§3.7Κατὰ λόγον δὲ συμβαίνει καὶ τὸ κατὰ πρόσθεσιν μὲν μὴ εἶναι δοκεῖν ἄπειρον οὕτως ὥστε παντὸς ὑπερβάλλειν μεγέθους, ἐπὶ τὴν διαίρεσιν δὲ εἶναι (περιέχεται γὰρ ἡ ὕλη ἐντὸς καὶ τὸ ἄπειρον, περιέχει δὲ τὸ εἶδος)· εὐλόγως δὲ καὶ τὸ ἐν μὲν τῷ ἀριθμῷ εἶναι ἐπὶ μὲν τὸ ἐλάχ ιστον πέρας ἐπὶ δὲ τὸ πλεῖον ἀεὶ παντὸς ὑπερβάλλειν πλήθους, ἐπὶ δὲ τῶν μεγεθῶν τοὐναντίον ἐπὶ μὲν τὸ ἔλαττον παντὸς ὑπερβάλλειν μεγέθους ἐπὶ δὲ τὸ μεῖζον μὴ εἶναι μέγεθος ἄπειρον.
And it turns out reasonably that, on the one hand, in the direction of addition there does not seem to be an infinite such as to exceed every magnitude, while on the other hand, in the direction of division there is; (for the matter is contained inside, as is the infinite, while the form contains); and it is reasonable also that in number there is a limit in the direction of the minimum, but in the direction of the greater it always exceeds every multiplicity, while in magnitudes, on the contrary, in the direction of the less it exceeds every magnitude, but in the direction of the greater there is no infinite magnitude.
αἴτιον δʼ ὅτι τὸ ἔν ἐστιν ἀδιαίρετον, ὅ τι περ ἂν ἓν ἦ (οἷον ἄνθρωπος εἶς ἄνθρωπος καὶ οὐ πολλοί), ὁ δʼ ἀριθμός ἐστιν ἕνα πλείω καὶ πόσʼ ἄττα, ὥστʼ ἀνάγκη στῆναι ἐπὶ τὸ ἀδιαίρετον (τὸ γὰρ τρία καὶ δύο παρώνυμα ὀνόματά ἐστιν, ὁμοίως δὲ καὶ τῶν ἄλλων ἀριθμῶν ἕκαστος), ἐπὶ δὲ τὸ πλεῖον ἀεὶ ἔστι νοῆσαι· ἄπειροο γὰρ αἱ διχοτομίαι τοῦ μεγέθους.
The cause is that the one is indivisible, whatever may be one (for example, one man is one man and not many), whereas number is more than one and a certain quantity of them, so that it is necessary to stop at the indivisible (for 'three' and 'two' are derivative names, and likewise also each of the other numbers), but in the direction of the greater it is always possible to think [of more]; for the bisections of a magnitude are infinite.
ὥστε δυνάμει μὲν ἔστιν, ἐνεργείᾳ δʼ οὔ ἀλλʼ ἀεὶ ὑπερβάλλει τὸ λαμβανόμενον παντὸς ὡρισμένου πλήθους.
So that it exists potentially, but not in actuality; rather, what is taken always exceeds every limited multiplicity.
ἀλλʼ οὐ χωριστὸς ὁ ἀριθμὸς οὗτος, οὐδὲ μένει ἡ ἀπειρία ἀλλὰ γίγνεται, ὥσπερ καὶ ὁ χρόνος καὶ ὁ ἀριθμὸς τοῦ χρόνου.
But this number is not separate, nor does the infinity remain, but it comes to be, just as also time and the number of time [do].
ἐπὶ δὲ τῶν μεγεθῶν τοὐναντίον ἐστί· διαιρεῖται μὲν γὰρ εἰς ἄπειρα τὸ συνεχές, ἐπὶ δὲ τὸ μεῖζον οὐκ ἔστιν ἄπειρον.
But in the case of magnitudes, it is the contrary; for the continuous is divided into infinites, but in the direction of the greater there is no infinite.
ὅσον γὰρ ἐνδέχεται δυνάμει εἶναι, καὶ ἐνεργείᾳ ἐνδέχεται τοσοῦτον εἶναι.
For as much as it is possible to be potentially, so much is it also possible to be in actuality.
ὥστε ἐπεὶ ἄπειρον οὐδέν ἐστι μέγεθος αἰσθητόν, οὐκ ἐνδέχεται παντὸς ὑπερβολὴν εἶναι ὡρισμένου μεγέθους· εἴη γὰρ ὄν τι τοῦ οὐρανοῦ μεῖζον.
So that since no sensible magnitude is infinite, it is not possible for there to be an exceeding of every limited magnitude; for [otherwise] there would be something greater than the heaven.
τὸ δʼ ἄπειρον οὐ ταὐτὸν ἐν μεγέθει καὶ κινήσει καὶ χρόνῳ, ὡς μία τις φύσις, ἀλλὰ τὸ ὕστερον λέγεται κατὰ τὸ πρότερον, οἷον κίνησις μὲν ὅτι τὸ μέγεθος ἐφʼ οὗ κινεῖται ἢ ἀλλοιοῦται ἢ αὐξάνεται, ὁ χρόνος δὲ διὰ τὴν κίνησιν.
But the infinite is not the same in magnitude, motion, and time, as if it were some single nature, but the posterior is so called according to the prior; for example, motion [is called infinite] because the magnitude over which it moves or alters or increases [is infinite], and time is so called because of the motion.
νῦν μὲν οὖν χρώμεθα τούτοις, ὕστερον δὲ ἐροῦμεν καὶ τί ἐστιν ἕκαστον, καὶ διότι πᾶν μέγεθος εἰς μεγέθη διαιρετόν.
For now indeed we use these things, but later we shall say also what each of them is, and why every magnitude is divisible into magnitudes.
οὐκ ἀφαιρεῖται δʼ ὁ λόγος οὐδὲ τοὺς
μαθηματικοὺς τὴν θεωρίαν, ἀναιρῶν οὕτως εἶναι ἄπειρον ὥστε ἐνεργείᾳ εἶναι ἐπὶ τὴν αὔξησιν ἀδιεξίτητον· οὐδὲ γὰρ νῦν δέονται τοῦ ἀπείρου(οὐ γὰρ χρῶνται), ἀλλὰ μόνον εἶναι ὅσην ἂν βούλωνται πεπερασμένην· τῷ δὲ μεγίστῳ μεγέθει τὸν αὐτὸν ἔστι τετμῆσθαι λόγον ὁπηλικονοῦν μέγεθος ἕτερον.
And our argument does not deprive even the mathematicians of their contemplation, by denying that the infinite exists in such a way as to be in actuality untraversable in the direction of increase; for even now they do not need the infinite (for they do not use it), but only that there be a finite magnitude as large as they wish; and it is possible for another magnitude of any size whatever to be divided in the same ratio as the greatest magnitude.
ὥστε πρὸς μὲν τὸ δεῖξαι ἐκείνοις οὐδὲν διοίσει τὸ εἶναι ἐν τοῖς οὖσιν μεγέθεσιν.
So that for their demonstrations it will make no difference that the infinite exists among existing magnitudes.
ἐπεὶ δὲ τὰ αἴτια διῄρηται τετραχῶς,
φανερὸν ὅτι ὡς ὕλη τὸ ἄπειρον αἴτιόν ἐστι, καὶ ὅτι τὸ μὲν εἶναι αὐτῷ στέρησις, τὸ δὲ καθʼ αὐτὸ ὑποκείμενον τὸ συνεχὲς καὶ αἰσθητόν.
And since causes are distinguished in four ways, it is clear that the infinite is a cause as matter, and that its being is privation, while the subject in itself is the continuous and sensible.
φαίνονται δὲ πάντες καὶ οἱ ἄλλοι ὡς ὕλῃ χρώμενοι τῷ ἀπείρῳ· διὸ καὶ ἄτοπον τὸ περι· έχον ποιεῖν αὐτὸ ἀλλὰ μὴ περιεχόμενον.
And all the others also appear to use the infinite as matter; wherefore also it is absurd to make it that which contains rather than that which is contained.
§3.8Λοιπὸν δʼ ἐπελθεῖν καθʼ οὓς λόγους τὸ ἄπειρον εἶναι δοκεῖ οὐ μόνον δυνάμει ἀλλʼ ὡς ἀφωρισμένον· τὰ μὲν γάρ ἐστιν αὐτῶν οὐκ ἀναγκαῖα, τὰ δʼ ἔχει τινὰς ἑτέρας ἀληθεῖς ἀπαντήσεις.
It remains to go over the arguments according to which the infinite seems to exist not only potentially but as something determined [separately/actually]; for some of them are not necessary, while others have certain other true answers.
οὔτε γὰρ ἵνα ἡ γένεσις μὴ ἐπιλείπῃ, ἀναγκαῖον ἐνεργείᾳ ἄπειρον εἶναι σῶμα αἰσθητόν· ἐνδέχεται γὰρ τὴν θατέρου φθορὰν θατέρου εἶναι γένεσιν, πεπερασμένου ὄντος τοῦ παντός.
For neither in order that generation may not fail is it necessary for a sensible body to be infinite in actuality; for it is possible for the destruction of one thing to be the generation of another, while the universe is finite.
ἔτι τὸ ἄπτεσθαι καὶ τὸ πεπεράνθαι ἕτερον.
Furthermore, to touch and to be limited are different.
τὸ μὲν γὰρ πρός τι καὶ τινός (ἅπτεται γὰρ πᾶν τινός), καὶ τῶν πεπερασμένων τινὶ συμβέβηκεν, τὸ δὲ πεπερασμένον οὐ πρός τι· οὐδʼ ἅψασθαι τῷ τυχόντι τοῦ τυχόντος ἔστιν.
For the former is relative and of something (for everything that touches touches something), and it belongs accidentally to some of the limited things, whereas the limited is not relative; nor is it possible for any chance thing to touch any chance thing.
τὸ δὲ τῇ νοήσει πιστεύειν ἄτοπον· οὐ γὰρ ἐπὶ τοῦ πράγματος ἡ ὑπεροχὴ καὶ ἡ ἔλλειψις, ἀλλʼ ἐπὶ τῆς νοήσεως.
And to trust in thought is absurd; for the excess and the defect are not in the thing, but in the thought.
ἕκαστον γὰρ ἡμῶν νοήσειέν ἄν τις πολλαπλάσιον ἑαυτοῦ αὔξων εἰς ἄπειρον· ἀλλʼ οὐ διὰ τοῦτο ἔξω τίς ἐστιντοῦ τηλικούτου μεγέθους ὃ ἔχομεν, ὅτι νοεῖ τις, ἀλλʼ ὅτι ἔστι· τοῦτο δὲ συμβέβηκεν.
For anyone might think each of us as many times larger than himself, increasing us to infinity; but it is not because someone thinks so that anyone is outside of so great a magnitude as we have, but because he is [of that size]; and this is accidental.
ὁ δὲ χρόνος καὶ ἡ κίνησις ἄπειρά ἐστι καὶ ἡ νόησις οὐχ ὑπομένοντος τοῦ λαμβανομένου.
But time and motion are infinite, and so is thought, because what is taken does not remain.
μέγεθος δὲ οὔτε τῇ καθαιρέσει οὔτε τῇ νοητικῇ αὐξήσει ἔστιν ἄπειρον.
But magnitude is not infinite either by diminution or by mental increase.
ἀλλὰ περὶ μὲν τοῦ ἀπείρου, πῶς ἔστι καὶ πῶς οὐκ ἔστι καὶ τί ἐστιν, εἴρηται.
Concerning the infinite, then, how it exists and how it does not exist, and what it is, has been said.