§2.1.5Δεῖ δὲ μὴ λανθάνειν ὅτι πολλάκις συμβαίνει διαμαρτάνειν καὶ μὴ ὑπάρχειν τὸ δεικνύμενον πρῶτον καθόλου, ᾗ δοκεῖ δείκνυσθαι καθόλου πρῶτον.
But we must not fail to notice that it often happens that an error occurs and that which is demonstrated does not belong as primary and universal, in the way in which it seems to be demonstrated as universal and primary.
ἀπατώμεθα δὲ ταύτην τὴν ἀπάτην, ὅταν ἢ μηδὲν ᾖ λαβεῖν ἀνώτερον παρὰ τὸ καθʼ ἕκαστον, ἢ ᾖ μέν, ἀλλʼ ἀνώνυμον ᾖ ἐπὶ διαφόροις εἴδει πράγμασιν, ἢ τυγχάνῃ ὂν ὡς ἐν μέρει ὅλον ἐφʼ ᾧ δείκνυται·
We are deceived by this deception when either we cannot grasp anything higher besides the particulars, or there is something but it is nameless for things that differ in species, or that of which it is demonstrated happens to be a whole which is as a part.
τοῖς γὰρ ἐν μέρει ὑπάρξει μὲν ἡ ἀπόδειξις, καὶ ἔσται κατὰ παντός, ἀλλʼ ὅμως οὐκ ἔσται τούτου πρώτου καθόλου ἡ ἀπόδειξις.
For the demonstration will indeed belong to the parts, and will be of every, but nevertheless the demonstration will not be of this as primary and universal.
λέγω δὲ τούτου πρώτου, ᾗ τοῦτο, ἀπόδειξιν, ὅταν ᾖ πρώτου καθόλου.
And I call a demonstration of this as primary, insofar as it is this, when it is of a primary universal.
εἰ οὖν τις δείξειεν ὅτι αἱ ὀρθαὶ οὐ συμπίπτουσι, δόξειεν ἂν τούτου εἶναι ἡ ἀπόδειξις διὰ τὸ ἐπὶ πασῶν εἶναι τῶν ὀρθῶν.
If, then, someone were to demonstrate that perpendicular lines do not meet, the demonstration might seem to be of this, because it is true of all perpendicular lines.
οὐκ ἔστι δέ, εἴπερ μὴ ὅτι ὡδὶ ἴσαι γίνεται τοῦτο, ἀλλʼ ᾗ ὁπωσοῦν ἴσαι.
But it is not, if indeed this does not happen because they are equal in this way, but insofar as they are equal in any way whatsoever.
καὶ εἰ τρίγωνον μὴ ἦν ἄλλο ἢ ἰσοσκελές, ᾗ ἰσοσκελὲς ἂν ἐδόκει ὑπάρχειν.
And if there were no other triangle than the isosceles, it would seem to belong to it insofar as it is isosceles.
καὶ τὸ ἀνάλογον ὅτι καὶ ἐναλλάξ, ᾗ ἀριθμοὶ καὶ ᾗ γραμμαὶ καὶ ᾗ στερεὰ καὶ ᾗ χρόνοι, ὥσπερ ἐδείκνυτό ποτε χωρίς, ἐνδεχόμενόν γε κατὰ πάντων μιᾷ ἀποδείξει δειχθῆναι· ἀλλὰ διὰ τὸ μὴ εἶναι ὠνομασμένον τι ταῦτα πάντα ἓν, ἀριθμοί μήκη χρόνοι στερεά, καὶ εἴδει διαφέρειν ἀλλήλων, χωρὶς ἐλαμβάνετο.
And also that proportionals alternate, insofar as they are numbers and lines and solids and times, just as it used to be demonstrated separately, though it was possible to demonstrate it of all of them by a single demonstration; but because there was no single named thing that was all of these—numbers, lengths, times, solids—and because they differ in species from one another, they were taken separately.
νῦν δὲ καθόλου δείκνυται· οὐ γὰρ ᾗ γραμμαὶ ἢ ᾗ ἀριθμοὶ ὑπῆρχεν, ἀλλʼ ᾗ τοδί, ὃ καθόλου ὑποτίθενται ὑπάρχειν.
But now it is demonstrated universally; for it did not belong to them insofar as they are lines or insofar as they are numbers, but insofar as they are this thing which they assume to belong universally.
διὰ τοῦτο οὐδʼ ἄν τις δείξῃ καθʼ ἕκαστον τὸ τρίγωνον ἀποδείξει ἢ μιᾷ ἢ ἑτέρᾳ ὅτι δύο ὀρθὰς ἔχει ἕκαστον, τὸ ἰσόπλευρον χωρὶς καὶ τὸ σκαληνὲς καὶ τὸ ἰσοσκελές, οὔπω οἶδε τὸ τρίγωνον ὅτι δύο ὀρθαῖς, εἰ μὴ τὸν σοφιστικὸν τρόπον, οὐδὲ καθʼ ὅλου τριγώνου, οὐδʼ εἰ μηδὲν ἔστι παρὰ ταῦτα τρίγωνον ἕτερον.
For this reason, even if someone should demonstrate of each triangle, either by one demonstration or by different ones, that each has two right angles—namely, the equilateral, the scalene, and the isosceles separately—he does not yet know of the triangle that it has two right angles, except in a sophistical way, nor does he know it of the universal triangle, even if there is no other triangle besides these.
οὐ γὰρ ᾗ τρίγωνον οἶδεν, οὐδὲ πᾶν τρίγωνον, ἀλλʼ ἢ κατʼ ἀριθμόν· κατʼ εἶδος δʼ οὐ πᾶν, καὶ εἰ μηδὲν ἔστιν ὅ οὐκ οἶδεν.
For he does not know it insofar as it is a triangle, nor of every triangle except in number; but in species he does not know it of every triangle, even if there is nothing of which he does not know.
Πότ᾿ οὖν οὐκ οἶδε καθόλου, καὶ πότʼ οἶδεν ἀπλῶς;
When, then, does he not know universally, and when does he know unqualifiedly?
δῆλον δὴ ὅτι εἰ ταὐτὸν ἦν τριγώνῳ εἶναι καὶ ἰσοπλεύρῳ ἢ ἑκάστῳ ἢ πᾶσιν.
Clearly, if to be a triangle were the same as to be an equilateral triangle, either each of them or all of them.
εἰ δὲ μὴ ταὐτὸν ἀλλʼ ἕτερον, ὑπάρχει δʼ ᾗ τρίγωνον, οὐκ οἶδεν.
But if they are not the same but different, and the attribute belongs to it insofar as it is a triangle, he does not know it.
πότερον δʼ ᾗ τρίγωνον ἢ ᾗ ἰσοσκελὲς ὑπάρχει;
But does it belong to it insofar as it is a triangle or insofar as it is isosceles?
καὶ πότε κατὰ τοῦθʼ ὑπάρχει πρῶτον;
And when does it belong to this primarily?
καὶ καθόλου τίνος ἡ ἀπόδειξις;
And of what is the demonstration universal?
δῆλον ὅτι ὅταν ἀφαιρουμένων ὑπάρχῃ πρώτῳ.
Clearly when, after abstraction, it belongs to the first thing.
οἷον τῷ ἰσοσκελεῖ χαλκῷ τριγώνῳ ὑπάρξουσι δύο ὀρθαί, ἀλλὰ καὶ τοῦ χαλκοῦν εἶναι ἀφαιρεθέντος καὶ τοῦ ἰσοσκελές.
For example, to the bronze isosceles triangle there will belong two right angles, but also when its being bronze is removed and when its being isosceles is removed.
ἀλλʼ οὐ τοῦ σχήματος ἢ πέρατος.
But not when figure or limit is removed.
ἀλλʼ οὐ πρώτων.
But these are not the first.
τίνος οὖν πρώτου;
Of what first thing, then?
εἰ δὴ τριγώνου, κατὰ τοῦτο ὑπάρχει καὶ τοῖς ἄλλοις, καὶ τούτου καθόλου ἐστὶν ἡ ἀπόδειξις.
If, indeed, of triangle, it belongs to the others in virtue of this, and the demonstration is universal of this.