N the first place, then, as is evident to all, fire and earth and water and air are bodies. And every sort of body possesses solidity, and every solid must necessarily be contained in planes; and every plane rectilinear figure is composed of triangles; and all triangles are originally of two kinds, both of which are made up of one right and two acute angles; one of them has at either end of the base the half of a divided right angle, having equal sides, while in the other the right angle is divided into unequal parts, having unequal sides. These, then, proceeding by a combination of probability with demonstration, we assume to be the original elements of fire and the other bodies; but the principles which are prior to these God only knows, and he of men who is the friend God. And next we have to determine what are the four most beautiful bodies which are unlike one another, and of which some are capable of resolution into one another; for having discovered thus much, we shall know the true origin of earth and fire and of the proportionate and intermediate elements. And then we shall not be willing to allow that there are any distinct kinds of visible bodies fairer than these. Wherefore we must endeavour to construct the four forms of bodies which excel in beauty, and then we shall be able to say that we have sufficiently apprehended their nature. Now of the two triangles, the isosceles has one form only; the scalene or unequal-sided has an infinite number. Of the infinite forms we must select the most beautiful, if we are to proceed in due order, and any one who can point out a more beautiful form than ours for the construction of these bodies, shall carry off the palm, not as an enemy, but as a friend. Now, the one which we maintain to be the most beautiful of all the many triangles (and we need not speak of the others) is that of which the double forms a third triangle which is equilateral; the reason of this would be long to tell; he who disproves what we are saying, and shows that we are mistaken, may claim a friendly victory.

τὰ τρίγωνα

Πρῶτον μὲν δὴ πῦρ καὶ γῆ καὶ ὕδωρ καὶ ἀὴρ ὅτι σώματά ἐστι͵ δῆλόν που καὶ παντί· τὸ δὲ τοῦ σώματος εἶδος πᾶν καὶ βάθος ἔχει. τὸ δὲ βάθος αὖ πᾶσα ἀνάγκη τὴν ἐπίπεδον περιειληφέναι φύσιν· ἡ δὲ ὀρθὴ τῆς ἐπιπέδου βάσεως ἐκ τριγώνων συνέστηκεν. [53d] τὰ δὲ τρίγωνα πάντα ἐκ δυοῖν ἄρχεται τριγώνοιν͵ μίαν μὲν ὀρθὴν ἔχοντος ἑκατέρου γωνίαν͵ τὰς δὲ ὀξείας· ὧν τὸ μὲν ἕτερον ἑκατέρωθεν ἔχει μέρος γωνίας ὀρθῆς πλευραῖς ἴσαις διῃρημένης͵ τὸ δ΄ ἕτερον ἀνίσοις ἄνισα μέρη νενεμημένης. ταύτην δὴ πυρὸς ἀρχὴν καὶ τῶν ἄλλων σωμάτων ὑποτιθέμεθα κατὰ τὸν μετ΄ ἀνάγκης εἰκότα λόγον πορευόμενοι· τὰς δ΄ ἔτι τούτων ἀρχὰς ἄνωθεν θεὸς οἶδεν καὶ ἀνδρῶν ὃς ἂν ἐκείνῳ φίλος ᾖ. [53e] δεῖ δὴ λέγειν ποῖα κάλλιστα σώματα γένοιτ΄ ἂν τέτταρα͵ ἀνόμοια μὲν ἑαυτοῖς͵ δυνατὰ δὲ ἐξ ἀλλήλων αὐτῶν ἄττα διαλυόμενα γίγνεσθαι· τούτου γὰρ τυχόντες ἔχομεν τὴν ἀλήθειαν γενέσεως πέρι γῆς τε καὶ πυρὸς τῶν τε ἀνὰ λόγον ἐν μέσῳ. τόδε γὰρ οὐδενὶ συγχωρησόμεθα͵ καλλίω τούτων ὁρώμενα σώματα εἶναί που καθ΄ ἓν γένος ἕκαστον ὄν. τοῦτ΄ οὖν προθυμητέον͵ τὰ διαφέροντα κάλλει σωμάτων τέτταρα γένη συναρμόσασθαι καὶ φάναι τὴν τούτων ἡμᾶς φύσιν ἱκανῶς εἰληφέναι. [54a] τοῖν δὴ δυοῖν τριγώνοιν τὸ μὲν ἰσοσκελὲς μίαν εἴληχεν φύσιν͵ τὸ δὲ πρόμηκες ἀπεράντους· προαιρετέον οὖν αὖ τῶν ἀπείρων τὸ κάλλιστον͵ εἰ μέλλομεν ἄρξεσθαι κατὰ τρόπον. ἂν οὖν τις ἔχῃ κάλλιον ἐκλεξάμενος εἰπεῖν εἰς τὴν τούτων σύστασιν͵ ἐκεῖνος οὐκ ἐχθρὸς ὢν ἀλλὰ φίλος κρατεῖ· τιθέμεθα δ΄ οὖν τῶν πολλῶν τριγώνων κάλλιστον ἕν͵ ὑπερβάντες τἆλλα͵ ἐξ οὗ τὸ ἰσόπλευρον τρίγωνον ἐκ τρίτου συνέστηκεν. [54b] διότι δέ͵ λόγος πλείων· ἀλλὰ τῷ τοῦτο ἐλέγξαντι καὶ ἀνευρόντι δὴ οὕτως ἔχον κεῖται φίλια τὰ ἆθλα.