGW, 2:92, translated in Kepler, Optics, 108.









15. Kepler’s analysis of conic sections is a story of a failure both to mathematically establish an a priori system and to apply it to find “the causes for the magnitude of refraction” (Kepler, KGW, 2:110, translated in Kepler, Optics, 93); see also Buchdahl, “Methodological Aspects.” Yet this failure, while delineating the limits of Kepler’s new science, exposes the essential features of his epistemology and scientific method: see Simon, Kepler astronome astrologue. 16. Kepler, KGW, 2:104, translated in Kepler, Optics, 123. 17. Klein, Greek Mathematical Thought, 120. 18. Ibid., 121. 19. The general framework for my argument is set in ibid., 121–23. See also Lachterman, Ethics of Geometry, 121, where he suggests that what characterizes modern episteme is the “equation of constructibility with the existence or objective reality of mathematical concepts.” 20. See Klein, Greek Mathematical Thought, 118–20. 21. Kepler, KGW, 2:61, translated in Kepler, Optics, 73. 22. Kepler, KGW, 2:62 (my translation). 23. Ibid. 24. See also Pecham, John Pecham and the Science of Optics, 91. 25. Kepler refers to Witelo’s bk. 5, proposition 36: Witelo, Witelonis Perspectivae liber quintus, 119. 26. Ibid., 119–20. 27. Kepler, KGW, 2:62. 28. Alhacen, De aspectibus, in Risner, Opticae thesaurus, 131. 29. Witelo, Witelonis Perspectivae liber quintus, 18 and 100. 30. Kepler, KGW, 2:63. 31. Aristotle, Physics 2.2.194a10–11, in Complete Works, 1:331. 32. Witelo, Witelonis Perspectivae liber secundus et liber tertius, 46 and 240.













33. Ibid., 46 and 241.This is in contrast to Aristotle’s dictum, “The mathematical sciences will not . . . be sciences of sensibles.” Aristotle, Metaphysics 13.3.1078a3–4, in Complete Works, 2:1704. 34. That this inverted understanding of the Aristotelian formula was still accepted in the early seventeenth century is seen from the opening lines of Scheiner’s preface to his Oculus: “Physici quam Optici circa visibilia, & organum visus versantur; modo tamen diverso. Geometria enim, teste Philosopho, l.2 Phys. t.20 de Physica linea considerat, sed non quatenus est Physici: Perspectiva autem mathematicam quidem lineam, sed non quatenus Physica est.” See also Dear, Discipline and Experience, 51–57. 35. Aristotle, Metaphysics 13.3.1078a22– 26, in Complete Works, 2:1704–5. 36. My interpretation of Aristotelian “abstraction” runs counter to Cleary, “On the Terminology of ‘Abstraction’ in Aristotle.” Cleary argues that the context of Aristotelian abstraction is that of a logical operation and not a mental process, as the Thomistic commentators would have it. While this is good enough for the sake of my argument, I still insist, and I think Cleary has to admit the same (see ibid., 36–44), that there is a psychological dimension to Aristotle’s use of the term “abstraction.” 37. Aristotle, On the Soul (De anima) 3.8.432a7–9, in Complete Works, 1:686–87. 38. Ibid., 3.3.428b18–26, in Complete Works, 681. 39. Ibid., 3.3.428b2–4, in Complete Works, 681. 40. Quoted and translated in Babbitt, “Nicole Oresme: The Limits of Imagination,” 63. 41. Kepler, KGW, 2:21. 42. Ibid., 20. 43. Ibid., 22.

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44. Although here Kepler separates his treatment of colors—i.e., the material effects—from the quantitative aspect, it is clear from the other propositions that these aspects are connected; see also Kepler’s letter to Maestlin, August 1599, discussed in chapter 1—there, Kepler tries to develop a theory of colors as depending on light’s angle of refraction. 45. Sabra, Theories of Light, 71–78. 46. Lindberg, Theories of Vision, 80. 47. Kepler, KGW, 2:25. 48. “Now we know . . . that nature acts along the shortest paths in everything she does. . . . Everything that is seen in any mirror is comprehended by sight along the shortest possible line” (Witelo, Witelonis Perspectivae liber quintus, 101). 49. Kepler, KGW, 2:71. 50. Ibid., 27–28 (my translation). 51. Ibid., 31. 52. In another proposition Kepler states, “Atqui haec opera non sunt formae consilio utentis aut finem respicientis, sed materiae suis Geometricis necessitatibus astrictae” (ibid., 71). 53. Ibid., 73. 54. Ibid., 38. 55. Ibid. 56. This analysis of the medieval theory of cognition follows Smith, “Getting the Big Picture,” and Tachau, Vision and Certitude. 57. Aristotle, On the Soul (De anima) 2.7.418b14–22, in Complete Works, 1:666. 58. Ibid., 3.7.431a15–19, in Complete Works, 1:685. 59. Aristotle, Metaphysics 1.1.980a25–27, in Complete Works, 2:1552. 60. Aristotle, On the Soul (De anima) 3.3.429a4–5, in Complete Works, 1:682. 61. Ibid., 3.3.428a11–12, in Complete Works, 1:680. 62. I follow here some suggestions made by Rosen, “Thought and Touch: A

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Note on Aristotle’s De Anima,” in Quarrel Between Philosophy and Poetry, 119–26. 63. Aristotle, On the Soul (De anima) 2.12.424a18–21, in Complete Works, 1:674. 64. Ibid., 3.12.434b23–27, in Complete Works, 1:691. 65. Ibid., 3.12.434b27–435a10, in Complete Works, 1:691. 66. Ibid., 3.8.432a1, in Complete Works, 1:686. 67. Ibid., 3.7.431b14–432a6, in Complete Works, 1:686. 68. Ibid., 2.11.423b4–7, in Complete Works, 1:673. 69. Ibid., 3.3.427b11–12; 428a11, in Complete Works, 1:680. 70. Aquinas, Scriptum super libros Sententiarum, 2:329–37. 71. Albertus Magnus, Commentarii in II Sententiarum, 241–42. 72. Albertus Magnus, De anima, lib. II, tract 3, cap. 15, in Opera Omnia, 7:122. 73. Roger Bacon, Roger Bacon’s Philosophy of Nature, 94. 74. Ibid., 180. 75. Ibid. 76. Oresme, Expositio, 209. 77. Ibid. 78. I use the text as it appears in McCluskey, “Nicole Oresme.” 79. Ibid., 135. 80. Ibid., 137. 81. Melanchthon, “Liber de Anima,” in Opera quae supersunt omnia, 13:109. 82. Ibid. In asserting that light is a quality, Melanchthon follows other sixteenth-century Aristotelian commentators, e.g., Suarez, Commentaria, 554. 83. Melanchthon, “Liber de Anima.” 84. Ibid., 111. 85. Ibid., 110. 86. Avicenna, the great Muslim philosopher, wrote that the eye is like a mirror and the “visible object is like the thing reflected in the mirror by the









mediation of air or another transparent body . . . if a mirror should possess a soul, it would see the image that is formed on it” (Avicenna, Le livre de science, 2:60). Albertus Magnus embraced Avicenna’s description and designated the eye a speculum animatum, asserting, “You know, however, that species is visible in the eye, the same as in a mirror, that is a clean and polished [surface] repeatedly receiving and representing the form, just as the thing” (De anima, 7:122), and see also Anzulewicz, De forma resultante in speculo. For Dante’s association of mirror and the eye, see, for instance, Akbari, Seeing Through the Veil, 130ff. 87. For a further elaboration of the cognitive premises of medieval optics, see Smith, “Getting the Big Picture.” 88. Kepler, KGW, 2:39 (my translation). 89. Ibid. 90. Ibid., 39–40. 91. Ibid., 40. 92. Ibid., 21. 93. Ibid. 94. Ibid., 23. 95. Ibid. 96. Ibid., 41. 97. Ibid. 98. Ibid., 41–42. 99. Ibid., 45. 100. Ibid. 101. See my examination of Kepler’s discussion of light as the image of the Trinity, above, chap. 2, “Geometrical Images for Common Notions.” 102. Kepler, KGW, 6:222, translated in Kepler, Harmony of the World, 303. 103. Kepler, KGW, 2:40. For further elaboration on this sentence and for the manner in which rays are associated with the motion of light, see above, “Kepler’s New Mathematics: From Abstraction to Representation.” 104. Kepler, KGW, 6:102, translated in Kepler, Harmony of the World, 143–44.







105. Kepler, KGW, 6:107, translated in Kepler, Harmony of the World, 150. As early as 1597, in a letter to Maestlin, Kepler remarked that geometrical entities are perceived by the mind just as the eye perceives colors, and the ears perceive sounds (KGW, 13:113). 106. Kepler, KGW, 6:101, translated in Kepler, Harmony of the World, 140. 107. Kepler, KGW, 6:233, translated in Kepler, Harmony of the World, 316. 108. Kepler, KGW, 6:232 (my translation). Aiton, Duncan, and Field suggest a somewhat different translation of the first sentence: Kepler, Harmony of the World, 315. 109. Kepler, KGW, 6:233, translated in Kepler, Harmony of the World, 316. 110. Kepler, KGW, 6:219, translated in Kepler, Harmony of the World, 299. 111. Kepler, KGW, 6:223, translated in Kepler, Harmony of the World, 304. 112. Kepler, KGW, 6:105, translated in Kepler, Harmony of the World, 146–47. 113. Kepler, KGW, 6:223, translated in Kepler, Harmony of the World, 304. 114. Kepler, KGW, 6:105, translated in Kepler, Harmony of the World, 147. 115. Kepler, KGW, 6:223, translated in Kepler, Harmony of the World, 303–4.

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1. Smith, Life and Letters of Sir Henry Wotton, 2:205–6. 2. Reeves, Painting the Heavens, describes several such exchanges and argues for a more elaborate effect of painters’ observation of the secondary light of the moon on the astronomers’ formulation of these problems. 3. Galilei, “Letters on the Sunspots,” in Discoveries and Opinions of Galileo, 127; and Galilei, “Assayer,” in Controversy on the Comets, 215. For an extensive analysis of Galileo’s involvement with the visual arts, see Panofsky,

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Galileo as a Critic of the Arts; Edgerton, “Galileo”; Edgerton, Mirror, the Window, and the Telescope, esp. 151–67; Lefèvre, “Galileo Engineer”; and Bredekamp, Galilei der Künstler. 4. For general discussions of the relationship of perspective and the mathematization of the scientific enterprise, see especially Kemp, Science of Art; Field, Invention of Infinity; and Camerota, La prospettiva del Rinascimento. 5. Ackerman, “Alberti’s Light,” 86. See also Federici-Vescovini, “La prospettiva del Brunelleschi.” 6. Crombie, “Science and the Arts in the Renaissance.” 7. My presentation of the medieval pyramid of vision is based on Alhacen (Ibn al-Haytham), De aspectibus. This text supplied the paradigm of optical theory in the Middle Ages to the Islamic world as well as to the Christian West. Cf. Lindberg, “Alhazen’s Theory of Vision,” and Smith, “What Is the History of Medieval Optics Really About?,” esp. 181–88. 8. Alhacen, De aspectibus, in Risner, Opticae thesaurus, 269; and also Lindberg’s explanation of Alhacen’s argument in Theories of Vision, 74–78. 9. “Quod est quolibet puncto cuiuslibet corporis colorati et illuminati cum quolibet lumine, exeunt lux et color super quamlibet lineam recta, quae poterit extendi ab illo puncto” (Alhacen, De aspectibus, in Risner, Opticae thesaurus, 10). 10. “In quolibet puncto superficiei visus transeunt in eodem tempore formae omnium punctorum, quae sunt in superficiebus omnium visibilium oppositorum in illo tempore: et forma unius puncti tantum transit recte per diaphanitatem tunicarumm visus: et est punctum, quod est apud extremitatem perpendicularis exeuntis ab illo puncto superficiei visus: et formae

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omnium punctorum reliquorumm refringuntur apod illud punctum superficiei visus, et transeunt per diaphonitatem tunicarum visus secundum lineas declinates ad superficie visus” (ibid., 9). 11. Alberti, On Painting, 40–41. My reading disputes and I think disproves such readings as that of Ackerman, “‘Alberti’s Light,’” 66. I prefer the older translation by Cecil Grayson (Battista, On Painting) over Rocco Sinisgalli’s new one (Il nuovo “De Pictura”) for stylistic reasons. Although Sinisgalli provides a more extensive and up-todate critical edition and convincing arguments that Alberti’s Italian version predated the Latin one, Grayson’s translation serves my purposes better. 12. Alberti, On Painting, 40–41. The verb recipi can be translated either as “to be drawn back,” which supports my interpretation better, or as “to be received.” The context points to Grayson’s translation (“going back”), since the rays that “go out towards the surfaces in front of them” (exeuntes . . . ad oppositam superficiem effluant) cannot originate from the visible surface and therefore must go back into the eye that transmitted them in the first place. 13. Ibid. 14. Ibid., 41. 15. Theisen, “Liber de visu.” Historians of art, like Panofsky and White, did point to Euclid as the source for Alberti’s optical ideas. Their disregard of the medieval perspectivist tradition, however, caused them also to disregard Alberti’s overlooking it as a germane historiographical problem. White, Birth and Rebirth of Pictorial Space; Panofsky, “Die Perspektive als ‘symbolische Form.’” 16. Theisen, “Liber de visu,” 62. 17. Ibid. 18. Alberti, On Painting, 42. Euclid, unlike Alberti, does not divide the rays













into “extrinsic,” “median,” and centric, and does not emphasize the importance of the perpendicular ray. These modifications and qualifications can be ascribed to Alberti’s need to adjust an abstract Euclidean geometrical model to the art of painting. 19. Gouwens, “Perceiving the Past.” 20. Wright, “Alberti’s De pictura”; Baxandall, Giotto and the Orators, 121– 39. See also Katz, Leon Battista Alberti; Gilbert, “Antique Frameworks”; Lee, “Ut pictura poesis”; Spencer, “Ut rhetorica pictura.” 21. Baxandall, Giotto and the Orators, 121–39. See also Puttfarken, Discovery of Pictorial Composition. 22. Alberti, On Painting, 54. 23. Alberti’s embracing of proportio owes much to Quintilian but is also well situated within the major trends of fifteenth-century humanistic thought. Cf. Nicholas of Cusa, Nicholas of Cusa on Learned Ignorance, 50; or the discussion of Lorenzo Valla’s notion of proportion in Trinkhaus, “Humanism and Greek Sophism: Protagoras in the Renaissance,” in Scope of Renaissance Humanism, 169–91. 24. The most poignant expression of Alberti’s criticism of human affairs is expressed in his satirical play Momus. For Alberti not as a critic of Renaissance humanism but as the fabricator and producer of different personae and social roles a humanist could pursue, see Grafton, Leon Battista Alberti. 25. Garin, “Il pensiero di Leon Battista Alberti” and “Leon Battista Alberti e il mondo dei morti.” See also Whitfield’s critique of Garin in “Momus and the Language of Irony” and Kircher, “Dead Souls.” 26. Alberti, Use and Abuse of Books, 18. 27. Ibid., 18–19. 28. Jarzombek, On Leon Battista Alberti, 63–64.







29. The original collection has come down to us in two sections, one dedicated to Alberti’s friend the mathematician and doctor Paolo Toscanelli, the other to Leonardo Bruni and Poggio Bracciolini. See Marsh’s introduction to Alberti, Dinner Pieces. 30. Alberti, Dinner Pieces, 68. 31. Alberti, Opera inedita, 233–34, translated in Alberti, Dinner Pieces, 216. 32. Alberti, On Painting, 36. 33. See Greenstein, “On Alberti’s ‘Sign.’” 34. Alberti, On Painting, 49. 35. Ibid., 68. 36. Ibid., 99. 37. Ibid. 38. Ibid., 102. 39. Leonardo da Vinci, Leonardo da Vinci’s Paragone, 195–97. 40. Ibid., 187. 41. Ibid., 197. 42. Ibid., 199. 43. Ibid., 185–87. 44. Kemp, Leonardo da Vinci, 298. 45. For further discussion of the prevailing tension in Leonardo’s natural philosophy between mathematical order and the mutability of nature, see Gal and Chen-Morris, “Nature’s Drawing,” esp. 433–35; Macagno, “Transformation Geometry in the Manuscripts of Leonardo da Vinci”; and Jeanneret, Perpetual Motion, 50–70. 46. For instance, in the Codex Trivulzianus, fol. 34v, Leonardo comments, “A point is not part of a line. The water of a river, which you touch, is the last of that which has passed and the first of that to come. So are things in the present” (Leonardo da Vinci, Sublimations of Leonardo da Vinci, 241). For Leonardo’s fascination with the motions of water, see Macagno, “Analogies in Leonardo’s Studies of Flow Phenomena” and “Mechanics of Fluids in the Madrid Codices.”

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47. Kemp, Leonardo da Vinci, 338; see also Zwijnenberg, Writings and Drawings of Leonardo da Vinci. 48. Kemp, Leonardo da Vinci, 302–3. 49. I follow here Kemp’s assertion that after 1508 Leonardo’s attitude toward artificial perspective and the power of painting to depict reality had changed dramatically: “His late attitude to painter’s perspective of the orthodox kind cannot be judged so much from what he said about it after 1508 but from the fact that he said very little about it at all. . . . [H]e certainly was not disposed to conduct the kind of extended expositions of linear perspective, which had featured so prominently in his Milanese notebooks. . . . From 1508 onwards he was engrossed in studying the infinite variables of the visible world, its illusions, ambiguities, deceptions and fleeting subtleties—all of which disrupted the linear stability of artificial perspective” (ibid., 332). 50. The refraction of light at the eye’s surface creates a wider visual angle than that assumed by Alberti’s perspective, with the effect that “the eye judges the size of an object as being larger than that which is shown in the painter’s perspective” (quoted in ibid., 331). 51. See also Fehrenbach, “Der oszillierende Blick”; and Bell, “Sfumato and Acuity in Painting.” 52. Leonardo da Vinci, Leonardo on the Human Body, 332–34. 53. See, e.g., Clark, Vanities of the Eye, esp. 55–56, 83–90. 54. Calvin, Institutes of the Christian Religion 1.11.12; and see also Spelman, “Calvin and the Arts.” 55. Elkins, Poetics of Perspective, 166. On the association of geometry in general and perspective in particular with melancholic humors, see Klibansky, Panofsky, and Saxl, Saturn and

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Melancholy, 317–38; Mazzio, “Three Dimensional Self.” 56. Elkins, Poetics of Perspective, 167. 57. Lomazzo, Tracte, 185. 58. Quoted and translated in Summers, Michelangelo and the Language of Art, 52. Similar opinions and assertion of the illusionary nature of paintings in general and perspectival paintings in particular can be found in many sixteenth-century writers; cf. Pino’s comment that painting “fa apparere quello che non e” (Barocchi, Trattati d’arte, 1:115), and Hollanda’s comments on Michaelangelo, De la pintura antigua, 22. 59. Hollanda, Four Dialogues on Painting, 73–74. 60. Agrippa von Nettesheim, Of the vanitie and vncertaintie of artes and sciences, 35. 61. Cited and translated in Maiorino, Portrait of Eccentricity, 62–63. 62. See the extensive discussion of this artistic self-awareness in Stoichita, Self-Aware Image. 63. David Hockney suggested, against accepted views, that the use of optical instruments by Renaissance painters was pervasive, explaining some of the optical qualities of their art. See Hockney, Secret Knowledge. See also Gorman, “Art, Optics and History.” 64. For further discussion of Kepler’s theory of cognition, see Escobar, “Kepler’s Theory of the Soul”; for the role of recognition in Kepler’s theory of cognition, see Claessens, “Imagination as Self-Knowledge.” 65. “Visio igitur fit per picturam rei visibilis ad album retinae et cauum parietem” (KGW, 2:153). This impression is even strengthened in Kepler’s 1611 Dioptrice, where Kepler refers to the retina as painted with colored rays of the visible things. The word he chooses for the instruments that do the actual painting is pencilli (little

brushes). Thus, Alpers concludes, “Artists’ brushes paint a picture of the world outside the eye on the opaque screen of the retina in the back of the eye.” Alternatively, in Kepler’s own words, “The tunic of the retina is painted with the colored rays of visible things” (Retiformis tunica pingitur a radijs coloratis rerum visibilium) (Kepler, Dioptrice, in KGW, 4:372); and see the discussion in Alpers, Art of Describing, 33–41. 66. KGW, 2:153. 67. “Cum hactenus Imago fuerit Ens rationale, iam figurae rerum vere in papyro existentes, seu alio pariete, picturae dicantur” (Kepler, KGW, 2:174). The definition is problematic since nowhere before does Kepler supply the term “rational entity” with a definition. Thus, the contrast between these two modes of imaging is not at all clear. I have amended Donahue’s translation (Kepler, Optics, 210) on two points: he translates Ens rationale as “being of reason,” which is somewhat vague and does not convey Kepler’s attempt at an exact definition. I suggest that rationale should be taken as referring to ratios and their calculation. Secondly, Donahue translates pariete as “surface,” and I suggest that “screen” is more appropriate in this context. 68. For Kepler’s differentiation between real pictures and images, see Malet, “Keplerian Illusions.” See also Smith, “Ptolemy, Alhazen, and Kepler.” 69. Pecham, Perspectiva communis,170. 70. Ibid., 159. 71. Ibid., 161. 72. Ibid., 171. 73. Kepler, KGW, 2:60. 74. Ibid., 143. 75. Ibid., 162. 76. Classical optics is made up of three parts: optics itself, dealing with questions of perspective, the propagation of light, and so on; and catoptrics,









divided in two: plane mirrors and reflecting surfaces, and burning mirrors and refraction. 77. Kepler, KGW, 2:164. 78. Ibid., 163. 79. Ibid. 80. Kepler, KGW, 2:77, translated in Kepler, Optics, 90 (translation amended). 81. See Kepler, KGW, 2:164: “But if you place a paper, say, if you insert a paper between the lens and the eye . . . now the image is not seen hanging in the air, but fixed on the paper. Because the paper strikes the eyes more clearly, it stabilizes them on the place of the image . . . the paper is seen principally, and the image secondarily. For not only mathematical dimensions create the image, but also and much more colors or illumination and physical causes.” 82. Kepler, KGW, 2:185, translated in Kepler, Optics, 221. It is easier to turn an inverted picture (where up is down and right is left) to its correct position than to reverse a mirror image (where down is down and up is up, but right is left) to its correct position. In the case of an inverted picture, all lines from the visible object to the screen on which the picture appears pass through one central point. 83. See Tachau, Vision and Certitude and “Problem of the Species in Medio.” 84. See especially Oresme, De causis mirabilium. While Oresme’s main concern in this treatise is to disprove supernatural events, a side effect is the undermining of sense-experience at large and visual perception in particular as mediated and unreliable cognitive instruments. Not only do sick eyes and obscure media affect the visual process, but there is no criterion to differentiate between species of a real object and those species stored in the human memory. Therefore, one

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cannot distinguish between a picture in a dream or any other delusion of an object perceived in the past and a present experience of a real external object. 85. Alberti critically associated the mirroring quality of painting with the myth of Narcissus (Alberti, On Painting, 62). See also Baskins, “Echoing Narcissus”; Barbieri, L’inventore della pittura; and for the general theme of painting and mirroring in the Renaissance, see Summers, Vision, Reflection, and Desire, 112–54. 86. Alberti, On Painting, 64. 87. See, e.g., Roger Bacon’s assertion that “Species substantie non est tantum ipsius forme, sed materie et totius compositi” (Bacon, Roger Bacon’s Philosophy of Nature, 28). 88. Kepler, KGW, 2:144.







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1. See, e.g., Baltrušaitis, Anamorphic Art. For the epistemological implications of anamorphosis, see Massey, Picturing Space, Displacing Bodies, esp. 23–36, and “Configuring Spatial Ambiguities.” 2. Kaufmann, “Eloquent Artist.” 3. Of course, this had significant social implications: the granting of a letter of majesty in 1595 by Rudolf II according to which painting was thenceforth considered an art and not a craft, thus releasing the painters from the restrictions of the guild system. Another social consequence was the ennobling of artists, such as Spranger and Arcimboldo. 4. Cicero, De oratore 3.6.21. 5. Cave, Cornucopian Text, xviii. 6. Barocchi, Trattati d’arte, 3:270. 7. For the meaning of “grotesque” and “grylli” as bizarre novelties and ridiculous images in the late sixteenth

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century in general and Arcimboldo’s circle in particular, see Kaufmann, Arcimboldo, esp. 103–9. 8. See, e.g., Gregorio Comanini’s comment that in fact “tutte l’arti rassomigliatrici, overo imitanti, hanno per proprio et adeguato fine il diletto. Et essendo la pittura una di queste, conviene dire che il diletto e non l’utile il suo fin sia” (Barocchi, Trattati d’arte, 3:248). 9. Screech, Laughter at the Foot of the Cross. 10. Kaufmann, “Arcimboldo’s Serious Jokes,” 79–80. 11. Comanini, Il figino overo del fine della pittura, 3:264–65, translated in Maiorino, Portrait of Eccentricity, 79. 12. Translated in Maiorino, Portrait of Eccentricity, 75. 13. Owens, “Allegorical Impulse,” 68. 14. For the emblematic tradition in early modern Europe, see Praz, Studies in Seventeenth-Century Imagery; Manning, Emblem; Watson, Achille Bocchi. See Pinkus, Picturing Silence, for a postmodern interpretation of early modern emblems. For the association of emblems and jokes in early modern Europe, see Bowen, “Two Literary Genres.” 15. I call this worldview emblematic, after Ashworth. His main essays on this subject are “Natural History and the Emblematic World View” and “Emblematic Natural History of the Renaissance”; see also Chen-Morris, “From Emblems to Diagrams.” 16. Alciati, Emblematum liber . . . , A1v (“ad lectorem”). 17. Stoichita, Self-Aware Image, 163–65. 18. The treatise, however, had already won imperial approval and privilege in 1598. 19. Evans, Rudolf II and His World, 206. 20. Ibid., 111. 21. For further discussion of Khunrath’s images, see Forshaw, “Alchemy in













the Amphitheatre” and “Curious Knowledge and Wonder-Working Wisdom.” 22. Maier, Atalanta fugiens. In my interpretation and translation of the following citations, I follow De Jong, Michael Maier’s “Atalanta Fugiens,” unless specified otherwise. 23. “Accommodata partim oculis & intellectui, figuris cupro incosos, adjectisque sententiis, Epigrammatis & notis, partim auribus & recreationi animi plus minus 50 Fugis Musicalibus trium Vocum, . . . singulari jucunditate videnda, legenda, meditanda, intelligenda, dijudicanda, canenda & audienda” (my translation). 24. “Accipe ovum & igneo percute gladio.” 25. Est avis in mundo sublimior omnibus, Ovum Cujus ut inquiras, cura sit una tibi. Albumen luteum circumdat molle vitellum, Ignito (ceu mos) cautus id ense petas; Vulcano Mars addat opem; pullaster & inde Exortus, ferri victor & ignis erit. 26. “Fac ex mare & foemina circulum, inde quadrangulum, hinc triangulum, fac circulum & habebis lap. Philosophorum.” 27. Foemina masque unus fiat tibi circulus, ex quo Surgat, habens aequum forma quadrata latus. Hinc Trigonum ducas, omni qui parte rotundam In Sphaeram redet: Tum Lapis ortus erit. Si res tanta tuae non mox venit ob via menti, Dogma Geometrae si capis, omne facies.” 28. See Yates, Theatre of the World, esp. 42–79. 29. These three elements correspond to Dee’s conception of three grades







in the advent of the initiate toward perfection and wisdom. The realm of pneumaticus (the spiritual) is divided into three levels, starting with the philosophos, associated with the element of water and having a “taste of the fundamental truths of natural knowledge.” The sophos follows, associated with the element of air. Dee explores the “celestial influences” and “the reasons for the rise, condition, and the decline of other things.” The last stage is the adeptiuus, who is associated with the element of fire and aspires to explore and understand “the supernatural virtues and metaphysical influences.” See Josten, “Translation of John Dee’s Monas Hieroglyphica,” 114–21, and Clulee, John Dee’s Natural Philosophy, 81–82. 30. For some initial treatments of serious play in early modern Europe, see Barolsky, Infinite Jest; Colie, Paradoxia Epidemica; Findlen, “Jokes of Nature and Jokes of Knowledge”; Gordon, Humanist Play and Belief; Kaufmann, “Arcimboldo’s Serious Jokes.” 31. For the role of visual images in Renaissance mnemonic techniques, see Yates, Art of Memory, and Bolzoni, Gallery of Memory. 32. Fludd and Maier not only shared the same Paracelsian and Rosicrucian sentiments, as well as the same publisher, De Bry in Frankfurt, but also probably met during Maier’s frequent trips to England. Maier dedicated his first publication to Sir William Paddy, King James’s physician and Fludd’s friend: see Yates, Rosicrucian Enlightenment, 80–82; Evans, Rudolf II and His World, 205–6; Tilton, Quest for the Phoenix, esp. 27–28, 87–112. Some scholars have cast doubt on a personal connection between Maier and Fludd; see Figala and Neumann, “Michael Maier (1569–1622),” 45; Moran, Alchemical World, 107–8.

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33. A discussion of the tripartite division of the soul is in Fludd, Utriusque cosmi maioris scilicet et minoris, tract. I, sect. I, lib. X, vol. 2:204–21. 34. “Therefore, the soul is seized in this way by some occult and spiritual power, as the similitudes of the body are portrayed in place of the body, just as one would see in a mirror” (ibid.; my translation). 35. “Each sense has one kind of object which it discerns, and never errs in reporting that. . . . Such objects are what we call the special objects of this or that sense” (Aristotle, On the Soul [De anima] 2.6.418a14–18, in Complete Works, 1:665). “Indeed, so often the soul is deceived and led astray by corporeal vision . . . the sense is deceived concerning the true and specific object, since what seems to appear in fantasy, it believes to be the effect of the body itself ” (Fludd, Utriusque cosmi maioris scilicet et minoris, 2:204). 36. “Immo vero inter omnes alios sensus maxime visus, quamvis praestantissimus, decipitur” (Fludd, Utriusque cosmi maioris scilicet et minoris, 2:204). 37. “Concerning the second grade of the vision of the soul, that is in the imagination or phantasy . . . in these the soul seems sometimes beguiled, suffering and aggravated” (ibid.). 38. Ibid., 205–6. 39. “Whence, often the images of corporeal things are present in a dream so distinctly, just as the body itself appears when one is awake, so much so that it is not possible to distinguish between the vision of those who sleep and the true cogitations of those who are awake, as [those who are sleeping] are moved by the always present flesh, and against their usual conduct, and the accepted mores, they see themselves mating, and what naturally is accumulated, is emitted through the genitals; and indeed the physicians

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name this action ‘nightly pollution’” (ibid., 206–7). 40. Ibid., 207. 41. “Qualiscunque tamen illa visorum natura sit, procul dubio corpus non est. Non enim corpora visa illas imagines in spiritu faciunt, nec eam vim habent, ut aliquid spirituale forment; sed ipse spiritus in seipso celeritate mira id praestat, utpote spiritualis, intellectualis atque rationalis” (ibid., 208). 42. See also Burnett, “Cosmogonic Experiments of Robert Fludd.” 43. Fludd, Mosaicall Philosophy, 1. 44. Ibid. 45. Ibid., 2. 46. Ibid., 3. 47. Ibid., 4. 48. Ibid., 5. 49. Ibid. 50. Ibid., 6. 51. Ibid. 52. Ibid., 7. 53. Ibid., 8. 54. Ibid., 56. 55. Ibid., 57. 56. Ibid. 57. Ibid., 129. 58. Ibid., 72. 59. Ibid., 152. 60. Ibid., 129. 61. Ibid. 62. Ibid. 63. Ibid., 152. 64. Ibid. 65. Ibid., 152–53. 66. Ibid., 153. 67. Ibid. 68. Ibid. 69. Ibid., 153–54. 70. Ibid., 157. 71. Ibid., 158. 72. Ibid., 168. 73. Ibid., 171. 74. Fludd, Robert Fludd and His Philosophical Key, 69–70. 75. Ibid., 87.











76. Ibid., 88. 77. On Protestant criticism of the allegorical mode of reading the book of nature, see Harrison, Bible, Protestantism, and the Rise of Natural Science. 78. See Kepler’s handwritten comment concerning the emblem of a falling Icarus, discussed above, introduction, “Epistemological Transgressions,” fig. 1. 79. Kepler to Michael Maestlin, June 1/11, 1598, KGW 13:225. For a more detailed discussion of Kepler’s globe, its emblematic plan (a device to serve beverages to the courtiers), and its entertaining and playful intentions, see Mosley, “Objects of Knowledge,” esp. 202–9. 80. See, e.g., Wotton’s discussion with Kepler on the difference between painters’ practice and that of the mathematicians, related above, chap. 4, “The ‘Illiberal” Drawings of Kepler.” The issue of painters’ practices is commented upon further in Kepler’s correspondence with Schickhardt: see Panofsky, “Die Perspektive als ‘symbolische Form,’” 295–96; Frangenberg, “Angle of Vision.” For Kepler’s debates with alchemists and Hermetic philosophers concerning visual representations, see, e.g., Kepler to Joachim Tanckius, May 12, 1608, KGW, 16:154–65 (discussed below); see also ibid., 4:245–46, for Kepler’s Tertius interveniens (Frankfurt, 1610), where he discusses the theory of signatures. For a Jungian treatment of Kepler’s 1617 polemic with Fludd, see Pauli, “Der Einfluss archetypischer Vorstellungen”; for further analysis, see Westman, “Nature, Art, and Psyche.” See Pantin, “Kepler’s Epitome,” for the way Kepler applies images and figures rhetorically and didactically. 81. Kepler, appendix to book 5 of Harmonices mundi, KGW, 6:374, quoted and translated in Copenhaver,















“Natural Magic, Hermeticism, and Occultism,” 283. 82. Kepler, KGW, 8:413; Kepler, Tychonis Brahei Dani Hyperaspistes, 186: “διαγραμμα ψευδες” or “ψεδογραφημα.” See also Kepler, KGW, 8:309: “[T]his entire schema is to be seen as means for the understanding of this new demonstration, indeed not to deliver the true cause concerning this matter.” 83. For Kepler’s critique of Aristotelian abstraction, see Chen-Morris, “Optics, Imagination,” and the discussion in chapter 3 above. For other treatments of Kepler’s philosophy, with an emphasis on his theory of archetypes, see Martens, Kepler’s Philosophy; for a more metaphysical treatment of Kepler’s notion of vision, see Samsonow, Die Erzeugung des Sichtbaren. 84. Kepler, Harmony of the World, 303–4. 85. “[D]emonstratio vero illa ex sensibilibus diagrammatis nunquam habetur, etsi iis adiuvetur: nec ex collectionibus oritur multorum sensilium in unum Axioma, sed a priori comparatur” (Kepler, KGW, 6:222). See also the discussion in chap. 3. 86. Ibid., 19:328, quoted and translated in Grafton, Defenders of the Text, 186–87. 87. Kepler, KGW, 13:132, quoted and translated in Grafton, Defenders of the Text, 189. 88. Kepler, New Astronomy, 62. 89. Ibid., 64. 90. Ibid., 66. 91. “Diese Imagination de signaturis sey nichts anders dann ein lustige Fantasey mussiger Kopffe / die nit feyren konnen / und gern etwas zu dichten haben” (Kepler, KGW, 4:245– 46), quoted and translated in Walker, “Kepler’s Celestial Music,” 55–56. 92. “Dasz Gott selber / da er wegen seiner allerhochsten gute nicht feyren konnen / mit den signaturis rerum

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also gespielt / und sich selbst in der Welt abgebildet habe: Also dasz es einer ausz meinem Gedancken ist / Ob nicht die gantze Natur und alle himmlische zierligkeit / in der Geometria symbolisiert sey. . . . Wie nun Gott der Schopffer gespielt / also hat er auch die Natur / als sein Ebenbild lehren spielen / und zwar eben das Spiel / das er jhr vorgespielet,” quoted and translated in Walker, “Kepler’s Celestial Music,” 56. 93. Ibid. 94. “Quinam uero ex ueteribus hominis staturam compossuerit ex notis sonorum duplicibus, non memini me legere. Vincit hac lectione Reinhardus” (Kepler, KGW, 14:154). 95. “Diuiduntur enim interualla illa in tonos majores minores et semitonia, id est in 9/8 10/9 16/15 quod aures probant, et Ptolemaeus aurium iudicij assertor” (ibid., 155). 96. Kepler, KGW, 6:99. 97. “He [the Creator] filled up all the intervals of 4/3 with the interval of 9/8, leaving a fraction over, and the interval this fraction expressed was in a ratio of 256 to 243” (Plato, Timaeus 36b, in Collected Dialogues, 1166). 98. Kepler, KGW, 6:99. 99. Ibid., 14:156. 100. Ibid. 101. Kepler uses the “golden section” or “divine proportion” as a justification for the acceptance of the pentagon as a geometrical correspondence with harmonic ratios. The regular pentagon is indeed constructible, but its sides are incommensurable with the diameter even in square: (side of the pentagon)2 = (1/2 radius)2 (10-2√5). However, using the golden section, Kepler can show that the side of an inscribed decagon is to the radius of the circle as the greater part to the whole in the golden section. The square on the side of a pentagon is equal to the square

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on the side of the decagon inscribed in the same circle plus the square on the radius. Moreover, the side of the pentagon is to the line joining two of its vertices as the greater part to the whole in the golden section. Since the pentagon contains the divine proportion within itself, and not only in relation to the radius of the circle in which it is inscribed (like the decagon), Kepler regards the pentagon as the archetypal figure of this proportion and thus of generation in general. This botanical analogy is a repeated theme with Kepler in the context of the application of geometry to natural processes: from his 1602 “More Certain Fundamentals of Astrology” through his speculations regarding the geometrical structure of a snowflake to Harmonices mundi. 102. Kepler, KGW, 14:157. 103. Ibid. 104. Ibid., 157–58. 105. “Atque hic excursus esto, quem mihi extorsit prurigo a reinhardi speculationibus concitata” (ibid., 158). 106. Ibid. 107. Ibid. 108. Ibid., 159. 109. Ibid. 110. Ibid., 161. 111. Ibid. 112. Ibid. 113. Ibid. 114. Ibid. 115. Ibid., 162. 116. Ibid. 117. Ibid., 162–63. 118. Ibid., 163. 119. Ibid. This proportion will be the subject of Kepler’s later investigations in the Hamonices mundi. 120. Ibid. An account of the usage of the word sphaera in the Renaissance is given in Jardine, “Significance of the Copernican Orbs.”

chapter 6











1. See also Rotman, Signifying Nothing, esp. 78–86; and Ostashevsky, “Crooked Figures.” 2. Shakespeare, King Lear, 1.2.33–36. 3. Browne, Booke, E3v. Also Harington, Vlysses upon Aiax, A2–A3: “You . . . are seene for a spectacle of follie, to those that cannot see without their spectacles.” 4. As Jay L. Halio, the editor of the New Cambridge Shakespeare King Lear, comments, “Spectacles are a symbol of what Gloucester does need. He does not see through Edmund’s plot.” Gloucester does not see Edmund’s invisible intentions but only the external, superficial actions that any eye can see with no aid. In order to see beyond this, one needs an artificial aid. Gloucester shows himself entirely credulous (114, note to line 35); see also Heilman, Great Stage, 45, 154. 5. Shakespeare, King Lear, 5.3.259–64. 6. Kepler, Ad Vitellionem, 298. 7. For further elaboration of this theme in the first decades of the seventeenth century, see Gal and Chen-Morris, “Empiricism Without the Senses.” 8. Pomponazzi, De naturalium effectuum causis, 22, quoted in Randall, “Introduction to Pomponazzi,” 277. 9. Kepler, KGW, 2:265–66, translated in Kepler, Optics, 319. 10. Kepler, Mysterium cosmographicum, 122 (translation amended). 11. Ibid., 65. 12. Dante’s words in Paradiso 142–45 are an allusion to Aristotle’s assertion that the first principle, which is the immaterial unmoved mover, produces motion in the movable heavens through being loved: “[T]he one is unmovable and the other is not. Thus it produces motion by being loved, and it moves the other moving things” (Aristotle,











Metaphysics 12.7.1072b3–4, in Complete Works, 1694). 13. Kepler, “Johannes Kepler’s On the More Certain Fundamentals of Astrology,” 91. 14. Ibid. 15. For the cultural importance of astrology in the Renaissance, see QuinlanMcGrath, Influences. For the debate on astrology in the Renaissance and especially the importance of Pico della Mirandola’s Disputationes adversus astrologiam divinatricem, see Rutkin, Astrology, Natural Philosophy and the History of Science; Rabin, “Kepler’s Attitude Toward Pico”; Valcke, “Jean Pic de la Mirandole et Johannes Kepler”; Broecke, Limits of Influence, esp. chap. 3; Westman, “Kepler’s Early Physical-Astrological Problematic” and the extensive discussion in Westman, Copernican Question, esp. 84–105. 16. “Coeli praeter communem luminis & motusinfluentiam nullam peculiarem vim esse” (Liebler, Epitome philosophiae naturalis, 235, translated in Westman, “Kepler’s Early PhysicalAstrological Problematic,” 232). See also Methuen, Kepler’s Tübingen, 193–97. 17. Kepler, “Johannes Kepler’s On the More Certain Fundamentals of Astrology,” 97. 18. Ibid., 96. 19. Ibid., 97. 20. Ibid. 21. Ibid., as translated in Field, “Lutheran Astrologer.” 22. Ibid. 23. Kepler, KGW, 2:16. 24. Ibid., 39. 25. Leonardo da Vinci, Treatise on Painting, 1:76. 26. Erasmus, Adagia, 405, 2nd chilias, 4th centuria, no. 38, quoted and discussed in Panofsky, “Nebulae in Pariete.”

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27. Kepler, Ad Vitellionem, 8. 28. Ibid., 31. 29. Kepler, New Year’s Gift, 5. 30. “[S]ine tamen materia aut soliditatis dimensione” (Kepler, KGW, 2:27–28). See also the discussion above in chapter 3. 31. Du Laurens, A Discourse of the Preservation of the Sight. 32. Ibid., 92. 33. Ibid., 87. 34. Kepler, KGW, 4:245–46: For Kepler’s sense of play and mode of inquiry, see also Hallyn, Poetic Structure of the World, esp. 163–202; Jaeger, “Johannes Kepler”; and Findlen, “Between Carnival and Lent.” 35. Baron, Origins of the Infinitesimal Calculus, 108–16. 36. This short and playful treatise should be read against the backdrop of Kepler’s ongoing contention with Epicurean atomism. See Boner, “Kepler v. the Epicureans.” 37. Lucretius, De rerum natura 2.688–99. For an otherwise brilliant reading of Kepler’s New Year’s Gift, see Tiffany, Toy Medium, 97ff. The reference to Lucretius is on p. 99. 38. Kepler, New Year’s Gift, 7. 39. Ibid. Kepler quite explicitly rejects clouds as the embodiment of nothing. He initially approaches the addressee of his treatise and comments, “accept with unclouded brow this enrichment by nothing.” He then dismisses Aristophanes’s Clouds: “Away with that panderer to vulgar scorn and ignorance, Aristophanes; what need have I of Socrates, the theme of his play?” 40. Ibid. 41. Ibid., 31. 42. Ibid., 27. 43. Ibid., 33. 44. Kepler discusses the “earthly soul,” or “animated faculty,” in the context of his astrological theories, especially in his De fundamentis astrologiae certioribus.

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See Field, “Lutheran Astrologer.” For Kepler’s discussion of this animated faculty in Harmonices mundi, see Schwaetzer, Si nulla esset in terra anima; and in the context of Kepler’s polemics with Robert Fludd, see Boner, “Kepler’s Living Cosmology.” 45. For the Renaissance association of melancholy and mathematics, see above, chap. 4, n55. 46. Kepler, KGW, 18:143. 47. Ibid., 14:158. 48. Kepler, Kepler’s “Somnium,” 16. For an extended version of my interpretation, see Chen-Morris, “Shadows of Instruction.” 49. Kepler, Kepler’s “Somnium,” 75–76. 50. Ibid., 63. 51. Throughout the last century, Kepler’s Somnium has attracted much scholarly attention. It was read as a clear indication of Kepler’s science as “Janus faced; it looks to the future, and yet it does not break with the past” (Cassirer, “Mathematical Mysticism and Mathematical Science,” 348–49). Simon, Kepler astronome astrologue, developed this line of research, adjusting to it structuralist insights. His conclusions were applied to Kepler’s Somnium in Reiss, Discourse of Modernism, chap. 4; Hallyn, Poetic Structure of the World, esp. 253–80; Paxson, “Kepler’s Allegory of Containment”; Spiller, “To Depart from the Earth with Such Writing”; and Campbell, Wonder and Science, 133–43. Others read the Somnium against the backdrop of Kepler’s involvement in his mother’s witch trial; see especially Nicolson, “Kepler, the Somnium, and John Donne,” and Lear’s introduction to Kepler’s Dream. For a reading of the Somnium as a thought experiment with allegorical undertones, see Aït-Touati, Fictions of the Cosmos, 17–44. My analysis situates the Somnium in the context of













Kepler’s epistemology, taking its cue from an ongoing interest in Keplerian philosophy of science: Westman, “Kepler’s Theory of Hypothesis”; Jardine, “Forging of Modern Realism”; Samsonow, Die Erzeugung des Sichtbaren; Kozhamthadam, Discovery of Kepler’s Laws; Martens, Kepler’s Philosophy. 52. For Kepler’s mode of reading the classics, see Grafton, Defenders of the Text, 178–204, and Commerce with the Classics, 185–224. More recently and with a special emphasis on Kepler’s treatment of enigmas, see Jardine and Segonds, “Kepler as Reader and Translator of Aristotle.” 53. Lucian, A True Story, in Lucian, vol. 1; Plutarch, Concerning the Face Which Appears in the Orb of the Moon, in Moralia, vol. 12. 54. Kepler leads his reader to expect his Somnium to have political significance, alluding to Macrobius’s definition of somnia as special dreams that serve to illuminate covert political realities. See Macrobius, Commentary on the Dream of Scipio, 87–92. 55. For Kepler’s humanist mode of reading histories and mythologies, see Grafton, Defenders of the Text, 174–203. On possible political allegories related to the Jesuit order alluded to in the Somnium, see Reeves, Evening News, 37–51. 56. Kepler, Kepler’s “Somnium,” 36. 57. Johannes Kepler, “On Giving Astrology Sounder Foundations,” translated in Field, “Lutheran Astrologer,” 230–31. 58. Quoted in Baumgardt, Johannes Kepler, 155. 59. Kepler, Kepler’s “Somnium,” 82. 60. Ibid., 41–42. 61. Kepler ends this note with an ironic and bitter remark that “in like manner most people expiate their love of science by being poor and incurring









the hatred of the ignorant rich.” This is not only an autobiographical remark but also a plea for a change in the status of scholars. Kepler’s vision of the relationship between money and knowledge is depicted in the frontispiece of the Rudolfine Tables, in which the imperial eagle drops coins of gold over the astronomical temple and its dwellers. 62. Ibid., 45. 63. Ibid., 35. 64. Ibid., 52–53. 65. Kepler states this methodological principle at the end of his geometrical explanation of the place of the image in the mirror: “Therefore it is not the occult nature of light, not the mind of universal nature, but the breadth of the sense of vision alone, that harmonizes with the causes for the sense of vision’s placing the image on the perpendicular” (Ad Vitellionem, 86). 66. Kepler, Kepler’s “Somnium,” 53. 67. Ibid., 89. 68. For Lucianic humor and rhetoric, see Jones, Culture and Society in Lucian, 52–55; and Branham, Unruly Eloquence. For an extensive commentary on A True Story, see Georgiadou and Larmour, Lucian’s Science Fiction Novel. For Lucian’s influence on Renaissance humanism, see Robinson, Lucian and His Influence in Europe, 95–197, and Marsh, Lucian and the Latins, esp. 148–210. 69. Lucian, True Story, 357. 70. Ibid., 249. 71. Ibid., 253. 72. A similar stand can be found in Jewish literature of Lucian’s time, for instance Mishnah Hagigah 2.1, discussed above, introduction, “Curiosity and the Threshold of Perception.” 73. Out of all the sages, only Plato arrived at the Isle of the Blessed, for the following reason: “It was said that he was living in his imaginary city

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under the constitution and the laws that he himself wrote.” The rest of the philosophers either changed their ways (Diogenes the cynic got married and even danced and indulged in tomfoolery), or got stuck on their way because of their intellectual complications: “None of the Stoics was there— they were said to be still on the way up the steep hill of virtue. . . . They said the Academicians wanted to come but were still holding off and debating, for they could not arrive at a conclusion even on the question whether such an island existed.” 74. As Teiresias tells Menippus in another of Lucian’s satiric dialogues: “The life of the common sort is best, and you will act more wisely if you stop speculating about heavenly bodies and discussing final causes and first causes, spit your scorn at those clever syllogisms, and counting all that sort of thing nonsense, make it always your sole object to put the present to good use and to hasten on your way, laughing a great deal and taking nothing seriously” (Lucian, Menippus, or the Descent into Hades, in Lucian 4:107–9). 75. Plutarch, Face on the Moon, 920b, 35. 76. Plutarch rejects the mathematicians’ assertion that the face on the moon is an optical illusion (920c–921f); then the Stoic theory that it is the result of the moon being a mixture of air and gentle fire (921f–922f); as well as the Aristotelian theory that the earth is in the center of the world and heavy things move toward it (923f–925a). 77. Ibid., 944a, 207. 78. Ibid., 223. 79. In this, Plutarch follows Plato’s Phaedrus, where myth complements rational discourse. After giving a logical proof of the soul’s immortality, Socrates proceeds to tell an allegory of the soul as winged steeds and winged chariot traveling in the heavens. He

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justifies this myth by saying, “As to the soul’s immortality we have said enough. . . . What manner of thing it is would be a long tale to tell, and most assuredly a god alone could tell it, but what it resembles, that a man might tell in a briefer compass” (Plato, Phaedrus 246a, in Collected Dialogues, 493). 80. For the role Plutarch’s treatise played in the scientific and theological debates on the nature of the moon in the early seventeenth century, especially in the context of Galileo’s telescopic observations of the moon, see Reeves, Painting the Heavens, and Casini, “Il Dialogo di Galileo e la luna di Plutarco.” 81. Lucian, True Story, 341. 82. Kepler, Kepler’s “Somnium,” 28–29. 83. See, e.g., Plutarch, Face on the Moon, 73: “So far beneath the stars is she [the moon] that the distances cannot be expressed, but you mathematicians in trying to calculate it run short of numbers”; and ibid., 107: “I am ashamed to confute a mathematical proposition, the foundation, as it were, on which rests the subject of catoptrics. Yet it must be said the proposition, ‘all reflection occurs at equal angles,’ is neither self-evident nor an admitted fact.” 84. In explaining the name of the hero of his story—Duracotus—Kepler comments that “[the] sound of this word came to me from a recollection of names of a similar sound in the history of Scotland, a land that looks out over the Icelandic ocean” (Kepler, Kepler’s “Somnium,” 87). Kepler expects that after this note, his reader will find it futile to allegorize on the hero’s name and to find in it allusions to deep mysteries. 85. Ibid., 99. 86. For instance, Kepler points out that the meaning of human words for the “nocturnal luminary” derives from appearances, and these words





do not express its true nature: “What we earth dwellers call earth, I chose to call ‘Volva’ from the point of view of the people of the moon. For our nocturnal luminary is called in Hebrew Lebbana, from its white color; in the Etruscan Language, Luna. . . . In Greek, Selene, from Selas, meaning ‘white sheen,’ since that is how it looks to us who live on the earth” (ibid., 78–79). See also Kepler’s introduction to Astronomia nova: “We acquire most of our information, both in quality and in quantity, through the sense of sight, it is impossible for us to abstract our speech from this ocular sense. Thus, many times each day we speak in accordance with the sense of sight, although we are quite certain that the truth of the matter is otherwise” (Kepler, New Astronomy, 59). 87. Kepler, Kepler’s “Somnium,” 14. 88. Ibid., 57. 89. Ibid., 58. 90. Ibid., 58–60. 91. Ibid., 63. For further elaboration on Kepler’s utilization of humor, see Gerlach, Humor und Witz; Findlen, “Between Carnival and Lent”; Jardine, “God’s Ideal Reader.” 92. Kepler, Kepler’s “Somnium,” 85. 93. Ibid., 88. 94. Ibid., 97. 95. Ibid. 96. Ibid., 101. 97. Ibid., 106. 98. Ibid., 124. 99. See, e.g., in Kepler, “Johannes Kepler’s On the More Certain Fundamentals of Astrology,” 103: “It is very foolish, however, to look here for those particular matters [political and military], the number and quality of which the curious seek in the almanacs. For what I have said about meteorology should hold here as well: nothing can be sought from astrology than a certain driving force





of dispositions; whatever will happen in human affairs is in the power of people’s free will, which is the image of God and not the offspring of nature.” See also Field, “Lutheran Astrologer.” 100. Kepler, Kepler’s “Somnium,” 129. 101. Ibid., 106. 102. Ibid., 145. 103. For Kepler, what saves the retinal picture from absurdity is the geometrical regularity of the inversion of the retinal picture; and see the discussion above, chap. 4, “Kepler’s Pictures and the Limits of Human Imagination.” 104. For serious jokes and serious play, see the discussion above, chap. 5, “Allegories and Jokes.”

postscript

1. Alberti, De pictura, 52–54. 2. Verville, L’Histoire veritable, fol. 2. 3. Quoted in Maiorino, Leonardo da Vinci, 25. 4. Swerdlow, “Derivation and First Draft of Copernicus’s Planetary Theory,” 439. 5. Sidney, Defence of Poesie, B4v–C1r. 6. Alpers, Art of Describing, 138.

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———. “Kepler’s Theory of Hypothesis and Trinkhaus, Charles. The Scope of the ‘Realist Dilemma.”’ Studies in the Renaissance Humanism. Ann Arbor: History and Philosophy of Science 3 University of Michigan Press, 1983. (1972): 233–64. Valcke, Louis. “Jean Pic de la Mirandole ———. “Nature, Art, and Psyche: Jung, Pauli, et Johannes Kepler: De la and the Kepler-Fludd Polemic.” In mathématique à la physique.” Occult and Scientific Mentalities in Rinascimento, 2nd ser., 36 (1996): the Renaissance, edited by Brian 275–96. Vickers, 177–229. Cambridge: Veltman, Kim H. “Leonardo and the Cambridge University Press, 1984. Camera Obscura.” In Studi vinciani White, John. The Birth and Rebirth of in memoria di Nando de Toni, 81–92. Pictorial Space. 2nd ed. London: Brescia: Ateneo di scienze lettere ed Faber and Faber, 1967. arti: Centro ricerche leonardiane, Whitfield, J. H. “Momus and the Language 1986. of Irony.” In The Languages of Verville, Béroalde de. L’Histoire veritable, ou Literature in Renaissance Italy, le Voyage des princes fortune. Paris: edited by Peter Hainsworth, Valerio C. de La Tour, 1610. Lucchesi, Christina Roaf, David Voelkel, James R. The Composition Robey, and J. R. Woodhouse, 31–43. of Kepler’s “Astronomia Nova.” Oxford: Clarendon Press, 1988. Princeton: Princeton University Witelo. Witelonis Perspectivae liber quintus: Press, 2001. Book V of Witelo’s “Perspectiva”; ———. Johannes Kepler and the New An English Translation, with Astronomy. New York: Oxford Introduction and Commentary and University Press, 1999. Latin Edition of the First Catoptrical Walker, D. P. “Kepler’s Celestial Music.” In Book of Witelo’s “Perspectiva.” Edited Studies in Musical Science in the and translated by A. Mark Smith. Late Renaissance, 34–62. London: Studia Copernicana 23. Wroclaw: Warburg Institute; Leiden: E. J. Brill, Ossolineum; Warsaw: Polish 1978. Academy of Sciences Press, 1983. Walsh, P. G. “The Rights and Wrongs of ———. Witelonis Perspectivae liber Curiosity (Plutarch to Augustine).” secundus et liber tertius: Books II Greece and Rome, 2nd ser., 35 (1988): and III of Witelo’s Perspectiva; A 73–85. Critical Latin Edition and English Watson, Elizabeth. Achille Bocchi and Translation with Introduction, the Emblem as Symbolic Form. Notes, and Commentaries. Edited Cambridge: Cambridge University and translated by Sabetai Unguru. Press, 1993. Studia Copernicana 28. Wroclaw: Westman, Robert S. The Copernican Ossolineum; Warsaw: Polish Question: Prognostication, Skepticism, Academy of Sciences Press, 1991. and Celestial Order. Berkeley: Wright, D. R. Edward. “Alberti’s De pictura: University of California Press, 2011. Its Literary Structure and Purpose.” ———. “The Copernican Question Revisited: Journal of the Warburg and A Reply to Noel Swerdlow and John Courtauld Institutes 47 (1984): 52–71. Heilbron.” Perspectives on Science 21 Wyss, Beat. “Der Dolch am linken (2013): 100–136. Bildrand: Zur Interpretation von ———. “Kepler’s Early Physical-Astrological Pieter Bruegels Landschaft mit Problematic.” Journal for the History dem Sturz des Ikarus.” Zeitschrift of Astronomy 32 (2001): 227–36.

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INDEX

Page numbers in italics refer to illustrations. abstraction, Kepler’s inversion of traditional notions of, 21, 67, 92 Adagia (Erasmus), 167 Ad Vitellionem paralipomena (Kepler) appendix to first chapter of, 76, 86 on Aristotle’s medium of vision, 86–90, 96 basis in a priori reason as goal of, 38, 49, 68 catoptrics, traditional, critique of, 67–70, 68 on color, source and transmittal of, 42, 86–91 on conic sections, 64–67, 65, 199n15 and coupling of optics and astronomy, 9–12, 24–25 decision to publish ahead of astronomical research, 3–4, 8, 9, 46 definition of astronomy in, 25–27 on distortion in optical images, 116 epigrams to, 33–34 as first account of physical phenomena in geometric terms, 47, 49, 51, 53 focus on camera obscura, 46–47 frontispiece of, 24–25, 26 on geometry: consonance with pure motion of light, 45, 53–54, 58–62, 72–77, 91, 95–96; as true language of creation, 53 on instruments of observation as reliable source of truth, 34–35, 49, 54, 58–62 isolation of element of motion in, 73–75

Kepler’s effort to reestablish astronomy as physical science in, 25–27 on light: as motion without substance, 53–54, 72–73; as ordering principle of universe, 46–47; spherical radiation characteristic of, 52–53; study of, as best window into true nature of reality, 53 on lunar disk diminution during solar eclipse, 159–60 and mathematics, reconceptualization of as representational language, 66–67 and math/physics boundary, erasure of, 48–49, 72–77 and new scientific language, 76 new starting point for study of optics in, 49 new theory of cognition developed in, 63, 67, 76 overthrow of Aristotelian theories in, 76, 86–91 overview of claims in, 1 on retinal pictura vs. artistic pictura, 113 and presumption in measuring the heavens, 54 purging of occult causes from optics in, 75 on reflection, 73, 75 on refraction, 73–75 as rewriting of optical theory, 38 on sphere as perfection sought by radiated virtues, 52 on study of physical motion as study

Ad Vitellionem paralipomena (continued) of shadows and reflections, 1, 166–69 subsumption of vision under artificial observation in, 24–25 on Sun, central place of as theologically appropriate, 53 transgression of heaven/earth boundary in, 49, 62 and validity of astronomical measurements, demonstration of, 62 on vision: geometry as correction for, 161–62; instruments as correction for, 161, 166–69, 184 visual economy outlined in, 166–69 and vocabulary of optical analysis, purification of, 49–51 Witelo as target of, 3–4, 8, 9–10, 24, 59, 67, 69–70 Alberti, Leon Battista de’Pasti medal of, 104 on gap between realm of virtue and reality, 104, 203n24 on painter’s practice, 102–7, 111, 120, 186 on painting as liberal art, 107 on points as signs, 105 on scholar’s life, 103–4 theory of vision in, 99–102, 105–6, 120, 121, 202n15, 203n18 veil technique used by, 106 on visual pyramid, 100–102, 105–6, 120, 186 Albertus Magnus, 80, 81–82, 83, 201n86 alchemy. See also emblems, alchemical in court of Rudolf II, 131 on fallen state of world, 137 on recollection of lost knowledge, 136–37 Alciati, Andrea, 130–31 Alfons, Sven, 123 Alhacen. See Ibn al-Haytham Alpers, Svetlana, 6, 99 Ambassadors (Holbein), 111 Amphitheatrum sapientiae aeternae (Khunrath), 131–33, 132 Anaritius (al-Nairizi), 81 ancient texts, Kepler’s approach to, 7–8, 149–50, 151, 158

236

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Apollonius, 64, 65 Apuleius, 12–13 Aquinas, Thomas, 27, 29, 80–81, 83 Arcimboldo, Giuseppe, 111, 121, 123–25, 124, 126–30, 129 Aristotelian theory of vision denial of existence of light, 77 distinction between geometry and optics , 70, 71 importance of eye in, 9 influence on humanist optics, 85–86 influence on medieval optics, 79–82 Kepler’s overthrow of, 76, 86–91 limitations of, as issue in pre-Keplerian astronomy, 54–55, 59 Oresme’s critique of, 83–84 primacy of direct perception in, 9, 56, 161 vision in, 77–79, 89, 120 transparent medium of sight in, 77, 79, 81 Aristotle. See also De anima (Aristotle) on astronomy as subalternate science, 27 cognitive origin of mathematical entities in, 94 Copernicus’s reliance on, 1 on errors of cognition, 71 four elements of, 44 on imagination, processing of sense data by, 71, 78 on mathematical language, 67 on optics as subalternate science, 27 on sense perception, 9, 56, 77, 78, 79, 138, 161, 208n35 on thinking process, 78–79 touch as basic metaphor for sensation in, 78 vision as metaphor for intellection in, 77–78 art and science, early modern association of, 98–99 Arte of English Poesie (Puttenham), 50 artificial perspective in alchemical emblems, as metaphor, 131–34 and anxieties about validity of perception, 187–88 assumed links to medieval optics, 98–100

and early modern association of art and science, 98–99 vs. imaginative vision, late-Renaissance debate on, 111–12 magical power of imitation in, as 15thcentury focus, 107–9 power to deceive, 16th-century criticisms of, 110–12 16th-century association of with melancholy, 111 The Art of Describing (Alpers), 6 astrology, Kepler on, 150, 164-5, 215n99 Astronomiæ instauratæ mechanica (Tycho), 54 Astronomia nova (Kepler) on geometry in astronomy, 30 on light as source of planetary movement, 40 on nature as text, 22, 149 on speech dependence on the sense of sight, 215n86 astronomical aiming, as issue in Brahe’s instrumentation, 54–55 astronomical observations, molding of into facts, Kepler on, 30–31 astronomy. See also Copernican astronomy; epistemological barrier between astronomy and visual experience; Kepler’s astronomy definition of in Kepler’s Ad Vitellionem, 25–27 and melancholy, Kepler on, 168–69 in pre-Keplarian thought, as subalternate science, 27–29 and risk of fantastical invention, 168–69 theory of vs. practice, Kepler on, 25–27, 30 Atalanta fugiens (Maier), 133–37, 134 Augustine, 80 Avicenna, 200–201n86 Bacon, Francis, 97 Bacon, Roger, 32, 82, 84, 120 Barbaro, Daniele, 111–12 Barker, Peter, 15 Baxandall, Michael, 102 Bernegger, Martin, 171–72, 180 Brahe, Tycho

and astronomical aiming, as issue, 54–55 and comet of 1577, 16 correspondence, 55 death of, 3 eclipse measurements: distortion of, 54, 55, 161; Kepler’s on, 37, 59–62, 161, 184–85 as exile, 2 focus on improved instruments and data, 2, 54, 55 Kepler’s defense of against Galileo, 149 in Kepler’s Somnium, 174, 176 Kepler’s work under, 2–3 and limits of Aristotelian paradigm, 54–55 lunar theory, Kepler’s disproving of, 36–37 and role of theory in data interpretation, 30–31 Ursus and, 15–16, 55 Bruno, Giordano, 2, 16–18 Burckhardt, Jacob, 103 Calvin, John, 110 camera obscura. See also instruments of observation; pinhole images and Brahe’s eclipse measurements, 54, 55, 59–62 focus on in Ad Vitellionem, 46–47 and geometry as means of capturing astronomical truth, 36 issues surrounding, in pre-Keplerian optics, 57–58 Kepler’s drawing using, 97–98, 113 Kepler’s experiments with, in Somnium, 181–82 as place of revealed truth not available to senses, 36–37, 167–69, 188–89 role in shaping modern modes of perception, 6 stage as, in Shakespeare’s Midsummer Night’s Dream, 35–36 validation of as instrument of observation, 35 Carbone, Lodovico, 29 Castiglione, Baldesar, 50 Catoptrics (Euclid), 68–69 catoptrics, Kepler on

237

critique of traditional catoptrics, 67–70, 68, 73, 76 perpendicular in, 74–75, 76, 91, 117–18 celestial influences on earthly matters. See also epistemological barrier between astronomy and visual experience Kepler’s demonstrations of, 164 light as medium of, 43–45 Cicero, 126 Cigoli, 98 Clark, Stuart, 6–7 Clavius, Christopher, 9, 150 Cleary, John J., 199n36 cognition. See also mind, human Aristotle’s theory of, 71 in Kepler, 63, 67, 76, 90–91 in Melanchthon’s model of vision, 86 colors harmonic proportions in, 42 Kepler on, 42, 86–91 Comanini, Gregorio, 128 Commentaria in Alciati Emblemata (Franciscus), 18, 19 Commentaries on the Theory of Mars . . . or the Key to Universal Astronomy (Kepler), decision to publish after Ad Vitellionem, 3–4, 8, 9, 46 Commentary on Euclid’s Geometry (alNairizi), 81 Concerning the Face Which Appears in the Orb of the Moon (Plutarch), 173–74, 178–81, 185, 214n76, 214n83 conic sections, Kepler on, 64–67, 65, 199n15 Copernican astronomy Kepler’s Somnium as argument for, 174, 175–76, 182–83, 184, 185 and perception as basis of knowledge, 1–2, 187–88 scholarship on, 1 correspondence of Kepler to Herwart von Hohenburg, 3 to Maestlin, 38–39, 41 to Matthias Bernegger, 171–72, 175 open letter to Archduke Ferdinand, 36 to Tanckius, 22, 150–56, 153 Corro, Antonio de, 15 Crary, Jonathan, 6 Creation

238

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and divine playfulness, Kepler on, 150 Fludd on, 143–44, 146 Crombie, Alistair, 4, 5 Cusanus, Nicolaus, 52, 186–87 De anima (Aristotle) on imagination, 71 Kepler’s critique of, 86 on light ray, as fictional construct, 70, 77 Melanchthon’s commentary on, 85–86 Oresme’s commentary on, 83, 84 on senses, infallibility of, 138, 208n35 on value of direct visual perception, 9 De commodis litterarum atque incommodis (Alberti), 107 Dee, John, 2, 131–32, 136, 207n29 De fundamentis astrologiae certioribus (Kepler), 43, 163–64 De luce (Grosseteste), 52 De multiplicatione specierum (Bacon), 82 De partibus animalium (Aristotle), 10–11 de’Pasti, Matteo, 104 De pictura (Alberti), 99–100, 102, 105 De revolutionibus orbium coelestium (Copernicus), 1 Descartes, René, 98 Dioptrice (Kepler), 204–5n65 direct visual perception. See vision, direct divine proportion, Kepler on, 152, 210n101 Donne, John, 189 Dürer, Albrecht, 24, 99, 111, 162, 171 early modern culture art theory in, 194n2 association of art and science in, 98–99 optics of, 6 earthly soul, Kepler on, 166, 212n44 Ecclesiastes, 12, 14, 15, 19 eclipses Brahe’s measurements of: distortion in, 54, 55, 161; Kepler’s on, 37, 59–62, 161, 184–85 as divine invitation to human inquiry, 33 Kepler’s prediction of 1600 eclipse, 36–37 elements, Kepler’s modification of Aristotelian scheme of, 44 Emblemata (Alciati), 130–31

emblems components of, 130 fragmented contexts of signification in, 130–31, 133 and gap between eternal forms and changing physical reality, 131 Kepler’s interest in visual representation and, 148 Kepler’s rejection of truth value of, 22, 147, 148 physical world as, in Fludd, 141, 146, 147 as 16th-century vogue, 131 emblems, alchemical Fludd’s theory of visual cognition and, 138–40 and gap between signs and concepts, 137–38, 140 mathematics in, as symbols, 131–37 as means of regaining lost knowledge, 137–38 power of to reveal underlying reality, 131, 140 spiritual meaning underlying all objects in, 131 unification of divergent truths within, 137 as vogue in Rudolf II’s court, 131 Empedocles, 176 epistemological barrier between astronomy and visual experience in Jewish tradition, 12 in Judeo-Christian tradition, 12, 14–15 Kepler’s overcoming of, 19–20, 30–35, 43 in medieval and classical thought, 10–12, 12–14 in Renaissance thought, 14–15 16th-century astronomers on, 15–18 Erasmus, 62, 167 Euclid, 68–69, 101–2, 202n15 Evans, R. J. W., 131 Fabricius, David, 33 the Fall alchemical views on, 137 lack of impact on physical world, in Kepler, 147–48 and veiling of truth, in Fludd, 141–42, 147

Ferdinand, Archduke, 36 Feselius, Philippus, 150 Ficino, Marsilio, 136 Field, J. V., 5 “Flora” (Gherardino), 128 Fludd, Robert alchemists of Rudolf ’s court and, 138, 207n32 on Creation, 143–44, 146 on dreams that reveal soul itself, 140 on emblems as doorway to underlying reality, 140 on fallen mind, unavailability of higher truth in, 141–42, 146 on God, nature of, 144 Kepler’s rejection of emblematic worldview of, 22, 147, 148, 154, 157–58 on nature as text, 141, 143–47, 157–58 non-mathematical use of mathematics in, 145–46, 148 physical world as emblem in, 141, 146, 147 on Platonic heptacord, 144–45 theory of visual cognition in, 138–40 weather glass experiment of, 141–47, 142 Franciscus Sanctius, 18, 19 Galilei, Galileo, 9, 98, 149, 195n25 Garin, Eugenio, 103 geometrical perspective. See artificial perspective geometry as accurate reflection of physical world, 33, 42, 93–94, 147–48, 169–72 in astronomy: Kepler’s resolution of epistemological issues in, 162–66; usefulness in capturing truth, 22–23, 30–31, 31–35, 36 as coeternal with mind of God, 93–94 cognitive origin of in Aristotle, 94 consonance with pure motion of light, 45, 53–54, 58–62, 72–77, 91, 95–96 as corrective to human vision, 98, 116–22, 161–62, 171, 177, 188–89 divine origin of, in Kepler, 90 and harmonic theory, Kepler on, 155–56 human mind as pre-structured to perceive, 91–96, 98, 116–17, 147–48, 149, 156, 165–66, 171

239

geometry (continued) and imagination’s role in cognitive leaps, 163 Kepler’s Ad Vitellionem as first attempt to account for physical phenomenon in terms of, 47, 49, 51, 53 Kepler’s analysis of Sun’s heating power and, 164 Kepler’s relocation of from abstraction to human mind, 90–96 Kepler’s relocation of from abstraction to physical world, 63, 66–67, 168 limited uses of in pre-Keplerian lower sciences, 16, 27–30 as link between Kepler’s optics and celestial physics, 47 as new platform for observing universe, 2, 188–89 non-mathematical use of in Fludd, 145–46, 148 of planetary motion, Keplers’ association of with geometry of light, 40 in pre-Keplerian ontology, as removed from reality, 95, 96 religious symbolism of for Kepler, 51–52 as true language of creation, in Kepler, 53 Gerson, Levi ben, 58 Gherardino, Filippo, 128 God Fludd on nature of, 150 light as embodiment of creative power of, 45 mind of, as structural model for nature and mind of man, 33, 42, 93–94, 147–48 playfulness of, Kepler on, 150 Goldstein, Bernard, 15 Grafton, Anthony, 203n24, 209nn86–87, 213nn52, 213n55 Grosseteste, Robert, 27, 32, 52 Halio, Jay L., 211n4 Harmonices mundi (Kepler), 40, 151 harmonic proportions human mind’s predilection to search for, 92 motion as requirement for human perception of, 92–93

240

MEASURING SHADOWS Index

in motions of and distance between planets, 40, 41–42, 45 as pervasive form throughout universe, 42–43 harmonic theory Kepler on, 151, 155–56 and planetary motion, Kepler on, 156–57 Hermathena, 126 Hermes Trismegistus, 151 Hockney, David, 204n63 Hohenburg, Herwart von, 3, 37 Holbein, Hans, 111 Horace, 34, 195nn24–25 humanism debate on existence and nature of light, 85–86 on metaphor, 50 Ibn al-Haytham (Alhacen) influence of, 4, 80 Kepler’s critique of catoptrics of, 69–70, 73 on mathematical language, 67 on visual cone, 100 Icarus, as symbol of barrier between astronomy and visual experience, 18, 18, 19 image(s). See also pinhole images artistic vs. accurate, as issue, 97–98 virtual vs. real, in medieval optics, 113–15 image (pictura) in Kepler’s theory of vision accuracy of, as issue, 98 as actively-created by mind, 116–22 in causal chain of visual experience, 113 distortion and inversion of by eye, 113, 115, 116; eye’s correction of, 118–20, 121, 185, 205n82, 215n103; Kepler’s experiments on, 116–17, 205n81; and potential for illusion, 116, 121–22 distortion of by instruments, 116 epistemological value of: as guaranteed by physical laws, 118–20, 121, 184–85, 215n103; manipulations required to establish, 115, 120–21; necessity of establishing, 113–15 factors affecting, 116 vs. image per se, 118–20

as interface between mind and environment, 113 Kepler’s experiments on, 116–17, 205n81 Renaissance theory of painting and, 98, 99, 113, 115 imagination in application of geometry to astronomical calculations, 162–66, 171–72 and art, 110–12 Fludd on dangers of, 139–40 Kepler’s redefinition of function of, 115 necessity of discipline in, Kepler on, 171–72, 189 pre-Keplerian conception of as obstacle to clear perception, 78, 95 as pre-structured to perceive geometry of nature, 91–92, 95, 158, 163, 165–66 Institutio oratorio (Quintilian), 102 instruments of observation. See also camera obscura astronomical aiming as issue in, 54–55 as corrective to human vision, Kepler on, 161, 166–69, 173, 184 limitations of, Brahe and, 54–55 need for new theory of, Kepler’s recognition of, 55 pre-Keplerian views on distortions inherent in, 56–57, 115 as reliable vehicles of knowledge, Kepler on, 34–35, 49, 54, 58–62, 182, 188–89 validation of in Kepler’s optics, 34–35, 49, 58–62 intellectual vision, Fludd on, 138, 139, 140 Intercoenales (Alberti), 104 Jamnitzer, Wentzel, 99 Jarzombek, Mark, 104 John Chrysostom (saint), 12 John the Evangelist (saint), 80 Kaestlin, Michael, 29 Kaufmann, Thomas DaCosta, 126 Kemp, Martin, 204n49 Kepler, Johannes. See also other specific topics drawings from camera obscura image, 97–98, 113

on Icarus metaphor of human limitations, 18, 19, 20 as imperial mathematician, 3 as victim of Counter-Reformation, 2 work under Brahe, 2–3 Kepler’s astronomy on divine invitation to human inquiry, 33, 35 effort to establish as physical science, 25–27, 29–30 establishing physical grounds of as goal, 3, 25–27, 29–30 and extraterrestrial platform for viewing universe: geometry as source of, 2, 188–89 Somnium on, 174, 175–76, 182–83, 184, 185; a priori basis of, 20, 177 Kepler’s optics. See also vision, Kepler’s theory of central focus on media and refraction of light in, 32–33 and direct contact of seer and seen, rejection of, 88, 90, 98 as geometrical science, 47 historiography on, 4–7 isolation of element of motion in, 73–75 material objects as geometric objects in, 72–73 and math/physics boundary, erasure of, 48–49, 72–77 overthrow of Aristotelian theories in, 76, 86–91 overthrow of perspectivist optics, 49–50 a priori basis of, as goal, 38, 49, 68 psychological effects on vision, inclusion of, 115, 116–22 refocusing of inquiry onto geometry of light, 31–35, 58–62 as reframing of old questions, 5, 6 and validation of indirect visual experience, 20 (See also shadow) and validation of instruments of observation, 34–35, 49, 54, 58–62 “The Key to a Deeper Astronomy” (Kepler), 30 Khunrath, Heinrich, 131–33, 132 al-Kindi, 52 King Lear (Shakespeare), 159–60, 169, 211n4

241

knowledge abstract, Kepler on thirsting of human mind for, 45–46 for its own sake, Kepler on, 45 sense perception as basis of: Copernican rejection of, 1–2; Kepler on, 45–46; pre-Keplerian anxieties about, 187–88 Lazius, Wolfgang, 123 Leonardo da Vinci anxieties about validity of perception in, 187–88 on artificial perspective: magical power of imitation of, 107–9, 111; and mathematics as guarantor of certitude, 109 blending of art and science in, 99 and camera obscura, 58 on distortions in perspectival art, 110, 204n49 on flux of experience, epistemological futility created by, 109–10, 204n49 imagination, stimulation of by confused things, 167 on language, limitations of, 108–9 and Renaissance goal of complete transparency, 186 Liber de visu (Euclid), 101–2 The Librarian (Arcimboldo), 123–25, 124 light Aquinas on types of, 80–81 Aristotelian denial of existence of, 77 biblical references to, 80: and humanist optics, 85; and scholastic optics, 80 as carrier of information, 47 as divine agent ordering universe, 46–47 as embodiment of God’s creative power, 45 Kepler on nature of, 38–47, 87–91 medieval debate on existence and nature of, 79–85 as medium of heavenly influences on earth, 43–45, 164 motion of, as consonant with geometry, 45, 53–54, 58–62, 72–77, 91, 95–96 as motion without substance, 53–54, 72–73, 87–88, 167

242

MEASURING SHADOWS Index

as ordering principle of universe, 46–47 Oresme on types of, 83 as source and carrier of visual information, Kepler on, 42, 86–91, 95–96 source of color in, 42 as source of life in sublunary region, 43–44, 45 as source of motion in universe, 38–40 spherical radiation characteristic of, 52–53 study of, as best window into true nature of reality, 53 Lindberg, David C., 4–5, 58, 73 Lomazzo, Giovanni Paolo, 111 Lucian on claims of astronomical knowledge, 13, 62, 214n74 De bello civili, 149 A True Story, 173–74, 177–78, 180, 185, 214n73 Lucretius, 169 lunar orbit, Kepler’s disproving of Brahe’s theory on, 36–37 Luther, Martin, 14–15, 19–20, 110 Maestlin, Michael, 38–39, 55, 60, 184 Maier, Michael, 131, 133–37, 134, 207n32 mathematically-based observation Aristotle’s rejection of, 11 as corrective to human vision, 161–62, 177, 188–89 usefulness in capturing astronomical truth, 22–23, 30–31, 31–35, 36 mathematics. See also geometry in alchemical emblems, as metaphor, 131–36 distinction between physics and: erasure of in Kepler’s optics, 48–49, 72–77; in pre-Keplerian science, 48–49, 70–72 Kepler’s reconceptualization of, 66–67 non-mathematical use of in Fludd, 145–46, 148 Maurolyco, Francesco, 9, 68 medieval perspectiva Ad Vitellionem as effort to move beyond, 4, 9

and bond between perceiver and perceived, 5, 8, 10–11 boundary between mathematics and physics in, 48–49, 70–72 debate on nature of vision in, 82–85 and epistemological barrier between astronomy and visual experience, 10–12 geometry as twice-removed from reality in, 95, 96 Kepler’s critique of catoptrics in, 67–70, 68 Kepler’s historicization of, 7–8 Kepler’s optics as overturning of, 49, 67 medieval theory of human cognition and, 5 on mental concepts, limited applicability to reality, 71–72, 95 and Renaissance understanding of optics, 8 secondary value of mediated visual perception in, 8–9, 113–15, 118, 120 on sight, 77–79 on species vs. images, 113–15, 118, 120 on transparent medium of sight, 82, 83, 84 on visual cone, 100 medium of sight in Aristotelian optics, 77, 79, 81 in humanist optics, 85 Kepler’s critique of, 86–87, 88–90, 96 perspectivists on, 82, 83, 84 Melencolia I (Dürer), 111, 162, 171 melancholy gap between intellect and reality as Renaissance source of, 162, 171 Kepler on astronomers and, 168–69, 171 16th-century association of artificial perspective with, 111 Melanchthon, Philipp, 85–86 Melissus of Samos, 28 memory of lost knowledge, alchemical works on, 136–37 metabasis in classical and medieval thought, 27–28 Kepler and, 29–30, 53–54, 162, 167 metaphor humanist perspective on, 50

Kepler’s purging of from optics, 49–51 Meteorology (Aristotle), 83 middle sciences, 27 A Midsummer Night’s Dream (Shakespeare), 35–36 mind, human. See also cognition creation of visual images and, 116–22 as pre-structured to perceive geometry of nature, 91–96, 98, 116–17, 147–48, 149, 156, 165–66, 171 shaping of perception by, 94–96 structural affinity with physical world and mind of God, 33, 42, 93–94, 147–48 mind of God, as structural model for physical world and mind of man, 33, 42, 93–94, 147–48 mirrors. See also reflection curved, Kepler on foci of, 65–66 pre-Keplerian optics on, 56–57 Momus (Alberti), 203n24 Monas Hieroglyphica (Dee), 131–32 motion, in perception of geometric proportions in nature, 92–93 moving force (“soul”) of Sun, 38–40 Mulerius, 16 Mysterium cosmographicum (Kepler) on application of geometry to astronomical calculations, 162 central claim of, 2–3 on goals of astronomical investigations, 196n42 image of heavenly globe in, 148 on light as source of planetary movement, 39–40, 41, 46 on light as vehicle for contemplating diving order, 45 on planetary motion, 166 al-Nairizi (Anaritius), 81 nature. See physical world Neoplatonism, Kepler and, 5 New Science, Kepler and, 5 On the Animals (Aristotle), 84 On the Soul (Aristotle), 84 Optical Part of Astronomy (Kepler), 176, 180

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optics. See also Aristotelian optics; catoptrics; epistemological barrier between astronomy and visual experience; Kepler’s optics; vision distinction between mathematics and, in pre-Keplerian science, 70–72 early modern, and painterly style, 6 Kepler’s refocusing of onto geometry of light, 31–35, 45, 58–62 in pre-Keplarian thought, 27, 28–29, 58, 60, 197n3 Optics (Kepler), 25 Oresme, Nicole, 13–14, 71–72, 83–84, 120, 205–6n84 painting Alberti on, 102–7, 111, 120, 186 of late Renaissance: and blurring of line between real and fantastic, 112; debate on realism vs. imagination in, 111–12; emphasis on rhetorical dimension, 112; turn to allegorical grotesques, 125–30, 129; unity of arts and sciences as goal in, 126–27 of Renaissance (See also artificial perspective): pictura in, vs. Kepler’s image (pictura), 98, 99, 113, 115; use of optical instruments in, 204n63 as transitional ground between physical and spiritual, in early modern Europe, 194n2 Panofsky, Erwin, 98–99 Paracelsus, 137 parallaxes, Kepler on, 161–62 Parmenides, 28 Pecham, John, 48, 56–57, 57–58, 59, 60, 113–15 perception. See sense perception; vision Perspectiva communis (Pecham), 48, 56–57, 57–58, 60, 113–15 perspective, geometrical. See artificial perspective Phaedrus (Plato), 214n79 physical sciences and correct fundamental principles, importance of, 28–29 Kepler’s effort to classify astronomy as, 25–27, 29–30

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as lower science, in pre-Keplarian thought, 27–29 physical world feminine and masculine principles in, Kepler on, 151–53, 153 in Fludd, as text, 141, 143–47, 157–58 mathematical structure of, in Kepler, 33, 42, 93–94, 147–48, 169–72; as key postulate, 95; mind as prestructured to perceive, 91–96, 98, 116–17, 147–48, 149, 156, 165–66, 171; as rejection of allegorical readings of nature, 147–48, 157–58; and structural affinity with mind of God and mind of man, 33, 42, 93–94, 147–48 physics as basis of astronomy, Kepler on, 25–27, 29–30 distinction between mathematics and: erasure of in Kepler’s optics, 48–49, 72–77; in pre-Keplerian science, 48–49, 70–72 Piero della Francesca, 107–8 pinhole images. See also camera obscura complexity of rays forming, 63–64 issues surrounding: Kepler’s resolution of, 59–62; in pre-Keplerian optics, 57–58 Kepler’s book-and-thread model of, 59–60, 99 Kepler’s geometric principles for, 37, 125–26 limitations of direct vision in viewing of, 60–61 mathematics needed to decipher, 64 planets light as source of motion in, 38–40 order of: basis in five regular bodies, 38, 40, 41–42, 45, 162, 163, 196n48; harmonic proportions in, 40, 41–42, 45, 196n43 Plato, 10, 151, 155, 214n79 Platonism, 5, 78, 95 Pliny the Elder, 176 Plotinus, 52 Plutarch, 12–13, 173–74, 178–81, 185, 214n76, 214n83

Politics (Aristotle), 71–72 Pomata, Gianna, 191n3 Pontormo, 111 Posterior Analytics (Aristotle), 27 Prague, as intellectual center, 2 Problemata (Pseudo-Aristotle), 57 Pseudo-Aristotle, 57 psychological effects, in Kepler’s optics, 115, 116–22 Ptolemy, 151 Puttenham, George, 50 Pythagoras, 151 Pythagoreans, 151, 154 Quintilian, 102 Rabin, Sheila, 5 Ramus, Petrus, 4 ray(s) Alberti on, 100–101 in humanist optics, 85 light as, in Kepler, 63, 72, 87–88 medieval perspectivism on, 82, 100 Pecham on, 114–15 in pre-Keplerian physics, as fiction, 70, 72, 77, 79, 80–81, 83–84 reflection. See also catoptrics, Kepler on and light’s influence on planets, 44 role of perpendicular, in pre-Keplerian accounts, 67–70, 68 role of perpendicular. in Kepler, 74–75, 76, 91, 117–18 Reformation, on artificial perspective’s power to deceive, 110–12 Reinhardus, Kepler’s critique of, 150–56 Renaissance. See also painting on applying artificial constructs to nature, 162 on celestial influences on earthly matter, 164 epistemological barrier between astronomy and visual experience in, 14–15 ideal of complete transparency in, 2, 186–87, 188–89 intellectual authority of ancient texts in, 7–8 and metaphor, views on, 50

optics of, 8, 76 Risner, Fredericus, 4 Rosen, Stanley, 180 Rubens, Peter Paul, 98 Rudolf II (Holy Roman Emperor) court of: alchemical emblems as vogue in, 131; Kepler at, 1–2, 3; painters in, 112; unity of arts and sciences as goal of artists in, 126–27 raising of painting from craft to art status, 206n3 Schegk, Jacob, 28 Scheiner, Christopher, 98, 195n25 scholars, Kepler on status of, 213n61 scholastic philosophers on mental concepts, limited applicability to reality, 71–72 scholarship on, 5 theories of vision in, 32, 79–82, 149 on unknowability of heavens, 13 science(s), higher vs. lower, in pre-Keplerian thought, 27–29 Scripture Fludd on truth of as obscured from fallen minds, 141–42 in Kepler, as source of moral, not physical truths, 149–50 sense perception, Aristotle on, 9, 56, 77–79, 138, 161, 208n35. See also vision series and structured sets, importance of in Kepler, 64–67 shadow. See also camera obscura new significance of in Kepler’s optics, 1, 20–23, 31–35, 98, 166–69, 172, 173, 185, 188–89 and risk of visual error, 172 in Shakespeare’s A Midsummer Night’s Dream, 35–36 Shakespeare, William, 35–36, 159–60, 169, 211n4 Sidereal message (Galileo), 177 sight. See vision signatures, Kepler on, 150 smell, harmonic proportions in, 42 Smith, Mark A., 5 snowflakes, Kepler on geometrical pattern underlying, 169–71

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Somnium (Kepler), 172–85 as assault on Renaissance goal of complete transparency, 186 on camera obscura: experiments with, 181–82; as metaphor for Somnium text itself, 185; as metaphor for truth received through instrumentation, 182, 184 Copernicanism as central message of, 174, 175–76, 182–83, 184, 185 daemon instructor in, 181; description of moon by, 174, 182–84, 185, 215n86; as lunar shadow creature, 172–73; as representative of mathematical perspective, 173, 181, 182–83, 185 on direct vision/empiricism, limitations of, 173–74, 175–77, 178, 180, 184, 185 Duracotus subplot in, 174, 175, 181, 182, 214n84 as effort to convert reader’s mode of vision, 173, 184, 185 on geometry as corrective to direct observation, 177 on Kepler’s integrated method, 176 on language, arbitrary nature of, 181 on Lucian’s A True Story, 173–74, 177–78, 180, 185 on Neoplatonic supersensual perception, limitations of, 173–74, 176–77, 178–80, 181 on optical instruments as reliable vehicles of knowledge, 173, 182, 183, 184–85 on Plutarch’s Concerning the Face Which Appears in the Orb of the Moon, 173–74, 178–80, 180–81, 185 political overtones of, 213n54 on reason, power of, 175, 176–77, 179 scholarship on, 212–13n51 on shadows as vehicles of knowledge, 173 structure of, 180, 181, 185 as sugar-coated pill for public, 174–75 on true science, goals of, 176–77 species Bacon on multiplication of, 82 vs. images, medieval theories on, 113–15, 118, 120

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Kepler’s redefining of, 32 sphere history of as image in religious speculation, 52 as image of Trinity, 51 motion implied in, 51, 52 steganography, Verville on, 25 Stephenson, Bruce, 196n43 Straker, Stephen M., 99, 191–92n11 Strena, seu De nive sexangula (Kepler), 169–71 Sun heating by, as celestial influence on earth, 164 as source of motion in universe, 38–40 Syndey, Philip, 188 Tanckius, Joachim, Kepler letter to, 22, 150–56, 153 Tesauro, Emanuele, 195n25 Tiffany, Daniel, 169 touch, harmonic proportions in, 42 Trinity, sphere as image of, 51 A True Story (Lucian), 173–74, 177–78, 180, 185, 214n73 Uccello, Paolo, 107–8 University of Tübingen, 28–29 Ursus, Reimarus, 15–16, 55 A Valediction: of Weeping (Donne), 189 Valla, Lorenzo, 50 Vanity of the Eyes (Clark), 6–7 Vertumnus (Emperor Rudolf II) (Arcimboldo), 128, 129 Vervelle, François de, 25 vision. See also medium of sight Alberti’s theory of, 99–102, 105–6, 120, 121, 202n15, 203n18 Aristotelian theory of, 9, 56, 77–79, 89, 120, 138, 161, 208n35 Avicenna on, 200–201n86 as basis of knowledge: Copernican rejection of, 1–2; Kepler on, 45–46; Renaissance anxieties about, 187–88 Fludd’s theory of, 138–40 humanist theories of, 95–96

Kepler’s effort to create new type of, 20, 173, 184, 185 medieval perspectivism on, 82–85 scholastic theories of, 77–82, 201n86 vision, direct. See also epistemological barrier between astronomy and visual experience Kepler’s Somnium on limitations of, 173–74, 175–77, 178, 180, 184, 185 limitations of, in viewing pinhole images, 60–61 new status of in Kepler’s optics, 59, 62, 63, 188–89 prioritization of, in pre-Keplerian optics, 9, 56–57, 161 vision, Kepler’s theory of, 86–91. See also image (pictura) in Kepler’s theory of vision accuracy of perceived image as issue in, 98 anatomy of eye and, 116–17, 204–5n65 and correction of vision through geometry, 161–62, 184–85 and correction of vision through instrumentation, 161, 166–69, 173, 184 and mind’s a priori geometric structuring of perceptions, 91–96, 98 and severing of visual sensation and actual occurrences, 161–62 and validity of human perception, 96 vision, mediated. See also epistemological barrier between astronomy and visual experience correction of through instrumentation, 161, 166–69, 173, 184 geometry in correction of, 98, 116–22, 161–62, 171, 177, 188–89 and imagination, necessity of disciplining, 171–72, 189 Kepler’s validation of, 20, 21, 87, 98, 191n3 and necessity of guiding mind, 120, 176–77 pre-Keplerian concerns about, 8–9, 56–57, 113–15, 118, 120–21, 161, 187–88 visual errors Fludd on, 138–39, 208n35

Kepler on, 158, 172 Renaissance anxieties about, 187–88 visual pyramid (cone) Alberti on, 100–102, 105–6, 120, 186 Euclid on, 102 Leonardo da Vinci on, 110 medieval perspectivism on, 100 visual representations of information. See also emblems Kepler’s interest in, 148 real diagrams as expressions of a priori principles, 149 and real diagrams vs. pseudodiagrams, Kepler on, 148–49, 153–54 Le Voyage des Princes Fortunez (Vervelle), 25 Wacker, Johannes, 169 Witelo distinction between mathematics and physics in, 71, 72 Kepler’s critique of catoptrics of, 69–70, 73, 76 as target of Ad Vitellionem, 3–4, 8, 9–10, 24, 59, 67 Wotton, Henry, 97–98, 113 Wright, D. R. Edward, 102 Zabarella, Jacopo, 28 Zwingli, Ulrich, 110

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