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Concerning which things, although you have heard frequent discourses, and have likewise read several arguments and disputes committed to writing upon the same subjects, it will not be amiss for me also to give a short account, after a brief repetition of Plato’s own words. God, said he, in the first place withdrew one part from the whole; which done, he took away the double of that; then a third part, sesquialter in proportion to the second, and triple to the first; then a fourth part, double to the second; next a fifth part, being the triple of the third; then a sixth, eight times the first; and lastly a seventh, being twenty-seven times the first. This done, he filled up the duple and triple intervals, retrenching also from thence certain other particles, and placing them in the midst of those intervals; so that in every interval there might be two medieties, the one exceeding and being exceeded by one and the same part of the extremes, the other exceeding and being exceeded by the same number. Now in regard that from these connections in the first spaces there arose the intervals of sesquialters, sesquiterces, and sesquioctaves, he filled up all the sesquiterce intervals with sesquioctaves, leaving a part of each, so that the interval left of the part might bear the numerical proportion of 256 to 243. Timaeus, p. 35 B.
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Here the question will be first concerning the quantity, next concerning the order, and in the third place concerning the force and virtue of the numbers. As to the quantity, we are to consider which he takes in the double and triple intervals. As to the order, whether they are to be placed in one row, according to the direction of Theodorus, or (as Crantor will have them) in the form of a Λ, placing the unit at the top, and the duples and triples apart by themselves in two several files. Lastly, we are to examine of what use and virtue they are in the structure and composition of the soul.
