131 



It is in virtue of this complex scheme of relation between direc- 

 tions, that we are enabled to conceive the possibility of reaching the 

 same point by different tracks from a common starting-point. We 

 are indeed so much in the habit of thinking of points as marked out 

 by physical phenomena (as by the letters in a geometrical illustra- 

 tion), that it is by no means obvious where the difficulty of the con- 

 ception lies. But it must be remembered that points in geometry 

 are distinguished solely by position, while the position of a given 

 point is determined by the nature of the track by which it is reached 

 from a point antecedently known. It is plain, therefore, that there 

 would be no means of identifying points attained by tracks differing 

 in any respect from each other, if the precise combination of distance 

 and direction by which they were respectively attained were the 

 ultimate test of their position. But now the knowledge of the fun- 

 damental scheme of relationship above explained makes us regard 

 the space traversed in each successive instant of time in the track 

 by which the position of a point is determined (and consequently 

 the whole space traversed in the entire track), as equivalent to a 

 certain distance in the direction of each of the two coordinates of 

 the scale. The aggregate character (in respect of distance and 

 direction) of the space traversed in different tracks (by which the 

 position of the terminal points is governed) will thus be made to 

 depend on the aggregate distance advanced in the direction of the 

 two coordinates, a question to be tried by simple superposition. 

 When the distance advanced in the direction of each coordinate is 

 the same, the positions finally attained will be recognized as iden- 

 tical, and the points will coincide whatever may be the amount of 

 intermediate divergence in the tracks by which they have actually 

 been reached. 



From the same principle it may be shown, that a straight line may 

 be drawn from a given point to any other point in space. Because 

 the space traversed in the track by which the second point must be 

 supposed to have been determined, will be equivalent in distance and 

 direction to a certain distance in each of the two standard directions of 

 the system. Now inasmuch as the series of directions intermediate 

 between anypairof transverse directions includes individuals partaking 

 in every conceivable proportion of the nature of both the transverse 

 directions between which they lie, it will always be possible to select 

 one of the series a certain distance in which will be equivalent to 

 given distances in each of the two transverse directions, and there- 

 fore the distances in the direction of the coordinates of the system 

 under consideration, into which the space traversed in the original 

 track has been resolved, may again be exchanged for an equivalent 

 distance in a single direction duly related to each of the coordinates ; 

 in other words, the same position may be attained by motion in a 

 single continuous direction as by a track of any other description, or 

 what amounts to the same thing, a straight line may be drawn from 

 a given point to a point determined by a track of any other de- 

 scription. 



As soon as a straight line is known as lying in a single continuous 



b 2 



