458 
PRESIDENTIAL ADDRESS TO THE 
[784 
i is the before-mentioned imaginary or V — 1 of ordinary analysis. The letters i, j, &c., 
are such that every binary combination i 2 , ij, ji, &c., (the ij being in general not = ji), 
is equal to a linear function of the letters, but under the restriction of satisfying 
the associative law: viz. for each combination of three letters ij .k is — i.jk, so that 
there is a determinate and unique product of three or more letters ; or, what is the 
same thing, the laws of combination of the units i, j, k, are defined by a multiplication 
table giving the values of i 2 , ij, ji, &c. ; the original units may be replaced by linear 
functions of these units, so as to give rise, for the units finally adopted, to a multi 
plication table of the most simple form ; and it is very remarkable, how frequently in 
these simplified forms we have nilpotent or idempotent symbols (i 2 = 0, or i 2 = i, as the 
case may be), and symbols i, j, such that ij —ji = 0 ; and consequently how simple are 
the forms of the multiplication tables which define the several systems respectively. 
I have spoken of this multiple algebra before referring to various geometrical 
theories of earlier date, because I consider it as the general analytical basis, and the 
true basis, of these theories. I do not realise to myself directly the notions of the 
addition or multiplication of two lines, areas, rotations, forces, or other geometrical, 
kinematical, or mechanical entities ; and I would formulate a general theory as follows : 
consider any such entity as determined by the proper number of parameters a, b, c (for 
instance, in the case of a finite line given in magnitude and position, these might be 
the length, the coordinates of one end, and the direction-cosines of the line considered 
as drawn from this end) ; and represent it by or connect it with the linear function 
ai + bj + ck + &c., formed with these parameters as coefficients, and with a given set of 
units, i, j, k, &c. Conversely, any such linear function represents an entity of the kind 
in question. Two given entities are represented by two linear functions ; the sum of 
these is a like linear function representing an entity of the same kind, which may 
be regarded as the sum of the two entities ; and the product of them (taken in a 
determined order, when the order is material) is an entity of the same kind, which 
may be regarded as the product (in the same order) of the two entities. We thus 
establish by definition the notion of the sum of the two entities, and that of the 
product (in a determinate order, when the order is material) of the two entities. The 
value of the theory in regard to any kind of entity would of course depend on the 
choice of a system of units, i, j, k, ..., with such laws of combination as would give a 
geometrical or kinematical or mechanical significance to the notions of the sum and 
product as thus defined. 
Among the geometrical theories referred to, we have a theory (that of Argand, 
Warren, and Peacock) of imaginaries in plane geometry ; Sir W. R. Hamilton’s very 
valuable and important theory of Quaternions ; the theories developed in Grassmann’s 
Ausdehnungslehre, 1841 and 18G2 ; Clifford’s theory of Biquaternions ; and recent extensions 
of Grassmann’s theory to non-Euclidian space, by Mr Homersham Cox. These different 
theories have of course been developed, not in anywise from the point of view in 
which I have been considering them, but from the points of view of their several 
authors respectively. 
The literal symbols x, y, &c., used in Boole’s Laws of Thought (1854) to represent 
things as subjects of our conceptions, are symbols obeying the laws of algebraic com-