XXXI. (95) 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 the three and the four. (96) 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.

XXXII. (97) The number seven displays also another beauty which it possesses, and one which is most sacred to think of. For as it consists of three and four, it displays in existing things a line which is free from all deviation and upright by nature. And in what way it does so I must show. The rectangular triangle, which is the beginning of all qualities, consists of the Numbers{5}{this discussion about numbers is not very intelligible; but here Philo is probably referring to the problem of Euclid on the subject of the square of the hypothenuse. Thus, if 3 and 4 represent the sides containing the angle, and 5 the side subtending it, we get (3×3)+(4×4) = 9+16= 25; 5×5= 25.} and four, and five; and the three and the four, which are the essence of the seven, contain the right angle; for the obtuse angle and the acute angle show irregularity, and disorder, and inequality; for one may be more acute or more obtuse than another. But a right angle does not admit of comparison, nor is one right angle more a right angle than another: but one remains similar to another, never changing its peculiar nature. But if the right-angled triangle is the beginning of all figures and of all qualities, and if the essence of the number seven, that is to say, the numbers three and four together, supply the most necessary part of this, namely, the right angle, then seven may be rightly thought to be the fountain of every figure and of every quality. (98) And besides what has been already advanced, this also may be asserted that three is the number of a plane figure, since a point has been laid down to be, according to a unit, and a line according to the number two, and a plane superficies according to the number three. Also, four is the number of a cube, by the addition of one to the number of a plane superficies, depth being added to the superficies. From which it is plain that the essence of the number seven is the foundation of geometry and trigonometry; and in a word, of all incorporeal and corporeal substances.