XXXVII. (107) But this number is not only a perfecter of things, but it is also, so to say, the most harmonious of numbers; and in a manner the source of that most beautiful diagram which describes all the harmonies, that of fourths, and that of fifths, and the diapason. It also comprises all the proportions, the arithmetical, the geometrical, and moreover the harmonic proportion. And the square consists of these numbers, six, eight, nine, and twelve; and eight bears to six the ratio of being one third greater, which is the diatessaron of harmony. And nine bears to six the ratio of being half as great again, which is the ratio of fifths. And twelve is to six, in a twofold proportion; and this is the same as the diapason. (108) The number seven comprises also, as I have said, all the proportions of arithmetrical 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 number. 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. (109) But there are two ways of judging of harmonic proportion. 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. (110) 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 subtracted 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. (111) 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 the boundaries of every thing. For what portion of all the things on earth is there which is not fond of seven; being subdued by an affection and longing for the seventh. (112) Accordingly men say, that the heaven is girdled with seven circles, the names of which are as follows; the arctic, the antarctic, the summer tropic, the winter tropic, the equinoctial, the zodiac, and last of all the galaxy. For the horizon is something which affects ourselves, in proportion as any one has acute vision, or the contrary; our sensation cutting off at one time a lesser, and at another time a greater circumference. (113) The planets too, and the corresponding host of fixed stars, are arrayed in seven divisions, displaying a very great sympathy with the air and the earth. For they turn the air towards the times, that are called the seasons of the year, causing in each of them innumerable changes by calm weather, and pleasant breezes, and clouds, and irresistible blasts of wind. And again, they make rivers to overflow and to subside, and turn plains into lakes; and again, on the contrary, they dry up the waters: they also cause the alterations of the seas, when they receded, and return with a reflux. For at times, when the tide recedes on a sudden, an extensive line of shore occupies what is usually a wide gulf of sea; and in a short time afterwards, the waters are brought back, and there appears a sea, sailed over, not by shallow boats, but by ships of exceeding great burden. And they also give increase and perfection to all the terrestrial animals and plants which produce fruit, endowing each with a nature to last a long time, so that new plants may flourish and come to maturity; ùthe old ones having passed away, in order to provide an abundant supply of necessary things.
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