ADDRESS. 35 
cases, difficult to draw the line; we do in ordinary mathematics use 
“symbols not denoting quantities, which we nevertheless combine in 
_ the way of addition and multiplication, a +b, and ab, and which may be 
“such as not to obey the commutative law ab =a, in particular this is or 
“may be so in regard to symbols of operation; and it could hardly be said 
that any development whatever of the theory of such symbols of opera- 
tion did not belong to ordinary algebra. But I do separate from ordinary 
‘mathematics the system of multiple algebra or linear associative algebra, 
‘developed in the valuable memoir by the late Benjamin Peirce, 
‘linear Associative Algebra’ (1870, reprinted 1881 in the American 
Journal of Mathematics, vol. iv. with notes and addenda by his son, C. S. 
Peirce) ; we here consider symbols A, B, &c. which are linear functions of 
a determinate number of letters or units 7, J, k, l, &e.. with coefficients 
which are ordinary analytical magnitudes, real or imaginary (viz. the 
coefficients are in general of the form # + ty, where iis the before-men- 
tioned imaginary or ./—1 of ordinary analysis). The letters i, 7, &c., 
are such that every binary combination 7, ij, ji, &e. (the ij being in 
general not = 7), is equal to a linear function of the letters, but under 
the restriction of satisfying the associative law: viz. for each eombina- 
tion of three letters ij.k is = ijk, 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, , are defined by a multiplication 
table giving the values of #, 4, jt, &e.; the original units may be replaced 
by linear functions of these units, so as to give rise, for the units finally 
adopted, toa multiplication table of the most simple form; and it is very 
remarkable, how frequently in these simplified forms we have nilpotent or 
idempotent symbols (i? =0, or #?=7 as the case may be), and symbols #, fs 
such that 7j=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 
o myself directly the notions of the addition or multiplication of two 
» areas, rotations, forces, or other geometrical, kinematical, or 
/mechanical entities; and I would formulate a general theory as follows : 
jeonsider any such entity as determined by the proper number of para- 
‘meters a, b, c, (for instance, in the case of a finite line given in magni- 
ude and position, these might be the length, the coordinates of one end, 
nd the direction-cosines of the line considered as drawn from this end) ; 
nd represent it by or connect it with the linear function ai+bj+ck+ &e. 
ormed with these parameters as coefficients, and with a given set of 
nits, 7, j, k, &e. Conversely, any such linear function represents an 
ntity of the kind in question. Two given entities are represented by 
wo linear functions ; the sum of these is a like linear function representing 
entity of the same kind, which may be regarded as the sum of the 
WO entities ; and the product of them (taken in a determined order, when 
D2 
