FOUNDATION OF ALGEBRA. 179 



to future addition, be considered as a new zero. We are now to assume 

 that, 



1. Parallelism and sameness of direction are meant to be identical 

 terms ; that is to say, the two directions conceivable on any one of two 

 parallels are severally the same as the two directions on the other. 



2. Every simple symbol represents a line given in length and direction : 

 thus a = b means that the lines a and b, equal in length, have also the 

 same direction. And the process implied in + a is the transference 

 of a point from the position to a given length in a given direction. 



We can now find the necessary meaning of (0 + a) + b ; necessary, 

 on the supposition that the technical algebra is 

 to become logical on the explanation of the sym- 

 bols before us. Let 0A and 02? represent the 

 lines symbolized by a and b : if then we take 

 A, at which we arrive by the process + a, as 

 a new zero, and proceed with it in the same 

 manner as in performing + b on the old zero, 



we draw AC parallel and equal to 02?, whence 0C being symbolized 

 by c, we have with reference to the first zero, 



+ c = (0 + a) + b = (0 + b) + a. 



I need not further dwell on the connection of addition and subtraction 

 in arithmetic with the processes called by the same names in this ex- 

 planation. I shall only here suggest that perhaps the words direct zero 

 process and inverse aero process might occasionally be found useful *. 



The usual method of defining the process of addition by reference 

 to the diagonal of a parallelogram is convenient, but destructive of all 

 true analogy. The fundamental theorem of statics suffers from the same 

 method of statement. 



I now proceed to the process of multiplication, which will readily 

 be seen to be connected with unity in precisely the same manner as is 

 addition with zero. If b be formed from unity by the train of processes 

 + 1 + 1 + 1, we consider a as a new unit, and let the symbol ba represent 



* In my Calculus of Functions (sect. 12, 13, 17) will be found some analogies connecting 

 simple addition with zero, and multiplication with unity. 



