Chapter XXVIII: 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_.

THE END.

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Transcriber’s Note

The Greek word καὶ (“and”) was twice abbreviated to a single ligature character (ϗ̀ ) (pp. xvii, xviii).

“Heraclitus” is twice spelled “Heracl_e_itus” (p. ii) / “Heracl_ei_tean” (p. vi).

At xiii.24, a quotation from Constantin Ritter has no closing quotation mark. This has been added after the first sentence, where the following spacing seems suspiciously large, indicating a possible dropped ending mark.

At 4.41, the phrase “_s. lubenter_ verò...” does not agree with the version available at the Perseus repository “_ego_ vero istinc sicut a domino agresti ac furioso profugi.” A note there offers “_libenter_ vero...” as an alternative.

At 37.22, the editor translates “ἐπ’ αὐτοφώρῳ λαβεῖν” as “to catch in the _fact_”. This is usually rendered as _act_, though either might be construed as correct.

173.37: The passage given at footnote 186 on p. 173 seems to confuse feminine and neuter forms. This is the orininal passage: προβάλλοντά τε ἀλλήλοις ἀεί, διατριβὴν τῆς πεττείας πολὺ _χαριεστέραν_ πρεσβυτῶν διατρίβοντα, φιλονικεῖν ἐν _ταῖς_ τούτων ἀξίαισι σχολαῖς.

304.35: The Myth of Er is recounted here, however, the printer included two characters (given as X here), which are blotted by hand. These were removed, and a comma has been added, viz., “a brave man, Er[XX], the son of Armenius....”

431.3: On p. 431 of the brief Appendix, the editor includes a mathematical discussion which erroneous prints the cube of 3, 4, 5, and 6 as the cube _root_ of each: ∛3 ∛4, ∛5, ∛6. They should have appeared as 3^3, 4^3, 5^3 and 6^3.

Errors deemed most likely to be the typesetter’s have been corrected, and are noted here. The references are to the page and line in the original.

vii.1 The most obvious arrangemen[t] Restored.

ix.21 who taught entirely πρ[ο/ὸ]ς δόξαν Replaced.

xi.36 τὸ αὐτὸ καθ[’] αὑτὸ Added.

xxi.17 ἐν αὑτῷ [μήδε/μηδὲ] πολυπραγμονεῖν Replaced.

xxi.8 (φρ[ονή/όνη]σις) Replaced.

xxi.31 ἡ διαλεκτικ[η/ή] Replaced.

xxi.3 ἐπίκουρον [ὄ/ὂ]ν τῷ λογιστικῷ φύσει Replaced.

xvi.26 are ever inferior to it.[’/”] Replaced.

xxiii.1 παρ[α/ά]δειγμα Replaced.

xxv.12 (ἀριστοκρ[άτι/ατί]α) Replaced.

xxv.39 liberality and magnanimity[.] Added.

10.39 ὃς ἀνθρώπους ἐκέκαστο Κλεπτοσύνῃ θ’, ὅρκ[ῴ/ῳ] Replaced.

22.37 διανοε[ι/ῖ] καταχεῖν μου τ[ω/ῶ]ν Replaced.

29.39 The term γραῶν [υ/ὕ]θλος or μῦθος Replaced.

42.34 Op. et Di. [ï/v]. 231 Replaced.

43.33 The ἀγ[υ/ύ]ρται were a species Replaced.

45.38 θεοὺς οὐδαμ[ω/ῶ]ς νομίζου Replaced.

45.40 ἀρίστων ε[ἷ/ἶ]ναι ποιητῶν Replaced.

45.41 καὶ ἀλλων ἄλλων πολλακισμυρίων Replaced.

45.43 πειρώμεθα[.] Added.

47.38 ὥς γ[’] ἐμο[ι/ὶ] δοκεῖ Added /
Replaced.

63.41 Far from it[.] Added.

64.36 εἶναι ἐν [ᾦ/ᾧ] δεῖ Replaced.

79.1 the husbandman is[ ]only a husbandman Inserted.

81.31 καὶ προ[ς/σ]ῳδίας Replaced.

90.38 Πλουτοίη δ[ε/ὲ] Μίδεω Replaced.

93.9 What are you speaking about? said he[,/.] Replaced.

163.34 what we are to say to him[./,]—are we Replaced.

167.42 as it has form, mode, and reason of Probable.
existence,[)]

168.34 νυκτ[έρι/ερί]δα ὁ εὐνοῦχος Replaced.

169.33 ἴσχουσι πλαν[ω/ώ]μενοι Replaced.

171.40 who has moderate desires[./,] Replaced.

172.38 ἄρ’ ἂν ο[ἶ/ἷ]ός τ’ εἴη ἐπιστήμης Replaced.

178.43 τοῦτο οὔτε ξυγγένεια ο[ἴ/ἵ]α τε ἐμποιεῖν Replaced.

180.11 call this wisdom and[,] apply himself Removed.

183.39 ὑπὸ δ[ε/ὲ] τῶν τεχνῶν Replaced.

183.40 καὶ ἀποτεθρυμμένοι[ διὰ] τὰς βαναυσίας Restored.
τυγχάνουσιν.

187.40 an elegant phrase equivalent to χα[ιρεῖ/ίρει]ν Replaced.

187.41 [ἑ/ἐ]ώντων Replaced.

206.38 τέχνας μ[ε/ὲ]ν μυρίας Replaced.

219.35 τῶν δ[ε/ὲ] ἀληθινῶν πολὺ Replaced.

231.31 it is now clear, p[a/e]rhaps, Replaced.

240.31 the amount of their property[./,] Replaced.

245.4 must necessarily be neglected[.] Added.

246.40 ἐν μάχῃ πλουσίῳ ἐσ[τ/κ]ιατροφήκοτι Replaced.

266.8 a[t/s] the democratic man Replaced.

266.41 πρόσωπον ἡδὺ τ[ἂ/ἀ]ν δόμοισι Replaced.

266.42 λυπηρά[τίς·/· τίς] γὰρ Misplaced
colon.

276.37 who are filling w[i]th unreal Inserted.

293.36 is rather this last[.] Added.

295.39 in our private misfortu[u/n]es Inverted.

330.40 ὁ λόγος ἡμῖν ο[ὖ/ὗ]τος Replaced.

336.30 and inter[e]fere with them unseasonably Removed.

322.35 it was said by the Syracusans to[ be] the best Probably.
governed

323.39 forth to the god[o/d]ess Replaced.

329.41 _[E/É]tudes sur le Timée_ Replaced.

344.41 κατ’ [ἔκεινα/ἐκεῖνα] γέγονε. Replaced.

356.35 φύσει[πάντι/παντὶ] κεῖται. Replaced.

356.36 [ε/ἐ]κμαγεῖον Replaced.

356.38 τρίτας [ουσίας/οὐσίας] Replaced.

357.35a πῦρ μ[ἐ/ὲ]ν ἑκάστοτε Replaced.

357.35b ὸ πεπυρωμἐνον μ[ἐ/έ]ρος Replaced.

362.39 [ὅ/ὄ]γκου μέγα ἀποτελέσειεν Replaced.

381.33 The old editions have [ἄ/ἅ]μα Replaced.

386.36 κεφ[άλη/αλὴ] Replaced.

391.39 ἕνεκ[ά/α], φησ[ι/ί] Replaced.

415.35 τὴν τοῦ θεάτρου δι[ανοί/άνοι]ν Replaced.

424.42 πρὸς [ἐ/ἑ]κατέρου τὴν χρῆσιν Replaced.