§3.5#1Χωριστὸν μὲν οὖν εἶναι τὸ ἄπειρον τῶν αἰσθητῶν, αὐτό τι ὂν ἄπειρον, οὐχ οἷόν τε.
It is impossible, then, for the infinite to be separate from sensible things, being itself some infinite thing.
εἰ γὰρ μήτε μέγεθός ἐστιν μήτε πλῆθος, ἀλλʼ οὐσία αὐτό ἐστι τὸ ἄπειρον καὶ μὴ συμβεβηκός, ἀδιαίρετον ἔσται (τὸ γὰρ διαιρετὸν ἢ μέγεθος ἔσται ἢ πλῆθος)·
For if it is neither magnitude nor multitude, but the infinite itself is substance and not an accident, it will be indivisible (for what is divisible will be either a magnitude or a multitude).
εἰ δὲ τοιοῦτον, οὐκ ἄπειρον, εἰ μὴ ὡς ἡ φωνὴ ἀόρατος.
And if it is such, it is not infinite, except in the way that voice is invisible.
ἀλλʼ οὐχ οὕτως οὔτε φασὶν εἶναι οἱ φάσκοντες εἶναι τὸ ἄπειρον οὔτε ἡμεῖς ζητοῦμεν, ἀλλʼ ὡς ἀδιεξίτητον.
But neither do those who claim that there is an infinite say it exists in this way, nor do we investigate it as such, but rather as what cannot be gone through.
εἰ δὲ κατὰ συμβεβηκὸς ἔστιν τὸ ἄπειρον, οὐκ ἂν εἴη στοιχεῖον τῶν ὄντων, ᾗ ἄπειρον, ὥσπερ οὐδὲ τὸ ἀόρατον τῆς διαλέκτου, καίτοι ἡ φωνή ἐστιν ἀόρατος.
But if the infinite exists by accident, it would not be an element of beings, qua infinite, just as the invisible is not an element of speech, although voice is invisible.
ἔτι πῶς ἐνδέχεται εἶναί τι αὐτὸ ἄπειρον, εἴπερ μὴ καὶ ἀριθμὸν κοὶ μέγεθος, ὧν ἐστι καθʼ αὐτὸ πάθος τι τὸ ἄπειρον;
Furthermore, how is it possible for there to be something which is itself infinite, if indeed there is not also number and magnitude, of which the infinite is an attribute in itself?
ἔτι γὰρ ἧττον ἀνάγκη ἢ τὸν ἀριθμὸν ἢ τὸ μέγεθος.
For there is still less necessity for [the infinite itself to exist] than for number or magnitude.
φανερὸν δὲ καὶ ὅτι οὐκ ἐνδέχεται εἶναι τὸ ἄπειρον ὡς ἐνεργείᾳ ὄν καὶ ὡς οὐσίαν καὶ ἀρχήν·
It is also manifest that it is not possible for the infinite to exist as being in actuality, and as substance and principle.
ἔσται γὰρ ὁτιοῦν αὐτοῦ ἄπειρον τὸ λαμβανόμενον, εἰ μεριστόν (τὸ γὰρ ἀπείρῳ εἶναι καὶ ἄπειρον τὸ αὐτό, εἴπερ οὐσία τὸ ἄπειρον καὶ μὴ καθʼ ὑποκειμένου), ὥστʼ ἢ ἀδιαίρετον ἢ εἰς ἄπειρα διαιρετόν·
For any part of it that is taken will be infinite, if it is divisible (for to be infinite and the infinite are the same, if indeed the infinite is substance and not predicated of a subject), so that it is either indivisible or divisible into infinites.
πολλὰ δʼ ἄπειρα εἶναι τὸ αὐτὸ ἀδύνατον (ἀλλὰ μὴν ὥσπερ ἀέρος ἀὴρ μέρος, οὕτω καὶ ἄπειρον ἀπείρου, εἴ γε οὐσία ἐστὶ καὶ ἀρχή)· ἀμέριστον ἄρα καὶ ἀδιαίρετον.
But it is impossible for the same thing to be many infinites (yet surely, just as a part of air is air, so also a part of the infinite is infinite, if indeed it is substance and principle); therefore it is without parts and indivisible.
ἀλλʼ ἀδύνατον τὸ ἐντελεχείᾳ ὂν ἄπειρον· ποσὸν γάρ τι εἶναι ἀναγκαῖον.
But it is impossible for the infinite existing in actuality [to be indivisible]; for it must of necessity be some quantity.
κατὰ συμβεβηκὸς ἄρα ὑπάρχει τὸ ἄπειρον.
Therefore the infinite belongs [to things] by accident.
ἀλλʼ εἰ οὕτως, εἴρηται ὅτι οὐκ ἐνδέχεται αὐτὸ λέγειν· ἀρχήν, ἀλλʼ ᾧ συμβέβηκε, τὸν ἀέρα ἢ τὸ ἄρτιον.
But if so, it has been said that it is not possible to call it a principle, but rather that to which it is an accident, such as air or the even.
ὥστε ἀτόπως ἂν ἀποφαίνοιντο οἱ λέγοντες οὕτως ὥσπερ οἱ Πυθαγόρειοί φασιν· ἅμα γὰρ οὐσίαν ποιοῦσι τὸ ἄπειρον καὶ μερίζουσιν.
So those who speak in this way, as the Pythagoreans say, would express themselves absurdly; for they simultaneously make the infinite a substance and divide it.
ἀλλʼ ἴσως αὕτη μὲν καθόλου ἡ ζήτησις,
εἰ ἐνδέχεται ἄπειρον καὶ ἐν τοῖς μαθηματικοῖς εἶναι καὶ ἐν τοῖς νοητοῖς καὶ μηδὲν ἔχουσι μέγεθος· ἡμεῖς δʼ ἐπισκοποῦμεν περὶ τῶν αἰσθητῶν καὶ περὶ ὧν ποιμωύμεθα τὴν μέθοδον, ἆρʼ ἔστιν ἐν αὐτοῖς ἢ οὐκ ἔστι σῶμα ἄπειρον ἐπὶ τὴν αὔξησιν.
But perhaps this investigation is general, whether it is possible for the infinite to exist also in mathematical objects and in intelligible objects and those having no magnitude; but we are examining concerning sensible things and those about which we conduct our inquiry, whether there is among them or not an infinite body in respect of growth.
λογικῶς μὲν οὖν σκοπουμένοις ἐκ τῶν τοιῶνδε δόξειεν ἂν οὐκ εἶναι· εἰ γάρ ἐστι σώματος λόγος τὸ ἐπιπέδῳ ὡρισμένον, οὐκ ἂν εἴη σῶμα ἄπειρον, οὔτε νοητὸν οὔτε αἰσθητόν (ἀλλὰ μὴν οὐδʼ ἀριθμὸς οὕτως ὡς κεχωρισμένος καὶ ἄπειρος· ἀριθμητὸν γὰρ ἀριθμὸς ἢ τὸ ἔχον ἀριθμόν· εἰ οὖν τὸ ἀριθμητὸν ἐνδέχεται ἀριθμῆσαι, καὶ διεξελθεῖν ἂν εἴη δυνατὸν τὸ ἄπειρον)· φυσικῶς δὲ μᾶλλον θεωροῦσιν ἐκ τῶνδε. οὔτε γὰρ σύνθετον οἶόν τε εἶναι οὔτε ἀπλοῦν.
If, then, we consider the matter logically, it would seem from such arguments not to exist: for if the definition of a body is 'that which is bounded by a surface', there could not be an infinite body, whether intelligible or sensible (nor indeed could there be a number in this way, as separated and infinite; for number is what is countable or what has number; if, then, it is possible to count what is countable, it would also be possible to go through the infinite); but to those who consider the matter more from the standpoint of physics, [it would seem so] from the following: for it is possible for it to be neither composite nor simple.
σύνθετον μὲν οὖν οὐκ ἔσται τὸ ἄπειρον σῶμα, εἰ πεπερασμένα τῷ πλήθει τὰ στοιχεῖα.
An infinite body, then, will not be composite, if the elements are finite in multitude.
ἀνάγκη γὰρ πλείω εἶναι, καὶ ἰσάζειν ἀεὶ τἀναντία, καὶ μὴ εἶναι ἓν αὐτῶν ἄπειρον (εἰ γὰρ ὁπυσῳοῦν λείπεται ἡ ἐν ἑνὶ σώματι δύναμις θατέρου, οἷον εἰ τὸ πῦρ πεπέρανται, ὁ δʼ ἀὴρ ἄπειρος, ἔστιν δὲ τὸ ἴσον πῦρ τοῦ ἴσου ἀέρος τῇ δυνάμει ὁποσαπλασιονοῦν, μόνον δὲ ἀριθμόν τινα ἔχον, ὅμῶς φανερὸν ὅτι τὸ ἄπειρον ὑπερβαλεῖ καὶ φθερεῖ τὸ πεπερασμένον· ἕκαστον δʼ ἄπειρον εἶναι ἀδύνατον· σῶμα μὲν γάρ ἐστιν τὸ πάντη ἔχον διάστασιν, ἄπειρον δὲ τὸ ἀπεράντως διεστηκός, ὥστε τὸ ἄπειρον σῶμα πανταχῇ ἔσται διεστηκὸς εἰς ἄπειρον.
For they must be more than one, and the contraries must always be equalized, and one of them must not be infinite (for if the power in one body falls short in any degree whatever of the power of the other—for instance, if fire is finite and air infinite, and an equal amount of fire is any number of times greater in power than an equal amount of air, provided it has only some definite ratio—nevertheless it is manifest that the infinite will exceed and destroy the finite; and it is impossible for each to be infinite; for a body is that which has dimension in every direction, and the infinite is that which is infinitely extended, so that the infinite body will be extended everywhere to infinity.