Arguments for Indivisible Lines and Their Refutation — Aristotle, On Indivisible Lines 1–10, Greek & English

§1

ἎΡΑ γ’ εἰσὶν ἄτομοι γραμμαί, καὶ ὅλως ἐν ἅπασι τοῖς ποσοῖς ἐστί τι ἀμερές, ὥσπερ ἔνιοί φασιν; Εἰ γὰρ ὁμοίως ὑπάρχει τό τε πολὺ καὶ τὸ μέγα καὶ τὰ ἀντικείμενα τούτοις, τό τε ὀλίγον καὶ τὸ μικρόν, τὸ δ' ἀπείρους σχεδὸν διαιρέσεις ἔχον οὐκ ἔστιν ὀλίγον ἀλλὰ πολύ, φανερὸν ὅτι πεπερασμένας ἕξει τὰς διαιρέσεις τὸ ὀλίγον καὶ τὸ μικρόν· εἰ δὲ πεπερασμέναι αἱ διαιρέσεις, ἀνάγκη τι εἶναι ἀμερὲς μέγεθος, ὥστε ἐν ἅπασιν ἐνυπάρξει τι ἀμερές, ἐπείπερ καὶ τὸ ὀλίγον καὶ τὸ μικρόν.

Are there indeed indivisible lines, and in general is there in all quantities something indivisible, as some people say? For if ‘many’ and ‘great’ and their opposites, ‘few’ and ‘small’, exist in the same way, and that which has almost infinite divisions is not ‘few’ but ‘many’, it is clear that ‘few’ and ‘small’ will have finite divisions; and if the divisions are finite, there must be some indivisible magnitude, so that an indivisible thing will exist in all things, since it does so in ‘few’ and ‘small’.

§2

Ἔτι εἰ ἔστιν δέα γραμμῆς,¹ ἡ δ' ἰδέα πρώτη τῶν συνωνύμων, τὰ δὲ μέρη πρότερα τοῦ ὅλου τὴν φύσιν, διαιρετὴ ἂν εἴη αὐτὴ ἡ γραμμή, τὸν αὐτὸν δὲ τρόπον καὶ τὸ τετράγωνον καὶ τὸ τρίγωνον καὶ τὰ ἄλλα σχήματα, καὶ ὅλως ἐπίπεδον αὐτὸ καὶ σῶμα· συμβήσεται γὰρ πρότερ’ ἅττα εἶναι τούτων.

Furthermore, if there is an Idea of line, and the Idea is prior to its homonyms, while parts are by nature prior to the whole, the line itself would be divisible, and in the same way also the square, the triangle, and the other figures, and generally the plane itself and the body; for it will result that there are certain things prior to them.

§3

Ἔτι εἰ σώματός ἐστι στοιχεῖα, τῶν δὲ στοιχείων μηδὲν πρότερον, τὰ δὲ μέρη τοῦ ὅλου πρότερα, ἀδιαίρετον ἂν εἴη τὸ πῦρ καὶ ὅλως τῶν τοῦ σώματος στοιχείων ἕκαστον, ὥστ’ οὐ μόνον ἐν τοῖς νοητοῖς ἀλλὰ καὶ ἐν τοῖς αἰσθητοῖς ἐστί τι ἀμερές.

Furthermore, if there are elements of body, and nothing is prior to elements, while parts are prior to the whole, fire and generally each of the elements of body would be indivisible; so that not only in intelligible but also in sensible things there is something indivisible.

§4

Ἔτι δὲ κατὰ τὸν Ζήνωνος λόγον ἀνάγκη τι μέγεθος ἀμερὲς εἶναι, εἴπερ ἀδύνατον μὲν ἐν πεπερασμένῳ χρόνῳ ἀπείρων ἄψασθαι, καθ’ ἕκαστον ἁπτόμενον, ἀνάγκη δ' ἐπὶ τὸ ἥμισυ πρότερον ἀφικνεῖσθαι τὸ κινούμενον, τοῦ δὲ μὴ ἀμεροῦς πάντως ἔστιν ἥμισυ.

Furthermore, according to Zeno’s argument, there must be some indivisible magnitude, if indeed it is impossible in a finite time to touch infinitely many things, touching each one individually, and it is necessary for the moving object to reach the half first, and of that which is not indivisible there is always a half.

§5

Εἰ δὲ καὶ ἅπτεται τῶν ἀπείρων ἐν πεπερασμένῳ χρόνῳ τὸ ἐπὶ τῆς γραμμῆς φερόμενον, τὸ δὲ θᾶττον ἐν τῷ ἴσῳ χρόνῳ πλεῖον διανύει, ταχίστη δ' τῆς διανοίας κίνησις, κἂν ἡ διάνοια τῶν ἀπείρων ἐφάπτοιτο² καθ’ ἕκαστον ἐν πεπερασμένῳ χρόνῳ, ὥστε εἰ τὸ καθ’ ἕκαστον ἅπτεσθαι τὴν διάνοιαν ἀριθμεῖν ἐστίν, ἐνδέχεται ἀριθμεῖν τὰ ἄπειρα ἐν πεπερασμένῳ χρόνῳ.

But if that which is carried along the line does touch infinitely many things in a finite time, and the swifter traverses more in the same time, and the movement of thought is the swiftest, then thought too would touch infinitely many things individually in a finite time, so that if for thought to touch each thing individually is to count, it is possible to count infinitely many things in a finite time.

§6

Εἰ δὲ τοῦτο ἀδύνατον, εἴη ἄν τις ἄτομος γραμμή. Ἔτι καὶ ἐξ ὦν αὐτοὶ οἱ ἐν τοῖς μαθήμασι λέγουσιν, εἴη ἄν τις ἄτομος γραμμή, ὡς φασίν, εἰ σύμμετροί εἰσὶν αἱ τῷ αὐτῷ μέτρῳ μετρούμεναι· ὅσαι δ' εἰσὶ σύμμετροι, πᾶσαί εἰσι μετρούμεναι. Εἴη γὰρ ἄν τι μῆκος πᾶσαι μετρηθήσονται. Τοῦτο δ’ ἀνάγκη ἀδιαίρετον εἶναι.

And if this is impossible, there would be some indivisible line. Furthermore, from what the mathematicians themselves say, there would be some indivisible line, as they assert, if those lines are commensurable which are measured by the same measure; and as many as are commensurable are all measured. For there would be some length by which all will be measured. And this must be indivisible.

§7

Εἰ γὰρ διαιρετόν, καὶ τὰ μέρη μέτρου τινὸς ἔσται· σύμμετρα γὰρ τῷ ὅλῳ. Ὤστε μέρους τινὸς εἴη διπλασία τὴν ἡμίσειαν,³ ἐπειδὴ τοῦτ’ ἀδύνατον ἂν εἴη μέτρον.

For if it is divisible, its parts also will belong to some measure; for they are commensurable with the whole. Thus, the half would be double of some part, since this would be impossible for a measure.

§8

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

And likewise also those measured once by it, just like all lines composed of the measure, are composed of indivisible parts. And the same will happen also in planes; for all those arising from rational lines are commensurable with one another, so that their measure will be indivisible.

§9

Ἀλλὰ μὴν εἴ τι τμηθήσεται μέτρον τινὰ τεταγμένην καὶ ὡρισμένην γραμμήν, οὐκ ἔσται οὔτε ῥητὴ οὔτ’ ἄλογος, οὔτε τῶν ἄλλων οὐδεμία ὦν νῦν δὴ εἴρηται, οἷον ἀποτομὴν ἐκ δυοῖν ὀνομάτοιν· ἀλλὰ καθ’ αὐτὰς μὲν οὐδέ τινας ἕξουσι φύσεις, πρὸς ἀλλήλας δὲ ἔσονται ῥηταὶ καὶ ἄλογοι.

But indeed, if any measure divides some ordered and defined line, it will be neither rational nor irrational, nor any of the other things just mentioned, such as an apotome or a bimedial; rather, in themselves they will have no nature of any kind, but in relation to one another they will be rational and irrational.

§10

Ἣ πρῶτον μὲν⁴ οὐκ ἀνάγκη τὸ ἀπείρους ἔχον διαιρέσεις μὴ εἶναι μικρὸν καὶ ὀλίγον· καὶ γὰρ τόπον καὶ μέγεθος καὶ ὅλως τὸ συνεχὲς μικρὸν μὲν λέγομεν, καὶ ἐφ’ ὦν μὲν ἁρμόττει τὸ ὀλίγον, οὐ μὴν ἀλλ’ ἀπείρους διαιρέσεις φαμὲν ἔχειν. Ἔτι δ' εἰ ἐν τῷ συνθέτῳ γραμμαί, κατὰ τούτων τῶν ἀτόμων λέγεται τὸ μικρόν, καὶ ἄπειροι στιγμαὶ ἐνυπάρχουσιν.

Or, first of all, it is not necessary that that which has infinite divisions is not small and few; for we do call place and magnitude and generally the continuous ‘small’ (and ‘few’ where it fits), and yet we say they have infinite divisions. Moreover, if there are lines in the composite, ‘small’ is said of these indivisible things, and infinite points exist in them.

Syntax notes4 items

968a2ἰδέα γραμμῆς
An argument applying the Platonic Theory of Ideas to geometric concepts. Under the premise that there exists an Idea of line, and based on the principle that parts are prior to the whole, it would lead to the absurdity that the Idea itself is divisible, hence the necessity of indivisible lines. The genitive γραμμῆς is a genitive of definition/apposition.
968a5κἂν ἡ διάνοια τῶν ἀπείρων ἐφάπτοιτο
The potential optative (ἄν with the optative ἐφάπτοιτο) is used in the apodosis of a conditional construction. The particle κἄν is a contraction of καὶ ἄν, which, following the 'if' clause of the previous sentence, introduces a potential conclusion: 'even so, thought would touch infinitely many things'.
968b1μέρους τινὸς εἴη διπλασία τὴν ἡμίσειαν
A highly difficult passage widely suspected of corruption in the manuscript tradition. In context, it demonstrates the absurdity resulting from assuming that a measure (the minimal unit) is divisible: it suggests that its 'half' (τὴν ἡμίσειαν, here treated as the subject despite the accusative form) would be 'double' (διπλασία) of some smaller part, which is a mathematical and logical contradiction for an ultimate measure.
968b1Ἣ πρῶτον μὲν
The spelling Ἣ in the manuscripts is likely a corruption or misaccentuation of the disjunctive conjunction ἢ ('or'), introducing the refutation. This marks the transition from the arguments supporting indivisible lines to the systematic refutation of those arguments.

Translated by Gemini 3.5 Flash (published 2026-05-22) · Revision history (2, latest 2026-09-21)

2026-09-21 AI review — source 2

urn:cts:greekLit:tlg0086.tlg019.humanitext-grc1:1-10