Benjamin Peirce. ; 175 
practical study, with good results. His own thought was 
turned especially to the logic that underlies all similar systems, 
and to the limits and the extensions of fundamental processes 
in mathematics. 
At the first session of the National Academy of Sciences, in 
1864, he read a paper on the elements of the mathematical the- 
ory of quality. Between 1866 and 1870 various papers were 
read to that Academy, or to this Academy, on “ Linear Alge- 
bra,” “ Algebras,” “ Limitations and Conditions of Associated 
Linear Algebras,” ‘Quadruple Linear Associative Algebra,” 
etc. These papers were not printed in form as read, but instead 
in 1870-71 appeared his “ Linear Associative Alge ebra. 
His own feeling about this contribution to science is ex- 
pressed in the salutatory to his friends: ‘‘This work has been 
the pleasantest mathematical effort of my life. Pi no other 
ized by Professor Peire ne, generous and calc » 
Such an eee can ie come from one who has made a 
special study of the laws of Saab ease. thought. To some 
mathematicians, ain other men of science, it may yet be a 
question, if the time, has come for them to say with entire 
certainty whether this work is to share the fate of Plato’s 
barren Areonations about numbers, or to become the solid 
basis of a wide extension of the laws of our thinking. Those 
who have thought most on the course which contemporary 
inathematical science is taking will probably agree that the new 
ground thus broken can hardly f, fail to bring forth precious 
fruit in the future by adding to the powers of mathematics as 
an instrument. 
In any case, the Associative Algebra can never lose its value 
as an important and most beautiful addition to apioes Mathemat- 
ics, and must ever remain a monument to the comprehensive 
ener of thought and analytical genius of its author. 
ofessor Peirce defines mathematics as the science which 
iplination. i is no further defined than as an BE bo dis- 
tributive with reference to addition; but the only algebras 
whe ta: are those whose mul Gplenea is seer? he 
