112 HISTORY OF MECHANICS. 



altogether new, when, in 1747, D'Alembert pub- 

 lished his views on the subject ; in which he main- 

 tained that, instead of one kind of curve only, there 

 were an infinite number of different curves, which 

 answered the conditions of the question. The pro- 

 blem, thus put forward by one great mathematician, 

 was, as usual, taken up by the others, whose names 

 the reader is now so familiar with in such an 

 association. In 1748, Euler not only assented to 

 the generalization of D'Alembert, but held that 

 it was not necessary that the curves so introduced 

 should be defined by any algebraical condition 

 whatever. From this extreme indeterminateness 

 D'Alembert dissented; while Daniel Bernoulli, 

 trusting more to physical and less to analytical rea- 

 sonings, maintained that both these generalizations 

 were inapplicable in fact, and that the solution was 

 really restricted, as had at first been supposed, to 

 the form of the trochoid, and to other forms deriv- 

 able from that. He introduced, in such problems, 

 the "Law of Coexistent Vibrations," which is of 

 eminent use in enabling us to conceive the results of 

 complex mechanical conditions, and the real import 

 of many analytical expressions. In the mean time, 

 the wonderful analytical genius of Lagrange had 

 applied itself to this problem. He had formed the 

 Academy of Turin, in conjunction with his friends 

 Saluces and Cigna; and the first memoir in their 

 Transactions was one by him on this subject: in 

 this and in subsequent writings he has established, 



