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.

XXXIII. And such great sanctity is there in the number seven, that it has a pre-eminent rank beyond all the other numbers in the first decade. For of the other numbers, some produce without being produced, others are produced but have no productive power themselves; others again both produce and are produced. But the number seven alone is contemplated in no part. And this proposition we must confirm by demonstration. Now the number one produces all the other numbers in order, being itself produced absolutely by no other; and the number eight is produced by twice four, but itself produces no other number in the decade, Again, four has the rank of both, that is, of parents and of offspring; for it produces eight when doubled, and it is produced by twice two. But seven alone, as I said before, neither produces nor is produced, on which account other philosophers liken this number to Victory, who had no mother, and to the virgin goddess, whom the fable asserts to have sprung from the head of Jupiter: and the Pythagoreans compare it to the Ruler of all things. For that which neither produces, nor is produced, remains immovable. For generation consists in motion, since that which is generated, cannot be so without motion, both to cause production, and to be produced. And the only thing which neither moves nor is moved, is the Elder, Ruler, and Lord of the universe, of whom the number seven may reasonably be called a likeness. And Philolaus gives his testimony to this doctrine of mine in the following words:-"For God," says he "is the ruler and Lord of all things, being one, eternal, lasting, immovable, himself like to himself, and different from all other beings."

XXXIV. Among the things then which are perceptible only by intellect, the number seven is proved to be the only thing free from motion and accident; but among things perceptible by the external senses, it displays a great and comprehensivo power, contributing to the improvement of all terrestrial things, and affecting even the periodical changes of the moon.

And

in what manner it does this, we must consider. The number seven when compounded of numbers beginning with the unit, makes eight-and-twenty, a perfect number, and one equalised in its parts. And the number so produced, is calculated to reproduce the revolutions of the moon, bringing her back to the point from which she first began to increase in a manner perceptible by the external senses, and to which she returns by waning. For she increases from her first crescent-shaped figure, to that of a half circle in seven days; and in seven more, she becomes a full orb; and then again she turns back, retracing the same path, like a runner of the diaulos, receding from an orb full of light, to a half circle again in seven days, and lastly, in an equal number she diminishes from a half circle to the form of a crescent; and thus the number before mentioned is completed. And the number seven by those persons who are in the habit of employing names with strict propriety is called the perfecting number; because by it, everything is perfected. And any one may receive a confirmation of this from the fact, that every organic body has three dimensions, length, depth, and breadth; and four boundaries, the point, the line, the superficies, and the solid; and by these, when combined, the number seven is made up.

But it would be impossible for bodies to be measured by the number seven, according to the combination of the three 'dimensions, and the four boundaries, if it did not happen that the ideas of the first numbers, one, two, three and four, in in which the number ten is founded, comprised the nature of the number seven. For the aforesaid numbers have four boundaries, the first, the second, the third, the fourth, and three intervals. The first interval being that between one and two; the second, that between two and three; the third, that between three and four.

XXXV. And besides what has been already said, the growth of men from infancy to old age, when measured by the number seven, displays in a most evident manner its perfecting power; for in the first period of seven years, the putting forth of the teeth takes place. And at the end of the second period of the same length, he arrives at the age of puberty: at the end

This refers to the Greek games. "The straight race was called στάδιον οι δρόμος. In the δίαυλος δρόμος the runners turned round the goal, and came back to the starting place."-Smith in v. Stadium.

of the third period, the growth of the beard takes place. The fourth period sees him arrive at the fulness of his manly strength. The fifth seven years is the season for marriage. In the sixth period he arrives at the maturity of his understanding. The seventh period is that of the most rapid improvement and growth of both his intellectual and reasoning powers. The eighth is the sum of the perfection of both. In the ninth, his passions assume a mildness and gentleness, from being to a great degree tamed. In the tenth, the desirable end of life comes upon him, while his limbs and organic senses are still unimpaired for excessive old age is apt to weaken and enfeeble them all.

And Solon, the Athenian lawgiver, described these different, ages in the following elegiac verses :

In seven years from th'earliest breath,
The child puts forth his hedge of teeth;
When strengthened by a similar span,
He first displays some signs of man.
As in a third, his limbs increase,
A beard buds o'er his changing face.
When he has passed a fourth such time,
His strength and vigour's in its prime.
When five times seven years o'er his head
Have passed, the man should think to wed;
At forty two, the wisdom's clear

To shun vile deeds of folly or fear:
While seven times seven years to sense
Add ready wit and eloquence.

And seven years further skill admit

To raise them to their perfect height.

When nine such periods have passed,

His powers, though milder grown, still last;
When God has granted ten times seven,

The aged man prepares for heaven,

XXXVI. Solon therefore thus computes the life of man by the aforesaid ten periods of seven years. But Hippocrates the physician says that there are seven ages of man, infancy, childhood, boyhood, youth, manhood, middle age, old age; and that these too, are measured by periods of seven, though not It is hardly necessary to remind the reader of the description of the seven ages of man in Shakespeare. As You Like It, Act II. sc. 7.

in the same order. And he speaks thus ; "In the nature of man there are seven seasons, which men call ages; infancy, childhood, boyhood, and the rest. He is an infant till he reaches his seventh year, the age of the shedding of his teeth. He is a child till he arrives at the age of puberty, which takes place in fourteen years. He is a boy till his beard begins to grow, and that time is the end of a third period of seven years. He is a youth till the completion of the growth of his whole body, which coincides with the fourth seven years. Then he is a man till he reaches his forty-ninth year, or seven times seven periods. He is a middle aged man till he is fiftysix, or eight times seven years old; and after that he is an old man.

And it is also affirmed for the particular praise of the number seven, that it has a very admirable rank in nature, because it is composed of three and four. And if any one doubles the the third number after the unit, he will find a square; and if he doubles the fourth number, he will find a cube. And if he doubles the seventh from both, he will both a cube and a square; therefore, the third number from the unit is a square in a double ratio. And the fourth number, eight, is a cube. And the seventh number, being sixty-four, is both a cube and a square at the same time; so that the seventh number is really a perfecting one, signifying both equalities, the plane superficies by the square, according to the connection with the number three, and the solid by the cube according to its relationship to the number four; and of the numbers three and four, are composed the number seven.

XXXVII. 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. The number seven comprises also, as I have said, all the proportions of arith

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 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. 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. 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. 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. 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

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