Visual Culture and Mathematics in the Early Modern Period

Ingrid Alexander-Skipnes

• Éditeur: Routledge
• Année: 2017
• ISBN:
  9781317192053
• Édition: 1
• Sujet: Arte
• Sous-sujet: Storia dell'arte rinascimentale

During the early modern period there was a natural correspondence between how artists might benefit 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 contribution by offering new interdisciplinary approaches that not only investigate perspective but also examine how mathematics enriched aesthetic theory and the human mind. The contributors explore the portrayal of mathematical activity and mathematicians 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.

1
Introduction

Ingrid Alexander-Skipnes

Mathematics has had a significant role in the cultural production of early modern Europe. It can also be argued that ideas from the world of visual representation led to advances in mathematics. Visual culture and mathematics were more closely connected in the fifteenth, sixteenth and seventeenth centuries than they are today. In our specialist-focused universities, these areas of study are commonly thought to be worlds apart and are rarely brought together.¹ In the early modern period, however, the study of visual culture together with mathematics was by no means considered impractical, as it might be thought of today, as these fields shared common concerns and approaches. While specialization has worked against a more holistic approach, recently there has been a growing interest in the interdisciplinary relationships between the two areas.

The importance of geometry was emphasized already in antiquity. Plato had insisted that the study of geometry was necessary in order to comprehend higher things, as the inscription “Let no one ignorant of geometry enter here” above the portal of his academy reaffirmed. For Aristotle, geometry held pride of place within the mathematical sciences because of its irrefutable proofs. In Book 35 of his Natural History, Pliny the Elder praised the skill of the artist Pamphilus, who was “highly educated in all branches of learning, especially arithmetic and geometry, without the aid of which he maintained art could not attain perfection.”² Euclid’s Elements and Optics and manuscripts of the work of Archimedes were fundamentals texts for artists. Book XIII of Euclid’s Elements was of particular interest for artists as it dealt with the geometry of rectilinear and circular figures. Archimedes’ work on circular and spherical geometry showed his familiarity with π; his balancing the lever and even his work with parabolic mirrors were of interest to early modern artists and engineers.

The hunt for Greek mathematical texts was an important part of a revival of interest in the culture of antiquity. There was a renewed confidence that what was excellent in Greek literature, architecture, art and mathematics could be recovered and strengthen the skills and knowledge of the ancient Greek world. A chair in Greek studies was established at the University of Florence in 1397, when Manuel Chrysoloras was invited to teach Greek. Tracking down manuscripts became a lively pursuit. Emissaries were sent out to explore Byzantium to find manuscripts, sometimes without patronage. Florence quickly became a vibrant center for the acquisition, translation and the dissemination of classical texts. Some mathematical texts were brought to Italy by humanists such as the Sicilian Giovanni Aurispa, who brought back a cache of some 238 Greek manuscripts from his second voyage in the East (1421–23) that included a manuscript of the Mathematical Collection of Pappus.³

One story that demonstrates the eagerness with which mathematical manuscripts were sought is that of Rinuccio da Castiglione and his purported acquisition of a manuscript by Archimedes.⁴ The story is told by Ambrogio Traversari, a Florentine monk and Greek scholar, who in 1424 was anxious to obtain a copy of an Archimedes manuscript which Rinuccio claimed to be in possession of. Intrigued by the rumor that had spread of the existence of the Archimedean text, Traversari tried in vain to see the manuscript. He invited Rinuccio to his monk’s cell, but the manuscript hunter babbled on incoherently on topics as varied as the perfidy of the Greeks to denouncing Tuscany’s hostility to learning.⁵ In the end, Traversari never saw the manuscript of Archimedes, and it is doubtful whether Rinuccio had it after all.

Much of the knowledge of mathematics was transmitted through and beyond the confines of a university education and into cultural circles. Humanist courts enjoyed the presence of mathematicians where other scientists, along with poets, painters, musicians and philosophers mingled comfortably, mostly due to the heterogeneity of their interests. Furthermore, an understanding of mathematics, feigned or learned, held a certain prestige in the humanist milieu.

In the fifteenth century, mathematically minded artists like Leon Battista Alberti and Piero della Francesca had frequented humanist courts where mathematics was central in a revival of interest in Greek culture. Interestingly, both artists were at the papal court of Pope Nicholas V (r. 1447–55) who not only commissioned one of the first translations of Archimedes but also was one of the few popes who lent his Greek mathematical texts. Thus the pope assisted in the spread of interest in Greek mathematics throughout the Italian peninsula. Leonardo da Vinci investigated questions of proportionality and optics extensively, and his drawings as well as his notes offer insight into his knowledge and use of mathematics, particularly geometry.

According to Giorgio Vasari, Piero della Francesca wrote “many” treatises on mathematics. The three known treatises reveal his knowledge of both Euclid and Archimedes. His Trattato d’abaco (Abacus treatise) covers arithmetic, algebra and geometry, while the Libellus de quinque corporibus regularibus (Short book on the five regular solids) goes further into a study of the Archimedean solids.⁶ Piero paraphrased parts of Euclid’s Optics in his treatise, De prospectiva pingendi (On perspective for painting), where his investigations into visual angles attempt to characterize the proportional relationships that Euclid had left undefined.⁷ Furthermore, it has been shown that he made a copy of an Archimedean text.⁸ Luca Pacioli, mathematician and com-patriot of Piero della Francesca, published, among several mathematical texts, one of the first Latin editions of Euclid’s Elements (1509). Leonardo da Vinci drew the illustrations for Pacioli’s important De divina proportione (On Divine Proportion).

On the other side of the Alps, the sixteenth-century German painter and engraver Albrecht Dürer contributed to the role that Greek mathematics played for visual artists in northern Europe through his study of proportion and perspective. He was instrumental in the dissemination of Italian theories of perspective in northern Europe, which had wide-reaching effects on other fields such as cartography and mathematics.⁹ Dürer purchased a copy of Euclid’s Elements in Italy and wrote a treatise on mathematics, Underweysung der Messung mit dem Zirckel und Richtscheyt (A Course in the Art of Measurement with Compass and Ruler) and Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion). Dürer demonstrated his profound interest in geometric figures and instruments in a memorable way in his engraving Melencolia I (1514). In their writings, Albrecht Dürer, Piero della Francesca and Leonardo da Vinci had revitalized early modern interest in the so-called Archime-dean solids, polyhedra with surfaces made up of plane polygonal surfaces whose sides make up their edges and the corners their vertices.

Linear perspective, rediscovered in fifteenth-century Florence, developed out of a need to depict a three-dimensional space on a two-dimensional surface.¹⁰ Linear perspective was arguably the most important technique for representation at the disposal of artists and architects of the early modern period. The Florentine architect and engineer Filippo Brunelleschi is credited with the invention, or rather rediscovery, of linear perspective. He had studied with the mathematician Paolo dal Pozzo Toscanelli. Brunelleschi invented a technique, essentially mathematical, whereby objects projected on a surface acquire a three-dimensional appearance. According to his biographer, Antonio di Tuccio Manetti, Brunelleschi painted two demonstrations of perspective (now lost), one of a view of the Baptistery seen from the cathedral door and the other from the Palazzo della Signoria, viewed a short distance away. His famous demonstration, which involved a perspectival construction and showed how perspective would work, took place in front of the Florence cathedral. Linear perspective is remarkably illustrated in Masaccio’s Trinity fresco (c. 1426) in Santa Maria Novella. Filippo Brunelleschi’s ground-breaking discovery and his important work in proportion for building as well as his adoption of mathematics would play an important role in the development of the period’s architecture. In his De pictura (1435), Leon Battista Alberti elaborated on the importance of linear perspective and optics. His call for a more naturalistic treatment of painting may have been inspired by a trip he made to northern Europe.

Studies in the geometry of vision, or optics, had their origins in ancient Greece. Euclid wrote on optics. His ideas were advanced by Ptolemy (c. 100–170) and further expanded on by Galen. Further experiments on the nature of light and its reception by the eye by Ibn al-Haytham (Alhazen, 965–c. 1040) remained central to the understanding of vision in the Middle Ages. Artistic practice and theory, which contributed to a better understanding of the sensory perception of light, color and form, had much in common with the scientific interest in empirical discovery. As a practical and theoretical tool, mathematics not only informed painters, draftsmen, architects, musician and philosophers, but it was also a way to connect with the classical past and unlock the mysteries of the natural world. Irrational ratios like the golden section and irrational numbers like π had their own cultural currency. Artists turned to mathematics to resolve questions of proportion and vision and to arrive at a clearer understanding of nature, while mathematicians sought to analyze natural phenomena; the behavior of numbers could be seen as an aesthetic question as much as a utilitarian one.

In the seventeenth century, it was believed that nature was mathematical in structure. A quest that had dogged natural philosophers in the seventeenth century was how to bring a quantity into a hitherto qualitative study of nature.¹¹ Galileo Galilei, Johannes Kepler and Isaac Newton advanced the study of optics through their study of astronomy. Galileo famously wrote:

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.¹²

The study of light and theories of vision advanced in the seventeenth century, particularly through the work of Kepler, who had uncovered laws of planetary motion, were grounded in mathematical patterns.¹³ Newton’s work on light and color, and reflection and refraction, went even further. Uncovering nature’s mysteries through mathematics had a parallel with the studies early modern artists engaged in. The discovery of the advances made in optics and the invention of the telescope in the Netherlands played a role in the naturalistic way northern European artists described reality.¹⁴ Mirrors and lenses could provide unusual optical effects and create spatial tensions within a painting. Architects also could achieve unusual light effects by manipulating forms and geometric shapes. Perspective, the advent of devices like the microscope and the telescope, optics and mathematics brought together representation and a new understanding of nature in unprecedented ways.

This volume has its genesis in two sessions I organized at the 100th College Art Association annual conference held in Los Angeles in February 2012. The interest in the conference papers and the lively discussions that ensued suggested to me that, in spite of the fact that visual culture and mathematics are rarely studied together in our specialist-focused universities, there exist common areas of interest that bring them together.

This volume consists of eight chapters that explore ways in which visual culture and mathematics interacted in the period from the fifteenth to the seventeenth centuries in Europe. The present volume makes no claim to cover the topic comprehensively or provide a parallel history of visual culture or the history of mathematics in the period but rather complements earlier studies such as Martin Kemp’s The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat, J. V. Field’s The Invention of Infinity: Mathematics and Art in the Renaissance and, more recently, Mark Peterson’s Galileo’s Muse: Renaissance Mathematics and the Arts, Alexander Marr’s Between Raphael and Galileo: Mutio Oddi and the Mathematical Culture of Late Renaissance Italy, and Robert Felfe, Naturform und bildnerische Prozesse: Elemente einer Wissensgeschichte in der Kunst des 16. und 17. Jahrhunderts.¹⁵

The contributors represent a broad range of disciplines that include art history, architectural history, mathematics, history of science, philosophy and economics. Together the chapters explore three main areas of focus—the role mathematics played in the period’s art theory; painters and the language of mathematics (painters express their mathematical knowledge); how Euclid offered not only practical solutions for structure, the representation of space and line for architects, painters and draftsmen, but how his geometry and optics engaged the viewer. The chapters cover a broad geographical area that includes Italy, Germany, Switzerland, the Netherlands, France and England.

The first section of the book, “The Mathematical Mind and the Search for Beauty,” addresses the role that mathematics played in defining Renaissance aesthetics, theories of vision and proportion in the search for beauty and harmony. It explores mathematical principles that enhanced both the liberal and the mechanical arts. This section opens with an chapter by John Hendrix, “Renaissance Aesthetics and Mathematics,” that examines the writings of Leon Battista Alberti, Nicolas Cusanus, Marsilio Ficino, Piero della Francesca and Luca Pacioli. Hendrix explores the underlying mathematical theories of these writers, derived from ancient authors such as Plato and Vitruvius, which contained concepts that could link nature, the human mind and the divine mind and which resulted in a kind of aesthetics and artistic creation. He returns to a concept mentioned earlier, that is, that mathematics played a different role than it does today. For Hendrix, mathematics played a fundamental role in defining the human mind as a microcosm of the cosmos and in cultural definitions of beauty and harmony.

In the next chapter, “Design Method and Mathematics in Francesco di Giorgio’s Trattati,” Angeliki Pollali addresses the interplay between Euclidean geometry and arithmetical solutions in two richly illustrated treatises by the fifteenth-century Sienese architect, sculptor and painter. In the following chapter, Matthew Landrus informs us that Leonardo helped his friend and improved Francesco’s method of technical illustrations. Earlier scholarship has emphasized Francesco’s use of arithmetic proportions derived from the human body, although Trattati I is informed by geometry. Pollali traces the scholarly preference for arithmetic back to the historiography of Renaissance ar...

Table des matières

• Cover
• Title
• Copyright
• Contents
• List of Figures
• Acknowledgments
• 1 Introduction
• PART I The Mathematical Mind and the Search for Beauty
• PART II Artists as Mathematicians
• PART III Euclid and Artistic Accomplishment
• Bibliography
• List of Contributors
• Index