A. 





B 

 B r 





C~ 



D 



XT 













FROM NUMBER TO QUATERNION. 71 



found in arithmetic. We will now exhibit those of the properties 

 which are fundamental. 



Addition (indicated by sign + ) 



First, operation uniform or single valued, i.e., having one result 

 only ; this property may be expressed in the following manner : 

 If a — a' and b = b' a + b = a' + b' 

 Second, commutative, i.e., a+b=b+a 

 Third, associative, i.e., a + b + c = a + (b + c) 

 Fourth, a + o — a a, b, c, a, b' being numbers. 



The geometrical operation on straight 

 lines which corresponds to addition of 

 their tensors can be defined as follows: 

 Let us propose to add AB and CD 

 rig. 2. (Fig. 2). 



From a point O on a straight line OX and in the standard direc- 

 tion take a length OB' = AB and then, always in the same direction 

 a length B'D' = CD; or in other terms : let the point O move from 

 O to the point B' such that OB' = AB and then from B' to the 

 point D' such that B'D' = CD ; OD' is the sum AB + CD. 



And this definition is easily extended to any number of lines. 

 From it one can see that all the fundamental properties of the 

 addition are obvious. 



Subtraction (indicated by sign — ) 

 Definition. — It is the operation which, being given the sum of 

 two tensors and one of them, consists in finding the other. 



It is the inverse operation of addition* 

 Let OD' = AB + CD 



then by definition OD'— CD = AB 



Geometrically it consists in displacing the movable point from 

 O to D' ( Fig. 2) in the standard direction, and from D' to B' in the 



* We call inverse operation with regard to another operation considered 

 as the direct one, a second operation which — all the terms but one being 

 given and the result of the first operation — consists iu finding the miss- 

 ing term. 



