crews through the lower pillars.

[Illustration: FIG. 4.]

_Pendulum._--Suppose that we have a body P (fig. 4) at rest, and that it
is material, that is to say, has "mass." And for simplicity let us
consider it a ball of some heavy matter. Let it be free to move
horizontally, but attached to a fixed point A by means of a spring. As
it can only move horizontally and not fall, the earth's gravity will be
unable to impart any motion to it. Now it is a law first discovered by
Robert Hooke (1635-1703) that if any elastic spring be pulled by a
force, then, within its elastic limits, the amount by which it will be
extended is proportional to the force. Hence then, if a body is pulled
out against a spring, the restitutional force is proportional to the
displacement. If the body be released it will tend to move back to its
initial position with an acceleration proportioned to its mass and to
its distance from rest. A body thus circumstanced moves with harmonic
motion, vibrating like a stretched piano string, and the peculiarity of
its motion is that it is isochronous. That is to say, the time of
returning to its initial position is the same, whether it makes a large
movement at a high velocity under a strong restitutional force, or a
small movement at a lower velocity under a smaller restitutional force
(see MECHANICS). In consequence of this fact the balance wheel of a
watch is isochronous or nearly so, notwithstanding variations in the
amplitude of its vibrations. It is like a piano string which sounds the
same note, although the sound dies away as the amplitude of its
vibrations diminishes.

[Illustration: FIG. 5.]

A pendulum is isochronous for similar reasons. If the bob be drawn aside
from D to C (fig. 5), then the restitutional force tending to bring it
back to rest is approximately the force which gravitation would exert
along the tangent CA, i.e.

BC displacement BC
g cos ACW = g -- = g ------------------.
OC length of pendulum

Since g is constant, and the length of the pendulum does not vary, it
follows that when a pendulum is drawn aside through a small arc the
force tending to bring it back to rest is proportional to the
displacement (approximately). Thus the pendulum bob under the influence
of gravity, if the arc of swing is small, acts as though instead of
being acted on by gravity it was acted on by a spring tending to drag it
towards D, and therefore is isochro