REPORT ON CERTAIN BRANCHES OF ANALYSIS. Z^t 



a + b \/ — 1 and a — b \^ — 1 may be considered as repre- 

 senting respectively a single line, equal in magnitude to 

 j\/(a^ + b^) *, and affected by the sign cos 9 + -/ — 1 sin 9 in 

 one case, and by the sign cos 9 — -v/ — 1 sin 9 in the other ; or 

 as denoting the same lines through the medium of the opera- 

 tions denoted in the one case by +, and in the other by — , 

 upon the two lines at right angles to each other, which are de- 

 noted by a and b \^ —1. 



We have spoken of the signs of operation + and — , as di- 

 stinguished from the same signs when used as signs of affection, 

 and we have also denominated a + b ^ — 1, and a — 6 V — 1, 

 the sum and difference of a and b '^ —\, though they can no 

 longer be considered to be so in the arithmetical or geometrical 

 sense of those terms ; but it is convenient to explain the mean- 

 ing of the same sign by the same term, though they may be 

 used in a sense which is not only very remote from, but even 

 totally opposed to f, their primitive signification; and such a 

 licence in the use both of signs and of phrases is a necessary 

 consequence of making their interpretation dependent, not upon 

 previous and rigorous definitions as is the case in arithmetical 

 algebra, but upon a combined consideration of their symbolical 

 conditions, and the specific nature of the quantities represented 

 by the symbols. It is this necessity of considering all the re- 

 sults of symbolical algebra as admitting of interpretation sub- 

 sequently to their formation, and not in consequence of any 

 previous definitions, which places all those results in the same 

 relation to the whole, as being equally the creations of the 

 same general principle : and it is this circumstance which jus- 



• The arithmetical quantity fjip? -\- h^) has been called the modulus of 

 a + b/^ — 1 by Cauchy, in his Cours d' Analyse, and elsewhere. It is the single 

 unaffected magnitude which is included in the affected magnitude a+h V — 1 '• 

 conversely the affected magnitude (cos 6 + -v/— 1 sin 6) aJc^ + 6^ is reducible 

 to the equivalent quantity a + 6 V — !> if cos 6 = — „ , and therefore 



sin 6 = — ==^ . 



^a^ + 62 



+ The sum of a and — 6, or o + (— 6), is identical with the difference of a 

 and b, or with a — b. The term operation, also, which is applied generally to 

 the fact of the transition from the component members of an expression to the 

 final symbolical result, will only admit of interpretation when the nature of the 

 process which it designates can be described and conceived. In all other cases 

 we must regard the final result alone. Thus, if a and b denote lines, we can 

 readily conceive the process by which we form the results a + 6 and a — 6, at 

 least when a is greater than b. But when we interpret o + 6 V— 1 to mean a 

 determinate single line with a determinate position, we are incapable of con- 

 ceiving any process or operation through the medium of which it is obtained. 



q2 



