f
distance. Thus:

Time 1 Distance 1
" 2 " 4
" 3 " 9
" 4 " 16

So it is obvious that the distance moved through by a body whose motion
is uniformly accelerated increases as the square of the time.

But, you are asking, what has all this to do with a revolving body? As
soon as your minds can be started from a state of rest, you will
perceive that it has everything to do with a revolving body. The
centripetal force, which acts upon a revolving body to draw it to the
center, is a constant force, and under it the revolving body must move
or be deflected through distances which increase as the squares of the
times, just as any body must do when acted on by a constant force. To
prove that a revolving body obeys this law, I have only to draw your
attention to Fig. 2. Let the equal arcs, A B and B C, in this figure
represent now equal times, as they will do in case of a body revolving
in this circle with a uniform velocity. The versed sines of the angles,
A O B and A O C, show that in the time, A C, the revolving body was
deflected four times as far from the tangent to the circle at A as it
was in the time, A B. So the deflection increased as the square of the
time. If on the table already given, we take the seconds of arc to
represent equal times, we see the versed sine, or the amount of
deflection of a revolving body, to increase, in these minute angles,
absolutely so far as appears up to the fifteenth place of decimals, as
the square of the time.

The standard from which all computations are made of the distances
passed through in given times by bodies whose motion is uniformly
accelerated, and from which the velocity acquired is computed when the
accelerating force is known, and the force is found when the velocity
acquired or the rate of acceleration is known, is the velocity of a body
falling to the earth. It has been established by experiment, that in
this latitude near the level of the sea, a falling body in one second
falls through a distance of 16.083 feet, and acquires a velocity of
32.166 feet per second; or, rather, that it would do so if it did not
meet the resistance of the atmosphere. In the case of a falling body,
its weight furnishes, first, the inertia, or the resistance to motion,
that has to be overcome, and affords the measure of this resistance,
and, second, it furnishes the measure of the attraction of the earth, or
the f