FOUNDATION OF ALGEBRA. 183 



But if = imply sameness of revolution, it is not true that e 2 *^- 1 = 1, 

 except in length. 



The interpretation of A^~ x might be easily attained from prescribed 

 definitions, and from their necessary result 



e ev=i — C os 9 + sin 9 \/ - 1 ; 



nor would this step be logically objectionable. It would, however, be more 

 satisfactory if something like an a priori interpretation, or simple explana- 

 tion, could be given. I do not consider the following as complete, but it is, 

 as far as it goes, of a new character. 



Conformably to definitions, we must have 



{(log a, 0)V=I}^ . {log a, 6}-' = (- log a, - 9), 



where by (log «, 9) is meant a line of the length a, and amount of revolu- 

 tion 9. Now we cannot suppose that the first operation changes the sign of 

 log a only, and the second that of 9 only : for this would be to make the 

 operation ( y~ l mean different things in different places. We must pro- 

 pose some operation of permanent form, which being twice performed will 

 make the alteration required. 



From the definitions, it follows that 



(log a, 0) x (0, 9) = (log a, 9), 



whence (log a, 9) must be the product of two functions, one of a and 

 the other of 9, the first of which is known, being e loga or a, and the second 

 of which must be of the form E e , since by definition 



(0, 9) x (0, 9') = (0, 9 + ff). 



Hence aJE e , or a(0, l) e , is the representative of a line a, inclined at an 

 angle 9. If then we make cos 9 and sin 9 mean nothing more than the pro- 

 jecting factors of a length inclined at the angle 9 upon the axis of the unit 

 line and its perpendicular, we have 



(cos 1 + \/ - 1 sin l) e = cos 9 + \/ - 1 sin 9. 



The definition does not differ from that of cos 9 and sin 9 in geometry, 

 and this equation is an a priori property of these functions, deducible im- 



