72 G. FLEURI. 



opposite direction and so that translation D'B' be equal to transla- 

 tion CD ; OB' is the result. 



But if OD' < CD the operation although possible geometrically 

 does not correspond to any result, for 7-15 for instance has no 

 signification in the case we are just considering. 



Multiplication (indicated by sign x or by sign .) 

 First, uniform if a = a' and b = b' ab = a'b' 

 Second, commutative ab = b a 

 Third, associative a b c = a (b c) 

 Fourth, distributive with regard to addition and subtraction 



(a + b — c) d = ab + bd — cd 

 Fifth, axl=a 



Sixth, a x = 



a, b, c, d, a', b' being numbers. 



Geometrically — Being given two lines AB and CD the operation 

 consists in the construction of a fourth proportional OF 

 OF = jOD 



By definition OF = AB x CD 

 and the definition is easily extended to any number of factors. 



Division (indicated by sign -r or an horizontal bar between the 



numbers). 

 An operation which, the product of two tensors and one of them 

 being given, consists in finding the other tensor. It is the inverse 

 operation of multiplication. 



By definition OF : AB = CD 



Geometrically it comes to the construction of a fourth propor- 

 tional CD 



OF CD 

 AB 1 



as for multiplication so that there is in fact no difference between 

 multiplication and division. Owing to the introduction of frac- 

 tions in arithmetic we knew already that fact, for divisions by 



