Aristotle, On Indivisible Lines

LCL 307: 418-419

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968 a

Ἔτι δὲ κατὰ τὸν τοῦ Ζήνωνος λόγον ἀνάγκη τι 20μέγεθος ἀμερὲς εἶναι, εἴπερ ἀδύνατον μὲν ἐν πεπερασμένῳ χρόνῳ ἀπείρων ἅψασθαι, καθ᾿ ἕκαστον ἁπτόμενον, ἀνάγκη δ᾿ ἐπὶ τὸ ἥμισυ πρότερον ἀφικνεῖσθαι τὸ κινούμενον, τοῦ δὲ μὴ ἀμεροῦς πάντως ἔστιν ἥμισυ. εἰ δὲ καὶ ἅπτεται τῶν ἀπείρων 25ἐν πεπερασμένῳ χρόνῳ τὸ ἐπὶ τῆς γραμμῆς φερόμενον, τὸ δὲ θᾶττον ἐν τῷ ἴσῳ χρόνῳ πλεῖον διανύει, ταχίστη δ᾿ ἡ τῆς διανοίας κίνησις, κἂν ἡ 968 bδιάνοια τῶν ἀπείρων ἐφάπτοιτο καθ᾿ ἕκαστον ἐν πεπερασμένῳ χρόνῳ, ὥστε εἰ τὸ καθ᾿ ἕκαστον ἅπτεσθαι τὴν διάνοιαν ἀριθμεῖν ἐστίν, ἐνδέχεται ἀριθμεῖν τὰ ἄπειρα ἐν πεπερασμένῳ χρόνῳ. εἰ δὲ τοῦτο ἀδύνατον, εἴη ἄν τις ἄτομος γραμμή.

5Ἔτι καὶ ἐξ ὧν αὐτοὶ οἱ ἐν τοῖς μαθήμασι λέγουσιν, εἴη ἄν τις ἄτομος γραμμή, ὡς φασίν, εἰ σύμμετροί εἰσιν αἱ τῷ αὐτῷ μέτρῳ μετρούμεναι· ὅσαι δ᾿ εἰσὶ μετρούμεναι, πᾶσαί εἰσι σύμμετροι. εἴη γὰρ ἄν τι μῆκος, ᾧ πᾶσαι μετρηθήσονται. 10τοῦτο δ᾿ ἀνάγκη ἀδιαίρετον εἶναι. εἰ γὰρ διαιρετόν, καὶ τὰ μέρη μέτρου τινὸς ἔσται. σύμμετρα γὰρ τῷ ὅλῳ. ὥστε μέρους τινὸς εἴη διπλασίαν τὴν ἡμίσειαν· ἐπεὶ δὲ τοῦτ᾿ ἀδύνατον, <ἀδιαίρετον> ἂν εἴη μέτρον.

Ὡσαύτως δὲ καὶ αἱ μετρούμεναι ἅπαξ ὑπ᾿ αὐτοῦ, ὥσπερ πᾶσαι αἱ ἐκ τοῦ μέτρου σύνθετοι γραμμαί, 15ἐξ ἀμερῶν σύγκεινται. τὸ δ᾿ αὐτὸ συμβήσεται κἀν τοῖς ἐπιπέδοις· πάντα γὰρ τὰ ἀπὸ τῶν ῥητῶν γραμμῶν σύμμετρα ἀλλήλοις, ὥστε ἔσται τὸ μέτρον


On Indivisible Lines

Again, according to the argument of Zeno, there must be some magnitude without parts, since it is impossible to touch an infinite number of things in a finite time, when touching each of them, and that which moves must first reach half-way, and half clearly belongs to that which is not without parts. But if anything travelling along a line touches an infinite series in a finite time, secondly if the faster it travels the greater the space it covers in the same time, and lastly if the movement of thought is the quickest movement, then even thought must touch an infinite series one by one in a finite time. If, then, thought touching the series one by one is counting, then it must be possible to count an infinite series in finite time. If this is impossible, then there must exist an indivisible line.

The next argument, we are told, is used by the mathematicians to prove that the indivisible line must exist, if we admit that “commensurate” lines are those which are measured by the same unit, and all the lines measured are “commensurate.” For there must be some length by which they are all measured. And this must be incapable of division. For if it is divisible, then its parts can also be expressed in the terms of some unit. For they are commensurate with the whole. So that the measurement of each part would be double its half; since this is impossible the unit of measurement must itself be indivisible.

Again, just as the lines built up from the unit of measurement are all composed of units without parts, so also must those be which are once measured by it. The same thing will also happen in plane figures; for all the squares on rational lines are commensurable with each other, so that their unit of measurement

DOI: 10.4159/DLCL.aristotle-divisible_lines.1936