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Plato's TIMAEUS : Triangles

Timaeus 53c - 55d  * Greek Fonts

ELPENOR EDITIONS IN PRINT

Page 3

The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. And the third body is made up of 120 triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle. The one element [that is, the triangle which has its hypotenuse twice the lesser side] having generated these figures, generated no more; but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining their right angles in a centre, and forming one equilateral quadrangle. Six of these united form eight solid angles, each of which is made by the combination of three plane right angles; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases. There was yet a fifth combination which God used in the delineation of the universe.

Now, he who, duly reflecting on all this, enquires whether the worlds are to be regarded as indefinite or definite in number, will be of opinion that the notion of their indefiniteness is characteristic of a sadly indefinite and ignorant mind. He, however, who raises the question whether they are to be truly regarded as one or five, takes up a more reasonable position. Arguing from probabilities, I am of opinion that they are one; another, regarding the question from another point of view, will be of another mind.

δεύτερον δὲ ἐκ μὲν τῶν αὐτῶν τριγώνων͵ κατὰ δὲ ἰσόπλευρα τρίγωνα ὀκτὼ συστάντων͵ μίαν ἀπεργασαμένων στερεὰν γωνίαν ἐκ τεττάρων ἐπιπέδων· καὶ γενομένων ἓξ τοιούτων τὸ δεύτερον αὖ σῶμα οὕτως ἔσχεν καὶ γενομένων ἓξ τοιούτων τὸ δεύτερον αὖ σῶμα οὕτως ἔσχεν τέλος. [55b] τὸ δὲ τρίτον ἐκ δὶς ἑξήκοντα τῶν στοιχείων συμπαγέντων͵ στερεῶν δὲ γωνιῶν δώδεκα͵ ὑπὸ πέντε ἐπιπέδων τριγώνων ἰσοπλεύρων περιεχομένης ἑκάστης͵ εἴκοσι βάσεις ἔχον ἰσοπλεύρους τριγώνους γέγονεν. καὶ τὸ μὲν ἕτερον ἀπήλλακτο τῶν στοιχείων ταῦτα γεννῆσαν͵ τὸ δὲ ἰσοσκελὲς τρίγωνον ἐγέννα τὴν τοῦ τετάρτου φύσιν͵ κατὰ τέτταρα συνιστάμενον͵ εἰς τὸ κέντρον τὰς ὀρθὰς γωνίας συνάγον͵ ἓν ἰσόπλευρον τετράγωνον ἀπεργασάμενον· [55c] ἓξ δὲ τοιαῦτα συμπαγέντα γωνίας ὀκτὼ στερεὰς ἀπετέλεσεν͵ κατὰ τρεῖς ἐπιπέδους ὀρθὰς συναρμοσθείσης ἑκάστης· τὸ δὲ σχῆμα τοῦ συστάντος σώματος γέγονεν κυβικόν͵ ἓξ ἐπιπέδους τετραγώνους ἰσοπλεύρους βάσεις ἔχον. ἔτι δὲ οὔσης συστάσεως μιᾶς πέμπτης͵ ἐπὶ τὸ πᾶν ὁ θεὸς αὐτῇ κατεχρήσατο ἐκεῖνο διαζωγραφῶν.

Ἃ δή τις εἰ πάντα λογιζόμενος ἐμμελῶς ἀποροῖ πότερον ἀπείρους χρὴ κόσμους εἶναι λέγειν ἢ πέρας ἔχοντας͵ [55d] τὸ μὲν ἀπείρους ἡγήσαιτ΄ ἂν ὄντως ἀπείρου τινὸς εἶναι δόγμα ὧν ἔμπειρον χρεὼν εἶναι͵ πότερον δὲ ἕνα ἢ πέντε αὐτοὺς ἀληθείᾳ πεφυκότας λέγειν ποτὲ προσήκει͵ μᾶλλον ἂν ταύτῃ στὰς εἰκότως διαπορήσαι. τὸ μὲν οὖν δὴ παρ΄ ἡμῶν ἕνα αὐτὸν κατὰ τὸν εἰκότα λόγον πεφυκότα μηνύει θεόν͵ ἄλλος δὲ εἰς ἄλλα πῃ βλέψας ἕτερα δοξάσει.

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Reference address : https://www.ellopos.net/elpenor/physis/plato-timaeus/triangles.asp?pg=3