16 G. H. KNIBBS. 



Leibniz'[1646 - 1716], Newton's contemporary and co-discoverer 

 in the calculus, established a notation which has, except for certain 

 purposes in dynamics, considerable advantages over the fluxional 

 notation of Newton. Among the first to appreciate the dis- 

 coveries of Leibniz at their true worth, were Jakob and Johann 

 Bernoulli, and Daniel Bernoulli, the son of the latter ; all of whom 

 like Newton, shewed the great power of the calculus in its 

 physical applications. 



The English mathematicians Roger Cotes [1682 - 1716], Brook 

 Taylor [1685-1731], and Colin Maclaurin [1698-1746], im- 

 mediately following Newton, though men of genius, suffer by 

 comparison with their illustrious predecessor ; and compared 

 with their successors on the continent, they fare equally badly. 

 The first third of the 18th century had not closed before the 

 mathematical brilliancy of the Swiss, Leonhard Euler [1707 - 

 1783], had manifested itself. Euler, and his no less illustrious 

 contemporaries, Lagrange [1736 — 1813], and Laplace [1749 — 

 1827], developed the analytical calculus with consummate genius, 

 firmly establishing it as a science independent of geometry. Euler 

 and Lagrange created also the calculus of variations; and to the 

 former astronomy is indebted for the method of variation of 

 arbitrary constants, by means of which he attacked the problem of 

 planetary perturbations, and gave approximate solutions for cer- 

 tain cases of the problem of three bodies. Lagrange and Laplace 

 are remarkable as being complementary in their talents; the former 

 delighted in the abstract and general ; his mastery of the calculus 

 was unrivalled ; his fertility of invention matchless ; though 

 Laplace surpassed him in ingenuity of the practical use of mathe- 

 matics, and in intuition of physical truths. It has been happily 

 said that " Lagrange saw in the problems of Nature so many 

 occasions for analytical triumphs, while Laplace regarded analy- 

 tical triumphs as the means for solving the problems of Nature." 

 Lagrange's 'Mecanique Analytique ' and Laplace's 'Mecanique 

 Celeste' and 'Theorie Analytique des Probabilites' are monuments 

 of the highest genius. 



