452 BENJAMIN PEIRCE. 



ever remain a monument to the comprehensive grasp of thought and 

 analytical genius of its author. 



Professor Peirce defines mathematics as .the science which draws 

 necessary conclusions. Algebra is formal mathematics. Addition 

 is taken to express a mixture, or mere union of elements, indepen- 

 dently of any mutual action which might arise if they were to be rhixed 

 in reality. From this definition, the commutative character of addition 

 necessarily follows. Multiplication is no further defined than as an 

 operation distributive with reference to addition ; but the only algebras 

 treated are those whose multiplication is associative. The subject is 

 further limited to linear algebras, that is, to such as contain only a 

 finite number of lineally independent expressions ; so that every quan- 

 tity considered may be put under the form, 



at 



-j- ^" -|- ck -j- etc. 



where i, j, k, are peculiar units, limited in number ; while a, b, c, are 

 scalars, — a term borrowed from the language of quaternions, but here 

 used in a modified sense to include, not merely the reals, but also the 

 imaginaries, of ordinary algebra. A variety of highly general theorems 

 are given, extending to all linear associative algebras. The author 

 next introduces the conception of a pure algebra, as contradistinguished 

 from one which is virtually equivalent to a combination of several. 

 Methods are developed for finding all such pure algebras of any order. 

 Finally, he obtains the complete series of multiplication-tables of these 

 algebras up to the fifth order, together with the most important class 

 of the sixth order. They are in number as follows : — 



Single Algebras 2 



Double " 3 



Triple " 5 



Quadruple" 18 



Quintruple " 70 



Sextuple " 65 



Professor Peirce never made any extended study of the possible 

 applications of his algebras ; he was far from thinking, however, that 

 their utility was dependent upon finding interpretations for them ; on 

 the contrary, he showed that certain of them could be advantageously 

 employed, without any interpretation, in the treatment of partial dif- 

 ferential equations like that of Laplace. 



He read to this Academy in May, 1875, a memoir "On the 

 Uses and Transformations of Linear Algebra," which is, we believe, 

 his only published addition to the principal treatise. He had also 



