De Opificio Mundi

and the clearest proof of this is afforded by the numbers already spoken of. In the seventh number increasing immediately from the unit in a twofold ratio, namely, the number sixty-four, is a square formed by the multiplication of eight by eight, and it is also a cube by the multiplication of four and four, four times. And again, the seventh number from the unit being increased in a threefold ratio, that is to say, the number seven hundred and twenty-nine, is a square, the number seven and twenty being multiplied by itself; and it is also a cube, by nine being multiplied by itself nine times.

And in every case a man making his beginning from the unit, and proceeding on to the seventh number, and increasing in the same ratio till he comes to the number seven, will at all times find the number, when increased, both a cube and a square. At all events, he who begins with the number sixty-four, and combines them in a doubling ratio, will make the seventh number four thousand and ninety-six, which is both a square and a cube, having sixty-four as its square root, and sixteen as its cube root.

And we must also pass on to the other species of the number seven, which is contained in the number ten, and which displays an admirable nature, and one not inferior to the previously mentioned species. The number seven consists of one, and two and four, numbers which have two most harmonious ratios, the twofold and the fourfold ratio; the former of which affects the diapason harmony, while the fourfold ratio causes that of the double diapason. It also comprehends other divisions, existing in some kind of yoke-like combination. For it is divided first of all into the number one, and the number six; then into the two and the five; and last of all, into

And the proportion of these numbers is a most musical one; for the number six bears to the number one a six-fold ratio, and the six-fold ratio causes the greatest possible difference between existing tones; the distance namely, by which the sharpest tone is separated from the flattest, as we shall show when we pass on from numbers to the discussion of harmony. Again, the ratio of four to two displays the greatest power in harmony, almost equal to that of the diapason, as is most evidently shown in the rules of that art. And the ratio of four to three effects the first harmony, that in the thirds, which is the diatessaron.