36 rREPORT—1883. 
the order is material) is an entity of the same kind, which may be re- 
garded 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, 7,j,/. . 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 atheory (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 1862 ; 
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 2, 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 combination (the distributive, commutative, and associative 
laws) but which are such that for any one of them, say 2, we have x—2?=0, 
this equation not implying (as in ordinary algebra it would do) either 
#=0 or else z=1. In the latter part of the work relating to the Theory 
of Probabilities there is a difficulty in making out the precise meaning of 
the symbols, and the remarkable theory there developed has, it seems to 
me, passed out of notice, without having been properly discussed. A paper 
by the same author, ‘Of Propositions numerically definite ’ (‘ Camb. Phil. 
Trans.’ 1869) is also on the borderland of logic and mathematics. It 
would be out of place to consider other systems of mathematical logic, 
but I will just mention that Mr. C. 8. Peirce in his ‘ Algebra of Logic’ 
(American Math. Journal, vol. ili.) establishes a notation for relative 
. terms, and that these present themselves in connection with the systems 
of units of the linear associative algebra. 
Connected with logic, but primarily mathematical and of the highest 
importance, we have Schubert’s ‘ Abziihlende Geometrie’ (1878). The 
general question is, How many curves or other figures are there which satisfy 
given conditions ? for example, How many conics are there which touch 
each of five given conics? The class of questions, in regard to the conic 
was first considered by Chasles, and we have his beautiful theory of the 
characteristics x, v, of the conics which satisfy four given conditions ; 
questions relating to cubics and quartics were afterwards ‘considered by 
Maillard and Zeuthen; and in the work just referred to the theory has 
become a very wide one. The noticeable point is that the symbols used 
by Schubert are in the first instance, not numbers, but mere logical 
symbols: for example, a letter g denotes the condition that a line shall cut 
