§1.2.14#2Πάλιν ἐν τῷ μέσῳ σχήματι δεδείχθω τὸ παντὶ τῷ Β ὑπάρχον.
Again, in the middle figure, let it be demonstrated that [A] belongs to all B.
οὐκοῦν ἡ μὲν ὑπόθεσις ἦν μὴ παντὶ τῷ Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Α παντὶ τῷ Γ καὶ τὸ Γ παντὶ τῷ Β· οὕτω γὰρ ἔσται τὸ ἀδύνατον.
Therefore, the assumption was that A does not belong to all B, and A has been taken to belong to all Γ, and Γ to all B; for in this way there will be the impossibility.
τοῦτο δὲ τὸ πρῶτον σχῆμα, τὸ Α παντὶ τῷ Γ καὶ τὸ παντὶ τῷ Β. ὁμοίως δὲ καὶ εἰ τινὶ δέδεικται ὑπάρχον· ἡ μὲν γὰρ ὑπόθεσις ἦν μηδενὶ τῷ Β τὸ ὑπάρχειν, εἴληπται δὲ τὸ Α παντὶ τῷ Γ καὶ τὸ Γ τινὶ τῷ Β. εἰ δὲ στερητικὸς ὁ συλλογισμός, ἡ μὲν ὑπόθεσις τὸ τινὶ τῷ Β ὑπάρχειν, εἴληπται δὲ τὸ μηδενὶ τῷ Γ καὶ τὸ Γ παντὶ τῷ Β, ὥστε γίνεται τὸ πρῶτον σχῆμα.
And this is the first figure, [namely] A belonging to all Γ and [Γ] to all B. And similarly also if it has been demonstrated to belong to some; for the assumption was that A belongs to no B, and A has been taken to belong to all Γ and Γ to some B. But if the syllogism is negative, the assumption is that [A] belongs to some B, and [A] has been taken to belong to no Γ and Γ to all B, so that the first figure is produced.
καὶ εἰ μὴ καθόλου ὁ συλλογισμός, ἀλλὰ τὸ Α τινὶ τῷ Β δέδεικται μὴ ὑπάρχειν, ὡσαύτως.
And if the syllogism is not universal, but A has been demonstrated not to belong to some B, likewise.
ὑπόθεσις μὲν γὰρ παντὶ τῷ Β τὸ ὑπάρχειν, εἴληπται δὲ τὸ μηδενὶ τῷ Γ καὶ τὸ Γ τινὶ τῷ Β· οὕτω γὰρ τὸ πρῶτον σχῆμα.
For the assumption is that [A] belongs to all B, and [A] has been taken to belong to no Γ and Γ to some B; for in this way the first figure is produced.
Πάλιν ἐν τῷ τρίτῳ σχήματι δεδείχθω τὸ Α παντὶ τῷ Β ὑπάρχειν.
Again, in the third figure, let it be demonstrated that A belongs to all B.
οὐκοῦν ἡ μὲν ὑπόθεσις ἦν μὴ παντὶ τῷ Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Γ παντὶ τῷ Β καὶ τὸ Α παντὶ τῷ Γ· οὕτω γὰρ ἔσται τὸ ἀδύνατον.
Therefore, the assumption was that A does not belong to all B, and Γ has been taken to belong to all B and A to all Γ; for in this way there will be the impossibility.
τοῦτο δὲ τὸ πρῶτον σχῆμα.
And this is the first figure.
ὡσαύτως δὲ καὶ εἰ ἐπὶ τινὸς ἡ ἀπόδειξις· ἡ μὲν γὰρ ὑπόθεσις μηδενὶ τῷ Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Γ τινὶ τῷ Β καὶ τὸ Α παντὶ τῷ Γ. εἰ δὲ στερητικὸς ὁ συλλογισμός, ὑπόθεσις μὲν τὸ Α τινὶ τῷ Β ὑπάρχειν, εἴληπται δὲ τὸ Γ τῷ μὲν Α μηδενί, τῷ δὲ Β παντί τοῦτο δὲ τὸ μέσον σχῆμα.
And likewise also if the demonstration is of a particular [proposition]; for the assumption was that A belongs to no B, and Γ has been taken to belong to some B and A to all Γ. But if the syllogism is negative, the assumption is that A belongs to some B, and Γ has been taken to belong to no A, but to all B; and this is the middle figure.
ὁμοίως δὲ καὶ εἰ μὴ καθόλου ἡ ἀπόδειξις.
And similarly also if the demonstration is not universal.
ὑπόθεσις μὲν γὰρ ἔσται παντὶ τῷ Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Γ τῷ μὲν Α μηδενί, τῷ δὲ Β τινί· τοῦτο δὲ τὸ μέσον σχῆμα.
For the assumption will be that A belongs to all B, and Γ has been taken to belong to no A, but to some B; and this is the middle figure.
Φανερὸν οὖν ὅτι διὰ τῶν αὐτῶν ὅρων καὶ δεικτικῶς ἔστι δεικνύναι τῶν προβλημάτων ἕκαστον.
Therefore, it is clear that it is possible to demonstrate each of the problems ostensively through the same terms.
ὁμοίως δʼ ἔσται καὶ δεικτικῶν ὄντων τῶν συλλογισμῶν εἰς ἀδύνατον ἀπάγειν ἐν τοῖς εἰλημμένοις ὅροις, ὅταν ἡ ἀντικειμένη πρότασις τῷ συμπεράσματι ληφθῇ.
And likewise also, when the syllogisms are ostensive, it will be possible to lead to impossibility in the terms taken, whenever the premise opposite to the conclusion is taken.
γίνονται γὰρ οἱ αὐτοὶ συλλογισμοὶ τοῖς διὰ τῆς ἀντιστροφῆς, ὥστʼ εὐθὺς ἔχομεν καὶ τὰ σχήματα διʼ ὧν ἕκαστον ἔσται.
For the same syllogisms are produced as those through conversion, so that we also immediately have the figures through which each will be.
δῆλον οὖν ὅτι πᾶν πρόβλημα δείκνυται κατʼ ἀμφοτέροος τοὺς τρόπους, διά τε τοῦ ἀδυνάτου καὶ δεικτικῶς, καὶ οὐκ ἐνδέχεται χωρίζεσθαι τὸν ἕτερον.
It is clear, therefore, that every problem is demonstrated in both ways, both through impossibility and ostensively, and it is not possible for the one to be separated from the other.