86 G. FLEURI. 



Quaternions — Calculus op Quaternions. 



Being now convinced of the impossibility of treating space 

 quantities by algebra we will seek for some other method. The 

 first method of generalization which presents itself to one's mind 

 is to work with space vectors as we have worked with plane vectors 

 or complex quantities. It is easily found that those vectors can 

 be adced by the rule of the parallelepiped — an obvious generaliza- 

 tion ci the rule of the parallelogram — and therefore that they can 

 be represented by expressions 



a i 4- bj + c k 



where a, b, c are scalar and i, j, k three linear units along three 

 rectangular axes, and Servois (thirty years before Hamilton) was 

 successful so far. But when we come to try the combination of 

 vectors by multiplication, we are soor stopped by difficulties which 

 took Hamilton fifteen years to overcome. Since the difficulty 

 comes from multiplication and since — after the general rule of 

 generalization — we must find complex quantities as a particular 

 case of space quantities, it is natural to look to multiplication of 

 complex quantities to start our generalization from.* 



We have seen that to multiply by a complex quantity OA 

 (operating on OB for instance) we have only to construct (on OB) 

 a triangle OBP directly similar to triangle 10 A where 0I=l o 

 and OP being the product (see Fig. 10) we have OP = OB . OA 



or OP = OB . — = OB . — 



Therefore, to multiply by vector OA is a double operation com- 



OA 



posed first of a multiplication by -— — 



f Another reflection may be suggested which will also induce one to 

 examine the multiplication of plane vectors. When we operate on a 

 scalar quantity by the extraction of root in algebra, a quantity of differ- 

 ent nature may be obtained ; why should it not be the case for the 

 multiplication of vectors ? Whether it is or not, the new kind of quantity 

 must be a generalization of a complex quantity therefore it is natural to 

 come to that kind of quantity again. 



