And that, if the infinite in no way exists, many impossibilities occur, is clear: for time will have a beginning and an end, and magnitudes will not be divisible into magnitude, and number will not be infinite. If of the infinite one part is in actuality, and the other in potentiality, and it is impossible for that which is now actually part of the infinite not to have been previously in potentiality, it is impossible, therefore, for the infinite always to be partly in actuality and partly in potentiality, but if it was first in potentiality, and later in actuality. If, just as time is endless with respect to the future, so too it is without beginning with respect to the past, then time will be ungenerated with respect to the future just as it is with respect to the past. But if it is impossible for time to be ungenerated with respect to the future, in which it has not yet come to be, while it has come to be with respect to the past, then time is not without beginning. For that which is without beginning has no generation, but time does. If nothing can be actually infinite, then neither the dichotomies of magnitude already taken are infinite, nor the number already taken; for both are in actuality. And if they are not infinite, they are then finite; and if they are finite, then they have a beginning, the one of being dichotomized, the other of being numbered. 20. From the same argument. The infinite exists by addition, and it exists by subtraction. That magnitude is not actually infinite has been said, but it is so by division. It remains, therefore, that the infinite is potential. But one must not take what is potential as if, for example, it is possible for this to be a statue, it will also be a statue, so also something infinite, which will be in actuality; but since being in potentiality has many senses, just as a day and a contest exist by one thing after another always coming to be, so too does the infinite. For in these cases it exists both potentially and actually. For in general the infinite exists in this way, by one thing after another always being taken, and what is taken is always finite, but always another and another. If nothing is actually infinite, it is clear that everything that exists in actuality is always finite; and being finite, it necessarily has a beginning, and having a beginning, it is necessarily generated, and being generated, there was then a time when it was not. This applies to every infinite spoken of by addition and subtraction, by potentiality and by division. 21. From the same argument. Nor if something is outside, is this infinite; but that which receives what is from outside becomes greater than what it was before receiving what was from outside. And that which becomes greater than itself by the addition of what is from outside is always finite. If the infinite is always in addition and subtraction, then the infinite is always in generation. How then is that which always has its being in becoming without beginning? For that which is generated is not without beginning. 22. From the same argument. The dichotomies of magnitude are infinite. So that in potentiality
Ὅτι δέ, εἰ μή ἐστιν ἄπειρον οὐδαμῶς, πολλὰ ἀδύνατα συμβαίνει, δῆλον· τοῦ τε γὰρ χρόνου ἔσται τις ἀρχὴ καὶ τε λευτή, καὶ τὰ μεγέθη οὐ διαιρετὰ εἰς μέγεθος, καὶ ἀριθμὸς οὐκ ἔσται ἄπειρος. Eἰ τοῦ ἀπείρου τὸ μέν ἐστιν ἐνεργείᾳ, τὸ δὲ δυνάμει, ἀδύνατον δὲ τὸ νῦν ἐνεργείᾳ ὂν τοῦ ἀπείρου μὴ πρότερον εἶναι δυνάμει, ἀδύνατον ἄρα τὸ ἄπειρον ἀεὶ τὸ μὲν εἶναι ἐνεργείᾳ, τὸ δὲ δυνάμει, ἀλλὰ εἰ πρότερον μὲν δυνάμει, ὕστερον δὲ ἐνεργείᾳ. Eἰ, ὥσπερ κατὰ τὸ μέλλον ἀτελεύτητος ὁ χρόνος, οὕτως καὶ κατὰ τὸ παρεληλυθὸς ἄναρχος, ἔσται ἄρα ὁ χρόνος ὥσπερ ἀγένητος κατὰ τὸ μέλλον οὕτως καὶ κατὰ τὸ παρελη λυθός. Eἰ δὲ τοῦτο ἀδύνατον ἀγένητον εἶναι τὸν χρόνον κατὰ τὸ μέλλον, καθ' ὃ οὔπω ἦν γεγονώς, γεγονὼς δὲ κατὰ τὸ παρεληλυθός, οὐκ ἄρα ἄναρχος ὁ χρόνος. Τοῦ μὲν γὰρ ἀνάρ χου οὐκ ἔστι γένεσις, τοῦ δὲ χρόνου ἐστίν. Eἰ οὐδὲν δύναται ἐνεργείᾳ εἶναι ἄπειρον, οὐκ ἄρα αἱ ἤδη ληφθεῖσαι διχοτομίαι τοῦ μεγέθους ἄπειροί εἰσιν, οὔτε ὁ ἤδη ληφθεὶς ἀριθμός· ἀμφότεροι γὰρ ἐνεργείᾳ εἰσίν. Eἰ δὲ οὐκ εἰσὶν ἄπειροι, πε περασμένοι ἄρα· εἰ δὲ πεπερασμένοι, καὶ ἀρχὴν ἄρα ἔχουσι, τὸ μὲν τοῦ διχοτομεῖσθαι, ὁ δὲ τοῦ ἀριθμεῖσθαι. κ. Ἐκ τοῦ αὐτοῦ λόγου. Τὸ ἄπειρον ἔστι μὲν προσθέσει, ἔστι δὲ ἀφαιρέσει. Τὸ δὲ μέγεθος ὅτι μὲν κατ' ἐνέργειαν οὐκ ἔστιν ἄπειρον, εἴρηται, διαιρέσει δέ ἐστι. Λείπεται οὖν δυνάμει εἶναι τὸ ἄπειρον. Oὐ δεῖ δὲ τὸ δυνάμει ὂν λαμβάνειν, ὥσπερ, εἰ δυνατὸν τουτὶ ἀνδριάντα εἶναι, ὡς καὶ ἔσται τουτὶ ἀνδριάς, οὕτω τι καὶ ἄπειρον, ὃ ἔσται ἐνεργείᾳ· ἀλλ' ἐπεὶ πολλαχῶς τὸ δυνάμει εἶναι, ὥσπερ ἡμέρα ἐστὶ καὶ ὁ ἀγών, τῷ ἀεὶ ἄλλο καὶ ἄλλο γίνεσθαι, οὕτω καὶ τὸ ἄπειρον. Καὶ γὰρ ἐπὶ τούτων ἐστὶ καὶ δυνάμει καὶ ἐνεργείᾳ. Ὅλως μὲν γὰρ οὕτως ἐστὶ τὸ ἄπειρον, τῷ ἀεὶ ἄλλο καὶ ἄλλο λαμβάνεσθαι, καὶ τὸ λαμ βανόμενον ἀεὶ πεπερασμένον εἶναι, ἀλλ' ἀεὶ ἕτερον καὶ ἕτερον. Eἰ κατ' ἐνέργειαν οὐκ ἔστι τι ἄπειρον, δῆλον ὅτι πᾶν τὸ ἐνεργείᾳ ὂν ἀεὶ πεπερασμένον ἐστί· πεπερασμένον δὲ ὂν ἐξ ἀνάγκης ἀρχὴν ἔχει, ἀρχὴν δὲ ἔχον ἐξ ἀνάγκης ἐστὶ γενητόν, γενητὸν δὲ ὂν ἦν ἄρα ποτὲ ὅτε οὐκ ἦν. Τοῦτο ἐπὶ παντὸς ἀπείρου προσθέσει τε καὶ ἀφαιρέσει, δυνάμει τε καὶ διαιρέσει λεγομένου. κα. Ἐκ τοῦ αὐτοῦ λόγου. Oὐδ' εἴ τι ἔξω ἐστί, τοῦτο ἄπειρόν ἐστιν· ἀλλὰ τὸ δε χόμενον τὸ ἔξωθεν μεῖζον γίνεται οὗ ἦν πρὸ τοῦ δέξασθαι τὸ ἔξωθεν. Τὸ δὲ τῇ προσθήκῃ τοῦ ἔξωθεν μεῖζον ἑαυτοῦ γιγνό μενον ἀεὶ πεπερασμένον ἐστίν. Eἰ τὸ ἄπειρον ἀεὶ ἐν προσθήκῃ καὶ ἀφαιρέσει, ἀεὶ ἄρα ἐν γενέσει τὸ ἄπειρον. Πῶς οὖν ἄναρχον τὸ ἀεὶ ἐν τῷ γί νεσθαι τὸ εἶναι ἔχον; Γενητὸν γὰρ ἄναρχον οὐκ ἔστιν. κβ. Ἐκ τοῦ αὐτοῦ λόγου. Ἄπειροι αἱ διχοτομίαι τοῦ μεγέθους. Ὥστε δυνάμει