But since almost every heresy (that has sprung up) through the arithmetical art has discovered measures of hebdomads and certain projections of Æons, each rending the art differently, while whatever variation prevailed was in the names merely; and (since) Pythagoras became the instructor of these, first introducing numbers of this sort among the Greeks from Egypt, it seems expedient not to omit even this, but, after we have given a compendious elucidation, to approach the demonstration of those things that we propose to investigate.

Arithmeticians and geometers arose, to whom especially Pythagoras first seems to have furnished principles. And from numbers that can continually progress ad infinitum by multiplication, and from figures, these derived their first principles, [308] as capable of being discerned by reason alone; for a principle of geometry, as one may perceive, is an indivisible point. From that point, however, by means of the art, the generation of endless figures from the point is discovered. For the point being drawn into length becomes a line, after being thus continued, having a point for its extremity. And a line flowing out into breadth begets a surface, and the limits of the surface are lines; but a surface flowing out into breadth becomes body. And when what is solid has in this manner derived existence from, altogether, the smallest point, the nature of a huge body is constituted; and this is what Simon expresses thus: "The little will be great, being as a point, and the great illimitable." Now this coincides with the geometrical doctrine of a point.

But of the arithmetical [309] art, which by composition contains philosophy, number became a first principle, which is an indefinable and incomprehensible (entity), comprising in itself all the numbers that can go on ad infinitum by aggregation. But the first monad became a principle, according to substance, of the numbers, which (principle) is a male [310] monad, pro-creating paternally all the rest of the numbers. Secondly, the duad is a female number, which by the arithmeticians is also itself denominated even. Thirdly, the triad is a male number; this also it has been the usual custom of arithmeticians to style odd. In addition to all these, the tetrad is a female number; and this same, because it is feminine, is likewise denominated even. All the numbers therefore, taken generically, are four -- number, however, as regards genus, is indefinite -- from which, according to their system, is formed the perfect number -- I mean the decade. For one, two, three, four, become ten -- as has been previously proved -- if the proper denomination be preserved, according to substance, for each of the numbers. This is the sacred quaternion, according to Pythagoras, having in itself roots of an endless nature, that is, all other numbers; for eleven, and twelve, and the rest, derive the principle of generation from the ten. Of this decade -- the perfect number -- there are called four parts -- number, monad, power, cube -- whose connections and mixtures take place for the generation of increase, according to nature completing the productive number. For when the square is multiplied into itself, it becomes a biquadratic; but when the square is multiplied into a cube, it becomes the product of a quadratic and cube; but when a cube is multiplied into a cube, it becomes the product of cube multiplied by cube. Wherefore all the numbers are seven; so that the generation of things produced may be from the hebdomad -- which is number, monad, power, cube, biquadratic, product of quadratic multiplied by cube, product of cube multiplied by cube.

Of this hebdomad Simon and Valentinus, having altered the names, detailed marvellous stories, from thence hastily adopting a system for themselves. For Simon employs his denominations thus: Mind, Intelligence, Name, Voice, Ratiocination, Reflection; and He who stood, stands, will stand. And Valentinus (enumerates them thus): Mind, Truth, Word, Life, Man, Church, and the Father, reckoned along with these, according to the same principles as those advanced by the cultivators of arithmetical philosophy. And (heresiarchs) admiring, as if unknown to the multitude, (this philosophy, and) following it, have framed heterodox doctrines devised by themselves.

Some indeed, then, attempt likewise to form the hebdomads from the medical [311] (art), being astonished at the dissection of the brain, asserting that the substance of the universe and the power of procreation and the Godhead could be ascertained from the arrangement of the brain. For the brain, being the dominant portion of the entire body, reposes calm and unmoved, containing within itself the spirit. Such an account, then, is not incredible, but widely differs from the conclusions which these (heretics) attempt to deduce from it. For the brain, on being dissected, has within it what may be called a vaulted chamber. And on either side of this are thin membranes, which they term little wings. Now these are gently moved by the spirit, and in turn propel towards the cerebellum the spirit, which, careering through a certain blood-vessel like a reed, advances towards the pineal gland. And near this is situated the entrance of the cerebellum, which admits the current of spirit, and distributes it into what is styled the spinal marrow. But from them the whole frame participates in the spiritual energy, inasmuch as all the arteries, like a branch, are fastened on from this blood-vessel, the extremity of which terminates in the genital blood-vessels, whence all the (animal) seeds proceeding from the brain through the loin are secreted (in the seminal glands). The form, however, of the brain is like the head of a serpent, respecting which a lengthened discussion is maintained by the professors of knowledge, falsely so named, as we shall prove. Six other coupling ligaments grow out of the brain, which, traversing round the head, and having their termination in (the head) itself, hold bodies together; but the seventh (ligament) proceeds from the cerebellum to the lower parts of the rest of the frame, as we have declared.

And respecting this there is an enlarged discussion, whence both Simon and Valentinus will be found both to have derived from this source starting-points for their opinions, and, though they may not acknowledge it, to be in the first instance liars, then heretics. Since, then, it appears that we have sufficiently explained these tenets likewise, and that all the reputed opinions of this earthly philosophy have been comprised in four books; it seems expedient to proceed to a consideration of the disciples of these men, nay rather, those who have furtively appropriated their doctrines. [312]

---

Footnotes:

[308] Sextus Empiricus, adv. Geom., 29 et seq. (See book vi. chap. xviii. of The Refutation.)

[309] The observations following have already been made in book i. of The Refutation.

[310] Some read arsis.

[311] The Abbe Cruice refers to Censorinus (De Die Natali, cap. vii. et xiv.), who mentions that two numbers were held in veneration, the seventh (hebdomad) and ninth (ennead). The former was of use in curing corporeal disease, and ascribed to Apollo; the latter healed the diseases of the mind, and was attributed to the Muses.

[312] At foot of ms. occur the words, "Fourth Book of Philosophumena."