Even phenomena, which on the surface seem casual and exceptional, or, on the other hand, maintain themselves with a uniform persistency, may be due to the remote influence of periodicity. Take for example, the
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 seconds be the longest period, the others are , , 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 push "in tune"
with the oscillations of the stone, so as always to secure a favourable moment for a push. If the pushes are out of tune, some increase the oscillations, but others check them. But when they are in tune, after a time all the pushes are favourable. The word "resonance"
comes from considerations of sound: but the phenomenon extends far beyond the region of sound. The laws of absorption and emission of light depend on it, the "tuning" of receivers for wireless telegraphy, the comparative importance of the influences of planets on each other's motion, the danger to a suspension bridge as troops march over it in step, and the excessive vibration of some ships under the rhythmical beat of their machinery at certain speeds. This coincidence of periodicities may produce steady phenomena when there is a constant association of the two periodic events, or it may produce violent and sudden outbursts when the association is fortuitous and temporary.
Again, the characteristic and constant periods of vibration mentioned above are the underlying causes of what appear to us as steady excitements of our senses. We work for hours in a steady light, or we listen to a steady unvarying sound. But, if modern science be correct, this steadiness has no counterpart in nature. The steady light is due to the impact on the eye of a countless
number of periodic waves in a vibrating ether, and the steady sound to similar waves in a vibrating air. It is not our purpose here to explain the theory of light or the theory of sound. We have said enough to make it evident that one of the first steps necessary to make mathematics a fit instrument for the investigation of Nature is that it should be able to express the essential periodicity of things. If we have grasped this, we can understand the importance of the mathematical conceptions which we have next to consider, namely, periodic functions.