38 HISTORY OF ACOUSTICS. 



still these do not deprive Bernoulli of the merit of having pointed 

 out the principle of Coexistent Vibrations, or divest that principle of 

 its value in physical science. 



Daniel Bernoulli's Memoir, of which we speak, was published at a 

 period when the clouds which involve the general analytical treatment 

 of the problem of vibrating strings, were thickening about Euler and 

 D'Alembert, and darkening into a controversial hue ; and as Bernoulli 

 ventured to interpose his view, as a solution of these difficulties, which, 

 in a mathematical sense, it is not, we can hardly be surprised that he 

 met with a rebuff. The further prosecution of the different modes of 

 vibration of the same body need not be here considered. 



The sounds which are called Grave Harmonics, have no analogy 

 with the Acute Harmonics above-mentioned ; nor do they belong to 

 this section ; for in the case of Grave Harmonics, we have one sound 

 from the co-operation of two strings, instead of several sounds from 

 one string. These harmonics are, in fact, connected with beats, of 

 which we have already spoken ; the beats becoming so close as to pro- 

 duce a note of definite musical quality. The discovery of the Grave 

 Harmonics is usually ascribed to Tartini, who mentions them in 1*754 ; 

 but they are first noticed 6 in the work of Sorge On tuning Organs, 

 1 744. He there expresses this discovery in a query. " Whence 

 comes it, that if we tune a fifth (2 : 3), a third sound is faintly heard, 

 the octave below the -lower of the two notes ? Nature shows that 

 with 2 : 3, she still requires the unity, to perfect the order 1, 2, 3." 

 The truth is, that these numbers express the frequency of the vibra- 

 tions, and thus there will be coincidences of the notes 2 and 3, which 

 are of the frequency 1, and consequently give the octave below the 

 sound 2. This is the explanation given by Lagrange, 7 and is indeed 

 obvious. 



J 



CHAPTER V. 



PROBLEM OF THE SOUNDS OF PIPES. 



T was taken for granted by those who reasoned on sounds, that the 

 sounds of flutes, organ-pipes, and wind-instruments in general, con- 



c Chladni. Acoust. p. 254. T Mem. Tur. i. p. 104. 



