DEVELOPMENT OF MATHEMATICAL THOUGHT. G53 



had airived at any finality in his .speculations, and, 

 beyond occasional hints which have only subsequently 

 become intelligilile, the love of finish exhibited in all his 

 published writings prevented liini from giving to tiic 

 world the suggestive ideas which evidently formed the 

 groundwork of his mathematical labours. There is no 

 doubt that — like Goethe in a very different sphere — 

 Gauss anticipated individually the developments in the 

 sphere of mathematical thought down to the end of the 

 century. The interpretation of the complex quantity 

 had been given by Wessel, Buee, and Argand ^ in the 

 early years of the century ; but it remained unnoticed 

 till it received the sanction of Gauss in a celebrated 

 memoir referring to the theory of numbers, and iniiil in 



through his father with the 

 speculations of the j'outliful Gauss, 

 and as Lobatchevsky was a pupil 

 of another student friend of Oauss 

 in the person of Prof. Bartels, it is 

 not unlikely that the interest which 

 these thinkers took in the subject 

 can be originally traced to the same 

 source. (See Dr Halsted's ad- 

 dress on Lobatchevsky, ' Neomonic 

 Series,' voh i., 1894). A complete 

 bibliography of the earlier papers, 

 referring to the so-called "non- 

 Euclidean" literature down to 

 1878, is given by Dr Halstcd in 

 the first two vols, of the ' Ameri- 

 can Journal of Mathematics ' : the 

 most recent publications are those 

 of the Hon. B. A. W. Russell in 

 his work, 'The Foundations of 

 Geometry' (1897) and his ex- 

 cellent article on " Non-Euclidean 

 Geometry " in the 28th vol. of the 

 ' Ency. Brit.' See also Klein's litho- 

 graphed lectures on ' Nicht-Euk- 

 lidische Geometrie,' Gottingen, 

 189.3. 



' The first somewhat exhaustive 

 historical statement as to the 



geometrical representation of the 

 complex or imaginary quantity wa.s^ 

 given bj' Hankel in the above- 

 mentioned work (see above, note, 

 p. 645), p. 82. He there says, after 

 discussing the claims of others, — 

 notably of Gauss, — that Argand in 

 his 'Essai' of the j-ear 1806 (re- 

 edited by Hoiiel, 1874) "had so 

 fully treated of the whole theory 

 that later nothing essentially new 

 was added, and that, except a 

 publication of still earlier date 

 were found, Argand must be con- 

 sidered the true founder of the 

 representation of complex (juan- 

 tities in the plane." Such an 

 earlier publication has indeed been 

 met with in a tract by Caspar 

 Wessel, which was presented to 

 the Danish Academy in 1797, and 

 puljlished in 1799. Having been 

 overlooked, like Argand's 'Essai,' 

 it has now been republished at 

 Copenhagen, 1897, with the title 

 ' Essai sur la representation de la 

 direction ' (see ' Eucyk. Math. 

 Wissenschaften,' vol. i. p. 155). 



