De institutione musica II.6: Quid sint quadrati numeri deque his speculatio.
A square number is one which has grown by a double measurement into equal sides, as twice two, three times three, four times four, five times five, six times six; the description of which is this:
The natural number set above is, then, the side of the squares described below. For squares are naturally consecutive when they follow one another in the series set beneath, as 4, 9, 16, and the rest. If, therefore, I take a lesser consecutive square from a greater consecutive square, what remains will be as much as is joined from the sides of both squares. As if I take 4 from 9, 5 are left, which is made up of two and three, which are the sides of both squares. Likewise I take 9 from that which is written to the number 16; seven are left, which is made up of three and four, which are the sides of the aforesaid squares; and it is the same in the rest. But if the squares are not consecutive, and one is passed over between them, the half of what remains is what is made from the sides of both. As if I take 4 from the square of 16, 12 are left. The half of these 12 is the number that comes from the sides of both. And the sides of both are two and four, which joined make six. And in the rest the manner is the same. But if two are passed over, a third part of what remains will be what the sides of both join to make. As if I take 4 from 25, two squares being passed over, 21 are left. Their sides are two and five, which make seven, which is the third part of the number 21. And this is the rule: if three are passed over, the fourth part of what remains when the lesser is subtracted from the greater is what is made from the sides of both; if four are passed over, the fifth; and the parts will come with a name of number one more than the number of those passed over.