The Essential Galileo (59 page)

S
IMP.
    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.

S
ALV.
    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 velocities
²⁰
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.

S
AGR.
    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.

S
ALV.
    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.

S
AGR.
    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

S
ALV.
    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.

S
AGR.
    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.

S
ALV.
    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 through 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.

S
AGR.
    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.

S
ALV.
    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.