§2.1.26Ἐπεὶ δʼ ἡ κατηγορικὴ τῆς στερητικῆς βελτίων, δῆλον ὅτι καὶ τῆς εἰς τὸ ἀδύνατον ἀγούσης.
Since affirmative demonstration is better than negative, it is clear that it is also better than demonstration leading to the impossible.
δεῖ δʼ εἰδέναι τίς ἡ διαφορὰ αὐτῶν.
But we must know what the difference between them is.
ἔστω δὴ τὸ Α μηδενὶ ὑπάρχον τῷ Β, τῷ δὲ Γ τὸ Β παντί ἀνάγκη δὴ τῷ Γ μηδενὶ ὑπάρχειν τὸ Α. οὕτω μὲν οὖν ληφθέντων δεικτικὴ ἡ στερητικὴ ἄν εἴη ἀπόδειξις ὅτι τὸ Α τῷ Γ οὐχ ὑπάρχει.
Let, then, A belong to no B, and let B belong to all C; it is necessary, then, for A to belong to no C. If, then, they are assumed in this way, the negative demonstration that A does not belong to C would be ostensive.
ἡ δʼ εἰς τὸ ἀδύνατον ὦδ’ ἔχει. εἰ δέοι δέξαι ὅτι τὸ Α τῷ Β οὐχ ὑπάρχει ληπτέον ὑπάρχειν, καὶ τὸ Β τῷ Γ, ὥστε συμβαίνει τὸ Α τῷ Γ ὑπάρχειν.
But demonstration leading to the impossible is as follows: if it were necessary to show that A does not belong to B, we must assume that it does belong, and B to C, so that it results that A belongs to C.
τοῦτο δʼ ἔστω γνώριμον καὶ ὁμολογούμενον ὅτι ἀδύνατον.
And let this be known and agreed to be impossible.
οὐκ ἄρα οἆόν τε τὸ τῷ Β ὑπάρχειν.
Therefore, it is not possible that [A] belongs to B.
εἰ οὖν τὸ Β τῷ Γ ὁμολογεῖται ὑπάρχειν, τὸ Α τῷ Β ἀδύνατον ὑπάρχειν.
If, then, it is agreed that B belongs to C, it is impossible for A to belong to B.
οἱ μὲν οὖν ὅροι ὁμοίως τάττονται, διαφέρει δὲ τὸ ὁποτέρα ἄν ᾖ γνωριμωτέρα ἡ πρότασις ἡ στερητική, πότερον ὅτι τὸ Α τῷ Β οὐχ ὑπάρχει ἢ ὅτι τὸ Α τῷ Γ. ὅταν μὲν οὖν ᾖ τὸ συμπέρασμα γνωριμώτερον ὅτι οὐκ ἔστιν, ἡ εἰς τὸ ἀδύνατον γίνεται ἀπόδειξις, ὅταν δʼ ἡ ἐν τῷ συλλογισμῷ, ἡ ἀποδεικτική.
Now, the terms are arranged in a similar way, but they differ in which of the two negative premises is better known: whether that A does not belong to B, or that A does not belong to C. When, therefore, the conclusion that it is not the case is better known, the demonstration leading to the impossible comes to be; but when the premise in the syllogism [is better known], the ostensive demonstration [comes to be].
φύσει δὲ προτέρα ἡ ὅτι τὸ Α τῷ Β ἢ ὅτι τὸ Α τῷ Γ. πρότερα γάρ ἐστι τοῦ συμπεράσματος ἐξ ὦν τὸ συμπέρασμα· ἔστι δὲ τὸ μὲν Α τῷ Γ μὴ ὑπάρχειν συμπέρασμα, τὸ δὲ Α τῷ Β ἐξ οὗ τὸ συμπέρασμα.
But by nature, that A does not belong to B is prior to that A does not belong to C. For those things from which the conclusion is derived are prior to the conclusion; and that A does not belong to C is the conclusion, while that A does not belong to B is that from which the conclusion is derived.
οὐ γὰρ εἰ συμβαίνει ἀναιρεῖσθαί τι, τοῦτο συμπέρασμά ἐστιν, ἐκεῖνα δὲ ἐξ ὧν, ἀλλὰ τὸ μὲν ἐξ οὗ συλλογισμός ἐστιν ὅ ἄν οὕτως ἔχῃ ὥστε ἢ ὅλον πρὸς μέρος ἢ μέρος πρὸς ὅλον ἔχειν, αἱ δὲ τὸ Α Γ καὶ Γ προτάσεις οὐκ ἔχουσιν οὕτω πρὸς ἀλλήλας.
For it is not the case that if something results in being refuted, this is the conclusion, and those are the premises from which it is derived; rather, that from which a syllogism proceeds is that which is so related as to be either a whole to a part or a part to a whole, whereas the premises A-C and C-B are not so related to one another.
εἰ οὖν ἡ ἐκ γνωριμωτέρων καὶ προτέρων κρείττων, εἰσὶ δʼ ἀμφότεραι ἐκ τοῦ μὴ εἶναί τι πισταί, ἀλλʼ ἡ μέν ἐκ προτέρου ἡ δʼ ἐξ ὑστέρου, βελτίων ἀπλῶς ἄν εἴη τῆς εἰς τὸ ἀδύνατον ἡ στερητικὴ ἀπόδειξις, ὥστε καὶ ἡ ταύτης βελτίων ἡ κατηγορικὴ δῆλον ὅτι καὶ τῆς εἰς τὸ ἀδύνατόν ἐστι βελτίων.
If, therefore, the demonstration from things better known and prior is superior, and both are trustworthy from something not being the case, but the one proceeds from what is prior and the other from what is posterior, the negative ostensive demonstration would be simply better than that leading to the impossible; so that, since the affirmative is better than this, it is clear that it is also better than that leading to the impossible.
§2.1.27Ἀκριβεστέρα δʼ ἐπιστήμη ἐπιστήμης καὶ προτέρα ἥ τε τοῦ ὅτι καὶ διότι ἡ αὐτή, ἀλλὰ μὴ χωρὶς τοῦ ὅτι τῆς τοῦ διότι, καὶ ἡ μὴ καθʼ ὑποκειμένου τῆς καθʼ ὑποκειμένου, οἷον ἀριθμητικὴ ἁρμονικῆς, καὶ ἡ ἐξ ἐλαττόνων τῆς ἐκ προσθέσεως, οἷον γεωμετρίας ἀριθμητική.
An intellectual discipline is more precise than another and prior to it, if it is simultaneously the science of the fact and of the reason, and not the science of the fact separately from that of the reason; and that which is not of a subject-matter than that which is of a subject-matter, as arithmetic is to harmonics; and that which proceeds from fewer elements than that which proceeds from addition, as arithmetic is to geometry.
λέγω δʼ ἐκ προσθέσεως, οἷον μονὰς οὐσία ἄθετος, στιγμὴ δὲ οὐσία θετός· ταύτην ἐκ προσθέσεως.
By "from addition" I mean, for example, a unit is a substance without position, whereas a point is a substance with position; the latter is from addition.
§2.1.28Μία δʼ ἐπιστήμη ἐστὶν ἡ ἑνὸς γένους, ὅσα ἐκ τῶν πρώτων σύγκειται καὶ μέρη ἐστὶν ἢ πάθη τούτων καθʼ αὑτά.
A science is single if it is of a single genus—namely, of all things that are composed of the first principles and are parts or essential attributes of these.
ἑτέρα δʼ ἐπιστήμη ἐστὶν ἑτέρας, ὅσων αἱ ἀρχαὶ μήτʼ ἐκ τῶν αὐτῶν μήθʼ ἅτεραι ἐκ τῶν ἑτέρων.
And one science is different from another if their principles are neither derived from the same principles nor are one set derived from the other.
τούτου δὲ σημεῖον, ὅταν εἰς τὰ ἀναπόδεικτα ἔλθῃ· δεῖ γὰρ αὐτὰ ἐν τῷ αὐτῷ γένει εἶναι τοῖς ἀποδεδειγμένοις.
A sign of this is when one arrives at the indemonstrables; for these must be in the same genus as the things demonstrated.
σημεῖον δὲ καὶ τούτου, ὅταν τὰ δεικνύμενα διʼ αὐτῶν ἐν ταὐτῷ γένει ὦσι καὶ συγγενῆ.
And a sign of this is also when the things shown through them are in the same genus and are cognate.