129 



ence ; and it may with equal legitimacy proceed to exemplify in 

 like manner any further construction which may be found necessary 

 in the course of demonstration. 



The question of motion has commonly been considered so essen- 

 tially distinct from that of position, that all reference to the former 

 subject has rigorously been excluded from the field of geometrical 

 inquiry. But the position of every point must ultimately be deter- 

 mined by motion from points antecedently known, and to the inci- 

 dents of motion we should accordingly look for the original source 

 of the relations of position. Now motion (in as far as it influences 

 position) admits of variation in two ways ; viz. in the direction of 

 the motion at each indivisible instant of time, and in the length of 

 the track accomplished in a finite period ; whence it has been said 

 by Sir John Herschel that space (which is primarily known as the 

 receptacle of motion) is reducible in ultimate analysis to distance 

 and direction. 



The relations of extent are simply those of equal, greater, and less, 

 with respect to which it will be necessary only to define the test by 

 which they are respectively to be demonstrated in concrete figure. 

 The relations of direction are of a much more complicated nature. 

 The different phases of this elementary attribute of motion are di- 

 stinguished, not, like those of colour, by a permanent character inde- 

 pendently cognizable in each individual, but more like musical notes, 

 by their relative position on a peculiar scale which may be made to 

 rest on any individual as an arbitrary basis. 



The scale by which directions are compared is founded on the 

 elementary relations of opposition and transverseness. In whatever 

 direction we suppose ourselves to be traversing space, we recognize 

 the possibility of returning to the same position from whence we set 

 out by motion in a different direction, the relation of which to the 

 original is that of opposition ; or the two may be classed together as 

 the positive and negative modifications of a common direction. 



Again, if we fix our thoughts upon any given direction, we find a 

 series of others in each of which it is possible to traverse space with- 

 out advance in the original direction or in the one opposed to it. 

 The directions so marked out by negation of progress in a certain 

 direction are said to be transverse to the normal or direction in which 

 no progress is made by the observer while advancing in the direction 

 of any of the transverse series. If now we start afresh from any of 

 the individuals of the latter series, it will be found that the series 

 includes the opposite direction, as well as one direction and its 

 opposite transverse to the former two. Every other individual of 

 the series will be recognized as partaking in different proportions of 

 the nature of these coordinates, or transverse directions, adopted as 

 the basis of the scale. In other words, it will be found that distance 

 in any intermediate direction is essentially composed of distance in 

 the direction of each of the coordinates in different proportions, vary- 

 ing from all of the one and none of the other, to all of the latter and 

 none of the former, with every modification arising from taking each 

 of the coordinates in both a positive and a negative sense. 



B 



