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The Mathematician Sees Double: Egyptian in Eratosthenes

Marquis Berrey

            Stephens 2003 and others have shown the ability of Ptolemaic court poets to "see double" by manipulating the symbols and narratives from classical Egyptian ideology of kingship to portray the bicephalous Ptolemaic monarchy.  Natural scientists also dedicated treatises to Ptolemaic rulers and participated in the discourses of the court.  Among them was Eratosthenes of Cyrene, who produced both poetry and scientific texts for Ptolemy III Euergetes.  SH 397 suggested that Eratosthenes' poem Hermes (interwoven with science) might have told of Thoth's invention of letters, since interpretatio graeca equates Thoth with Hermes.  Might his court science have seen double too?  The text on the doubling of the cube ascribed to Eratosthenes (Eutocius Arch.Sph.Cyl.II 88.3-96 Heiberg) is his most extensive piece of extant science.  While earlier scholarship debated its authenticity (Knorr 1989, Geus 2002, Vitrac 2008), Netz 2009 has indirectly argued for authenticity by contextualizing it within contemporary Alexandrian aesthetics: the text shows a characteristically Alexandrian concern for belatedness.  The text's belatedness provides insight into the Ptolemaic court scientist's appropriation of Egyptian symbols.  I argue that the text shows Egyptian symbols only of contemporary Ptolemaic Egypt because of the agonistic context of Hellenistic Greek science.

            The text details Eratosthenes' solution to a classic problem of proportionality.  The text itself is double: the letter begins with an address to Ptolemy, followed by a history of the problem, the usefulness of Eratosthenes' instrumental solution, and a geometric proof; the dedication (including an 18-line epigram) reverses the order.  Read as halves, a difference in the utility of Eratosthenes' innovation stands out: the letter highlights (90.13-27) preserving the proportions of altars and temples and enlarging war machines; the epigram highlights (96.10-13) preserving the measurements of animal-pens, granaries, and wells.  The letter's audience is the king; the dedication's audience are the scribes of the temple economy.

            There are continuities between Eratosthenes' mathematics and classical Egyptian mathematics in both purpose and rhetorical style of proof.  For example, pRhind contains problems on measuring produce in granaries of cylindrical and orthogonal shapes (nos.41-46) and problems on distributing them (nos.69-78).  The practical problems confronting scribes would be in adjusting for different granaries' shapes and preserving the proportions for those measurements (Imhausen 2003): Eratosthenes' solution to the problem of proportionality provides answers to these practical scribal needs.  Moreover, the rhetorical form of the entire Eratosthenic text supports a diversity of proof styles from a formal Euclidean proof to a mechanical proof similar to Archimedes PE and Meth.  Recent scholarship on the history of mathematical proof has focused on the comparative role of the rhetoric of mathematical persuasion within a given social context (Chemla 2012).  The proceduralism of Eratosthenes' mechanical proof is comparable to the algorithmic style of certain Egyptian proofs, so Eratosthenes' proof styles may vary to persuade different social audiences.

            Yet it is not clear that Eratosthenes references the mathematics of the classical Egyptian past.  Problems of proportionality in calculating the measures of granaries and their distribution existed within contemporary Ptolemaic society as much as within the classical Egyptian past: the mathematical purpose is explicitly not belated.  The rhetorical form of Eratosthenes' mechanical proof is located ambiguously between the styles of classical Egyptian proofs and contemporary non-Euclidean Greek proofs.  Most importantly, the egotism of Eratosthenes' solution and the denigration of past mathematicians in both the letter and epigram (88.13-90.11; 96.16-19) are located within the agonistic context of Hellenistic Greek science: the text's concern for scientific belatedness is exclusively Greek.   In Hellenistic science past innovators of scientific knowledge rarely represent the best current state of that knowledge, whereas Greeks thought Egyptian science fixed and nearly unchangeable (Lang 2012: 128-34).  Eratosthenes' science thus references contemporary Ptolemaic Egypt, not its classical Egyptian past.  Eratosthenes' letter to Ptolemy III Euergetes can be best contextualized within a political ideology of the monarch's care for his current subjects, a presentist scientific contribution of the Ptolemaic court scientist.

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Scientific Modes of Perception and Expression

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