446 Cambridge Philosophical Society. 



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 leached. 



From the same principle it may be shoviTi, 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 any pair of 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 

 direction, it becomes the most obvious means of marking the direc- 

 tion so exhibited throughout a finite extent of line. The series of 

 directions transverse to a given normal may then be represented by 

 two straight lines crossing each other at right angles, and an inde- 

 finite number of other straight lines diverging from the point of 

 intersection, and dividing the plane surface round that point into as 

 man}' parts as there are diverging lines. If now we take two of 

 these lines, like the hands of a clock, and suppose one to remain 

 fixed while the other revolves from left to I'ight, it will pass success- 

 ively through all the directions intermediate between left and front, 

 while the quantity of plane surface intercepted between the hands 

 abutting on the point of intersection will continually increase as the 

 difference in their direction becomes greater, or in proportion as 

 distance in the direction of the moveable hand contains a greater 

 proportion of distance in the direction transverse to that of the fixed 

 one. Thus we are taught a new mode of estimating the relation 

 between the direction of straight lines diverging from a common 

 point ; not by a proportion which addresses itself to the understand- 



