DEVELOPMENT OF MATHEMATICAL THOUGHT. 715 



has been introduced into mathematical writings which 

 has not a little puzzled outsiders, and even exposed 

 the logically rigorous deductions of mathematicians to 

 the ridicule — not to say the contempt — of eminent 

 philosophical authorities. The complete parallelism or 

 correspondence of geometrical with algebraical notions 

 — the possibility of expressing the former with perfect 

 accuracy by the latter, and of retranslating the latter 

 into the former, and this in more than one way, accord- 

 ing to the choice of the space element (point, line, 

 sphere), led to the habit of using purely geometrical pre- 

 sentable ideas as names for algebraical relations which 

 had been generalised by the addition of more than 

 a limited number of variables. Thus the conception 

 of curvature, easily defined for a plane curve, and 54. 



"^ ^ ^ Curvature 



extended by Gauss to surfaces, was, by adding a third cf space, 

 variable in the algeljraic formula, applied to space. 

 We are then told that it is necessary to understand 

 what is meant by the curvature of space, this being a 

 purely algebraical relation, not really presentable, but 

 only formed by analogy from the geometrically present- 

 able relations of geometrv on a surface. In a similar 



the different points of origin of this 

 most recent mathematical specula- 

 tion, which are to be found in 

 the mathematical literature of all 

 the principal nations, have been 

 put in the true light and brought 

 into connection. In fact, here, 

 as in sevei-al other subjects, his 

 publications, including his litho- 

 graphed lectures on non-Euclidean 

 geometry (delivered at Guttingen, 

 1893-94), serve as the best guide 

 through the labyrinth and contro- 

 versies of this intricate subject. 

 See especially his article "Ueber 



die so-genannte nicht-Euclidische 

 Geometrie" in vol. iv., "Math. Ann.,' 

 1871. In this paper he connects the 

 independent researches of Cayley 

 (following Laguerre, ' Nouv. Ann. 

 de Math.,' 1853), who in his sixth 

 memoir on Quantics showed how 

 metrical geometry can be included 

 in projective geometry by refer- 

 ring figures to a fundamental fixed 

 figure in space called by him the 

 "Absolute," with the independent 

 researches of Lobatchevski, Bolyai, 

 Riemann, and Beltrami. 



