§3#1Ἀξιῶ γὰρ, ἔφην, ἐκμαθόντας ὅσα περὶ τῆς ἀποδεικτικῆς μεθόδου γέγραπται τοῖς πρὸ ἡμῶν, ἐπ’ ἄλλων πραγμάτων πρότερον αὐτῆς λαβεῖν πεῖραν, εἰ ὄντως εὑρίσκει τι τῶν ζητουμένων· εἶναι δὲ τῶν τοιούτων πραγμάτων φύσιν ἱκανὴν μαρτυρῆσαι τοῖς εὑροῦσιν αὐτὰ, καθάπερ ἐπὶ τῷδε, τὴν δοθεῖσαν εὐθεῖαν, ἢ τὰ ἐπιταχθέντα μέρη διελεῖν.
For I said that I expect those who have thoroughly learned all that has been written about the demonstrative method by our predecessors, to first take trial of it on other matters, to see if it truly finds any of the things being sought; and that the nature of such matters is sufficient to bear witness to those who have found them, just as in this case, to divide a given straight line, or the ordered parts.
ἴσως δ’ οὐ νοοῦσιν οἱ παντάπασιν ἀμαθεῖς, ὅ τί ποτε λέγεται.
Perhaps those who are completely ignorant do not understand what is being said.
ἐγὼ δὲ καθάπερ ὅλοις αὐτοῖς ἐπιχειρήσω σαφέστερον εἰπεῖν.
But I will attempt to speak more clearly, as if to them as a whole.
εὐθεῖαν μὲν δοθεῖσαν ὁ λόγος λέγει τὴν προτεθεῖσαν ἐπί τινος ἀκριβῶς ὁμαλοῦ ἐπιπέδου, τὰ δ’ ἐπιταχθέντα μέρη κατὰ τὴν βουλὴν τοῦ προλαβόντος· εἴ τις εἰς πέντε διελεῖν ἴσα τὴν προτεθεῖσαν εὐθεῖαν ἢ ἑπτὰ κελεύσειεν ἤτοι εἰς εἴκοσιν ἢ εἰς ἑκατόν·
Now, by "given straight line," the discourse means that which is set forth on some accurately flat plane, and the "ordered parts" according to the will of the one who prescribed it beforehand; if someone should order to divide the set-straight-line into five equal parts, or seven, or indeed twenty, or a hundred.
ἐὰν γὰρ εὕρῃς μέθοδον μίαν, ἢ παρ’ ἑτέρου διδαχθεὶς χρώμενος, εἰς ὅσα περ ἂν ἐθέλῃς μέρη τὴν προτεθεῖσαν εὐθεῖαν διορίσεις, αὐτὸ μαρτυρήσει σοι τὸ πρᾶγμα, φαίνοιτο ἅπαντα τὰ μέρη τοῖς οὕτω διαιρεθεῖσιν ἀκριβῶς ἴσα, καὶ τὰ τοιαῦτα πάντα προβλήματα βεβαίως εὑρημένα δι’ αὐτῶν τῶν ἐναργῶς φαινομένων, οἷον ὅταν περιγράψαι τῷ δοθέντι τετραγώνῳ κύκλον ἐπιταχθῶμεν· κατὰ ταῦτα δὲ καὶ τῷ δοθέντι κύκλῳ τετράγωνον περιγράψαι, ἢ ἐγγράψαι· πάλιν περὶ τὸ δοθὲν πεντάγωνον, ἰσόπλευρόν τε καὶ ἰσογώνιον, κύκλον περιγράψαι.
For if you find one method, or use it after being taught by another, you will define the set-straight-line into as many parts as you wish, and the matter itself will bear witness to you, since all the parts will appear accurately equal to those who have divided them so, and all such problems will be securely found through the very things that are clearly apparent, as when we are ordered to circumscribe a circle about a given square; and in accordance with these, also to circumscribe or inscribe a square for a given circle; again, to circumscribe a circle about a given pentagon that is equilateral and equiangular.
ἐὰν γάρ τις ἕκαστον τῶν τοιούτων δι’ ἧς ἔμαθεν μεθόδου περιγράψαι παραχρῆμα δυνηθῇ, πρὸς αὐτοῦ τοῦ πράγματος ἐμαρτυρήθη τὸ ζητούμενον εὑρηκώς.
For if anyone is able immediately to circumscribe each of such things through the method he learned, his having found the thing sought is witnessed by the matter itself.
οὐ μὴν, εἰ γέγονεν ἢ ἀγέννητος ὁ κόσμος ἐστὶ, δύναται τὸ πρᾶγμα μαρτυρῆσαι αὐτὸ, καθάπερ οὐδὲ εἰ πεπερασμένον ἢ ἄπειρον τὸ πᾶν, ἢ πόσος ὁ τῶν κυμάτων ἀριθμός· οὐδὲν γὰρ τῶν τοιούτων ζητημάτων ὑπ’ αὐτοῦ τοῦ δοκοῦντος εὑρίσκεσθαι πράγματος ἐπικρίνεται· καθάπερ ἐὰν ἐπιταχθεὶς τῷ κύκλῳ δωδεκάπλευρον, ἰσόγωνόν τε καὶ ὀρθογώνιον περιγράψαι, ἢ ἐγγράψαι, εὐθέως τοῦτο ποιήσοις.
However, whether the universe is generated or ungenerated, the matter itself cannot bear witness, just as neither whether the whole is finite or infinite, nor how great is the number of waves; for none of such investigations is judged by the very matter that seems to be found; whereas, if ordered to circumscribe or inscribe a dodecagon, equiangular and right-angled, for a circle, you would immediately do this.
καὶ γὰρ ἐγγραφόμενόν τε καὶ περιγραφόμενον ἐναργῶς ὁρᾶται, ὥσπερ γε καὶ ὁ κύκλος ἐγγραφόμενός τε καὶ περιγραφόμενος τῷ τοιούτῳ πολυγώνῳ.
For both what is inscribed and what is circumscribed are clearly seen, just as the circle itself is seen inscribed and circumscribed in such a polygon.
ὅταν γ’ οὖν εὕρωμεν ἀποδεικτικὴν μέθοδον, ἐπὶ τὸ ζητούμενον ἡμᾶς ἄγει, μαρτυρεῖταί τε πρὸς αὐτοῦ τοῦ πράγματος ἐναργῶς, καὶ βασάνους μικρὰς ἐκ τοῦδε τῆς κατ’ αὐτῆς ἀληθείας ἕξομεν, ὡς τολμήσασθαί ποτε χρήσασθαι κᾀπὶ τῶν δυναμένων αὐτὸ μαρτυρεῖν ἐναργῶς.
Therefore, when we find a demonstrative method, it leads us to the thing sought, and is clearly witnessed by the matter itself, and we shall have from this small tests of the truth concerning it, so as to dare at some time to use it also upon things that can witness it clearly.
οὐκοῦν ἀξιῶ τινα περὶ τῶν μεγίστων πραγμάτων εὐθέως ἀποκινδυνεύειν· εἰ δὲ δὴ μέγιστά ἐστι τὰ πρὸς εὐδαιμονίαν ἄγοντα, προγυμνάσασθαί γε πρότερον ἐν ὕλαις πραγμάτων, ἐλεγχούσαις μὲν τὸ ψεῦδος οἰόμενον εὑρηκέναι τὰ ζητούμενα, μαρτυρούσαις δὲ τοῖς ἀληθῶς εὑρηκόσιν.
Therefore, I do not expect anyone to immediately risk danger concerning the greatest matters; but if indeed the greatest are those leading to happiness, I expect them to have previously trained themselves in the materials of matters that refute the falsehood of those who think they have found the things sought, and witness to those who have truly found them.
ἡ τοιαύτη δὲ μέθοδος ἡ κατὰ γεωμετρίαν ἐστὶν, ἀριθμητικήν τε καὶ λογιστικὴν, ἀστρονομίαν τε καὶ ἀρχιτεκτονίαν· ἐν γὰρ ὀνόματι τῷ τῆς ἀρχιτεκτονίας ὑπογράφω καταγραφὰς ὡρολογιῶν, καὶ κλεψυδρῶν, ὑδροσκοπιῶν τε καὶ μηχανημάτων ἁπάντων, ἐν οἷς ἐστι καὶ τὰ πνευματικὰ προσαγορευόμενα.
Such a method is that in geometry, arithmetic, calculation, astronomy, and architecture; for under the name of architecture I sketch out the designs of sundials, water-clocks, hydroscopes, and all machines, among which are also those called pneumatic.
μαρτυρεῖ γοῦν ἅπαντα ταῦτα τοῖς εὑρίσκουσιν αὐτὰ, καθάπερ καὶ τὴν ἀστρονομίαν, καὶ γὰρ ταῦτα πρὸς αὐτῶν τῶν ἐναργῶς φαινομένων ἐπικρίνεται, βασανίζεταί τε καὶ μαρτυρεῖται, εἴ γε δὴ τῶν ἐναργῶς φαινομένων ἐστὶν ἔκλειψις ἡλίου καὶ σελήνης, ὅσα τε κατὰ τοὺς ἀπλανεῖς καὶ πλανήτας ἀστέρας ὁρᾶται.
At any rate, all these bear witness to those who find them, just as astronomy also does; for these too are judged, tested, and witnessed by the very things that are clearly apparent, if indeed among those things clearly apparent are the eclipse of the sun and the moon, and all that is seen concerning the fixed stars and planets.
ὅτι δὲ δάκνονται τὴν ψυχὴν ἅπαντες ἐπὶ τοῖς λόγοις τούτοις, ὅσοι προβεβηκότες ἤδη κατὰ τὴν ἡλικίαν οὐκ ἔτι καιρὸν ἔχουσι χρόνῳ πολλῷ βασανίσαι τὴν ἀποδεικτικὴν μέθοδον ἐπὶ πραγμάτων ἐναργῶς αὐτὴν μαρτυρῆσαι δυναμένων· ἤτοι γὰρ, ὡς ἔφην, διὰ φιλαυτίαν, ἢ διὰ δοξοσοφίαν, ἢ φιλοτιμίαν, ἢ φιλοδοξίαν, ἢ ἀλαζονείαν, ἢ χρηματισμὸν, ἔνιοι μὲν ἑαυτοὺς, ἔνιοι δ’ ἄλλους ἀναπείθουσιν ἐπίστασθαί τι βέβαιον.
But all those who are already advanced in age and no longer have the opportunity over a long time to test the demonstrative method upon matters capable of witnessing it clearly, are bitten in their soul by these words; for either, as I said, through self-love, or conceit, or love of honor, or love of glory, or boastfulness, or love of gain, some persuade themselves, and others persuade others, that they know something secure.
οὐδὲν οὖν ἐστι θαυμαστὸν, ἐν ταῖς ἰδίαις ἕκαστον διατριβαῖς τοὺς φοιτητὰς ἀναπείθειν, ἐνίους μὲν ὀνώδεις φύσει, τινὰς.
It is therefore nothing wonderful that in their own schools each persuades his pupils, some of whom are naturally like asses, and some...