§13.7#2φανερὸν δὲ καὶ ὅτι οὐκ ἐνδέχεται, εἰ ἀσύμβλητοι πᾶσαι αἱ μονάδες, δυάδα εἶναι αὐτὴν καὶ τριάδα καὶ οὕτω τοὺς ἄλλους ἀριθμούς.
It is also clear that, if all the units are inassociable, it is impossible for there to be the Dyad-itself, the Triad-itself, and so on with the other numbers.
ἄν τε γὰρ ὦσιν ἀδιάφοροι αἱ μονάδες ἄν τε διαφέρουσαι ἑκάστη ἑκάστης, ἀνάγκη ἀριθμεῖσθαι τὸν ἀριθμὸν κατὰ πρόσθεσιν, οἷον τὴν δυάδα πρὸς τῷ ἑνὶ ἄλλου ἑνὸς προστεθέντος, καὶ τὴν τριάδα ἄλλου ἑνὸς πρὸς τοῖς δυσὶ προστεθέντος, καὶ τὴν τετράδα ὡσαύτως· τούτων δὲ ὄντων ἀδύνατον τὴν γένεσιν εἶναι τῶν ἀριθμῶν ὡς γεννῶσιν ἐκ τῆς δυάδος καὶ τοῦ ἑνός.
For whether the units are non-different or each different from every other, number must be numbered by addition, e.g. the dyad by another 'one' being added to the 'one', and the triad by another 'one' being added to the two, and the tetrad likewise; and this being so, it is impossible for the generation of numbers to be as they generate them from the dyad and the One.
μόριον γὰρ γίγνεται ἡ δυὰς τῆς τριάδος καὶ αὕτη τῆς τετράδος, τὸν αὐτὸν δὲ τρόπον συμβαίνει καὶ ἐπὶ τῶν ἐχομένων.
For the dyad becomes a part of the triad, and this of the tetrad, and the same thing happens in the case of the succeeding numbers.
ἀλλʼ ἐκ τῆς δυάδος τῆς πρώτης καὶ τῆς ἀορίστου δυάδος ἐγίγνετο ἡ τετράς, δύο δυάδες παρʼ αὐτὴν τὴν δυάδα· εἰ δὲ μή, μόριον ἔσται αὐτὴ ἡ δυάς, ἑτέρα δὲ προσέσται μία δυάς.
But the tetrad was generated from the first dyad and the indefinite dyad—two dyads apart from the Dyad-itself; otherwise, the Dyad-itself will be a part, and another single dyad will be added.
καὶ ἡ δυὰς ἔσται ἐκ τοῦ ἑνὸς αὐτοῦ καὶ ἄλλου ἑνός· εἰ δὲ τοῦτο, οὐχ οἷόν τʼ εἶναι τὸ ἕτερον στοιχεῖον δυάδα ἀόριστον· μονάδα γὰρ μίαν γεννᾷ ἀλλʼ οὐ δυάδα ὡρισμένην.
And the dyad will be from the One-itself and another 'one'; but if this is so, it is not possible for the other element to be the indefinite dyad; for it generates a single unit, but not a definite dyad.
ἔτι παρʼ αὐτὴν τὴν τριάδα καὶ αὐτὴν τὴν δυάδα πῶς ἔσονται ἄλλαι τριάδες καὶ δυάδες;
Again, apart from the Triad-itself and the Dyad-itself, how will there be other triads and dyads?
καὶ τίνα τρόπον ἐκ προτέρων μονάδων καὶ ὑστέρων σύγκεινται;
And in what way are they composed of prior and posterior units?
πάντα γὰρ ταῦτʼ ἄτοπά ἐστι καὶ πλασματώδη, καὶ ἀδύνατον εἶναι πρώτην δυάδα, εἶτʼ αὐτὴν τριάδα.
For all these things are absurd and fictional, and it is impossible for there to be a first dyad, and then the Triad-itself.
ἀνάγκη δʼ, ἐπείπερ ἔσται τὸ ἓν καὶ ἡ ἀόριστος δυὰς στοιχεῖα.
Yet this is necessary, if the One and the indefinite dyad are to be elements.
εἰ δʼ ἀδύνατα τὰ συμβαίνοντα, καὶ τὰς ἀρχὰς εἶναι ταύτας ἀδύνατον. —εἰ μὲν οὖν διάφοροι αἱ μονάδες ὁποιαιοῦν ὁποιαισοῦν, ταῦτα καὶ τοιαῦθʼ ἕτερα συμβαίνει ἐξ ἀνάγκης· εἰ δʼ αἱ μὲν ἐν ἄλλῳ διάφοροι αἱ δʼ ἐν τῷ αὐτῷ ἀριθμῷ ἀδιάφοροι ἀλλήλαις μόναι, καὶ οὕτως οὐθὲν ἐλάττω συμβαίνει τὰ δυσχερῆ.
And if the consequences are impossible, it is also impossible for these to be the principles.—If, then, the units are different, any one from any other, these and other such consequences follow of necessity; but if those in a different number are different, while those in the same number are alone non-different from one another, even so the difficulties that follow are no less.
οἷον γὰρ ἐν τῇ δεκάδι αὐτῇ ἔνεισι δέκα μονάδες, σύγκειται δὲ καὶ ἐκ τούτων καὶ ἐκ δύο πεντάδων ἡ δεκάς.
For example, in the Decad-itself there are ten units, and the decad is composed both of these and of two pentads.
ἐπεὶ δʼ οὐχ ὁ τυχὼν ἀριθμὸς αὐτὴ ἡ δεκὰς οὐδὲ σύγκειται ἐκ τῶν τυχουσῶν πεντάδων, ὥσπερ οὐδὲ μονάδων, ἀνάγκη διαφέρειν τὰς μονάδας τὰς ἐν τῇ δεκάδι ταύτῃ.
And since the Decad-itself is not just any chance number, nor is it composed of any chance pentads, just as it is not of any chance units, it is necessary that the units in this decad differ.
ἂν γὰρ μὴ διαφέρωσιν, οὐδʼ αἱ πεντάδες διοίσουσιν ἐξ ὧν ἐστὶν ἡ δεκάς· ἐπεὶ δὲ διαφέρουσι, καὶ αἱ μονάδες διοίσουσιν.
For if they do not differ, neither will the pentads differ of which the decad consists; but since they do differ, the units will also differ.
εἰ δὲ διαφέρουσι, πότερον οὐκ ἐνέσονται πεντάδες ἄλλαι ἀλλὰ μόνον αὗται αἱ δύο, ἢ ἔσονται;
And if they differ, will there be no other pentads contained in it but only these two, or will there be?
εἴτε δὲ μὴ ἐνέσονται, ἄτοπον· εἴτʼ ἐνέσονται, ποία ἔσται δεκὰς ἐξ ἐκείνων;
And if they are not contained, it is absurd; but if they are, what sort of decad will be formed from them?
οὐ γὰρ ἔστιν ἑτέρα δεκὰς ἐν τῇ δεκάδι παρʼ αὐτήν.
For there is no other decad in the Decad-itself apart from itself.
ἀλλὰ μὴν καὶ ἀνάγκη γε μὴ ἐκ τῶν τυχουσῶν δυάδων τὴν τετράδα συγκεῖσθαι· ἡ γὰρ ἀόριστος δυάς, ὥς φασι, λαβοῦσα τὴν ὡρισμένην δυάδα δύο δυάδας ἐποίησεν· τοῦ γὰρ ληφθέντος ἦν δυοποιός. —ἔτι τὸ εἶναι παρὰ τὰς δύο μονάδας τὴν δυάδα φύσιν τινά, καὶ τὴν τριάδα παρὰ τὰς τρεῖς μονάδας, πῶς ἐνδέχεται;
Moreover, it is also necessary that the tetrad is not composed of any chance dyads; for the indefinite dyad, as they say, taking the definite dyad, made two dyads; for it was the doubler of what was taken.—Again, how is it possible for the dyad to be some nature apart from its two units, and the triad apart from its three units?
ἢ γὰρ μεθέξει θατέρου θατέρου, ὥσπερ λευκὸς ἄνθρωπος παρὰ λευκὸν καὶ ἄνθρωπον (μετέχει γὰρ τούτων), ἢ ὅταν ᾖ θατέρου θάτερον διαφορά τις, ὥσπερ ὁ ἄνθρωπος παρὰ ζῷον καὶ δίπουν.
For either one will participate in the other, just as 'white man' is apart from 'white' and 'man' (for it participates in these), or when one is a certain difference of the other, just as 'man' is apart from 'animal' and 'two-footed'.
ἔτι τὰ μὲν ἁφῇ ἐστὶν ἓν τὰ δὲ μίξει τὰ δὲ θέσει· ὧν οὐδὲν ἐνδέχεται ὑπάρχειν ταῖς μονάσιν ἐξ ὧν ἡ δυὰς καὶ ἡ τριάς· ἀλλʼ ὥσπερ οἱ δύο ἄνθρωποι οὐχ ἕν τι παρʼ ἀμφοτέρους, οὕτως ἀνάγκη καὶ τὰς μονάδας.
Again, some things are one by contact, others by mixture, others by position; none of which can belong to the units of which the dyad and triad consist; but just as two men are not some one thing apart from both, so it must also be with the units.