§13.2#1ὅτι μὲν τοίνυν ἔν γε τοῖς αἰσθητοῖς ἀδύνατον εἶναι καὶ ἅμα πλασματίας ὁ λόγος, εἴρηται μὲν καὶ ἐν τοῖς διαπορήμασιν ὅτι δύο ἅμα στερεὰ εἶναι ἀδύνατον, ἔτι δὲ καὶ ὅτι τοῦ αὐτοῦ λόγου καὶ τὰς ἄλλας δυνάμεις καὶ φύσεις ἐν τοῖς αἰσθητοῖς εἶναι καὶ μηδεμίαν κεχωρισμένην·
Therefore, that it is impossible for them to exist in sensible things at least, and that the argument is a mere fiction, has been stated in the difficulties, namely that it is impossible for two solids to exist in the same place at the same time, and further, that by the same argument, the other powers and natures would also exist in sensible things and none of them would be separated.
—ταῦτα μὲν οὖν εἴρηται πρότερον, ἀλλὰ πρὸς τούτοις φανερὸν ὅτι ἀδύνατον διαιρεθῆναι ὁτιοῦν σῶμα· κατʼ ἐπίπεδον γὰρ διαιρεθήσεται, καὶ τοῦτο κατὰ γραμμὴν καὶ αὕτη κατὰ στιγμήν, ὥστʼ εἰ τὴν στιγμὴν διελεῖν ἀδύνατον, καὶ τὴν γραμμήν, εἰ δὲ ταύτην, καὶ τἆλλα.
These things, then, have been said before, but in addition to these, it is clear that it is impossible for any body whatever to be divided; for it will be divided along a plane, and this along a line, and this along a point, so that if it is impossible to divide a point, it is also impossible to divide a line, and if a line is impossible, so too are the others.
τί οὖν διαφέρει ἢ ταύτας εἶναι τοιαύτας φύσεις, ἢ αὐτὰς μὲν μή, εἶναι δʼ ἐν αὐταῖς τοιαύτας φύσεις;
What difference, then, does it make whether these are such natures, or themselves are not, but such natures exist in sensible things?
τὸ αὐτὸ γὰρ συμβήσεται· διαιρουμένων γὰρ τῶν αἰσθητῶν διαιρεθήσονται, ἢ οὐδὲ αἱ αἰσθηταί.
For the same result will follow: for when the sensible things are divided, they will be divided, or else not even the sensible things will be divided.
ἀλλὰ μὴν οὐδὲ κεχωρισμένας γʼ εἶναι φύσεις τοιαύτας δυνατόν.
But indeed, neither is it possible for such natures to exist separated.
εἰ γὰρ ἔσται στερεὰ παρὰ τὰ αἰσθητὰ κεχωρισμένα τούτων ἕτερα καὶ πρότερα τῶν αἰσθητῶν, δῆλον ὅτι καὶ παρὰ τὰ ἐπίπεδα ἕτερα ἀναγκαῖον εἶναι ἐπίπεδα κεχωρισμένα καὶ στιγμὰς καὶ γραμμάς (τοῦ γὰρ αὐτοῦ λόγου)·
For if there are to be solids besides the sensible ones, separate from them and prior to them, it is clear that besides the planes it is necessary that there be other separate planes and points and lines (for the same argument applies).
εἰ δὲ ταῦτα, πάλιν παρὰ τὰ τοῦ στερεοῦ τοῦ μαθηματικοῦ ἐπίπεδα καὶ γραμμὰς καὶ στιγμὰς ἕτερα κεχωρισμένα (πρότερα γὰρ τῶν συγκειμένων ἐστὶ τὰ ἀσύνθετα· καὶ εἴπερ τῶν αἰσθητῶν πρότερα σώματα μὴ αἰσθητά, τῷ αὐτῷ λόγῳ καὶ τῶν ἐπιπέδων τῶν ἐν τοῖς ἀκινήτοις στερεοῖς τὰ αὐτὰ καθʼ αὑτά, ὥστε ἕτερα ταῦτα ἐπίπεδα καὶ γραμμαὶ τῶν ἅμα τοῖς στερεοῖς τοῖς κεχωρισμένοις· τὰ μὲν γὰρ ἅμα τοῖς μαθηματικοῖς στερεοῖς τὰ δὲ πρότερα τῶν μαθηματικῶν στερεῶν).
And if these exist, again besides the planes, lines, and points of the mathematical solid, there will be other separate ones (for the incomposite is prior to the composite; and if prior to sensible bodies there are non-sensible bodies, by the same argument, prior to the planes in the motionless solids are those planes themselves by themselves, so that these planes and lines are different from those which exist together with the separate solids; for some exist together with the mathematical solids, and others are prior to the mathematical solids).
πάλιν τοίνυν τούτων τῶν ἐπιπέδων ἔσονται γραμμαί, ὧν πρότερον δεήσει ἑτέρας γραμμὰς καὶ στιγμὰς εἶναι διὰ τὸν αὐτὸν λόγον· καὶ τούτων τῶν ἐκ ταῖς προτέραις γραμμαῖς ἑτέρας προτέρας στιγμάς, ὧν οὐκέτι πρότεραι ἕτεραι.
Again, therefore, of these planes there will be lines, prior to which there must be other lines and points by the same argument; and prior to the points in those prior lines, there must be other prior points, prior to which there are no longer others.
ἄτοπός τε δὴ γίγνεται ἡ σώρευσις (συμβαίνει γὰρ στερεὰ μὲν μοναχὰ παρὰ τὰ αἰσθητά, ἐπίπεδα δὲ τριττὰ παρὰ τὰ αἰσθητά—τά τε παρὰ τὰ αἰσθητὰ καὶ τὰ ἐν τοῖς μαθηματικοῖς στερεοῖς καὶ τὰ παρὰ τὰ ἐν τούτοις—γραμμαὶ δὲ τετραξαί, στιγμαὶ δὲ πενταξαί·
And so this accumulation becomes absurd; for there turn out to be solids of only one kind besides the sensible ones, but planes of three kinds besides the sensible ones (those besides the sensible ones, those in the mathematical solids, and those besides those in these), lines of four kinds, and points of five kinds.
ὥστε περὶ ποῖα αἱ ἐπιστῆμαι ἔσονται αἱ μαθηματικαὶ τούτων;
So, about which of these will the mathematical sciences be?
οὐ γὰρ δὴ περὶ τὰ ἐν τῷ στερεῷ τῷ ἀκινήτῳ ἐπίπεδα καὶ γραμμὰς καὶ στιγμάς· ἀεὶ γὰρ περὶ τὰ πρότερα ἡ ἐπιστήμη)·
For surely not about the planes, lines, and points in the motionless solid; for science is always about what is prior.
ὁ δʼ αὐτὸς λόγος καὶ περὶ τῶν ἀριθμῶν· παρʼ ἑκάστας γὰρ τὰς στιγμὰς ἕτεραι ἔσονται μονάδες, καὶ παρʼ ἕκαστα τὰ ὄντα, τὰ αἰσθητά, εἶτα τὰ νοητά, ὥστʼ ἔσται γένη τῶν μαθηματικῶν ἀριθμῶν.
And the same argument applies also to numbers; for besides each point there will be other units, and besides each existing thing, first the sensible ones and then the intelligible ones, so that there will be genera of mathematical numbers.
ἔτι ἅπερ καὶ ἐν τοῖς ἀπορήμασιν ἐπήλθομεν πῶς ἐνδέχεται λύειν;
Furthermore, how is it possible to solve those difficulties which we went through in the discussion of difficulties?
περὶ ἃ γὰρ ἡ ἀστρολογία ἐστίν, ὁμοίως ἔσται παρὰ τὰ αἰσθητὰ καὶ περὶ ἃ ἡ γεωμετρία·
For the objects about which astronomy is concerned will exist, in like manner as those of geometry, besides the sensible ones.
εἶναι δʼ οὐρανὸν καὶ τὰ μόρια αὐτοῦ πῶς δυνατόν, ἢ ἄλλο ὁτιοῦν ἔχον κίνησιν;
But how is it possible for a heaven and its parts, or anything else having motion, to exist?
ὁμοίως δὲ καὶ τὰ ὀπτικὰ καὶ τὰ ἁρμονικά· ἔσται γὰρ φωνή τε καὶ ὄψις παρὰ τὰ αἰσθητὰ καὶ τὰ καθʼ ἕκαστα, ὥστε δῆλον ὅτι καὶ αἱ ἄλλαι αἰσθήσεις καὶ τὰ ἄλλα αἰσθητά· τί γὰρ μᾶλλον τάδε ἢ τάδε;
And in like manner also optics and harmonics; for there will be voice and sight besides the sensible and individual things, so that it is clear that the other senses and other sensible things will also exist; for why should these be so rather than those?