OPERATIVE DIVISION 223 



(7) This suggests a most interesting operative geometry which 

 I can recommend to fellow amateurs, and which is slightly 

 different from Cartesian geometry because in it curves re- 

 present, not functions resulting from operations, but the 

 operations themselves . Thus, with angular co-ordinates, a + bo 

 is a straight line, and e is an exponential curve, and so on ; 

 and itself is the straight line given by the equation y = x in 

 Cartesian geometry. In fact the application of this operative 

 notation to geometry, and especially the use of what may be 

 called the geometrical operation, furnish beautiful proofs of many 

 trigonometrical and Cartesian propositions — almost rivalling 

 quaternions in this respect. I propose to write a paper upon 

 the subject some day, but at present give only the shortest 

 possible statement as an introduction to the proper theme 

 of this article. 



(8) We now proceed to operative division. If <f)(x) = x( x ) x ^(*) 

 and <f>(x) and ^(x) are known and we wish to find the value 

 of x( x )> then we can often do so by the rule of ordinary 

 division. In similar cases we can also generally find the form 

 of the operation % when <f> = [%] [yfr] by a similar rule of opera- 

 tive division. Suppose for instance that we have given the 

 equation [co 8 -f bo + a]x = and we wish to transfer the 

 origin to the left or negative side by putting x = y — p. We 

 then have to find the form of 



[co 8 + bo + a][o - p]y ; 

 which can be done easily enough by Taylor's theorem, or by 

 ordinary simplification, or by the means used for Horner's 

 method. But operative division gives perhaps the quickest 



<b 

 and most easily remembered process. For, since ^=X-i' 



we have 



co* + bo + a 

 [co. + bo + «][o -£] - ; 



+ p 



and we proceed as follows : 



+ />]co 8 + bo + a[co 8 + (b - 2cp)o + (a - bp + cp l ) 



CO* + 2CpO + Cp* 



(b - 2cp)o + (a - cp l ) 

 (b — 2cp)o j- (b — 2cp)p 



a — bp + cp l 



a - bp + cp l 



