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[36a] After that He went on to fill up the intervals in the series of the powers of 2 and the intervals in the series of powers of 3 in the following manner1: He cut off yet further portions from the original mixture, and set them in between the portions above rehearsed, so as to place two Means in each interval, —one a Mean which exceeded its Extremes and was by them exceeded by the same proportional part or fraction of each of the Extremes respectively2; the other a Mean which exceeded one Extreme by the same number or integer as it was exceeded by its other Extreme. And whereas the insertion of these links formed fresh intervals in the former intervals, that is to say, intervals of 3:2 and 4:3 and 9:8, He went on to fill up the 4:3 intervals with 9:8 intervals.


1 Lit. “the double intervals and the triple intervals.” See the preceding note.

2 The “harmonic Mean.”

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