DIFFERENT SOUNDS OF THE SAME STEIXG. 37 



had noticed 1 that when a string vibrates, one which is in unison witb 

 it vibrates without being touched. He was also aware that this was 

 true if the second string was an octave or a twelfth below the first. 

 This was observed as a new fact in England in 1674, and comrnuni- 



O ' 



cated to the Royal Society by Wallis. 2 But the later observers as- 

 certained farther, that the longer string divides itself into two, or into 

 three equal parts, separated by nodes, or points of rest; this they 

 proved by hanging bits of paper on different parts of the string. The 

 discovery so modified was again made by Sauveur 3 about 1700. The 

 sounds thus produced in one string by the vibration of another, have 

 been termed Sympathetic Sounds. Similar sounds are often pro- 

 duced by performers on stringed instruments, by touching the string 

 at one of its aliquot divisions, and are then called the Acute Harmo- 

 nics. Such facts were not difficult to explain on Taylor's view of the 

 mechanical condition of the string ; but the difficulty was increased 

 when it was noticed that a sounding body could produce these differ- 

 ent notes at the same time. Mersenue had remarked this, and the fact 

 was more distinctly observed and pursued by Sauveur. The notes 

 thus produced in addition to the genuine note of the string, have been 

 called Secondary Notes ; those usually . heard are, the Octave, the 

 Twelfth, and the Seventeenth above the note itself. To supply a mode 

 of conceiving distinctly, and explaining mechanically, vibrations which 

 should allow of such an effect, was therefore a requisite step in 

 acoustics. 



This task was performed by Daniel Bernoulli in a memoir pub- 

 lished in 1755." He there stated and proved the Principle of the co- 

 existence of small vibrations. It was already established, that a string 

 might vibrate either in a single swelling (if we use this word to ex- 

 press the curve between two nodes which Bernoulli calls a venire), or 

 in two or three or any number of equa" swellings with immoveable 

 nodes between. Daniel Bernoulli showed further, that these nodes 

 might be combined, each taking place as if it were the only one. 

 This appears sufficient to explain the coexistence of the harmonic 

 sounds just noticed. D'Alembert, indeed, in the article Fundamental 

 in the French Encyclopedic, and Lagrange in his Dissertation on 

 Sound in the Turin Memoirs, 5 offer several objections to this explana- 

 tion ; and it cannot be denied that the subject has its difficulties ; but 



1 Harm. lib. iv. Prop. 28 (1636). 2 Ph. Tr. 1677, April. 3 A. P. 1701. 

 * Berlin Mem. 1753, p 147. & T. i. pp. 64, 103. 



