De institutione musica II.30: De maiore parte toni, in quibus minimis numeris constet.
The remaining part, then, which is the greater, is called apotome by the Greeks, but by us it can be called decisio. For nature brings it about that, whenever something is cut so that it is not divided into equal parts, by as much as the lesser part is less than the half, by so much the greater part, which is increased, exceeds the half. As much, therefore, as the lesser semitone is less than a whole half of the tone, so much does the apotome exceed the whole half of the tone. And since we have taught that the semitone stands principally in 256 and 243, let us now prove in what least numbers that which is called the apotome can stand. If 243 could receive an eighth part, when a sesquioctave number was compared with it, then the relation of 256 to the sum of the sesquioctave, compared with the least number, would show the apotome by necessary reasoning. But now, since the eighth part is shown to be lacking to it, let both numbers be multiplied by eight. And from 243 multiplied eight times the number 1,944 is made.
If its own eighth part is added to this, that is, 243, they will make 2,187. Again, let 256 grow by eight; they will therefore make 2,048. And let this be placed in the middle of the terms written above:
The third term, therefore, holds to the first the proportion of a tone, the second to the first that of the lesser semitone, and the third to the second that of the apotome. And in the same numbers the proportion of the first and of the apotome appears to stand, since the space of the semitone is contained in the least numbers 256 and 243. And 1,944 and 2,048 are in the same proportion as 243 to 256 for this reason, that 256 and 243 have been multiplied by eight. For if one number multiplies any two numbers, those which arise from that multiplication will be in the same proportion in which were the numbers which the former number multiplied.