§14.6ἀπορήσειε δʼ ἄν τις καὶ τί τὸ εὖ ἐστὶ τὸ ἀπὸ τῶν ἀριθμῶν τῷ ἐν ἀριθμῷ εἶναι τὴν μῖξιν, ἢ ἐν εὐλογίστῳ ἢ ἐν περιττῷ.
And one might also raise the question what the good is that comes from numbers because the mixture is in a number, either in one that is easy to calculate or in an odd number.
νυνὶ γὰρ οὐθὲν ὑγιεινότερον τρὶς τρία ἂν ᾖ τὸ μελίκρατον κεκραμένον, ἀλλὰ μᾶλλον ὠφελήσειεν ἂν ἐν οὐθενὶ λόγῳ ὂν ὑδαρὲς δὲ ἢ ἐν ἀριθμῷ ἄκρατον ὄν.
For in fact honey-water is no more healthy if it is mixed three times three, but it would be more beneficial if it were diluted in no ratio at all, than if it were unmixed though in a numerical ratio.
ἔτι οἱ λόγοι ἐν προσθέσει ἀριθμῶν εἰσὶν οἱ τῶν μίξεων, οὐκ ἐν ἀριθμοῖς, οἷον τρία πρὸς δύο ἀλλʼ οὐ τρὶς δύο. τὸ γὰρ αὐτὸ δεῖ γένος εἶναι ἐν ταῖς πολλαπλασιώσεσιν, ὥστε δεῖ μετρεῖσθαι τῷ τε Α τὸν στοῖχον ἐφʼ οὗ ΑΒΓ καὶ τῷ Δ τὸν ΔΕΖ·
Further, the ratios of mixtures are in the addition of numbers, not in multiplying numbers; for instance, 'three to two,' but not 'three times two.' For there must be the same genus in multiplications, so that the row on which ABC stands must be measured by A, and DEF by D; so that all must be measured by the same.
ὥστε τῷ αὐτῷ πάντα. οὔκουν ἔσται πυρὸς ΒΕΓΖ καὶ ὕδατος ἀριθμὸς δὶς τρία.
Therefore there cannot be a number 'twice three' of fire, as represented by BEGZ, and of water.
—εἰ δʼ ἀνάγκη πάντα ἀριθμοῦ κοινωνεῖν, ἀνάγκη πολλὰ συμβαίνειν τὰ αὐτά, καὶ ἀριθμὸν τὸν αὐτὸν τῷδε καὶ ἄλλῳ.
¦1 But if all things must share in number, it is necessary that many of the same things should happen, and that the same number should belong to this thing and to another.
ἆρʼ οὖν τοῦτʼ αἴτιον καὶ διὰ τοῦτό ἐστι τὸ πρᾶγμα, ἢ ἄδηλον;
Is this, then, the cause, and is the thing because of this, or is it unclear?
οἷον ἔστι τις τῶν τοῦ ἡλίου φορῶν ἀριθμός, καὶ πάλιν τῶν τῆς σελήνης, καὶ τῶν ζῴων γε ἑκάστου τοῦ βίου καὶ ἡλικίας·
For instance, there is a certain number of the revolutions of the sun, and again of those of the moon, and indeed of the life and age of each of the animals.
τί οὖν κωλύει ἐνίους μὲν τούτων τετραγώνους εἶναι ἐνίους δὲ κύβους, καὶ ἴσους τοὺς δὲ διπλασίους;
What, then, prevents some of these from being squares and others cubes, and some equal and others double?
οὐθὲν γὰρ κωλύει, ἀλλʼ ἀνάγκη ἐν τούτοις στρέφεσθαι, εἰ ἀριθμοῦ πάντα ἐκοινώνει.
For nothing prevents it, but they must rotate within these limits, if all things shared in number.
ἐνεδέχετό τε τὰ διαφέροντα ὑπὸ τὸν αὐτὸν ἀριθμὸν πίπτειν· ὥστʼ εἴ τισιν ὁ αὐτὸς ἀριθμὸς συνεβεβήκει, ταὐτὰ ἂν ἦν ἀλλήλοις ἐκεῖνα τὸ αὐτὸ εἶδος ἀριθμοῦ ἔχοντα, οἷον ἥλιος καὶ σελήνη τὰ αὐτά.
And it was possible for things that differ to fall under the same number; so that if the same number happened to belong to certain things, those things which have the same form of number would be the same as one another, for example, the sun and the moon would be the same.
ἀλλὰ διὰ τί αἴτια ταῦτα;
But why are these things causes?
ἑπτὰ μὲν φωνήεντα, ἑπτὰ δὲ χορδαὶ ἡ ἁρμονία, ἑπτὰ δὲ αἱ πλειάδες, ἐν ἑπτὰ δὲ ὀδόντας βάλλει (ἔνιά γε, ἔνια δʼ οὔ), ἑπτὰ δὲ οἱ ἐπὶ Θήβας.
There are seven vowels, the harmony has seven strings, the Pleiades are seven, some animals shed their teeth at seven years (some do, others do not), and the leaders against Thebes were seven.
ἆρʼ οὖν ὅτι τοιοσδὶ ὁ ἀριθμὸς πέφυκεν, διὰ τοῦτο ἢ ἐκεῖνοι ἐγένοντο ἑπτὰ ἢ ἡ πλειὰς ἑπτὰ ἀστέρων ἐστίν;
Is it, then, because the number is by nature such, that therefore those leaders were seven, or that the Pleiades consist of seven stars?
ἢ οἱ μὲν διὰ τὰς πύλας ἢ ἄλλην τινὰ αἰτίαν, τὴν δὲ ἡμεῖς οὕτως ἀριθμοῦμεν, τὴν δὲ ἄρκτον γε δώδεκα, οἱ δὲ πλείους·
Or was it because of the gates, or some other cause, in the case of the leaders, while we count the constellation in this way, and the Great Bear as twelve, while others count more?
ἐπεὶ καὶ τὸ ΞΨΖ συμφωνίας φασὶν εἶναι, καὶ ὅτι ἐκεῖναι τρεῖς, καὶ ταῦτα τρία· ὅτι δὲ μυρία ἂν εἴη τοιαῦτα, οὐθὲν μέλει (τῷ γὰρ Γ καὶ Ρ εἴη ἂν ἓν σημεῖον)·
Since they also say that X, PS, Z are concordant, and because those concords are three, these also are three; but it does not matter that there might be ten thousand such things (for there might be one sign for G and R).
εἰ δʼ ὅτι διπλάσιον τῶν ἄλλων ἕκαστον, ἄλλο δʼ οὔ, αἴτιον δʼ ὅτι τριῶν ὄντων τόπων ἓν ἐφʼ ἑκάστου ἐπιφέρεται τῷ σίγμα, διὰ τοῦτο τρία μόνον ἐστὶν ἀλλʼ οὐχ ὅτι αἱ συμφωνίαι τρεῖς, ἐπεὶ πλείους γε αἱ συμφωνίαι, ἐνταῦθα δʼ οὐκέτι δύναται.
But if it is because each of these is double of the others, and others are not, and the cause is that, there being three places of articulation, one is added in each case to sigma, it is because of this that there are only three, and not because the concords are three; for indeed the concords are more, but here there can no longer be more.
ὅμοιοι δὴ καὶ οὗτοι τοῖς ἀρχαίοις Ὁμηρικοῖς, οἳ μικρὰς ὁμοιότητας ὁρῶσι μεγάλας δὲ παρορῶσιν.
These people, then, are like the ancient Homeric commentators, who see small similarities but overlook great ones.
λέγουσι δέ τινες ὅτι πολλὰ τοιαῦτα, οἷον αἵ τε μέσαι ἡ μὲν ἐννέα ἡ δὲ ὀκτώ, καὶ τὸ ἔπος δεκαεπτά, ἰσάριθμον τούτοις, βαίνεται δʼ ἐν μὲν τῷ δεξιῷ ἐννέα συλλαβαῖς, ἐν δὲ τῷ ἀριστερῷ ὀκτώ· καὶ ὅτι ἴσον τὸ διάστημα ἔν τε τοῖς γράμμασιν ἀπὸ τοῦ Α πρὸς τὸ Ω, καὶ ἀπὸ τοῦ βόμβυκος ἐπὶ τὴν ὀξυτάτην ἐν αὐλοῖς, ἧς ὁ ἀριθμὸς ἴσος τῇ οὐλομελείᾳ τοῦ οὐρανοῦ.
And some say that there are many such cases, for example, that of the middle strings one is nine and the other eight, and that the epic verse consists of seventeen syllables, which is equal in number to these, and is scanned with nine syllables in the right half and eight in the left; ¦1 and that the interval is equal both in the letters from A to Omega, and from the lowest note of the flute to the highest, the number of which is equal to the whole system of the heaven.
ὁρᾶν δὲ δεῖ μὴ τοιαῦτα οὐθεὶς ἂν ἀπορήσειεν οὔτε λέγειν οὔθʼ εὑρίσκειν ἐν τοῖς ἀϊδίοις, ἐπεὶ καὶ ἐν τοῖς φθαρτοῖς.
But we must observe that no one would find difficulty either in stating or in finding such things in eternal things, since they are found even in perishable things.
ἀλλʼ αἱ ἐν τοῖς ἀριθμοῖς φύσεις αἱ ἐπαινούμεναι καὶ τὰ τούτοις ἐναντία καὶ ὅλως τὰ ἐν τοῖς μαθήμασιν, ὡς μὲν λέγουσί τινες καὶ αἴτια ποιοῦσι τῆς φύσεως, ἔοικεν οὑτωσί γε σκοπουμένοις διαφεύγειν (κατʼ οὐδένα γὰρ τρόπον τῶν διωρισμένων περὶ τὰς ἀρχὰς οὐδὲν αὐτῶν αἴτιον)·
But the praised properties in numbers, and their contraries, and in general the things in mathematics, in the way in which some speak of them and make them causes of nature, seem, to those who study them in this way, to slip away; for in none of the ways defined concerning the principles is any of them a cause.
ἔστιν ὡς μέντοι ποιοῦσι φανερὸν ὅτι τὸ εὖ ὑπάρχει καὶ τῆς συστοιχίας ἐστὶ τῆς τοῦ καλοῦ τὸ περιττόν, τὸ εὐθύ, τὸ ἰσάκις ἴσον, αἱ δυνάμεις ἐνίων ἀριθμῶν·
Yet, in the way they treat them, they make it clear that the good is present, and that the odd, the straight, the equal-times-equal, and the powers of certain numbers, belong to the column of the beautiful.
ἅμα γὰρ ὧραι καὶ ἀριθμὸς τοιοσδί· καὶ τὰ ἄλλα δὴ ὅσα συνάγουσιν ἐκ τῶν μαθηματικῶν θεωρημάτων πάντα ταύτην ἔχει τὴν δύναμιν.
For the seasons and a certain number of this kind are simultaneous; and indeed all the other things which they collect from mathematical theorems have this same value.
διὸ καὶ ἔοικε συμπτώμασιν· ἔστι γὰρ συμβεβηκότα μέν, ἀλλʼ οἰκεῖα ἀλλήλοις πάντα, ἓν δὲ τῷ ἀνάλογον·
Therefore they are like coincidences; for they are accidents, but they are all appropriate to one another, and are one by analogy.
ἐν ἑκάστῃ γὰρ τοῦ ὄντος κατηγορίᾳ ἐστὶ τὸ ἀνάλογον, ὡς εὐθὺ ἐν μήκει οὕτως ἐν πλάτει τὸ ὁμαλόν, ἴσως ἐν ἀριθμῷ τὸ περιττόν, ἐν δὲ χροιᾷ τὸ λευκόν.
For in each category of being there is the analogue, as the straight in length, so is the level in breadth, perhaps the odd in number, and the white in color.
—ἔτι οὐχ οἱ ἐν τοῖς εἴδεσιν ἀριθμοὶ αἴτιοι τῶν ἁρμονικῶν καὶ τῶν τοιούτων (διαφέρουσι γὰρ ἐκεῖνοι ἀλλήλων οἱ ἴσοι εἴδει· καὶ γὰρ αἱ μονάδες)· ὥστε διά γε ταῦτα εἴδη οὐ ποιητέον.
Further, the numbers which are in the forms are not causes of harmonic relations and the like (for those numbers, even when equal in form, differ from one another; for their units also differ); so that for these reasons at least, one must not make forms.
τὰ μὲν οὖν συμβαίνοντα ταῦτά τε κἂν ἔτι πλείω συναχθείη·
These, then, are the consequences, and even more might be brought together.
ἔοικε δὲ τεκμήριον εἶναι τὸ πολλὰ κακοπαθεῖν περὶ τὴν γένεσιν αὐτῶν καὶ μηδένα τρόπον δύνασθαι συνεῖραι τοῦ μὴ χωριστὰ εἶναι τὰ μαθηματικὰ τῶν αἰσθητῶν, ὡς ἔνιοι λέγουσι, μηδὲ ταύτας εἶναι τὰς ἀρχάς.
But the fact that they have much trouble about the generation of numbers, and can in no way construct a consistent theory, seems to be a proof that mathematical things are not separate from sensible things, as some say, nor are they the principles.