80 



G. FLEURI. 



therefore to construct the product OP of two complex quantities 

 OA and OB we have simply to construct on OB a triangle OBP 

 directly similar to the triangle OIA formed with 1 and OA; 

 the side OP of the triangle is the product in question. 



We see at once, first, that the multiplication of complex quanti- 

 ties is uniform and we have already seen, second, that it is com- 

 mutative, i.e., a a . bo = o . a a 



Third, We have 



a a . bo . c — {oub c) a ,oj r definition of multiplication 



— In h r\ i /a i \ associativity of multiplication 



- yw . v o; a + ^ + y; and addition of scalar* quantities 



= a a . (b c) a + definition of multiplication 

 that is to say the multiplication of complex quantities is associative. 



Fourth, As for directive quantities we should easily see from 

 the arithmetical definition of multiplication that the corresponding 

 operation on complex quantities is distributive ; but a geometrical 

 demonstration is easily given. 



Consider (a a + bo) . c 

 LetOA = a a AB = &g OA + AB = OB 



Draw OA' making an angle a + y with 

 OX and take OA' = a c = OA . c then 

 draw A'B' of argument /3 + y and such 

 that A'B' = b c = AB . c; finally join OB'. 



The two triangles OAB - OA'B' being 

 similar by construction give OB' = OB . c 

 and argument OB' = argument OB -1- y 



But OB' = OA' + A^B 7 therefore 



K + b p) c y = OB' = a a . c y + bp.c y 

 Fifth, a a . 1 = a a 

 Sixth, a a . o = o 



Division. — The definition given remains always the same and 

 can be expressed by the equality 



11. 



* Scalar, i.e., quantities of index zero or tt. 



