TENDENCIES IN MATHEMATICAL SCIENCE 325 



problems which had defied the feebler tools of the earlier mathe- 

 matics. One obstacle after another has been gradually sur- 

 mounted by the invention of new and more powerful methods of 

 ever increasing generality, just as increasingly powerful telescopes 

 have revealed unnumbered new suns ; and no boundary or limit 

 to this evolutionary progress can be foreseen or imagined. On 

 the other hand, as the new world has been gradually settled, 

 civilized, and cultivated, so the fields of mathematics which were 

 opened up in the eighteenth century have been critically ex- 

 amined in the nineteenth, with much revision of fundamentals. 



The main features of eighteenth century mathematics were : 

 the working out of the differential and integral calculus into 

 substantially the form they have ever since retained ; the begin- 

 nings of differential equations as a natural outgrowth of integral 

 calculus, and the beginnings of the calculus of variations; the 

 systematic application of the new ideas to mechanics, and in par- 

 ticular to celestial mechanics. The century was also notable for 

 important discoveries in astronomy and physics, including for ex- 

 ample that of the aberration of light ; a vigorous attack on " the 

 problem of three bodies " ; and the earlier telescopic work of the 

 Herschels, culminating in the discovery of a new planet, Uranus. 



Among the leading mathematicians of the period were Mac- 

 laurin of Scotland, various members of the Swiss Bernoulli family, 

 Euler also a native of Switzerland, Lagrange of Italy, and in 

 France, Clairaut, d'Alembert, and Laplace. In spite of the unique 

 supremacy of Newton, the absence of Britons from this list is 

 notable. The bitter personal controversy between Newton's ad- 

 herents and those of Leibnitz produced or aggravated an un- 

 fortunate division between the English and the continental 

 mathematicians. For the former, persistence in Newton's in- 

 ferior notation became a matter of national pride, and progress 

 was correspondingly retarded. Of the mathematicians named 

 above, the Bernoullis and Euler on the continent and Maclaurin 

 in Scotland bore a leading part in the systematization of the 

 calculus, while Lagrange and Laplace were preeminent in the 

 development of analytical and celestial mechanics respectively. 



