The Essential Galileo

by Galilei, Galileo, Finocchiaro, Maurice A.

  SIMP. So far as I remember, Aristotle5 inveighs against the ancient view that a vacuum is a necessary prerequisite for motion and that the latter could not occur without the former. In opposition to this view Aristotle shows that it is precisely the phenomenon of motion, as we shall see, which renders untenable the idea of a vacuum. His procedure is the following. He begins with two assumptions. The first concerns bodies of different weights moving in the same medium; the second, one and the same body [106] moving in different media. In the first case he supposes bodies of different weights to move in one and the same medium with different speeds that stand to one another in the same ratio as the weights; so that, for example, a body that is ten times as heavy as another will move ten times as rapidly as the other. In the second case he assumes that the speeds of one and the same body moving in different media are in inverse ratio to the densities of these media; thus, for instance, if the density of water were ten times that of air, the speed in air would be ten times greater than in water. From this second supposition, he gives the following demonstration: since the thinness of a vacuum differs infinitely from that of any medium filled with matter however rare, any body that moves in a plenum through a certain space in a certain time ought to move through a vacuum instantaneously; but instantaneous motion is an impossibility; it is therefore impossible that the existence of a vacuum should result from the existence of motion.

  SALV. The argument is, as you see, ad hominem; that is, it is directed against those who thought the vacuum a prerequisite for motion. Now if I admit the argument to be conclusive and concede also that motion cannot take place in a vacuum, the assumption of a vacuum considered absolutely and not with reference to motion is not thereby invalidated. But to tell you what the ancients might possibly have replied and in order to better understand just how conclusive Aristotle’s demonstration is, we may, in my opinion, deny both of his assumptions. And as to the first, I greatly doubt that Aristotle ever tested by experiment whether it be true that two stones, one weighing ten times as much as the other, if allowed to fall at the same instant from a height of, say, one hundred cubits, would so differ in speed that when the heavier had reached the ground, the other would not have fallen more than ten cubits.

  SIMP. His language would seem to indicate that he had tried the experiment, because he says, “we see the heavier”; now the word see shows that he had made the experiment.

  SAGR. But I, Simplicio, who have made the test can assure you [107] that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound, provided both are dropped from a height of two hundred cubits.

  SALV. But, even without further experiment, it is possible to prove clearly, by means of a short and conclusive argument, that a heavier body does not move more rapidly than a lighter one, provided both bodies are of the same material—in short, such as those mentioned by Aristotle. But tell me, Simplicio, whether you admit that each falling body acquires a definite speed fixed by nature, a velocity that cannot be increased or diminished except by the use of force or resistance.

  SIMP. There can be no doubt but that one and the same body moving in a single medium has a fixed velocity that is determined by nature and that cannot be increased except by the addition of some impetus or diminished except by some resistance that retards it.

  SALV. If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion?

  SIMP. You are unquestionably right.

  SALV. But if this is true, and if a large stone moves with a speed of, say, eight units while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight units. But the two stones when tied together make a stone larger than that which before moved with a speed of eight units. Hence the heavier body moves with less speed than the lighter—an effect that is contrary to [108] your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly.

  SIMP. I am all at sea because it appears to me that the smaller stone when added to the larger increases its weight, and by adding weight I do not see how it can fail to increase its speed or, at least, not to diminish it.

  SALV. Here again you are in error, Simplicio, because it is not true that the smaller stone adds weight to the larger.

  SIMP. This is, indeed, quite beyond my comprehension.

  SALV. It will not be beyond you once I have shown you the equivocation under which you are laboring. Note that it is necessary to distinguish between heavy bodies in motion and the same bodies at rest. A large stone placed in a balance not only acquires additional weight by having another stone placed upon it, but even by the addition of a handful of hemp its weight is augmented six to ten ounces according to the quantity of hemp. But if you tie the hemp to the stone and allow them to fall freely from some height, do you believe that the hemp will press down upon the stone and thus accelerate its motion, or do you think the motion will be retarded by a partial upward pressure? One always feels the pressure upon his shoulders when he prevents the motion of a load resting upon him; but if one descends just as rapidly as the load would fall, how can it gravitate or press upon him? Do you not see that this would be the same as trying to strike a man with a lance when he is running away from you with a speed that is equal to, or even greater, than that with which you are following him? You must therefore conclude that, during free and natural fall, the small stone does not press upon the larger and consequently does not increase its weight as it does when at rest.

  SIMP. But what if we should place the larger stone upon the smaller?

  [109] SALV. Its weight would be increased if the larger stone moved more rapidly; but we have already concluded that when the small stone moves more slowly it retards to some extent the speed of the larger, so that the combination of the two, which is a heavier body than the larger of the two stones, would move less rapidly—a conclusion that is contrary to your hypothesis. We infer therefore that large and small bodies move with the same speed provided they are of the same specific gravity.

  SIMP. Your discussion is really admirable; yet I do not find it easy to believe that a birdshot falls as swiftly as a cannon ball.

  SALV. Why not say a grain of sand as rapidly as a grindstone? But, Simplicio, I trust you will not follow the example of many others who divert the discussion from its main intent and fasten upon some statement of mine that lacks a hairsbreadth of the truth and, under this hair, hide the fault of someone else that is as big as a ship’s cable. Aristotle says that “an iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit.” I say that they arrive at the same time. You find, on making the experiment, that the larger outstrips the smaller by two inches; that is, when the larger has reached the ground, the other is short of it by two inches. Now you would not hide behind these two inches the ninety-nine cubits of Aristotle, nor would you mention my small error and at the same time pass over in silence his very large one. Aristotle declares that bodies of different weights, in the same medium, travel (in so far as their motion depends upon gravity) with speeds that are proportional to their weights; this he illustrates by means of bodies in which it is possible to perceive the pure and unadulterated effect of gravity, disregarding other considerations such as shape and certain extremely small disturbances; these influences are greatly dependent upon the medium, which modifies the simple effect of gravity alone. Thus we observe that gold, the densest of all substances, when beaten out into a very thin leaf, goes floating through the air; the same thing happens with stone when ground into a very fine powder. But if you wish to maintain the general proposition, you will have to show that the [110] s
ame ratio of speeds is preserved in the case of all heavy bodies, and that a stone of twenty pounds moves ten times as rapidly as one of two; and I claim that this is false and that, if they fall from a height of fifty or a hundred cubits, they will reach the ground at the same moment.

  SIMP. Perhaps the result would be different if the fall took place not from a few cubits but from some thousands of cubits.

  SALV. If this were what Aristotle meant, you would burden him with another error, which would amount to a lie. For there is no such sheer height available on earth, and so it is clear that Aristotle could not have made the experiment; yet he wishes to give us the impression of his having performed it when he speaks of such an effect as one which we see.

  SIMP. In fact, Aristotle does not employ this principle but uses the other one, which is not, I believe, subject to these same difficulties.

  SALV. But the other is as false as this one. And I am surprised that you yourself do not see the fallacy and do not perceive this. For if it were true that, in media of different densities and different resistances, such as water and air, one and the same body moved in air more rapidly than in water in proportion as the density of water is greater than that of air, then it would follow that any body that falls through air ought also to fall through water. But this conclusion is false inasmuch as many bodies that descend in air not only do not descend in water, but actually rise.

  SIMP. I do not understand the necessity of your inference; and in addition I will say that Aristotle discusses only those bodies that fall in both media, not those that fall in air but rise in water.

  SALV. The defense which you advance for the Philosopher is such that he himself would have certainly avoided it, so as not to aggravate his first mistake. But [111] tell me now whether the density of the water, or whatever it may be that retards the motion, bears a definite ratio to the density of air, which retards it less; and if so fix a value for it at your pleasure.

  SIMP. Such a ratio does exist. Let us assume it to be ten. Then, for a body that falls in both these media, the speed in water will be ten times slower than in air.

  SALV. I shall now take one of those bodies that fall in air but not in water, say a wooden ball, and I shall ask you to assign to it any speed you please for its descent through air.

  SIMP. Let us suppose it moves with a speed of twenty units.

  SALV. Very well. Then it is clear that this speed bears to some smaller speed the same ratio as the density of water bears to that of air; and the value of this smaller speed is two units. Thus really if we follow exactly the assumption of Aristotle, we ought to infer that the wooden ball that falls in air (a substance ten times less resisting than water) with a speed of twenty units would fall in water with a speed of two, instead of coming to the surface from the bottom as it does; unless perhaps you wish to reply, which I do not believe you will, that the rising of the wood through the water is the same as its falling with a speed of two units. But since the wooden ball does not go to the bottom, I think you will agree with me that we can find a ball of another material, not wood, which does fall in water with a speed of two.

  SIMP. Undoubtedly we can; but it must be of a substance considerably heavier than wood.

  SALV. That is it exactly. But if this second ball falls in water with a speed of two units, what will be its speed of descent in air? If you hold to the rule of Aristotle you must reply that it will move at the rate of twenty units; but twenty is the speed which you yourself have already assigned to the wooden ball; hence this and the other heavier ball will each move through air with the same speed. But now how does the Philosopher harmonize this result with his other, namely, that bodies of different weight move through the same medium with different speeds—speeds that are proportional to their weights? But without going into the matter more deeply, how [112] have these common and obvious properties escaped your notice? Have you not observed that two bodies that fall in water, one with a speed a hundred times as great as that of the other, will fall in air with speeds so nearly equal that one will not surpass the other by as much as one-hundredth part? Thus, for example, an egg made of marble will descend in water one hundred times more rapidly than a hen’s egg, while in air falling from a height of twenty cubits the one will fall short of the other by less than four inches. A heavy body that sinks through ten cubits of water in three hours will traverse ten cubits of air in one or two pulse beats. And if the heavy body be a ball of lead it will easily traverse the ten cubits of water in less than double the time required for ten cubits of air.

  And here, Simplicio, I am sure you understand that there is no room for hairsplitting or reply. We conclude, therefore, that the argument does not show anything against the existence of a vacuum. If it did, it would only do away with vacuums of considerable size, which neither I nor, in my opinion, the ancients ever believed to exist in nature; but they might possibly be produced by force, as may be gathered from various experiments whose description would here occupy too much time.

  SAGR. Seeing that Simplicio is silent, I will take the opportunity of saying something. You have clearly demonstrated that bodies of different weights do not move in one and the same medium with velocities proportional to their weights but that they all move with the same speed; and here we understand of course that they are of the same substance or at least of the same specific gravity, certainly not of different specific gravities, for I hardly think you would have us believe a [113] ball of cork moves with the same speed as one of lead. And you have clearly demonstrated that one and the same body moving through differently resisting media does not acquire speeds that are inversely proportional to the resistances. As a result, I am curious to learn what are the ratios actually observed in these two cases.

  [§10.3 Day I: The Pendulum]6

  [127] SIMP. I had thought the previous experiments left something to be desired; but now I am fully satisfied.

  SALV. The things set forth by me up to this point are new—in particular, my saying that differences of weight, even when very great, are without effect in changing the speed of falling bodies, so that as far as weight is concerned they all fall with equal speed. This idea is, I say, so new, and at first glance so remote from fact, that if we do not have the means of making it just as clear as sunlight, it had better not be mentioned; but having once allowed it to pass my lips, I must neglect no experiment or argument to corroborate it.

  SAGR. Not only this, but also many other of your views are so far removed from the commonly accepted opinions and doctrines that if you were to publish them, you would stir up a large number of antagonists; for human nature is such that men do not look with favor upon discoveries—either of truth or falsity—in their own field, when made by someone other than themselves. They call him an innovator of doctrine, an unpleasant title, by which they hope to cut those knots which they cannot untie, and by subterranean mines they seek to destroy structures which patient artisans have built with customary tools. [128] But as for ourselves who have no such thoughts, the experiments and arguments which you have thus far adduced are fully satisfactory; however, if you have any experiments that are more direct or any arguments that are more convincing, we will hear them with pleasure.

  SALV. The experiment made to ascertain whether two bodies differing greatly in weight will fall from a given height with the same speed offers some difficulty. For if the height is considerable, the retarding effect of the medium, which must be penetrated and thrust aside by the falling body, will be greater in the case of the small momentum of the very light body than in the case of the great force of the very heavy body. Thus, in a long distance, the light body will be left behind; and if the height be small, one may well doubt whether there is any difference, and whether it will be observable even if there is.

  It occurred to me, therefore, to repeat many times the fall through a small height in such a way that I might accumulate all those small intervals of time that elapse between the arrival of the heavy and light bodies respectively at their
common terminus, so that this sum makes an interval of time that is not only observable, but easily observable. In order to employ the slowest speeds possible and thus reduce the change which the resisting medium produces upon the simple effect of gravity, it occurred to me to allow the bodies to fall along a plane slightly inclined to the horizontal; for in such a plane, just as well as in a vertical plane, one may discover how bodies of different weight behave. Besides this, I also wished to rid myself of the resistance that might arise from contact of the moving body with the aforesaid inclined plane.

  Accordingly, I took two balls, one of lead and one of cork, the former more than a hundred times heavier than the latter, and suspended them by means of two equal fine threads, each four or five cubits long. Pulling each ball aside from the perpendicular, I let them go at the same instant, and they, falling along the circumferences of circles having these equal strings for radii, passed beyond the perpendicular and returned along the same path. This free oscillation repeated a hundred times showed clearly [129] that the heavy ball maintains so nearly the period of the light ball that neither in a hundred swings nor even in a thousand will the former anticipate the latter by as much as a single moment, so perfectly do they keep step. We can also observe the effect of the medium which, by the resistance which it offers to motion diminishes the oscillation of the cork more than that of the lead, but without altering the frequency of either; even when the arc traversed by the cork did not exceed five or six degrees and that of the lead fifty or sixty, the swings were performed in equal times.

  SIMP. If this be so, why is not the speed of the lead greater than that of the cork, seeing that the former traverses sixty degrees in the same interval in which the latter covers scarcely six?

  SALV. But what would you say, Simplicio, if both covered their paths in the same time when the cork, drawn aside through thirty degrees, traverses an arc of sixty, while the lead pulled aside only two degrees traverses an arc of four? Would not then the cork be proportionately swifter? And yet experiment shows that this is what happens. For note this.