Chapter XVIII.—Pythagoras' System of Numbers.

Pythagoras, then, declared the originating principle of the universe to be the unbegotten monad, and the generated duad, and the rest of the numbers. And he says that the monad is the father of the duad, and the duad the mother of all things that are being begotten—the begotten one (being mother) of the things that are begotten. And Zaratas, the pupil of Pythagoras, was in the habit of denominating unity a father, and duality a mother. For the duad has been generated from the monad, according to Pythagoras; and the monad is male and primary, but the duad female (and secondary). And from the duad, again, as Pythagoras states, (are generated) the triad and the succeeding numbers up to ten. For Pythagoras is aware that this is the only perfect number—I mean the decade—for that eleven and twelve are an addition and repetition of the decade; not, however, that what is added ^[653] constitutes the generation of another number. And all solid bodies he generates from incorporeal (essences). For he asserts that an element and principle of both corporeal and incorporeal entities is the point which is indivisible. And from a point, he says, is generated a line, and from a line a surface; and a surface flowing out into a height becomes, he says, a solid body. Whence also the Pythagoreans have a certain object of adjuration, viz., the concord of the four elements. And they swear in these words:—

"By him who to our head quaternion gives,

A font that has the roots of everlasting nature." ^[654]

Now the quaternion is the originating principle of natural and solid bodies, as the monad of intelligible ones. And that likewise the quaternion generates, ^[655] he says, the perfect number, as in the case of intelligibles (the monad) does the decade, they teach thus. If any, beginning to number, says one, and adds two, then in like manner three, these (together) will be six, and to these (add) moreover four, the entire (sum), in like manner, will be ten. For one, two, three, four, become ten, the perfect number. Thus, he says, the quaternion in every respect imitated the intelligible monad, which was able to generate a perfect number.