DEVELOPMENT OF MATHEMATICAL THOUGHT. 709 



knowledge of the higher mathematical relations, but 

 also to reveal the uncertainty and absence of rigorous 

 definition of the foundations of arithmetic and of geo- 

 metry. Accordingly we find these great thinkers con- 

 tinually interrupting their more advanced researches by 

 examinations of the principles. This feeling of un- 

 certainty had led, ever since the end of the eighteenth 

 century, to many isolated attacks and half-philosophical 

 discussions by various writers in this country and 

 abroad. Many of them remained long unrecognised ; 

 such were the suggestive writings of Hamilton, De 

 Morgan, Peacock in England, Bolzano l in Bohemia, 



52. 



Examina- 

 tion of 

 foundations. 



1 The merits of Bernhard Bolzano 

 (1781-1848) as one of the earliest 

 representatives of the critical period 

 of mathematics were recognised 

 after a long interval of neglect by 

 Hankel in his article on " Limit " 

 mentioned above. This philosophi- 

 cal mathematician published many 

 years before Cauchy a tract on the 

 Binomial Theorem (Prague, 1816), 

 in which he gives, in Hankel's 

 opinion, the first rigid deduction of 

 various algebraical series. " Bol- 

 zano's notions as to convergency of 

 series are eminently clear and 

 correct, and no fault can be found 

 with his development of those series 

 for a real argument (which he 

 everywhere presupposes) ; in the 

 preface he gives a pertinent criti- 

 cism of earlier developments of the 

 Binomial Theorem, and of the un- 

 restricted use of infinite series, 

 which was then common. In fact, 

 he has everything that can place 

 him in this respect on the same level 

 with Cauchy, only not the art pecu- 

 liar to the French of refining their 

 ideas and communicating them in the 

 most appropriate and taking man- 

 ner. So it came about that Bolzano 

 remained unknown and was soon 



forgotten ; Cauchy was the happy 

 one who was praised as a reformer of 

 the science, and whose elegant writ- 

 ings were soon widely circulated." 

 (Hankel, loc. cit., p. 210.) Follow- 

 ing on this statement of Hankel 

 and a remark of Prof. H. A. 

 Schwarz, who looks upon Bolzano 

 as the inventor of a line of reason- 

 ing further developed by Weier- 

 strass ('Journal fiir Mathematik,' 

 vol. Ixxiv. p. 22, 1872), Prof. O. 

 Stolz published in 1881 ('Math. 

 Ann.,' vol. xviii. p. 255) an account 

 of the several writings of Bolzano, 

 beginning in the year 1810, in so far 

 as they referred to the principles of 

 the Calculus. " All these writings 

 are remarkable inasmuch as they 

 start with an unbiassed and acute 

 criticism of the contributions of the 

 older literature" (loc. cit., p. 257). 

 A posthumous tract by Bolzano, 

 ' Paradoxieen des Unendlichen,' 

 was republished in 1889 in 'Wis- 

 senschaftliche Classiker,' vol. ii., 

 Berlin (Meyer and Miiller). As 

 stated above, Hankel was also one 

 of the first to draw attention to 

 the originality and importance of 

 Hermann Grassmann's work. 



