On the Symbol v'—l in Geometry. 173 



same in either case, although certain hints are furnished by 

 proposing the inquiry as a theorem, which may facilitate our 

 discovery of a method of research adapted to the purpose in 

 view. 



As another instance, we may propose to prove the theorem 



l-SeiV-l 4- e-t^^X — l _ J!_ 4. ^ _ 



2 {€ +e l ~ 1 l.* + 1.2.3.4 -' 



or we may propose to find u and v such that 



fi 2 4 



1 -IT? + TTsTir* ---«'{-' + «-'}• 



Treating this proposition as a theorem, we shall have to 

 establish the identity of value of the two sides of the equation ; 

 and treating it as a problem to find the values of u and v, 

 which will fulfill the identity : and the two processes differ 

 not the least in logical character, and are only varied in the 

 order of their subordinate details. From both forms of the 

 proposition, we learn, however, that the series cannot be con- 

 verted into the form v{uf + u~ 6 }, whilst u and v are real 

 numbers positive or negative; and that the condition can only 

 be fulfilled by the co-existence of the incompatible operations 

 expressed by V being performed upon — . In other words, 

 the operations which give the value in a finite algebraic form 

 are inconsistent with each other ; and the imposed conditions 

 are therefore incongruous. 



Precisely the same remark applies to Demoivre's as to Eu- 

 ler's theorem, viz. that (cos and sin being abbreviations for 

 two specified series) 



u 6 = cos + v sin is the problem, and 



d ^- 1 = cos + v' — I . sin is the theorem. 



Also from this the same conclusion respecting congruity is 

 deducible in the same way. 



It is too familiarly known to require being insisted on here, 

 that the coefficient of V — 1 in a result, does in general fur- 

 nish some information as to the method by which the relative 

 magnitudes in the data may be so modified as to render the 

 proposed problem soluble, or so that the enunciated theorem 

 shall be expressible by means of congruous operations. For if 

 that coefficient can be reduced by any such relations amongst 

 the data to zero, the expression itself disappears in conse- 

 quence ; thus giving the extreme case of solvability of the pro- 

 position composed of data of the specified kind. And, again, 

 if that coefficient can be made to take the form V —y, the 

 symbol V — \ will not appear in the result at all, since v / — y 



