De institutione musica II.9: De proportione numerorum, qui ab aliis metiuntur.
If the difference of two numbers measures them exactly, the numbers which their difference measures are in the same proportion in which are those numbers according to which the difference measures them. Let the numbers be 50 and 55. These are compared with each other in the sesquidecimal relation, and their difference is five, which is the tenth part of the number 50. This five, therefore, measures 50 ten times and 55 eleven times. According to the numbers 10 and 11, then, 55 and 50 are measured by their own difference, that is five, and they are composed in the sesquidecimal comparison of 11 to 10. The numbers which their own difference has exactly measured are therefore in the same proportion as those according to which their own difference has measured them. But if any difference of numbers measures those numbers whose difference it is in such a way that the plurality of numbers exceeds the same measure, and the excess is the same in both, and the measure of the difference is smaller than the plurality of the numbers, the numbers will have a greater proportion to each other if what remains after the measuring is taken away from them than they had whole, when their own difference measured them. Let there be two numbers, 53 and 58. Let five, which is their difference, measure them. Five measures 53 ten times up to 50 and leaves three. Again it measures 58 eleven times up to 55 and in it again leaves three. Let three then be taken from both; there are 50 and 55, which are set out in this manner:
In this it is manifest that there is a greater proportion between 50 and 55 than between 53 and 58. For in lesser numbers a greater proportion is always found; which we shall demonstrate a little later. But if that measuring of the difference goes beyond the multitude of the numbers and passes both by the same plurality of number, the numbers measured above will be in a lesser proportion, with the addition of the sum by which the measuring goes beyond both, than they were before, when their own difference measured them. Let the numbers be 48 and 53. Their difference is five. Let five measure 48 ten times, making 50. The number 50 therefore exceeds 48 by two. Let the same measure 53 eleven times, making 55, which again exceeds the number 53 by the same two. Let two be added to both and let them be set out in this manner:
Compared, then, with the addition of the two by which the difference measuring them goes beyond, 50 and 55 are in a lesser proportion than the numbers 48 and 53, which the same difference of five has measured. Greater and lesser proportions are understood in this way. A half part is greater than a third, a third part is greater than a fourth, a fourth part is greater than a fifth, and so on in the same manner. Whence it comes about that the sesquialter proportion is greater than the sesquitertian, and the sesquitertian surpasses the sesquiquartan. And the same in the rest. Hence it happens that in lesser numbers the proportion of superparticular numbers always seems greater. This appears in the natural number. Let the natural number be set out: 1, 2, 3, 4, 5. Two is double to the unit, three is sesquialter to two, four is sesquitertian to three. Three and four are the greater numbers, three, two and one the lesser. In the greater, therefore, the lesser proportion is contained, and in the lesser the greater. Hence it appears that if an equal plurality is added to any numbers containing a superparticular proportion, the proportion is greater before the increase of the equal plurality than after the equal plurality has been added to them.