§32Ἡ δὲ ἄτομος οὐκ ἄπειρος, ὥστε ἔξει πέρας.
But the indivisible line is not infinite, so that it will have a limit.
Διαιρετὴ ἄρα· τὸ γὰρ πέρας ἄλλο καὶ οὗ πέρας.
Therefore it is divisible; for the limit is other than that of which it is the limit.
Ἣ ἔσται τις οὔτ’ ἄπειρος οὔτε πεπερασμένη γραμμὴ παρὰ ταύτας.
Or there will be some line other than these, which is neither infinite nor finite.
Ἕτι οὐκ ἐν ἁπάσῃ γραμμῇ στιγμὴ ἔσται.
Furthermore, there will not be a point in every line.
Ἐν μὲν γὰρ τῇ ἀτόμῳ οὐκ ἔστιν· εἰ μὲν γὰρ μία μόνη, ὑπάρξει γραμμή, εἶτα στιγμή· εἰ δὲ πλείους, διαιρετὴ ἡ γραμμή.
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.
§33Εἰ μὲν οὖν ἐν τῇ ἀτόμῳ μὴ ἐνυπάρχει στιγμή, οὐδ’ ὅλως ἐν γραμμῇ ἔσται· αἱ γὰρ ἄλλαι ἐκ τῶν ἀτόμων.
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.
Ἔτι ἢ μηθὲν τῶν στιγμῶν ἔσται μεταξὺ ἢ γραμμή· εἰ δὲ μεταξὺ γραμμή, ἐν ἀπάσαις δὲ πλείους στιγμαί, οὐκ ἔσται ἄτομος.
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.
Ἔτι οὐχ ἀπάσης ἔσται γραμμῆς τετράγωνον· ἔξει γὰρ μῆκος καὶ πλάτος, ὥστε διαιρετόν, ἐπεὶ τὸ μέν, τὸ δέ τι, §34Εἰ δὲ τὸ τετράγωνον, καὶ ἡ γραμμή.
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.
Ἔτι τὸ πέρας τῆς γραμμῆς στιγμὴ ἔσται, ἀλλ' οὐ γραμμή.
Furthermore, the limit of a line will be a point, but not a line.
Πέρας μὲν γάρ, τὸ ἔσχατον δὲ ἡ ἄτομος.
For it is a limit, but the indivisible line is the extreme.
Εἰ γὰρ στιγμή, τὸ πέρας τῇ ἀτόμῳ ἔσται στιγμή, καὶ ἔσται γραμμὴ γραμμῆς στιγμῇ μείζων.
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.
Εἰ δ' ἐνυπάρχει τῇ ἀτόμῳ ἡ στιγμή, διὰ τὸ ταὐτὸ πέρας τῶν συνεχουσῶν γραμμῶν, ἔσται τι πέρας τῆς ἀμεροῦς, §35Ὄλως τε τί διοίσει στιγμὴ γραμμῆς;
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?
Οὐδὲν γὰρ ἴδιον ἔξει ἡ ἄτομος γραμμὴ παρὰ τὴν στιγμὴν πλὴν τοὔνομα.
For the indivisible line will have nothing peculiar to itself compared to the point, except the name.
Ἕτι ὁμοίως μένει ἐπίπεδον καὶ σῶμά ἐστιν ἄτομον.
Furthermore, similarly, an indivisible plane and body remain.
Ἑνος γὰρ ὄντος ἀδιαιρέτου καὶ τἆλλα συνακολουθήσει διὰ τὸ θάτερον διῃρῆσθαι κατὰ θάτερον.
For if one thing is indivisible, the others will follow, because the one is divided in accordance with the other.
§36Σῶμα 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.
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.
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.
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.
Δῆλον δ' ἐκ τούτων ὅτι οὐδ’ ἂν ἐκ στιγμῶν εἴη γραμμή.
And from this it is clear that a line would not be composed of points either.
§37Σχεδὸν γὰρ οἱ πλεῖστοι τῶν λόγων οἱ αὐτοὶ ἁρμόσουσιν.
For almost most of the same arguments will apply.
Ἀνάγκη γὰρ διαιρεῖσθαι τὴν στιγμήν, ὅταν ἢ ἐκ περιττῶν τέμνηται ἴσα ἢ ἐξ ἀρτίων τὰ ἄνισα.
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.
Καὶ τὸ τῆς γραμμῆς μέρος μὴ εἶναι γραμμήν, μηδὲ τὸ τοῦ ἐπιπέδου ἐπίπεδον.
And also that a part of a line would not be a line, nor a part of a plane a plane.
§38Καὶ γραμμὴ δὲ γραμμῆς στιγμῇ εἶναι μείζων· ἐξ ὦν γὰρ σύγκειται, τούτοις καὶ ὑπερέξει.
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.
Τοῦτο δ' ὅτι ἀδύνατον, ἔκ τε τῶν ἐν τοῖς μαθήμασι δῆλον, καὶ ἔτι συμβήσεται τὴν στιγμὴν ἐν χρόνῳ δὴ εἶναι τὸ φερόμενον, εἴπερ τὴν μείζω μὲν ἐν πλείονι χρόνῳ, τὴν δ’ ἴσην ἐν ἴσῳ, ἡ δὲ τοῦ χρόνου ὑπεροχὴ χρόνος.
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.
§39Ἁλλ' ἴσως καὶ ὁ χρόνος ἐστὶν ἐκ τῶν νῦν, καὶ τοῦ αὐτοῦ λόγου λέγειν ἄμφω.
But perhaps time also is composed of 'nows', and to speak of both belongs to the same argument.
Εἰ δὴ τὸ νῦν ἀρχὴ καὶ πέρας τοῦ χρόνον 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.
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.
§40Ἔτι ἡ μὲν γραμμὴ μέγεθός τι, δὲ τῶν στιγμῶν σύνθεσις οὐδὲν ποιεῖ μέγεθος διὰ τὸ μηδ’ ἐπὶ πλείω τόπον ἔχειν.
Furthermore, while a line is a certain magnitude, the composition of points makes no magnitude because they do not occupy more space.
Ὄταν γὰρ ἐπὶ γραμμὴν γραμμὴ τεθῇ καὶ ἐφαρμόσῃ, οὐδὲν γίνεται μεῖζον τὸ πλάτος.
For when a line is placed upon a line and coincides, the width does not become any larger.
Ἐν δὲ τῇ γραμμῇ καὶ στιγμαὶ ἐνυπάρχονσιν, οὐδ’ ἂν αἱ στιγμαὶ πλείω κατέχοιεν τόπον, ὥστε οὐκ ἂν ποιοῖεν μέγεθος.
And since points also are present in a line, neither would points occupy more space, so that they would not make a magnitude.
§41Ἔτι 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.
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.
Εἰ γάρ τι ἐστὶν ἢ θάτερον μή ἐστιν, οὐκ ἂν ὅλον ὅλου ἅπτοιτο.
For if there is some part, or if one of the two does not exist, it would not touch as a whole.