Table Of ContentVisual 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
