490 Prof. Donkin o?i the Geometrical Interpretation 



of operation, or (to adopt Sir W. Hamilton's convenient 

 term) the operand, is in general not represented by any sym- 

 bol at all, but is to be iniderstood. Thus the symbol ab 

 represents the successive performance of the operations 6, a 

 upon the operand. Neither operations nor operand are de- 

 Jined, further than definition is involved in the assumed laws 

 of the combination of symbols. If we assume ab = ba, we limit 

 to a certain extent the choice of meanings for the operatio7is 

 a and b ; and if we assume that the arithmetical equation 

 7w + w = 5 shall be true in symbolical algebra (where m and n 

 are absolute numbers, and s their sum), we limit the choice 

 of the operand', for let it be a ; then the above equation sig- 

 nifies 7wa4-wa = 5a, and the meaning of a must be such that 

 this may be capable of an intelligible and consistent interpre- 

 tation. It is not necessary for our present purpose to attempt 

 a complete discussion of the laws of the usual algebraical 

 symbols. The following remarks, however, may serve to elu- 

 cidate the processes to be explained hereafter. 



The symbols + and — are used in algebra in two essen- 

 tially different ways. First, they are used, as in arithmetic, 

 to indicate addition and subtraction. Thus « + i denotes a 

 certain resultant formed by separately performing the opera- 

 tions a and b on the operand, and adding the result of the 

 latter operation to that of the former. The meaning of ad- 

 ditiofi cannot be defined till the nature of the operand is de- 

 termined. The same may be said of a — b, mutatis mutandis. 

 The equations 



a+{b + c) = a-}-b ■\-c, &c. 



suggested by arithmetic as convenient assumptions, suggest 

 in their turn further assumptions ; namely, that if + and — 

 should ever be used as true symbols of operation, so that -Ka 

 should denote the result of performing the operation + upon 

 the operand Zi, and +— « should denote the performance of 

 the successive operations a, — , + upon an operand (not ex- 

 pressed), then the laws of their combinations should be 



Such combinations do not really occur at all in thej^r^^ use 

 of these symbols. The expression a—{b — c) only suggests, 

 but does not really contain the combination . 



We now come to the second way in which + and — are 

 used, namely, as true symbols of operation, defined by the 

 laws of combination just stated. 



Let us for the present distinguish this use of the symbols 

 by inclosing them in brackets. Then the following equations 



