J. V. Field’s *The Invention of Infinity: Mathematics and Art in the Renaissance* examines the relationship between art (primarily painting) and the mathematics of perspective from the end of the Middle Ages to the late seventeenth century. The study is a synthetic work that integrates a variety of mathematics with the history of art from Giotto to La Hyre. Additional topics investigated include social history, calendars, sundials, theatrical architecture, and fresco.

Field’s technical knowledge of a variety of disciplines is refreshingly competent and clear. The discussions reflect not only an ability to grasp old mathematical texts but also demonstrate an understanding of the technical aspects of art. *The Invention of Infinity* examines in detail a number of artistic works, mathematical thinkers, and particular treatises related to the development of perspective. The main thrust of the study concerns the growth in technical understanding of the mathematics of perspective (as well as other topics such as conic sections), but the reader will also gain insight into the history of art and fine art.

Field displays both artistic and mathematical sensitivity. A painting’s value as art is often irrelevant to the consistent employment of mathematical ideas about perspective. Field makes this quite clear. In the analysis of Masaccio’s *Trinity* fresco, which was personally examined in detail, the author concludes that “As visual illusionism the *Trinity* undoubtedly works…” (59), and more importantly, that “a picture so impressively visually correct as the *Trinity* can turn out to be mathematically faulty is a warning against confusing artist with mathematician.” (61)

After nodding to the “learned tradition” of the Middle Ages, Field anchors the history in a clear description of Alberti’s construction, which was a geometric technique allowing the creation of pictures in correct perspective, particularly those with elaborate pavements (*pavimentos*) in paintings such as Veneziano’s *Madonna and Child* as well as in a variety of other painting subjects, including military landscapes such as Paolo Uccello’s *Rout of San Romano*. The diagrams of the basic distance point construction and the “extravagant version” of this construction are plain to even the mathematically untrained. The only notable exception to the clarity of presentation is why an ideal viewing *point* of a particular painting might exist at all (rather than, say, an ideal range of viewing).

The reader discovers one of the more interesting sections of the study in Chapters 4 and 5, “Piero Della Francesca’s Mathematics” and “Piero Della Francisca’s Perspective Treatise.” (62–114) Francesca was both an artist and a mathematician, and three of his treatises on perspective have survived. Field treats the reader to a detailed analysis of each, with numerous examples explained, many of which show the historian of mathematics translating the language of the past into present terms so that those unacquainted with manners of fifteenth century expression can nonetheless understand.

Of particular interest are Francesca’s diagrams which, as the author expresses it, “invit[e] us to visualize what the solid actually look[s] like.” (73) Of equal interest is Field’s investigation into social history, namely the evolution from mathematical treatises directed to the artist with little or no mathematical interest *per se* to the “infiltration” of practical mathematics into the upper classes. This interest was due in part to the need to design elaborate military defenses in response to the role of artillery and small arms in the sixteenth century. As the officer corps in every European state was dominated by some collective aristocracy, mathematics became a “polite” topic of conversation.

In Chapter 7, “The Professionals Move In” Field describes the transition to treatises on perspective by mathematicians “for mathematicians.” (161) Under the influence of the need for calendar reform, the popularity of sundial problems, and the beginning of serious editing of Greek mathematics in the sixteenth century, the analysis of perspective experienced a decisive shift from practical problems and manuals for the artist to serious constructions and proofs. Field discusses Federico Commandino’s editing of the ancients, Giovanni Benedetti’s entirely new manner of “jump[ing] straight into three dimensions” (162) and Guidobaldo del Monte “at last seeing the three-dimensional ‘pyramid of vision’, with its vertex in the eye.” (176)

Much of the latter third of Field’s investigation concerns conics, particularly those of Benedetti and Girard Desargues. Desargues is given lengthy treatment in Chapter 8, “Beyond the Ancients”. This section will strike the reader as more difficult, reflecting Desargues’ confusing notation, cryptic manner, and tendency of “trying to do too much at once.” (203) In Desargues one finds rigorous proofs of perspective ideas, that with the skilled interpretation that Field provides can be understood visually in three dimensions. (See the diagram for two triangles in perspective on page 221.)

One of the most useful aspects of Chapter 8 and, indeed, much of the study is the author’s clear and often delightfully chatty guidance through the labyrinth of non-modern (as in pre-19th century) mathematical texts in Latin, Italian, French, and German. One of the more gratifying of the author’s analyses is that of the 1649 painting, *Geometry*, by Laurent de La Hyre in which we see paintings within paintings and diagrams from Euclidean proofs. Field presents a series of observations and deductions which allow for the precise identification of the centric point of this painting. This is no small feat and one quite rarely accomplished.

Despite the conceptual significance of Desargues’ work (in that it founded a projective geometry rather than merely a mathematization of perspective), his contributions “dropped out of sight for nearly two centuries before the main results were rediscovered independently.” (224) Thus, his work did not enter “the mainstream mathematical tradition” (226) except for a notion of space transmitted by one of his readers, Blaise Pascal. Field makes the very interesting claim that, “it is possibly under the influence of Desargues’ rigorous mathematical treatment of infinity in geometry that Pascal was able to accept the idea of infinite space as an entity in its own right…For Pascal explicitly, as by implication for Desargues, geometry has become the science of space”. (227) Whether such a claim is true or not would require a good deal more investigation.

One of the frustrating aspects of the book is its dependence on the reader to provide some of the connective tissue among the disparate topics (however interesting) covered. The author assumes that it is quite clear that the history of the ideas is flowing from early attempts at creating perspective painting to notions of infinity from rigorous mathematics. This assumption is misplaced. If one is motivated to read this book by curiosity about the evolution of infinity’s “invention” one will be sorely disappointed. It is a topic not explicitly mentioned until page 195!

Towards the end of this study, Field asserts that “Desargues is the first mathematician to get the idea of infinity properly under control.” (196) This may indeed be true given our present knowledge, but the interested reader looks in vain for an open and clear discussion of infinity. One’s interest is peeked by the comment, “[Desargues] states that a set of parallel lines is a set of concurrent lines whose meeting point lies at an infinite distance, and then goes on to give a rigorous discussion of the mathematical properties of the ‘point at infinity’ at which such parallel lines meet.” (196) One looks in vain for this discussion of Desargues’ understanding of infinity or of his place within that long, arduous, and still incomplete conquest of the infinite that has instilled fear and awe since the age of the Pre-Socratics. Perhaps one’s curiosity regarding these historical and mathematical issues might be assuaged by a perusal of Field’s works listed in the bibliography (J. V. Field and J. J. Gray, 1987, *The Geometrical Work of Girard Desargues*).

The bibliography is one of the more disappointing parts of the book, listing a mere thirty-six works, of which nine are Field’s.

The juxtaposition of many of this study’s topics often raise (and sometimes answer) very interesting questions about the relationship between mathematics and art, social change, history, technology, theater and astronomy. In his discussion of manuals, which attempt to show how one might construct perspective-based drawings, Field points out that their intent was not to explain the meaning behind the process (that is, the mathematics) but merely the process itself.

The evolution of the book takes the reader from practice to theory but I do not think that, thereby, Field implies that practice is inferior to concept or proof. There is, however, a tinge of the Whig theory of history in *The Invention of Infinity*. Of course, the growth in the Western canon of mathematics is undeniable and impressive. Yet Field apparently cannot resist acerbically remarking on the popular (and sometimes scholarly) view of the Renaissance as a period of brilliant diversity of knowledge. In the context of the significance of geometry to art the author rails,

There looms again the spectre of the sentimental notion of the Renaissance Man (very rarely Woman) whose wonderfully universal knowledge gave rise to works beyond the reach of the petty specialists of the present day. Even in subtler forms, any such notion is rubbish. There is now more to know. Most twelve year olds in the Western world have more information about almost anything scientific than Dante did. (233–234)

One wonders if Field is perhaps creating a straw man. Dante is usually seen as a medieval figure rather than a man of the Renaissance. The parenthetical nod to political correctness is gratuitous. Further, given Field’s profession, one wonders about how much contact the author has had with twelve or even eighteen year olds. That information is available often fails to imply that young people or even people in general know or desire to know much about it. Working with secondary students in the modern Western world, certainly in the United States, engenders a less sanguine perspective than Field’s.

*The Invention of Infinity* gives the reader great insight into the practice of mathematics in history. As such it is instructive to anyone attempting to understand old texts. As an example of interdisciplinary work, this book is excellent. It would be a useful text or case study in an art history class, a class on the Renaissance, or a class in the history of mathematics. Field’s study is particularly helpful in that it demonstrates the possibility of sympathetically approaching what are traditionally rather different disciplines. The author is technically impressive and at the same time a lover of art. In that sense, Field represents the best in the old idea of the Renaissance Man despite the disparaging remarks about such romantic notions.

Peter K. Benbow is Headmaster of the Eureka School of Santa Barbara. He studied the history of mathematics under John North at Oxford.