ON GRAPHIC METHODS IN MECHANICAL SCIENCE, 379 



the measure of the angle which the radius vector, or direction of the 

 polar distance, makes with the line Oi Oo. The direction in which the 

 distance is measured from any one point to any other point can always 

 be found and stated numerically with reference to the two fixed points 

 by drawing a parallel to the direction in question through one of the fixed 

 points. The direction can then always be stated in terms of distance from 

 the other of the two fixed points at which the arc cuts the radius vector. 

 (2) Just as when dealing with only one fixed point, so when dealing 

 with two we may remove Oi to an infinite distance, and obtain, as 

 before, a straight line for the arc, and from this the distances to A, B, 

 C, D, E, can be measured (fig. 7). 



Fig. 7. 



D 



O 



o 



O a ■ D ; a 



oc 



The distances from Oj are now measured along a straight line, which 

 is the base line of the single-point method, the line through O, O2, which 

 may be called the vertical line, being perpendicular to it. If we next sup- 

 pose O2 to be also removed to an infinite distance, magnitudes previously 

 measured along the base line may now be measured from any fixed ver- 

 tical line, and parallel to the base line, since all parallel lines pass through 

 the same point when it is at an infinite distance. 



This is the Cartesian system of co-ordinates, the distance of any point 

 from the base and vertical lines being respectively the ordinate and 

 abscissa of the point. The direction in which any distance is measured 

 between any two points on the surface can be stated just as easily as with 

 polar co-ordinates by the ratio of the distance between the points measured 

 parallel to the vertical axis and the distance measured parallel to the 

 horizontal axis. This is, of course, the same thing as taking the ratio of 

 distances between the points, measured respectively in the directions of 

 the two fixed points, which are now, however, imaginary. 



Except for the matter of convenience, the use of two points, or the 

 use of polar co-ordinates or of Cartesian co-ordinates, might with equal 

 advantage be employed, and it is quite possible that just as Descartes 

 invented and developed, the last-mentioned system, so some other method 

 might be found still more serviceable for graphical purposes. Thus, for 

 instance, it does not matter if what corresponds to the base line and 

 vertical line are not at right angles to each other, or the distances are 

 taken in directions parallel instead of perpendicular to them, which is m 

 fact the method of oblique co-ordinates. Or we need not use the idea of 

 direct measure of lengths at all ; we may employ with the two fixed points 

 two angles as in fig. 8 and find A as readily ; for we know that the solu- 

 tion of a triangle can be found from two angles and the included side as 

 easily as with three given sides. 



The general conclusion arrived at is that, no matter what system may 



