196 THIRD REPORT 1833. 



by the independent signs * + and — , which no longer denote 

 operations, though they may denote affections of quantity. It 

 appears likewise that + c is identical with c, but that — c is a 

 quantity of a different nature from c : the interpretation of its 

 meaning must depend upon the joint consideration of the spe- 

 cific nature of the magnitude denoted by a, and of the symbolical 

 conditions which the sign — , thus used, is required to satisfyj-. 

 In a similar manner, the result of the operation, or rather 

 the operation itself, of extracting the square root of such a 

 quantity as « — 6 is impossible, unless a is greater than b. To 

 remove the limitation in such cases, (an essential condition in 

 symbolical algebra,) we assume the existence of a sign such 

 as -v/ — I ; so that if we should suppose b =■ a -\- c, we should 



get V {a — b) = '/{« — (« + (?)} = V [a — a—c) = V [ — c) 



= V' — \ cX- In a similar manner, in order to make the ope- 

 ration universally applicable, when the ?«"' root of a — 6 is 

 required, we assume the existence of a sign v^ — 1, for which, 

 as will afterwards appear, equivalent symbolical forms can al- 

 ways be found, involving a/ — 1 and numerical quantities. 



By assuming, therefore, the independent existence of the 

 signs +, — , X/\, and v^ — 1, (1)", and ( — 1)''§, we shall obtain 

 a symbolical result in all those operations, which we call addi- 

 tion, subtraction, multiplication, division, extraction of roots, 

 and raising of powers, though their meaning may or may ?iot be 

 identical with that which they possess in arithmetic. Let us 

 now inquire a little further into the assumptions which deter- 

 mine the symbolical character and relation of these funda- 

 mental operations. 



The operations called addition and subtraction are denoted 

 by the signs + and — . 



They are the inverse of each other. 



* That is, not preceded by other symbols as in the expressions a — c and 

 a + c. 



\ Amongst these conditions, the principal is, that if — c be subjected to 

 the operation denoted by the sign — , it will become identical with + c: thus, 

 a — ( — c) ■:= a-\- c. It does not follow, however, that the sign — thus used, 

 must necessarily admit of interpretation. 



X The symbolical form, however, of this and of similar signs is not arbi- 

 trary, but dependent upon the general laws of symbolical combination. 



§ I do not assert the necessity of considering such signs as V — 1. (1)", 

 ( — 1)», as forming essentially a part of the earliest and most fundamental as- 

 sumptions of algebra : the necessity for their introduction will arise when 

 those operations with which they are connected are first required to be con- 

 sidered, and will in all cases be governed by the general principle above men- 

 tioned. 



