§2.1.29Πλείους δʼ ἀποδείξεις εἶναι τοῦ αὐτοῦ ἐγχωρεῖ οὐ μόνον ἐκ τῆς αὐτῆς συστοιχίας λαμβάνοντι μὴ τὸ συνεχὲς μέσον, οἷον τῶν Β τὸ Γ καὶ Δ καὶ Ζ, ἀλλὰ καὶ ἐξ ἑτέρας.
It is possible for there to be multiple demonstrations of the same thing, not only by assuming from the same lineage a non-continuous middle term—as, for example, C, D, and Z for B—but also from a different lineage.
οἷον ἔστω τὸ Α μεταβάλλειν, τὸ δʼ ἐφʼ ᾧ Δ κινεῖσθαι, τὸ δὲ Β ἥδεσθαι, καὶ πάλιν τὸ ἠρεμίζεσθαι.
For example, let A be 'to change', D 'to be moved', B 'to enjoy oneself', and again [H] 'to become calm'.
ἀληθὲς οὖν καὶ τὸ Δ τοῦ Β καὶ τὸ τοῦ Δ κατηγορεῖν· ὁ γὰρ ἡδόμενος κινεῖται καὶ τὸ κινούμενον μεταβάλλει.
It is true, then, to predicate D of B, and [A] of D; for the one who enjoys oneself is moved, and that which is moved changes.
πάλιν τὸ τοῦ Η καὶ τὸ Η τοῦ Β ἀληθὲς κατηγορεῖν· πᾶς γὰρ ὁ ἡδόμενος ἠρεμίζεται καὶ ὁ ἠρεμιζόμενος μεταβάλλει.
Again, it is true to predicate [A] of H, and H of B; for everyone who enjoys oneself becomes calm, and the one who becomes calm changes.
ὥστε διʼ ἑτέρων μέσων καὶ οὐκ ἐκ τῆς αὐτῆς συστοιχίας ὁ συλλογισμός.
Thus, the syllogism proceeds through different middle terms and not from the same lineage.
οὐ μὴν ὥστε μηδέτερον κατὰ μηδετέρου λέγεσθαι τῶν μέσων· ἀνάγκη γὰρ τῷ αὐτῷ τινι ἄμφω ὑπάρχειν.
Yet, this is not so that neither of the middle terms is said of the other; for it is necessary for both to belong to some same thing (namely, B).
ἐπισκέψασθαι δὲ καὶ διὰ τῶν ἄλλων σχημάτων ὁσαχῶς ἐνδέχεται τοῦ αὐτοῦ γενέσθαι συλλογισμόν.
One must also investigate through the other figures in how many ways a syllogism of the same thing can come to be.
§2.1.30Τοῦ δʼ ἀπὸ τύχης οὐκ ἔστιν ἐπιστήμη διʼ ἀποδείξεως.
But there is no scientific knowledge through demonstration of what is by chance.
οὔτε γὰρ ὡς ἀναγκαῖον οὔθʼ ὡς ἐπὶ τὸ πολὺ τὸ ἀπὸ τύχης ἐστίν, ἀλλὰ τὸ παρὰ ταῦτα γινόμενον· ἡ δʼ ἀπόδειξις θατέρου τούτων.
For what is by chance is neither as necessary nor as for the most part, but is that which happens contrary to these; whereas demonstration is of one or the other of these.
πᾶς γὰρ συλλογισμὸς ἢ διʼ ἀναγκαίων ἢ διὰ τῶν ὡς ἐπὶ τὸ πολὺ προτάσεων· καὶ εἰ μὲν αἱ προτάσεις ἀναγκαῖαι, καὶ τὸ συμπέρασμα ἀναγκαῖον, εἰ δʼ ὡς ἐπὶ τὸ πολύ, καὶ τὸ συμπέρασμα τοιοῦτον.
For every syllogism is either through necessary premises or through those that are for the most part; and if the premises are necessary, the conclusion is also necessary, but if they are for the most part, the conclusion is also such.
ὥστʼ εἰ τὸ ἀπό τύχης μήθʼ ὡς ἐπὶ τὸ πολύ μήτʼ ἀναγκαῖον, οὐκ ἄν εἴη αὐτοῦ ἀπόδειξις.
So that if what is by chance is neither for the most part nor necessary, there would not be demonstration of it.
§2.1.31Οὐδὲ διʼ αἰσθήσεως ἔστιν ἐπίστασθαι.
Nor is it possible to have scientific knowledge through perception.
εἰ γὰρ καὶ ἔστιν ἡ αἴσθησις τοῦ τοιοῦδε καὶ μὴ τοῦδέ τινος, ἀλλʼ ασθάνεσθαί γε άναγκκαῖον τόδε τι καὶ πού καὶ νῦν.
For even if perception is of what is of such a character and not of a particular this, yet perceiving at least must be of a particular this, at a certain place, and now.
τὸ δὲ καθόλου καὶ ἐπὶ πᾶσιν ἀδύνατον αἰσθάνεσθαι· οὐ γὰρ τόδε οὐδὲ νῦν· οὐ γὰρ ἄν ἦν καθόλου· τὸ γὰρ ἀεὶ καὶ πανταχοῦ καθόλου φαμὲν εἶναι.
But it is impossible to perceive the universal and what is over all cases; for it is not a particular this nor now—for then it would not be universal; for we say that the universal is what is always and everywhere.
ἐπεὶ οὖν αἱ μὲν ἀποδείξεις καθόλου, ταῦτα δʼ οὐκ ἔστιν αἰσθάνεσθαι, φανερὸν ὅτι οὐδʼ ἐπίστασθαι διʼ ασθήσεως ἔστιν, ἀλλὰ δῆλον ὅτι καὶ εἰ ἦν αἰσθάνεσθαι τὸ τρίγωνον ὅτι δυσὶν ὀρθαῖς ἴσας ἔχει τὰς γωνίας, ἐζητοῦμεν ἄν ἀπόδειξιν καὶ οὐχ ὥσπερ φασί τινες ἠπιστάμεθα· αἰσθάνεσθαι μὲν γὰρ ἀνάγκη καθʼ ἕκαστον, ἡ δʼ ἐπιστήμη τὸ τὸ καθόλου γνωρίζειν ἐστίν.
Since, therefore, demonstrations are universal, and these cannot be perceived, it is clear that neither is it possible to have scientific knowledge through perception, but it is clear that even if it were possible to perceive that the triangle has its angles equal to two right angles, we would still seek a demonstration and not, as some say, already have scientific knowledge of it; for perceiving is necessarily of particulars, whereas scientific knowledge is the recognition of the universal.
διὸ καὶ εἰ ἐπὶ τῆς σελήντης ὄντες ἑωρῶμεν ἀντιφράττουσαν τὴν γῆν, οὐκ ἄν ᾔδειμεν τὴν αἰτίαν τῆς ἐκλείψεως.
Therefore, even if we were on the moon and saw the earth obstructing [the sun], we would not know the cause of the eclipse.
ᾔσθανόμεθα γὰρ ἄν ὅτι νῦν ἐκλείπει, καὶ οὐ διότι ὅλως· οὐ γὰρ ἦν τοῦ καθόλου αἴσθησις.
For we would perceive that it is now undergoing an eclipse, but not why it is so overall; for there was no perception of the universal.
οὐ μὴν ἀλλʼ ἐκ τοῦ θεωρεῖν τοῦτο πολλάκις συμβαῖνον τὸ καθόλου ἄν θηρεύσαντες ἀπόδειξιν εἶχομεν· ἐκ γὰρ τῶν καθʼ ἕκαστα πλεῖὁνων τὸ καθόλου δῆλον.
Nevertheless, from observing this happening many times, by hunting for the universal we would have a demonstration; for from many particulars the universal becomes clear.
τὸ δὲ καθόλου τμιον, ὅτι δηλοῖ τὸ αἴτιον· ὥστε περὶ τῶν τοιούτων ἡ καθόλου τιμιωτέρα τῶν αἰσθήσεων καὶ τῆς νοήσεως, ὅσων ἕτερον τὸ αἴτιον· περὶ δὲ τῶν πρώτων ἄλγλος λόγος.
And the universal is valuable because it reveals the cause; so that concerning such things (where the cause is other than themselves) the universal [cognition] is more valuable than perceptions and intuition; but concerning the primary things there is another account.
Φανερὸν οὖν ὅτι ἀδύνατον τῷ αἰσθάνεσθαι ἐπίστασθαί τι τῶν ἀποδεικτῶν, εἰ μή τις τὸ αἰσθάνεσθαι τοῦτο λέγει, τό ἐπιστήμην ἔχειν διʼ ἀποδείξεως.
It is clear, then, that it is impossible by perceiving to have scientific knowledge of any of the demonstrable things, unless one calls this 'perceiving': namely, having scientific knowledge through demonstration.
ἔστι μέντοι ἕνια ἀναγόμενα εἰς αἰσθήσεως ἔκλειψιν ἐν τοῖς προβλήμασιν.
There are, however, some problems that are referred to a failure of perception.
ἕνια γὰρ εἰ ἑωρῶμεν οὐκ ἄν ἐζητοῦμεν, οὐχ ὡς εἰδότες τῷ ὁρᾶν, ἀλλʼ ὡς ἔχοντες τὸ καθόλου ἐκ τοῦ ὁοᾶν.
For some things we would not seek if we saw them, not as knowing by seeing, but as having the universal from seeing.
οἶον εἰ τὴν ὕαλον τετρυπημέντην ἑωρῶμεν καὶ τὸ φῶς διιόν, δῆλον ἄν ἦν καὶ διὰ τί καίει, τῷ ὁρᾶν μὲν χωρὶς ἐφʼ ἑκάστης, νοῆσαι δʼ ἄμα ὅτι ἐπὶ πασῶν οὕτως.
For example, if we saw the glass to be perforated and the light passing through it, it would also be clear why it burns, by seeing individually in each case, but simultaneously understanding that it is thus in all cases.