106 MULTIPLE ALGEBRA. 



It may be that in some cases the fact that only one kind of product 

 is known in ordinary algebra has led those to whom the problem 

 presented itself in the form of finding a new algebra to adopt this 

 characteristic derived from the old. Perhaps the reason lies deeper 

 in a distinction like that in arithmetic between concrete and abstract 

 numbers or quantities. The multiple quantities corresponding to 

 concrete quantities such as ten apples or three miles are evidently 

 such combinations as ten apples + seven oranges, three miles north- 

 ward + five miles eastward, or six miles in a direction fifty degrees 

 east of north. Such are the fundamental multiple quantities from 

 Grassmann's point of view. But if we ask what it is in multiple 

 algebra which corresponds to an abstract number like twelve, which 

 is essentially an operator, which changes one mile into twelve miles, 

 and 81,000 into $12,000, the most general answer would evidently 

 be : an operator which will work such changes as, for example, that 

 of ten apples + seven oranges into fifty apples + 100 oranges, or that 

 of one vector into another. 



Now an operator has, of course, one characteristic relation, viz., its 

 relation to the operand. This needs no especial definition, since it 

 is contained in the definition of the operator. If the operation 

 is distributive, it may not inappropriately be called multiplication, 

 and the result is par excellence the product of the operator and 

 operand. The sum of operators qua operators, is an operator which 

 gives for the product the sum of the products given by the operators 

 to be added. The product of two operators is an operator which is 

 equivalent to the successive operations of the factors. This multi- 

 plication is necessarily associative, and its definition is not really 

 different from that of the operators themselves. And here I may 

 observe that Professor C. S. Peirce has shown that his father's 

 associative algebras may be regarded as operational and matricular.* 



Now the calculus of distributive operators is a subject of great 

 extent and importance, but Grassmann's view is the more compre- 

 hensive, since it embraces the other with something besides. For 

 every quantitative operator may be regarded as a quantity, i.e., as 

 the subject of mathematical operation, but every quantity cannot 

 be regarded as an operator ; precisely as in grammar every verb may 

 be taken as substantive, as in the infinitive, while every substantive 

 does not give us a verb. 



Grassmann's view seems also the most practical and convenient. 

 For we often use many functions of the same pair of multiple 

 quantities, which are distributive with respect to both, and we need 

 some simple designation to indicate a property of such fundamental 



* Amer. Journ. Math., vol. iv, p. 221. 



