First Complete Latin Philo

The ninth age languishes, then the cold limbs slacken
The never-dying virtue of the mind.
But whoever is able to reach the tenth goal,
Will have already fallen by a timely death.

Thus Solon, by the ten septenaries just mentioned, reckons the span of human life. But Hippocrates the physician says there are seven ages: of the infant, the boy, the adolescent, the youth, the man, the elder, and the old man. He measures these by septenaries of years, though not consecutively. For he speaks thus: "In the nature of man there are seven periods, which they call ages: infant, boy, adolescent, and the rest. He is an infant until he loses his teeth within the seventh year. A boy until the emission of seed, at the doubled seventh year. An adolescent until the growth of the beard, at the thrice-seventh year. A youth until the whole body grows, at the four-times-seventh year. A man until the forty-ninth year, at the seven-times-seventh. An elder until the fifty-sixth, at the eight-times-seventh. Whatever follows thereafter belongs to old age."

This is also brought forward for the special commendation of the septenary: that it possesses a marvelous order in nature, consisting of three and four. If one squares the third number from unity, he will find a quadrangular [square] number; but the fourth, a cube. From both combined, he finds both a cube and a square. Therefore, the third from unity, in a double ratio, is a square. But the fourth, namely eight, is a cube; and the seventh, sixty-four, is both a cube and a square, so that the seventh number is altogether perfect, promising both equalities: the superficial through the triangle, according to its kinship with the ternary; and the solid through the cube, according to the familiarity it has with the quaternary. Moreover, from the ternary and the quaternary the septenary is formed.

It is, however, not only perfective, but to say it in a word, most harmonic, and in a way the source of a most beautiful diagram, which contains all harmonies—namely, the fourth, the fifth, and the octave—and likewise all proportions—namely, the arithmetic, the geometric, and furthermore the harmonic. This table consists of these numbers: six, eight, nine, twelve. Eight to six is in the sesquitertian ratio, which is the harmony of the fourth. Nine to six is sesquialteran, which is the fifth. But twelve to six is in the double ratio, which is the octave. It also contains (as I said) all proportions: the arithmetic indeed in these numbers: six, nine, twelve. For just as the middle exceeds the first by three, so is it exceeded by the last by the same amount. The geometric in these four numbers: six, eight, nine, twelve. For in the same ratio as eight is to six, so is twelve to nine; it is the sesquitertian ratio. Furthermore, the harmonic in these three: six, eight, twelve. There is a twofold test of harmonic proportion. One, when the last has the same ratio to the first as that which the difference by which the last exceeds the middle has to the difference by which the first is exceeded by the middle. This will become most evident from the numbers just proposed, which are six, eight, twelve. For the last is double the first, and the parts by which these numbers exceed each other are likewise double. For twelve exceeds eight by four, and eight exceeds six by two. But four is the double of two. The second test of harmonic proportion is when the middle term exceeds and is exceeded by the extremes by an equal portion [of those extremes]; for eight, standing in the middle