This rich book is a study of style in the fields of Hellenistic mathematics and Alexandrian poetry. Following up The Shaping of Deduction (1999) and The Transformation of Mathematics (2004), N. has provided another groundbreaking and provoking study of ancient Greek mathematics. N. claims that Hellenistic mathematics is characterised by a certain ‘ludic’ style which compares to Alexandrian poetry. The book thus consists of two unequal parts: the first three chapters describe ludic style in mathematics, especially in Archimedes, whereas the last chapter contextualises the findings by searching for stylistic parallels in Hellenistic poetry. N. tells a simple, but compelling story: Hellenistic mathematics was like Hellenistic poetry, playful, surprising, ludic, until the overriding pedagogical intent of later times suppressed this. The volume provides a coherent reading of two areas of Greek literature that have, to my knowledge, never been put side by side.

N. distinguishes three types of mathematical treatises, ludic, survey and pedagogic. By way of a long series of detailed studies of Hellenistic mathematical works both extant and lost (N. discusses almost all the works of Archimedes plus, e.g., Hipparchus, Aristarchus, Apollonius, Diocles, Hypsicles, Eratosthenes and some passages of Euclid), he stresses the ludic as a typical feature of Hellenistic mathematics. The ludic comprises features such as the non-utilitarian, playful, even carnivalesque character of calculation, playful opacity of argument, fascination with size, as well as rhetorical and narrative aspects in presentation and arrangement. Surprise is what these authors were aiming at, at the level both of the proof and of the treatise. Archimedes emerges as a Hellenistic master-narrator (throughout, N. understands the arrangement of propositions in a mathematical book as a form of narrative), Hellenistic mathematics as an elitist practice meant to ‘dazzle and overwhelm’ the reader (p. 54). In this context, one waits for the notion of mathematics as a ‘game’ which, however, is never applied. In many cases, N. looks at well-known facts in a new way, for example, at the calculations of the earth's circumference as a symptom of the Hellenistic ludic fascination with size. The ludic, according to N., has a complicated alliance with personality: it utilises the mathematician's voice in different ways. These are discussed with respect to intertextuality, three levels of which N. distinguishes in mathematical writing. Everywhere, N. sees ‘mosaic’ structures of argument, that is, structures that surprisingly juxtapose arguments from different fields, and claims the resulting hybrid objects as typical for the Hellenistic ludic style. Before N. turns to poetry, he looks at points of contact between Hellenistic mathematics and Alexandrian poetry, for example, Eratosthenes' poem on finding the two mean proportionals, Archimedes' Sand-reckoner, Hipparchus' commentary on Aratus, etc. Much of this is understood in the terms of the ‘mock-serious’. This first part includes many important asides and excursuses on, for example, the ‘triumph of the impersonal’ in later Greek mathematics and scientific naming.

For the second part, ‘The Poetic Interface’, N. shifts gear and, in order to show that poetic practices are on the one hand complementary to those of mathematics and on the other parallel to them, he discusses Hellenistic poetry mainly by commenting upon two recent surveys, M. Fantuzzi & R. Hunter (2004) and G.O. Hutchinson (1988). The most compelling analysis in this part is his description of the ‘carnival of erudition’ (p. 175), analogous in function and, partly, even formally, to the ‘carnival of calculation’. In the remainder of the book, N. searches for duality, juxtaposition, ‘bifurcations’, mosaics, contrasts, etc., in Hellenistic authors such as Aratus, Nicander, Posidippus, Callimachus (along the way, he suggests that Callimachus fr. 191.60 f. Pf. be read in the light of Euclid, Elem. 11.14), Theocritus, Apollonius, Asclepiades and Lycophron. Of course, Hellenistic poetry is full of these. N. comes into his own when he takes up these stock judgements on Hellenistic poetry such as its ‘erudite’ character, and tries to fill it with new life in the light of the mathematical writing. His analysis of Hellenistic poetic display of knowledge as carnivalesque, due to its asymptotic character, i.e. the frustration of closure and its tendency to perform ‘a certain holiday away from sublimity while still being directly engaged with it’ (p. 209), is compelling. The chapter is a running, certainly self-conscious and sometimes ironic, commentary on Hutchinson and Fantuzzi & Hunter that brings out the ‘ludic’ in Hellenistic poetry through brief examples. Here N. discovers precisely the same structures as those that determined his description of Hellenistic mathematical literature. The main question, however, is what these stylistic parallels actually mean, a question which N. leaves open.

From the beginning, N. keeps his project firmly within the confines of style, that is, of formal analysis. He succeeds admirably in showing that there are many close parallels between the style of mathematical writing and Hellenistic poetry, all involving some form or other of the ‘ludic’. In many ways, the two fields match so perfectly in N.'s presentation that the reader may be inclined to assume that the categories of description of mathematics were inspired by the canonical research on Hellenistic poetry. The book is more successful when it discusses mathematics, to which it brings a new approach: one might say that N. has managed to contextualise the notion of beauty in Hellenistic mathematics.

There are, however, some gaps in the argument. I will conclude by pointing out three: first, visual art has too much to contribute to questions of style, size and narrative that one can afford to ignore it entirely (at the same time, this is why P. Galison's metaphor of ‘critical opalescence’ [p. 228] does not apply). N. could have made his account even more convincing had he integrated some passages on visual material. As the visual material would have shown, the story must be more complex than the pattern ‘a is influenced by b = Archimedes reads Callimachus’, which is, more or less, what N. advocates, shifting the burden of explanation from historians of mathematics to those of poetry (although he occasionally hints at the court context as a possibly unifying frame of the two fields). Such explanation would have profited from bringing more comparanda to the comparison. At one point, N. chooses to understand the parallels of style as homology (p. 230). At least taken in the way P. Bourdieu does (e.g. in Distinction, transl. R. Nice [1984], pp. 175–208), the concept has the potential to explain the parallel, a path N. does not follow.

Second, much recent research on Hellenistic poetry has tried to arrive, through analysing form, at hypothetical reconstructions of the meaning of these forms for their respective primary audiences. Antiquarianism, for example, lends itself to certain political implications; intertextuality in Hellenistic poetry, of which N. makes too little, comes with evocations of a fictitious, ideologically loaded past; ethnological lore presents imaginary spaces that refer readers to their actual worlds, etc. To be sure, N. occasionally comments on mathematical texts in this vein, for example, when he briefly puts Archimedes' Cattle-problem in a court context. Third, N. operates with the questionable category of the ‘Hellenistic aesthetic’, as if times per se could arrive at aesthetic choices. It is human subjects who make such choices, and for specific reasons that one can try, at least, to describe in terms of social groups, contexts and motivations.

Despite occasionally tending towards exaggeration (the reader is always ‘shocked’, juxtapositions are usually ‘jarring’, debates ‘rage’, etc.), N. has kept his account amazingly lucid; on occasion, however, the ludic prevails. The book could, therefore, serve a purpose perhaps not anticipated by N., namely to provide a brilliant and much-needed introduction to Hellenistic mathematics for scholars of literature and, at the same time, a context-check to historians of mathematics. Due to the masterful survey of Hellenistic mathematics and its many facets on display here, the volume deserves to be read alongside such classics as Fantuzzi & Hunter's Tradition and Innovation.

I noticed few slips (the historian of Egyptian mathematics' name is Imhausen, the nymph's name Britomartis).