Galileo Galilei


Source: Two New Sciences, including centers of gravity and force of percussion (Discorsi e dimonstrazioni mathematiche, 1632), translated with introduction and notes by Stillman Drake, U of Wisconsin Press, 1974.  Selections from the Third Day, pp. 153-162, 165-167.
 
 

On Naturally Accelerated Motion1


    Those things that happen which relate to equable motion 2  have been considered in the preceding book; next, accelerated motion is to be treated of.
    And first, it is appropriate to seek out and clarify the defintion that best agrees with that [accelerated motion] which nature employs.  Not that there is anything wrong with inventing at pleasure some kind of motion and theorizing about its consequent properties, in the way that some men have derived spiral and conchoidal lines from certain motions, though nature makes no use of these [paths]; and by pretending these, men have laudably demonstrated their essentials from assumptions [ex suppositione].  But since nature does employ a certain kind of acceleration fro descending heavy things, we decided to look into their properties so that we might be sure that the definition of accelerated motion which we are about to adduce agrees with the essence of naturally accelerated motion.  And at length, after continual agitation of mind, we are confindent that this has been found, chiefly for the very powerful reason that the essentials succesively demonstrated by us correspond to, and are seen to be in agreement with, that which physical experiments [naturalia experimenta] show forth to the senses. 3   Further, it is as though we have been led by the hand to the investigation of naturally accelerated motion by consideration of the custom and procedure of nature herself in all her other works, in the performance of which she habitually employs the first, simplest, and easiest means.  And indeed, no one of judgment believes that swimming or flying can be accomplished in a simpler or easier way than that which fish and birds employ by natural instinct.
    Thus when I consider that a stone, falling from rest at some height, successively acquires new increments of speed, why should I not believe that those additions are made by the simplest and most evident rule?  For if we look into this attentively, we can discover no simpler addition and increase than that which is added on always in the same way.  We easily understand that the closest affinity holds between time and motion, and thus equable and uniform motion is defined through uniformities of times and spaces; and indeed, we call movement equable, when in equal times equal spaces are traversed. 4   And by this same equality of parts of time, we can percieve the increase of swiftness to be made simply, conceiving mentally that this motion is uniformly and continually accelerated in the same way whenever, in any equal times, equal additions of swiftness are added on. 5
    Thus, taking any equal particles of time whatever, in the first instance in which the moveable departs from rest and descent is begun, the degree of swiftness acquired in the first and second little parts of time [together] is double the degree that the moveable acquired in the first little part[of time]; and the degree that it gets in three little parts of time is triple; and in four, quadruple that same degree [acquired] in the first particle of time.  So, for clearer understranding, if the moveable were to continue in its motion at the degree of momentum of speed acquired in the first little part of time, and were to extend its motion successively and equably with that degree, this movement would be twice as slow as [that] at the degree of speed obtained in two little parts of time.  And thus it is seen that we shall not depart much from the correct rule if we assume that intensification of speed is made according to the extension of time; from which the definition of the motion of which we are going to treat may be put thus:

[DEFINITION]

I say that that motion is equably or uniformly accelerated which, abandoning rest, adds on to itself equal momenta of swiftness in equal times.


Sagr. Just as it would be unreasonable for me to oppose this, or any other definition whatever assigned by any author, all [definitions] being arbitrary, so I may, without offense, doubt whether this definition, conceived and assumed in the abstract, is adapted to, suitable for, and verified in the kind of accelerated motion that heavy bodies in fact employ in falling naturally. And since it seems that the Author promises us that what he has defined is the natural motion of heavy bodies, I should like to hear you remove certain doubts that disturb my mind, so that I can then apply myself with better attention to the propositions that are expected, and their demonstrations.

Salv. It will be good for you and Simplicio to propound the difficulties, which I imagine will be the same ones that occurred to me when I first saw this treatise, and that our Author himself put to rest for me in our discussions, or that I removed for myself by thinking them out.

Sagr. I picture to myself a heavy body falling. 6  It leaves from rest: that is, from the deprivation of any speed whatever, and enters into motion in which it goes accelerating according to the ratio of increase of time from its first instant of motion. It will have obtained, for example, eight degrees of speed in eight pulse-beats, of which at the fourth beat it will have gained four; at the second [beat], two; and at the first, one. Now, time being infinitely divisible, what follows from this? The speed being always diminished in this ratio, there will be no degree of speed, however small (or we might say, žno degree of slowness, however greatÓ), such that the moveable will not be found to have this [at some time] after its departure from infinite slowness, that is, from rest. Thus if the degree of speed that it had at four beats of time were such that, maintaining this uniformly, it would run two miles in one hour, while with the degree of speed that it had at the second beat it would have made one mile an hour, it must be said that in instants of time closer and closer to the first [instant] of its moving from rest, it would be found to be so slow that, continuing to move with this slowness, it would not pass a mile in an hour, nor in a day, nor in a year, nor in a thousand [years], and it would not pass even one span in some still longer time. Such events I find very hard to accommodate in my imagination, when our senses show us that a heavy body in falling arrives immediately at a very great speed. 7

Salv. This is one of the difficulties that gave me pause at the outset; but not long afterward I removed it, and its removal was effected by the same experience that presently sustains it for you.
    You say that it appears to you that experience shows the heavy body, having hardly left from rest , entering into a very considerable speed; and I say that this same experience makes it clear to us that the first impetuses of the falling body, however heavy it may be, are very slow indeed. Place a heavy body on some yielding material, and leave it until it has pressed as much as it can with its mere weight. It is obvious that if you now raise it one or two braccia, and then let it fall on the same material, it will make a new pressure on impact, greater than it made by its weight alone. This effect will be caused by the falling moveable in conjunction with the speed gained in fall, and will be greater and greater according as the height is greater from which the impact is made; that is, according as the speed of the striking body is greater. The amount of speed of a falling body, then, we can estimate without error from the quality and quantity of its impact.
    But tell me gentlemen; if you let a sledge fall on a pole from a height of four braccia, and it drives this, say, four inches into the ground, and will drive it much less from a height of two braccia, and still less from a height of one, and less yet from a span only; if finally it is raised but a single inch, how much more will it accomplish than if it were placed on top [of the pole] without striking it at all? Certainly very little. And its effect would be quite imperceptible if it were lifted only the thickness of a leaf. Now, since the effect of impact is governed by the speed of a given percussent, who can doubt that its motion is very slow and minimal when its action is imperceptible? You now see how great is the force of truth, when the same experience that seemed to prove one thing at first glance assures us of the contrary when it is better considered.
    But without restricting ourselves to this experience, though no doubt it is quite conclusive, it seems to me not difficult to penetrate this truth by simple reasoning. We have a heavy stone, held in the air at rest. It is freed from support and set at liberty; being heavier than air, it goes falling downward, not with uniform motion, but slowly at first and continually accelerated thereafter. Now, since speed may be increased or diminished in infinitum, what argument can persuade me that this moveable, departing from infinite slowness (which is rest), enters immediately into a speed of ten degrees rather than into one of four, or into the latter before a speed of two, or one, or one-half, or one one-hundredth? Or, in short, into all the lesser [degrees] in infinitum?
    Please hear me out. I believe you would not hesitate to grant me that the acquisition of degrees of speed by the stone falling from the state of rest may occur in the same order as the diminution and loss of those same degrees when, driven by impelling force, the stone is hurled upward to the same height. But if that is so, I do not see how it can be supposed that in the diminution of speed in the ascending stone, consuming the whole speed, the stone can arrive at rest before passing through every degree of slowness.

Simp. But if the degrees of greater and greater tardity are infinite, it will never consume them all, and this rising heavy body will never come to rest, but will move forever while always slowing downůsomething that is not seen to happen.8

Salv. This would be so, Simplicio, if the moveable were to hold itself for any time in each degree; but it merely passes there, without remaining beyond an instant. And since in any finite time [tempo quanto], however small, there are infinitely many instants, there are enough to correspond to the infinitely many degrees of diminished speed. It is obvious that this rising heavy body does not persist for any finite time in any one degree of speed 9, for if any finite time is assigned, and if the moveable had the same degree of speed at the first instant of that time and also at the last, then it could likewise be driven upward with this latter degree [of speed] through as much space [again], just as it was carried from the first [instant] to the second; and at the same rate it would pass from the second to a third, and finally, it would continue its uniform motion in infinitum.

Sagr. From this reasoning, it seems to me that a very appropriate an answer can be deduced for the question agitated among philosophers as to the possible cause of acceleration of the natural motion of heavy bodies. For let us consider that in the heavy body hurled upwards, the force [virtu] impressed upon it by the thrower is continually diminishing, and that this is the force that drives it upward as long as this remains greater than the contrary force of its heaviness; then when these two [forces] reach equilibrium, the moveable stops rising and passes through a state of rest. Here the impressed impetus is [still] not annihilated, but merely that excess has been consumed that it previously had over the heaviness of the moveable, by which [excess] it prevailed over this [heaviness] and drove [the body] upward. The diminutions of this alien impetus then continuing, and in consequence the advantage passing over to the side of the heaviness, descent commences, though slowly because of the opposition of the impressed force, a good part of which still remains in the moveable. And since this continues to diminish, and comes to be overpowered in ever-greater ratio by the heaviness, the continual acceleration of the motion arises therefrom. 10

Simp. The idea is clever, but more subtle than sound; for if it were valid, it would explain only those natural motions which had been preceded by violent motion, in which some part of the external impetus still remained alive. But where there is no such residue, and the moveable leaves from long-standing rest, the whole argument loses its force.

Sagr. I believe you are mistaken, and that the distinction of cases made by you is superfluous, or rather, is idle. For tell me: can the thrower impress on the projectile sometimes much force, and sometimes little, so that it may be driven upward a hundred braccia, or twenty, or four, or only one?

Simp. No doubt he can.

Sagr. No less will the force impressed be able to overcome the resistance of heaviness by so little that it would not raise [the body] more than an inch. And finally, the force of projection may be so small as just to equal the resistance of the heaviness, so that the moveable is not thrown upward, but merely sustained. Thus, when you support a rock in your hand, what else are you doing but impressing on it just as much of that upward impelling force as equals the power of its heaviness to draw it downward? And do you not continue this force of yours, keeping it impressed through the whole time that you support [the rock] in your hand? Does the force perhaps diminish during the length of time that you support the rock? Now, as to this sustaining that prevents the fall of the rock, what difference does it make whether it comes from your hand, or a table, or a rope tied to it? None whatever. You must conlude, then Simplicio, that it makes no difference at all whether the fall of the rock is preceded by a long rest, or a short one, or one only momentary, and that the rock always starts with just as much of the force contrary to its heaviness as was needed to hold it at rest.

Salv. The present does not seem to me to be an opportune time to enter into the investigation of the cause of the acceleration of natural motion, concerning which various philosophers have produced various opinions, some of them reducing this to approach to the center; others to the presence of successively less parts of the medium [remaining] to be divided; and others to a certain extrusion by the surrounding medium which, in rejoining itself behind the moveable, goes pressing and continually pushing it out. Such fantasies, and others like them, would have to be examined and resolved, with little gain.11  For the present, it suffices our Author that we understand him to want us to investigate and demonstrate some attributes [passiones] of a motion so accelerated (whatever be the cause of its acceleration) that the momenta of its speed go increasing, after its departure from rest, in that simple ratio with which the continuation of time increases, which is the same as to say that in equal times, equal additions of speed are made. And if it shall be found that the events that then shall have been demonstrated are verified in the motion of naturally falling and accelerated heavy bodies, we may deem that the definition assumed includes that motion of heavy things, and that it is true that their acceleration goes increasing as the time and the duration of motion increases.

Sagr. By what I now picture to myself in my mind, it appears to me that this could perhaps be defined with greater clarity, without varying the concept, [as follows]: Uniformly accelerated motion is that in which the speed goes increasing according to the increase of space traversed. 12  Thus for example, the degree of speed acquired by the moveable in the descent of four braccia would be double that which it had after falling through the space of two, and this would be the double of that resulting in the space of the first braccio. For there seems to me to be no doubt that the heavy body coming from a height of six braccia has, and strikes with, double the impetus that it would have from falling three braccia, and triple that which it would have from two, and six times that had in the space of one.

Salv. It is very comforting to have had such a companion in error, and I can tell you that your reasoning has in it so much of the plausible and probable, that our Author himself did not deny to me, when I proposed it to him, that he had labored for some time under the same fallacy. But what made me marvel then was to see revealed, in a few simple words, to be not only false but impossible, two propositions which are so plausible that I have propounded them to many people, and have not found one who did not freely concede them to me.

Simp. Truly, I should be one of those who concede them. That the falling heavy body vires acquirat eundo [acquires force in going], the speed increasing in the ratio of the space, while the momentum of the same percussent is double when it comes from double height, appear to me as propositions to be granted without repugnance or controversy.

Salv. And yet they are as false and impossible as [it is] that motion should be made instantaneously, and here is a very clear proof of it. When speeds have the same ratio as the spaces passed or to be passed, those spaces come to be passed in equal times; if therefore the speeds with which the falling body passed the space of four braccia were the doubles of the speeds with which it passed the first two braccia, as one space is double the other space, then the times of those passages are equal 13; but for the same moveable to pass the four braccia and the two in the same time cannot take place except in instantaneous motion. But we see that the falling heavy body makes its motion in time, and passes the two braccia in less [time] than the four; therefore it is false that its speed increases as the space.
    The other proposition is shown to be false with the same clarity. For that which strikes being the same body, the difference and momenta of the impacts must be determined only by the difference of the speeds; if therefore the percussent coming from a double height delivers a blow of double momentum, it must strike with double speed; but double speed passes the double space in the same time, and we see the time of descent to be longer from the greater height.

Sagr. Too evident and too easy is this [reasoning] with which you make hidden conclusions manifest. This great facility renders the conclusions manifest. This great facility renders the conclusions less prized than when they were under seeming contradiction. I think that people generally will little esteem ideas gained with so little trouble, in comparison with those over which long and unresolvable altercations are waged.

Salv. Things would not be so bad if men who show with great brevity and clarity the fallacies of propositions that have commonly been held to be true by people in general received only such bearable injury as scorn in place of thanks. What is truly unpleasant and annoying is a certain other attitude that some people habitually take. Claiming, in the same studies, at least parity with anyone that exists, these men see that the conclusions they have been putting forth as true are later exposed by someone else, and shown to be false by short and easy reasoning. I shall not call their reaction envy, which then usually transforms itself into rage and hatred against those who reveal such fallacies, but I do say that they are goaded by a desire to maintain inveterate errors rather than to permit newly discovered truths to be accepted. This desire sometimes induces them to write in contradiction to those truths of which they themselves are only too aware in their own hearts, merely to keep down the reputations of other men in the estimation of the common herd of little understanding. I have heard from our Academician not a few such false conclusions, accepted as true and [yet] easy to refute; and I have kept a record of some of these.

Sagr. And you must not keep them from us, but must share them with us some time, even if we need a special session for the purpose. But now, taking up our thread again, it seems to me that we have at this point fixed the definition of uniformly accelerated motion, of which we shall treat in the ensuing discussion; and it is this:

[DEFINITION]

We shall call that motion equably or uniformly accelerated which, abandoning Rest, adds on to itself equal momenta of swiftness in equal times.


 
 

PROPOSITION I. THEOREM I 14

The time in which a certain space is traversed by a moveable in uniformly accelerated movement from rest is equal to the time in which the same space would be traversed by the same moveable carried in uniform motion whose degree of speed is one-half the maximum and final degree of speed of the previous, uniformly accelerated, motion.


    Let line AB represent the time in which the space CD is traversed by a moveable in uniformly accelerated movement from rest at C. Let EB, drawn in any way upon AB, represent the maximum and final degree of speed increased in the instants of the time AB. All the lines reaching AE from single points of the line AB and drawn parallel to BE will represent the increasing degrees of speed after the instant A.15  Next, I bisect BE at F, and I draw FG and AG parallel to BA and BF; the parallelogram AGFB will [thus] be constructed, equal to the triangle AEB, its side GF bisecting AE at I.

    Now if the parallels in triangle AEB are extended as far as IG, we shall have the aggregate of all parallels contained in the quadrilateral equal to the aggregate of those included in triangle AEB 16, for those in triangle IEF are matched by those contained in triangle GIA, while those which are in the trapezium AIFB are common. Since each instant and all instants of time AB correspond to each point and all points of line AB, from which points the parallels drawn and included within triangle AEB represent increasing degrees of the increased speed, while the parallels contained within the parallelogram represent in the same way just as many degrees of speed not increased but equable, it appears that there are just as many momenta of speed consumed in the accelerated motion according to the increasing parallels of triangle AEB, as in the equable motion according to the paraliels of the parallelogram GB. For the deficit of momenta in the first half of the accelerated motion (the momenta represented by the parallels in triangle AGI falling short) is made up by the momenta represented by the parallels of triangle IEF.
    It is therefore evident that equal spaces will be run through in the same time by two moveables, of which one is moved with a motion uniformly accelerated from rest, and the other with equable motion having a momentum one-half the momentum of the maximum speed of the accelerated motion; which was [the proposition] intended.

PROPOSITION II. THEOREM II

If a moveable descends from rest in uniformly accelerated motion, the spaces run through in any times whatever are to each other as the duplicate ratio of their times; that is are as the squares of those times. 17


    Let the flow of time from some first instant A be represented by the line AB, in which let there be taken any two times, AD and AE. Let HI be the line in which the uniformly accelerated moveable descends from point H as the first beginning of motion; let space HL be run through in the first time AD, and HM be the space through which it descends in time AE. I say that space MH is to space HL in the duplicate ratio of time EA to Time AD. Or let us say that spaces MH and HL have the same ratio as do the squares of EA and AD.

    Draw line AC at any angle with AB. From points D and E draw the parallels DO and EP, of which DO will represent the maximum degree of speed acquired at instant D of time AD, and PE the maximum degree of speed acquired at instant E of time AE. Since it was demonstrated above that as to spaces run through those are equal to one another of which one is traversed by a moveable in uniformly accelerated motion from rest, and the other is traversed in the same time by a moveable carried in equable motion whose speed is one-half the maximum acquired in the accelerated motion, it follows that spaces MH and LH are the same that would be traversed in times EA and DA in equable motions whose speeds are as the halves of PE and OD. Therefore if it is shown that these spaces MH and LH are in the duplicate ratio of the times EA and DA, what is intended will be proved.
    Now in Proposition IV of Book I [žOn Uniform Motion,Ó above] it was demonstrated that the spaces run through by moveables carried in equable motion have to one another the ratio compounded from the ratio of speeds and from the ratio of times, since the ratio of one-half PE to one-half OD, or of PE to OD, is that of AE to AD. Hence the ratio of spaces run through is the duplicate ratio of the times; which was to be demonstrated.
    It also follows from this that this same ratio of spaces is the duplicate ratio of the maximum degrees of speed; that is, of lines PE and OD, since PE is to OD as EA is to DA.

COROLLARY I

From this it is manifest that if there are any number of equal times taken successively from the first instant or beginning of motion, say AD, DE, EF,and FG, in which spaces HL, LM, MN, and NI are traversed, then these spaces will be to one another as are the odd numbers from unity, that is as 1, 3, 5, 7; but this is the rule [ratio] for excesses of squares of lines equally exceeding one another [and] whose [common] excess is equal to the least of the samelines, or, let us say, of the squares successively from unity. Thus when the degrees of speed are increased in equal times according to the simple series of natural numbers, the spaces run through in the same times undergo increases according with the series of odd numbers from unity. 19
 

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