718 



SCIENTIFIC THOUGHT. 



56. 



Klein's 

 exposition. 



land and Von Staudt in Germany. It was reserved for 

 Prof. Felix Klein of Gottingen to show how the gener- 

 alised notions of distance introduced into geometry by 

 Cay ley and Von Staudt opened out an understanding of 

 the three geometries of Euclid, of Lobatchevski, and of 

 Eiemann.^ We have to go back to the purely projective 

 properties of space to understand these different possi- 

 bilities. Lobatchevski attacked the problem practically, 

 Eiemann analytically, Klein geometrically. Through the 

 labours of Klein the subject has arrived at a certain 

 finality. And what was still wanting after he had 

 written his celebrated memoir (which was approved and 



^ See the note ou p. 714, above ; 

 also ' Math. Ann.,' vol. iv. p. 573, 

 and vol. vi. p. 112. Prof. Klein 

 — following a usage iu mathe- 

 matical language — distinguishes 

 three different geometries, the 

 hyperbolic, the elliptic, and the 

 parabolic geometry, corresponding 

 to the possession by the straight 

 line at infinity of two real or two 

 imaginary (that is, none) or two 

 coincident points. The whole 

 matter turns upon the fact that, 

 although metrical relations of 

 figures are in general changed 

 by projection, there is one metri- 

 cal relation — known in geometry 

 as the " anharaionic ratio" (in 

 German Doppdve7-hdltniss)— which 

 in all projective transformations 

 remains unchanged. As this an- 

 harmonic ratio of points or lines 

 can be geometrically constructed 

 without reference to measure- 

 ment (Von Staudt, 'Geometrie 

 der Lage,' 1847 and 1857), a 

 method is thus found by which, 

 starting from a purely descriptive 

 property or relation, distance and 

 angles — i.e., metrical quantities — 

 can be defined. Some doubts have 



j been expressed whether, starting 

 from the purely projective pro- 

 perties of space and building up 

 geometry in this way (arriving at 

 the metrical properties by the 

 construction suggested by Von 

 Staudt), the ordinary idea of 

 distance and number is not tacitly 

 introduced from the beginning. 

 This may be of philosophical, 

 but is not of mathematical, 

 importance, as the main object 

 in the mathematical treatment is 

 to gain a starting - point from 

 which the several possible con- 

 sistent systems of geometry can 

 be deduced and taken into view 

 together. See on this point, 

 inter alia, Cayley's remarks in 

 the appendix to vol. ii. of ' Col- 

 lected Works ' (p. 604 sqq), also 

 Sir R. S. Ball's paper (quoted 

 there), and more recently the dis- 

 cussion on the subject in Mr 

 Bertrand Russell's ' Essay on the 

 Foundations of Geometry' (1897, 

 p. 31, &c. ; p. 117, &c.) See 

 also the same author's article on 

 non-Euclidean Geometry in the 

 supplement of the ' Ency. Brit.,' 

 vol. xxviii. 



