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Now Plato laid down this for a position, that the intervals of sesquialters, sesquiterces, and sesquioctaves having once arisen from these connections in the first spaces, the Deity filled up all the sesquiterce intervals with sesquioctaves, leaving a part of each, so that the interval left of the part should bear the numerical proportion of 256 to 243.Timaeus, p. 36 A. From these words of Plato they were constrained to enlarge their numbers and make them bigger. Now there must be two numbers following in order in sesquioctave proportion. But the six does not contain a sesquioctave; and if it should be cut up into parts and the units bruised into fractions, this would strangely perplex the study of these things. Therefore the occasion itself advised multiplication; so that, as in changes in the musical scale, the whole scheme was extended in agreement with the first (or base) number. Eudorus therefore, imitating Crantor, made choice of 384 for his first number, being the product of 64 multiplied by 6; which way of proceeding the number 64 led them to having for its sesquioctave 72. But it is more agreeable to the words of Plato to introduce the half of 384. For the remainder of that will bear a sesquioctave proportion in those numbers which Plato mentions, 256 and 243, if we make use of 192 for the first number. But if the same number be made choice of doubled, the remainder (or leimma) will have the same proportion, but the numbers will be doubled, i.e. 512 and 486. For 256 is in sesquiterce proportion to 192, as 512 to 384. Neither was Crantor’s reduction of the proportions to this number without reason, which made his followers willing to pursue it; in regard that 64 is both the square of the first cube, and the cube of the first square; and being multiplied by 3, the first odd and trigonal, and the first perfect and sesquialter number, it produces 192, which also has its sesquioctave, as we shall demonstrate.
