[490] Plat. Epinom. pp. 983-984.
[491] Plat. Epinom. p. 984.
[492] Plat. Epinom. pp. 984 D-985 D.
[493] Plat. Epinom. pp. 982 D, 983 C.
Improving effects of the study of Astronomy in this spirit.
Next, the Athenian intimates that the Greeks have obtained their astronomical knowledge, in the first instance, from Egypt and Assyria, but have much improved upon what they learnt (p. 987): that the Greeks at first were acquainted only with the three φοραὶ — the outer or sidereal sphere (Ἀπλανὴς), the Sun, and the Moon — but unacquainted with the other five or planetary φοραὶ, which they first learned from these foreigners, though not the names of the planets (p. 986): that all these eight were alike divine, fraternal agents, partakers in the same rational nature, and making up altogether the divine Κόσμος: that those who did not recognise all the eight as divine, consummately rational, and revolving with perfectly uniform movement, were guilty of impiety (p. 985 E): that these kosmical, divine, rational agents taught to mankind arithmetic and the art of numeration (p. 988 B): that soul, or plastic, demiurgic, cognitive force (p. 981 C), was an older and more powerful agent in the universe than body — but that there were two varieties of soul, a good and bad, of which the good variety was the stronger: the good variety of soul produced all the good movements, the bad variety produced all the bad movements (p. 988 D, E): that in studying astronomy, a man submitted himself to the teaching of this good soul and these divine agents, from whom alone he could learn true wisdom and piety (pp. 989 B-990 A): that this study, however, must be conducted not with a view to know the times of rising and setting of different stars (like Hesiod) but to be able to understand and follow the eight περιφοράς (p. 990 B).
Study of arithmetic and geometry: varieties of proportion.
To understand these — especially the five planetary and difficult περιφορὰς — arithmetic must also be taught, not in the concrete, but in the abstract (p. 990 C, D), to understand how much the real nature of things is determined by the generative powers and combination of Odd and Even Number. Next, geometry also must be studied, so as to compare numbers with plane and solid figures, and thus to determine proportions between two numbers which are not directly commensurable. The varieties of proportion, which are marvellously combined, must be understood — first arithmetical and geometrical proportions, the arithmetical proportion increasing by equal addition (1 + 1 = 2), or the point into a line — then the geometrical proportion by way of multiplication (2 × 2 = 4; 4 × 2 = 8), or the line raised into a surface, and the surface raised into a cube. Moreover there are two other varieties of proportion (τὸ ἡμιόλιον or sesquialterum, and τὸ ἐπίτριτον or sesquitertium) both of which occur in the numbers between the ratio of 6 to 12 (i.e. 9 is τὸ ἡμιόλιον of 6, or 9 = 6 + 6/2; again 8 is, τὸ ἐπίτριτον of 6, or 8 = 6 + 6/3). This last is harmonic proportion, when there are three terms, of which the third is as much greater than the middle, as the middle is greater than the first (3 : 4 : 6) — six is greater than four by one-third of six, while four is greater than three by one-third of three (p. 991 A).
When the general forms of things have thus been learnt, particular individuals in nature must be brought under them.
Lastly, having thus come to comprehend the general forms of things, we must bring under them properly the visible individuals in nature; and in this process interrogation and cross-examination must be applied (p. 991 C). We must learn to note the accurate regularity with which time brings all things to maturity, and we shall find reason to believe that all things are full of Gods (p. 991 D). We shall come to perceive that there is one law of proportion pervading every geometrical figure, every numerical series, every harmonic combination, and all the celestial rotations: one and the same bond of union among all (p. 991 E). These sciences, whether difficult or easy, must be learnt: for without them no happy nature will be ever planted in our cities (p. 992 A). The man who learns all this will be the truly wise and happy man, both in this life and after it; only a few men can possibly arrive at such happiness (p. 992 C). But it is these chosen few who, when they become Elders, will compose our Nocturnal Council, and maintain unimpaired the perpetual purity of the Platonic City.