§2.1.23Δεδειγμένων δὲ τούτων φανερὸν ὅτι, ἐάν τι τὸ αὐτὸ δυσὶν ὑπάρχῃ, οἷον τὸ Α τῷ τε Γ καὶ τῷ Δ, μὴ κατηγορουμένου θατέρου κατὰ θατέρου, ἢ μηδαμῶς ἢ μὴ κατὰ παντός, ὅτι οὐκ ἀεὶ κατὰ κοινόν τι ὑπάρξει.
These things having been shown, it is clear that if the same thing belongs to two things, for example, A to both C and D, while neither is predicated of the other either not at all or not of every instance, it will not always belong to them in virtue of something common.
οἷον τῷ ἰσοσκελεῖ καὶ τῷ σκαληνεῖ τὸ δυσὶν ὀρθαῖς ἴσας ἔχειν κατὰ κοινόν τι ὑπάρχει (ᾗ γὰρ σχῆμά τι, ὑπάρχει, καὶ οὐχ ᾗ ἕτερον), τοῦτο δʼ οὐκ ἀεὶ οὕτως ἔχει.
For example, having angles equal to two right angles belongs to the isosceles and the scalene in virtue of something common (for it belongs to them in so far as they are a certain shape, and not in so far as they are different), but this is not always the case.
ἔστω γὰρ τὸ Β καθʼ ὃ τὸ Α τῷ Γ Δ ὑπάρχει.
For let B be that in virtue of which A belongs to C and D.
δῆλον τοίνυν ὅτι καὶ τὸ Β τῷ Γ καὶ Δ κατʼ ἄλλο κοινόν, κἀκεῖνο καθʼ ἕτερον, ὥστε δύο ὅρων μεταξὺ ἄπειροι ἂν ἐμπίπτοιεν ὅροι.
It is clear, then, that B also belongs to C and D in virtue of another common term, and that one in virtue of another still, so that infinitely many terms would fall between two terms.
ἀλλʼ ἀδύνατον.
But this is impossible.
κατὰ μὲν τοίνυν κοινόν τι ὑπάρχειν οὐκ ἀνάγκη ἀεὶ τὸ αὐτὸ πλείοσιν, εἴπερ ἔσται ἄμεσα διαστήματα.
Therefore, it is not necessary that the same thing always belongs to several things in virtue of something common, if indeed there are to be immediate intervals.
ἐν μέντοι τῷ αὐτῷ γένει καὶ ἐκ τῶν αὐτῶν ἀτόμων ἀνάγκη τοὺς ὅρους εἶναι, εἴπερ τῶν καθ᾿ αὑτὸ ὑπαρχόντων ἔσται τὸ κοινόν· οὐ γὰρ ἦν ἐξ ἄλλου γένους εἰς ἄλλο διαβῆναι τὰ δεικνύμενα.
However, the terms must be in the same genus and derived from the same indivisibles, if indeed the common term is to belong to those things which belong in themselves; for it was not possible for demonstrated things to pass from one genus to another.
Φανερὸν δὲ καὶ ὅτι, ὅταν τὸ Α τῷ Β ὑπάρχῃ. εἰ μὲν ἔστι τι μέσον, ἔστι δεῖξαι ὅτι τὸ τῷ Β ὑπάρχει, καὶ στοιχεῖα τούτου ἔστι ταὐτὰ καὶ τοσαῦθʼ ὅσα μέσα ἐστίν· αἱ γὰρ ἄμεσοι προτάσεις στοιχεῖα, ἢ πᾶσαι ἢ αἱ καθόλου.
And it is also clear that, when A belongs to B, if there is some middle term, it is possible to show that it belongs to B, and the elements of this demonstration are the same as and as many as the middle terms; for the immediate premises are elements, either all of them or the universal ones.
εἰ δὲ μὴ ἔστιν, οὐκέτι ἔστιν ἀπόδειξις, ἀλλʼ ἡ ἐπὶ τὰς ἀρχὰς ὀδὸς αὕτη ἐστίν.
But if there is none, there is no longer a demonstration, but this is the path toward the principles.
ὁμοίως δὲ καὶ εἰ τὸ Α τῷ Β μὴ ὑπάρχει, εἰ μὲν ἔστιν ἢ μέσον ἢ πρότερον ᾧ οὐχ ὑπάρχει, ἔστιν ἀπόδειξις, εἰ δὲ μή, οὐκ ἔστιν, ἀλλʼ ἀρχή, καὶ στοιχεῖα τοσαῦτʼ ἔστιν ὅσοι ὅροι· αἱ γὰρ τούτων προτάσεις ἀρχαὶ τῆς ἀποδείξεώς εἰσιν.
Similarly also if A does not belong to B: if there is either a middle term or something prior to which it does not belong, there is a demonstration, but if not, there is none, but it is a principle, and there are as many elements as there are terms; for the premises of these are the principles of the demonstration.
καὶ ὥσπερ ἔνιαι ἀρχαί εἰσιν ἀναπόδεικτοι, ὅτι ἐστὶ τόδε τοδὶ καὶ ὑπάρχει τόδε τῳδί, οὕτω καὶ ὅτι οὐκ ἔστι τόδε τοδὶ οὐδʼ ὑπάρχει τόδε τῳδί, ὥσθʼ αἱ μὲν εἶναί τι, αἱ δὲ μὴ εἰναί τι ἔσονται.
And just as some principles are indemonstrable, that this is this and this belongs to this, so also that this is not this nor does this belong to this, so that some will be that something is, and others that something is not.
Ὅταν δὲ δέῃ δείξαι, ληπτέον ὃ τοῦ Β πρῶτον κατηγορεῖται.
When it is necessary to demonstrate, one must take what is first predicated of B.
ἔστω τὸ Γ, καὶ τούτου ὁμοίως τὸ Δ. καὶ οὕτως ἀεὶ βαδίζοντι οὐδέποτʼ ἐξωτέρω πρότασις οὐδʼ ὑπάρχον λαμβάνεται τοῦ Α ἐν τῷ δεικνύναι, ἀλλʼ ἀεὶ τὸ μέσον πυκνοῦται, ἕως ἀδιαίρετα γέντται καὶ ἕν.
Let this be C, and similarly of this, D. And to one who always proceeds in this way, no premise or belonging term is ever taken outside of A in the course of demonstrating, but the middle is always condensed until they become indivisible and one.
ἔστι δʼ ἓν ὅταν ἄμεσον γένηται, καὶ μία πρότασις ἁπλῶς ἡ ἄμεσος.
And it is one when it becomes immediate, and the immediate premise is simply one premise.
καὶ ὥσπερ ἐν τοῖς ἄλλοις ἡ ἀρχὴ ἀπλοῦν, τοῦτο δʼ οὐ ταὐτὸ πανταχοῦ, ἀλλʼ ἐν βάρει μὲν μνᾶ, ἐν δὲ μέλει δίεσις, ἄλλο δʼ ἐν ἄλλῳ, οὕτως ἐν συλλογισμῷ τὸ ἓν πρότασις ἄμεσος, ἐν δʼ ἀποδείξει καὶ ἐπιστήμῃ ὁ νοῦς.
And just as in other things the principle is something simple, though this is not the same everywhere, but in weight it is the mina, in music the diesis, and another thing in another field, so in a syllogism the 'one' is the immediate premise, and in demonstration and science it is intellect.
ἐν μὲν οὖν τοῖς δεικτικοῖς συλλογισμοῖς τοῦ ὑπάρχοντος οὐδὲν ἔξω πίπτει, ἐν δὲ τοῖς στερητικοῖς, ἔνθα μὲν ὃ δεῖ ὑπάρχειν, οὐδὲν τούτου ἔξω πίπτει, οἷον εἰ τὸ Α τῷ Β διὰ τοῦ Γ μή (εἰ γὰρ τῷ μὲν Β παντὶ τὸ Γ, τῳ δὲ Γ μηδενὶ τὸ Α)· πάλιν ἂν δέῃ ὅτι τῷ Γ τὸ οὐδενὶ ὑπάρχει, μέσον ληπτέον τοῦ καὶ Γ, καὶ οὕτως ἀεὶ πορεύσεται.
In demonstrative syllogisms, then, nothing of what belongs falls outside; whereas in negative syllogisms, in the one case, nothing falls outside of that which must belong, for example, if A does not belong to B through C (for if C belongs to every B, and A to no C); again, if it is necessary to show that A belongs to no C, a middle term must be taken between A and C, and thus it will always proceed.
ἐὰν δὲ δέῃ δεῖξαι ὅτι τὸ Δ τῷ Ε οὐχ ὑπάρχει τῷ τὸ Γ τῷ μὲν Δ παντὶ ὑπάρχειν, τῷ δὲ Ε μηδενί, τοῦ Ε οὐδέποτʼ ἔξω πεσεῖται· τοῦτο δʼ ἐστὶν ᾧ δεῖ ὑπάρχειν.
But if it is necessary to show that D does not belong to E by virtue of C belonging to every D and to no E, it will never fall outside of E; and this is that which must belong.
ἐπὶ δὲ τοῦ τρίτου τρόπου, οὔτε ἀφʼ οὗ δεῖ οὔτε ὃ δεῖ στερῆσαι οὐδέποτʼ ἔξω βαδιεῖται.
And in the third figure, it will never go outside either of that from which it must belong or of that which must be denied.