176 Benjamin Peirce. 
sions; so that every quantity considered may be put under the 
form, 
at + bj) + ck + ete. 
where 7, 7, &, are peculiar units, limited in number; while a, 2, 
c, are scalars,—a term borrowed from the language of quaterni- 
ons, but here used ina 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 Ne ORT ENP 
Double ‘* oes 3 
Triple ee Sewoe des ay Cer CeCe CES Ce eos 5 
Quadruple “ ser : 18 
Quintuple ‘“ cto 
Sextuple “ 65 
Professor Peirce never made any extended study of the possi- 
ble applications of his algebras; he was far from thinking, 
however, that their utility was dependent upon finding inter- 
pretations for then; on the contrary, he showed that certain of — 
advantageously employed, without any interpre- 
tation, in the treatment of partial differential equations like 
that of Laplace. 
He read to this Academy in May, 1875, a memoir “On the 
Uses and Transformation of Linear Algebra,” which is, we be- 
lieve, his only published addition to the principal treatise. He 
had also made some progress in the investigation of the laws of 
non-associative algebras. 
Professor Peirce could not fail to be interested in all ques- . 
tions that concern the equilibrium, the history, and the devel- 
opment of the solar system. At first he was loth to accept the 
nebular hypothesis in any form. But the results of his studies 
led him, at last, to defend its main propositions as the true laws 
of creation. ; 
he rings of Saturn are of prime import in any explanation 
ory of the rings. He announced, as the result of his analysis, 
ot be solid, that a fluid ring could not 
