﻿Elements of Geometry. 19 



The fundamental notions give rise to those o£ surfaces and 

 lines. Surfaces are connected with the fact that a sensation 

 may be unaltered by movement in any of a certain group 

 of directions (which are said to be in a surface and to cha- 

 racterize it), while it may be altered by any movement in 

 any direction not in this group (directions away from the 

 surface) . Of lines there are two kinds, which will be termed 

 respectively " edges" and " scratches."" Edges arise from 

 the fact that the group of directions characteristic of a 

 surface may change suddenly at some part of it. It is 

 a matter of convenience whether the parts characterized by 

 different directions are spoken of as different surfaces or 

 as parts of the same surface : we shall adopt the second 

 alternative. Scratches arise from the fact that, while the 

 directions characterizing a surface are unaltered, the sen- 

 sation the occurrence of which distinguishes "in the 

 surface" from "out of the surface" may change suddenly. 

 Some, but not all lines, are such that the whole of them 

 lies along a single direction. Points are of little importance 

 in the earlier stages of geometry ; they arise from the fact 

 that two lines may have a part in common. Two points, 

 both on the same line, are termed the ends of the part of 

 the line between those points. 



The recognition of surfaces and lines is the first step 

 towards geometry. Euclid attempts to give an account 

 of them in Defs. 1, 2, 3, 5 of Book I., which are the least 

 successful part of his treatise. The account given of them 

 here is no better than Euclid's for the purpose of conveying 

 a notion of them to one who does not possess it already ; 

 but since there are no such persons, the objection is not 

 serious. But our account is better in drawing attention to the 

 notions that are fundamental in geometry and in not assuming 

 familiarity with conceptions, such as length, which are 

 necessarily subsequent. 



4. Some surfaces, but not all, when subjected to muscular 

 force undergo only such changes as can be compensated by 

 a suitable movement of the whole body ; if such a movement 

 is made, the group of directions characterizing the surfaces 

 is restored. In other words, such surfaces can move without 

 alteration of form ; they provide the original and crude con- 

 ception of a rigid body. By means of the motions of rigid 

 bodies, it is sometimes possible to bring parts of two pre- 

 viously distinct surfaces info contiguity, so that (here is 

 nothing between those parts. In particular, edges, or parts of 

 edges, can often be brought into such contiguity. Scratches 



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