THEORY OF APPROXIMATION. 97 



the scale of a balance. How do we know that two pound 

 weights 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 certainly and 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. We may, in fact, 

 assume with much probability that bodies gravitate in 

 entire independence of each other. Had not this been 

 true the calculations of physical astronomy would have 

 been almost infinitely more complex than they actually 

 are, and the progress of knowledge would have been 

 vastly slower. 



The science of the spectrum again is much simplified by 

 the fact that elements do not apparently interfere with 

 each other in the production of light. The spectrum of 

 sodium chloride is the spectrum of sodium superposed 

 upon that of chlorine. Were it otherwise, we should 

 have as many distinct spectra as there are distinct com- 

 pounds in chemistry, and the subject would be almost 

 hopelessly complex. The spectrum of a substance would 

 then no more enable us to tell its components than the 

 appearance of a new mineral indicates its composition. 

 But it would probably be too early to assert the entire 

 absence of any joint spectra. There is so much yet 

 unexplained in the subject that some effects due to the 

 mutual action of elements may possibly be discovered, 

 and the independence wiU then be only approximate. 



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 distinctly employed by 

 Daniel Bemouilli in the theory of sound, under the title of 

 'The Principle of the Coexistence of Small Vibrations.' 



VOL. II. H 



