§1.2.5#1Τὸ δὲ κύκλῳ καὶ ἐξ ἀλλήλων δείκνυσθαί ἐστι τὸ διὰ τοῦ συμπεράσματος καὶ τοῦ ἀνάπαλιν τῇ κατηγορίᾳ τὴν ἑτέραν λαβόντα πρότασιν συμπεράνασθαι τὴν λοιπήν, ἣν ἐλάμβανεν ἐν θατέρῳ συλλογισμῷ.
To prove circularly and reciprocally is, by means of the conclusion and by taking one of the premises with its attribution reversed, to conclude the remaining premise which one assumed in the other syllogism.
οἷον εἰ ἔδει δεῖξαι ὅτι τὸ τῷ Γ παντὶ ὑπάρχει, ἔδειξε δὲ διὰ τοῦ Β, πάλιν εἰ δεικνύοι ὅτι τὸ τῷ Β ὑπάρχει, λαβὼν τὸ μὲν Α τῷ ὑπάρχειν τὸ δὲ Γ τῷ Β· πρότερον δʼ ἀνάπαλιν ἔλαβε τὸ Β τῳ Γ ὑπάρχον.
For instance, if it were necessary to show that A belongs to every C, and one showed it through B, and then again should show that A belongs to B, by assuming that A belongs to C and C to B; whereas previously, conversely, one assumed B belonging to C.
ἢ εἰ τὸ τῷ Γ δεῖ δείξαι ὑπάρχον, εἰ λάβοι τὸ Α κατὰ τοῦ Γ, ὃ ἦν συμπέρασμα, τὸ δὲ Β κατὰ τοῦ Α ὑπάρχειν· πρότερον δʼ ἐλήφθη ἀνάπαλιν τὸ Α κατὰ τοῦ Β. ἄλλως δʼ οὐκ ἔστιν ἐξ ἀλλήλων δεῖξαι.
Or if it is necessary to show that B belongs to C, if one should assume A of C, which was the conclusion, and B to belong to A; whereas previously, conversely, A was assumed of B. In no other way is it possible to prove reciprocally.
εἴτε γὰρ ἄλλο μέσον λήψεται, οὐ κύκλῳ· οὐδὲν γὰρ λαμβάνεται τῶν αὐτῶν· εἴτε τούτων τι, ἀνάγκη θάτερον μόνον·
For if another middle term is assumed, it will not be circular (for none of the same terms is assumed); and if one of these is assumed, it must be only one of them.
εἰ γὰρ ἄμφω, ταὐτὸν ἔσται συμπέρασμα, δεῖ δʼ ἕτερον.
For if both are assumed, the conclusion will be the same, whereas it ought to be different.
Ἐν μὲν οὖν τοῖς μὴ ἀντιστρέφουσιν ἐξ ἀναποδείκτου τῆς ἑτέρας προτάσεως γίνεται ὁ συλλογισμός· οὐ γὰρ· ἔστιν ἀποδεῖξαι διὰ τούτων τῶν ὅρων ὅτι τῷ μέσῳ τὸ τρίτον ὑπάρχει ἢ τῷ πρώτῳ τὸ μέσον.
In those terms which do not convert, then, the syllogism is produced from one of the premises being unproven; for it is not possible to prove through these terms that the third belongs to the middle term, or the middle to the first.
ἐν δὲ τοῖς ἀντιστρέφουσιν ἔστι πάντα δεικνύναι διʼ ἀλλήλων, οἷον εἰ τὸ Α καὶ τὸ καὶ τὸ Γ ἀντιστρέφουσιν ἀλλήλοις.
But in those which convert, it is possible to prove all reciprocally, as, for instance, if A, B, and C convert with one another.
δεδείχθω γὰρ τὸ Α Γ διὰ μέσου τοῦ Β, καὶ πάλιν τὸ Α διά τε τοῦ συμπεράσματος καὶ διὰ τῆς Β Γ προτάσεως ἀντιστραφείσης, ὡσαύτως δὲ καὶ τὸ Β Γ διά τε τοῦ συμπεράσματος καὶ τῆς Α Β a προτάσεως ἀντεστραμμένης.
For let A belonging to C be proved through the middle B, and then again A belonging to B through the conclusion and through the premise BC converted; and similarly also B belonging to C through the conclusion and through the premise AB converted.
δεῖ δὲ τήν τε Γ Β καὶ τὴν Β Α πρότασιν ἀποδεῖξαι· ταύταις γὰρ ἀναποδείκτοις κεχρήμεθα μόναις.
But it is necessary to prove both the premise CB and the premise BA; for we have used these alone as unproven.
ἐὰν οὖν ληφθῇ τὸ Β παντὶ τῷ Γ ὑπάρχειν καὶ τὸ Γ παντὶ τῷ Α συλλογισμὸς ἔσται τοῦ Β πρὸς τὸ Α. πάλιν ἐὰν ληφθῇ τὸ μὲν Γ παντὶ τῷ Α, τὸ δὲ παντὶ τῷ Β. παντὶ τῷ Β τὸ Γ ἀνάγκη ὑπάρχειν.
If, therefore, B should be assumed to belong to every C, and C to every A, there will be a syllogism of B with respect to A. Again, if C should be assumed to belong to every A, and A to every B, C must belong to every B.
ἐν ἀμφοτέροις δὴ τούτοις τοῖς συλλογισμοῖς ἡ Γ Α πρότασις εἴληπται ἀναπόδεικτος· αἰ γὰρ ἕτεραι δεδειγμέναι ἦσαν.
In both of these syllogisms, indeed, the premise CA has been assumed as unproven; for the others had been proved.
ὥστʼ ἂν ταύτην ἀποδείξωμεν, ἅπασαι ἔσονται δεδειγμέναι διʼ ἀλλήλων.
So that if we prove this, all will have been proved reciprocally.
ἐὰν οὖν ληφθῇ τὸ Γ παντὶ τῷ Β καὶ τὸ Β παντὶ τῷ Α ὑπάρχειν, ἀμφότεραί τε αἱ προτάσεις ἀποδεδειγμέναι λαμβάνονται, καὶ τὸ Γ τῷ Α ἀνάγκη ὑπάρχειν.
If, therefore, C should be assumed to belong to every B, and B to every A, both premises are assumed as proved, and C must belong to A.
φανερὸν οὖν ὅτι ἐν μόνοις τοῖς ἀντιστρέφουσι κύκλῳ καὶ διʼ ἀλλήλων ἐνδέχεται γίνεσθαι τὰς ἀποδείξεις, ἐν δὲ τοῖς ἄλλοις ὡς πρότερον εἴπομεν.
It is clear, then, that only in those terms which convert is it possible for proofs to be made circularly and reciprocally, but in the others as we said before.