Abstract
Historians of ancient philosophy and theological writers often come up against the puzzling issue of understanding the meaning of the term hypostasis used by different ancient authors. One could hardly expect that the same issue would be of interest for historians of ancient mathematics. Indeed, altogether absent from the works of Euclid, Archimedes, and Apollonius, and scarcely appearing in a nonmathematical context in the works of Heron and Nicomachus, the term hypostasis and its cognates appear 127 times in the six books of Diophantus’ Arithmetica preserved in Greek. This chapter examines Diophantus’ use of the term hypostasis and argues in favour of interpreting it as a term for numbers qua specific, individual entities. It is composed of three parts. The first part discusses the different statuses of numbers in a worked-out problem according to Diophantus’ general method, and the relevant issue of the Diophantine conception of an arithmetical problem; the second part investigates all instances of the term within Diophantus’ text; and the third part surveys briefly the testimonies of the Byzantine commentators of the Arithmetica, which provide further evidence supporting the interpretation proposed in this paper.
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Notes
- 1.
- 2.
- 3.
Quotations from the Arithmetica without mentioning the editor refer to Tannery’s edition.
- 4.
See Smith (1994, p. 33). Ancient Greek literature provides a variety of usages of the verb hyphistêmi, both nontechnical and technical (especially philosophical and theological). The range of its meanings, according to the Liddel and Scott Greek-English lexicon, include ‘give substance to’, ‘cause to subsist’, ‘treat as subsisting’, ‘subsist’, and ‘exist’ (A 4, B IV.2). The foregoing meanings are given also by the Patristic Greek lexicon of G.W.H. Lampe, which also has ‘cause to exist’, ‘make’, ‘exist as a substance’, and ‘exist as an entity’; see Lampe (1961, A 6, 7, B 2).
- 5.
Hyphistêmi functions here the same way as all other verbs used by Diophantus to state positions (tassô, plassô, estô).
- 6.
For an in-depth discussion of the philosophical and other usages of the term, the reader is referred to the book by Romano and Taormina (1994) and the references included therein.
- 7.
See Liddell and Scott (1996), s.v. hypostasis (B III.2).
- 8.
(Lampe 1961, A 7.a).
- 9.
See Khazdan (1991), s.v. Hypostasis (ii, 966).
- 10.
Problem 42 in the Arabic book IV is the only place in which we encounter a clause that seems relevant. See Rashed (1984, 3, 83); Sesiano (1982, 120.1361–1362). In regard to the Greek original behind this clause, Sesiano writes that it “must have been something like anatrechomen epi tas hypostaseis.”
- 11.
The verb hyphistêmi appears twice in Metrica, and once in Dioptra, with the meaning of assigning a concrete measure or identity to a line segment. The occurrences in Metrica are “Similarly, by positing (hypostêsômetha) the diameter AB 14 units” (Heron 1903, 74.26–27); “And the side of <the square of> 63 is approximately 71⁄2+1⁄4 + 1⁄8 + 1⁄16. Therefore, to find the area I have to posit (hypostêsamenon) so great the perpendicular” (Heron 1903, 28.1–3). The term also occurs (as either noun or verb) 15 times in the pseudo-Heronian Definitiones.
- 12.
This was also the point of view of Tannery, who in his “Index Graecitatis apud Diophantum” (s.v. hypostasis) defines hypostasis as “numeri quaesiti valor vel numericus vel expressus in x” (Tannery 1893–1895, ii, 285).
- 13.
From several instances we encounter within the Arithmetica the phrase is combined with verbs such as erchomai (come to, arrive at), dierchomai (go through, pass through), and poiein (to make, to do).
- 14.
See problems I.7–10, II.11, IV.18, and IV.37.
- 15.
See problems 15–17, 21–22, 27–28 in the fifth book, and problems 5, 7–11, 15, 17 in the sixth book.
- 16.
The translation is based on Allard’s edition of the text. In Tannery’s edition (78.14–19), the last word arithmos is replaced by its abbreviation.
- 17.
In Allard’s edition the sign for leipsis is missing.
References
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Acknowledgements
I thank Jeffrey Oaks, Stathis Psillos, Paul Kalligas, Michalis Sialaros, and Vassilios Karakostas, who kindly read this paper and suggested improvements. Above all, however, I express my gratitude to Kostas Gavroglu. Our friendship and collaboration for more than 20 years have been for me a unique source of inspiration and encouragement.
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Christianidis, J. (2015). The Meaning of Hypostasis in Diophantus’ Arithmetica . In: Arabatzis, T., Renn, J., Simões, A. (eds) Relocating the History of Science. Boston Studies in the Philosophy and History of Science, vol 312. Springer, Cham. https://doi.org/10.1007/978-3-319-14553-2_21
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