30 HISTOEY OF ACOUSTICS. 



tceiith vibration of the other, while all the intermediate vil rations of 

 the two tones would, in various degrees, disagree with each other ; 

 and thus every such cycle, of fifteen and sixteen vibrations, might be 

 heard as a separate beat of sound. Now, Sauveur wished to take a 

 case in which these beats were so slow as to be counted, 6 and in which 

 the ratio of the vibrations of the notes was known from a knowledge 



o 



of their musical relations. Thus if the two notes form an interval of 

 a semitone, their ratio will be that above supposed, fifteen to sixteen ; 

 and if the beats be found to be six in a second, we know that, in that 

 time, the graver note makes ninety and the acuter ninety-six vibra- 

 tions. In this manner Sauveur found that an open organ-pipe, five 

 feet long, gave one hundred vibrations in a second. 



Sauveur's other method is more recondite, and approaches to a 

 mechanical view of the question. 8 He proceeded on this basis ; a 

 string, horizontally stretched, cannot be drawn into a mathematical 

 straight line, but always hangs in a very flat curve, or festoon. Hence 

 Sauveur assumed that its transverse vibrations may be conceived to be 

 identical with the lateral swingings of such a festoon. Observing that 

 the string C, in the middle of a harpsichord, hangs in such a festoon 

 to the amount of l-323rd of an inch, he calculates, by the laws of 

 pendulums, the time of oscillation, aiid finds it l-122nd of a second. 

 Thus this C, his fixed note, makes one hundred and twenty -two vibra- 

 tions in a second. It is curious that this process, seemingly so arbi- 

 trary, is capable of being justified on mechanical principles ; though 

 we can hardly give the author credit for the views which this 

 justification implies. It is, therefore, easy to understand that it 

 agreed with other experiments, in the laws which it gave for the 

 dependence of the tone on the length and tension. 



The problem of satisfactorily explaining this dependence, on mecha- 

 nical principles, naturally pressed upon the attention of mathemati- 

 cians when the law of the phenomena was thus completely determined 

 by Mersenne and Sauveur. It was desirable to show that both the 

 circumstances and the measure of the phenomena were such as known 

 mechanical causes and laws would explain. But this problem, as 

 might be expected, was not attacked till mechanical principles, and 

 the modes of applying them, had become tolerably familiar. 



As the vibrations of a string are produced by its tension, it appeared 

 tc be necessary, in the first place, to determine the law of the tension 



5 Ac. Sc. Hist. 1700, p. 131. c Ac. S>.: Hist. 1718 



