It is, indeed, a rather strange thing to hear; but the name that we, at any rate, give it—one that people would never approve, from inexperience in the matter—is astronomy; people are ignorant that he who is truly an astronomer must be wisest, not he who is an astronomer in the sense understood by Hesiod and all the rest of such writers, the sort of man who has studied settings and risings; but the man who has studied the seven Footnote 1i.e. of the sun, the moon, and the five planets; cf. 987 B. With the astronomy and mathematics of the rest of the Epinomis cf. Plato, Laws, vii. 818-820. out of the eight orbits, each travelling over its own circuit in such a manner as could not ever be easily observed by any ordinary nature, that did not partake of a marvellous nature. As to this, we have now told, and shall tell, as we profess, by what means and in what manner it ought to be learnt; and first let us make the following statement.
The moon travels through its orbit very swiftly, bringing first the month and full-moon; and in the second place we must remark the sun, with his turning motion through the whole of his orbit, and with him his satellites. But to avoid repeating again and again the same things on the same subjects in our discussion, the other courses of these bodies that we have previously described are not easily understood: we must rather prepare our faculties, such as they may possibly be, for these matters; and so one must teach the pupil many things beforehand, and continually strive hard to habituate him in childhood and youth. And therefore there will be need of studies: the most important and first is of numbers in themselves; not of those which are corporeal, but of the whole origin of the odd and the even, and the greatness of their influence on the nature of reality. When he has learnt these things, there comes next after these what they call by the very ridiculous name of geometry, Footnote 2Which means literally measuring the earth; this developed into the arithmetical calculation of squares, cubes, roots, etc. Cf. the account Plato gives (Theaet. 147 D ff.) of quadrangular
and equilateral
numbers, showing how the terms of geometry had to be used for arithmetic. As there was no number equal (or like
) to the square
root of 2, recourse was had to the geometrical symbol of the diagonal of a square whose side is 1; and similarly cubic
roots were reckoned with the aid of stereometry. when it proves to be a manifest likening Footnote 3Likening here means comparing in an exact manner,
so as to obtain a ratio or proportion between numbers not directly commensurable; cf. Plato, Laws, viii. 820. of numbers not like one another by nature in respect of the province of planes; and this will be clearly seen by him who is able to understand it to be a marvel not of human, but of divine origin. And then, after that, the numbers thrice increased and like to the solid nature, and those again which have been made unlike, he likens by another art, namely, that which its adepts called stereometry; and a divine and marvellous thing it is to those who envisage it and reflect, how the whole of nature is impressed with species and class according to each analogy, as power and its opposite Footnote 4Power
is multiplication, its opposite
is extension: 1 point doubled gives the beginning of a line; multiplying 2 by 2 gives 4 as a square surface, and by 2 again, 8 as the cube. So (see below) we proceed from 1 to 8.
continually turn upon the double.