Dialogues

raising the power and extracting the root (as in the translation). Numbers are called “like and unlike” (ὁμοιοῦντές τε καὶ ἀνομοιοῦντες) when the factors or the sides of the planes and cubes which they represent are or are not in the same ratio: e.g. 8 and 27 = 2³ and 3³; and conversely. “Waxing” (αὔξοντες) numbers, called also “increasing” (ὑπερτελεῖς), are those which are exceeded by the sum of their divisors: e.g. 12 and 18 are less than 16 and 21. “Waning” (φθίνοντες) numbers, called also “decreasing” (ἐλλιπεῖς) are those which succeed the sum of their divisors: e.g. 8 and 27 exceed 7 and 13. The words translated “commensurable and agreeable to one another” (προσήγορα καὶ ῥητά) seem to be different ways of describing the same relation, with more or less precision. They are equivalent to “expressible in terms having the same relation to one another,” like the series 8, 12, 18, 27, each of which numbers is in the relation of 1¹⁄₂ to the preceding. The “base,” or “fundamental number, which has ⅓ added to it” (1⅓) = ⁴⁄₃ or a musical fourth. Ἁρμονία is a “proportion” of numbers as of musical notes, applied either to the parts or factors of a single number or to the relation of one number to another. The first harmony is a “square” number (ἴσην ἰσάκις); the second harmony is an “oblong” number (προμήκη), i.e. a number representing a figure of which the opposite sides only are equal. Ἀριθμοὶ ἀπὸ διαμέτρων = “numbers squared from” or “upon diameters”; ῥητῶν = “rational,” i.e. omitting fractions, ἀῤῥήτων, “irrational,” i.e. including fractions; e.g. 49 is a square of the rational diameter of a figure the side of which = 5 ∶ 50, of an irrational diameter of the same. For several of the explanations here given and for a good deal besides I am indebted to an excellent article on the Platonic Number by Dr. Donaldson (Proceedings of the Philological Society, vol. I p. 81 and following).

The conclusions which he draws from these data are summed up by him as follows. Having assumed that the number of the perfect or divine cycle is the number of the world, and the number of the imperfect cycle the number of the state, he proceeds: “The period of the world is defined by the perfect number 6, that of the state by the cube of that number or 216, which is the product of the last pair of terms in the Platonic Tetractys;140 and if we take this as the basis of our computation, we shall have two cube numbers (αὐξήσεις δυνάμεναί τε καὶ δυναστευόμεναι), viz. 8 and 27; and the mean proportionals between these, viz. 12 and 18, will furnish three intervals and four terms, and these terms and intervals stand related to one another in the sesqui-altera ratio, i.e. each term is to the preceding as ³⁄₂. Now if we remember that the number 216 = 8 × 27 = 3³ + 4³ + 5³, and 3² + 4² = 5², we must admit that this number implies the numbers 3, 4, 5, to which musicians attach so much importance. And if we combine the ratio ⁴⁄₃ with the number 5, or multiply the ratios of the sides by the hypotenuse, we shall by first squaring and then cubing obtain two expressions, which denote the ratio of the two last pairs of terms in the Platonic Tetractys, the former multiplied by the square, the latter by the cube of the number 10, the sum of the first four digits which constitute the Platonic Tetractys.” The two ἁρμονίαι he elsewhere explains as follows: “The first ἁρμονία is ἴσην ἰσάκις ἑκατὸν τοσαυτάκις, in other words (⁴⁄₃ × 5)² = 100 × ²²⁄_3². The second ἁρμονία, a cube of the same root, is described as 100 multiplied (α) by the rational diameter of 5 diminished by unity, i.e., as shown above, 48: (β) by two incommensurable diameters, i.e. the two first irrationals, or 2 and 3: and (γ) by the cube of 3, or 27. Thus we have (48 + 5 + 27) 100 = 1000 × 2³. This second harmony is to be the cube of the number of which the former harmony is the square, and therefore must be divided by the cube of 3. In other words, the whole expression will be: (1), for the first harmony,⁴⁰⁰⁄₉: (2), for the second harmony,⁸⁰⁰⁰⁄₂₇.”

The reasons which have inclined me to agree with Dr. Donaldson and also with Schleiermacher in supposing that 216 is the Platonic number of births are: (1) that it coincides with the description of the number given in the first part of the passage (ἐν ᾧ πρώτῳ⁠ ⁠… ἀπέφησαν): (2) that the number 216 with its permutations would have been familiar to a Greek mathematician, though unfamiliar to us: (3) that 216 is the cube of 6, and also the sum of 3³, 4³, 5³, the numbers 3, 4, 5 representing the Pythagorean triangle, of which the sides when squared equal the square of the hypotenuse (3² + 4² = 5²): (4) that it is also the period of the Pythagorean Metempsychosis: (5) the three ultimate terms or bases (3, 4, 5) of which 216 is composed answer to the third, fourth, fifth in the musical scale: (6) that the number 216 is the product of the cubes of 2 and 3, which are the two last terms in the Platonic Tetractys: (7) that the Pythagorean triangle is said by Plutarch (de Iside et Osiride, 373 E), Proclus (super prima Euclid IV p. 111), and Quintilian (De musica III p. 152) to be contained in this passage, so that the tradition of the school seems to point