The Essential Galileo (58 page)

When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner that is exceedingly simple and rather obvious to everybody? If now we examine the matter carefully, we find no addition or increment simpler than that which repeats itself always in the same manner. This we readily understand when we consider the intimate relationship between time and motion: first, uniformity of motion is defined by and conceived through equal times and equal spaces, and so we call a motion uniform when equal distances are traversed during equal time intervals; then in a similar manner, we may, through equal time intervals, conceive additions of speed as taking place with equal simplicity, [198] and so we may picture to our mind a motion as uniformly and continuously accelerated when, during any equal intervals of time whatever, equal increments of speed are given to it. Thus, if any number of equal intervals of time are considered, counting from the time at which the moving body left its position of rest and began to descend, the amount of speed acquired during the first two time intervals will be double that acquired during the first time interval alone; the amount added during three of these time intervals will be triple; and during four, quadruple that of the first time interval. To put the matter more clearly, if a body were to continue its motion with the same degree of speed which it had acquired during the first time interval and were to retain this same speed uniformly, then its motion would be twice as slow as that which it would have if its velocity had been acquired during two time intervals.

And thus, it seems, we shall not be far wrong if we put the degree of speed as proportional to the time elapsed. Hence the definition of motion which we are about to discuss may be stated as follows:
A motion is said to be uniformly accelerated when, starting from rest, it acquires equal increments of speed during equal time intervals
.

S
AGR.
    Although I can offer no rational objection to this or indeed to any other definition devised by any author whomsoever, since all definitions are arbitrary, I may nevertheless without offense be allowed to doubt whether such a definition as the above, established in an abstract manner, corresponds to and describes that kind of accelerated motion which we meet in nature in the case of freely falling bodies. And since the Author apparently maintains that the motion described in his definition is that of freely falling bodies, I would like to clear my mind of certain difficulties in order that I may later apply myself more earnestly to the propositions and their demonstrations.

S
ALV.
    It is well that you and Simplicio raise these difficulties. They are, I imagine, the same which occurred to me when I first saw this treatise, and which were removed either by discussion with the Author himself, or by turning the matter over in my own mind.

S
AGR.
    When I think of a heavy body falling from rest, that is, starting with zero speed and [199] gaining speed in proportion to the time from the beginning of the motion, such a motion would, for instance, in eight beats of the pulse acquire eight degrees of speed, having acquired four degrees at the end of the fourth beat, two at the end of the second, and one at the end of the first. Now since time is divisible without limit, it follows from all these considerations that if the earlier speed of a body is less than its present speed in a constant ratio, then there is no degree of speed however small (or, one may say, no degree of slowness however great) with which we may not find this body traveling after starting from infinite slowness, i.e., from rest. So if the speed which the body had at the end of the fourth beat was such that, if kept uniform, it would traverse two miles in an hour, and if keeping the speed which it had at the end of the second beat, it would traverse one mile an hour, we must infer that, as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it would not traverse a mile in an hour, or in a day, or in a year, or in a thousand years; indeed, it would not traverse a palm in an even greater time. This is a phenomenon that baffles the imagination, while our senses show us that a heavy falling body suddenly acquires great speed.

S
ALV.
    This is one of the difficulties which I also experienced at the beginning, but which I shortly afterwards removed; and the removal was effected by the very experiment that creates the difficulty for you. You say the experiment appears to show that immediately after a heavy body starts from rest it acquires a very considerable speed; and I say that the same experiment makes clear the fact that the initial motions of a falling body, no matter how heavy, are very slow and gentle. Place a heavy body upon a yielding material, and leave it there without any pressure except that owing to its own weight. It is clear that if one lifts this body a cubit or two and allows it to fall upon the same material, it will, with this impulse, exert a new and greater pressure than that caused by its mere weight; and this effect is brought about by the weight of the falling body together with the velocity acquired during the fall, an effect that will be greater and greater according to the height of the fall, that is, according as the velocity of the falling body becomes greater. From the quality and intensity of the blow we are thus enabled to accurately estimate the speed of a falling body. But tell me, gentlemen, is it not true that if a sledgehammer be allowed to fall upon a stake from a height of [200] four cubits and drives it into the earth, say, four inches, then coming from a height of two cubits it will drive the stake a much smaller distance, and from the height of one cubit still less, and from a height of one palm even less? Finally, if the block be lifted only one inch, how much more will it accomplish than if merely laid on top of the stake without percussion? Certainly very little. If it be lifted only the thickness of a leaf, the effect will be altogether imperceptible. And since the effect of the blow depends upon the velocity of the striking body, can anyone doubt that the motion is very slow and the speed extremely small whenever the effect is imperceptible? See now the power of truth: the same experiment that at first glance seemed to show one thing, when more carefully examined assures us of the contrary.

But without depending upon the above experiment, which is doubtless very conclusive, it seems to me that it ought not to be difficult to establish such a fact by reasoning alone. Imagine a heavy stone held in the air at rest; the support is removed and the stone set free; then since it is heavier than the air, it begins to fall, and not with uniform motion but slowly at the beginning and with a continuously accelerated motion. Now since velocity can be increased and diminished without limit, what reason is there to believe that such a moving body starting with infinite slowness, that is, from rest, immediately acquires a speed of ten degrees rather than a speed of four, or of two, or of one, or of a half, or of a hundredth; or, indeed, of any of the infinite number of smaller values? Pray listen. I hardly think you will refuse to grant that the gain of speed of the stone falling from rest follows the same sequence as the diminution and loss of this same speed when, by some impelling force, the stone is thrown to its former elevation; but if this is so, I do not see how you can doubt that the ascending stone, diminishing in speed, must before coming to rest pass through every possible degree of slowness.

S
IMP.
    But if the number of degrees of greater and greater slowness is limitless, they will never be all exhausted; therefore, such an ascending heavy body will never reach rest, but will continue to move without limit always at a slower rate. But this is not the observed fact.

S
ALV.
    This would happen, Simplicio, if the moving body were to maintain its speed for any length of time at each degree of velocity. But it merely passes each point without delaying more than an instant; and since [201] each time interval (however small) may be divided into an infinite number of instants, these will always be sufficient to correspond to the infinite degrees of diminished velocity. That such a heavy rising body does not remain for any length of time at any given degree of velocity is evident from the following: some time interval having been assigned, if the body moves with the same speed in the last as in the first instant of that time interval, it could from this second degree of elevation be in like manner raised through an equal height, just as it was transferred from the first elevation to the second, and for the same reason it would pass from the second to the third and would finally continue in uniform motion forever.

S
AGR.
    From these considerations it appears to me that we may obtain a proper solution of the problem discussed by philosophers, namely, what causes the acceleration in the natural motion of heavy bodies. Since, as it seems to me, the force impressed by the agent projecting the body upwards diminishes continuously, this force, so long as it was greater than the contrary force of gravity, impelled the body upwards; when the two are in equilibrium the body ceases to rise and passes through the state of rest in which the impressed impetus is not destroyed, but only its excess over the weight of the body has been consumed—the excess that caused the body to rise. Then as the diminution of the external impetus continues, and gravity gains the upper hand, the fall begins, but slowly at first on account of the opposition of the impressed force, a large portion of which still remains in the body; but as this continues to diminish, it also continues to be more and more overcome by gravity, and hence the continuous acceleration of motion results.

S
IMP.
    The idea is clever, yet more subtle than sound. For even if the argument were conclusive, it would explain only the case where a natural motion is preceded by a violent motion in which there still remains active a portion of the external force; but where there is no such remaining portion and the body starts from an antecedent state of rest, the cogency of the whole argument fails.

S
AGR.
    I believe that you are mistaken and that this distinction between cases which you make is superfluous or, rather, nonexistent. But, tell me, cannot a projectile receive from the projector either a large or a small force, and thus be thrown to a height of a hundred cubits, as well as twenty or four or one?

[202] S
IMP.
    Undoubtedly, yes.

S
AGR.
    So this impressed force may exceed the resistance of gravity so slightly as to raise it only an inch; and finally the force of the projector may be just large enough to exactly balance the resistance of gravity, so that the body is not lifted at all but merely sustained. When you hold a stone in your hand, do you do anything but give it a force impelling it upwards equal to the power of gravity drawing it downwards? And do you not continuously impress this force upon the stone as long as you hold it in the hand? Does it perhaps diminish with the time during which you hold the stone? And what does it matter whether this support that prevents the stone from falling is furnished by one's hand, or by a table, or by a rope from which it hangs? Certainly nothing at all. You must conclude, therefore, Simplicio, that it makes no difference whatever whether the fall of the stone is preceded by a period of rest that is long, short, or instantaneous, provided only that the fall does not begin as long as the stone is acted upon by a force opposed to its weight and sufficient to hold it at rest.

S
ALV.
    The present does not seem to be the proper time to investigate the cause of the acceleration of natural motion, concerning which various opinions have been expressed by various philosophers. That is, some explain it by attraction to the center; others reduce it to the gradual decrease of the amount of medium to be overcome; still others attribute it to a certain pressure of the surrounding medium, which closes in behind the falling body and drives it from one position to another. Now, all these fantasies, and others too, ought to be examined; but it is not really worth while. At present it is the purpose of our Author merely to investigate and to demonstrate some of the properties of an accelerated motion such that (whatever the cause of this acceleration may be) the moments of its velocity go on increasing after departure from rest in simple proportionality to the time, which is the same as saying that in equal time intervals the body receives equal increments of velocity; and if we find that the properties to be demonstrated later are realized in freely falling and accelerated bodies, we may conclude that the assumed definition includes such a motion of falling bodies, and that it is true [203] that their speed goes on increasing as the time and the duration of the motion.

S
AGR.
    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.

S
ALV.
    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.