704 



SCIENTIFIC THOUGHT. 



confided many separate and lengthy investigations. It 

 was through one of these that a test-case, in which exist- 

 ing mathematical definitions broke down, was published 

 in 1872. It forms a kind of era in the history of 



middle of the sixties, delivered 

 lectures at the University of 

 Leipsic upon " Complex numbers 

 and their functions," starting in a 

 characteristic manner with that ex- 

 tended algebra which Cauchy and 

 Riemann had used to such good 

 purpose. The first part of these 

 lectures was published in 1867. 

 In the preface Hankel says : " In 

 the natural sciences we witness in 

 recent times the distinct tendency 

 to ascend from the world of em- 

 pirical detail to the great principles 

 which govern everything special and 

 connect it into a whole i.e., the 

 desire for a philosophy of nature, 

 not forced upon us from outside, 

 but naturally evolved out of the 

 subject itself. Also in the domain 

 of mathematics a similar want 

 seems to make itself generally felt 

 a want which has always been 

 alive in England. " Had the author 

 not been prematurely taken away, 

 there is no doubt that he would 

 have still more largely contributed 

 to the revolution of mathematical 

 ideas now in progress. As it 

 is, he made one further import- 

 ant contribution, of which more 

 hereafter. In Italy Prof. Ulisse 

 Dini began to lecture in the year 

 1871 to 1872 on the theory of 

 functions, and published his lec- 

 tures in 1878. A translation was 

 brought out in German (1892) by 

 Prof. Liiroth and Mr A. Schepp, 

 in which many of the modern 

 developments are utilised. In 

 France we owe to M. Jules Tannery 

 a valuable introduction to the 

 theory of functions of one variable, 

 based upon a series of lectures 

 delivered in the Ecole Normale 

 in 1883, in which, as he says 



(Preface, p. vii), he collected 

 the labours of Cauchy, Abel, Le- 

 jeune Dirichlet, Riemann, Ossian 

 Bonnet, Heine, Weierstrass, and 

 others ; after which he considers 

 that nothing essential need be 

 added in the way of elucidation of 

 the foundations of the theory. 

 M. Emil Borel published in 1898 

 ' Lectures on the Theory of Func- 

 tions,' the first of a series of 

 text - books dealing with various 

 aspects of the theory of functions, 

 in which he largely refers to the 

 labours of Weierstrass. Before 

 Weierstrass's theory had become 

 known, however, M. Me"ray had al- 

 ready entered upon an exposition 

 of the foundations of analysis on 

 lines which had much analogy with 

 those adopted by Weierstrass. In 

 England the late Prof. Clifford had 

 occupied himself in various memoirs 

 with the theories of Riemann ; but 

 we owe the first comprehensive 

 treatise, embracing the work of 

 Riemann as well as that of Weier- 

 strass, to Prof. Forsyth ('Theory 

 of Functions of a Complex Vari- 

 able,' Cambridge, 1893). Almost 

 simultaneously Professors Harkness 

 and Morley published a ' Treatise on 

 the Theory of Functions,' and in 

 1898 an 'Introduction to the 

 Theory of Analytic Functions,' in 

 which they in the main adopted the 

 point of view of Weierstrass. A 

 very original thinker, whose in- 

 dependent researches reach back to 

 the year 1872, and who played an 

 important part in the investigation 

 of many obscure points, was the 

 late Prof. Paul Du Bois-Reymond, 

 who published in 1882 the first 

 part of his ' Allgemeine Func- 

 tionentheorie,' containing the 



