Lecture 14 — No sensible body is infinite

90 That in general it is impossible for the infinite to be acted upon by the finite or to act upon it may be shown as follows.

91 Let A be an infinite, B a finite, C the time of a given movement produced by one in the other. Suppose, then, that A was heated, or impelled, or modified in any way, or caused to undergo any sort of movement whatever, by in the time C. Let D be less than B; and, assuming that a lesser agent moves a lesser patient in an equal time, call the quantity thus modified by D, E. Then, as D is to B, so is E to some finite quantum. We assume that the alteration of equal by equal takes equal time, and the alteration of less by less or of greater by greater takes the same time, if the quantity of the patient is such as to keep the proportion which obtains between the agents, greater and less. If so, no movement can be caused in the infinite by any finite agent in any time whatever. For a less agent will produce that movement in a less patient in an equal time, and the proportionate equivalent of that patient will be a finite quantity, since no proportion holds between finite and infinite.

92 Nor, again, can the infinite produce a movement in the finite in any time whatever. Let A be an infinite, B a finite, C the time of action. In the time C, D will produce that motion in a patient less than B, say F. Then take E, bearing the same proportion to D as the whole BF bears to F. E will produce the motion in BF in the time C. Thus the finite and infinite effect the same alteration in equal times. But this is impossible; for the assumption is that the greater effects it in a shorter time. It will be the same with any time that can be taken, so that there will no time in which the infinite can effect this movement.

93 And, as to infinite time, in that nothing can move another or be moved by it. For such time has no limit, while the action and reaction have.

94 Nor can infinite be acted upon in any way by infinite. Let A and B be infinites, CD being the time of the action A of upon B. Now the whole B was modified in a certain time, and the part of this infinite, E, cannot be so modified in the same time, since we assume that a less quantity makes the movement in a less time. Let E then, when acted upon by A, complete the movement in the time D. Then, as D is to CD, so is E to some finite part of B. This part will necessarily be moved by A in the time CD. For we suppose that the same agent produces a given effect on a greater and a smaller mass in longer and shorter times, the times and masses varying proportionately. There is thus no finite time in which infinites can move one another. Is their time then infinite? No, for infinite time has no end, but the movement communicated has.

95 If therefore every perceptible body possesses the power of acting or of being acted upon, or both of these, it is impossible that an infinite body should be perceptible.

96 All bodies, however, that occupy place are perceptible. There is therefore no infinite body beyond the heaven. Nor again is there anything of limited extent beyond it. And so beyond the heaven there is no body at all. For if you suppose it an object of intelligence, it will be in a place—since place is what 'within' and 'beyond' denote—and therefore an object of perception. But nothing that is not in a place is perceptible.

137. After showing that a sensible body is not infinite with a reason based on local motion, the Philosopher here shows the same thing with a reason based on action and passion, which follow upon every motion. Concerning this he does two things:

first he demonstrates the proposition;

secondly, he excludes an objection, at all bodies, however (275b6; [144]).

138. With regard to the first, he gives the following argument: No infinite body has active or passive power or both; but every sensible body has active or passive power or both. Therefore, no sensible body is infinite. Then with regard to this he does two things:

first he proves the major premise;

secondly, he presents the minor and conclusion, at if therefore every perceptible body (275b4; [143]).

About the first he does two things:

first, he proposes what he intends and says that it is clear from what will be said that not only is it impossible for something infinite to be moved locally but that universally it is impossible for something infinite to be acted upon or to act upon a finite body.

Secondly, at let A be an infinite (275a2), he proves his proposition.

First he shows that the infinite is not acted upon by the finite, at nor, again, can the infinite (275a2; [139]);

secondly, that the finite is not acted upon by the infinite (275a14; [140]);

thirdly, that the infinite is not acted upon by the infinite, at nor can infinite (275a24; [142]).

139. He says therefore first (275a2) that if an infinite body is acted upon by a finite, let A be an infinite body and B a finite body and, since every motion occurs in time, let G be the time in which B moves or A has been moved. If, therefore, we posit that A, which is the infinite body, is altered by B, which is the finite body, say heated or carried, i.e., moved locally, or affected in any other way, e.g., cooled or moistened, or moved in any way, in time G, let us take one part of the mover B, i.e., a part D (and it makes no difference, so far as the proposition is concerned, if D be some other body less than B). Now it is clear that a smaller body moves a smaller mobile in an equal time (supposing, of course, that there is in the smaller body less power—which must be said, if it is a body of like parts—hence the lesser power moves a smaller body in an equal time). Therefore, let E be a body which is altered or any other way moved by D in the time G, taking E as a part of the infinite whole A. But since both D and B are finite, and since any two finite bodies are mutually proportionate, then, according to the ratio of D to B, let there be taken the proportion of E to any other larger finite body, for example, F.

Having posited these preliminaries, he makes some suppositions.

The first of these is that an altering cause which is equal in magnitude and power will alter an equal body in equal time.

A second is that a smaller altering body will alter a smaller in equal time, the result being that one moved body will be less than the other moved body according to a ratio of something greater to something less, i.e., in the same proportion that the larger moving body exceeds the smaller moving body.

From these preliminaries, therefore, he concludes that the infinite cannot be moved by any finite in any time. For something less than the infinite will in an equal time be moved by that body which is less than the body moving the infinite; in other words, E, which is less than A, will be moved by D, which is less than B, according to our suppositions. But what is "analogous" to E, i.e., in the same ratio to E, as B to D, is finite, for it cannot be said that A, which is infinite, is to E, as B is to D, because the infinite has no proportion to the finite. Now on the assumption that something finite is to E as B is to D, then commutatively B is to that finite, as D is to E. But D moves E in time G; therefore B moves the finite in time G. But G was the time in which it was supposed that B moved the infinite whole A. Therefore the finite will move a finite and an infinite in the same time.

140. Then, at the infinite (275a14), he proves that an infinite body does not move a finite body in any time.

First he shows that it does not move it in finite time;

secondly, not in infinite time, at and, as to infinite (275a22; [141]).

He says therefore first (275a14) that neither will an infinite body move a finite body in any time, namely, determinate time. For if the contrary should be the case, let Abe the infinite body, and B or BZ the finite body moved by it, and G the time in which it is being moved. Let D be a finite part of the infinite body A. And because a lesser moves a smaller in equal time, then a finite body D in time G moves Z, a body smaller than B, but a part of B. Now because the whole BZ is proportionate to Z, let it be taken that the whole BZ is to Z, as E is to D, each of which is part of the infinite. Therefore, commutatively, E is to BZ in the same proportion as D is to Z. But D moves Z in time G; therefore E will move BZ in time G. But in time G, BZ was being moved by the infinite body A. It follows, therefore, than an infinite and a finite are altering or somehow moving one and the mane mobile in the same amount of time. But this is impossible — for it was supposed above that a greater mover moves an equal mobile in less time, because it moves more swiftly. Consequently, it is impossible for the finite to be moved by the infinite in time G; and the same follows no matter what finite time is taken. Hence there is no finite time possible in which the infinite moves the finite.

141. Then, at and, as to infinite (275a22), he shows that this cannot occur in infinite time. For it is not possible that in an infinite time something shall have moved or shall have been moved—because infinite time has no end, whereas every action or passion does have an end, for nothing acts or is acted upon except in order to reach some end. What remains, therefore, is that an infinite does not move a finite in infinite time.

142. Then, at nor can infinite (275a24), he proves that the infinite does not move the infinite. And he says that an infinite cannot undergo anything from an infinite with respect to any species of motion at all. Otherwise let A be the infinite body which is acting, and B the infinite body acted upon, and DG the time in which B underwent something from A, and let E be a part of the infinite mobile B. Now, since the entire B has been modified by A in the entire time DG, it is clear that E, which is part of B, was not being moved in this whole time. For we must suppose that a smaller mobile is moved in less time by the same mover—for to the extent that a mobile is more overcome by a mover, the more swiftly is it moved by it. So let E, which is less than B, be moved by A in a time D which is part of the whole time GD. Now D is proportionate to GD, since both are finite. Let us assume, therefore, that E has the same ratio to some larger part of the infinite mobile as D has to GD. Then that finite mobile greater than E must be moved by A in time GD, for we must suppose that a larger and a smaller mobile are moved in more and less time when the same mover is acting, in such a way that the division of the mobiles corresponds to the ratio of the times. Since, therefore, the ratio of that finite to E equals the ratio of the entire time DG to D, then commutatively, we must say that the ratio of the entire time DG to that larger finite mobile is as the ratio of time D to mobile E. But E is moved by A in time D; therefore, that greater finite mobile will be moved by A in time DG. Hence the finite and the infinite will be moved in the same amount of time—which is impossible. And the same impossibility follows whatever be the finite time assumed. Consequently it is impossible for an infinite to be moved by an infinite in finite time.

It remains, therefore, that if it is moved, it is moved in infinite time. But that, too, is impossible, as was proved above, because infinite time has no end, but everything which is moved has an end to its motion—for although the whole motion of the heaven does not have an end, one revolution does.

It is therefore plain that the infinite has neither active nor passive power.

143. Then, at if therefore (275b4), assuming the minor premise, he draws the conclusion and says that every sensible body has active or passive power or both. He says sensible body here to differentiate from mathematical body, so that the former means every natural body which, as such, is apt to cause motion or be moved. Thus he concludes that it is impossible for a sensible body to be infinite.

144. Then, at all bodies, however (275b6), he excludes a certain objection—for someone could say that there is outside the heavens an intelligible body which is infinite. And he says that all bodies in place are sensible. For they are not mathematical bodies, because these do not have place except in a metaphorical sense, as is said in On Generation I. Now place is not needed except for motion, as is said in Physics IV, and only sensible and natural bodies are subject to motion—for mathematical things are outside of motion. Consequently, it is plain that all bodies in place are sensible. From this he concludes that there is not an infinite body outside the heaven; and indeed, more universally, that no body exists outside the heaven, either absolutely, i.e., namely, an infinite body, or in a certain respect (or up to a certain point), i.e., a finite body. Since bodies are either finite or infinite, it follows that no body at all exists outside the heaven. For if you should say that this body is intellectual, it will follow that it is in a place on account of your assuming that it is outside the heaven—because "outside" and "within*"* imply place. Consequently, it follows that if there is a body outside the heaven, then, whether it is finite or infinite, it is sensible, since there is no sensible body which does not exist in a place—for even the heaven is somehow in place, as is plain from Physics IV.

So it is manifest according to these words that no intelligible body, finite or infinite, is outside the heaven, because outside of signifies place, and nothing is in place except a sensible body. It is also manifest that no infinite sensible body exists outside the heavens, for it was shown above that no sensible body is infinite. But the fact that no sensible finite body exists outside the heavens he does not prove here but supposes it, unless perhaps it is proved by the fact that every sensible body is in a place, and all places are contained within the heavens and determined by the three local motions mentioned above, namely, those around the middle, from the middle, and to the middle.