tso 



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. Within the sphere of three directions so related to each 

 other we arc 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, 

 and in the conception of intermediate directions arising from the 

 combination of transverse coordinates in different proportions, a 

 uniform scale by which, when applied to known directions in space, 

 the position of any other direction may be accurately defined inde- 

 pendent (it must be observed) of any reference to the notion of 

 angular magnitude, of which as yet no mention has been made. 



When two directions only are known in a system, they must be 

 considered as members of the scries transverse to a common normal ; 

 and one of the two being identified with the first coordinate of the 

 scale, the position of the second will be completely determined by 

 the proportion in which it partakes of the nature of the second co- 

 ordinate or transverse direction of the series. 



The directions commonly adopted as the basis of the scale, are the 

 up and down, fore and aft, and right and left lines marked out (in 

 any given position of the observer in a system) by the constitution 

 of his bodily frame ; and thus (in any given position of our bodies) 

 a particular direction is denned in our thoughts by the proportion in 

 which it partakes of the nature of those coordinates, that is to say, 

 by the proportion in which distance in the direction in question is 

 essentially composed of distance up or down, distance to the front 

 or rear, and of distance to the right or left. 



For the sake of simplifying the question, we will now confine our 

 thoughts to motion in a plane surface, or to directions having refer- 

 ence to two transverse coordinates. Now although, in the actual 

 apprehension of a figured system, the observer must be supposed to 

 traverse the entire outline, and thus continually to change his place, 

 yet he must be capable of doing so without rotation on his own axis, 

 as he would otherwise acquire no notion of the configuration of his 

 track in the external system. He will accordingly carry with him 

 throughout the fundamental conceptions of front and back, right and 

 left, and by reference to these coordinates will be able to compare 

 and to identify directions in any part of the system. 



