476 THE PRINCIPLES OF SCIENCE. [CHAP. 



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 ap- 

 proximation. 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 approxi- 

 mately true. A wave never is exactly the same when 

 another wave is interfering with it, but the less the dis- 

 placement 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 disturb- 

 ances, and was able to show that certain summation 01 

 resultant tones ought to be heard, and experiments subse- 

 quently devised for the purpose showed that they might 

 be heard. 



Throughout the mechanical sciences the Principle of the 

 Superposition of Small Motions is of fundamental im- 

 portance, 1 and it may be thus explained. Suppose 



1 Thomson and Tait's Natural Philosophy, voL i. p. 60. 



