256 G. H. KNIBBS. 



It has been seriously proposed by some mathematicians that 

 the nature of space itself^ should be ascertained by experiment. 2 

 Any essay of that character would necessarily have to rely upon 

 a better definition than that a straight line is the shortest distance 

 between two points. Euclid's definition, although perhaps not 

 sufficiently explicit, goes to the heart of the matter, viz. "a 

 straight line is that which lies evenly between its extreme points," 3 

 a definition which Henrici states "must be meaningless to anyone 

 who has not the notion of straightness in his mind." 4 Probably 

 no one who ever had any capacity for geometrical thinking ever 

 failed instantly to grasp what was intended by Euclid's definition, 

 whereas the quantitative definition which relies upon shortness is 

 not only not obvious, but is meaningless unless the idea of straight- 

 ness is already in the mind. The apodictic certainty with which 

 straightness is conceptually distinguished from even infinitesimal 

 curvature, confirms the opinion that it is intuitive. 5 As an 

 elementary definition is Euclid's susceptible of improvement 1 ? 



The test of straightness indicated in Euclid's definition, des- 

 patches at once the doubts raised in the non-euclidean geometry, 

 even to beings limited to perception in two-dimensional space. 6 If 

 it be possible to rotate a line round itself as axis, either one point 



1 Not of space filled with some medium whose physical properties may 

 properly form the subject of inquiry, but space per se. 



* Euclid and some other geometers have been supposed guilty of the 

 philosophical immorality of making assumptions which ' Experience ' 

 might shew to be invalid. It is not clear upon what ratiocinative' basis 

 the objectors propose to interpret ' Experience/ 



3 Euclid's definition is :— "Ev(9eia ypa/xfirj Zcttlv, tjtls !£ urov tols 4<f> 

 eavTyjs o-^/xetoi5 kcitou," which is translated by Simson, " Eecta linea 

 est, quse ex aequo suis interjicitur punctis." Proclus explains e£ tcrov 

 as being stretched between its extremities, rj J71-' aKpw rera/jLtvY]. The 

 reader interested in discussion on modern proposals to supersede Euclid's 

 definition, will find "Euclid and his modern rivals" — by C. L. Dodgson. 

 MacmillaD, 1879 — well worth reading. 



4 Encyc. Brit, x., 376. 



5 Chrjstal states, loc. cit., p. 641. "We have by generalisation from 

 experience (!) ideas more or less refined .... of a geometrical straight 

 line, etc." See previous footnote 3, p. 247. 



6 Say the sphere or pseudo-sphere dwellers of Helmholtz, vide On the 

 origin and significance of geometrical axioms. Trans. Atkinson, pp. 34 

 -44,60-68. 



