THE HISTORY OF MATHEMATICS 399 



there are certain names in the English school which have lived to the 

 present day. Brook Taylor who was born in 1685 gave the funda- 

 mental series for the expansion of a function which is known by his 

 name, followed a little later by Colin Maclaurin with the particular 

 case which is named after him. The latter also determined the attrac- 

 tion of an ellipsoid and introduced the idea of equi-potential surfaces. 

 De Moivre, of French ancestry but English birth and training, intro- 

 duced the use of the imaginary into Trigonometry and thus prepared 

 the way for the great development of the theory of functions of a com- 

 plex variable which took place in the nineteenth century. Roger Cotes 

 must also be mentioned if only for the high opinion Newton had of 

 his abilities : he was only thirty-four years old when he died. 



On the continent the Bernouilli brothers, friends and admirers of 

 Leibnitz, were largely responsible for realising the power of the 

 calculus and making it known. They applied it to many physical 

 problems but perhaps their greatest influence came through their teach- 

 ing abilities. Most of those in this period who achieved distinction 

 were their pupils and several of their descendents were well known as 

 mathematicians during the eighteenth century. But the most able men 

 of this period were undoubtedly Clairaut and d'Alembert. The former 

 produced the first theory of the motion of the moon developed from 

 the differential equations of motion : in it he showed that the theoretical 

 motion of its perigee, which in the Principia was obtained to only half 

 the observed value, agreed with observation when we proceed to a 

 higher approximation. There was an interesting development in this 

 connection. Clairaut at first thought that it would be necessary to make 

 an addition to the Newtonian law in order to produce agreement and 

 it was only when he carried his work further that he saw such an 

 addition to be unneccessary. A similar addition was examined by 

 Newcomb and others as a possibility which might explain the deviation 

 of the perihelion of Mercury from its observed value, and it is only 

 within the last five years that such an addition has been deduced from 

 the relativity theory by Einstein. Clairaut also obtained the well 

 known formula for the variation of gravity due to the shape of the 

 earth. 



D'Alembert is best known by his work on dynamics in which he 

 showed how the equations of motion of a rigid body could be written 

 down: the principle is still known by his name. He also reached the 

 well known wave-equation, a partial differential one of the second 

 order, in several physical investigations and showed how a solution 

 may be obtained. This was pioneer work but its further development 

 was not carried forward to any extent by him. 



The great period of continental activity which began in the middle 

 of the eighteenth century and contained the names of Euler, Lagrange, 



