Small Scale Versus Large Scale
Table of Contents
I believe that a falling body:
- acquires force [vires] in its descent.
- has a speed that increases in proportion to the space
- has a momentum [momento] that is doubled when it falls from a doubled height
And yet they are as false and impossible as that motion should be completed instantaneously.
If the velocities are in proportion to the spaces traversed, or to be traversed, then these spaces are traversed in equal intervals of time.
If, therefore, the velocity with which the falling body traverses a space of eight feet were double that with which it covered the first four feet (just as 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 fall eight feet and four feet in the same time is possible only in the case of instantaneous [discontinuous] motion.
But observation shows us that the motion of a falling body occupies time, and less of it in covering a distance of four feet than of eight feet; therefore it is not true that its velocity increases in proportion to the space.
The falsity of the other proposition may be shown with equal clearness.
Consider a single striking body the difference of momentum in its blows can depend only upon difference of velocity.
For if the striking body falling from a double height were to deliver a blow of double momentum, it would be necessary for this body to strike with a doubled velocity.
But with this doubled speed it would traverse a doubled space in the same time-interval; observation however shows that the time required for fall from the greater height is longer.
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 certain conclusions which later are quickly and easily shown by another to be false.
I do not describe such a feeling as one of 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.
I have heard from our Academician many such fallacies held as true but easily refutable; some of these I have in mind.
Our definition of uniformly accelerated motion is: A motion is said to be equally or uniformly accelerated when, starting from rest, its momentum (celeritatis momenta) receives equal increments in equal times.
With this definition, Galileo makes a single assumption: 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 we mean 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.
This speed is that which would be acquired by the same body falling from C to B.
Your assumption appears to me so reasonable that it ought to be conceded without question, provided of course there are no chance or outside resistances, and that the planes are hard and smooth, and that the figure of the moving body is perfectly round, so that neither plane nor moving body is rough.
All resistance and opposition having been removed, my reason 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 equal momenta [impeti eguali].
Fig. 45
Imagine this page to represent a vertical wall, with a nail driven into it.
From the nail, let there be suspended a lead bullet of 2 ounces through a fine vertical thread AB 6 feet long.
On this wall, draw a horizontal line DC, at right angles to the vertical thread AB, which hangs about two finger-breadths 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 the string; from this we may rightly infer that the ball in its descent through the arc CB acquired a momentum [impeto] on reaching B, which 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 finger-breadths in order that the thread, again carrying the bullet through the arc CB, may strike upon the nail E when the bullet 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 momentum [impeto] which previously starting at the same point B carried the same body through the arc BD to the horizontal CD.
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 if the nail were placed nearer B than to the intersection of AB with the horizontal CD) then the thread leaps over the nail and twists itself about it.
Fig. 46
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 [momento] acquired by the fall through the arc CB is the same as that gained by fall through the arc DB.
But the momentum [momento] acquired at B, owing to fall through CB, is able to lift the same body [mobile] through the arc BD; therefore, the momentum acquired in the fall BD 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 [momenti] which 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, IB are equal.
Sagr. The argument seems to me so conclusive and the experiment so well adapted to establish the hypothesis that we may, indeed, consider it as demonstrated.
We are going to apply this principle mainly in motions which occur on plane surfaces, and not on curved, along which acceleration varies in a manner greatly different from that which we have assumed for planes.
So that, although the above experiment shows us that the descent of the moving body through the arc CB confers upon it momentum [momento] just sufficient to carry it to the same height through any of the arcs BD, BG, BI, we are not able, by similar means, to show that the event would be identical in the case of a perfectly round ball descending along planes whose inclinations are respectively the same as the chords of these arcs. It seems likely, on the other hand, that, since these planes form angles at the point B, they will present an obstacle to the ball which has descended along the chord CB, and starts to rise along the chord BD, BG, BI.
In striking these planes some of its momentum [impeto] will be lost and it will not be able to rise to the height of the line CD; but this obstacle, which interferes with the experiment, once removed, it is clear that the momentum [impeto] (which gains in strength with 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 inferences from it correspond to and agree perfectly with experiment.
Galileo assumed this single principle passes next to the propositions which he clearly 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 the mean of the highest speed and the speed just before acceleration began.
Let us represent by the line AB the time in which the space CD is traversed by a body which 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 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 which will be equal in area to the triangle AEB, since the side GF bisects the side AE at the point I.
For if the parallel lines in the triangle AEB are extended to GI, then the sum of all the parallels contained in the quadrilateral is equal to the sum 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.
Each and every instant of time in the time-interval AB has its corresponding point on the line AB.
From this line, points parallels drawn in and limited by the triangle AEB represent the increasing values of the growing speed.
Parallels contained within the rectangle represent the values of a speed which is not increasing, but constant, it appears, in like manner, that the momenta [momenta] assumed by the moving body may also 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 momenta may lack in the first part of the accelerated motion (the deficiency of the momenta being represented by the parallels of the triangle AGI) is made up by the momenta represented by the parallels of the triangle IEF.
Fig. 47
Hence equal spaces will be traversed in equal times by two bodies, one of which, starting from rest, moves with a uniform acceleration, while the momentum of the other, moving with uniform speed, is one-half its maximum momentum under accelerated motion.
q. e. d.