eight of the odd numbers beginning with unity],[1] the addition of which (1, 3, 5, 7, 9, 11, 13, 15) in succession generates squares;[2] and of the inequality of the number 56, consisting of seven of the even numbers beginning with 2 (2, 4, 6, 8, 10, 12, 14), which produce the numbers that are not squares.[3]
Again, according to another way of indicating, the number 120 consists of four numbers—of one triangular, 15; of another, a square, 25; of a third, a pentagon, 35; and of a fourth, a hexagon, 45. The 5 is taken according to the same ratio in each mode. For in triangular numbers, from the unity 5 comes 15; and in squares, 25; and of those in succession, proportionally. Now 25, which is the number 5 from unity, is said to be the symbol of the Levitical tribe. And the number 35 depends also on the arithmetic, geometric, and harmonic scale of doubles—6, 8, 9, 12; the addition of which makes 35. In these days, the Jews say that seven months' children are formed. And the number 45 depends on the scale of triples—6, 9, 12, 18—the addition of which makes 45; and similarly, in these days they say that nine months' children are formed.
Such, then, is the style of the example in arithmetic. And let the testimony of geometry be the tabernacle that was constructed, and the ark that was fashioned,—constructed in most regular proportions, and through divine ideas, by the gift of understanding, which leads us from things of sense to intellectual objects, or rather from these to holy things, and to the holy of holies. For the squares of wood indicate that the square form, producing right angles, pervades all, and points out security. And the length of the structure was three hundred cubits, and the breadth fifty, and the height thirty; and above,
- ↑ The clause within brackets has been suggested by Hervetus to complete the sense.
- ↑ That is, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 120; and 1 + 3 = 4 + 5 = 9 + 7 = 16 + 9 = 25 + 11 = 36 + 13 = 49 + 15 = 64, giving us the numbers 4, 9, 16, 25, 36, 49, 64, the squares of 2, 3, 4, 5, 6, 7, 8.
- ↑ ἑτερομήϰεις the product of two unequal factors, i.e. 2 + 4 + 6 + 8 + 10 + 12 + 14 = 56; and 2 + 4 = 6 = 3 × 2, 6 + 4 = 10 = 5 × 2, and so on.
