determinate accidents should follow upon a determinate form; and among these accidents is quantity. So every natural body has a greater or smaller determinate quantity. Hence it is impossible for a natural body to be infinite. The same appears from movement; because every natural body has some natural movement; whereas an infinite body could not have any natural movement; neither direct, because nothing moves naturally by a direct movement unless it is out of its place; and this could not happen to an infinite body, for it would occupy every place, and thus every place would be indifferently its own place. Neither could it move circularly; forasmuch as circular motion requires that one part of the body is necessarily transferred to a place occupied by another part, and this could not happen as regards an infinite circular body: for if two lines be drawn from the centre, the farther they extend from the centre, the farther they are from each other; therefore, if a body were infinite, the lines would be infinitely distant from each other; and thus one could never occupy the place belonging to any other.
The same applies to a mathematical body. For if we imagine a mathematical body actually existing, we must imagine it under some form, because nothing is actual except by its form; hence, since the form of quantity as such is figure, such a body must have some figure, and so would be finite; for figure is confined by a term or boundary.
Reply to Objection 1: A geometrician does not need to assume a line actually infinite, but takes some actually finite line, from which he subtracts whatever he finds necessary; which line he calls infinite.
Reply to Objection 2: Although the infinite is not against the nature of magnitude in general, still it is against the nature of any species of it; thus, for instance, it is against the nature of a bicubical or tricubical magnitude, whether circular or triangular, and so on. Now what is not possible in any species cannot exist in the genus; hence there cannot be any infinite magnitude, since no species of magnitude is infinite.
Reply to Objection 3: The infinite in quantity, as was shown above, belongs to matter. Now by division of the whole we approach to matter, forasmuch as parts have the aspect of matter; but by addition we approach to the whole which has the aspect of a form. Therefore the infinite is not in the addition of magnitude, but only in division.
Reply to Objection 4: Movement and time are whole, not actually but successively; hence they have potentiality mixed with actuality. But magnitude is an actual whole; therefore the infinite in quantity refers to matter, and does not agree with the totality of magnitude; yet it agrees with the totality of time and movement: for it is proper to matter to be in potentiality.
Whether an infinite multitude can exist?
Objection 1: It seems that an actually infinite multitude is possible. For it is not impossible for a potentiality to be made actual. But number can be multiplied to infinity. Therefore it is possible for an infinite multitude actually to exist.
Objection 2: Further, it is possible for any individual of any species to be made actual. But the species of figures are infinite. Therefore an infinite number of actual figures is possible.
Objection 3: Further, things not opposed to each other do not obstruct each other. But supposing a multitude of things to exist, there can still be many others not opposed to them. Therefore it is not impossible for others also to coexist with them, and so on to infinitude; therefore an actual infinite number of things is possible.
On the contrary, It is written, "Thou hast ordered all things in measure, and number, and weight" (Wis. 11:21).
I answer that, A twofold opinion exists on this subject. Some, as Avicenna and Algazel, said that it was impossible for an actually infinite multitude to exist absolutely; but that an accidentally infinite multitude was not impossible. A multitude is said to be infinite absolutely, when an infinite multitude is necessary that something may exist. Now this is impossible; because it would entail something dependent on an infinity for its existence; and hence its generation could never come to be, because it is impossible to pass through an infinite medium.
A multitude is said to be accidentally infinite when its existence as such is not necessary, but accidental. This can be shown, for example, in the work of a carpenter requiring a certain absolute multitude; namely, art in the soul, the movement of the hand, and a hammer; and supposing that such things were infinitely multiplied, the carpentering work would never be finished, forasmuch as it would depend on an infinite number of causes. But the multitude of hammers, inasmuch as one may be broken and another used, is an accidental multitude; for it happens by accident that many hammers are used, and it matters little whether one or two, or many are used, or an infinite number, if the work is carried on for an infinite time. In this way they said that there can be an accidentally infinite multitude.
This, however, is impossible; since every kind of multitude must belong to a species of multitude. Now the species of multitude are to be reckoned by the species of numbers. But no species of number is infinite; for every number is multitude measured by one. Hence it is impossible for there to be an actually infinite multitude, either absolute or accidental. Likewise multitude in nature is created; and everything created is comprehended under some clear intention of the Creator; for no agent acts aimlessly. Hence everything created must be comprehended in a certain number. Therefore it is impossible for an actually infinite multitude to exist, even accidentally. But a potentially infinite multitude is possible; because the increase of multitude follows upon the division of magnitude; since the more a thing is divided, the greater number of things result. Hence, as the infinite is to be found potentially in the division of the continuous, because we thus approach matter, as was shown in the preceding article, by the same rule, the infinite can be also found potentially in the addition of multitude.
Reply to Objection 1: Every potentiality is made actual according to its mode of being; for instance, a day is reduced to act successively, and not all at once. Likewise the infinite in multitude is reduced to act successively, and not all at once; because every multitude can be succeeded by another multitude to infinity.
Reply to Objection 2: Species of figures are infinite by infinitude of number. Now there are various species of figures, such as trilateral, quadrilateral and so on; and as an infinitely numerable multitude is not all at once reduced to act, so neither is the multitude of figures.
Reply to Objection 3: Although the supposition of some things does not preclude the supposition of others, still the supposition of an infinite number is opposed to any single species of multitude. Hence it is not possible for an actually infinite multitude to exist.
THE EXISTENCE OF GOD IN THINGS (FOUR ARTICLES)
Since it evidently belongs to the infinite to be present everywhere, and in all things, we now consider whether this belongs to God; and concerning this there arise four points of inquiry:
(1) Whether God is in all things?
(2) Whether God is everywhere?
(3) Whether God is everywhere by essence, power, and presence?
(4) Whether to be everywhere belongs to God alone?
Whether
