J 



ON MUSICAL INSTRUMENTS. 317 



tiga\ions j<^atingto acustics. It must not be denied that these propositions 

 contain some very inconclusive reasoning respecting the nature of the 

 motions constituting sound, hy which the determination of a particular 

 case is erroneously extended into a general solution of the problem. The 

 velocity is, however, truly calculated, because it is in fact independent of 

 the particular nature of the vibration, and all that is wanting to generalise 

 the proposition is the remark, that if the velocity of sound is the same in 

 all cases, it must be such as the calculation indicates. An error nearly 

 similar was committed by Brook Taylor,* who in the year 1714 investi- 

 gated the time occupied by the vibration of a string or cord upon a 

 particular supposition, which he considered as a necessary condition, but 

 which in fact confined the inquiry to a limited case. It happens, however, 

 that the same determination of the frequency of vibration is equally true 

 in all possible cases. Sauveur obtained, about the same time, a similar con- 

 clusion from reasoning still less accurate : his merits with respect to the 

 theory of acustics in general are, however, by no means contemptible. 

 Lagranget and Euler^ have corrected and much extended the investi- 

 gations of Newton, and of Taylor, and Bernoulli and Dalembert|| have 

 also materially contributed to the complete examination and discussion of 

 the subject. 



About the year 1750, Daniel Bernoulli succeeded in obtaining a solution 

 of a problem still more difficult than those which relate to the motions of 

 cords : he determined the frequency of the vibrations of an elastic rod 

 fixed at one end, as well as the relations of its subordinate sounds. The 

 solution is not indeed absolutely general, but it may perhaps be adapted to 

 all possible cases, by considering the effect of a combination of various 

 sounds produced at the same time. Euler has also great merit in extend- 

 ing and facilitating the mathematical part of this investigation, although he 

 has committed several mistakes respecting the mechanical application of it, 

 some of which he has himself corrected, and others have been noticed by 

 Riccati and Chladni. 



The grave harmonics produced by the combination of two acute sounds 

 were noticed about the same time by Romieu and by Tartini, but first by 

 Romieu : ^[ their existence is not only remarkable in itself, but particularly 

 as it leads to some interesting consequences respecting the nature of sound 

 and hearing in general. 



Bernoulli has also investigated, in a very ingenious manner, the sounds 

 produced by the air in pipes of various forms, although confessedly on 

 suppositions deviating in some measure from the truth : the results of his 



* De Motu Nervi Tensi, Ph. Tr. 1713, xxviii. 26. Methodus Incrementorum, 

 Lond. 1715. t Mel. de Turin, i. ii. & UL 



J Hist, et Mem. de Berlin, 1748, 1753, 1759, p. 185, &c. ; 1765, p. 355. Nov. 

 Com. Petr. ix. xvii. xix. ActaPetr. 1779, p. 2; 1780, p. 2; 1781, p. 1. Mel. de 

 Turin, vol. iii. See Lect. XXXI. 



|| Hist, et Mem. de Berlin, 1747, 1750, 1753, 1763. Opuscula, i. & iv. 



IT Mem. de 1'Acad. de Montpellier, 1751. See Tartini, Trattato diMusica, Pad. 

 1754; and Mercadier de Belesta, Systeme de Musique, Paris, 1776; or Matthew 

 Young's Enquiry into the principal Phenomena of Musical Strings, Dublin, 1784, 

 p. 2, sect. vi. The existence of the grave harmonic was first noticed by Sorge, 

 Anweisung zur Stimmung der Orgelwerke, &c. Hamburg, 1744. 



