§13.9#1ἀπορήσειε δʼ ἄν τις καὶ ἐπεὶ ἁφὴ μὲν οὐκ ἔστιν ἐν τοῖς ἀριθμοῖς, τὸ δʼ ἐφεξῆς, ὅσων μὴ ἔστι μεταξὺ μονάδων (οἷον τῶν ἐν τῇ δυάδι ἢ τῇ τριάδι), πότερον ἐφεξῆς τῷ ἑνὶ αὐτῷ ἢ οὔ, καὶ πότερον ἡ δυὰς προτέρα τῶν ἐφεξῆς ἢ τῶν μονάδων ὁποτεραοῦν. —ὁμοίως δὲ καὶ περὶ τῶν ὕστερον γενῶν τοῦ ἀριθμοῦ συμβαίνει τὰ δυσχερῆ, γραμμῆς τε καὶ ἐπιπέδου καὶ σώματος.
And one might also raise a difficulty, since, although there is no contact in numbers, there is succession among those units between which there is nothing (such as those in the Dyad or the Triad), whether there is anything successive to the One itself or not, and whether the Dyad is prior to the successive things or to either of the units. —Similarly, difficulties also occur concerning the genera posterior to number, namely, the line, the plane, and the body.
οἱ μὲν γὰρ ἐκ τῶν εἰδῶν τοῦ μεγάλου καὶ τοῦ μικροῦ ποιοῦσιν, οἷον ἐκ μακροῦ μὲν καὶ βραχέος τὰ μήκη, πλατέος δὲ καὶ στενοῦ τὰ ἐπίπεδα, ἐκ βαθέος δὲ καὶ ταπεινοῦ τοὺς ὄγκους· ταῦτα δέ ἐστιν εἴδη τοῦ μεγάλου καὶ μικροῦ.
For some generate them from the species of the Great and the Small: for example, lengths from the long and short, planes from the wide and narrow, and volumes from the deep and shallow; and these are species of the Great and Small.
τὴν δὲ κατὰ τὸ ἓν ἀρχὴν ἄλλοι ἄλλως τιθέασι τῶν τοιούτων.
And other thinkers set up the principle corresponding to the One in different ways for these things.
καὶ ἐν τούτοις δὲ μυρία φαίνεται τά τε ἀδύνατα καὶ τὰ πλασματώδη καὶ τὰ ὑπεναντία πᾶσι τοῖς εὐλόγοις.
And in these also there appear countless impossibilities, fictitious assertions, and things contrary to all reasonable views.
ἀπολελυμένα τε γὰρ ἀλλήλων συμβαίνει, εἰ μὴ συνακολουθοῦσι καὶ αἱ ἀρχαὶ ὥστʼ εἶναι τὸ πλατὺ καὶ στενὸν καὶ μακρὸν καὶ βραχύ (εἰ δὲ τοῦτο, ἔσται τὸ ἐπίπεδον γραμμὴ καὶ τὸ στερεὸν ἐπίπεδον· ἔτι δὲ γωνίαι καὶ σχήματα καὶ τὰ τοιαῦτα πῶς ἀποδοθήσεται; ), ταὐτό τε συμβαίνει τοῖς περὶ τὸν ἀριθμόν· ταῦτα γὰρ πάθη μεγέθους ἐστίν, ἀλλʼ οὐκ ἐκ τούτων τὸ μέγεθος, ὥσπερ οὐδʼ ἐξ εὐθέος καὶ καμπύλου τὸ μῆκος οὐδʼ ἐκ λείου καὶ τραχέος τὰ στερεά. —πάντων δὲ κοινὸν τούτων ὅπερ ἐπὶ τῶν εἰδῶν τῶν ὡς γένους συμβαίνει διαπορεῖν, ὅταν τις θῇ τὰ καθόλου, πότερον τὸ ζῷον αὐτὸ ἐν τῷ ζῴῳ ἢ ἕτερον αὐτοῦ ζῴου.
For they turn out to be separated from one another, unless their principles also imply one another, so that the wide and narrow is also long and short (but if this is so, the plane will be a line and the solid a plane; and further, how will angles and figures and such things be accounted for?); and the same thing happens as to those who speak about number; for these are attributes of magnitude, but magnitude does not consist of them, just as length does not consist of straight and curved, nor solids of smooth and rough. —Common to all these is what happens to be a puzzle about species considered as genus when one posits universals, namely, whether 'Animal itself' is in the animal or is other than 'animal'.
τοῦτο γὰρ μὴ χωριστοῦ μὲν ὄντος οὐδεμίαν ποιήσει ἀπορίαν· χωριστοῦ δέ, ὥσπερ οἱ ταῦτα λέγοντές φασι, τοῦ ἑνὸς καὶ τῶν ἀριθμῶν οὐ ῥᾴδιον λῦσαι, εἰ μὴ ῥᾴδιον δεῖ λέγειν τὸ ἀδύνατον.
For if it is not separate, this will cause no difficulty; but if it is separate, as those who say these things assert, of the One and of numbers, it is not easy to solve—unless indeed one should call the impossible 'not easy'.
ὅταν γὰρ νοῇ τις ἐν τῇ δυάδι τὸ ἓν καὶ ὅλως ἐν ἀριθμῷ, πότερον αὐτὸ νοεῖ τι ἢ ἕτερον; —οἱ μὲν οὖν τὰ μεγέθη γεννῶσιν ἐκ τοιαύτης ὕλης, ἕτεροι δὲ ἐκ τῆς στιγμῆς (ἡ δὲ στιγμὴ αὐτοῖς δοκεῖ εἶναι οὐχ ἓν ἀλλʼ οἷον τὸ ἕν) καὶ ἄλλης ὕλης οἵας τὸ πλῆθος, ἀλλʼ οὐ πλήθους· περὶ ὧν οὐδὲν ἧττον συμβαίνει τὰ αὐτὰ ἀπορεῖν.
For when one conceives the One in the Dyad, and in number generally, does one conceive the One itself or something else? —So some generate magnitudes from such matter, but others from the point (and the point seems to them to be not the One but like the One) and from other matter such as plurality, but not plurality; concerning which, none the less, the same difficulties occur.
εἰ μὲν γὰρ μία ἡ ὕλη, ταὐτὸ γραμμὴ καὶ ἐπίπεδον καὶ στερεόν (ἐκ γὰρ τῶν αὐτῶν τὸ αὐτὸ καὶ ἓν ἔσται)· εἰ δὲ πλείους αἱ ὗλαι καὶ ἑτέρα μὲν γραμμῆς ἑτέρα δὲ τοῦ ἐπιπέδου καὶ ἄλλη τοῦ στερεοῦ, ἤτοι ἀκολουθοῦσιν ἀλλήλαις ἢ οὔ, ὥστε ταὐτὰ συμβήσεται καὶ οὕτως· ἢ γὰρ οὐχ ἕξει τὸ ἐπίπεδον γραμμὴν ἢ ἔσται γραμμή. —ἔτι πῶς μὲν ἐνδέχεται εἶναι ἐκ τοῦ ἑνὸς καὶ πλήθους τὸν ἀριθμὸν οὐθὲν ἐπιχειρεῖται· ὅπως δʼ οὖν λέγουσι ταὐτὰ συμβαίνει δυσχερῆ ἅπερ καὶ τοῖς ἐκ τοῦ ἑνὸς καὶ ἐκ τῆς δυάδος τῆς ἀορίστου.
For if the matter is one, line, plane, and solid will be the same (for from the same things the same and single thing will result); but if the matters are more than one, and one is for the line, another for the plane, and another for the solid, either they imply one another or they do not, so that the same results will follow even in this way; for either the plane will not contain a line, or it will be a line. —Further, how it is possible for number to be from the One and plurality is not attempted; but in whatever way they speak, there occur the same difficulties as for those who generate number from the One and the indefinite Dyad.
ὁ μὲν γὰρ ἐκ τοῦ κατηγορουμένου καθόλου γεννᾷ τὸν ἀριθμὸν καὶ οὐ τινὸς πλήθους, ὁ δʼ ἐκ τινὸς πλήθους, τοῦ πρώτου δέ (τὴν γὰρ δυάδα πρῶτόν τι εἶναι πλῆθος), ὥστε διαφέρει οὐθὲν ὡς εἰπεῖν, ἀλλʼ αἱ ἀπορίαι αἱ αὐταὶ ἀκολουθήσουσι, μῖξις ἢ θέσις ἢ κρᾶσις ἢ γένεσις καὶ ὅσα ἄλλα τοιαῦτα.
For one generates number from the predicated universal and not from a particular plurality, while another from a particular plurality, but the first one (for the Dyad is the first plurality); so that there is, so to speak, no difference, but the same difficulties will follow: mixture, or position, or blending, or generation, and all other such things.
μάλιστα δʼ ἄν τις ἐπιζητήσειεν, εἰ μία ἑκάστη μονάς, ἐκ τίνος ἐστίν·
And most of all, one might inquire: if each unit is one, from what does it consist?
οὐ γὰρ δὴ αὐτό γε τὸ ἓν ἑκάστη.
For surely each is not the One itself.
ἀνάγκη δὴ ἐκ τοῦ ἑνὸς αὐτοῦ εἶναι καὶ πλήθους ἢ μορίου τοῦ πλήθους.
It must then consist of the One itself and plurality, or of a part of plurality.
τὸ μὲν οὖν πλῆθός τι εἶναι φάναι τὴν μονάδα ἀδύνατον, ἀδιαίρετόν γʼ οὖσαν· τὸ δʼ ἐκ μορίου ἄλλας ἔχει πολλὰς δυσχερείας· ἀδιαίρετόν τε γὰρ ἕκαστον ἀναγκαῖον εἶναι τῶν μορίων (ἢ πλῆθος εἶναι καὶ τὴν μονάδα διαιρετήν) καὶ μὴ στοιχεῖον εἶναι τὸ ἓν καὶ τὸ πλῆθος (ἡ γὰρ μονὰς ἑκάστη οὐκ ἐκ πλήθους καὶ ἑνός)· ἔτι οὐθὲν ἄλλο ποιεῖ ὁ τοῦτο λέγων ἀλλʼ ἢ ἀριθμὸν ἕτερον· τὸ γὰρ πλῆθος ἀδιαιρέτων ἐστὶν ἀριθμός.
Now to say that the unit is some plurality is impossible, since it is at least indivisible; but to say that it consists of a part of plurality involves many other difficulties; for it is necessary either that each of the parts be indivisible (or else the unit will be a plurality and divisible), and that the One and plurality be not elements, (since each unit does not consist of plurality and the One); further, he who says this does nothing else than produce another number; for a plurality of indivisibles is a number.
ἔτι ζητητέον καὶ περὶ τοὺς οὕτω λέγοντας πότερον ἄπειρος ὁ ἀριθμὸς ἢ πεπερασμένος.
Further, we must also investigate, regarding those who speak in this way, whether number is infinite or finite.