The edited volume of Ingrid Alexander-Skipnes, a lecturer in art history at the Kuntsgeschichtliches Institut of the Albert-Ludwigs-Universität Freiburg and an associate professor emerita of art history at the University of Stavanger, Norway, is a collection of essays on the cultural history of mathematics and art, with a focus on the early modern period. The title is very appropriate, since visual culture and mathematics have mutually enriched each other from prehistoric times to the present day. From medieval times and the Renaissance, mathematics became the key support of innovations in the arts and of the emergent modern science and thence of modern technology and engineering. This book covers this historical period. It promotes new models of inquiry and new narratives of early modern art and its history, with a focus on mathematics. It is organized in nine chapters covering how mathematics developed in Europe between 1400 and 1800, in relation to painting, sculpture and architecture, with special reference to religious and/or ritual practices.

The origin of this collection is the presentations and discussions in two sessions organized by the editor during the 100th Annual Conference of the College Art Association held in Los Angeles in 2012. Chapter 1 is a generous introduction by the editor, explaining the range and the scope of each of the eight chapters commissioned from experts in various specialities. The chapters are organized in three parts. Part I, ‘The mathematical mind and the search for beauty’, is divided into three chapters; Part II, ‘Artists as mathematicians’, has two chapters; Part III, ‘Euclid and artistic accomplishments’, has three chapters.

Renaissance is a moment of definition of new aesthetics and the introduction of new elements for beauty and harmony, such as new theories of vision and of proportion. John Hendrix, a professor at Roger Williams University in Bristol, Rhode Island, wrote Chapter 2 on ‘Renaissance aesthetics and mathematics’. From the beginning of his chapter, he clarifies that he uses the word ‘aesthetics’ to mean philosophy of art. He examines the contributions of Leon Battista Alberti, Nicolas Cusanus, Marsilio Ficino, Piero della Francesca and Luca Paccioli. These scholars relied much on the ancient writers Plato and Vitruvius. The author does not discuss their approaches to mathematics, but the way they were influenced by their theories and how they were decisive in the formulation of a new philosophy of aesthetics. Hendrix offers a long analysis of Alberti's De re aedificatoria, with many references to Vitruvius. Some paragraphs are devoted to Plato's Timaeus. Chapter 3 on ‘Design methods and mathematics in Francesco de Giorgio's Trattati’ is authored by Angeliki Pollali, an assistant professor of art history at Derec–the American College of Greece. She claims that Francesco's intention was to establish a correspondence between geometrical and numerical methods, and proceeds with a detailed study of his works. The final chapter of Part I is written by Matthew Landrus, a research fellow in the history of art at Wolfson College, University of Oxford. In this chapter on ‘Mathematical and proportion theories in the work of Leonardo da Vinci and contemporary artists/engineers at the turn of the sixteenth century’, Landrus shows that Leonardo favoured visual solutions to numerical ones and briefly discusses how many of his contemporaries used proportional geometry in pictorial, mechanical and architectural projects.

In Part II, Chapter 5, on ‘Dürer's Underweysung der Messung and the geometric construction of alphabets’, is by Rangsook Yoon, an art historian and curator of many exhibitions who has devoted much of her research to Dürer's work. In this chapter, she concludes, based on the techniques and theoretical comments of Dürer and contemporaries, that everything, of nature or of human invention, can be explained through mathematics, including the letters of an alphabet. The next chapter, on ‘The mathematical use of ϕ and π in the paintings of Piero della Francesca’, is authored by Perry Brooks, who teaches Italian Renaissance art at the State University of New York at Stony Brook. The author claims that Piero della Francesca was fascinated by proportions and by irrational numbers and goes through a detailed study of how ϕ and π were dominant features in his paintings.

In Part III, Chapter 7, on ‘The point and its line: an early modern history of movement’, is written by Caroline O. Fowler, a postdoctoral associate in the physical history of art at Yale University. She points out how, in the period covered by this collection of essays, scholarly concerns with movement were influential in making the line the foundation of artistic practice. The next chapter, on ‘Between a golden ratio and a semiperfect solid: Fra Luca Pacioli and the portrayal of mathematical humanism’, is authored by Renzo Baldasso, an assistant professor at Arizona State University, and John Logan, an independent researcher in art history and the history of mathematics. The authors' focus is the interpretation of Euclidean figures in the famous portrait attributed to Jacopo de’ Barbari. The final offering, Chapter 9, by Ingrid Akexander-Skipnes, the editor, is on ‘Mathematical imagination in Raphael's School of Athens', and offers a detailed analysis of the painting, showing how a distinguished painter contributed, in a unique and pioneering way, to synthetic pictorial history of mathematics.

The book closes with a bibliography of around five hundred items, details of the contributors and a helpful index. This book is an important scholarly contribution to the history of early modern art and its relation to science and mathematics.