462 Mr. H. Wedgwood on a System of Geometry, 



investigation, the directions diverging from any point in the 

 system must be understood as defined by reference to the stand- 

 ard directions of the observer's own body ; and a direction in the 

 external system at B will be the same with a given direction at 

 A when it occupies the same position in respect of up and down, 

 right and left, fore and aft. A straight line in the external 

 system extending directly in front of the observer, or directly to 

 his right when he is stationed at B, will be recognized as lying 

 in the same direction with a straight line directly in front or 

 directly to the right when stationed at A ; and so of straight 

 lines extending from A and B respectively, and lying in simUar 

 intermediate positions between fore and aft and right and left. 



Now let us examine the import of these conditions applied to 

 the case of straight lines AC, BD passing in parallel directions 

 through the points A and B respectively. The meaning of pa- 

 rallelism under the definition will be, that the line BD occupies 

 the same position among the standard directions in the body of 

 an observer moving from A to B without rotation in the system, 

 with that which was filled by the line AC when the observer was 

 at A. On thus comparing the directions of AC and BD by the 

 standard of his own bodily structure, the observer must pass from 

 A to B by some definite path, to which alone the relation of 

 parallelism would seem to refer. But what if he had gone by 

 any other track ? Would he then have found the line BD in the 

 same position among the lines of his own bodily standard ? 



Here we have the condition that the observer is brought to 

 the same point B by a straight line AB, and by a track of some 

 other nature ; and as points are known in geometry by position 

 alone, it must be supposed that the position attained by the 

 straight line AB is recognized as identical with that attained by 

 the other track. Let AB be supposed, for example, to extend 

 directly to the right of the observer in his original position at A, 

 whence it follows that A will lie at the distance BA directly to 

 his left when he arrives at B. Then if X be the point to which 

 he is brought by the other track, it is plain that he cannot iden- 

 tify the points B and X without being sensible, when situated 

 at X, that A is placed directly to his left at a distance equal to 

 BA. Thus it appears that an observer moving, under the neces- 

 sary conditions, from the same original position in a system at 

 A, will be similarly placed in the system at any second point B, 

 whether he attain that point by a straight line or by a track of 

 any other description ; and any fixed straight line in the system 

 passing through B will in either case occupy the same position 

 among the directions of his bodily scale. Thus the straight 

 lines AC and BD may be considered as lying in the same direc- 

 tions at A and B respectively, independent of any reference to 



