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Lucid Proof: Greek Mathematics and the Alexandrian Aesthetic. By REVIEL 
NETZ. Cambridge and New York: Cambridge University Press, 2009. Pp. xv + 
272. Cloth, $99.00. ISBN 978–0–521–89894–2.

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CJ Online 2010.02.01

Lucid Proof, Netz’s (N.) third book-length study in Greek mathematics, 
both complements and departs from his earlier work. N.’s first book 
analyzed the style of Greek mathematical treatises in a general, abstract 
way, covering the full temporal range of Greek scientific mathematical 
writing (5th c. BC – 6th c. AD). [[1]] The second, more culturally and 
historically directed, treated the renovation of mathematics in the Middle 
Ages. [[2]] The current project focuses on one period of Greek mathematical 
writing (3rd – 2nd c. BC) and aims to analyze the scientific style of 
writing within the context of contemporary (Hellenistic) literary (poetic) 
style.

N.’s premise, to ground Greek mathematical writings “not in the 
generalized polemical characteristics of Greek culture, but rather in a 
more precise interface between the aesthetics of poetry and of mathematics, 
operative in Alexandrian civilization” (p. x), reflects current trends 
across the discipline in investigating the role of aesthetic, social and 
political ideology in the formation and reception of texts. [[3]] Although 
the socio-political settings for poetry and science differ, science, like 
literature, responds to its socio-cultural heritage and is “the creature 
of its own age” (p. 241). In four packed chapters, N. analyzes Greek 
mathematical works with a deep sensitivity for style, and he situates them 
within the broader intellectual landscape of the early Hellenistic world. 
Needless to say, there is much math in the book, but N. is methodical and 
cautious. His careful analysis, bolstered by exegetic diagrams, makes the 
material accessible even to the mathematically challenged. Moreover, Greek 
is used sparingly, and is translated when it is. Herewith, my only major 
complaint: full Greek texts of significant passages would be salutary. 
Although the text is often “thick,” much resembling the mathematics 
that serves as the core of the study, pertinacious readers will be rewarded 
by a deeper understanding of an important set of Hellenistic texts.

N. makes the simple assumption that “people do the things they enjoy 
doing” (p. x) and that literary style reflects this enjoyment. His 
animated style proves the point. Hellenistic mathematics is a verbal, 
textual activity, produced by educated, non-professional elites for an 
audience of genteel amateurs, most of them also consumers of Hellenistic 
poetry. Such readers find no charms in prosaic compositions of mensuration 
and other mundanely practical topics. Consequently, mathematical texts are 
stylistically playful, subtle and sophisticated. In his analysis of 
Archimedes’ Spiral Lines (3–14), for example, N. shows that Archimedes 
incorporates intrigue, suspense, surprise, variety and sharp transitions, 
with increasing opaqueness that deliberately obfuscates the mathematics and 
disorients the reader. In the end, Archimedes draws together contradictory 
propositions into an elegant multi-dimensional, surprising narrative, 
manipulating both arithmetic and geometry, straddling the physical and 
abstract, to the reader’s amazement. In Spiral Lines, an exegesis of 
complex multi-dimensional geometric concepts, Archimedes spins a text of 
coiling layers of propositions and proofs that narratively parallels the 
very mathematics under study. Content and style are interdependent and 
complementary. Spiral Lines, in fact, is presented as a verbal spiral. N., 
ever cautious, does not make so blunt an observation, but like Archimedes 
and other Hellenistic mathematicians, he allows modern readers to fit 
together the puzzle pieces for themselves. Mathematics becomes an activity 
shared by author and reader.

In the first chapter, “The Carnival of Calculation,” N. offers a close 
reading of several mathematical works on large numbers whose results are 
open-ended: e.g., Archimedes’ Stomachion, a tangram game; Aristarchus’ 
On the Sizes and Distances of the Sun and Moon; Eratosthenes’ Sieve, an 
algorithm to find prime numbers; and Archimedes’ Sand Reckoner, a 
numerical system to express very large numbers. The authors employ a 
complex, mosaic structure, “via complex, thick structure of calculation, 
to unwieldy numbers” (p. 20), which provides N. with his “carnival of 
calculation.” Content and style are integrated according to four main 
themes: (1) bounding the unbounded (a prominent aim in early Greek 
cosmologic and geographic initiatives); (2) demonstrating the opaque, 
cognitive texture of calculation (evoking the abstruseness and erudition of 
contemporary literature); (3) engaging in non-utilitarian calculation (just 
as Hellenistic literature may focus on apolitical themes); and (4) the 
Hellenistic fascination with size, both the extremely small and the 
extremely large. Several treatises, through series of complex calculations, 
lead up to “fantastically rich numbers…, contributing to a sense of 
dazzlement, of the carnivalesque” (p. 58), a direct reflection of 
contemporary political and military culture: the Ptolemaic pompe, the 
colossus, huge warships (p. 60). The act of calculation does not simplify 
or solve, but shows the complexity of the problem.

The second chapter, “The Telling of Mathematics,” centers on the 
narrative technique of suspense and surprise as employed by mathematical 
writers. N. shows that, despite the ostensibly impersonal nature of Greek 
science, authorial voice is successfully modulated. Many mathematical works 
are cast as letters, whose writer–narrator–characters interact directly 
with addressees, usually prominent historical figures explicitly invoked in 
the prologues. Science, no less than history or drama, can be highly 
personal, as recent scholarship has shown. [[4]] N. raises cogent questions 
about readership and the expectations of those readers (40–3, 75–80). 
Authors, writing as much for themselves as for those with shared interests, 
present challenging mathematical riddles, offering no pedagogic 
intervention to explain the flow of the text. To do so would cheat the 
reader of the delight of discovery. Where is the fun in reading a text that 
gives away the answers? The puzzle is meant to tantalize and to be solved.

In the third chapter, “Hybrids and Mosaics,” N. explores variety in 
Hellenistic Greek mathematics. Authors juxtapose (seemingly) unrelated 
threads in compositional variation. On the surface, most treatises seem 
incongruous. But critical reading shows that authors balance the abstract 
with the concrete, geometry with arithmetic, mathematical approaches with 
mechanical, and they even present multiple proofs of the same proposition 
to create a richer reading experience. In two separate treatises, for 
instance, Archimedes offers three discrete proofs for the basic measurement 
of the parabola, while Eratosthenes emphasizes the multiplicity of his 
abstract and mechanical approaches to duplicating the cube. Variatio is 
likewise reflected in the very topics studied in Hellenistic geometry—the 
description, mensuration and understanding of complex planar and spherical 
shapes, such as Nicomedes’ cissoids and conchoids. Further, mathematical 
nomenclature is drawn from visual but mundane vocabulary (shells, locks). 
N. discusses parallels from contemporary medical terminology (pp. 157–9), 
providing another example of the unexpected juxtaposition of the sublime 
(scientific) with the mundane. N.’s discussion could be further advanced 
with evidence from geography and other scientific fields. Eratosthenes 
reduces the landmasses to easily recognizable geometrical shapes 
(rhomboids, triangles), and metaphors drawn from daily life are 
deliberately and vividly applied to maps by Strabo and his predecessors. 
[[5]]

N. begins to connect science to literature by linking Hellenistic 
mathematics with earlier literature. Archimedes’ Sand Reckoner plays on 
an ancient poetic trope that dates back to Homer (p. 165). Eratosthenes 
appeals to mythology in Doubling the Cube, and Homer, no longer the 
divinely authoritative source, serves as a foil in his Geography. 
Nicander’s Theriaca is composed in hexameters. Archimedes’ Cattle 
Problem suggests a literary setting (Odyssey 12) and may even respond to 
contemporary Sicilian politics (pp. 168–9).

The fourth chapter, “The Poetic Interface,” is probably of greatest 
interest to CJ’s audience. Here, N. investigates how poetic conventions 
complement and parallel scientific style, and how poets weave science into 
literature. It is perhaps no surprise that N. branches beyond mathematics 
but restricts himself to passages whose scientific content is unambiguous: 
e.g., Apollonius of Rhodes’ Argonautica, with its interface between 
“modern” and mythic geography, ethnography and medicine; Callimachus’ 
geographically relevant Hymn to Delos and astronomically charged Lock of 
Berenice; and Aratus’ Phaenomena, whose hexameters straddle astronomy, 
astrology and meteorology. N.’s analysis of theme and purpose is 
satisfying. For example, he shows that The Lock of Berenice demonstrates 
the features of mathematical style lucidly explored in the first three 
chapters: duality of meaning, bounding the unbounded, and the impossibility 
(unsolvability) of the task. Callimachus also effectively retains the 
scientific context of Conon’s original astronomical discovery (or, 
rather, declaration) while blurring the precise geometrical reference in 
utilizing a mathematically charged phrase: en grammaisin, to do something 
based on a diagrammatic representation in geometry (p. 179; see also 195). 
One might like to have seen even more analysis of scientifically charged 
vocabulary as used by the poets (technical terminology in Apollonius; 
medical vocabulary in Theocritus?). Likewise useful would have been further 
social contextualization of science in literature, along the lines of 
N.’s observation that Leto’s new-fangled upright position for giving 
birth, perhaps, derives from Herophilus’ theories in obstetrics (p. 194).

Lucid Proof is a welcome addition and valuable complement to the growing 
body of scholarship in Greek science, especially those works that 
investigate the nexus between science and literature, including Cuomo on 
practical mathematics and Romm on geography. [[6]] N. successfully 
contextualizes scientific activity within the Hellenistic intellectual 
landscape and personalizes these authors, men of lively intellect who 
approached mathematics with éclat and vivacity, spinning page-turning 
tales of suspense and mystery. Undoubtedly (or at least hopefully) this 
study will inspire some among the mathophobic to read these gems of 
Hellenistic mathematics with a fresh eye.

GEORGIA L. IRBY-MASSIE
College of William and Mary

[[1]] The Shaping of Deduction in Greek Mathematics (Cambridge, 1999).

[[2]] The Transformation of Mathematics in the Early Mediterranean 
(Cambridge, 2004).

[[3]] N. does not deny the polemical nature of Greek science, but merely 
adds a new layer to the interpretation of Hellenistic mathematics. For 
contentiousness in Greek science, see G.E.R. Lloyd, “Democracy, 
philosophy, and science in ancient Greece,” in J. Dunn (ed.), Democracy: 
the Unfinished Journey (Oxford, 1992) 41–56; T.E. Rhill, Greek Science 
(Oxford, 1999) repr. Cambridge, 2006.

[[4]] G.E.R. Lloyd, “Mathematics as a Model of Method in Galen,” in 
R.W. Sharples (ed.), Philosophy and the Sciences in Antiquity (Aldershot, 
2005) 110–30; “The Meno and the Mysteries of Mathematics,” Phronesis 
37 (1990) 166–83; Katherine Clarke, “In Search of the Author of 
Strabo’s Geography,” JRS 87 (1997) 92–110; Roger Brock, “Authorial 
voice and narrative management in Herodotus,” in Peter Derow and Ruth 
Parker (eds.), Herodotus and his world: essays from a conference in memory 
of George Forrest (Oxford, 2003) 3–16; Ismene Lada-Richards, “Authorial 
voice and theatrical self-definition in Terence and beyond: the 
‘Hecyra’ prologues in ancient and modern contexts.” G&R NS 51(2004) 
55–82.

[[5]] Klaus Geus, “Measuring the Earth and the Oikoumene,” in Richard 
J.A. Talbert and Kai Brodersen (eds.), Space in the Roman world: its 
perception and presentation (Münster, 2004) 11–26; Daniela Dueck, “The 
parallelogram and the pinecone: definition of geographical shapes in Greek 
and Roman geography on the evidence of Strabo,” AncSoc 35 (2005) 19–57.

[[6]] S. Cuomo, Ancient Mathematics (London, 2001); J. Romm, The Edges of 
the Earth in Ancient Thought: Geography, Exploration and Fiction 
(Princeton, 1992).


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