76 



G. FLEURI. 



At first sight, however, that definition may be puzzling, and 

 seems more arbitrary than it is in reality. For, there is a much 

 greater difference between the definition given in arithmetic for 

 the multiplication of whole numbers and that given for the 

 multiplication of fractions. And the best proof that such is the 

 case is that the definition given for the fractions properly under- 

 stood is still sufficient for directive quantities. 



To multiply a quantity (multiplicand) by an other (multiplier) 

 is to determine a third quantity derived from the multiplicand by 

 the same operation as was needed to derive the multiplier from 

 unity. 



Take for instance a x b^ 



multiplicand a multiplier b^ unit 1 



b^ is formed from the unit by multiplying the tensor of the unit 

 by b and turning the result through an angle tt ; then to obtain 

 our product we must multiply the tensor of a Q by b — result (ab) 

 and then turn it of angle ir result (ab)^ 



From our definition we see first that the multiplication of 

 directive quantities is uniform, since addition and multiplication 

 of absolute quantities are uniform. 



Second, now denoting by a, /3 and y any quantity o or it we can 

 write a a . bp - (ab) a + p = (ba)p + a = bp . a a 



from definition of multiplication, commutativity of addition and 

 multiplication of absolute quantities and again definition of 

 multiplication • therefore our operation is commutative. 



Third, a a (bp c y ) =a a ((b c)p + y ) 



= (<*>bc) a + p + y 



= a a .bp.c y 

 so that the multiplication of directive quantities is associative. 



Fourth, a a (bp + c y ) = a a bp + a a c y 



for bo + c being formed from unity by multiplying it by bo then 

 by c and adding the results we must multiply a a by bo then by 

 c y and add the results to get a a (ho + c ) 

 therefore the multiplication is distributive. 



