Reviel Netz,

*Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic.*Cambridge/New York: Cambridge University Press, 2009. Pp. xv, 255. ISBN 9780521898942. $99.00.

Reviewed by Anne Mahoney, Tufts University

The argument of this book is that the style of Hellenistic mathematics is "comparable to that of contemporary literature" (p. x): playful and complex. This is the third part of Netz's extended project on Greek mathematics and, like the earlier books, it is a beautiful, creative study of well-known texts from an unexpected point of view.^{1} The main character in this story is Archimedes, "a giant of mythical magnitude; perhaps, of all Greeks, the one to have achieved the most" (p. 198); here we see not only Archimedes the brilliant mathematician, dimly grasping ideas that would not be fully developed for almost two thousand years, but also Archimedes the writer, "master story-teller" (p. 229, the last words of the book) and craftsman of Greek prose. Netz argues that the way Archimedes and other Hellenistic mathematicians present their mathematics is designed to please the reader: in this period, mathematical texts are playful, personal, even poetic.

This may sound paradoxical to modern ears. As Netz points out, in our time "the people who read and write mathematics lead a life totally separate from that of the people who read and write poetry" (p. 228; perhaps slightly over-stated). The aesthetic concerns of 21st-century math and those of 21st-century poetry are quite different. But in the Hellenistic period (Netz focuses especially on 250 to 150 BC), the writers of mathematics seem to have picked up some aesthetic values, generic trends, from the writers of poetry: mathematicians and poets both participated in "the kind of verbal art favored in the Hellenistic world" (p. x). That is, the style or feel of a Hellenistic mathematical text is not so different from the feel of Hellenistic poetry, by Callimachus, say, or Posidippus.

Netz begins with a case study, the *Spiral Lines* of Archimedes. In this work, Archimedes defines a particular kind of spiral curve, then proves four major facts about its area and its relationship to circles circumscribing it. The work is framed as a letter to Dositheus, best known for being the addressee of this and several other works by Archimedes (p. 2). It starts with an introduction, in which Archimedes mentions several other sets of problems, but then readers "learn all of a sudden -- four Teubner pages into the introduction -- that this is going to be a study of spirals" (p. 3). From the very beginning, Archimedes raises readers' expectations, then goes off in a different direction. The introduction is densely written, and the proofs that lead up to the main goals are not motivated: "no effort is made to explain their evolving structure" (p. 4). Instead, Archimedes starts with theorems about *linear* motion, then goes on to some abstract geometric results about circles. Then come two results about proportions or ratios, "enormously opaque" (p. 6) and with no obvious relationship to spirals, or indeed to anything else we have seen so far. We are now roughly halfway through the work, and only now does Archimedes get back to spirals. First, Archimedes proves the *second* of the four results he set up as goals in the introduction, using some of the apparently irrelevant results about motion and about circles that have come before. Then he shows some additional results that follow from this fact. As Netz observes, "at this point, therefore, the reader is thoroughly disoriented: the next proofs can be about some further consequences of goal (ii), or about goal (iii), (i), or anything else" (p. 10). What actually does come next is a discussion of "bounding the spiral area between sectors of circles" (p. 10), with no particular reason why.

"Then we reach proposition 24 and now -- only now! -- the treatise as a whole makes sense, in a flash as it were" (p. 10). Archimedes suddenly pulls everything together, showing how all the apparently disparate, unrelated discussions about motion, circles, and parts of circles all actually do relate to the problems announced in the introduction. From here to the end, it isn't exactly smooth sailing, but the remaining propositions of the treatise are somewhat more obviously connected, and the last two things proved are the last two of the four goals from the introduction.

Netz brings out several features of this treatise that, he argues, are typical of Hellenistic mathematical writing in general (p. 12-14). First of all, there's a ton of calculation. Next, the treatise revels in straddling the boundaries between genres -- concrete physics and theoretical abstraction, geometry and arithmetic. Finally, the rhetorical structure is neither "axiomatic" (like a modern mathematical paper) nor "pedagogical" (like a textbook) but a "mosaic" (p. 13), in which the pieces come from different conceptual areas, in no obviously logical sequence, but ultimately come together. "Archimedes made a deliberate choice to produce a mystifying, obscure, 'jumpy' treatise. And it is clear why he should have done so: so as to inspire a reader with the shocking delight of discovering, in proposition 24, how things fit together; so as to have them stumble, with a gasp, into the final, very rich results of proposition 27-28" (p. 13-14). The next three chapters of the book take up each of these points, with examples and analyses from other Hellenistic mathematical works.

The final chapter, entitled "The Poetic Interface," looks at poetic texts rather than mathematical ones, to show the similarities between these two groups of writings. Netz looks at some texts that use science: Aratus's *Phaenomena*, of course, but also the geography in Apollonius Rhodius' *Argonautica*, astrology there and elsewhere, the fragments of didactic epics by Nicander, and the *Lithica* and *Iamatica*, "stone poems" and "recovery of health poems," from the New Posidippus. The next section of the chapter compares the florid calculations of Hellenistic mathematics with the piling-up of erudite details in Hellenistic poetry. Here Netz chooses Callimachus's *Hymn to Artemis* as a case study rather than something more overtly erudite like the *Aitia*, partly because the hymn is both complete and a whole lot shorter than the *Aitia*, but more importantly because even this non-didactic work displays the same "carnival of erudition" as the longer one (p. 206). The close reading of the hymn (p. 200-206) is particularly well done. Two more sections of the chapter show how the "mosaic" structure Netz finds in some mathematical works is also available to poets; here examples include Apollonius Rhodius and Theocritus. The concluding section brings the threads together, observing that in the Hellenistic period, in both science writing and poetry, we have "a certain carnivalesque play of detail, with its ironic self-undercutting; mosaic composition; narrative surprise; indeed a certain tendency to experiment with one's very generic boundaries" (p. 227); the period is marked by a "scientific-poetic program of multigeneric experimentation" (p. 229).

Although one may argue with some of the details here (for example, the idea explored p. 130 ff, that presenting a second, wholly different proof of a result "undercuts the very notion of a definitive proof" and demands a "suspension of disbelief granted ... by the reader"), the overall case is convincing. Netz gives both mathematical and poetic works the same kind of close reading, bringing out the poetry in the science and vice versa. Mathematical facts can be presented in any number of ways, with more or less explanation or motivation, more or less concern for clarity or elegance. Netz argues that mathematicians in the Hellenistic period chose to present their work in the same style as contemporary poets. This playful, exuberantly erudite, deliberately complicated style was part of the culture. As he puts it, "in some sense, Hellenistic mathematics could not have been otherwise" (p. 240). Readers interested in mathematics, Hellenistic literature, or the history of aesthetics will find this a valuable book.^{2}

**Notes:**

1. Netz's first general study is *The Shaping of Deduction in Greek Mathematics*, reviewed here as 2000.02.17; the second is *The Transformation of Mathematics in the Early Mediterranean*, reviewed as 2004.10.25. Reviews from a more mathematical point of view have appeared in *Mathematical Reviews* and can be found in MathSciNet: MR1683176 (2000f:01003) is a review of the former by J. L. Berggren and MR2072579 (2005m:01008) of the latter by Benno Artmann.

2. All the Greek is translated. So is much of the mathematics, into modern terminology and notation. There are 36 diagrams and a 7-page bibliography.

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