Refutation of Indivisible Lines from Geometry and Motion — Aristotle, On Indivisible Lines 22–31, Greek & English

§22

Ὄτι μὲν οὖν ἔκ γε τῶν εἰρημένων λόγων οὔτ’ ἀναγκαῖον ἀτόμους εἶναι γραμμὰς οὔτε πιθανόν, φανερόν. Ἔτι δὲ καὶ ἐκ τῶνδε γένοιτ’ ἂν φανερώτερον. Πρῶτον μὲν ἐκ τῶν ἐν τοῖς μαθήμασι δεικνυμένων καὶ τιθεμένων, ἃ οὐ δίκαιον ἢ πιστοτέροις λόγοις κινεῖν.¹

Therefore, from what has been said, it is clear that it is neither necessary nor plausible for there to be indivisible lines. Furthermore, it would become even clearer from the following. First, from what is demonstrated and assumed in mathematics, which it is not right to disturb except by more trustworthy arguments.

§23

Οὔτε γὰρ ὁ τῆς γραμμῆς οὔτε ὁ τῆς εὐθείας ὅρος ἐφαρμόσει τῇ ἀτόμῳ διὰ τὸ μήτε μεταξὺ τινῶν εἶναι μήτ’ ἔχειν μέσον. Ἔπειτα πᾶσαι αἱ γραμμαὶ σύμμετροι ἔσονται. Πᾶσαι γὰρ ὑπὸ τῶν ἀτόμων μετρηθήσονται, αἵ τε μήκει σύμμετροι καὶ αἱ δυνάμει.

For neither the definition of a line nor that of a straight line will apply to the indivisible, because it is neither between any things nor has a middle. Next, all lines will be commensurable. For all of them will be measured by the indivisibles, both those commensurable in length and those in power.

§24

Αἱ δὲ ἄτομοι σύμμετροι πᾶσαι μήκει· ἴσαι γάρ· ὥστε καὶ δυνάμει. Εἰ δὲ τοῦτο, διαιρετὸν ἔσται τὸ τετράγωνον. Ἔτι εἰ ἡ περὶ τὴν μείζω τὸ πλάτος ποιεῖ παραβαλλομένη, τὸ ἴσον τῶν ἀπὸ τῆς ἀτόμου καὶ τῆς ποδιαίας παραβαλλομένων περὶ τὴν δίπουν² ἔλαττον ποιήσει τὸ πλάτος τῆς ἀμεροῦς· ἔσται ἔλαττον τὸ περὶ τῆς ἀτόμου.

And the indivisible lines are all commensurable in length; for they are equal; so also in power. But if this is so, the square will be divisible. Furthermore, if a square applied to a larger line determines its width, then when an area equal to the squares on the indivisible line and the foot-long line is applied to the two-foot line, it will make the width smaller than that of the indivisible line; that is, the width in the case of the indivisible line will be smaller.

§25

Ἔτι εἰ ἐκ τριῶν δοθεισῶν εὐθειῶν συνίσταται τρίγωνον, καὶ ἐκ τῶν ἀτόμων συσταθήσεται. Ἐν ἅπαντι δὲ ἰσοπλεύρῳ ἡ κάθετος ἐπὶ μέσην πίπτει, ὥστε καὶ ἐπὶ τὴν ἄτομον. Ἔτι εἰ τὸ τετράγωνον τῶν ἁμερῶν διὰ μέσου ἐμπεσούσης καὶ καθέτου ἀχθείσης, ἡ τοῦ τετραγώνου πλευρὰ τὴν κάθετον δύναται καὶ τὴν ἡμίσειαν τῆς διαμέτρου,³ ὥστε οὐκ ἐλαχίστη.

Furthermore, if a triangle is constructed from three given straight lines, it will also be constructed from indivisible lines. And in every equilateral triangle, the perpendicular falls on the middle of the base, and so also on the indivisible line. Furthermore, if in a square of indivisibles, when a perpendicular is drawn through the middle, the side of the square is equal in power to the perpendicular and the half of the base, so it is not the smallest.

§26

Οὐδὲ διπλάσιον τὸ ἀπὸ τῆς διαμέτρου χωρίον ἔσται τοῦ ἀπὸ τῆς ἀτόμου. Ἀφαιρεθέντος γὰρ τοῦ ἴσον ἡ λοιπὴ ἔσται ἐλάσσων τῆς ἀμεροῦς. Εἰ γὰρ ἴσως τετραπλάσιον ἂν ἔγραψεν ἡ διάμετρος, ἄλλα δ’ ἄν τις καὶ ἕτερα τοιαῦτα συνάγοι· πᾶσι γὰρ ὡς εἰπεῖν ἐναντιοῦται τοῖς ἐν τοῖς μαθήμασιν.

Nor will the square on the diagonal be double the square on the indivisible line. For when an equal portion is subtracted, the remainder will be smaller than the indivisible. If indeed the diagonal should describe a fourfold area, one might also draw other such conclusions; for it is, so to speak, contrary to all things in mathematics.

§27

Πάλιν τοῦ μὲν ἀμεροῦς μία ἡ σύναψις, γραμμῆς δὲ δύο· καὶ γὰρ ὅλη ὅλης ἅπτεται, καὶ κατὰ τὸ πέρας ἐξ ἐναντίας. Ἔτι γραμμὴ προστεθεῖσα οὐ ποιεῖ μείζω τὴν ὅλην· τὰ γὰρ ἀμερῆ συντιθέμενα οὐ ποιήσει μεῖζον.

Again, the contact of the indivisible is single, while that of a line is twofold; for either whole touches whole, or they touch at the extremity from opposite sides. Furthermore, a line added does not make the whole larger; for indivisibles put together will not make anything larger.

§28

Ἔτι ἐκ δυοῖν ἀμεροῖν μηδὲν γίνεσθαι συνεχὲς διὰ τὸ πλείους διαιρέσεις ἔχειν ἅπαν τὸ συνεχές· ἅπασα δὲ γραμμὴ παρὰ τὴν ἄτομον συνεχὴς οὐκ ἄν εἴη γραμμὴ ἄτομος. Ἔτι εἰ ἅπασα γραμμὴ παρὰ τῆς ἀτόμου καὶ ἴσα καὶ ἄνισα διαιρεῖται, καὶ μὴ ἐκ τριῶν ἀτόμων καὶ ὅλως περιττῶν, ὥστ’ ἀδιαίρετος ἡ ἄτομος.

Furthermore, from two indivisibles nothing continuous can arise, because every continuous thing has more divisions; and since every line other than the indivisible is continuous, there would be no indivisible line. Furthermore, if every line other than the indivisible is divided into both equal and unequal parts, and is not composed of three indivisibles (or in general an odd number), then the indivisible line must be undivided.

§29

Ὁμοίως δὲ κἄν εἰ δίχα τέμνεται· πᾶσα γὰρ ἡ ἐκ τῶν περιττῶν. Εἰ δὲ δίχα μὲν μὴ πᾶσα τέμνεται ἀλλ’ ἡ ἐκ τῶν ἀρτίων, τὴν δὲ δίχα διαιρουμένην καὶ ὅσα δυνατὸν τέμνειν, διαιρεθήσεται καὶ οὕτως ἄτομος, ὅταν ἡ ἐκ τῶν ἀρτίων εἰς ἄνισα διαιρῆται.

Likewise even if it is cut in half; for this is the case with all those composed of an odd number. But if not every line is cut in half but only those composed of an even number, and if we divide the one that is cut in half as far as possible, the indivisible will also be divided in this way when the line composed of an even number is divided into unequal parts.

§30

Πάλιν εἰ τὸ κεκινημένον ἐν ᾦ χρόνῳ κινεῖται τὴν ὅλην ἐν τῷ ἡμίσει τὴν ἡμίσειαν κινηθήσεται, καὶ ἐν τῷ ἐλάττονι ἔλαττον ἢ τὴν ἡμίσειαν, ὥστ’ εἰ μὲν περιττῶν σύγκειται τῶν ἀτόμων τὸ μῆκος, ἀναιρεθήσεται ἡ μέση τομὴ τῶν ἀτόμων, εἴπερ ἐν τῷ ἡμίσει χρόνῳ τὸ ἥμισυ δίεισιν· ὁμοίως γὰρ ὅ τε χρόνος καὶ ἡ γραμμὴ τμηθήσεται.

Again, if that which is moved in a certain time moves the whole distance, in half the time it will move half the distance, and in less time less than half; so that if the length is composed of an odd number of indivisibles, the middle division of the indivisibles will be done away with, if indeed in half the time it traverses half the distance; for both time and line will be divided likewise.

§31

Ὤστε οὐδεμία τῶν συγκειμένων τμηθήσεται εἰς ἴσα καὶ ἄνισα, οὐδ’ ὁμοίως τοῖς χρόνοις τμηθήσονται. Οὐκ ἔσονται ἄτομοι γραμμαί. Τὰ δὲ τοῦ αὐτοῦ λόγου ἐστί, καθάπερ ἐλέχθη, τὸ πάντα ταῦτα ποιεῖν ἐξ ἀμερῶν. Ἔτι ἄπασα ἡ μὴ ἄπειρος δύο ἔχει πέρατα· γραμμὴ γὰρ ὥρισται τούτοις.

So none of the composite lines will be divided into equal and unequal parts, nor will they be divided in a manner like the times. There will be no indivisible lines. And it belongs to the same argument, as was said, to make all these things from indivisibles. Furthermore, every line that is not infinite has two limits; for a line is defined by these.

Syntax notes3 items

969b22ἃ οὐ δίκαιον ἢ πιστοτέροις λόγοις κινεῖν
The antecedent of the relative pronoun ἃ is the mathematical principles and demonstrations mentioned just before. Here, ἢ functions as "other than" or "except" after a comparative notion, meaning "which it is not right to disturb except by more trustworthy arguments." Some editors conjecture μὴ instead of ἢ.
970a1τὸ ἴσον τῶν ἀπὸ τῆς ἀτόμου καὶ τῆς ποδιαίας παραβαλλομένων περὶ τὴν δίπουν
This refers to the geometrical "application of areas" (parabole) in ancient Greek mathematics. It discusses the "width" (πλάτος) obtained when an area equal to the sum of the squares on an indivisible line and a foot-long line is applied (as a rectangle) to a two-foot line, showing the absurdity of it becoming smaller than the indivisible line.
970a1ἡ τοῦ τετραγώνου πλευρὰ τὴν κάθετον δύναται καὶ τὴν ἡμίσειαν τῆς διαμέτρου
The verb δύναμαι is used here in its mathematical sense, "to be equal in power to" or "to be equivalent to in square." Based on the Pythagorean theorem, it indicates that the square on the side of the square is equal to the sum of the squares on the perpendicular and on the half of the base (or diagonal).

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-grc1:22-31