504 
Proceedings of the Royal Society of Edinburgh. [Sess. 
1. Introductory illustration. — I was recently led back to the present 
work by facing a problem of mere laborious calculation in the method of 
least squares, and this method will probably serve as an indication of the 
kind of inquiry in which Multenions may be expected to be of practical 
utility. 
In the method, a supposed set of values of the unknowns is contemplated 
in various ways : — (1) The true unknowns are often symbolised ; (2) a 
particular set called the most probable set is mainly considered ; (3) a quite 
arbitrary set is sometimes treated of. In all these three ways we look upon 
a set as a whole, as a sort of individual belonging to a community of 
individuals. In Multenions we use a single symbol, a, to denote such a 
set, and call a a fictor. If the unknowns are denoted by a;, y, .... we may 
say that a = (x, y, . . . .). The particular value of a for which x = 1 and 
y — z — .... =0 is called a fictit (denoted by q) ; and the value for which 
x is any scalar and y=z= ... . =0 is said to be x times the fictit q just 
mentioned. More generally, any fictor a = (x, y, z, .... ) is put as the 
sum xq-f^/q-f. . . ., where q = (l, 0, 0, ....), i 2 = (0, 1, 0, . . . . ), etc. ; or 
(x, y, z, . . . .) = x(l, 0,0 .. . .) + y(0, 1, 0, . . . .) + etc. 
Multenions is based on fictits. 
2. The Laws, Symbols, and Parts of Multenions. — The laws differ 
from Grassmann’s (Ausdehnungslehre) in some important respects. The 
product of two units (fictits or multits) is never zero, so that fictit products 
are not “combinatorial.” There is but one mathematical meaning of 
product. Grassmann’s various products are “ parts ” of our one product. 
Hence we speak of combinatorial parts, but not of combinatorial products. 
Before enunciating what we are to consider as fundamental laws, it is 
well to state that in § 3, below, an extension of these laws will be considered. 
The extension is of the nature of an alternative or permissible simplifying 
restriction, and will, in the subsequent part of the paper, be accepted or 
rejected as we see fit. 
[The references to the odd number N in the fundamental laws might 
be omitted without any serious alteration of what follows in the present 
paper. We almost invariably suppose our symbols confined to a given 
“ multiplex ” of order n less than N. The reason for introducing N is to 
make quaternions a particular symmetrical case of our subject, multenions, 
namely, when N = 3. We shall generally, however, in our references 
to quaternions make the mathematically equivalent assumption that 
quaternions is a particular unsymmetrical case of our present subject 
