896 PROCEEDINGS OF THE AMERICAN ACADEMY 



universe, and placing by its side an ideal universe, its exact counter- 

 part, with which it can be compared and contrasted, and, by means of 

 curiously connecting fibres, form with it an organic whole, from which 

 modern analysis has developed her surpassing geometry. The letters 

 or units of the linear algebras, or to use the better term proposed by 

 Mr. Charles S. Peirce, the vids of these algebras, are fitted to perform 

 a similar function each in its peculiar way. This is their primitive and 

 perhaps will always be their principal use. It does not exclude the 

 possibility of some special modes of interpretation, but, on the contrary, 

 a higher philosophy', which believes in the capacity of the material 

 universe for all expressions of human thought, will find, in the utility 

 of the vids, an indication of their probable reality of interpretation. 

 Doctor Hermann Hankel's alternate numbers, with Professor Clifford's 

 applications to determinants, are a curious and interesting example of 

 the possible advantage to be obtained from the new algebras. Doctor 

 Spottiswoode in his fine, generous, and complete analysis of my own 

 treatise before the London Mathematical Society in November of 

 1872, has regarded these numbers as quite different from the algebras 

 discussed in my treatise, because they are neither linear nor limited. 

 But there is no difficulty in reducing them to a linear form, and, indeed, 

 my algebra (e,) is the simplest case of Hankel's alternate numbers, and 

 in any other case in which n is the number of the Hankel elements 

 employed, the complete number of vids of the corresponding linear 

 algebra is 2"— 1. The limited character of the algebras which I have 

 investigated may be regarded as an accident of the mode of discussion. 

 There is, however, a large number of unlimited algebras suggested by 

 the investiiiations, and Hankel's numbers themselves would have been 

 a natural generalization from the proposition of § 65 of my algebra.* 

 Another class of unlimited algebras, which would readily occur from 

 the inspection of those which are given, is that in which all the powers 

 of a vid are adopted as independent vids, and the highest power may 

 either be zero, or unity, or the vid itself, and the zero power of the 

 fundamental vid, i. e., unity itself may also be retained as a vid. But 

 I desire to draw especial attention to that class, which is also unlimited, 

 and for which, when it was laid before the mathematical society of 

 London in January of 1870, Professor Clifford proj^osed the appropriate 

 name of quadrates. 



* This remark is not intended as a foundation for a claim upon tlie Hankel 

 numbers, which were published in 1867, three years prior to tlie publication of 

 my own treatise. — B. P. 



