82 G. FLEURI. 



elements the definitions of the trigonometrical ratios and the 

 knowledge of the theorem on a rectangular triangle of sides a 

 (hypotenuse), b and c 



b = a cos (a.b) = a sin (c.a) 



c = a sin (a.b) = a cos (c.a) 

 which is immediately demonstrated — we easily see that a complex 

 quantity a a can be put under the form 



a (cos a + i sin a) 

 The definition of multiplication gives then almost at once the 

 expression of de Moivre's theorem 



( a(cos a + i sin a))™ = a n (cos na + i sin na)* 



Remark — By means of complex quantities introduced from the 

 very beginning of Algebra, as we have done, many singularities 

 can be explained, for instance the finding of two values in the 

 extraction of a square root ceases to be a case of exception since 

 the extraction of the nth root — as it is readily shown — is an oper- 

 ation ^-form n , — \, a+2k7r . . a+2k7r, 

 V a — a n (cos + ^ sin ) 



a x n n ' 



A more extensive knowledge of Algebra including the study of 

 series of an imaginary variable and of its condition of convergence 

 shows that any complex quantity a a can also be put under a 

 fourth form a e ia 



where e is the well known series. 



Let us add that when a complex quantity a a is written under 

 the form a a = a . l a = a e ia = a (cos a + i sin a) 



a being the tensor, 



1 = e ia = cos a + i sin a 

 is — according to Hamilton — the versor of the quantity, a fact 

 expressed by U« a = e ia in Hamilton's notation. 



* The ordinary demonstration given of that theorem is a survival from 

 bygone ages. It is logically a vicious circle as it consists in applying to 

 imaginary quantities rules of calculations only known for real quantities 

 without showing them to be applicable. 



