Lecture 9 — The need for treating of the infinity of the universe

48 This being clear, we must go on to consider the questions which remain. First, is there an infinite body, as the majority of the ancient philosophers thought, or is this an impossibility?

49 The decision of this question, either way, is not unimportant, but rather all-important, to our search for the truth. It is this problem which has practically always been the source of the differences of those who have written about nature as a whole. So it has been and so it must be;

50 since the least initial deviation from the truth is multiplied later a thousandfold. Admit, for instance, the existence of a minimum magnitude, and you will find that the minimum which you have introduced, small as it is, causes the greatest truths of mathematics to totter. The reason is that a principle is great rather in power than in extent; hence that which was small at the start turns out a giant at the end. Now the conception of the infinite possesses this power of principles, and indeed in the sphere of quantity possesses it in a higher degree than any other conception; so that it is in no way absurd or unreasonable that the assumption that an infinite body exists should be of peculiar moment to our inquiry. The infinite, then, we must now discuss, opening the whole matter from the beginning.

51 Every body is necessarily to be classed either as simple or as composite; the infinite body, therefore, will be either simple or composite. But it is clear, further, that if the simple bodies are finite, the composite must also be finite, since that which is composed of bodies finite both in number and in magnitude is itself finite in respect of number and magnitude: its quantity is in fact the same as that of the bodies which compose it. What remains for us to consider, then, is whether any of the simple bodies can be infinite in magnitude, or whether this is impossible. Let us try the primary body first, and then go on to consider the others.

52 The body which moves in a circle must necessarily be finite in every respect, for the following reasons.

53 If the body so moving is infinite, the radii drawn from the center will be infinite. But the space between infinite radii is infinite: and by the space between the radii I mean the area outside which no magnitude which is in contact with the two lines can be conceived as falling. This, I say, will be infinite: first, because in the case of finite radii it is always finite; and secondly, because in it one can always go on to a width greater than any given width; thus the reasoning which forces us to believe in infinite number, because there is no maximum, applies also to the space between the radii. Now the infinite cannot be traversed, and if the body is infinite the interval between the radii is necessarily infinite: circular motion therefore is an impossibility.

54 Yet our eyes tell us that the heavens revolve in a circle, and by argument also we have determined that there is something to which circular movement belongs.

94. After explaining the perfection of the universe and pointing out the parts that make it complete, the Philosopher here begins to inquire into its infinity, because, as is said in Physics III, some have attributed the notion of "perfect" to the infinite. Now something can be said to be infinite in three ways: in one way with respect to magnitude; in another with respect to number, and in a third way with respect to duration.

First, then, he asks whether the universe is infinite according to magnitude;

secondly, whether according to multitude, i.e., whether there is just one world, or an infinitude, or many, at we must now proceed to explain (L. 16; 276a18);

thirdly, whether it is infinite in duration, as though ever existing, at having established these distinctions (L. 22; 279b4).

About the first he does two things:

first he speaks in a prefatory manner about his intention;

secondly, he carries out his proposal, at the body which moves in a circle must necessarily (271b26; [99]).

About the first he does three things:

first he states his intention;

secondly, he assigns the reason for his intention, at the decision of this question (271b4; [96]);

thirdly, he decides upon a method of treatment, at every body is necessarily (271b17; [98]).

95. He says therefore first (271b1) that because it is now clear with respect to the foregoing, namely, that there is no motion contrary to circular motion, and as to the other things mentioned, we must now direct our attention to what remains. And first we must inquire whether there exists any body infinite in act with respect to magnitude, as very many of the early philosophers thought (i.e., all those who posited one material principle, such as fire, or air, or water, or something intermediate); or whether it is impossible that there be a body infinite in act, as was proved in Physics III, supposing, however, that there is no body other than the four elements, according to the opinion of others. Since, however, he has just now proved that there is another body besides the four elements, he therefore repeats this consideration in order that the search for the truth may be more universal.

96. Then, at the decision of this question (271b4), he gives a reason for his intention, from the diversity that happens on account of the aforesaid position.

And first he mentions this consequent diversity, and says that it makes no slight difference to the speculation of truth in natural philosophy whether things are this way or that, i.e., whether or not there exists a body that is infinite according to magnitude. Rather, it does make a difference with respect to the whole universe and every natural consideration. For what has just been said, was in the past, and will be in the future, the source of almost all the contradictions between those who have put forth anything about the whole nature of things. For those who posited one infinite principle assumed that all things come to be by a kind of separation from that principle: thus, on account of the infinitude of that principle, they said that the generation of things does not fail. It is as though someone said that from an infinite mass of dough, loaves of bread could be made ad infinitum. But those who posited finite principles said that things come to be ad infinitum through a reciprocal commingling and separating of the elements.

97. Then, at since the least initial deviation (271b8) he assigns the cause why such diversity follows from this: it is because one who makes a slight departure from the truth in his principles gets 10,000 times farther from the truth as he goes on. This is so because all things that follow depend on their principles. This is especially clear in an error at the crossroads: for one who at the beginning is only a slight distance from the right road gets very far away from it later on. And he gives, as an example of what he is talking about, the case of those who posited a smallest magnitude, as Democritus posited indivisible bodies. By thus introducing a least quantity, he overthrew the most important propositions of mathematics—for example, that any given line can be cut into two halves. The reason for this effect is that a principle, though small in stature, is nevertheless great in power, just as from a small seed a mammoth tree is produced. Hence it is that what is small in the beginning becomes multiplied in the end, because it reaches unto all that to which the power of the principle extends, whether this be true or false. Now the infinite has the nature of a principle (for all who have spoken about the infinite considered it a principle, as was said in Physics III); besides, the infinite has the greatest force with respect to quantity, because it exceeds every given quantity. If, therefore, a principle which is the least in quantity makes a great difference in what follows from it, then much more is this so of the infinite, which is outstanding not only in virtue of being a principle but also in quantity. Consequently, it is neither inappropriate nor unreasonable that a remarkable difference should follow in natural science from the assumption that some body is infinite. And therefore it must be discussed by resuming our consideration from the principle which we accepted above about the difference between simple and composite bodies.

98. Then, at every body is necessarily (271b17), he points out what order must be followed***,*** and says that of necessity every body is either a member of the simple group or of the composite group. Consequently an infinite body must be one or the other. Again, it is plain that if simple bodies are finite in multitude and magnitude, so too must composite body be—for a composite body has as much quantity as the quantity of the simple bodies of which it is composed. However, it has been shown above that simple bodies are finite in multitude, because there is no body other than the ones mentioned. It remains, therefore, to see whether any of the simple bodies is infinite in magnitude, or whether this is impossible. And this we shall show by first arguing from the first body, i.e., the one that is moved circularly; then we shall consider the remaining bodies, namely, those moved with a straight motion.

99. Then at the body which moves (271b26) he shows that there is not an infinite body.

First with reasons proper to the individual bodies;

secondly, with three general reasons applying to all, infinite weight is therefore impossible (L. 13; 274a16).

As to the first he does two things:

first he proves the proposition as to the body moved circularly;

secondly, as to the bodies moved with a straight motion, further, neither that which moves (L. 12; 273a7).

About the first he does two things:

first he proposes his intention and says that it is plain from what will be said that every circularly moved body must be finite (for this is the first of bodies).

100. Then, at if the body so moving is infinite (271b28) he proves his proposition with six arguments, the first of which is this:

if any body is infinite, it cannot be moved circularly; but the first body is moved circularly. Therefore, it is not infinite.

First, then, he proves the conditional proposition as follows: if a circularly moved body is infinite, then the straight lines proceeding from its center are infinite, for they are extended as far as the quantity of the body. But the distance between the infinite lines is infinite. Now someone might say that even if there are infinite lines from the center, yet the distance between them is finite, because every distance is measured according to a straight line, and a finite line can be drawn between two such radii, for example, very close to the center. But it is clear that beyond that line a greater straight line can be drawn between the lines we first mentioned. And therefore he says that he is not speaking of the distance that such lines measure, but that that distance is infinite which is measured by a line beyond which no greater line can be taken, and which touches each of the first lines. That this distance is infinite he proves in two ways.

First, because every such distance between any finite lines proceeding from the center is finite; for the ends of the lines proceeding from the center and of the finite line measuring the greatest distance between them must coincide.

Secondly, he proves the same point because it is possible, if the distance between two measured lines proceeding from the center is given, to take another distance which is greater, just as it is possible to take a number greater than a given number. Hence, just as the infinite is in numbers, so is it in this distance under discussion. From this he argues as follows: the infinite cannot be traversed, as was proved in Physics VI. But if a body be infinite, the distance between the lines proceeding from the center must be infinite, as was proved. But in order that circular motion occur, one line proceeding from the center must reach the position of another. Consequently, it could never happen that anything be moved circularly.

101. Secondly, at yet our eyes tell us that the heavens (272a5) he proves in two ways the destruction of the consequent.

First, because it is evident to sense that the heavens are moved circularly;

secondly, because it was proved above by reason that circular motion belongs to some body.

What remains, therefore, is that it is impossible for the circularly moved body to be infinite.