Contradictions of Indivisible Lines and Points — Aristotle, On Indivisible Lines 32–41, Greek & English

§32

Ἡ δὲ ἄτομος οὐκ ἄπειρος, ὥστε ἔξει πέρας. Διαιρετὴ ἄρα· τὸ γὰρ πέρας ἄλλο καὶ οὗ πέρας. Ἣ ἔσται¹ τις οὔτ’ ἄπειρος οὔτε πεπερασμένη γραμμὴ παρὰ ταύτας. Ἕτι οὐκ ἐν ἁπάσῃ γραμμῇ στιγμὴ ἔσται. Ἐν μὲν γὰρ τῇ ἀτόμῳ οὐκ ἔστιν· εἰ μὲν γὰρ μία μόνη, ὑπάρξει γραμμή, εἶτα στιγμή· εἰ δὲ πλείους, διαιρετὴ ἡ γραμμή.

But the indivisible line is not infinite, so that it will have a limit. Therefore it is divisible; for the limit is other than that of which it is the limit. Or there will be some line other than these, which is neither infinite nor finite. Furthermore, there will not be a point in every line. For in the indivisible line there is none; for if there is only one, there will exist first a line, and then a point; but if there are more, the line is divisible.

§33

Εἰ μὲν οὖν ἐν τῇ ἀτόμῳ μὴ ἐνυπάρχει στιγμή, οὐδ’ ὅλως ἐν γραμμῇ ἔσται· αἱ γὰρ ἄλλαι ἐκ τῶν ἀτόμων. Ἔτι ἢ μηθὲν τῶν στιγμῶν ἔσται μεταξὺ ἢ γραμμή· εἰ δὲ μεταξὺ γραμμή, ἐν ἀπάσαις δὲ πλείους στιγμαί, οὐκ ἔσται ἄτομος. Ἔτι οὐχ ἀπάσης ἔσται γραμμῆς τετράγωνον· ἔξει γὰρ μῆκος καὶ πλάτος, ὥστε διαιρετόν, ἐπεὶ τὸ μέν, τὸ δέ τι,² Εἰ δὲ τὸ τετράγωνον, καὶ ἡ γραμμή. Ἔτι τὸ πέρας τῆς γραμμῆς στιγμὴ ἔσται, ἀλλ' οὐ γραμμή. Πέρας μὲν γάρ, τὸ ἔσχατον δὲ ἡ ἄτομος. Εἰ γὰρ στιγμή, τὸ πέρας τῇ ἀτόμῳ ἔσται στιγμή, καὶ ἔσται γραμμὴ γραμμῆς στιγμῇ μείζων. Εἰ δ' ἐνυπάρχει τῇ ἀτόμῳ ἡ στιγμή, διὰ τὸ ταὐτὸ πέρας τῶν συνεχουσῶν γραμμῶν, ἔσται τι πέρας τῆς ἀμεροῦς, Ὄλως τε τί διοίσει στιγμὴ γραμμῆς; Οὐδὲν γὰρ ἴδιον ἔξει ἡ ἄτομος γραμμὴ παρὰ τὴν στιγμὴν πλὴν τοὔνομα. Ἕτι ὁμοίως μένει ἐπίπεδον καὶ σῶμά ἐστιν ἄτομον. Ἑνος γὰρ ὄντος ἀδιαιρέτου καὶ τἆλλα συνακολουθήσει διὰ τὸ θάτερον διῃρῆσθαι κατὰ θάτερον.

If, then, a point is not present in the indivisible line, neither will it be in a line at all; for the other lines are composed of indivisible ones. Furthermore, either there is nothing between points, or a line; but if a line is between them, and there are several points in all lines, it will not be indivisible. Furthermore, there will not be a square on every line; for it will have length and width, so that it is divisible, since the one is length and the other is width, and if the square is divisible, so also is the line. Furthermore, the limit of a line will be a point, but not a line. For it is a limit, but the indivisible line is the extreme. For if it is a point, the limit of the indivisible line will be a point, and one line will be larger than another by a point. And if the point is present in the indivisible line, because of the same limit of continuous lines, there will be some limit of the indivisible, And in general, how will a point differ from a line? For the indivisible line will have nothing peculiar to itself compared to the point, except the name. Furthermore, similarly, an indivisible plane and body remain. For if one thing is indivisible, the others will follow, because the one is divided in accordance with the other.

§36

Σῶμα οὐκ ἔσται ἀδιαίρετον διὰ τὸ εἶναι ἐν αὐτῷ βάθος καὶ πλάτος, οὐδ’ ἂν γραμμὴ εἴη ἀδιαίρετος· σῶμα μὲν γὰρ κατ’ ἐπίπεδον, ἐπίπεδον δὲ κατὰ γραμμήν.³ Ἐπεὶ δὲ οἵ τε λόγοι δι' ὦν ἐπιχειροῦσι πείθειν ἀσθενεῖς εἰσί, καὶ ψευδεῖς ἐναντίαι δόξαι πάσαι τοῖς ἰσχύουσι πρὸς πίστιν, φανερὸν ὅτι οὐκ ἄν εἴη γραμμὴ ἄτομος. Δῆλον δ' ἐκ τούτων ὅτι οὐδ’ ἂν ἐκ στιγμῶν εἴη γραμμή.

A body will not be indivisible because there is in it depth and width, nor would a line be indivisible; for a body is defined in terms of a plane, and a plane in terms of a line. And since the arguments by which they attempt to persuade are weak, and all contrary opinions are false to those that have power for conviction, it is clear that there would be no indivisible line. And from this it is clear that a line would not be composed of points either.

§37

Σχεδὸν γὰρ οἱ πλεῖστοι τῶν λόγων οἱ αὐτοὶ ἁρμόσουσιν. Ἀνάγκη γὰρ διαιρεῖσθαι τὴν στιγμήν, ὅταν ἢ ἐκ περιττῶν τέμνηται ἴσα ἢ ἐξ ἀρτίων τὰ ἄνισα. Καὶ τὸ τῆς γραμμῆς μέρος μὴ εἶναι γραμμήν, μηδὲ τὸ τοῦ ἐπιπέδου ἐπίπεδον.

For almost most of the same arguments will apply. For it is necessary for the point to be divided, when a line composed of an odd number of points is cut into equal parts, or one of an even number into unequal parts. And also that a part of a line would not be a line, nor a part of a plane a plane.

§38

Καὶ γραμμὴ δὲ γραμμῆς στιγμῇ εἶναι μείζων· ἐξ ὦν γὰρ σύγκειται, τούτοις καὶ ὑπερέξει. Τοῦτο δ' ὅτι ἀδύνατον, ἔκ τε τῶν ἐν τοῖς μαθήμασι δῆλον, καὶ ἔτι συμβήσεται τὴν στιγμὴν ἐν χρόνῳ δὴ εἶναι τὸ φερόμενον,⁴ εἴπερ τὴν μείζω μὲν ἐν πλείονι χρόνῳ, τὴν δ’ ἴσην ἐν ἴσῳ, ἡ δὲ τοῦ χρόνου ὑπεροχὴ χρόνος.

And also a line would be larger than another line by a point; for it will exceed by those things from which it is composed. And that this is impossible is clear from what is assumed in mathematics, and furthermore it will follow that the moving thing is in a point in a certain time, if indeed it traverses the larger distance in more time, and the equal in equal time, and the excess of time is time.

§39

Ἁλλ' ἴσως καὶ ὁ χρόνος ἐστὶν ἐκ τῶν νῦν, καὶ τοῦ αὐτοῦ λόγου λέγειν ἄμφω. Εἰ δὴ τὸ νῦν ἀρχὴ καὶ πέρας τοῦ χρόνου καὶ ἡ γραμμὴ στιγμῆς, μή ἐστι δὲ συνεχὴς ἡ ἀρχὴ καὶ τὸ πέρας ἀλλ' ἔχουσί τι μεταξύ, οὐκ ἂν εἴη οὔτε τὰ νῦν οὔτε στιγμαὶ ἀλλήλοις συνεχεῖς.

But perhaps time also is composed of ‘nows’, and to speak of both belongs to the same argument. If indeed the ‘now’ is the beginning and limit of time, and the point is of a line, but the beginning and the limit are not continuous but have something between them, then neither ‘nows’ nor points would be continuous with one another.

§40

Ἔτι ἡ μὲν γραμμὴ μέγεθός τι, δὲ τῶν στιγμῶν σύνθεσις οὐδὲν ποιεῖ μέγεθος διὰ τὸ μηδ’ ἐπὶ πλείω τόπον ἔχειν. Ὄταν γὰρ ἐπὶ γραμμὴν γραμμὴ τεθῇ καὶ ἐφαρμόσῃ, οὐδὲν γίνεται μεῖζον τὸ πλάτος. Ἐν δὲ τῇ γραμμῇ καὶ στιγμαὶ ἐνυπάρχουσιν, οὐδ’ ἂν αἱ στιγμαὶ πλείω κατέχοιεν τόπον, ὥστε οὐκ ἂν ποιοῖεν μέγεθος.

Furthermore, while a line is a certain magnitude, the composition of points makes no magnitude because they do not occupy more space. For when a line is placed upon a line and coincides, the width does not become any larger. And since points also are present in a line, neither would points occupy more space, so that they would not make a magnitude.

§41

Ἔτι εἰ ἅπαντα ἅπτεται παντὸς ἢ ὅλον ὅλου ἢ τινὶ τινὸς ἢ ὅλον τινός, ἡ δὲ στιγμὴ ἀμερὴς ὅλως ἅπτοιτο. Τὸ δ’ ὅλον ὅλου ἁπτόμενον ἀνάγκη ἓν εἶναι. Εἰ γάρ τι ἐστὶν ἢ θάτερον μή ἐστιν,⁵ οὐκ ἂν ὅλον ὅλου ἅπτοιτο.

Furthermore, if everything touches everything else either as a whole touching a whole, or a part a part, or a whole a part, and the point, being partless, must touch as a whole, then that which touches as a whole must be one. For if there is some part, or if one of the two does not exist, it would not touch as a whole.

Syntax notes5 items

970b32Ἣ ἔσται
Read as the conjunction ἤ ("or else", "otherwise") rather than the relative pronoun ἥ. It introduces the absurd alternative that if one rejects the previous conclusion that the indivisible line is finite and thus divisible, there must exist a third type of line which is neither infinite nor finite.
970b33ἐπεὶ τὸ μέν, τὸ δέ τι
A subordinate clause introduced by the conjunction ἐπεί, where the predicates or nouns μῆκος and πλάτος are omitted. It should be understood as "since the one is [length] and the other is [width]," showing that a square is divisible into elements of two different dimensions.
971a1σῶμα μὲν γὰρ κατ’ ἐπίπεδον, ἐπίπεδον δὲ κατὰ γραμμήν
A verb or participle such as ὁρίζεται ("is defined/bounded") or διαιρεῖται ("is divided") is omitted after the preposition κατά. It expresses the dimensional definition or boundary relation between geometrical objects: "for a body is [defined] by a plane, and a plane by a line."
971a1τὴν στιγμὴν ἐν χρόνῳ δὴ εἶναι τὸ φερόμενον
An accusative with infinitive construction functioning as the subject of the impersonal verb συμβήσεται, with τὸ φερόμενον ("the moving thing") as its subject. It shows the absurd consequence that "the moving thing would be in a point in a certain time," leading to the impossibility of motion if space is composed of indivisible points.
971a1Εἰ γάρ τι ἐστὶν ἢ θάτερον μή ἐστιν
A conditional clause meaning "for if there is some [part], or if one of the two does not exist." It explains the logical requirement that for two things to touch as a whole (i.e., to coincide completely and become one), there must be no distinct parts, and both things must actually exist.

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

2026-09-22 AI review — source 1
2026-09-21 AI review — source 1

urn:cts:greekLit:tlg0086.tlg019.humanitext-grc1:32-41