M. Duhamel on the Multiple Sounds of Bodies. 417 



perposition of an indefinite number of curves, having for their 

 bases, some the entire length of the string, and others the half, 

 the third, the quarter, &c. If these different curves were taken 

 separately for the figure of the string, the fundamental note 

 would be heard for the first, the octave for the second, the 

 twelfth for the third, &c. Now analysis shows that if the 

 ordinates of the initial figure were the sums of those which 

 correspond to any number of these curves, the ordinates vari- 

 able with the time would always be the sums of those which 

 should correspond to them, at the same instant, in the partial 

 motions answering to each of the curves, taken as initial figure. 

 Thence Daniel Bernouilli concludes that the movement of 

 each point being capable of being decomposed into a succes- 

 sion of others, which, if they existed singly, would give the 

 sounds 1, 2, 3, 4, &c., all these sounds must necessarily be 

 heard at once. He expresses himself as follows, in the par- 

 ticular case in which the octave alone is heard along with its 

 fundamental sound: — "This absolute movement of the point 

 D comprises really two periodical movements, one with relation 

 to the point C, and the other with relation to the point B. 

 The number of the first periodical returns will always be 

 double that of the second. The mind perceives each species of 

 these periodical returns^ and thence remarks two sounds, one of 

 •which is the octave of the other" 



This rather subtile explanation, however, did not satisfy 

 all geometers ; and in reality it is not an explanation, since 

 the fact in question is not referred to other admitted facts. 



Lagrange well observed, that this decomposition of the 

 movement was a purely geometrical conception, but one which 

 proved nothing relative to the sound produced ; he added, 

 that this sound could alone result from the absolute movement, 

 which was single. Daniel Bernouilli was much surprised at 

 these objections, which in no degree altered his views. The 

 following passage occurs in his memoir entitled Recherches 

 Physiques, Mecanigues, et Analytiques, sur le son, et sur les 

 tons des tuyaux d'orguesx — "I have given the explanation of 

 this phaenomenon with relation to strings in the Memoires 

 de Berlin o{ n 53. This explanation is so luminous and so 

 self-evident, that the celebrated M. de Lagrange cannot have 

 examined it with sufficient attention. Not content with reject- 

 ing it, he blames the learned M. Euler for having approved 

 it. Let I be the length of the string, tt the semicircle in the 

 radius = 1 ; can it be doubted that the equation 



. ntx o . Swa^ 

 ^=«sin-y- +bsin— T- 



PhiL Mag, S. 3. Vol. 84. No. 231. June 1849. 2 E 



