The Essential Galileo (52 page)

S
ALV.
    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.

S
IMP.
    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?

S
ALV.
    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.

Having pulled aside the pendulum of lead, say through an arc of fifty degrees, and set it free, it swings beyond the perpendicular almost fifty degrees, thus describing an arc of nearly one hundred degrees. On the return swing it describes a little smaller arc. And after a large number of such oscillations it finally comes to rest. Each oscillation, whether of ninety, fifty, twenty, ten, or four degrees, takes the same time. Accordingly, the speed of the moving body keeps on diminishing, since in equal intervals of time it traverses arcs that grow smaller and smaller.

Precisely the same things happen with the pendulum of cork suspended by a string of equal length, except that a smaller number of oscillations is required to bring it to rest, since on account of its lightness it is less able to overcome the resistance of the air. Nevertheless, the oscillations, whether large or small, are all performed in time intervals that are not only equal among themselves, but also equal to the period of the lead pendulum. Hence, if while the lead is traversing an arc of fifty degrees the cork covers one of only ten, it is true that the cork moves [130] more slowly than the lead; but on the other hand, it is also true that the cork covers an arc of fifty while the lead passes over one of only ten or six; thus, at different times, we have now the cork, now the lead, moving more rapidly. But if these same bodies traverse equal arcs in equal times, we may rest assured that their speeds are equal.

S
IMP.
    I hesitate to admit the conclusiveness of this argument because of the confusion that arises from your making both bodies move now rapidly, now slowly and now very slowly, which leaves me in doubt as to whether their velocities are always equal.

S
AGR.
    Allow me, if you please, Salviati, to say just a few words. Now tell me, Simplicio, whether you admit that one can say with certainty that the speeds of the cork and the lead are equal whenever both, starting from rest at the same moment and descending the same slopes, always traverse equal spaces in equal times?

S
IMP.
    This can neither be doubted nor gainsaid.

S
AGR.
    Now it happens, in the case of the pendulums, that each of them traverses now an arc of sixty degrees, now one of fifty, or thirty or ten or eight or four or two, etc.; and when they both swing through an arc of sixty degrees they do so in equal intervals of time; the same thing happens when the arc is fifty degrees or thirty or ten or any other number; and therefore we conclude that the speed of the lead in an arc of sixty degrees is equal to the speed of the cork when the latter also swings through an arc of sixty degrees; in the case of a fifty-degree arc these speeds are also equal to each other; so also in the case of other arcs. But this is not saying that the speed which occurs in an arc of sixty is the same as that which occurs in an arc of fifty; nor is the speed in an arc of fifty equal to that in one of thirty, etc.; but the smaller the arcs, the smaller the speeds; this is inferred from our sensibly seeing that one and the same moving body requires the same time for traversing a large arc of sixty degrees as for a small arc of fifty or even a very small arc of ten; all these arcs, indeed, are covered in the same interval of time. It is true therefore that [131] the lead and the cork each diminish their speed in proportion as their arcs diminish; but this does not contradict the fact that they maintain equal speeds in equal arcs.

My reason for saying these things has been rather because I wanted to learn whether I had correctly understood Salviati, than because I thought Simplicio had any need of a clearer explanation than that given by Salviati; like everything else of his, this is extremely lucid, and indeed such that when he solves questions that are difficult not merely in appearance, but in reality and in fact, he does so with reasons, observations, and experiments that are common and familiar to everyone. In this manner he has, as I have learned from various sources, given occasion to some highly esteemed professors for undervaluing his discoveries on the ground that they are commonplace and established upon a lowly and vulgar basis; as if it were not a most admirable and praiseworthy feature of the demonstrative sciences that they spring from and grow out of principles well known, understood, and conceded by all.

But let us continue with this light diet. If Simplicio is satisfied to understand and admit that the weight inherent in various falling bodies has nothing to do with the difference of speed observed among them, and that all bodies, insofar as their speeds depend upon it, would move with the same velocity, pray tell us, Salviati, how you explain the appreciable and evident inequality of motion. Please reply also to the objection urged by Simplicio—an objection in which I concur—namely, that a cannon ball falls more rapidly than a birdshot. Actually, this difference of speed is small as compared to the one I have in mind: that is, bodies of the same substance moving through a single medium, such that the larger ones will descend, during a single pulse beat, a distance which the smaller ones will not traverse in an hour, or in four, or even in twenty hours; as for instance in the case of stones and fine sand, and especially that very fine sand that produces muddy water and that in many hours will not fall through as much as two cubits, a distance which stones not very large will traverse in a single pulse beat.

S
ALV.
    The action of the medium in producing a greater retardation upon those bodies that have a smaller specific gravity has already been explained by showing that this results from a diminution of weight. But to explain how one and the same medium produces such different retardations in bodies [132] that are made of the same material and have the same shape, but differ only in size, requires a discussion more subtle than that by which one explains how a more expanded shape or an opposing motion of the medium retards the speed of the moving body. The solution of the present problem lies, I think, in the roughness and porosity that are generally and almost necessarily found in the surfaces of solid bodies. When the body is in motion these rough places strike the air or other ambient medium. The evidence for this is found in the humming that accompanies the rapid motion of a body through air, even when that body is as round as possible. One hears not only humming, but also hissing and whistling, whenever there is any appreciable cavity or elevation upon the body. We observe also that a round solid body rotating in a lathe produces a current of air. But what more do we need? When a top spins on the ground at its greatest speed, do we not hear a distinct buzzing of high pitch? This sibilant note diminishes in pitch as the speed of rotation slackens, which is evidence that these small wrinkles on the surface meet resistance in the air. There can be no doubt, therefore, that in the motion of falling bodies these irregularities strike the surrounding fluid and retard the speed; and this they do so much the more in proportion as the surface is larger, which is the case of small bodies as compared with larger.

S
IMP.
    Stop a moment please, as I am getting confused. For although I understand and admit that friction of the medium upon the surface of the body retards its motion and that, other things being equal, the larger surface suffers greater retardation, I do not see on what ground you say that the surface of the smaller body is larger. Besides, if, as you say, the larger surface suffers greater retardation, the larger solid should move more slowly, which is not the case. But this objection can be easily met by saying that, although the larger body has a larger surface, it has also a greater weight, in comparison with which the resistance of the larger surface is no more than the resistance of the small surface in comparison with its smaller weight; so the speed of the larger solid does not become less. I therefore see no reason for expecting any difference [133] of speed so long as the driving weight diminishes in the same proportion as the retarding power of the surface.

S
ALV.
    I shall answer all your objections at once. You will admit, of course, Simplicio, that if we take two equal bodies of the same material and same shape (bodies that would therefore fall with equal speeds), and if we diminish the weight of one of them in the same proportion as its surface (maintaining the similarity of shape), we would not thereby diminish the speed of this body.

S
IMP.
    This inference seems to be in harmony with your theory, which states that the weight of a body has no effect in either accelerating or retarding its motion.

S
ALV.
    I quite agree with you in this opinion, from which it appears to follow that if the weight of a body is diminished in greater proportion than its surface, the motion is retarded to a certain extent; and this retardation is greater and greater in proportion as the diminution of weight exceeds that of the surface.

S
IMP.
    This I admit without hesitation.

S
ALV.
    Now you must know, Simplicio, that it is not possible to diminish the surface of a solid body in the same ratio as the weight, and at the same time maintain similarity of shape. For since it is clear that in the case of a diminishing solid the weight grows less in proportion to the volume, if the volume diminishes more rapidly than the surface (and the same shape is maintained) then the weight must diminish more rapidly than the surface. But geometry teaches us that, in the case of similar solids, the ratio of the volumes is greater than the ratio of their surfaces; which, for the sake of better understanding, I shall illustrate by a particular case.

Take, for example, a cube two inches on a side, so that each face has an area of four square inches and the total area, i.e., the sum of the six faces, amounts to twenty-four square inches. Now imagine this cube to be sawed through three times so as to divide it into eight smaller cubes: each is one inch on the side; each face is one square inch; and the total [134] surface of each cube is six square inches, instead of twenty-four as in the case of the larger cube. It is evident that the surface of the little cube is only one-fourth that of the larger, namely, the ratio of six to twenty-four; but the volume of the smaller cube is only one-eighth that of the large one; the volume, and hence also the weight, diminishes therefore much more rapidly than the surface. If we now divide the little cube into eight others, we shall have, for the total surface of one of these, one and one-half square inches, which is one-sixteenth of the surface of the original cube; but its volume is only one-sixty-fourth. Thus, by two divisions, you see that the volume is diminished four times as much as the surface. And if the subdivision be continued until the original solid be reduced to a fine powder, we shall find that the weight of one of these smallest particles has diminished hundreds and hundreds of times as much as its surface. And this, which I have illustrated in the case of cubes, holds also in the case of all similar solids, where the volumes are to each other as the three-halves power of their surfaces.