DEVELOPMENT OF MATHEMATICAL THOUGHT. 643 



notably astronomy not infrequently also as objects of 

 mere curiosity without any practical purpose whatever. 

 In the latter part of the eighteenth century the need 

 was felt of putting the new science into a compre- 

 hensive system. The attempts to do this notably the 

 great text-books of Leonhard Euler in Germany and of 

 Lacroix in France revealed how uncertain were the 

 foundations and how paradoxical some of the apparent 

 conclusions of the reasoning which, in the hands of the 

 great inventors and masters, had led to such remarkable 

 results. 



As in other cases which we dealt with in former 

 chapters of this work, so also in the present instance we 

 may find a guide through the labyrinth of modern mathe- 

 matical thought in the terms of language around which 

 cluster the more recent doctrines. Two terms present n. 



Modern 



themselves which were rare or altogether absent in older ^y v i "" 

 treatises : these terms are the " complex quantity " and "bought, 

 the " continuous." To these we can add a third term 

 which we meet with on every page of the writings of 

 mathematicians since Newton and Leibniz, but which has 

 only very recently been subjected to careful analysis and 

 rigorous definition, the term " infinite." Accordingly we 

 may say that the range of mathematical thought during 



their labours are almost forgotten, 

 although in their elaborate treat- 

 ises there are to be found many 

 formulae which had to be redis- 

 covered when, fifty years later, 

 the general theory of forms and 

 substitutions began to be sys- 

 tematically developed, and proved 

 to be an indispensable instrument 

 in dealing with many advanced 

 mathematical problems. See on 



the latter subject an article by 

 Major MacMahon on " Combin- 

 ational Analysis" ('Proc., London 

 Math. Soc..' vol. xxviii. p. 5, &c. ), 

 as also the chapters on this subject 

 and on " Determinants " in the 

 first vol. of the ' Encyclopadie der 

 Mathematischen Wissenschaf ten ' 

 (Leipzig, 1898)., Also, inter alia, a 

 note by J. Muir in ' Nature,' vol. 

 Ixvii., 1903, p. 512. 



