THE INVENTION OF INFINITY

Mathematics and Art in the Renaissance

J. V. FIELD

Department of History of Art Birkbeck College, University of London

OXFORD NEW YORK TOKYO

OXFORD UNIVERSITY PRESS

1997 CONTENTS

Figure acknowledgements xi Introduction 1 1 Medieval mathematics and optics and the Renaissance style in art 4 The learned tradition 4 The science of sight 6 Naturalism in art 7 Giotto's mathematics 13 - Useful mathematics 14 Goldsmiths and others 16

2 Building, drawing and 'artificial perspective' 20 Reviving the ancient style 20 Brunelleschi's 'artificial perspective' 21 Alberti's construction (1435) 25 Other perspective constructions 29 Alberti's check: an earlier rule? 35 Brunelleschi's invention 40

3 Through the wall: Masaccio's Trinity fresco (c. 1426) 43 Looking for squares 45 The vault: orthogonal section 49 The vault: transverse section—and a viewing distance 51 The dimensions of the vaulted area 53 The figures 54 Explaining away the shapes of the abaci 55 Masaccio's mathematics 56 Visually right and mathematically wrong 59

4 Piero della Francesca's mathematics 62 The abacus treatise 62 Geometry in the abacus treatise 68 The short book on the five regular solids 76

5 Piero della Francesca's perspective treatise 80 Piero's preliminaries 81 Perspective of plane figures 86 Prisms and combinations of prisms 97 More difficult shapes 102

IX Contents

6 Practitioners and patricians 114 Perspective and the ungrateful apprentice 114 Perspective treatises for painters 117 Mathematics in the education of the upper classes 129 Vitruvian problems 131 The Teatro Olimpico 136

7 The professionals move in 143 Calendar reform 143 Editing perspective: Egnazio Danti 147 Editing the Ancients: Federico Commandino 150 Explaining practice: Giovanni Battista Benedetti 161 The view from point A: Guidobaldo del Monte 171

8 Beyond the ancients 178 Conies in practical texts 180 Commandino and the ancients 181 Kepler on conies 183 Sundial problems 186 Benedetti on conies 187 Girard Desargues 190 Desargues on perspective 192 Desargues' projective geometry 196 Mathematics for mathematicians 205

9 Fragmented perspectives 207 Pictures and treatises 207 Abraham Bosse 209 Laurent de La Hyre 214 Bosse's treatise of 1648 and Desargues' Perspective Theorem 220 Mathematical responses and another geometry 224 Blaise Pascal 226 Philippe de La Hire 227 The next generation and beyond 229

Appendix: The abacists' pet triangle, with sides 13, 14, 15 235

Bibliography 239

Index 241