304 LECTURE XXXIII. 



Physiol. Elliott on Vision and Hearing, 1780. Vicq d'Azyr ofc^e Hearing of 

 Birds, Hist, et Mem. de Paris, 1778, p. 381, H. 5. Galvani on do. Corn^Dn. vi. 

 O. 420. Scarpa de Auditu et Olfactu, fol. Patav. 1789. Comparetti de Aure 

 Interna, 4to, Paris, 1789. Home on the Membrana Tympani, Ph. Tr. 1800, p. 1. 

 Cooper on do. ibid. 1800, p. 151 ; 1801, p. 435. Gough on the Method of judging 

 of the Position of Sonorous Bodies, Manch. Mem. v. 622. Darwin's Zoonomia, ii. 

 487. Saunders's Anatomy of the Ear, 1806. Ramdohr, Magazin der Gess. Nat. 

 Freunde, Berlin, 1811, p. 389. Cuvier's Report on a Paper of Flourens, Annales 

 de Chimie, xxxix. 104. Muncke, Kastner's Archiv, vii. 1. Wollaston on Sounds 

 inaudible to certain Ears, Ph. Tr. 1820. Weber de Aure, Lips. 1820. Wheat- 

 stone, Journal of Science, 1827, vol. ii. Breschet, Recherches sur 1'Organe de 

 1'Ouie, 1836. Cyclopaedia of Anat. and Phys. art. Organ of Hearing, by Jones. 

 Lincke, Handbuch der Ohren heilkunde, Leipz. 1837. * 



LECTURE XXXIII. 



ON HARMONICS. 



THE philosophical theory of harmonics, or of the combinations of 

 sounds, was considered by the ancients as affording one of the most refined 

 employments 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 entirety 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 mo- 

 tions produced by one series of undulations being wholly indifferent with 

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

 necessarily 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 composition 

 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 determining 

 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 

 only. Thus, when two equal undulations, of equal frequency, coincide in 

 this manner, and when the particular motions are directed the same way 

 at the same time, the velocities in each direction are added together, and 



