ARISTOTLE AND EUCLID'S POSTULATES

Abstract

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 ἀγαγεῖνκαὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖνκαὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕεσθαι