250 Mr. De Morgan on the relative Signs of Coordinates. 



the axis of x. If we say that this angle shall be called posi- 

 tive when it is formed by the rotation of a line from the posi- 

 tive part of the axis of x towards that of the axis of ^, it is not 

 that, mutatis mutandis, the negative sign would not equally 

 apply to that angle, but because we always consider those 

 conventions which make the resulting equations most simple 

 as preferable, and even practically necessary. In the same 

 way this practical necessity may be shown to exist correla- 

 tivdy imth another hypothesis in the present case ; and it may 

 even be made obvious that without an implied selection be- 

 tween the two signs in (1.) many equations which are used as 

 imiversally true are not so in reality. The principle I as- 

 sume is, that continuity, and the universality of equations in 

 their most simple and usual forms, are desirable. 



\B' 



^ 'k. 



i^ 



A 



D' 



ID 



Let OA and OB be the directions in which x and ?/ are po- 

 sitive, and let P be a point whose coordinates are m and «, in 

 that quarter of space in which both coordinates are positive. 

 Now let moveable rectangular axes of ^ and v) (new coordi- 

 nates) first coincide with those of x and y, and then make a 

 complete revolution. Let these axes be 0«, 0/3, Oy, OS 

 (not drawn), and let 6 be the angle (the direction of revolu- 

 tion being A B C D) by which O a has separated from O A 

 to gain any position in question. Supposing that at the ini- 

 tial coincidence ^ and x are measured positive in the same 

 direction (certainly the most convenient outset), the condition 

 of continuity requires that f and )j should remain positive 

 until they become either nothing or infinite : the latter they 

 cannot become for the given point P. Hence — that is to say, 

 from this hypothesis, not of necessity, but of indisputable con- 

 venience, — it follows that when 0« lies in the quadrant A'OD', 



