Visual Culture and Mathematics in the Early Modern Period During the early modern period there was a natural correspondence between how artists might benefi t from the knowledge of mathematics and how mathematicians might explore, through advances in the study of visual culture, new areas of enquiry that would uncover the mysteries of the visible world. This volume makes its contri- bution by offering new interdisciplinary approaches that not only investigate perspec- tive but also examine how mathematics enriched aesthetic theory and the human mind. The contributors explore the portrayal of mathematical activity and mathema- ticians as well as their ideas and instruments, how artists displayed their mathematical skills and the choices visual artists made between geometry and arithmetic, as well as Euclid’s impact on drawing, artistic practice and theory. These chapters cover a broad geographical area that includes Italy, Switzerland, Germany, the Netherlands, France and England. The artists, philosophers and mathematicians whose work is discussed include Leon Battista Alberti, Nicholas Cusanus, Marsilio Ficino, Francesco di Giorgio, Leonardo da Vinci and Andrea del Verrocchio, as well as Michelangelo, Galileo, Piero della Francesca, Girard Desargues, William Hogarth, Albrecht Dürer, Luca Pacioli and Raphael. Ingrid Alexander-Skipnes is Lecturer in Art History at the Kunstgeschictliches Insti- tut at Albert-Ludwigs-Universität Freiburg, Germany. She is an Associate Professor Emerita, University of Stavanger, Norway. Visual Culture in Early Modernity Series Editor: Kelley Di Dio A forum for the critical inquiry of the visual arts in the early modern world, Visual Culture in Early Modernity promotes new models of inquiry and new narratives of early modern art and its history. The range of topics covered in this series includes, but is not limited to, painting, sculpture and architecture as well as material objects, such as domestic furnishings, religious and/or ritual accessories, costume, scientifi c/medical apparata, erotica, ephemera and printed matter. 51 Genre Imagery in Early Modern Northern Europe New Perspectives Edited by Arthur J. DiFuria 52 Material Bernini Edited by Evonne Levy and Carolina Mangone 53 The Enduring Legacy of Venetian Renaissance Art Edited by Andaleeb Badiee Banta 54 The Bible and the Printed Image in Early Modern England Little Gidding and the pursuit of scriptural harmony Michael Gaudio 55 Prints in Translation, 1450–1750 Image, Materiality, Space Edited by Suzanne Karr Schmidt and Edward H. Wouk 56 Imaging Stuart Family Politics Dynastic Crisis and Continuity Catriona Murray 57 Sebastiano del Piombo and the World of Spanish Rome Piers Baker-Bates 58 Early Modern Merchants as Collectors Edited by Christina M. Anderson 59 Visual Culture and Mathematics in the Early Modern Period Edited by Ingrid Alexander-Skipnes Visual Culture and Mathematics in the Early Modern Period Edited by Ingrid Alexander-Skipnes First published 2017 by Routledge 711 Third Avenue, New York, NY 10017 and by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2017 Taylor & Francis The right of the editor to be identifi ed as the author of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifi cation and explanation without intent to infringe. Library of Congress Cataloging in Publication Data Names: Alexander-Skipnes, Ingrid, editor. Title: Visual culture and mathematics in the early modern period / edited by Ingrid Alexander-Skipnes. Description: New York : Routledge, 2017. | Series: Visual culture in early modernity | Includes bibliographical references and index. Identifi ers: LCCN 2016034996 | ISBN 9781138679382 (alk. paper) Subjects: LCSH: Art—Mathematics. | Mathematics in art. Classifi cation: LCC N72.M3 V575 2017 | DDC 700/.46—dc23 LC record available at https://lccn.loc.gov/2016034996 ISBN: 978-1-138-67938-2 (hbk) ISBN: 978-1-315-56343-5 (ebk) Typeset in Sabon by Apex CoVantage, LLC Contents List of Figures vii Acknowledgments x 1 Introduction 1 INGRID ALEXANDER-SKIPNES PART I The Mathematical Mind and the Search for Beauty 9 2 Renaissance Aesthetics and Mathematics 11 JOHN HENDRIX 3 Design Method and Mathematics in Francesco di Giorgio’s T rattati 32 ANGELIKI POLLALI 4 Mathematical and Proportion Theories in the Work of Leonardo da Vinci and Contemporary Artist/Engineers at the Turn of the Sixteenth Century 52 MATTHEW LANDRUS PART II Artists as Mathematicians 69 5 Dürer’s U nderweysung der Messung and the Geometric Construction of Alphabets 71 RANGSOOK YOON 6 Circling the Square: The Meaningful Use of Φ and Π in the Paintings of Piero della Francesca 84 PERRY BROOKS vi Contents PART III Euclid and Artistic Accomplishment 111 7 The Point and Its Line: An Early Modern History of Movement 113 CAROLINE O. FOWLER 8 Between the Golden Ratio and a Semiperfect Solid: Fra Luca Pacioli and the Portrayal of Mathematical Humanism 130 RENZO BALDASSO AND JOHN LOGAN 9 Mathematical Imagination in Raphael’s School of Athens 150 INGRID ALEXANDER-SKIPNES Bibliography 177 List of Contributors 196 Index 199 Figures Cover: Raphael. School of Athens . Detail of Euclid and his pupils. Stanza della Segnatura, Vatican Palace. Photo copyright: Vatican Museums. 2.1 Leon Battista Alberti. Palazzo Rucellai, Florence. 12 2.2 Piero della Francesca (1415/20–1492). Legend of the True Cross: Finding of the Three Crosses and Verifi cation of the True Cross , c. 1452. San Francesco, Arezzo © 2016. 23 3.1 Codex II.I.141, folio 41 recto, detail. Biblioteca Nazionale Centrale, Florence. 33 3.2 Codex II.I.141, folio 41 verso. Biblioteca Nazionale Centrale, Florence. 35 3.3 Codex II.I.141, folio 22 recto. Biblioteca Nazionale Centrale, Florence. 38 3.4 Codex II.I.141, folio 22 verso, detail. Biblioteca Nazionale Centrale, Florence. 39 3.5 Codex II.I.141, folio 42 verso. Biblioteca Nazionale Centrale, Florence. 43 3.6 Codex II.I.141, folio 38 verso, detail. Biblioteca Nazionale Centrale, Florence. 46 5.1 Albrecht Dürer. Constructing Roman Alphabet, 1525, folio K2r . Woodcut in the U nderweysung der Messung . The George Khuner Collection, the Metropolitan Museum of Art, New York. 74 5.2 Albrecht Dürer. Determination of the Size of Lettering on High Buildings, 1525, folio K 1v . Woodcut in the Underweysung der Messung . The George Khuner Collection, the Metropolitan Museum of Art, New York. 77 6.1 Diagram I: Generation of φ from a square, via a compass-swing based on the diagonal from the midpoint of a side to a corner of the square. Diagram II: Division by φ, via two compass-swings, of the side of a right triangle, where the side to be divided and its adjacent side form a right angle and are in the ratio 2:1. Diagram III: Angular measures and φ-ratios in the regular pentagon. Diagram IV: The right triangle with sides of φ, φ, and 1, and the signifi cant angle of 51°50’. 85 6.2 Piero della Francesca. M ary Magdalene . Duomo, Arezzo. Detail of base with measurements. 88 viii Figures 6.3 Piero della Francesca. F lagellation . Galleria Nazionale delle Marche, Urbino. Hypothetical Floor Plan as drawn by B.A.R. Carter (with indication of Π-module distances provided by present author). 90 6.4 Piero della Francesca. F lagellation . Galleria Nazionale delle Marche, Urbino. Hypothetical Floor Plan of left side according to Welliver. 91 6.5 Piero della Francesca. R esurrection . Museo Civico, Sansepolcro. 94 6.6 Piero della Francesca. R esurrection. Museo Civico, Sansepolcro. 95 6.7 Piero della Francesca. M ary Magdalene . Duomo, Arezzo. 99 6.8 Piero della Francesca. R esurrection . Museo Civico, Sansepolcro. 100 7.1 Erhard Ratdolt. P reclarissimus liber elementorum . Library of Congress, Washington, DC. 115 7.2 Albrecht Dürer. U nderweysung der Messung . National Gallery of Art Library, Washington, DC. 117 7.3 Albrecht Dürer. M en Drawing a Lute . From “Unterweisung der Messung ,” Gedrückt zu Nuremberg: [s.n.], im 1525. Jar. “Institutiones geometricos.” Spencer Collection. 118 7.4 Leon Battista Alberti and Pier Francesco Alberti, “Defi nition of a Circle,” D e pictura, National Gallery of Art, Washington, DC. 124 8.1 Jacopo de’ Barbari (?). P ortrait of Luca Pacioli and Gentleman , 1495, 98 × 108 cm. Museo e Real Bosco Capodimonte, Naples. 131 8.2 Dodecahedron, Summa, and Cartiglio. Detail of Jacopo de’ Barbari. P ortrait of Luca Pacioli and Gentleman , 1495. Museo e Real Bosco Capodimonte, Naples. 135 8.3 Rhombicuboctahedron. Detail of Jacopo de’ Barbari. P ortrait of Luca Pacioli and Gentleman , 1495. Museo e Real Bosco Capodimonte, Naples. 136 8.4 Rhombicuboctahedron from Luca Pacioli Divina proportione (Venice: Paganino Paganini, 1509), Part III, Figure 36. Houghton Library, Cambridge MA, shelfmark Typ 525.09.669. 137 8.5 Arbor proportio et proportionalitas from Luca Pacioli Divina proportione (Venice: Paganino Paganini, 1509), Part III, Figure 62. Houghton Library, Cambridge MA, shelfmark Typ 525.09.669. 138 8.6 Slate tablet. Detail of Jacopo de’ Barbari. P ortrait of Luca Pacioli and Gentleman , 1495. Museo e Real Bosco Capodimonte, Naples. 139 8.7 Sum. Detail of Jacopo de’ Barbari. P ortrait of Luca Pacioli and Gentleman, 1495. Museo e Real Bosco Capodimonte, Naples. 139 8.8 Segments with numbers. Detail of Jacopo de’ Barbari. Portrait of Luca Pacioli and Gentleman , 1495. Museo e Real Bosco Capodimonte, Naples. 140 9.1 Raphael. School of Athens , c. 1509–10. Stanza della Segnatura, Vatican Palace. 151 9.2 Philosophy and the Liberal Arts. Gregor Reisch. Margarita Philosophica. Title page. Johann Schott and Michael Furter: Basel, 1508. Universitätsbibliothek Freiburg i. Br./Historische Sammlungen. 157 Figures ix 9.3 Raphael. S chool of Athens . Detail of Euclid and his pupils. Stanza della Segnatura, Vatican Palace. 160 9.4 Raphael. S chool of Athens . Detail of Euclid’s slate. Stanza della Segnatura, Vatican Palace. 161 9.5 Raphael. P hilosophy . Ceiling of the Stanza della Segnatura, Vatican Palace. 165 9.6 Raphael. Two Men Conversing on a Flight of Steps. (WA1846.191). A Study for the S chool of Athens . Silverpoint with white heightening on pink paper, 27.8 × 20 cm. 168