The Essential Galileo (62 page)

First we must recall the fact that on a plane of any inclination whatever a body starting from rest gains speed, or quantity of impetus, in direct proportion to the time, in agreement with the definition of naturally accelerated motion given by the Author. Hence, as he has shown in the preceding proposition, the distances traversed are proportional to the squares of the times and therefore to the squares of the speeds. Whatever impetus is gained at the first instant, the increments of speed during the same time will be respectively the same, since in each case the gain of speed is proportional to the time.

Let
AB
be an inclined plane whose height above the horizontal
BC
is the vertical
AC
. As we have seen above, the impetus of a body falling along the vertical
AC
is to the impetus of the same body along the incline
AB
as
AB
is to
AC
. On the incline
AB
, lay off
AD
, the third proportional to
AB
and
AC;
then the impetus along
AC
is to that along
AB
(i.e., along
AD
) as the length
AC
is to the length
AD.
Therefore, the body will traverse the space
AD
, along the incline
AB
, in the same time which it would take in falling the vertical distance
AC
(since the moments are in the same ratio as the distances); and the speed at
C
is to the speed at
D
as the distance
AC
is to the distance
AD
. But according to the definition of accelerated motion, the speed at
B
is to the speed of the same body at
D
as the time required to traverse
AB
is to the time required for
AD;
and according to the last corollary of the second proposition, the time for passing through the distance
AB
bears to the time for passing through
AD
the same ratio as the distance
AC
(the mean proportional between
AB
and
AD
) to
AD
. Accordingly the two speeds at
B
and
C
each bear to the speed at
D
the same ratio, namely, that of the distance
AC
to
AD;
hence they are equal. This is the theorem which I set out to prove.

From the above we are better able to demonstrate the following third proposition of the Author, in which proposition he employs the preceding principle:
The time required to traverse an incline is to that required to fall through the vertical height of the incline in the same ratio as the length of the incline to
[219]
its height.

For, according to the second corollary of the second proposition, if
AB
represents the time required to pass over the distance
AB
, the time required to pass the distance
AD
will be the mean proportional between these two distances and will be represented by the line
AC;
but if
AC
represents the time needed to traverse
AD
, it will also represent the time required to fall through the distance
AC
, since the distances
AC
and
AD
are traversed in equal times; consequently, if AB represents the time required for
AB
, then
AC
will represent the time required for A
C
. Hence, the times required to traverse
AB
and
AC
are to each other as the distances
AB
and
AC
.

By the same reasoning it can be shown that the time required to fall through
AC
is to the time required for any other incline
AE
as the length
AC
is to the length
AE;
therefore, by equidistance of ratios, the time of fall along the incline
AB
is to that along
AE
as the distance
AB
is to the distance
AE
, etc.
²⁷

One might, by applying this same theorem, as Sagredo will readily see, immediately demonstrate the sixth proposition of the Author. But let us end this digression here, which Sagredo has perhaps found rather tedious, though I consider it quite important for the theory of motion.

S
AGR.
    On the contrary it has given me great satisfaction, and indeed I find it necessary for a complete grasp of that principle.

S
ALV.
    I will now resume the reading of the text.

[§10.8 Day IV: The Parabolic Path of Projectiles]
²⁸

[268] S
ALV.
    Once more, Simplicio is here on time. So let us, without rest, take up the question of motion. Here is the text of our Author “On the Motion of Projectiles”:

In the preceding pages we have discussed the properties of uniform motion and of motion naturally accelerated along planes of all inclinations. I now propose to set forth those properties that belong to a body whose motion is compounded of two other motions, namely, one uniform and one naturally accelerated; these properties, well worth knowing, I propose to demonstrate in a rigorous manner. This is the kind of motion seen in a moving projectile; its origin I conceive to be as follows.

Imagine any particle projected along a horizontal plane without friction. Then we know, from what has been more fully explained in the preceding pages, that this particle will move along this same plane with a motion that is uniform and perpetual, provided the plane has no limits. But if the plane is limited and elevated, then the moving particle, which we imagine to be a heavy body, will on passing over the edge of the plane acquire, in addition to its previous uniform and enduring motion, a downward propensity due to its own weight; and so the resulting motion, which I call projection, is compounded of one that is uniform and horizontal and another that is downward and naturally accelerated. We now proceed to demonstrate some of its properties, the first of which is as follows.

[269]
Theorem 1, Proposition 1: A projectile that is carried by a uniform horizontal motion compounded with a naturally accelerated downward motion describes a path that is a semiparabola.

S
AGR.
    Here, Salviati, it will be necessary to stop a little while for my sake and, I believe, also for the benefit of Simplicio; for it so happens that I have not gone very far in my study of Apollonius
²⁹
and am merely aware of the fact that he treats of the parabola and other conic sections, without an understanding of which I hardly think one will be able to follow the proof of other propositions depending upon them. Since even in this first beautiful theorem the author finds it necessary to prove that the path of a projectile is a parabola, I imagine we shall have to deal with this kind of curve, and so it will be absolutely necessary to have a thorough understanding, if not of all the properties which Apollonius has demonstrated for these figures, at least of those that are needed for the present treatment.

S
ALV.
    You are quite too modest, pretending ignorance of facts which not long ago you acknowledged as well known—I mean at the time when we were discussing the strength of materials and needed to use a certain theorem of Apollonius that gave you no trouble.

S
AGR.
    I may have chanced to know it; or I may possibly have assumed it since it was needed only once in that discussion. But now when we have to follow all these demonstrations about such curves, we ought not, as they say, to swallow it whole, and thus waste time and energy.

S
IMP
. And then, even if Sagredo were, as I believe, well
equipped for all his needs, I do not understand even the elementary terms; for although our philosophers have treated the motion of projectiles, I do not recall their having described the path of a projectile except to state in a general way that it is always a curved line, unless the projection be vertically upwards. Thus, if [270] the little geometry I have learned from Euclid since our previous discussion does not enable me to understand the demonstrations that are to follow, then I shall be obliged to accept the theorems on faith, without fully comprehending them.

S
ALV.
    On the contrary, I desire that you should understand them from the Author himself, who, when he allowed me to see this work of his, was good enough to prove for me two of the principal properties of the parabola because I did not happen to have at hand the books of Apollonius. These properties, which are the only ones we shall need in the present discussion, he proved in such a way that no prerequisite knowledge was required. These theorems are, indeed, proved by Apollonius, but after many preceding ones, which would take a long time to follow. I wish to shorten our task by deriving the first property purely and simply from the mode of generation of the parabola and proving the second immediately from the first.

Beginning now with the first, imagine a right cone, erected upon the circular base
ibkc
with apex at
l
. The section
bac
of this cone made by a plane drawn parallel to the side
lk
is the curve that is called a parabola. The base of this parabola
bc
cuts at right angles the diameter
ik
of the circle
ibkc
, and the axis
ad
is parallel to the side
lk
. Now having taken any point
f
in the curve
bfa
, draw the straight line
fe
parallel to
bd
. Then, I say, the square of
bd
is to the square of
fe
in the same ratio as the axis
ad
is to the portion
ae
.

Now, through the point
e
pass a plane parallel to the circle
ibkc
, producing in the cone a circular section whose diameter is the line
geh
. Since
bd
is at right angles to
ik
in the circle
ibk
, the square of
bd
is equal to the rectangle formed by
id
and
dk;
so also in the upper circle that passes through the points
gfh
, the square of
fe
is equal to the rectangle formed by
ge
and
eh;
hence the square of
bd
is to the square of
fe
as the rectangle
id-dk
is to the rectangle
ge-eh
. And since the line
ed
is parallel to
hk
, the line
eh
, being parallel to
dk
, is equal to it; therefore the rectangle
id-dk
is to the rectangle
ge-eh
as [271]
id
is to
ge
, that is, as
da
is to
ae;
hence also the rectangle
id-dk
is to the rectangle
ge-eh
, that is, the square of
bd
is to the square of
fe
, as the axis
da
is to the portion
ae
. QED.

The other proposition necessary for this discussion we demonstrate as follows. Let us draw a parabola whose axis
ca
is prolonged upwards to a point
d;
from any point
b
draw the line
bc
parallel to the base of the parabola; if now the point
d
is chosen so that
da
equals
ca
, then, I say, the straight line drawn through the points
b
and
d
will be tangent to the parabola at
b
.