DEVELOPMENT OF MATHEMATICAL THOUGHT. 711 



same subject, through which it became more widely 

 known and attracted the attention of other than 

 purely mathematical writers. The small but eminently 

 suggestive volume of Hankel showed the necessity of 

 a revision and extension of the fundamental principles 

 and definitions l of general arithmetic and algebra as 



result of the investigation, that 

 Riemann has arrived at exactly 

 the same results. My starting- 

 point was the question, How must 

 a magnitude of several dimensions 

 be constituted, if solid bodies are 

 to move in it everywhere continu- 

 ously, monodromically, and as freely 

 as bodies move in real space?" On 

 receiving from Sobering a reply 

 with a copy of Riemann's paper, 

 Helmholtz wrote (18th May), "I 

 enclose a short exposition of that 

 which in my researches on the same 

 subject is not covered by Riemann's 

 work." A fuller paper, with the 

 title " On the Facts which lie at the 

 foundation of Geometry," appeared 

 in the 'Gottinger Nachrichten,' 

 June 3, 1868. See Helmholtz, 

 ' Wiss. Abhandl.,' vol. ii. pp. 610 

 and 618, &c. ; also 'H. von Helm- 

 holtz,' by Leo Koenigsberger (1903), 

 vol. ii. p. 138, &c. In another 

 lecture, " On the origin and mean- 

 ing of the Axioms of Geometry " 

 (1870, reprinted in abstract in 

 ' The Academy,' vol. i. ), as well as 

 in an article in vol. i. of ' Mind ' 

 (p. 301), he discussed "the philo- 

 sophical bearing of recent in- 

 quiries concerning geometrical 

 axioms and the possibility of 

 working out analytically other 

 systems of geometry with other 

 axioms than Euclid's " (reprinted in 

 vol. ii. of ' Vortriige und Reden '). 



1 In this treatise Hankel intro- 

 duced into German literature the 

 three terms "distributive," "asso- 

 ciative," and "commutative" to 

 define the three principles which 



govern the elementary operations 

 of arithmetic, and introduced fur- 

 ther what he calls the principle of 

 the permanence of formal rules 

 in the following statement : " If two 

 forms, expressed in the general 

 terms of universal arithmetic, are 

 equal to each other, they are to 

 remain equal if the symbols cease 

 to denote simple quantities ; hence 

 also if the operations receive a 

 different meaning." Hankel seems 

 to have been led to his definitions 

 by a study of French and English 

 writers, among whom he mentions 

 Servois (' Gergonne's Ann.,' v. p. 93, 

 1814) as having introduced the 

 terms "distributive" and "com- 

 mutative," and Sir W. R. Hamilton 

 as having introduced the term 

 " associative." He further says 

 (p. 15): "In England, where 

 investigations into the funda- 

 mental principles of mathematics 

 have always been treated with 

 favour, and where even the great- 

 est mathematicians have not 

 shunned the treatment of them 

 in learned dissertations, we must 

 name George Peacock of Cambridge 

 as the one who first recognised 

 emphatically the need of formal 

 mathematics. In his interesting 

 report on certain branches of 

 analysis, the principle of perma- 

 nence is laid down, though too 

 narrowly, and also without the 

 necessary foundation." Other 

 writings, of what he terms Pea- 

 cock's Cambridge school, such as 

 those of De Morgan, Hankel states 

 that he had not inspected ; mention- 



