389 



LECTURE XXXIir. 



ON HARMONICS. 



The philosopliical theory of harmonics, or of the combinations of sounds, 

 was considered by the ancients as affording one of the most refined em- 

 ployments of mathematical speculation ; nor has it been neglected in modern 

 times, but it has been in general either treated in a very abstruse and confused 

 manner, or connected entirely with the practice of music, and habitually 

 associated with ideas of mere amusement. We shall, however, find the 

 difficulties by no means insuperable, and the subject will appear to be 

 worthy of attention, not only on its own account, but also for the sake of 

 its analogy with many other departments of science. 



It appears both from theory and from experience, that the transmission 

 of one sound does not at all impede the passage of another through the same 

 medium. The ear too is capable of distinguishing, without difficulty, not 

 only two sounds at once, but also a much greater number. The motions 

 produced by one series of undulations being wholly indiffisrent with respect ^ 

 to the effect of another series, and each particle of the medium being neces- 

 sarily agitated by both sounds, its ultimate motion must always be the 

 result of the motions which would have been produced in it by the 

 separate sounds, combined according to the general laws of the composi- 

 tion of motion, which are the foundation of the principal doctrines of 

 mechanics. When the two sounds, thus propagated together, coincide 

 very nearly in direction, the motions belonging to each sound may 

 be resolved into two parts, the one in the common or intermediate direction, 

 and the other transverse to it; the latter portions will obviously be very small ; 

 they will sometimes destroy each other, and may always be neglected in determin- 

 ing the effect of the combination, since the ear is incapable of distinguishing 

 a difference in the directions of sounds which amounts to a very few degrees 



