ISO 



PROFESSOR DE MORGAN, ON THE 



the same operation on this new unit, or + a + a + a. Similarly, if by 

 the line 1 we mean a line having the length and 

 direction 1, and OA and OS by a and b, and if we 

 take Oias a new unit, and perform on it the opera- 

 tions by which we pass from 01 to 02?, that is, take 

 an angle AoC equal to 10 J?, and let 0C be in 

 length the result of the arithmetical operation on Oi 

 and OB, — then 0C must be represented by ab. The 

 processes of multiplication and division might be called 

 the direct and inverse unit processes. 



There is now nothing particular to be said about the four operations, 

 or the simple powers, with positive or negative, whole or fractional, real 

 exponents, or any combinations of them. The interpretation of a + b\/- 1 

 follows in the usual manner. 



In illustration of the propriety of considering symbols as functions of 

 zero or unity for purposes of addition or multiplication, it may be advanced 

 that unless we do so, we change the meaning of the terms direct and in- 

 verse as we proceed from the lower to the higher parts of the science. Un- 

 questionably, if ever we have a right to assume a clear conception of this 

 distinction, it is in the comparison of addition with subtraction, and of mul- 

 tiplication with division ; but for all that, a + x and a - x are not inverse 

 functions, considered with respect to x, though they are so with respect to a. 

 And similarly of ax and a -=- x. When we come to the symbol x", then, and 

 then only, do we begin to describe inversion correctly : for we usually 

 consider this as a function of x and not of n, when we assert Xn to be the 

 inverse. But if we considered this as a function of n, the inverse would be 

 log n : log x. 



The separation, as it is called, of the symbols of operation and quantity, 

 is a method of explaining technical algebra as simple in its character as the 

 preceding. Let the fundamental object of conception be (p (x — nh), n 

 being infinite, which stands in the place hitherto occupied by 0. Let* 

 2 (j> (x + ah) represent the train of operations by which we pass from 

 (\>(x — oo h) to cf> (x + a — lh), or 



* In the common method of treating this subject, the inverse symbol is made to precede 

 the direct one. Several adaptations of notation are necessary before we can exactly represent 

 the common methods. 



