Cambridge Philosophical Society. 445 



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 series 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 secohd 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 defined 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 abJe to compare 

 and to identify directions in any part of the system. 



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 pha?nomena (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 tliey 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 



