The works of Plato · Plato
APPENDIX.
(_See page 235._) 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 geometric 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. As to the elucidation, however, of the principal difficulties herein involved, we are still as far afield as ever;—nor has the united learning of Schneider, Trendelenburg, Böckh, Schleiermacher, or Cousin, removed any obstacle to their successful solution. To state their theories at length would take up much space without any adequate benefit;—and this must be our apology for substituting so short a notice for what we intended to be a lengthened statement. Many details however on this _vexata quæstio_ may be gathered by the patient reader from Schneider’s preface to vol. iii. of his edition of Plato,—Cousin’s note in his translation _ad locum_,—and his translation of Proclus, with notes _ad locum_. It must be confessed, however, that the question still remains open and unsolved—still, _dignus vindice nodus_.
Text from Project Gutenberg, public domain in the United States.