2 E. B. Wilson, 
algebras, that is, the point-, line-, and plane-analyses of Grassmann. 
These subjects occupied approximately half the time of the course. 
During the remaining half, he took up the theory of dyadics, which 
is in immediate and intimate connection with the theory of matrices, 
and concluded with C.S. Peirce’s thorem that any linear associative 
algebra may be put in quadrate, that is, in matricular form. This 
brief series of lectures by no means contained all of Gibbs’s ideas 
and developments in multiple algebra. Indeed he had published 
at a much earlier date some reflections and theories on the subject! 
which found no place in his course. An examination of the notes 
which he left at his death shows, however, that he followed his 
usual custom of not committing his results to paper except im so 
far as they were immediately needed for the lectures in his course. 
The reason for my being so bold at this time as to publish some 
of the most essential extracts from Gibbs’s lectures on multiple 
algebra is partly because they may be of interest to mathematicians 
who may be concerned with the theory of matrices or with mul- 
tiple algebra, and partly because I desire to make use of the ab- 
breviations which his notation and methods afford in discussing some 
geometric problems in connection with the theory of strains. If at 
any time in developing the multiple algebra I take the liberty of 
adding to what I find in Starkweather’s notes or in my own, or 
if I depart from the methods of Gibbs, I shall try to make the 
fact evident—not for the purpose of claiming any originality of my 
own, but that the reader may have as definite as possible an idea 
of what Gibbs did in his course on multiple algebra, in so far as 
I find it necessary or advisable to print it at this time. 
2. Preliminary notions and notations.—Let the primary elements 
of the algebra be denoted by Greek small letters, a, B, y,...- 
If the algebra is z-dimensional, any 7-4-1 of the elements will be 
connected by a linear relation, 
+ a40--08--cy--... 16 n--1- tense 
with scalar coefficients a, 6, c, ... not all of which are zero; and 
any element of the system may be expressed linearly with scalar 
coefficients in terms of any given z linearly independent elements, as 
(1) 6 = 00 bea 6 y-.. wu tO terme 

' On Multiple Algebra, an address before the section of mathematics 
and physics of the American Association for the Advancement of Science, 
by the Vice-President. Proceedings of the American Association for the 
Advancement of Science, volume 35, pp. 37-66. This address is reprinted 
in The Scientific Papers, volume 2, pp. 91-117. 
