An introduction to mathematics, by A. N. Whitehead.

170 INTRODUCTION TO MATHEMATICS
principle of resonance. Resonance arises
when two sets of connected circumstances
have the same periodicities. It is a dynamical law that the small vibrations of all bodies
when left to themselves take place in definite
times characteristic of the body. Thus a
pendulum with a small swing always vibrates
in some definite time, characteristic of its
shape and distribution of weight and length.
A more complicated body may have many
ways of vibrating; but each of its modes of
vibration will have its own peculiar "period."
Those periods of vibration of a body are
called its "free" periods. Thus a pendulum
has but one period of vibration, while a suspension bridge will have many. We get a
musical instrument, like a violin string, when
the periods of vibration are all simple submultiples of the longest; i.e. if t seconds be
the longest period, the others are 1t, it, and so
on, where any of these smaller periods may be
absent. Now, suppose we excite the vibrations of a body by a cause which is itself periodic; then, if the period of the cause is very
nearly that of one of the periods of the body,
that mode of vibration of the body is very
violently excited; even although the magnitude of the exciting cause is small. This
phenomenon is called "resonance." The
general reason is easy to understand. Any
one wanting to upset a rocking stone will