§52Ἔσται γραμμὴ γραμμῆς στιγμῇ μείζων.
A line will be larger than a line by a point.
Τοῦτο δ’ ἀδύνατον.
But this is impossible.
Ἀλλὰ καθ’ έαυτὴν μὲν οὐχ’ οἶόν τε, κατὰ συμβεβηκὸς δ' ἐνδέχεται στιγμὴν ἀπὸ γραμμῆς ἀφελεῖν, τῷ ἐνυπάρχειν ἐν τῇ ἀφαιρουμένῃ γραμμῇ.
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.
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.
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.
§53Αὕτη δ’ ἡ ἀφαίρεσις κατὰ συμβεβηκός.
And this subtraction is accidental.
Εἰ δὲ τὸ πέρας ἅπτεται, οὔτε πέρας ἢ αὐτοῦ ἢ τῶν ἐκείνου τινός.
But if the limit touches, it is not a limit either of itself or of any of those things.
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 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.
§54Ὁ 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.
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.
Οὐκ ἀληθὲς δὲ κατὰ στιγμὴν εἰπεῖν, οὐδ’ ὅτι ἐλάχιστον τῶν ἐκ γραμμῆς εἰς τὸ ἐλάχιστον τῶν ἐνυπαρχόντων εἴρηται.
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.
Τὸ δὲ ἐλάχιστον, ὦν ἐστὶν ἐλάχιστον, καὶ ἔλαττόν ἐστιν.
But the least is also smaller than those things of which it is the least.
§55Ἐν δὲ τῇ γραμμῇ οὐδὲν ἄλλο ἢ στιγμιαὶ καὶ γραμμαὶ ἐνυπάρχουσιν.
But in a line there are inherent nothing other than points and lines.
Ἡ δὲ γραμμὴ τῆς στιγμῆς οὐκ ἔστι μείζων· οὐδὲ γὰρ αὗ τὸ ἐπίπεδον τῆς γραμμῆς.
But a line is not larger than a point; for neither, again, is a plane larger than a line.
Ὥστ’ οὐκ ἔσται στιγμὴ τὸ ἐν γραμμῇ ἐλάχιστον.
So that a point will not be the least in a line.
Εἰ δὲ συμβλητὸν τῇ γραμμῇ ἡ στιγμή, τὸ δὲ ἐλάχιστον ἐν τρισὶ προσώποις, οὐκ ἔσται ἡ στιγμὴ τῶν ἐν τῇ γραμμῇ ἐλάχιστον.
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.
§56Καὶ ἄλλ’ ἄττα ἐνυπάρχει παρὰ τὰς στιγμὰς καὶ τὰς γραμμὰς ἐν τῷ μήκει· οὐ γὰρ ἐκ στιγμῶν.
And some other things are inherent in length besides points and lines; for it is not composed of points.
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.
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.
§57Ἔτι 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.
§58Ἔτι εἰς ὃ ἐλάχιστόν τι τῶν ἐν τῇ οἰκίᾳ, μήτε τῆς οἰκίας συμβαλλομένης πρὸς αὐτὸ λέγεται· ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων· οὐδὲ τὸ ἐν γραμμῇ ἐλάχιστον πρὸς γραμμὴν σνγκρινόμμενον ἔσται.
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.
Ὥστε οὐχ ἁρμόσει τὸ ἐλάχιστον, ἐπεὶ τὸ μὴ ὄν ἐν τῇ οἰκίᾳ μή ἐστι τῶν ἐν τῇ οἰκίια ἐλάχιστον.
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.
Ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων.
And likewise in other cases.
§59Ἐνδέχεται γὰρ στιγμὴν αὐτὴν καθ’ αὐτὴν εἶναι.
For it is possible for a point to exist by itself.
Οὐκ ἔσται κατὰ ταύτης ἀληθὲς εἰπεῖν ὅτι τὸ ἐν γραμμῇ ἐλάχιστον, ὅτι οὐκ ἔστιν ἡ στιγμὴ ἄρθρον ἀδιαίρετον.
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.
Τὸ μὲν γὰρ ἄρθρον ἀεὶ δυοῖν ὅρος, ἡ δὲ στιγμὴ καὶ μιᾶς γραμμῆς ὅρος ἐστίν.
For a joint is always the limit of two things, but a point is also the limit of a single line.
Ἔτι ἡ μὲν πέρας, τὸ δὲ διαίρεσίς ἐστι μᾶλλον.
Furthermore, the former is a limit, while the latter is rather a division.
§60Ἔτι ἡ γραμμὴ καὶ τὸ ἐπίπεδον ἄρθρα ἔσονται· ἀνάλογον γὰρ ἔχουσιν, ὅτι τὸ ἄρθρον διάφορόν πως ἐστίν, διὸ καὶ Ἐμπεδοκλῆς ἐποίησε διὸ δεῖ ὀρθῶς.
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.
Ἡ δὲ στιγμὴ καὶ τὸ ἐν τοῖς ἀκινήτοις.
And a point is also among motionless things.
Ἔτι οὐδεὶς ἔχει ἄπειρα ἄρθρα ἐν τῷ σώματι ἢ τῇ χειρί, στιγμνὰς δ’ ἀπείρους.
Furthermore, no one has infinite joints in their body or hand, but they have infinite points.
Ἔτι λίθου ἄρθρον οὐκ ἔστιν, οὐδ’ ἔχει, στιγμὰς δὲ ἔχει.
Furthermore, there is no joint of a stone, nor does it have one, but it has points.