make it quite clear. I shall have to preface
it by the explanation of two simple laws. The first of these is, that a
body acted on by a constant force, so as to have its motion uniformly
accelerated, suppose in a straight line, moves through distances which
increase as the square of the time that the accelerating force continues
to be exerted.

The necessary nature of this law, or rather the action of which this law
is the expression, is shown in Fig. 3.

[Illustration: Fig. 3]

Let the distances A B, B C, C D, and D E in this figure represent four
successive seconds of time. They may just as well be conceived to
represent any other equal units, however small. Seconds are taken only
for convenience. At the commencement of the first second, let a body
start from a state of rest at A, under the action of a constant force,
sufficient to move it in one second through a distance of one foot. This
distance also is taken only for convenience. At the end of this second,
the body will have acquired a velocity of two feet per second. This is
obvious because, in order to move through one foot in this second, the
body must have had during the second an average velocity of one foot per
second. But at the commencement of the second it had no velocity. Its
motion increased uniformly. Therefore, at the termination of the second
its velocity must have reached two feet per second. Let the triangle A B
F represent this accelerated motion, and the distance, of one foot,
moved through during the first second, and let the line B F represent
the velocity of two feet per second, acquired by the body at the end of
it. Now let us imagine the action of the accelerating force suddenly to
cease, and the body to move on merely with the velocity it has acquired.
During the next second it will move through two feet, as represented by
the square B F C I. But in fact, the action of the accelerating force
does not cease. This force continues to be exerted, and produces on the
body during the next second the same effect that it did during the first
second, causing it to move through an additional foot of distance,
represented by the triangle F I G, and to have its velocity accelerated
two additional feet per second, as represented by the line I G. So in
two seconds the body has moved through four feet. We may follow the
operation of this law as far as we choose. The figure shows it during
four seconds, or any other unit, of time, and also for any unit o