Lecture 12 — Various reasons why a body moving in a straight line is not infinite

61 Further, neither that which moves towards nor that which moves away from the center can be infinite.

62 For the upward and downward motions are contraries and are therefore motions towards contrary places. But if one of a pair of contraries is determinate, the other must be determinate also. Now the center is determined; for, from whatever point the body which sinks to the bottom starts its downward motion, it cannot go farther than the center. The center, therefore, being determinate, the upper place must also be determinate. But if these two places are determined and finite, the corresponding bodies must also be finite.

63 Further, if up and down are determinate, the intermediate place is also necessarily determinate. For, if it is indeterminate, the movement within it will be infinite; and that we have already shown to be an impossibility. The middle region then is determinate, and consequently any body which either is in it, or might be in it, is determinate.

64 But the bodies which move up and down may be in it, since the one moves naturally away from the center and the other towards it. From this alone it is clear that an infinite body is an impossibility;

65 but there is a further point. If there is no such thing as infinite weight, then it follows that none of these bodies can be infinite. For the supposed infinite body would have to be infinite in weight. (The same argument applies to lightness: for as the one supposition involves infinite weight, so the infinity of the body which rises to the surface involves infinite lightness.)

66 This is proved as follows. Assume the weight to be finite, and take an infinite body, AB, of the weight C. Subtract from the infinite body a finite mass, BD, the weight of which shall be E. E then is less than C, since it is the weight of a lesser mass. Suppose then that the smaller goes into the greater a certain number of times, and take BF bearing the same proportion to BD which the greater weight bears to the smaller. For you may subtract as much as you please from an infinite. If now the masses are proportionate to the weights, and the lesser weight is that of the lesser mass, the greater must be that of the greater. The weights, therefore, of the finite and of the infinite body are equal.

67 Again, if the weight of a greater body is greater than that of a less, the weight of GB will be greater than that of FB; and thus the weight of the finite body is greater than that of the infinite.

68 And, further, the weight of unequal masses will be the same, since the infinite and the finite cannot be equal.

69 It does not matter whether the weights are commensurable or not. If they are incommensurable the same reasoning holds. For instance, suppose E multiplied by three is rather more than C: the weight of three masses of the full size of BD will be greater than C. We thus arrive at the same impossibility as before.

70 Again we may assume weights which are commensurate; for it makes no difference whether we begin with the weight or with the mass. For example, assume the weight E to be commensurate with C, and take from the infinite mass a part BD of weight E. Then let a mass BF be taken having the same proportion to BD which the two weights have to one another. (For the mass being infinite you may subtract from it as much as you please.) These assumed bodies will be commensurate in mass and in weight alike.

71 Nor again does it make any difference to our demonstration whether the total mass has its weight equally or unequally distributed. For it must always be Possible to take from the infinite mass a body of equal weight to BD by diminishing or increasing the size of the section to the necessary extent. From what we have said, then, it is clear that the weight of the infinite body cannot be finite. It must then be infinite. We have therefore only to show this to be impossible in order to prove an infinite body impossible.

72 But the impossibility of infinite weight can be shown in the following way.

73 A given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time,

74 the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will take half as long over a given movement.

75 Further, a finite weight traverses any finite distance in a finite time.

76 It necessarily follows from this that infinite weight, if there is such a thing, being, on the one hand, as great and more than as great as the finite, will move accordingly, but being, on the other hand, compelled to move in a time inversely proportionate to its greatness, cannot move at all. The time should be less in proportion as the weight is greater. But there is no proportion between the infinite and the finite: proportion can only hold between a less and a greater finite time. And though you may say that the time of the movement can be continually diminished, yet there is no minimum.

77 Nor, if there were, would it help us. For some finite body could have been found greater than the given finite in the same proportion which is supposed to hold between the infinite and the given finite; so that an infinite and a finite weight must have traversed an equal distance in equal time. But that is impossible.

78 Again, whatever the time, so long as it is finite, in which the infinite performs the motion, a finite weight must necessarily move a certain finite distance in that same time.

110. After showing that the circularly moved body is not infinite, the Philosopher here shows the same for the body moved with a straight motion, whether from the middle [center] or to the middle [center].

First he proposes what he intends and says that just as the circularly moved body cannot be infinite, so, too, the body which is moved with a straight motion, whether from the middle or to the middle, cannot be infinite.

Secondly, he shows the proposition, at for the upward and downward motions (273a8; [111]),

and this first on the part of the places which are proper to such bodies;

secondly, on the part of heaviness and lightness, through which such bodies are moved to their proper places, at but there is a further point, at but there is a further point (273a22; [114]).

About the first he does two things:

first, he shows the proposition as to the extreme bodies, of which one is absolutely heavy, namely, earth, and the other absolutely light, namely, fire, at 111;

secondly, as to the intermediate bodies, which are air and water, at further, if up (273a15; [112]).

111. He proposes therefore first (273a8) that motions of the kind that are up and down, or from the middle and to the middle, are contrary motions. For contrary local motions are ones to contrary places, as has been said, and as was shown in Physics V. It remains, therefore, that the proper places to which such bodies are carried are contrary. Now, we could have at once concluded from this that such places are determinate: for contraries are things which are most distant; but places that are the greatest distance apart are determinate, for the greatest distance is such that none is greater, whereas in infinites a greater and greater distance is always possible. Hence if the places were infinite, contrariety of places would cease. However, Aristotle passes over this argument as manifest and proceeds by another tack. For it is true that if one contrary is determinate, so too the other, because contraries are members of one genus. But the middle of the world which is the midway terminus of a downward motion is determinate—for from whatever part of the heavens something is moved downward (which exists under the upper part facing the heavens) it can travel no farther in its journey from the heavens than the middle: for if it should go beyond the middle, it would now get closer to the heavens and thus would be moved upward. Accordingly, it is clear that the middle place is determinate. It is likewise clear from the aforesaid that the middle having been determined, i.e., the downward place, then the upward place is also necessarily determinate, because they are contraries. And if both are determinate, then the bodies which are apt to be in these places must be finite. Hence it is clear that the extreme bodies subject to straight motion are finite.

112. Then, at further, if up (273a15) he shows the same thing for the intermediate bodies.

First he proposes a conditional, namely, that if up and down are determinate, the intermediate place must be determinate. And he proves this with two arguments.

The first of which is this: if, when the extremes were determinate, the intermediate should not be determinate, it would follow that a motion from one extreme to the other would be infinite, on account of the infinite intermediate. But that this is impossible has been shown previously in the discussion about circular motion where it was pointed out that motion through the infinite cannot be completed. Consequently, the intermediate place is determinate. Thus, since the thing in place is commensurate with the place, it follows that the body actually existing in this place or that can exist there, is finite.

113. He gives the second argument at but the bodies (273a19) and it is this: a body that is moved up or down can reach the state of existing in such a place. This is clear from the fact that such a body is apt to be moved from the middle or to the middle, i.e., it has a natural inclination to this or that place. Now a natural inclination cannot be in vain, because God and nature do nothing in vain, as was had above. Consequently, whatever is naturally moved upward or downward can have its own motion terminated so as to be up or down. But this could not be, if the intermediate place were infinite. Consequently, the intermediate place is finite; so, too, is the body existing in it.

Therefore, in summary he concludes from the foregoing that it is clear that no body can be infinite.

114. Then, at but there is a further point (273a22) he shows that there is no infinite heavy or light body by an argument based on heaviness or lightness. It is this: if a heavy or a light body be infinite, then heaviness or lightness must be infinite. But this is impossible. Therefore, the first supposition [of non-infinity] must be true. With respect to this, then, he does two things:

first, he proves the conditional proposition, at (273a22);

secondly, he proves the destruction of the consequent, at but the impossibility of infinite (273b29; [119]).

As to the first he does two things:

first he proposes what he intends and says: if there is no infinite heaviness, none of these, i.e., no heavy body, will be infinite, for the heaviness of an infinite body must be infinite (273a22). And the same goes for a light body—for if the heaviness of a heavy body is infinite, the lightness, too, must be infinite, if one supposes some light body carried upward to be infinite.

115. Secondly, at this is proved (273a27), he proves what he had supposed.

First he presents the proof;

secondly, he dismisses some objections, at it does not matter whether the weights (273b10; [116]).

First, then, he presents an argument leading to an impossibility (66273a27) and it is this: if what was said above is not true, then suppose that the heaviness of an infinite body is finite, and let AB be the infinite body and G its finite heaviness. From this infinite body take away a finite part which is the magnitude BD, which is necessarily much less than the whole infinite body. Now the heaviness of this smaller body is less; consequently, the heaviness of BD is less than the heaviness G, which is the heaviness of the whole infinite body. Let this lesser heaviness be E. Now let E be a measure of the greater but finite heaviness G—for example, E is a third part of the whole G. Now take from the infinite body a part to be added to the finite body BD, according to the proportion by which G exceeds E, and let this exceeding body be BZ, in such a way that the ratio between the lesser heaviness E to the greater G is the same as that between the body BD and body BZ. That this can be done is proved by the fact that from an infinite body can be taken away as much as is needed, since, as is said in Physics II, the infinite is that whose quantity is such that, as much as is taken away, there always remains something beyond to be taken.

Therefore, with these presuppositions, he now argues to three incompatible consequences.

First he reasons in this manner. The ratio of heavy magnitudes is the same as the ratio of their heaviness—for we see that a larger body has more heaviness and a smaller body less. But the ratio of E to G, i.e., of the lesser to the more heavy is the same as that of BD to BZ, i.e., of the smaller body to the larger, was was supposed. Therefore, since E is the heaviness of BD, it will follow that G is the heaviness of the body BZ. But G was assumed to be the heaviness of the whole infinite body. Therefore the numerical value of the heaviness of the finite and of the infinite body will be the same. But this is unacceptable, because it will follow that the whole remainder of the infinity body will have no heaviness. Therefore, the first is impossible, namely, that the heaviness of an infinite body be finite.

Secondly, at again, if the weight of a greater (273b6), he leads to another unacceptable consequence. For since it is possible to take from an infinite body as much as one wishes, as has been said, let yet another part be taken from the infinite body and added to the body BZ. And let G be one finite body greater than the finite body BZ. Now the heaviness of larger body is greater, as was said above. Therefore, the heaviness of BI is greater than the heaviness G which was proved to be the heaviness of the body BZ. But it was assumed in the beginning that G was the heaviness of the whole infinite body. Therefore, the heaviness of a finite body will be greater than that of an infinite body. This is impossible. Therefore, the first is impossible, namely, that the heaviness of an infinite body be finite.

Thirdly, at and, further, the weight of unequal masses (273b8), he leads to the third incompatibility, namely, that the heaviness of unequal magnitudes would be the same. This clearly follows from the foregoing, because the infinite is not equal to the finite, because it is greater than it. Hence, since these conclusions are impossible, it is impossible for the heaviness of an infinite body to be finite.

116. Then, at it does not matter (273b10) he dismisses two objections against the foregoing argument:

first, the first;

secondly, the second, at nor again does it make any difference (273b23; [118]).

The first objection is that he had supposed in the preceding argument that the lesser heaviness E is a numerical measure of the greater heaviness G. Now this can be denied, for not every greater is measured by a smaller, because a line of 3 hands' length is not a measure of a line of hands' length.

But the Philosopher excludes this objection in two ways.

First, because it makes no difference, so far as the conclusion is concerned, whether the two heavinesses, namely, the greater and the less, in question are commensurate, so that the less measures the greater, or not, for the same reasoning holds in either case. For it is necessary that the lesser, taken a certain number of times, either measure or exceed the greater: for example, the product of 2 taken 3 times measures 6 [for 3 times 2 equals 6], while it does not measure 5, but exceeds it. Accordingly, if the heaviness E does not measure the heaviness G, let E be such that 3 times E measures a heaviness greater than the heaviness G. And so in this case the same impossibility as before results, because, if we had taken from the infinite body three magnitudes of quantity BD, the heaviness of such a magnitude will be 3 times that of heaviness E, which is assumed to be the heaviness of the body BD. But a heaviness that is 3 times E is greater (according to our assumptions) than the heaviness G which is the heaviness of the infinite body. Wherefore, the same impossibility as before follows, namely, that the heaviness of a finite body exceed that of an infinite body.

117. Secondly, at again (b) we may assume (273b15), he excludes the same objection in another way. And he says that we can assume in the demonstration under discussion that the two heavinesses are commensurate, in such a way that E is commensurate to G. For above we first took from the magnitude a part BD whose heaviness we called E, and this was grounds for saying that E does not measure G. But it makes no difference, so far as the proposition is concerned, whether we begin with the heaviness (by taking any part we want) or with the magnitude so taken. For example, we might begin with the heaviness and take a part of it, namely, E, which measures the whole, namely, G, and then we can take from the infinite body a part BD whose heaviness is E and proceed as above, so that as the heaviness E is to heaviness G, so the magnitude BD is to a greater magnitude BZ. This we can do, because, since the magnitude of the whole body is infinite, as much can be taken from it as we please. By taking the parts of the heaviness and of the magnitude in this way, it will follow that the magnitudes and the heavinesses will be mutually commensurate, i.e., the lesser heaviness will measure the greater, and the smaller magnitude the larger.

118. Then at nor again does it make any difference (273b23) he excludes a second objection. For he had supposed that the magnitudes are proportional to the heavinesses. Now this is true in bodies having similar parts, for where there is like heaviness throughout the whole, there must be more heaviness in the larger part. But in a body of unlike parts this is not necessarily so, because the heaviness of a smaller part could be greater than that of a larger part, just as a smaller part of earth is heavier than a larger part of water.

This objection he therefore excludes by saying that it makes no difference to the aforesaid demonstration whether the infinite magnitude in question is homogeneous, i.e., of similar parts, or heterogeneous, i.e., of dissimilar parts. For from an infinite body we can take as much as we wish, either adding or subtracting; hence we can assume certain parts to have a heaviness equal to the part taken first, namely BD, whether the parts taken later are larger or smaller in magnitude. For if we should first take BD as having 3 Cubits and having heaviness E, and then take many other parts, for example, of 10 cubits, to make an equal heaviness, it will be the same as if we had taken another equal part having the same heaviness. Consequently, the same impossibility follows.

Therefore, having presented his demonstration and excluded the objections thereto, he concludes from the foregoing that the heaviness of an infinite body cannot be finite. Therefore, it must be infinite. If then, as he will immediately prove, infinite heaviness is impossible, the consequence is that it is impossible for there to be an infinite body.

119. Then, at but the impossibility of infinite (273b29) he proves what he had supposed, namely, that there cannot be infinite heaviness. And in this he destroys the consequent of the previously posited conditional. Concerning this he does two things:

First, he proposes what he intends, and says that we must still show from what will follow that infinite heaviness is impossible.

120. Secondly, at a given weight moves (273b30) he proves the proposition.

First he lays down certain presuppositions;

secondly, he uses them in his argument, at it necessarily follows from this (274a3; [121]);

thirdly, he excludes an objection, at nor, if there were (274a10; [122]).

First, then, he presents three suppositions.

The first of these (273b30) is that, if such a heaviness, i.e., of some certain amount, moves so much, i.e., throughout a definite magnitude of space, in this time, i.e., a determined time, then necessarily as much and more, i.e., a greater heaviness that has as much and something more than a lesser, will move through as great a magnitude in less time—for by as much as a moving power is stronger, by that much is its motion swifter. Consequently, it will traverse an equal distance in less time, as is proved in Physics VI.

The second supposition is at the times being in inverse proportion (273b32) and follows from the first. For if a greater heaviness moves in less time, then the analogy, i.e., proportion between heavinesses and times is the same, but inversely so, i.e., if half the heaviness moves something in a certain amount of time, then double that amount moves in its half, i.e., in half the time.

The third supposition, at further, a finite (274a2), states that a finite heaviness moves through a finite magnitude of space in a certain finite time.

121. Then, at it necessarily follows from this (274a3), he argues from these premises. If an infinite heaviness should exist, two contradictories will follow: namely, that something would be moved according to it, and not moved. That it would be moved follows, indeed, from the first supposition—for if a certain heaviness moves in a certain amount of time, a greater will move more swiftly, i.e., in less time. Since, therefore, an infinite weight is greater than a finite, then, if a finite moves through a definite distance in a definite time, as the third supposition says, the consequence is that an infinite heaviness will move as much and more, i.e., either through a greater distance in the same time, or through an equal distance in less time, which is to be moved more swiftly. But that something is not moved according to infinite heaviness follows from the second supposition. For a thing must be moved in proportion to the greatness of the weight in inverse proportion, i.e., the greater weight will move in less time. But there can be no proportion between an infinite and a finite weight, although there is a proportion between less time and more time, provided the time is finite. Consequently, there can be no time given in which an infinite weight can move, but something will always be able to be taken as moved in less time than the time in which an infinite weight moves, for there can be taken no least time in which an infinite weight can move in the sense that it would be impossible for something to be moved in a lesser time. Now the reason why no such least time can be assumed is that since all time is divisible, as is any continuum, it is always possible to take a time smaller than any given time, i.e., a part of the divided time. Consequently, an infinite heaviness cannot exist.

122. Then, at nor, if there were (274a10), he excludes an objection. For someone could say that there is a least time, namely, an indivisible time, in which the infinite heaviness moves, just as some have posited certain minimum and indivisible magnitudes. But he excludes this objection:

first, he shows that an impossibility follows upon assuming a minimum time and that an infinite heaviness moves in that time;

secondly, he shows that the same impossibility follows if an infinite heaviness should move in any amount of time, even not the minimum, at again, whatever the time (274a13; [123]).

He says therefore first (274a10) that even if there were a minimum time, it would not help in escaping the impossibility that follows from the assumption of infinite heaviness. For although we posit a minimum time, we do not exclude a ratio of this time to a greater time, for this minimum time will be a part of a greater time, just as one is part of number, which allows it to have a ratio to every number. But an indivisible which is not part of a divisible has no proportion to it, just as a point is not a part of a line and therefore there is no proportion of a point to a line. So let us take another heaviness which is finite and as much heavier proportionally than the finite heaviness that moved something in more time than the infinite heaviness, as the minimum time of the infinite heaviness is less than the greater time of the other finite heaviness. For example, let E be the infinite heaviness, and B the minimum time in which it moves, and let G be the finite heaviness that moves something in more time D than time B. Then let F be the other heaviness which is greater than G in the proportion that D exceeds B. Then, since the lessening of time corresponds to the increasing of heaviness, it will follow that the heaviness F, which is finite, will move something in the same time as the infinite heaviness. But this is impossible. It should be noted that, just as there is no proportion between a point and a line, so there is none between an instant and time, because an instant is not a part of time. Consequently, the only way Aristotle's argument could be destroyed, would be by positing that an infinite heaviness should move in an instant. But that is impossible, as was proved in Physics VI, namely, that any motion should occur in an instant.

123. Then at again, whatever the time (274a18), he shows that the same impossibility follows in whatever time we assume an infinite heaviness to move. And this is what he says, namely, that if an infinite heaviness should move in any finite time whatever, even though it be not the minimum, it is still necessary that in that time a finite heaviness could move through a finite distance—for one will be able to take an excess of weight corresponding to a lessening of the time, as was said above. Consequently, it is clearly impossible for an infinite heaviness to exist; and the same argument holds for lightness.