﻿20 Dr. Norman Campbell on the 



can be brought into contiguity with edges, and, in a sense, 

 into contiguity with other scratches ; but the criterion of 

 contiguity in the last case is much less direct and requires 

 methods involving something other than the simple per- 

 ception of nothing between. 



The recognition of the possibility of contiguity is the 

 second step towards geometry and leads immediately to 

 the third, which consists in the establishment of a definite 

 criterion for a straight line. A crude criterion is provided 

 by direct perception : a young child knows the difference 

 between a straight and a bent line by simply looking at 

 them; the recognition seems to depend on the fact that 

 a straight line is all in one direction and is symmetrical 

 with regard to the unsymmetrical directions of left and 

 right or back and front. The crude criterion is stated as 

 well as it can be in Euclid's Def. 4. But contiguity 

 provides a much more stringent criterion, which in the 

 first instance is applicable only to edges and not to 

 scratches. Two edges are straight if, when two portions 

 of one are brought into contiguity with two portions of 

 the other, all the portions between these two portions are 

 also in contiguity, however the contiguity of the first pairs 

 of portions is effected. It appears as an experimental 

 fact, that if A, B and 0, D are two pairs of straight edges 

 according to this criterion, C is also straight if tested 

 against A ; accordingly an edge can be called straight 

 independently of the other member of the pair on which 

 the test is carried out. A scratch is straight if it can be 

 brought into complete contiguity with a straight edge, 

 Those facts are stated in Axiom 10. 



Other definitions of a straight line are sometimes offered : 

 e. g., (1) an axis of rotation, (2) the shortest distance 

 between two points, (3) the path of a ray of light. (1) is 

 almost equivalent to that stated here; (2) will be noticed 

 presently ; (3) is not accurately true (i. e., if it is adopted, 

 the familiar propositions about straight lines are not true), 

 but it is important as an approximation for comparatively 

 rough measurements. 



A plane surface (or, according to our usage, part of 

 a surface) is then defined as in Def. 7. It can also be 

 defined by the complete contiguity of three pairs of sur- 

 faces ; but the contiguity of surfaces is not easy to describe 

 accurately. Such a definition is, however, actually used in 

 making optical flats and surface plates ; if it were adopted, 

 it would still be necessary to introduce the fact that it 

 agrees with our definition, in order to measure angle. The 



