Impossibility of Subtracting Points and Line Definitions — Aristotle, On Indivisible Lines 52–60, Greek & English

§52

Ἔσται γραμμὴ γραμμῆς στιγμῇ μείζων.¹ Τοῦτο δ’ ἀδύνατον. Ἀλλὰ καθ’ έαυτὴν μὲν οὐχ’ οἶόν τε, κατὰ συμβεβηκὸς δ' ἐνδέχεται στιγμὴν ἀπὸ γραμμῆς ἀφελεῖν, τῷ ἐνυπάρχειν ἐν τῇ ἀφαιρουμένῃ γραμμῇ. Εἰ τοῦ ὅλου ἀφαιρουμένου καὶ ἡ ἀρχὴ καὶ τὸ πέρας ἀφαιρεῖται, γραμμῆς δ’ ἦν ἡ ἀρχὴ καὶ τὸ πέρας στιγμή, καὶ γραμμῆς ἐγχωρεῖ ἀφαιρεῖν καὶ στιγμὴν ἐνδέχοιτο.

A line will be larger than a line by a point. But this is impossible. But while in itself it is not possible, accidentally it is possible to subtract a point from a line, by its being inherent in the subtracted line. If, when the whole is subtracted, both the beginning and the end are also subtracted, and the beginning and end of a line were a point, then it would be permissible to subtract from a line, and it would be possible to subtract a point.

§53

Αὕτη δ’ ἡ ἀφαίρεσις κατὰ συμβεβηκός. Εἰ δὲ τὸ πέρας ἅπτεται, οὔτε πέρας ἢ αὐτοῦ ἢ τῶν ἐκείνου τινός. Ἡ δὲ στιγμή, ᾗ πέρας γραμμῆς, ἅπτεται, Ἧι μὲν οὖν γραμμῆς ἔσται στιγμὴ μείζων, ἡ δὲ στιγμὴ ἐκ στιγμῶν· τῶν γὰρ ἁπτομένων οὐδὲν ἀνὰ μέσον.

And this subtraction is accidental. But if the limit touches, it is not a limit either of itself or of any of those things. And the point, in so far as it is the limit of a line, touches. If, then, a line is to be larger than another line by a point, a point must be composed of points; for there is nothing between things that touch.

§54

Ὁ αὐτὸς λόγος καὶ ἐπὶ τῆς τομῆς, εἰ ἡ τομὴ στιγμῆς καὶ ἡ τομὴ ἅπτεταί τινος καὶ ἐπὶ στερεοῦ καὶ ἐπιπέδου· ὡσαύτως δὲ καὶ τὸ στερεὸν ἐξ ἐπιπέδων καὶ γραμμῶν. Οὐκ ἀληθὲς δὲ κατὰ στιγμὴν εἰπεῖν, οὐδ’ ὅτι ἐλάχιστον τῶν ἐκ γραμμῆς εἰς τὸ ἐλάχιστον τῶν ἐνυπαρχόντων εἴρηται. Τὸ δὲ ἐλάχιστον, ὦν ἐστὶν ἐλάχιστον, καὶ ἔλαττόν ἐστιν.²

The same argument holds also for division, if division is of a point and division touches something, both in the case of a solid and a plane; and likewise the solid would be composed of planes and lines. And it is not true to speak of a point, nor that the least of the things in a line is said in reference to the least of the inherent parts. But the least is also smaller than those things of which it is the least.

§55

Ἐν δὲ τῇ γραμμῇ οὐδὲν ἄλλο ἢ στιγμιαὶ καὶ γραμμαὶ ἐνυπάρχουσιν. Ἡ δὲ γραμμὴ τῆς στιγμῆς οὐκ ἔστι μείζων· οὐδὲ γὰρ αὗ τὸ ἐπίπεδον τῆς γραμμῆς. Ὥστ’ οὐκ ἔσται στιγμὴ τὸ ἐν γραμμῇ ἐλάχιστον. Εἰ δὲ συμβλητὸν τῇ γραμμῇ ἡ στιγμή, τὸ δὲ ἐλάχιστον ἐν τρισὶ προσώποις, οὐκ ἔσται ἡ στιγμὴ τῶν ἐν τῇ γραμμῇ ἐλάχιστον.

But in a line there are inherent nothing other than points and lines. But a line is not larger than a point; for neither, again, is a plane larger than a line. So that a point will not be the least in a line. But if a point is comparable with a line, and ‘the least’ is found among three terms, a point will not be the least of the things in a line.

§56

Καὶ ἄλλ’ ἄττα ἐνυπάρχει παρὰ τὰς στιγμὰς καὶ τὰς γραμμὰς ἐν τῷ μήκει· οὐ γὰρ ἐκ στιγμῶν. Εἰ δὲ τὸ ἐν τόπῳ ὂν ἡ στιγμὴ μῆκος ἢ ἐπίπεδον ἢ στερεὸν ἐκ τούτων τι, ἐξ ὦν δ’ ἐστὶν ἡ γραμμή, ἐκεῖνα ἐν τόπῳ (καὶ γὰρ ἡ γραμμή), καὶ μήτε σῶμα μήτ’ ἐπίπεδον μήτε ἐκ τούτων τι ἐνυπάρχει τῇ γραμμῇ, οὐκ ἔσται οὐθὲν ὅλως παρὰ τὰς στιγμὰς καὶ τὰς γραμμὰς ἐν τῷ μήκει.

And some other things are inherent in length besides points and lines; for it is not composed of points. But if that which is in space is a point, length, plane, or solid, or some of these, and those things of which a line is composed are in space (for a line is also in space), and neither a body nor a plane nor any of these is inherent in a line, there will be nothing at all in length besides points and lines.

§57

Ἔτι εἰ τοῦ ἐν τόπῳ ὄντος τὸ μεῖζον λεγόμενον μῆκος ἡ ἐπιφάνεια στερεόν, ἡ δὲ στιγμὴ ἐν τόπῳ, τὸ δ' ἐν τῷ μήκει ὑπάρχον παρὰ τὰς στιγμὰς καὶ τὰς γραμμὰς οὐθὲν τῶν προειρημένων, ὥστ’ οὐκ ἔσται ἡ στιγμὴ τῶν ἐνυπαρχόντων ἐλάχιστον.

Furthermore, if of that which is in space, that which is called ‘larger’ is length, surface, or solid, and a point is in space, but that which exists in length besides points and lines is none of the aforementioned, then a point will not be the least of the inherent things.

§58

Ἔτι εἰς ὃ ἐλάχιστόν τι τῶν ἐν τῇ οἰκίᾳ, μήτε τῆς οἰκίας συμβαλλομένης πρὸς αὐτὸ³ λέγεται· ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων· οὐδὲ τὸ ἐν γραμμῇ ἐλάχιστον πρὸς γραμμὴν συγκρινόμενον ἔσται. Ὥστε οὐχ ἁρμόσει τὸ ἐλάχιστον, ἐπεὶ τὸ μὴ ὄν ἐν τῇ οἰκίᾳ μή ἐστι τῶν ἐν τῇ οἰκίᾳ ἐλάχιστον. Ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων.

Furthermore, if whatever is the ‘least’ of the things in a house is not so called by comparison with the house itself, and likewise in other cases, then neither will the least in a line be compared with the line. So that ‘the least’ will not fit; since that which is not in the house is not the least of the things in the house. And likewise in other cases.

§59

Ἐνδέχεται γὰρ στιγμὴν αὐτὴν καθ’ αὐτὴν εἶναι. Οὐκ ἔσται κατὰ ταύτης ἀληθὲς εἰπεῖν ὅτι τὸ ἐν γραμμῇ ἐλάχιστον, ὅτι οὐκ ἔστιν ἡ στιγμὴ ἄρθρον ἀδιαίρετον.⁴ Τὸ μὲν γὰρ ἄρθρον ἀεὶ δυοῖν ὅρος, ἡ δὲ στιγμὴ καὶ μιᾶς γραμμῆς ὅρος ἐστίν. Ἔτι ἡ μὲν πέρας, τὸ δὲ διαίρεσίς ἐστι μᾶλλον.

For it is possible for a point to exist by itself. It will not be true to say of it that it is ‘the least in a line’, because a point is not an indivisible joint. For a joint is always the limit of two things, but a point is also the limit of a single line. Furthermore, the former is a limit, while the latter is rather a division.

§60

Ἔτι ἡ γραμμὴ καὶ τὸ ἐπίπεδον ἄρθρα ἔσονται· ἀνάλογον γὰρ ἔχουσιν, ὅτι τὸ ἄρθρον διάφορόν πως ἐστίν, διὸ καὶ Ἐμπεδοκλῆς ἐποίησε διὸ δεῖ ὀρθῶς. Ἡ δὲ στιγμὴ καὶ τὸ ἐν τοῖς ἀκινήτοις. Ἔτι οὐδεὶς ἔχει ἄπειρα ἄρθρα ἐν τῷ σώματι ἢ τῇ χειρί, στιγμὰς δ’ ἀπείρους. Ἔτι λίθου ἄρθρον οὐκ ἔστιν, οὐδ’ ἔχει, στιγμὰς δὲ ἔχει.

Furthermore, a line and a plane will be joints; for they have an analogous relation, because a joint is in some way different, which is why Empedocles also wrote... wherefore it must be right. And a point is also among motionless things. Furthermore, no one has infinite joints in their body or hand, but they have infinite points. Furthermore, there is no joint of a stone, nor does it have one, but it has points.

Syntax notes4 items

972a52γραμμὴ γραμμῆς στιγμῇ μείζων
In relation to the comparative μείζων, the genitive γραμμῆς is a genitive of comparison ('than a line'), and the dative στιγμῇ is a dative of measure of difference ('by a point').
972a54Τὸ δὲ ἐλάχιστον, ὧν ἐστὶν ἐλάχιστον, καὶ ἔλαττόν ἐστιν
The relative pronoun ὧν is a partitive genitive dependent on the superlative ἐλάχιστον ('the least of which'). In the main clause, the comparative ἔλαττον ('smaller') implies a comparison with those same objects (the antecedent of ὧν), which is grammatically omitted.
972b1μήτε τῆς οἰκίας συμβαλλομένης πρὸς αὐτὸ
This is a genitive absolute construction with the negative particle μήτε, expressing an attendant circumstance or negative condition ('without the house itself being compared to it') within the conditional clause introduced by εἰ.
972b1ἄρθρον ἀδιαίρετον
The term ἄρθρον normally means a 'joint' or 'articulation', but here refers to a logical or geometrical point of junction that connects two things by being a common limit (δυοῖν ὅρος). The author argues that a point (στιγμή) can be the limit (πέρας) of a single line, whereas a joint essentially requires two parts to connect, making them conceptually distinct.

Translated by Gemini 3.5 Flash (published 2026-05-22) · Revision history (3, latest 2026-09-21)

2026-09-21 AI review — source 3

urn:cts:greekLit:tlg0086.tlg019.humanitext-grc1:52-60