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 northward + 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 $1,000 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 qud 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 multiplication is necessarily associative, and its definition is not really dififerent 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.[1]
Now the calculus of distributive operators is a subject of great extent and importance, but Grassmann's view is the more comprehensive, since it embraces the other with something besides. For every quantitative operator may be regarded as a qucuitity, i.a, 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.
