as being impartible, to the monad; but a line, as the first interval, to the duad; and again, a superficies, as having a more abundant interval, to the triad; and a solid to the tetrad. They also called, as is evident from the testimony of Aristotle, the first length the duad; for it is not simply length, but the first length, in order that by this cee might signify cause. In a similar manner also, they denominated the first breadth, the triad; and the first depth the tetrad. They also referred to formal principles all psychical knowledge. And intellectual knowledge indeed, as being contracted according to impartible union, they referred to the monad; but scientific knowledge, as being evolved, and as proceeding from cause to the thing caused, yet through the inerratic, and always through the same things, they referred to the duad; and opinion to the triad, because the power of it is not always directed to the same thing, but at one time inclines to the true, and at another to the false. And they referred sense to the tetrad, because it has an apprehension of bodies ; for in the duad, indeed, there is one interval from one monad to the other; but in the triad there are two intervals from any one monad to the rest; and in the tetrad there are three. ‘They referred, therefore, to principles every thing knowable, viz. beings, and the gnostic powers of these. But they divided beings not according to breadth, but according to depth; into intelligibles, objects of science, objects of opinion, and sensibles. In a similar manner, also, they divided knowledge into intellect, science, opinion, and sense. The extremity, therefore, of the intelligible triad, or animal itself, as it is called by Plato in the Timeeus, is assumed from the division of the objects of knowledge, manifesting the intelligible order, in which forms them- selves, viz. the first forms and the principles of these, are contained, viz. the idea of the one itself, of the first length, which is the duad itself, and also the ideas of the first breadth and the first depth; (for in common
the term first is adapted to all of them), viz. to the triad itself, and the tetrad itself.
“ Again, the Pythagoreans and Plato did not denomi- nate idea from one thing, and ideal number from another. But since the assertion is eminently true, that all things are similar to number, it is evident that number, and especially every ideal number, was denomi- nated on account of its paradigmatic peculiarity. If any one, however, wishes to apprehend this from the appellation itself, it is easy to infer that idea was so eal from rendering as it were its participants similar to itself, and imparting to them form, order, beauty, and unity ; and this in consequence of always preserving the same form, expanding its own power to the infinity of particulars, and investing with the same species its eternal participants. Number also, since it imparts pro- portion and elegant arrangement to all things, was allotted this appellation. For the ancients, says Syri- anus,! call to adapt or compose apoat arsai, whence is derived apiOucs arithmos number. Hence avaporov anar- ston among the Greeks signifies 1ncomposite. Hence too, those Grecian sayings, you will adapt the balance, they placed number together with them, and also number and friendship. From all which number was called by the Greeks arithmos, as that which measures and orderly arranges all things, and unites them in amicable league.
“Farther still, some of the Pythagoreans discoursed
about inseparable numbers alone, i.e. numbers which
are inseparable from mundane natures, but others about
such as have a subsistence separate from the universe,
in which as paradigms they saw those numbers are con-
tained, which are perfected by nature. But others,
making a distinction between the two, unfolded their
doctrine in a more clear and perfect manner. If it be
requisite, however, to speak concerning the difference
of these monads, and their privation of difference, we
must say that the monads which subsist in quantity,
1 In Aristot. Metaphys. Lib. 13.
are by no means to be extended to essential numbers ; but when we call essential numbers monads, we must assert that all of them mutually differ from each other by difference itself, and that they possess a privation of difference from sameness. It is evident also, that those which are in the same order, are contained through mutual comparison, in sameness rather than in difference, but that those which are in different orders are con- versant with much diversity, through the dominion of difference.
‘‘ Again, the Pythagoreans asserted that nature pro-
duces sensibles by numbers; but then these numbers
were not mathematical but physical; and as they spoke
symbolically, it is not improbable that they demon-
strated every property of sensibles by mathematical
names. However, says Syrianus, to ascribe to them
a knowledge of sensible numbers alone, is not only
ridiculous, but highly impious. For they received
indeed, from the theology of Orpheus, the principles
of intelligible and intellectual numbers, they assigned
them an abundant progression, and extended their
dominion as far as to sensibles themselves.”
Again, their conceptions about mathematical and physical number, were as follow:
‘““As in every thing, according to the doctrine of
Aristotle, one thing corresponds to matter, and another
to form, in any number, as for instance the pentad, its
five monads, and in short its quantity, and the number
which is the subject of participation, are derived from
the duad itself; but its form, i.e. the pentad itself, is
from the monad: for every form is a monad, and
unites its subject quantity. The pentad itself, there-
fore, which is a monad, proceeds from the principal
monad, forms its subject quantity, which is itself form-
less, and connects it to its own form. For there are
two principles of mathematical numbers in our souls:
the monad, which comprehends in itself all the forms
of numbers, and corresponds to the monad in intel-
lectual natures; and the duad, which is a certain generative principle of infinite power, and which on this account, as being the image of the never-failing and intelligible duad, is called indefinite. While this pro- ceeds to all things, it is not deserted in its course by the monad, but that which proceeds from the monad con- tinually distinguishes and forms boundless quantity, gives a specific distinction to all its orderly progressions, and incessantly adorns them with forms. And as in mundane natures, there is neither any thing formless, nor any vacuum among the species of things, so like- wise in mathematical number, neither is any quantity left innumerable; for thus the forming power of the monad would be vanquished by the indefinite duad, nor does any medium intervene between the conse- quent numbers, and the well-disposed energy of the monad.
“Neither, therefore, does the pentad consist of sub-
stance and accident, as a white man; nor of genus and
difference, as man of animal and biped; nor of five
monads mutually touching each other, like a bundle
of wood; nor of things mingled, like a drink made from
wine and honey; nor of things sustaining position, as
stones by their position complete the house; nor lastly,
as things numerable, for these are nothing else than
particulars. But it does not follow that numbers them-
selves, because they consist of indivisible monads, have
nothing else besides monads, (for the multitude of
points in continued quantity is an indivisible multitude,
yet it is not on this account that there is a completion
of something else from the points themselves); but
this takes place because there is something in them
which corresponds to matter, and something which
corresponds to form. Lastly, when we unite the
triad with the tetrad, we say that we make seven.
The assertion, however, is not true: for monads con-
joined with monads, produce indeed the subject of the
number 7, but nothing more. Who then imparts the
heptadic form to these monads? Who is it also that gives the form of a bed to a certain number of pieces of wood? Shall we not say that the soul of the carpenter, from the art which he possesses, fashions the wood, so as to receive the form of a bed, and that the numerative soul, from possessing in herself a monad which has the relation of a principle, gives form and subsistence to all numbers? But in this only consists the difference, that the carpenter’s art is not naturally inherent in us, and requires manual operation, because it is conversant with sensible matter; but the numera- tive art is naturally present with us, and is therefore possessed by all men, and has an intellectual matter which it instantaneously invests with form. And this is that which deceives the multitude, who think that the heptad is nothing besides seven monads. For the imagination of the vulgar, unless it first sees a thing unadorned, afterwards the supervening energy of the adorner, and lastly, above all the thing itself, perfect and formed, cannot be persuaded that it has two natures, one formless, the other formal, and still further, that which beyond these imparts form; but asserts that the subject is one, and without generation. Hence, perhaps, the ancient theologists and Plato ascribed temporal generations to things without generation, and to things which are perpetually adorned, and regularly disposed, privation of order and ornament, the erron- eous and the boundless, that they might lead men to the knowledge of a formal and effective cause. It is, therefore, by no means wonderful, that though seven sensible monads are never without the heptad, these should be distinguished by science, and that the former should have the relation of a subject, and be analogous to matter, but the latter should correspond to species and form.
*¢ Again, as when water is changed into air, the water
does not become air, or the subject of air, but that
which was the subject of water becomes the subject of
air, so when one number unites itself with another, as for instance the triad with the duad, the species or forms of the two numbers are not mingled, except in their immaterial reasons (or productive principles), in which at the same time that they are separate, they are not impeded from being united, but the quantities of the two numbers which are placed together, become the subject of the pentad. The triad, therefore, is one, and also the tetrad, even in mathematical numbers: for though in the ennead or number nine, you may con- ceive a first, second, and third triad, yet you see one thing thrice assumed; and in short, in the ennead there is nothing but the form of the ennead in the quantity of nine monads. But if you mentally separate its sub- ject, (for form is impartible) you will immediately in- vest it with forms corresponding to its division; for our soul cannot endure to see that which is formless, un- adorned, especially as she possesses the power of invest- ing it with ornament.
“Since also separate numbers possess a demiurgic or
fabricative power, which mathematical numbers imitate,
the sensible world likewise contains images of those
numbers by which it is adorned; so that all things are
in all, but in an appropriate manner in each. ‘The
sensible world, therefore, subsists from immaterial and
energetic reasons, and from more ancient causes. But
those who do not admit that nature herself is full of
productive powers, lest they should be obliged to double
things themselves, these wonder how from things void
of magnitude and gravity, magnitude and gravity are
composed; though they are never composed from
things of this kind which are void of gravity and magni-
tude, as from parts. But magnitude is generated from
essentially impartible elements; since form and matter
are the elements of bodies; and still much more is it
generated from those truer causes which are considered
in demiurgic reasons and forms. Is it not therefore
necessary that all dimensions, and all moving masses,
must from these receive their generation? For either
bodies are unbegotten, like incorporeal natures; or of
things with interval, things without interval are the
causes; of partibles impartibles; and of sensibles and
contraries, things insensible and void of contact: and
we must assent to those who assert that things possess-
ing magnitude are thus generated from impartibles.
Hence the Pythagorean Eurytus, and his followers,
beholding the images of things themselves in numbers,
rightly attributed certain numbers to certain things,
according to their peculiarity. In consequence of this,
he said that a particular number is the boundary of this
plant, and again, another number of this animal; just as
of a triangle 6 is the boundary, of a square 9, and of a
cube 8. As the musician, too, harmonizes his lyre
through mathematical numbers, so nature through her
own natural numbers, orderly arranges, and modulates
her productions.
“Indeed, that numbers are participated by the
heavens, and that there is a solar number, and also
a lunar number, is manifest according to the adage,
even tothe blind. For the restitutions of the heavenly
bodies to their pristine state (aroxatacracets) would
not always be effected through the same things, and in
the same manner, unless one and the same number had
dominion in each. Yet all these contribute to the pro-
cession of the celestial spheres, and are contained by
their perfect number. But there is also a certain
natural number belonging to every animal. For things
of the same species would not be distinguished by
organs after the same manner, nor would they arrive
at puberty and old age about the same time, or generate,
nor would the feetus be nourished or increase, accord-
ing to regular periods, unless they were detained by
the same measure of nature. According to the best
of the Pythagoreans also, Plato himself, number is the
cause of better and worse generations. Hence though
the Pythagoreans sometimes speak of the squares and
cubes of natural numbers, they do not make them to
be monadic, such as the number 9, and the number 27;
but they signify through these names, from similitude,
the progression of natural numbers into, and dominion
about, generations. In like manner, though they call
them equal or double, they exhibit the dominion and
symphony of ideas in these numbers. Hence different
things do not use the same number, so far as they are
different, nor do the same things use a different number,
so far as they are the same.