The Essential Galileo

by Galilei, Galileo, Finocchiaro, Maurice A.

  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.

  SIMP. 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.

  SAGR. 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?

  SIMP. This can neither be doubted nor gainsaid.

  SAGR. 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.

  SALV. 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.

  SIMP. 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.

  SALV. 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.

  SIMP. 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.

  SALV. 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.

  SIMP. This I admit without hesitation.

  SALV. 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.

  Thus you see how much greater is the resistance, arising from contact of the surface of the moving body with the medium, in the case of small bodies than in the case of large. And when one considers that the irregularities on the very small surfaces of fine dust particles are perhaps no smaller than those on the surfaces of larger solids that have been carefully polished, one will see how important it is that the medium should be very fluid and offer no resistance to being thrust aside, easily yielding to a small force. You see, therefore, Simplicio, that I was not mistaken when, not long ago, I said that the surface of a small solid is comparatively greater than that of a large one.

  SIMP. I am quite convinced. And believe me, if I were again beginning my studies, I should follow the advice of Plato and start with the mathematical sciences, which proceed very cautiously and admit nothing as established until it has been rigorously demonstrated.

  SAGR. This discussion has afforded me great pleasure. But before proceeding further, I should like to hear the explanation …7

  [139] SALV. Let us see whether we cannot derive from the pendulum a satisfactory solution of all these difficulties. And first, as to the question whether one and the same pendulum really performs its oscillations, large, medium, and small, all in exactly the same time, I shall rely upon what I have already heard from our Academician. He has clearly shown that the time of descent is the same along all chords, whatever the arcs that subtend them, whether the arc is 180 degrees (corresponding to the whole diameter), 100 degrees, 60 degrees, 10 degrees, 2 degrees, 1/2 degree, or 4 minutes; it is understood, of course, that these chords all terminate at the lowest point of the circle, where it touches the horizontal plane.

  Now, if we consider descent along arcs instead of their chords, then (provided they do not exceed ninety degrees) experiment shows that they are all traversed in equal times; but these times are shorter for the arcs than for the chords, an effect that is all the more remarkable because at first glance one would think just the opposite to be true. For since the terminal points of the two motions are the same and since the straight line included between these two points is the shortest distance between them, it would seem reasonable that motion along this line should be executed in the shortest time; but this is not the case, for the shortest time—and therefore the most rapid motion—is that employed along the arc of which this straight line is the chord.

  As to the times of oscillation of bodies suspended by threads of different lengths, they bear to each other the same proportion as the square roots of the lengths of the thread; or one might say the lengths are to each other as the squares of the times. For example, if one wishes to make the oscillation time of one pendulum twice that of another, one must make its suspension thread four times [140] as long; in like manner, if one pendulum has a thread nine times as long as another, this second pendulum will execute three oscillations during each one of the first. From this it follows that the lengths of the suspending cords bear to each other the [inverse] ratio of the squares of the number of oscillations performed in the same time.

  SAGR. Then, if I understand you correctly, I can easily measure the length of a string whose upper end is attached at any height whatever even if this end were invisible and I could see only the lower extremity. For if I attach to the lower end of this string a rather heavy weight and give it a to-and-fro motion, and if I ask a friend to count the number of its oscillations while I, during the same time interval, count the number of oscillations of a pendulum which is exactly one cubit in length, then knowing the number of oscillations which each pendulum makes in the given interval of time one can determine the length of the string. Suppose, for example, that my friend counts 20 oscillations of the long cord during the same time in which I count 240 of my string, which is one cubit in length; taking the squares of the two numbers, 20 and 240, namely, 400 and 57,600, then, I say, the long string contains 57,600 units of length as compared to the 400 contained in my string; and since the length of my string is one cubit, I shall divide 57,600 by 400 and thus obtain 144. Accordingly, I shall call the length of the other string 144 cubits.

  SALV. Nor will you miss it by as much as a palm’s breadth, especially if you observe a large number of oscillations.

  SAGR. You give me frequent occasion to admire the wealth and profusion of nature when, from such common and even trivial phenomena, you derive facts that are not only striking and new but that are often far removed from what we would have imagined. Thousands of times I have observed, especially in churches, oscillations of lamps suspended by long cords and inadvertently set into motion. But the most I could infer from these observations was the improbability of the view of those who think that such oscillations are maintained by the medium, namely, the air; for, in that case, the air must needs have considerable judgment and little else to do but kill [141] time by pushing back and forth a hanging weight with perfect regularity. But I never dreamed of learning that one and the same body, when suspended from a string a hundred cubits long and pulled aside first through an arc of ninety degrees and then through one degree or half a degree, would employ the same time in passing through the least as through the largest of these arcs; indeed, it still strikes me as almost impossible. Now I am waiting to hear how these same simple phenomena can furnish solutions for those acoustical problems—solutions that will be at least partly satisfactory.

  SALV. First of all one must observe that each pendulum has its own time of oscillation so definite and determinate that it is not possible to make it move with any other period than that which nature has given it. For let anyone take in his hand a cord to which a weight is attached and try, as much as he
pleases, to increase or diminish the frequency of its oscillations; it will be time wasted. On the other hand, one can confer motion upon even a heavy pendulum that is at rest by simply blowing against it; by repeating these blasts with a frequency which is the same as that of the pendulum, one can impart considerable motion. Suppose that by the first puff we have displaced the pendulum from the vertical by, say, half an inch; then if we add a second puff after the pendulum has returned and is about to begin the second oscillation, we shall impart additional motion; and so on with other blasts provided they are applied at the right instant, and not when the pendulum is coming toward us, since in this case the blast would impede rather than aid the motion. Continuing thus with many impulses, we impart to the pendulum such impetus that a much greater force than that of a single blast will be needed to stop it.

  [§10.4 Day II: The Mathematics of

  Strength, Size, and Weight]8

  [151] SAGR. While Simplicio and I were awaiting your arrival, we were trying to recall that last consideration which you advanced as a principle and basis for the results you intended to obtain. This consideration dealt with the resistance which all solids offer to fracture, and which depends upon a certain cement that holds the parts glued together so that they yield and separate only under considerable pull. Later we tried to find the explanation for this coherence, seeking it mainly in the vacuum. This was the occasion of our many digressions, which occupied the entire day and led us far afield from the original subject. As I have already stated, that was the investigation of the resistance which solids offer to fracture.

  SALV. I remember it all very well. Resuming the thread of our discussion, whatever the nature of this resistance which solids offer to powerful pulling, there can at least be no doubt of its existence. And although this resistance is very great in the case of a direct pull, it is found, as a rule, to be less in the case of bending forces. Thus, for example, a rod of steel or of glass will sustain a longitudinal pull of a thousand pounds, whereas a weight of fifty pounds would be quite sufficient to break it if the rod were fastened at right angles into a vertical wall. It is this second type of resistance which we must consider, seeking to discover in [152] what proportion it is found in prisms and cylinders of the same material, whether alike or unlike in shape, length, and thickness. In this discussion I shall take for granted the well-known mechanical principle that has been shown to govern the behavior of a lever, namely, that the force bears to the resistance the inverse ratio of the distances that separate the fulcrum from the force and resistance respectively.