(c) The Irrational
Schol. lxii. in Eucl. Elem. x., Eucl. ed. Heiberg v. 450. 16–18
Τὸ θεώρημα τοῦτο Θεαιτήτειόν ἐστιν εὕρημα, καὶ μέμνηται αὐτοῦ ὁ Πλάτων ἐν Θεαιτήτῳ, ἀλλ᾿ ἐκεῖ μὲν μερικώτερον ἔγκειται, ἐνταῦθα δὲ καθόλου.Plat. Theaet. 147 d–148 b
ΘΕΑΙΤΗΤΟΣ. Περὶ δυνάμεών τι ἡμῖν Θεόδωρος ὅδε ἔγραφε, τῆς τε τρίποδος πέρι καὶ πεντέποδος [ἀποφαίνων]1 ὅτι μήκει οὐ σύμμετροι τῇ ποδιαίᾳ, καὶ οὕτω κατὰ μίαν ἑκάστην προαιρούμενος μέχρι τῆς ἑπτακαιδεκάποδος· ἐν δὲ ταύτῃ πως ἐνέσχετο. ἡμῖν οὖν εἰσῆλθέ τι τοιοῦτον, ἐπειδὴ ἄπειροι τὸ πλῆθος αἱ δυνάμεις ἐφαίνοντο, πειραθῆναι συλλαβεῖν εἰς ἕν, ὅτῳ πάσας ταύτας προσαγορεύσομεν τὰς δυνάμεις.
Euclid, Elements x., Scholium lxii., ed. Heiberg v. 450. 16–18
This theorem [Eucl. Elem. x. 9]a is the discovery of Theaetetus, and Plato recalls it in the Theaetetus, but there it arises in a particular case, here it is treated generally.Plato, Theaetetus 147 d–148 b
Theaetetus. Theodorusb was proving to us a certain thing about square roots, I mean the square roots of three square feet and five square feet, namely, that these roots are not commensurable in length with the foot-length, and he proceeded in this way, taking each case in turn up to the root of seventeen square feet; at this point for some reason he stopped.c Now it occurred to us, since the number of square roots appeared to be unlimited, to try to gather them into one class, by which we could henceforth describe all the roots.
- aThe enunciation is: The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length. But the squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.
- bTheodorus of Cyrene, claimed by Iamblichus (Vit. Pythag. 36) as a Pythagorean and said to have been Plato’s teacher in mathematics (Diog. Laert. ii. 103).
- cSeveral conjectures have been put forward to explain how Theodorus proved that √3̅, √5̅ . . . √1̅7̅ are incommensurable. They are summarized by Heath (H. G. M. i. 204-208). One theory is that Theodorus adapted the traditional proof (supra, p. 110) of the incommensurability of √2̅. Another, put forward by Zeuthen (“Sur la constitution des livres arithmétiques des Eléments d’Euclide et leur rapport à la question de 1’irrationalité” in Oversigt over det kgl, Danske videnskabernes Selskabs Forhandlinger, 1915, pp. 422 ff.), depends on the process of finding the greatest common measure as stated in Eucl. x. 2. If two magnitudes are such that the process of finding their G.C.M. never comes to an end, the two magnitudes are incommensurable. The method is simple in theory, but the geometrical application is fairly complicated, though doubtless not beyond the capabilities of Theodorus.