THE GEOMETRY OF MOTION 207 



wish to distinguish from P, and to do this we must give 

 what is termed direction to the distance OP, we must 

 determine, as it were, whether it runs north and south, 

 south-west and north-east, or upwards and downwards.^ 

 But even this is not enough. We must be also told the 

 sense of this direction, whether, for example, it be op or op' 

 (Fig. 8), or, say, runs from south-west to north-east or 

 north-east to south-west. Thus, if we want to plot our 

 position in space about a point O, we must do this by 



IP 



Fig. 8. 



measuring distances from O in given directions and with 

 given senses. We must know distance and bearing'^ from 

 O to determine fully a point P. To represent geometric- 

 ally the position of P with regard to O, we may draw a 

 piece of a straight line {op) having as many units of length 

 on our scale as there are units of distance from O to P, 

 the line having the same direction as this distance, and 

 having an arrow-head upon it to mark the sense. Such 

 a line marking the magnitude, direction, and sense of P's 

 position relative to O is termed a step. Such a step tells 



1 In the conceptual space which corresponds most closely to perceptual 

 space — so-called space of three dimensions — we require, in order- to mark the 

 relative position of all possible bodies, to start from thi-ee standard points 

 (which must not be in the same straight line) in order to fix direction. 

 Throughout this chapter we shall understand by the position of a point 1' 

 relative to another point O, the directed step OP, and by the motion of P 

 relative to O change in this directed step. A fuller account of Position will 

 be found in the chapter under that title contributed by the author to Clifford's 

 Common Sense of the Exact Sciences. 



^ With the signification in which the words are here used, a line has 

 direction but not beariftg. We must add to direction the conception of sense 

 before we form the idea of bearing. 



