FROM NUMBER TO QUATERNION. 93 



_ A T . aA , a 2 A 2 . o 8 A 3 . a 4 A 4 , a*A s , 



= ^-— ■ + $+•" 



+A(f-S+S+-) 



= cos a + A sin a 

 so that any quaternion with tensor a 



a (cos a + A sin a) 

 can be put under the form ae Aa 



comparable to form ae ia of plane complex quantities. But owing 

 to non-commutativity of multiplication of quaternions, we cannot 

 write e A.a xe BP =e ka + B& 



as Newton's binomial formula is no more true. 



Division. — It is the inverse operation of multiplication, but 

 owing to the non-commutativity of the latter operation two 

 definitions could be given : 



Being given two quaternions A and B, find a quaternion Q such 

 that A = Q . B 



or such that A = B . Q 



The first definition has been chosen. 



The geometrical interpretation is obvious from the definition of 



multiplication of quaternions, and it is easy to see from it that a 



quaternion AOB is equal to the quotient of the two vectors which 



form it. OB 



AOB = 



OA 



Conclusion. — We now come to the conclusion I have given at 

 the beginning of this paper, that is to say that the quaternion 

 calculus constitute a natural extension of algebra. 



I hope also that I have established a fact which would be an 

 advantage to have better known, to wit that Algebra is not scalar 

 but vectorial. 



The spreading of that truth would help singularly to the spread- 

 ing of quaternions. But independently of that fact, the historical 

 order for the introduction of the different kinds of algebraic 

 quantities is certainly not the logical one, and it is therefore fully 

 time that the teaching of such an important branch of mathematics 

 as algebra should be improved. 



