The Essential Galileo (61 page)

We took a piece of wooden molding or scantling, about twelve cubits long, half a cubit wide, and three inches thick; on its edge we cut a channel a little more than one inch in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball. [213] Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time required to make the entire descent. We repeated this experiment many times in order to measure the time with an accuracy such that the deviation between two measurements never exceeded one-tenth of a pulse beat. Having performed this operation and having assured ourselves of its reliability, we now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that for half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane, i.e., of the channel along which we rolled the ball. We also observed that the times of descent, for various inclinations of the plane, bore to one another precisely that ratio that, as we shall see later, the Author had predicted and demonstrated for them.

For the measurement of time, we employed a large vessel of water placed in an elevated position. To the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length. The water thus collected after each descent was weighed on a very accurate balance. The differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.

S
IMP.
    I would like to have been present at these experiments. But feeling confidence in the care with which you performed them, and in the fidelity with which you relate them, I am satisfied and accept them as true and most certain.

S
ALV.
    Then we can resume our reading and proceed.

[214]
Corollary 2: Secondly, it follows that, starting from any initial point
,
if we take any two distances, traversed in any time intervals whatsoever, these time intervals bear to one another the same ratio as one of the distances to the mean proportional of the two distances.

That is, if from the initial point
S
we take two distances
ST
and
SV
and their mean proportional is
SX
, the time of fall through
ST
is to the time of fall through
SV
as
ST
is to
SX;
and the time of fall through
SV
is to the time of fall through
ST
as
SV
is to
SX.
For since it has been shown that the spaces traversed are in the same ratio as the squares of the times; and since, moreover, the ratio of the space
SV
to the space
ST
is the square of the ratio
SV
to
SX;
it follows that the ratio of the times of fall through
SV
and
ST
is the ratio of the distances
SV
and
SX.

Scholium:
The above corollary has been proven for the case of vertical fall. But it holds also for planes inclined at any angle; for it is to be assumed that along these planes the velocity increases in the same ratio, that is, in proportion to the time, or, if you prefer, as the series of natural numbers.
²⁴

Here, Sagredo, I should like, if it be not too tedious to Simplicio, to interrupt for a moment the present reading in order to make some additions on the basis of what has already been proved and of what mechanical principles we have already learned from our Academician. This addition I make for the greater confirmation of the truth of the principle which we have considered above by means of probable arguments and experiments; and what is more important, for the purpose of deriving it geometrically, after first demonstrating a single lemma that is fundamental in the study of impetus.

S
AGR.
    If the advance which you propose to make is such as will confirm and fully establish these sciences of motion, I will gladly devote to it any length of time. Indeed, I shall not only be glad [215] to have you proceed, but I beg of you at once to satisfy the curiosity which you have awakened in me concerning this particular point. And I think that Simplicio is of the same mind.

S
IMP.
    Quite right.

S
ALV.
    Since then I have your permission, let us first of all consider this notable fact—that the momenta or speeds of one and the same moving body vary with the inclination of the plane. The speed reaches a maximum along a vertical direction, and for other directions it diminishes as the plane diverges from the vertical. Therefore the impetus, strength, energy, or, one might say, the momentum of descent of the moving body is diminished by the plane upon which it is supported and along which it rolls.

For the sake of greater clearness, erect the line
AB
perpendicular to the horizontal
AC;
next draw
AD, AE,AF
, etc., at different inclinations to the horizontal. Then I say that all the impetus of the falling body is along the vertical and is a maximum when it falls in that direction; the momentum is less along
DA
and still less along
EA
, and even less yet along the more inclined
FA.
Finally, on the horizontal
CA
the impetus vanishes altogether; the body finds itself in a condition of indifference as to motion or rest; it has no inherent tendency to move in any direction and offers no resistance to being set in motion. offers no resistance to being set in motion. For just as a heavy body or system of bodies cannot of itself move upwards, or recede from the common center toward which all heavy things tend, so it is impossible for any body of its own accord to assume any motion other than one that carries it nearer to the aforesaid common center. Hence, along the horizontal, by which we understand a surface every point of which is equidistant from this same common center, the body will have no impetus or momentum whatever.

[216] This change of impetus being clear, it is here necessary for me to explain something which our Academician wrote when in Padua, embodying it in a treatise on mechanics prepared solely for the use of his students, and proving it at length and conclusively when considering the origin and nature of that marvelous instrument, the screw. What he proved is the manner in which the impetus varies with the inclination of the plane, as for instance that of the plane
FA
, one end of which is elevated through a vertical distance
FC.
This direction
FC
is that along which the impetus of a heavy body and the momentum of descent become maximum; let us try to determine what ratio this momentum bears to that of the same body moving along the incline
FA
. This ratio, I say, is the inverse of that of the aforesaid lengths. This is the lemma preceding the theorem which I hope to demonstrate later.

It is clear that the impetus of a falling body is equal to the least resistance or force sufficient to hinder it and stop it. In order to measure this force or resistance, I propose to use the weight of another body. Let us place upon the plane
FA
a body
G
connected to the weight
H
by means of a string passing over the point
F;
then the body
H
will ascend or descend, along the perpendicular, the same distance which the body
G
moves along the incline
FA;
but this distance will not be equal to the rise or fall of
G
along the vertical, in which direction alone
G
, like other bodies, exerts its resistance. This is clear. For consider that the motion of the body
G
from
A
to
F
in the triangle
AFC
is made up of a horizontal component
AC
and a vertical component
CF;
and remember that this body experiences no resistance [217] to motion along the horizontal (because by such a motion the body neither gains nor loses distance from the common center of heavy things, which distance is constant along the horizontal); then it follows that resistance is met only in consequence of the body rising through the vertical distance
CF
. Since then the body
G
in moving from
A
to
F
offers resistance only in so far as it rises through the vertical distance
CF
, while the other body
H
must fall vertically an amount equivalent to the entire distance
FA;
and since this ratio is maintained whether the motion be large or small, the two bodies being tied together; hence, we are able to assert positively that in case of equilibrium (namely, when the two bodies are at rest) the momenta, the velocities, or their propensities to motion, i.e., the spaces that would be traversed by them in equal times, must be in the inverse ratio to their weights.
²⁵
This is what has been demonstrated in every case of mechanical motion. Thus, in order to hold the weight
G
at rest, one must give
H
a weight smaller in the same ratio as the distance
CF
is smaller than
FA
.

If we do this, namely, we let the ratio of the weight
G
to the weight
H
be the same as
FA
to
FC
, then equilibrium will occur, that is, the weights
H
and
G
will have equal moments and the two bodies will come to rest. And since we are agreed that the impetus, energy, momentum, or propensity to motion of a moving body is as great as the least force or resistance sufficient to stop it; and since we have found that the weight
H
is capable of preventing motion in the weight
G;
it follows that the lesser weight
H
, whose entire moment is along the perpendicular
FC
, will be an exact measure of the partial moment which the larger weight
G
exerts along the inclined plane
FA
. But the measure of the total moment of the body
G
is its own weight, since to prevent its fall it is only necessary to balance it with an equal weight, provided this second weight be free to move vertically. Therefore, the partial impetus or moment of
G
along the incline
FA
will bear to the maximum and total impetus of this same body
G
along the perpendicular
FC
the same ratio as the weight
H
to the weight
G;
this ratio is, by construction, the same which the height
FC
of the incline bears to the length
FA
. We have here the lemma which I proposed to demonstrate and which, as you will see, has been assumed by our Author in the second part of the sixth proposition of the present treatise.

S
AGR.
    From what you have shown thus far, it appears to me that one might infer, arguing by perturbed equidistance of ratios, that the moments of one and the same body moving along planes differently inclined but having the same vertical height, such as
FA
and
FI
,
²⁶
are to each other inversely as the lengths of the planes.

[218] S
ALV.
    Perfectly right. This point established, I pass to the demonstration of the following theorem:
If a body falls freely along smooth planes inclined at any angle whatsoever but of the same height, the speeds which it has when reaching the bottom are equal, provided that all impediments are removed.