378 REPOKT— 1892. 



point on the surface represents by its distance from two fixed points two 

 numerical facts. By the use of a third fixed point the power of represent- 

 ing results is in no way augmented, for the position of A is fixed by its 

 distance from Oi and O2, so that no fresh fact, independent of the two 

 previous ones, can be represented by its distance from O3. This is not a 

 mere accident, to be avoided by any geometrical device, but is the result 

 of using for a gra,phic diagram two dimensions of space in the form of a 

 surface, either plane or developable into a plane. Hereafter a device will 

 be noticed by which, representing artificially three dimensions of space, 

 a third independent variable may apparently, though not really, be repre- 

 sented. Hence it may be said that two conditions determine the position 

 of every point, and, conversely, every point represents two conditions, 

 and two only. 



Just as with one fixed point, when one dimension of space only was 

 employed, any number of numerical facts could be represented by taking 

 any required number of points, so with the use of two fixed points, or of 

 two dimensions in space, any number of facts could also be represented. 

 Now it does not matter in what way the two measurements are used, so 

 long as the position of the point on the surface is fixed by them ; and as 

 for practical purposes the use of the two points in the above manner is 

 not convenient (one condition, for instance, being that the sum of the 

 lengths must be equal to or greater than the distance between the fixed 

 points), other ways of attaining the same results are employed. 



(1) Using one point, say Oj, as a centre, an arc is drawn through O2 ; 

 a given distance measured round this arc from the latter to a point. A, is 

 taken for one of the numerical quantities, and through the point a so 

 formed a radius is drawn from the other fixed point, O,, and along this 

 radius the second quantity is measured, represented by O, A, fig. 6. 



Fig. 6. 



A 







B 



Q 



.4--- ---, c '-' 



1 I -r ^ 



0.0-' \ \ 



I 



o 



\ 



I 

 \ I 



fo 



\ 



I 



\ 



yd 



\ 



I 



w 



In this way any number of quantities — O, A, O, B, 0, C — nre re pre- < 

 sented. This method is what is known as the use of polar co-ordinates, 

 the distance from 0; being the polar distance, and the distance from O2 



