MULTIPLE ALGEBRA. 105 



excluded from good society among mathematicians. And if we admit 

 as suitable the notations used in this memoir (where it is noticeable 

 that the author rather avoids multiple algebra, and only uses it very 

 sparingly), we shall logically be brought to use a great deal more. 

 For example, if it is a good thing to write in our equations a single 

 letter to represent a matrix of n 2 numerical quantities, why not use 

 a single letter to represent the n quantities operated upon, as Grass- 

 mann and Hamilton have done ? Logical consistency seems to de- 

 mand it. And if we may use the sign )( to denote an operation by 

 which two sets of quantities are combined to form a third set, as is 

 the case in this memoir, why not use other signs to denote other 

 functional operations of which the result is a multiple quantity ? 

 If it be conceded that this is the proper method to follow where 

 simplicity of conception, or brevity of expression, or ease of trans- 

 formation is served thereby, our algebra will become in large part a 

 multiple algebra. 



We have considered the subject a good while from the outside ; we 

 have glanced at the principal events in the history of multiple algebra; 

 we have seen how the course of modern thought seems to demand its 

 aid, how it is actually leaning toward it, and beginning to adopt its 

 methods. It may be worth while to direct our attention more 

 critically to multiple algebra itself, and inquire into its essential 

 character and its most important principles. 



I do not know that anything useful or interesting, which relates 

 to multiple quantity, and can be symbolically expressed, falls outside 

 of the domain of multiple algebra. But if it is asked, what notions 

 are to be regarded as fundamental, we must answer, here as else- 

 where, those which are most simple and fruitful. Unquestionably, no 

 relations are more so than those which are known by the names 

 of addition and multiplication. 



Perhaps I should here notice the essentially different manner in 

 which the multiplication of multiple quantities has been viewed 

 by different writers. Some, as Hamilton, or De Morgan, or Peirce, 

 speak of the product of two multiple quantities, as if only one product 

 could exist, at least in the same algebra. Others, as Grassmann, speak 

 of various kinds of products for the same multiple quantities. Thus 

 Hamilton seems for many years to have agitated the question, what 

 he should regard as the product of each pair of a set of triplets, or 

 in the geometrical application of the subject, what he should regard 

 as the product of each pair of a system of perpendicular directed 

 lines.* Grassmann asks, What products, i.e., what distributive 

 functions of the multiple quantities, are most important ? 



*Phil. Mag. (3), vol. xxv, p. 490; North British Review, vol. xlv, (1866), p. 57. 



