PROBLEM OF VIBRATIONS OF STRINGS. 333 



author in whom I find an examination of the details 

 of this case, (Harmonicorum Liber, Paris, 1636.) 

 He asserts", that the differences and concords of 

 acute and grave sounds depend on the rapidity 

 of vibrations, and their ratio ; and he proves this 

 doctrine by a series of experimental comparisons. 

 Thus he finds 3 that the note of a string is as its 

 length, by taking a string first twice, and then four 

 times as long as the original string, other things 

 remaining the same. This, indeed, was known to 

 the ancients, and was the basis of that numerical 

 indication of the notes which the proposition ex- 

 presses. Mersenne further proceeds to show the 

 effect of thickness and tension. He finds (Prop. 7) 

 that a string must be four times as thick as another, 

 to give the octave below ; he finds, also (Prop. 8), 

 that the tension must be about four times as great 

 in order to produce the octave above. From these 

 proportions various others are deduced, and the law 

 of the phenomena of this kind may be considered 

 as determined. Mersenne also undertook to mea- 

 sure the phenomena numerically, that is, to deter- 

 mine the number of vibrations of the string in 

 each of such cases ; which at first might appear 

 difficult, since it is obviously impossible to count 

 with the eye the passages of a sounding string 

 backwards and forwards. But Mersenne rightly 

 assumed, that the number of vibrations is the same 

 so long as the tone is the same, and that the ratios 



2 L. i. Prop. 15. 3 L. ii. Prop. 6. 



