Approximate Independence of Small Effects.
One result of the theory of approximation possesses such importance in physical science, and is so often applied, that we may consider it separately. The investigation of causes and effects is immensely simplified when we may consider each cause as producing its own effect invariably, whether other causes are acting or not. Thus, if the body P produces x, and Q produces y, the question is whether P and Q acting together will produce the sum of the separate effects, x + y. It is under this supposition that we treated the methods of eliminating error (Chap. XV.), and errors of a less amount would still remain if the supposition was a forced one. There are probably some parts of science in which the supposition of independence of effects holds rigidly true. The mutual gravity of two bodies is entirely unaffected by the presence of other gravitating bodies. People do not usually consider that this important principle is involved in such a simple thing as putting two pound weights in the scale of a balance. How do we know that two pounds together will weigh twice as much as one? Do we know it to be exactly so? Like other results founded on induction we cannot prove it absolutely, but all the calculations of physical astronomy proceed upon the assumption, so that we may consider it proved to a very high degree of approximation. Had not this been true, the calculations of physical astronomy would have been infinitely more complex than they actually are, and the progress of knowledge would have been much slower.
It is a general principle of scientific method that if effects be of small amount, comparatively to our means of observation, all joint effects will be of a higher order of smallness, and may therefore be rejected in a first approximation. This principle was employed by Daniel Bernoulli in the theory of sound, under the title of The Principle of the Coexistence of Small Vibrations. He showed that if a string is affected by two kinds of vibrations, we may consider each to be going on as if the other did not exist. We cannot perceive that the sounding of one musical instrument prevents or even modifies the sound of another, so that all sounds would seem to travel through the air, and act upon the ear in independence of each other. A similar assumption is made in the theory of tides, which are great waves. One wave is produced by the attraction of the moon, and another by the attraction of the sun, and the question arises, whether when these waves coincide, as at the time of spring tides, the joint wave will be simply the sum of the separate waves. On the principle of Bernoulli this will be so, because the tides on the ocean are very small compared with the depth of the ocean.
The principle of Bernoulli, however, is only approximately true. A wave never is exactly the same when another wave is interfering with it, but the less the displacement of particles due to each wave, the less in a still higher degree is the effect of one wave upon the other. In recent years Helmholtz was led to suspect that some of the phenomena of sound might after all be due to resultant effects overlooked by the assumption of previous physicists. He investigated the secondary waves which would arise from the interference of considerable disturbances, and was able to show that certain summation of resultant tones ought to be heard, and experiments subsequently devised for the purpose showed that they might be heard.