I 



V34 



SCIENTIFIC THOUGHT. 



clearer enunciation of the fundamental conceptions of 

 the science, and though the ways in which they 

 approach the subject are different, a general consensus 

 seems to be within view as to the elementary definitions. 

 The main difficulty lies in the introduction into pure 

 arithmetic of the ideas which are forced upon us when 



Elemente der Functionenlehre " 

 {' Journal fiir Mathematik,' vol. 

 Ixxiv. p. 172, 1872). This paper 

 refers both to Weierstrass's and 

 Cantor's theories ; H. Kossak, in 

 the pamphlet referred to above 

 (p. 7i2, note). This contains the 

 principles of Weierstrass's theory ; 

 C. H. Meray, ' Nouveau Precis 

 d' Analyse infinitesimale ' (Paris, 

 1872). The first comprehensive 

 publication of Georg Cantor be- 

 longs to the year 1883, ' Grund- 

 la^en eiiier allgemeinen Mannig- 

 faltigkeitslehre ' (Leipzig, Teub- 

 ner). It was preceded by various 

 articles in the ' Journal fiir Mathe- 

 matik,' vol. Ixxvii. p. 257, vol. 

 Ixxxiv. p. 82, aud| ' Math. Ann.,' 

 vol. XV. p. 1, in which he had in- 

 troduced and defined several of the 

 terms and conceptions that have 

 since become generally accepted in 

 writings on this subject. These 

 earlier publications, by — or refer- 

 ring to — the pioneers in this new 

 province of mathematical thought, 

 were followed by a number of 

 further expositions by Cantor, 

 Dedekind, and Weierstrass. The 

 principal writings of Cantor have 

 been republished in the ' Acta 

 Mathematica,' vol. ii. Prof. Dede- 

 kind published in the year 1888 an 

 important pamphlet, ' Was sind 

 und was sollen die Zahlen,' and has 

 incorporated many of the results of 

 his researches in his later editions 

 of Dirichlet's 'Lectures ' ; whilst the 

 lines of reasoning peculiar to Weier- 



strass have become better known 

 through the writings of his pupils 

 and the collected edition of his 

 mathematical works which is now 

 in progress. A complete biblio- 

 graphy is given in three important 

 articles in vol. i. of the German 

 'Math. Encyc' by Profs. Schu- 

 bert (p. 1, &c.), Pringsheim (p. 

 48, &c.), and Schonflies (p. 184, 

 &c.) Important works, giving a 

 summary and analysis of these 

 various researches, now exist in 

 the mathematical and philosophical 

 literature of France, Germanj-, 

 Italy, and England. Like the non- 

 Euclidean geometry, the subject 

 has attracted considerable atten- 

 tion also outside purely mathe- 

 matical circles. Notably Cantor's 

 writings have been exhaustively 

 dealt with from a philosophical 

 point of view — in Germany by 

 Walter Brix (Wundt's ' Philoso- 

 phische Studien,' vol. v. p. 632, 

 vol. vi. p. 104 and 261), and by 

 B. Kerry, ' System einer Theorie 

 der Grenz-begriffe ' (Leipzig und 

 Wien, 1890) ; in France by M. 

 Louis Couturat, ' De I'lnfini ma- 

 thematique' (Paris, 1896); and 

 latterly in this country by Mr 

 Bertrand Kussell, ' The Principles 

 of Mathematics,' vol. i. (Cambridge, 

 1903). Italian mathematicians have 

 also dealt largely with the subject, 

 notably G. Peano, who published 

 an important work, ' Arithmetices 

 principia nova methodo exposita ' 

 (Turin, 1889). 



