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[990d] When he has learnt these things, there comes next after these what they call by the very ridiculous name of geometry,1 when it proves to be a manifest likening2 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

1 Which 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.

2 “Likening”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.

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