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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