Clement of Alexandria: STROMATA (MISCELLANIES), Part IV, Complete

Translated by Alexander Roberts and James Donaldson.

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On another principle, 120 is a triangular [3338] number, and consists of the equality [3339] of the number 64, [which consists of eight of the odd numbers beginning with unity], [3340] the addition of which (1, 3, 5, 7, 9, 11, 13, 15) in succession generate squares; [3341] 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 [3342]

Again, according to another way of indicating, the number 120 consists of four numbers--of one triangle, 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.

[3338] "Triangular numbers are those which can be disposed in a triangle, as 3 .:, 6, etc, being represented by the formula (x^2 + x)/2" (Liddell and Scott's Lexicon). Each side of the triangle of courses contains an equal number of units, the sum of which amounts to the number. [[136]Elucidation VI.]

[3339] This number is called equality, because it is composed of eight numbers, an even number; as fifty-six is called inequality, because it is composed of seven numbers, an odd number.

[3340] The clause within brackets has been suggested by Hervetus to complete the sense.

[3341] That is, 1+3+5+7+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.

[3342] eteromekeis, the product of two unequal factors, i.e., 2+4+6+8+10+12+14=56; and 2+4=6=3 x 2, 6+4=10=5 x 2, and so on.

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