Works of Philo of Alexandria

Philo — Works Vol. I, Yonge — Section 54

metrical proportion, from the numbers six, and nine, and twelve ; for as the number in the middle exceeds the first number by three, it is also exceeded by three by the last num- ber. And geometrical proportion is according to these four numbers. For the same ratio that eight bears to six, that also does twelve bear to nine. And this is the ratio of thirds. Harmonic ratio consists of three numbers, six, and eight, and twelve. But there are two ways of judging of harmonic pro- portion. One when, whatever ratio the last number bears to the first, the excess by which the last number exceeds the middle one is the same as the excess by which the middle number exceeds the first. And any one may derive a most evident proof of this from the numbers before mentioned, six, and eight, and twelve : for the last number is double the first. And again, the excess of twelve over eight is double the excess of eight over six. For the number twelve exceeds eight by four, and eight exceeds six by two ; and four is the double of two. And another test of harmonic proportion is, when the middle term exceeds and is exceeded by those on each side of it, by an equal portion ; for eight being the middle term, exceeds the first term by a third part ; for if six be sub- tracted from it, the remainder two is one third of the original number six : and it is exceeded by the last term in an equal proportion ; for if eight be taken from twelve, the remainder four is one third of the whole number twelve. XXXVIII. Let this then be premised, as of necessity it must, respecting the honourable qualities which this diagram or square has, and the name to which it is entitled, and the number seven unfolds an equal number of ideas, and even more in the case of incorporeal things, which are perceptible only by the intellect : and its nature extends also over every visible essence, reaching to both heaven and earth, which are le boundaries of every thing. For what portion of all the things earth is there which is not fond of seven ; being subdued by in affection and longing for the seventh. Accordingly men say,