Note 431.3 — APPENDIX.

The difficulty of the passage here alluded to, which has baffled the efforts of every critic, from Proclus and Iamblichus downwards, gave rise in ancient times to the trite proverb—Numeris Platonicis nihil obscurius.—nor has the ingenuity of modern speculators thrown much light on its obscurity. Baroccius, however, a Venetian nobleman, in 1566, gave to the world with more pretence than merit, a wordy dissertation professing to solve this knotty problem. The words of Plato are—ἔστι δὲ θείῳ μὲν γεννητῷ περίοδος, ἣν ἀριθμὸς περιλαμβάνει τέλειος, ἀνθρωπείῳ δὲ ἐν ᾧ πρώτῳ αὐξήσεις δυνάμεναί τε καὶ δυναστευόμεναι τρεῖς ἀποστάσεις, τέτταρας δὲ ὅρους λαβοῦσαι ὁμοιούντων τε καὶ ἀνομοιούντων καὶ αὐξόντων καὶ φθινόντων, πάντα προσήγορα καὶ ῥητὰ πρὸς ἄλληλα ἀπέφηναν· ὧν ἐπίτριτος πυθμὴν πεμπάδι συζυγεὶς δύο ἁρμονίας παρέχεται τρὶς αὐξηθείς, τὴν μὲν ἴσην ἰσάκις, ἑκατὸν τοσαυτάκις, τὴν δὲ ἰσομήκη μὲν, τῇ προμήκει δέ, ἑκατὸν μὲν ἀριθμῶν ἀπὸ διαμέτρων ῥητῶν πεμπάδος δεομένων ἑνὸς ἑκάστων, ἀῤῥητῶν δὲ δυεῖν, ἑκατὸν δὲ κύβων τριάδος· ξύμπας δὲ οὗτος, ἀριθμὸς γεωμετρικός, τοιούτου κύριος, ἀμεινόνων τε καὶ χειρόνων γενέσεων, &c. With reference to the former part, he says that Plato is alluding to ratios of greater or less inequality,—to which respectively belong multiples and submultiples, superparticulars and subsuperparticulars, &c., following the language of the schoolmen. Then again, as to the term ἐπίτριτος πυθμήν which had been variously rendered by the Latin radix, propago, fundus, and solidum, he conceives it to be a root or prime number, and fixes on the number seven as its representative (4 + 3 = 7). Further, by the two harmonics, he supposes Plato to mean the connexion between the square and the cube of TWELVE, which is itself made up of seven and five (ἐπίτριτος πυθμὴν πεμπάδι συζυγείς),—i. e. 144 and 1728,—twelve being a number often mentioned by Plato, as not only perfect, but the type of perfection,—the dodecahedron having given origin to the sphere, and being the type of the world,—as he has expressly stated in the Timæus. If this be granted, the 431geometric or mathematical—perfect or fatal—number may be considered to be the cube of 12 = 1728. This opinion, however, is strongly impugned by Schneider, who conceives it to be 216, i. e. ∛6,—which is made up of three other cubes; 27, 64, and 125, i. e. ∛3, ∛4, and ∛5. Others again have considered the root to be 9,—the cube-root of 729. So much for the ἐπίτριτος πυθμήν.