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- 52. 



Examina- 



certainty had led, ever since the end of the eighteenth turn of 



J foundati 



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, 



dations. 



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 

 peculiar to the French of refining 

 their ideas and communicating 

 them in the most appropriate and 

 taking manner. 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 writings were soon 

 widely circulated." (Hankel, loc. 

 cit., p. 210.) Following on this 

 statement of Hankel and a remark 

 of Prof. H. A. Schwarz, who looks 

 upon Bolzano as the inventor of a 

 line of reasoning further developed 

 by Weierstrass ('Journal fur 

 Mathematik,' vol. Ixxiv. p. 22, 

 1872), Prof. 0. 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 criti- 

 cism 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 Qrassmann's work. 



