§149οὐκοῦν δύο μὲν ὂν τὸ ἓν ποιήσειεν ἂν ταῦτα καὶ ἐν δυοῖν χώραιν ἅμα γένοιτο·
Therefore, if the one were two, it would do this and be in two places at the same time; but as long as it is one, will it not refuse to do so?
ἕως δʼ ἂν ᾖ ἕν, οὐκ ἐθελήσει; οὐ γὰρ οὖν.
It certainly will not.
ἡ αὐτὴ ἄρα ἀνάγκη τῷ ἑνὶ μήτε δύο εἶναι μήτε ἅπτεσθαι αὐτῷ αὑτοῦ.
The same necessity, therefore, applies to the one, namely, neither to be two nor to touch itself.
ἡ αὐτή.
The same.
ἀλλʼ οὐδὲ μὴν τῶν ἄλλων ἅψεται.
But indeed, neither will it touch the others.
τί δή;
Why so?
ὅτι, φαμέν, τὸ μέλλον ἅψεσθαι χωρὶς ὂν ἐφεξῆς δεῖ ἐκείνῳ εἶναι οὗ μέλλει ἅψεσθαι, τρίτον δὲ αὐτῶν ἐν μέσῳ μηδὲν εἶναι.
Because, we say, that which is going to touch, being separate, must lie next to that which it is going to touch, and nothing third must be between them.
ἀληθῆ.
True.
δύο ἄρα δεῖ τὸ ὀλίγιστον εἶναι, εἰ μέλλει ἅψις εἶναι.
Therefore, there must be at least two, if there is to be touching.
δεῖ.
There must.
ἐὰν δὲ τοῖν δυοῖν ὅροιν τρίτον προσγένηται ἑξῆς, αὐτὰ μὲν τρία ἔσται, αἱ δὲ ἅψεις δύο.
And if to the two limits (terms) a third is added next, they themselves will be three, and the touchings two.
ναί.
Yes.
καὶ οὕτω δὴ ἀεὶ ἑνὸς προσγιγνομένου μία καὶ ἅψις προσγίγνεται, καὶ συμβαίνει τὰς ἅψεις τοῦ πλήθους τῶν ἀριθμῶν μιᾷ ἐλάττους εἶναι.
And so always, as one is added, one touching is also added, and it results that the touchings are fewer by one than the multitude of the numbers.
ᾧ γὰρ τὰ πρῶτα δύο ἐπλεονέκτησεν τῶν ἅψεων εἰς τὸ πλείω εἶναι τὸν ἀριθμὸν ἢ τὰς ἅψεις, τῷ ἴσῳ τούτῳ καὶ ὁ ἔπειτα ἀριθμὸς πᾶς πασῶν τῶν ἅψεων πλεονεκτεῖ· ἤδη γὰρ τὸ λοιπὸν ἅμα ἕν τε τῷ ἀριθμῷ προσγίγνεται καὶ μία ἅψις ταῖς ἅψεσιν.
For by the same amount that the first two exceeded the touchings, so that the number was greater than the touchings, by this same amount every subsequent number also exceeds all the touchings; for henceforth, at the same time, one is added to the number and one touching to the touchings.
ὀρθῶς.
Correctly.
ὅσα ἄρα ἐστὶν τὰ ὄντα τὸν ἀριθμόν, ἀεὶ μιᾷ αἱ ἅψεις ἐλάττους εἰσὶν αὐτῶν.
Therefore, however many the existing things are in number, the touchings are always fewer than they by one.
ἀληθῆ.
True.
εἰ δέ γε ἓν μόνον ἐστίν, δυὰς δὲ μὴ ἔστιν, ἅψις οὐκ ἂν εἴη.
But if there is only one, and there is no dyad, there would be no touching.
πῶς γάρ;
For how could there be?
οὔκουν, φαμέν, τὰ ἄλλα τοῦ ἑνὸς οὔτε ἕν ἐστιν οὔτε μετέχει αὐτοῦ, εἴπερ ἄλλα ἐστίν.
And we say, do we not, that the others than the one are neither one nor participate in it, if indeed they are others?
οὐ γάρ.
Indeed.
οὐκ ἄρα ἔνεστιν ἀριθμὸς ἐν τοῖς ἄλλοις, ἑνὸς μὴ ἐνόντος ἐν αὐτοῖς.
Therefore, there is no number in the others, since one is not in them.
πῶς γάρ;
For how could there be?
οὔτʼ ἄρα ἕν ἐστι τὰ ἄλλα οὔτε δύο οὔτε ἄλλου ἀριθμοῦ ἔχοντα ὄνομα οὐδέν.
Therefore, the others are neither one nor two, nor have any name of any other number.
οὔ.
No.
τὸ ἓν ἄρα μόνον ἐστὶν ἕν, καὶ δυὰς οὐκ ἂν εἴη.
Therefore, the one alone is one, and a dyad could not be.
οὐ φαίνεται.
It does not appear so.
ἅψις ἄρα οὐκ ἔστιν δυοῖν μὴ ὄντοιν.
Therefore, there is no touching, since two do not exist.
οὐκ ἔστιν.
There is not.
οὔτʼ ἄρα τὸ ἓν τῶν ἄλλων ἅπτεται οὔτε τὰ ἄλλα τοῦ ἑνός, ἐπείπερ ἅψις οὐκ ἔστιν.
Therefore, neither does the one touch the others, nor do the others touch the one, since there is no touching.
οὐ γὰρ οὖν.
Indeed not.
οὕτω δὴ κατὰ πάντα ταῦτα τὸ ἓν τῶν τε ἄλλων καὶ ἑαυτοῦ ἅπτεταί τε καὶ οὐχ ἅπτεται.
In this way, then, according to all these things, the one both touches and does not touch both the others and itself.
ἔοικεν.
It seems so.
ἆρʼ οὖν καὶ ἴσον ἐστὶ καὶ ἄνισον αὑτῷ τε καὶ τοῖς ἄλλοις;
Is it then also equal and unequal to itself and to the others?
πῶς;
How?
εἰ μεῖζον εἴη τὸ ἓν ἢ τἆλλα ἢ ἔλαττον, ἢ αὖ τὰ ἄλλα τοῦ ἑνὸς μείζω ἢ ἐλάττω, ἆρα οὐκ ἂν τῷ μὲν ἓν εἶναι τὸ ἓν καὶ τἆλλα ἄλλα τοῦ ἑνὸς οὔτε τι μείζω οὔτε τι ἐλάττω ἂν εἴη ἀλλήλων αὐταῖς γε ταύταις ταῖς οὐσίαις;
If the one were greater or less than the others, or again the others greater or less than the one, would they not, by the very fact of the one being one and the others being other than the one, in respect of these very essences, be neither greater nor less than one another?
ἀλλʼ εἰ μὲν πρὸς τῷ τοιαῦτα εἶναι ἑκάτερα ἰσότητα ἔχοιεν, ἴσα ἂν εἴη πρὸς ἄλληλα· εἰ δὲ τὰ μὲν μέγεθος, τὸ δὲ σμικρότητα, ἢ καὶ μέγεθος μὲν τὸ ἕν, σμικρότητα δὲ τἆλλα, ὁποτέρῳ μὲν τῷ εἴδει μέγεθος προσείη, μεῖζον ἂν εἴη, ᾧ δὲ σμικρότης, ἔλαττον;
But if, in addition to being such as they are, each possessed equality, they would be equal to one another; whereas if one possessed greatness and the other smallness, or if the one possessed greatness and the others smallness, would not that to which the form of greatness was added be greater, and that to which smallness was added be less?
ἀνάγκη.
Necessary.