1907-8.] Algebra after Hamilton, or Multenions. 
503 
XXXIV. — Algebra after Hamilton, or Multenions. By Alex. 
MAulay, M.A., Professor of Mathematics and Physics, University 
of Tasmania, Hobart. Communicated by Professor C. G. Knott. 
(First MS. received December 1906. Read March 4, 1907. Supplement received from 
Tasmania with revision of first proof, June 1908.) 
Summary. 
Ever since I have learnt something of the meanings of Grassmann’s 
Ausdehnungslehre, and have at the same time learnt to regard the beauties 
of that system with something akin to awe, I have been persuaded that on 
the lines of Quaternion Algebra there is to be built a system very much 
like the Ausdehnungslehre, but an improvement thereon. Of course it will 
be matter for differing opinions whether what I call Multenions is really 
an improvement on the Ausdehnungslehre. I here record my own personal 
opinion that it is. 
I do not suppose that anybody will maintain that a multitude of 
different kinds of multiplication within the bounds of one method can be 
regarded as anything but a blemish, — a blemish that may be justified by 
necessity and utility. The Ausdehnungslehre seems to me to have this 
blemish, and Multenions not to have it. Whether along with the absence 
of the blemish there becomes present an additional difficulty of manipula- 
tion is questionable. I have not found it so, but this may be due to the 
fact that I have so long been in the habit of thinking through the 
quaternion machinery. 
I think these general remarks are all that can be usefully set down as 
a summary. The paper itself appears to be too condensed to admit of a 
true summary other than a mere table of contents. 
This paper is to be regarded as a preliminary outline of what is necessarily a large 
subject, which demands time and labour for due development. It is based on the work of 
Hamilton, Grassmann, and their followers. Their treatises will, however, be referred to 
but scantily. Chapter iv. of Octonions, which I propose to regard as virtually a part of 
the present paper, contains copious references to the source, Ausdehnungslehre, from which 
it was practically derived. 
List of some of the terms used below : — Fictor, fictit, fictorplex, continent fictorplex. 
Multenion, multit, multiplex, continent multiplex. Fictorlinity, fictorcolinity, multilinity. 
Replacement, retroplacement, rigid replacement, unireplacement. Complement, conjugate, 
reversate and reciprocal of a multenion. “ Sequence ” is used to prevent confusion with 
“ order ” in the technical sense. The conventions as to notation are as far as possible those 
of Quaternions as adopted by Hamilton. 
[April 1908]. — The lack, in Tasmania, of all mathematical literature bearing on Matrices, 
has made it impossible to check to what extent many of the results are novel or simply in 
different guise. 
