MULTIPLE ALGEBRA. 107 



importance in the algebra of such functions, and no advantage 

 appears in singling out a particular function to be alone called 

 the product. Even in quaternions, where Hamilton speaks of 

 only one product of two vectors (regarding it as a special case 

 of the product of quaternions, i.e., of operators), he nevertheless 

 comes to use the scalar part of this product and the vector part 

 separately. Now the distributive law is satisfied by each of these, 

 which therefore may conveniently be called products. In this 

 sense we have three kinds of products of vectors in Hamilton's 

 analysis. 



Let us then adopt the more general view of multiplication, and call 

 any function of two or more multiple quantities, which is distributive 

 with respect to all, a product, with only this limitation, that when 

 one of the factors is simply an ordinary algebraic quantity, its effect 

 is to be taken in the ordinary sense. 



It is to be observed that this definition of multiplication implies 

 that we have an addition both of the kind of quantity to which the 

 product belongs, and of the kinds to which the factors belong. Of 

 course, these must be subject to the general formal laws of addition. 

 I do not know that it is necessary for the purposes of a general 

 discussion to stop to define these operations more particularly, either 

 on their own account or to complete the definition of multiplication. 

 Algebra, as a formal science, may rest on a purely formal foundation. 

 To take our illustration again from mechanics, we may say that 

 if a man is inventing a particular machine, a sewing machine, a 

 reaper, nothing is more important than that he should have a precise 

 idea of the operation which his machine is to perform, yet when he is 

 treating the general principles of mechanics he may discuss the lever, 

 or the form of the teeth of wheels which will transmit uniform 

 motion, without inquiring the purpose to which the apparatus is to 

 be applied ; and in like manner that if we were forming a particular 

 algebra, a geometrical algebra, a mechanical algebra, an algebra 

 for the theory of elimination and substitution, an algebra for the 

 study of quantics, we should commence by asking, What are the 

 multiple quantities, or sets of quantities, which we have to consider ? 

 What are the additive relations between them ? What are the multi- 

 plicative relations between them ? etc., forming a perfectly defined 

 and complete idea of these relations as we go along; but in the 

 development of a general algebra no such definiteness of conception 

 is requisite. Given only the purely formal law of the distributive 

 character of multiplication, this is sufficient for the foundation of a 

 science. Nor will such a science be merely a pastime for an ingenious 

 mind. It will serve a thousand purposes in the formation of parti- 

 cular algebras. Perhaps we shall find that in the most important 



