PROPAGATION OF SOUND. 345 



though logically incomplete, was the great step in 

 the solution of the problem ; for mathematicians 

 could not but presume that his result was not re- 

 stricted to the hypothesis on which he had obtained 

 it ; and the extension of the solution required only 

 mere ordinary talents. The logical defect of his 

 solution was assailed, as might have been expected. 

 Cramer (professor at Geneva), in 1741, conceived 

 that he was destroying the conclusiveness of New- 

 ton's reasoning, by showing that it applied equally 

 to other modes of oscillation. This, indeed, contra- 

 dicted the enunciation of the 48th Prop, of the 

 Second Book of the Principia ; but it confirmed 

 and extended all the general results of the demon- 

 stration ; for it left even the velocity of sound unal- 

 tered, and thus showed that the velocity did not 

 depend mechanically on the type of the oscillation. 

 But the satisfactory establishment of this physical 

 generalization was to be supplied from the vast 

 generalizations of analysis, which mathematicians 

 were now becoming able to deal with. Accordingly 

 this task was performed by the great master of 

 analytical generalization, Lagrange, in 1759, when, 

 at the age of twenty-three, he and two friends pub- 

 lished the first volume of the Turin Memoirs. Euler, 

 as his manner was, at once perceived the merit of 

 the new solution, and pursued the subject on the 

 views thus suggested. Various analytical improve- 

 ments and extensions were introduced into the 



