§2.1.22#2Ὑπόκειται δὴ ἓν καθʼ ἑνὸς κατηγορεῖσθαι, αὐτὰ δὲ αὐτῶν, ὅσα μὴ τί ἐστι, μὴ κατηγορεῖσθαι συμβεβηκότα γάρ ἐστι πάντα, ἀλλὰ τὰ μὲν καθʼ αὐτά, τὰ δὲ καθʼ ἕτερον τρόπον·
It is laid down that one thing is predicated of one, and that those things themselves, as many as do not signify essence, are not reciprocally predicated of themselves; for they are all accidents, but some are in themselves, others in another way.
ταῦτα δὲ πάντα καθʼ ὑποκειμένου τινὸς κατηγορεῖσθαί φαμεν, τὸ δὲ συμβεβηκός οὐκ εἶναι ὑποκείμενόν τι· οὐδὲν γὰρ τῶν τοιούτων τίθεμεν εἶναι ὃ οὐχ ἕτερόν τι ὂν λέγεται ὃ λέγεται, ἀλλʼ αὐτὸ ἄλλου καὶ τοῦτο καθʼ ἑτέρου.
And we say that all these are predicated of some underlying subject, and that an accident is not an underlying subject; for we assume that there is nothing of such a kind which is said to be what it is said to be without being something else, but it is itself of another, and this is of another still.
οὔτʼ εἰς τὸ ἄνω ἄρα ἓν καθʼ ἑνὸς οὔτʼ εἰς τὸ κάτω ὑπάρχειν λεχθήσεται.
Neither in the upward direction then nor in the downward direction will one thing be said to belong to one infinitely.
καθʼ ὧν μὲν γὰρ λέγεται τὰ συμβεβηκότα, ὅσα ἐν τῇ οὐσίᾳ ἑκάστου, ταῦτα δὲ οὐκ ἄπειρα· ἄνω δὲ ταῦτά τε καὶ τὰ συμβεβηκότα, ἀμφότερα οὐκ ἄπειρα.
For that of which accidents are said, namely, as many as are in the substance of each thing, these are not infinite; and in the upward direction, both these and the accidents are not infinite.
ἀνάγκη ἄρα εἶναί τι οὗ πρῶτόν τι κατηγορεῖται καὶ τούτου ἄλλο, καὶ τοῦτο ἵστασθαι καὶ εἶναί τι ὃ οὐκέτι οὔτε κατʼ ἄλλου προτέρου οὔτε κατʼ ἐκείνου ἄλλο πρότερον κατηγορεῖται.
It is necessary, therefore, that there be something of which something is first predicated, and of this another, and that this should stop, and that there be something which is no longer predicated of another prior thing, nor another prior thing of it.
Εἷς μὲν οὖν τρόπος λέγεται ἀποδείξεως οὗτος, ἔτι δʼ ἄλλος, εἰ ὧν πρότερα ἄττα κατηγορεῖται, ἔστι τούτων ἀπόδειξις, ὧν δʼ ἔστιν ἀπόδειξις, οὔτε βέλτιον ἔχειν ἐγχωρεῖ πρὸς αὐτὰ τοῦ εἰδέναι, οὐτʼ εἰδέναι ἄνευ ἀποδείξεως, εἰ δὲ τόδε διὰ τῶνδε γνώριμον, τάδε δὲ μὴ ἴσμεν μηδὲ βέλτιον ἔχομεν πρός αὐτὰ τοῦ εὐδέναι, οὐδὲ τὸ διὰ τούτων γνώριμον ἐπιστησόμεθα.
Now this is said to be one way of demonstration, but there is still another, if indeed, of those things of which certain prior things are predicated, there is a demonstration of these, and of those of which there is a demonstration, it is not possible to be in a better state than knowing them, nor to know them without demonstration; and if this is known through these, and we do not know these nor are in a better state than knowing them, neither shall we understand that which is known through them.
εἰ οὖν ἔστι τι εἰδέναι διʼ ἀποδείξεως ἁπλῶς καὶ μὴ ἐκ τινῶν μηδʼ ἐξ ὑποθέσεως, ἀνάγκη ἵστασθαι τὰς κατηγορίας τὰς μεταξύ.
If then it is possible to know something through demonstration simply, and not from certain things nor from a hypothesis, it is necessary that the intermediate predications should stop.
εἰ γὰρ μὴ ἵστανται, ἀλλʼ ἔστιν ἀεὶ τοῦ ληφθέντος ἐπάνω, ἁπάντων ἔσται ἀπόδειξις· ὥστʼ εἰ τὰ ἄπειρα μὴ ἐγχωρεῖ διελθεῖν, ὧν ἔστιν ἀπόδειξις, ταῦτʼ οὐκ εἰσόμεθα διʼ ἀποδείξεως.
For if they do not stop, but there is always something above that which is taken, there will be a demonstration of all things; so that if it is not possible to traverse infinite things, of which there is a demonstration, we shall not know these through demonstration.
εἰ οὖν μηδὲ βέλτιον ἔχομεν πρὸς αὐτὰ τοῦ εἰδέναι, οὐκ ἔσται οὐδὲν ἐπίστασθαι διʼ ἀποδείξεως ἀπλῶς, ἀλλʼ ἐξ ὑποθέσεως.
If then we are not in a better state than knowing them, there will be no understanding anything through demonstration simply, but only from a hypothesis.
Λογικῶς μὲν οὖν ἐκ τούτων ἄν· τις πιστεύσειε περὶ τοῦ λεχθέντος, ἀναλυτικῶς δὲ διὰ τῶνδε φανερὸν συντομώτερον, ὅτι οὔτʼ ἐπὶ τὸ ἄνω οὔτʼ ἐπὶ τὸ κάτω ἄπειρα τὰ κατηγορούμενα ἐνδέχεται εἶναι ἐν ταῖς ἀποδεικτικαῖς ἐπιστήμαις, περὶ ὧν ἡ σκέψις ἐστίν.
Logically, then, one might believe in what has been said from these things, but analytically, it is clearer more concisely through these, that it is not possible for the predicated things to be infinite either in the upward or in the downward direction in the demonstrative sciences, with which our inquiry is concerned.
ἡ μὲν γὰρ ἀπόδειξίς ἐστι τῶν ὅσα ὑπάρχει καθʼ αὑτὰ τοῖς πράγμασιν.
For demonstration is of as many things as belong in themselves to things.
καθʼ αὑτὰ δὲ διττῶς· ὅσα τε γὰρ ἐκείνοις ἐνυπάρχει ἐν τῷ τί ἐστι, καὶ οἷς αὐτὰ ἐν τῷ τί ἐστιν ὑπάρχουσιν αὐτοῖς· οἷον τῷ ἀριθμῷ τὸ περιττόν, ὃ ὑπάρχει μὲν ἀριθμῷ, ἐνυπάρχει δʼ αὐτὸς ὁ ἀριθμὸς ἐν τῷ λόγῳ αὐτοῦ, καὶ πάλιν πλῆθος ἢ τὸ διαιρετὸν ἐν τῷ λόγῳ τῷ τοῦ ἀριθμοῦ ἐνυπάρχει.
And "in themselves" is twofold: for both as many as are in those things in their essence, and those for which they themselves belong to them in their essence; as the odd for number, which indeed belongs to number, but number itself is present in its definition, and again multiplicity or divisibility is present in the definition of number.
τούτων δʼ οὐδέτερα ἐνδέχεται ἄπειρα εἶναι, οὔθʼ ὡς τὸ περιττὸν τοῦ ἀριθμοῦ (πάλιν γὰρ ἂν τῷ περιττῷ ἄλλο εἴη ᾧ ἐνυπῆρχεν ὑπάρχοντι· τοῦτο δʼ εἰ ἔστι, πρῶτον ὁ ἀριθμὸς ἐνυπάρξει ὑπάρχουσιν αὐτῷ· εἰ οὖν μὴ ἐνδέχεται ἄπειρα τοιαῦτα ὑπάρχειν ἐν τῷ ἑνί, οὐδʼ ἐπὶ τὸ ἄνω ἔσται ἄπειρα· ἀλλὰ μὴν ἀνάγκη γε πάντα ὑπάρχειν τῷ πρώτῳ, οἷον τῷ ἀριθμῷ, κἀκείνοις τὸν ἀριθμόν, ὥστʼ ἀντιστρέφοντα ἔσται, ἀλλʼ οὐχ ὑπερτείνοντα)· οὐδὲ μὴν ὅσα ἐν τῷ τί ἐστιν ἐνυπάρχει, οὐδὲ ταῦτα ἄπειρα· οὐδὲ γὰρ ἂν εἴη ὁρίσασθαι.
And of these, neither is possible to be infinite; not as the odd of number (for again there would be something else to the odd in which it was present, belonging to it; and if this is so, number will first be present in those belonging to it; if then it is not possible for infinite such things to belong to the one, neither will they be infinite in the upward direction; but indeed, it is necessary that all things belong to the first, as to number, and number to them, so that they will be reciprocal, but not exceeding); nor indeed as many as are present in the essence, nor are these infinite; for otherwise it would not be possible to define.
ὥστʼ εἰ τὰ μὲν κατηγορούμενα καθʼ αὑτὰ πάντα λέγεται, ταῦτα δὲ μὴ ἄπειρα, ἵσταιτο ἂν τὰ ἐπὶ τὸ ἄνω, ὥστε καὶ ἐπὶ τὸ κάτω.
So that if all predicated things are said in themselves, and these are not infinite, the things in the upward direction would stop, so that those in the downward direction would also stop.
Εἱ δʼ οὕτω, καὶ τὰ ἐν τῷ μεταξύ δύο ὅρων ἀεὶ πεπερασμένα.
And if so, the things between two terms are also always finite.
εἰ δὲ τοῦτο, δῆλον ἤδη καὶ τῶν ἀποδείξεων ὅτι ἀνάγκη ἀρχάς τε εἶναι, καὶ μὴ πάντων εἶναι ἀπόδειξιν, ὅπερ ἔφαμέν τινας λέγειν κατʼ ἀρχάς.
And if this is so, it is already clear also concerning demonstrations that there must be principles, and that there is not demonstration of all things, which we said some people assert at the beginning.
εἰ γὰρ εἰσὶν ἀρχαί, οὔτε πάντʼ ἀποδεικτὰ οὔτʼ εἰς ἄπειρον οἷόν τε βαδίζειν· τὸ γὰρ εἶναι τούτων ὁποτερονοῦν οὐδὲν ἄλλο ἐστὶν ἢ τὸ εἶναι μηδὲν διάστημα ἄμεσον καὶ ἀδιαίρετον, ἀλλὰ πάντα διαιρετά.
For if there are principles, neither are all things demonstrable, nor is it possible to go to infinity; for that either of these should be is nothing else than that there is no immediate and indivisible interval, but all are divisible.
τῷ γὰρ ἐντὸς ἐμβάλλεσθαι ὅρον, ἀλλʼ οὐ τῷ προσλαμβάνεσθαι ἀποδείκνυται τὸ ἀποδεικνύμενον, ὥστʼ εἰ τοῦτʼ εἰς ἄπειρον ἐνδέχεται ἰέναι, ἐνδέχοιτʼ ἂν δύο ὅρων ἄπειρα μεταξὺ εἶναι μέσα.
For the thing demonstrated is demonstrated by inserting a term within, and not by adding one, so that if this is possible to go to infinity, there could be infinite middle terms between two terms.
ἀλλὰ τοῦτʼ ἀδύνατον, εἰ ἵστανται αἱ κατηγορίαι ἐπὶ τὸ ἄνω καὶ τὸ κάτω.
But this is impossible, if the predications stop in the upward and downward directions.
ὅτι δὲ ἵστανται, δέδεικται λογικῶς μὲν πρότερον, ἀναλυτικῶς δὲ νῦν.
And that they do stop has been shown logically before, and analytically now.