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ARISTOTLE AND EUCLID'S POSTULATES

Published online by Cambridge University Press:  08 November 2013

Fabio Acerbi
Fabio Acerbi*
Affiliation:
CNRS, UMR8560 Centre Alexandre Koyré, Paris
*
fabacerbi@gmail.com
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Extract

Book 1 of Euclid's Elements opens with a set of unproved assumptions: definitions (ὅροι), postulates, and ‘common notions’ (κοιναὶ ἔννοιαι). The common notions are general rules validating deductions that involve the relations of equality and congruence. The attested postulates are five in number, even if a part of the manuscript tradition adds a sixth, almost surely spurious (‘two straight lines do not contain a space’), that in some manuscripts features as the ninth, and last, common notion. The postulates are called αἰτήματα both in the manuscripts of the Elements and in the ancient exegetic tradition. It is not said, however, that this denomination is original, as it coincides with the nomen rei actae associated with the verbal form introducing the postulates themselves. Since antiquity it has been recognized that the postulates naturally split into two groups of quite different character: the first three are rules licensing basic constructions, the fourth and the fifth are assertions stating properties of particular geometric objects (see Procl. In Euc. 182–4 and 188–93). I transcribe postulates 1–3 in the text established by Heiberg (Euclidis Elementa 1.4.14–5.2):

ᾐτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν

καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖν

καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕεσθαι

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Type
Research Article
Information
The Classical Quarterly , Volume 63 , Issue 2 , December 2013 , pp. 680 - 685
Copyright
Copyright © The Classical Association 2013 

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Footnotes

In memoriam Ian Mueller

References

1 I shall refer to the following articles by the name of the author and the date of publication: Zeuthen, H.G., ‘Die geometrische Construction als “Existenzbeweis” in der antiken Geometrie’, Mathematische Annalen 47 (1896), 222–8CrossRefGoogle Scholar; Lee, H.D.P., ‘Geometrical method and Aristotle's account of first principles’, CQ 29 (1935), 113–24CrossRefGoogle Scholar; Gomez-Lobo, A., ‘Aristotle's hypotheses and the Euclidean postulates’, RMeta 30 (1977), 430–9Google Scholar; Mueller, I., ‘On the notion of a mathematical starting point in Plato, Aristotle, and Euclid’, in Bowen, A.C. (ed.), Science and Philosophy in Classical Greece (New York and London, 1991), 59–97.Google Scholar The references to the Aristotelian commentators are to the page(s) of the corresponding volumes in the Commentaria in Aristotelem Graeca series.

2 The case of a self-evident yet not conceded principle of course does not apply.

3 See Lee (1935), 114–17. As a matter of fact, these four pages are not much more than a cento of quotations from two of Heath's books.

4 See Gomez-Lobo (1977), whose interpretation, however, must be corrected on a crucial point: the verb ‘to be’ which often introduces the ἔκθεσις has a ‘presential’, not a copulative value: this makes it unnecessary to assign to the same verb a ‘predicative value with ellipsis of the predicate’ (p. 433) in Aristotelian passages such as An. post. Α 2, 72a18–24.

5 This evidence is of limited value since the whole proof was rewritten by Eutocius.

6 In the Elements, a semicircle is said to be ‘described on’ (forms of γράϕειν ἐπί) a diameter, as for instance in propositions 6.13, 13.13–16. The canonization process was not immediate: in his duplication of a cube, Philon περιέγραψε a semicircle ‘making use’ of a point and a line segment as centre and radius respectively (110.9–10 Marsden).

7 The peculiar form of the third postulate is strictly functional to its application in El. 1.2–3 – the two constructions that give to the basic operation of cutting off a segment from a straight line the epistemic status of a proposition (i.e. a statement supported by a proof) and that can be used in place of the postulate (i.e. an unproven statement) itself.

8 See e.g. the much-debated passage on the definition of ‘ratio’ at Top. Θ 3, 158b30–5.

9 This designation was apparently a rigid one before Aristotle, and the only one to be found in Plato.

10 At An. pr. Α 24, 41b21–2, An. post. Α 10, 76a41 and 76b20–1, Α 11, 77a30–1, Metaph. Κ 4, 1061b20.

11 Notice in it the three occurrences of forms of (προσ)λαμβάνειν: two of them are referred to as assumptions, one as an instantiation of an assumption.