The Essential Galileo

by Galilei, Galileo, Finocchiaro, Maurice A.

  SAGR. So far as I see at present, the definition might have been put a little more clearly perhaps without changing the fundamental idea. That is, uniformly accelerated motion is motion such that its speed increases in proportion to the space traversed; so that, for example, the speed acquired by a body in falling four cubits would be double that acquired in falling two cubits, and this latter speed would be double that acquired in the first cubit. For there is no doubt but that a heavy body falling from the height of six cubits has, and strikes with, an impetus double that which it had at the end of three cubits, triple that which it had at the end of two, and six times that which it had at the end of one.

  SALV. It is very comforting to me to have had such a companion in error. Moreover, let me tell you that your reasoning seems so highly likely and probable that our Author himself admitted, when I put it forward to him, that he had for some time shared the same fallacy. But what most surprised me was to see two propositions proven in a few simple words to be not only false but also impossible, even though they are so inherently likely that they have commanded the assent of everyone to whom I have presented them.

  SIMP. I am one of those who accept them. I believe that a falling body acquires force in its descent, its velocity increasing in proportion to the space, and that the moment of the same striking body is double when it falls from a double height. These propositions, it appears to me, ought to be conceded without hesitation or controversy.

  SALV. And yet they are as false and impossible as that motion should be completed instantaneously. Here is a very clear demonstration of it. When the velocities are in proportion to the spaces traversed or to be traversed, these spaces are traversed in equal intervals of time; if, therefore, the velocities20 with which the falling body traverses a space of four cubits were double the velocities with which it covered the first two cubits (since the one distance is double the other), then the time intervals required for these passages would be equal; but for one and the same body to move four cubits and two cubits in the same time is possible only in the case of [204] instantaneous motion; but observation shows us that the motion of a falling body takes time, and less of it in covering a distance of two cubits than of four cubits; therefore, it is false that its velocity increases in proportion to the space.

  The falsity of the other proposition may be shown with equal clearness. For if we consider a single striking body, the difference in the moment of its percussions can depend only upon a difference of velocity; thus, if the striking body falling from a double height were to deliver a percussion of double moment, it would be necessary for this body to strike with a double velocity; with this double speed it would traverse a double space in the same time interval; but observation shows that the time required for fall from the greater height is longer.

  SAGR. You present these recondite matters with too much evidence and ease. This great facility makes them less appreciated than they would be had they been presented in a more abstruse manner. For, in my opinion, people esteem more lightly that knowledge which they acquire with so little labor than that acquired through long and obscure discussion.

  SALV. If those who demonstrate with brevity and clearness the fallacy of many popular beliefs were treated with contempt instead of gratitude, the injury would be quite bearable. But on the other hand, it is very unpleasant and annoying to see men who claim to be peers of anyone in a certain field of study take for granted conclusions that later are quickly and easily shown by another to be false. I do not call such a feeling envy, which usually degenerates into hatred and anger against those who discover such fallacies; I would call it a strong desire to maintain old errors, rather than accept newly discovered truths. This desire at times induces them to unite against these truths, although at heart believing in them, merely for the purpose of lowering the esteem in which certain others are held by the unthinking crowd. Indeed, I have heard our Academician talk about many such false propositions, held as true but easily refutable; and I have even made a list of some of them.

  SAGR. You must not withhold them from us, but must tell us about them at the proper time, even though an extra session be necessary. [205] For now, continuing the thread of our discussion, it would seem that so far we have formulated the definition of the uniformly accelerated motion to be treated in what follows. It is this: A motion is said to be equally or uniformly accelerated when, starting from rest, its velocity receives equal increments in equal times.

  [§10.7 Day III: Laws of Falling Bodies]21

  SALV. This definition established, the Author assumes the truth of a single principle, namely: The speeds acquired by one and the same body moving down planes of different inclinations are equal when the heights of these planes are equal.

  By the height of an inclined plane he means the perpendicular let fall from the upper end of the plane upon the horizontal line drawn through the lower end of the same plane. Thus, to illustrate, let the line AB be horizontal, and let the planes CA and CD be inclined to it; then the Author calls the perpendicular CB the “height” of the planes CA and CD. He supposes that the speeds acquired by one and the same body descending along the planes CA and CD to the terminal points A and D are equal since the heights of these planes are the same, CB; and also it must be understood that this speed is that which would be acquired by the same body falling from C to B.

  SAGR. Your assumption appears to me so probable that it ought to be conceded without question, provided of course that there are no accidental or external resistances, and that the planes are hard and smooth and the shape of the moving body is perfectly round, so that neither plane nor moving body is rough. All resistance and opposition having been removed, my natural instinct tells me at once that a heavy and perfectly round ball descending along the lines CA, CD, CB would reach the terminal points A, D, B with the same impetus.

  SALV. What you say is very plausible. But, going beyond likelihood, I hope by experiment to increase its probability to such an extent that it shall be little short of a necessary demonstration. [206] Imagine this page to represent a vertical wall, with a nail driven into it; and from the nail let there be suspended a lead ball of one or two ounces by means of a fine vertical thread, AB, say two or three cubits long; on this wall draw a horizontal line DC, at right angles to the vertical thread AB, which hangs about two inches in front of the wall. Now bring the thread AB with the attached ball into the position AC and set it free; first it will be observed to descend along the arc CBD, to pass the point B, and to travel along the arc BD, till it almost reaches the horizontal CD, a slight shortage being caused by the resistance of the air and of the string; from this we may rightly infer that the ball in its descent through the arc CB acquired an impetus on reaching B that was just sufficient to carry it through a similar arc BD to the same height. Having repeated this experiment many times, let us now drive a nail into the wall close to the perpendicular AB, say at E or F, so that it projects out some five or six inches in order that the thread, again carrying the ball through the arc CB, may strike upon the nail E when the ball reaches B, and thus compel it to traverse the arc BG, described about E as center; from this we can see what can be done by the same impetus that, previously starting at the same point B, carried the same body through the arc BD to the horizontal CD. Now, gentlemen, you will observe with pleasure that the ball swings to the point G in the horizontal, and you would see the same thing happen if the obstacle were placed at some lower point, say at F, about which the ball would describe the arc BI, the rise of the ball always terminating exactly on the line CD. But when the nail is placed so low that the remainder of the thread below it will not reach to the height CD (which would happen [207] if the nail were placed nearer to B than to the intersection of AB with the horizontal CD), then the thread leaps over the nail and twists itself about it.

  This experiment leaves no room for doubt as to the truth of our supposition. For since the two arcs CB and DB are equal and similarly placed, the momentum acquired by the fall throug
h the arc CB is the same as that gained by fall through the arc DB; but the momentum acquired at B owing to fall through CB is able to lift the same body through the arc BD; therefore, the momentum acquired in the fall DB is equal to that which lifts the same body through the same arc from B to D; so, in general, every momentum acquired by fall through an arc is equal to that which can lift the same body through the same arc. But all these momenta that cause a rise through the arcs BD, BG, and BI are equal, since they are produced by the same momentum, gained by fall through CB, as experiment shows. Therefore, all the momenta gained by fall through the arcs DB, GB, and IB are equal.

  SAGR. The argument seems to me so conclusive and the experiment so well adapted to establish the postulate that we may, indeed, accept it as if it were demonstrated.

  SALV. I do not wish, Sagredo, that we trouble ourselves too much about this matter, especially since we are going to apply this principle mainly to motions that occur on plane surfaces, and not upon curved ones, along which acceleration varies in a manner greatly different from that which we have assumed for planes. Thus, although the above experiment shows us that the descent of the moving body through the arc CB confers upon it enough momentum to carry it to the same height through any of the arcs BD, BG, or BI, we are not able to show with similar evidence that the same would happen in the case of a perfectly round ball descending along planes whose inclinations are respectively the same as the chords of these arcs. Instead, since these planes form angles at the point B, it seems likely that they will present an obstacle to the ball that has descended along the chord CB and starts to rise along the chords BD, BG, or BI; in striking these planes, it will lose some of its impetus and will not be able to rise to the height of the line CD. But if one removes this obstacle, which is prejudicial to the experiment, it is clear to the intellect that the impetus (which gains [208] strength by the amount of descent) will be able to carry the body to the same height. Let us then, for the present, take this as a postulate, the absolute truth of which will be established when we find that the conclusions based on this hypothesis correspond to and agree perfectly with experiment. The Author having assumed this single principle, he passes next to the propositions which he conclusively demonstrates. The first of these is as follows.

  Theorem 1, Proposition 1:The time in which any space is traversed by a body starting from rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is one-half the highest and final speed reached during the previous uniformly accelerated motion.

  Let us represent by the line AB the time in which the space CD is traversed by a body that starts from rest at C and is uniformly accelerated; let the final and highest value of the speed gained during the interval AB be represented by the line EB, drawn at right angles to AB; draw also the line AE; then all lines drawn from equidistant points on AB and parallel to BE will represent the increasing values of the speed, beginning with the instant A. Let the point F bisect the line EB; draw FG parallel to BA, and GA parallel to FB, thus forming a parallelogram AGFB, whose area will be equal to that of the triangle AEB, and whose side GF bisects the side AE at the point I.

  Now, if the parallel lines in the triangle AEB are extended to GI, then the aggregate of all the parallels contained in the quadrilateral is equal to the aggregate of those contained in the triangle AEB; for those in the triangle IEF are equal to those contained in the triangle GIA, while those included in the trapezium AIFB are common. Furthermore, each and every instant of time in the time interval AB has its corresponding point on the line AB, from which points the parallels drawn in and limited by the triangle AEB represent the increasing values of the growing velocity; and the parallels contained within the rectangle represent the values of a speed that is not increasing but constant. [209] Hence it appears that the moments of speed acquired by the moving body may be represented, in the case of the accelerated motion, by the increasing parallels of the triangle AEB, and in the case of the uniform motion, by the parallels of the rectangle GB; for, what the moments of speed may lack in the first part of the accelerated motion (the deficiency of the moments being represented by the parallels of the triangle AGI) is made up by the moments represented by the parallels of the triangle IEF. Therefore, it is clear that equal spaces will be traversed in equal times by two bodies, one of which starts from rest and moves with uniform acceleration, while the other moves with a uniform speed whose moment is one-half the maximum moment of speed under the accelerated motion. QED.

  Theorem 2, Proposition 2: If a body falls from rest with a uniformly accelerated motion, then the spaces traversed are to each other as the squares of the time intervals employed in traversing them.

  Let the time beginning with any instant A be represented by the straight line AB, in which are taken any two time intervals AD and AE. Let HI represent the distance through which the body, starting from rest at H, falls with uniform acceleration. If HL represents the space traversed during the time interval AD, and HM that covered during the interval AE, then the space HM stands to the space HL in a ratio that is the square of the ratio of the time AE to the time AD; or we may say simply that the distances HM and HL are related as the squares of AE and AD.

  Draw the line AC making any angle whatever with the line AB; and from the points D and E, draw the parallel lines DO and EP; of these two lines, DO represents the greatest velocity attained during the time interval AD, while EP represents the maximum velocity acquired during the time AE. But it has just been proved that so far as distances traversed are concerned, it is precisely the same whether a body falls from rest with a uniform acceleration or whether it falls during an equal time interval with a constant speed that is one-half the maximum speed attained during the accelerated motion. It follows therefore that the distances HM and HL are the same as would be traversed during the time intervals AE and AD by uniform velocities equal to one-half those represented by EP and DO respectively. If, therefore, one can show that the distances HM and HL are in [210] the same ratio as the squares of the time intervals AE and AD, our proposition will be proven. But in the fourth proposition of the first section above,22 it has been shown that the spaces traversed by two bodies in uniform motion bear to one another a ratio that is equal to the product of the ratio of the velocities by the ratio of the times; and in the present case the ratio of the velocities is the same as the ratio of the time intervals, for the ratio of one-half EP to one-half DO, or of EP to DO, is the same as that of AE to AD; hence the ratio of the spaces traversed is the same as the squared ratio of the time intervals. QED.

  It also clearly follows that the ratio of the distances is the square of the ratio of the final velocities, that is, of the lines EP and DO, since these are to each other as AE to AD.

  Corollary 1: Hence it is clear that if we take any number of consecutive equal intervals of time, counting from the beginning of the motion, such as AD, DE, EF, FG, in which the spaces HL, LM, MN, NI are traversed, these spaces will bear to one another the same ratio as the series of odd numbers, 1, 3, 5, 7.

  For this is the ratio of the differences of the squares of the lines which exceed one another by equal amounts and whose excess is equal to the smallest of these same lines; or we may say that this is the ratio of the differences of the squares of the natural numbers beginning with unity. Therefore, whereas after equal time intervals the velocities increase as the natural numbers, the increments in the distances traversed during these equal time intervals are to one another as the odd numbers beginning with unity.

  SAGR. Please suspend the reading for a moment, since there just occurs to me an idea which I want to illustrate by means of a diagram in order that it may be clearer both to you and to me. Let the line AI represent the lapse of time measured from the initial instant A; through A draw the straight line AF making any angle whatever; join the terminal points I and F; divide the time AI in half at C; draw CB parallel to IF. Let us consider CB as the maximum value
of the velocity that increases from zero at the beginning in simple proportionality to the segments (inside the triangle ABC) of lines drawn parallel to BC; or what is the same thing, let us suppose the velocity to increase in proportion to the time; then I admit without question, in view of the preceding argument, that the space traversed by a body falling in the aforesaid manner will be equal to the space traversed by the [211] same body during the same length of time traveling with a uniform speed equal to EC, or half of BC. Further let us imagine that the body has fallen with accelerated motion so that at the instant C it has the velocity BC. It is clear that if the body continued to descend with the same speed BC, without acceleration, it would in the next time interval CI traverse double the distance covered during the interval AC with the uniform speed EC, which is half of BC. But since the falling body acquires equal increments of speed during equal increments of time, it follows that the velocity BC, during the next time interval CI, will be increased by an amount represented by the parallels of the triangle BFG, which is equal to the triangle ABC. Thus, if one adds to the velocity GI half of the velocity FG, the maximum increment of speed acquired by the accelerated motion and determined by the parallels of the triangle BFG, one will have the uniform velocity IN with which the same space would have been traversed in the time CI. And since this speed IN is three times as great as EC, it follows that the space traversed during the interval CI is three times as great as that traversed during the interval AC. Now, let us imagine the motion extended over another equal time interval IO, and the triangle extended to APO; it is then evident that if the motion continues during the interval IO, at the constant rate IF acquired by acceleration during the time AI, the space traversed during the interval IO will be four times that traversed during the first interval AC, because the speed IF is four times the speed EC. But if we enlarge our triangle so as to include FPQ, which is equal to ABC, still assuming the acceleration to be constant, we shall add to the uniform speed an increment RQ, equal to EC; then the value of the equivalent uniform speed during the time interval IO will be five times that during the first time interval AC; therefore, the space traversed will be quintuple that during the first interval AC. It is thus evident by this simple computation that a moving body starting from rest and acquiring velocity at a rate proportional to the time, will, during equal intervals of time, traverse distances that [212] are related to each other as the odd numbers beginning with unity, 1, 3, 5; or considering the total space traversed, that covered in double time will be quadruple that covered during unit time; in triple time, the space is nine times as great as in unit time. And in general the spaces traversed are in the squared ratio of the times, i.e., in the ratio of the squares of the times.