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Sisting of all these wholes, complete and free from age and sickness. 32C. We are here given one of the reasons why the Demiurge thought it better that the visible world should resemble its model in respect of uniqueness (3IB),! ' 1 52 THE WORLD'S BODY and cold things and whatever has strong powers '. 'Powers' (bV1laflc£(;) means the qualities or properties of bodies considered as having the 'power to act and be acted upon' (MJvafl£(; TOV nO£elV xat naaxc£v). Hotness is the property of fire that is manifest when fire makes something else hot or causes in sentient beings a sensation of heat.

The numbers in the third column are squares (c5vvap;et~), those in the second column are the roots of these squares. Square roots also were sometimes called c5vvap;et~. The underlying notion seems to be that any number (represented by a line) has, in itself and without the aid of any other factor, the power of multiplying itself or generating its own square by advancing as far as its own length into the second dimension. Hence a line is said c515vaa()at the square • 1 C~. ov. 48 THE WORLD'S BODY 31B-32c THE WORLD'S BODY plane figure it thus generates.

Taking the double and triple proportions, we have I I 4, 8 3, 9, 27 These are the two series that Plato takes later (35B) as the basis for the harmony of the World-Soul. Both series emanate from unity, in which all the' powers' concerned are conceived as gathered up. The series proceed through the first even, and the first odd, number to their squares and cubes. " He would certainly choose a progression of what was held to be the most perfect type. 4 Nicomachus, in his chapter on continuous geometrical proportion (ii, 24), repeats that this is the only proportion in the most proper sense (xve{w~ xaAovp;iv'YJ) and gives the same examples: 'the numbers proceeding from unity according to the double proportion' : I, 2, I, I, 2, 4, 8, 16, and the triple proportion: 32, 64 .

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