444 Cambridge Philosophical Society. 



with respect to Mliich 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. 



In like manner, as each intermediate direction is transverse to 

 the original normal, a secondary series of directions with a differ- 

 ent normal will arise from the combination of these coordinates 

 in every proportion, and the whole expanse of space around the 

 observer will be recognized as consisting of distance in every pos- 

 sible combination of proportions in the direction of three coordinates, 

 of which the first may be taken at pleasure in space, the second may 

 be identified with any of the series transverse to the first coordinate, 

 and the third will be the single direction transverse to each of the 

 former two. Witliin the sphere of three directions so related to each 

 other we are entirely shut in. Whatever may be the particular 

 direction in which the coordinates be laid, we can conceive no fourth 

 direction essentially differing in nature from the former three, 

 and therefore can conceive no possible direction which cannot be 

 derived from some combination of three coordinates, or in which 

 a given distance cannot be resolved into equivalent distances in the 

 direction of the three coordinates. 



We have thus in the relations of transverseness and opposition, 



