In the Pythagorean triangle was employed to construct the period of gestation for the microcosm or man: here it is used to construct two periods in the lifetime of the macrocosm or Universe, for that is what Plato means by the two ‘harmonies.’ The translation is: ‘of which, 4, 3 married with 5, yields two harmonies when thrice increased, the one equal an equal number of times, so many times 100, the other of equal length one way, but oblong:—on the one side, of 100 squares of rational diameters of five diminished by one each, or if of irrational diameters, by two: on the other of one hundred cubes of three.’ The antecedent of ὧν is , which I have already interpreted as the numbers 3, 4, 5. Of these numbers (ὧν) the i.e. 3, 4 (cf. Theo Smyrn. p. 80 ed. Hiller, Proclus l. c. II p. 37 ), is ‘married’ or ‘coupled’ with 5. That is to say, 3, 4, and 5 are multiplied together: whence we get 3 x 4 x 5=60. ‘Thrice increased’ is ‘three times multiplied by itself’; and 60 thrice increased is therefore 60 x 60 x 60 x 60. This sum, which is 12,960,000, yields two harmonies. One of the two harmonies is ‘equal an equal number of times, viz. so many times 100,’ in other words, it is a square (cf. Theaet. 147 E) each of whose sides is a certain number of times 100 (for τοσαυτάκις cf. τοσοῦτον in Alc. I 108 E), viz. of course 36 times 100, for 60 x 60 x 60 x 60=3600^{2}. See Fig. 4. The other harmony which 60 x 60 x 60 x 60 yields is a rectangle (with προμήκη cf. Theaet. 148 A), one of whose sides is one hundred cubes of 3, i.e. 2700, and the other the number which Plato describes in . What is that number? means (numerical) ‘squares of’ (cf. Procl. l. c. II p. 38. 9 et al.): the side in question is therefore ‘100 squares of’—what? Of the rational diameter of 5 etc. Now the ‘rational diameter of 5’ is the nearest rational number to the real diameter of a square whose side is 5 (Theo l. c. pp. 43 ff. and other authorities). The real diameter of a square whose side is 5 is √50. See Fig. 5. AC^{2}=5^{2} + 5^{2}=50 (by Pythagoras' famous εὕρημα Eucl. I 47): therefore AC=√50. And the nearest rational number to √50 is 7: for √49=7. Consequently 7 is ‘the rational diameter’ of 5. And 100 squares of 7=100 x 49=4900. But we are told to diminish the 100 squares by 1 each. Do so: 4900 - (1 x 100)=4800. This side is therefore 4800. The words give us an alternative way of reaching the number 4800. The construction is = (or of 100) ‘squares of irrational diameters of 5, wanting 2 each.’ Now the irrational diameter of 5 is √50. Square this and it becomes 50. 100 squares of 50= 5000. Subtract 2 from each square and you have 5000 - (2 x 100)=4800. The two sides of the oblong are therefore 4800 and 2700 (‘one hundred cubes of three’). The area is 4800 x 2700= 12,960,000 which is 60 x 60 x 60 x 60. See Fig. 6. Thus the arithmetical meaning of this part of Plato's Number may be expressed by us as follows:

(3 x 4 x 5)^{4}=3600^{2}=4800 x 2700.

In this explanation, which is defended at length in App. I, Pt i § 2, the most important novelty is my view of . Most, but not quite all, of the other expressions have been explained in the above way at one time or another, though never, as far as I have noticed, by any single critic. The meaning of was perfectly well known to ancient mathematicians: and Proclus fully understood the ‘rational’ and ‘irrational’ diameters of 5. The full explanation of is due to Barozzi, except that he did not multiply the sides. As regards , I believe that I have proved my view in App. I, Pt i § 2 and Pt iii. Here I will only say that just as in the increasing series 1, 60, 3600, 216000 the number 216000 or 60^{3} is the ‘third increase’ () of unity, so in the increasing series 60, 3600, 216000, 12960000, the number 12960000 or (as we express it, but as Plato, to whom ‘power’ means either ‘square’ or ‘root,’ never did or could express it, 60^{4}) is the third increase of 60.

συζυγείς. The metaphor is from marriage, and marriage, among the Pythagoreans, was usually expressed by multiplication. Thus 6, which is the product of the first male number 3 and the first female number 2, was called by them marriage. συζυγεῖσα also means ‘multiplied with’ in Proclus l. c. II p. 544 (App. I, Pt i § 2).

. The square and oblong may be regarded as ἁρμονίαι because in them, as in the number 216 above, all things are . Thus 12,960,000=(35 + 1) x 360,000, so that, as 35 is a ἁρμονία, 12,960,000 contains the portentous number of 360,000 ἁρμονίαι plus (1 x 360,000=) 360,000, each ἁρμονία thus having added to it, as before, the unit which is . The analogy between the Microcosm and the Macrocosm is thus preserved : see on 546 B above. So much for the arithmetical meaning of the term ἁρμονίας. In App. I, Pt ii § 5 I have given my reasons for connecting the two ἁρμονίαι with the myth of the Politicus. In that myth we are told how two cycles of equal and vast duration invariably succeed one another in the life of the Universe, a progressive and a retrogressive cycle. These two cycles are two Great Years, in the first of which ὁμοιότης prevails and the Universe is fresh and strong, while in the second, in which we are living now, ἀνομοιότης begins to assert itself and the Universe flags and wanes. Cf. 547 A note Here the first ἁρμονία, which is a square and therefore ὅμοιον, represents the progressive cycle, the cycle of όμοιότης, and the second ἁρμονία, which is an oblong, and therefore ἀνόμοιον (see above on 546 B line 12), stands for the retrogressive cycle, the cycle of ἀνομοιότης. If this identification is, as I believe, correct, each ἁρμονία represents a Great Year. The area or number of each harmony, according to Plato, is 12,960,000, and as Plato elsewhere says that the Great Year is measured (Tim. 39 D), i.e. by the diurnal revolutions of the heavens, we may take this number as denoting days. Converted into years, on the astronomical calculation of 360 days to the year, followed by Plato here and elsewhere, the number becomes 36,000 years, which was known in Ptolemaic astronomy as the magnus Platonicus annus. For the evidence on all these points, see App. I, Pt ii §§ 5, 6.

‘This whole number, a number measuring the earth, is lord of better and worse births.’ On its arithmetical side, γεωμετρικός means only that the number is reached by means of γεωμετρία and expressed in geometrical figures: but I have no doubt that Plato meant the word to bear another and profounder meaning, suitable to the real import of the two harmonies whereof this is the number. The number is , for it measures an aeon of the Universe, of which the Earth is part (cf. 555 A note and VI 511 B note): and indeed it is artistically right that the meaning of the two harmonies should be summed up at the climax of the whole in a single pregnant word. How do good and bad births depend upon this number? Because in the early days of our era, when God had but lately left the world, and ἀνομοιότης and ἀνωμαλία were young, Nature produced better children than . Plato in fact invites us to think of his city as having existed soon after the change to the aeon in which we now live, just as throughout Book VIII and part of IX the Ideal City is figured in the past. For more on this subject see App. I, Pt ii §§ 5—7. I know not what others will think, but to me it seems that the extraordinary range and elevation of its central ideas make the Platonic number worthy even of a writer who is full of ‘thoughts that wander through eternity.’ The connexion between the Human Child and the Divine, the Microcosm and the Macrocosm, has played no small part in the history of human thought, and the story of a Great Year, with the hope which it affords of the ἀποκατάστασις of all things (Acts 3. 21), has been and is, in its religious setting, the solace and support of many a ‘human child.’