Now the more readily to find out these means Eudorus hath taught us an easy method. For after you have proposed the extremities, if you take the half part of each and add them together, the product shall be the middle, alike in both duple and triple proportions, in arithmetical mediety. But as for sub-contrary mediety, in duple proportion, first having fixed the extremes, take the third part of the lesser and the half of the larger extreme, and the addition of both together shall be the middle; in triple proportion, the half of the lesser and the third part of the larger extreme shall be the mean. As for example, in triple proportion, let 6 be the least extreme, and 18 the biggest; if you take 3 which is the half of 6, and 6 which is the third part of 18, the product by addition will be 9, exceeding and exceeded by the same proportional parts of the extremes. In this manner the mediums are found out; and these are so to be disposed and placed as to fill up the duple and triple intervals. Now of these proposed numbers, some have no middle space, others have not sufficient. Being therefore so augmented that the same proportions may remain, they will afford sufficient space for the aforesaid mediums. To which purpose, instead of a unit they choose the six, as being the first number including in itself a half and third part, and so multiplying all the figures below it and above it by 6, they make sufficient room to receive the mediums, both in double and triple distances, as in the example below:— 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. 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.