338 Mr, Graves's Explaticttion of a remarkable Paradox 



tesimally imaginary. Though this course may seem to savour 

 of the system " ignotum 'per ignotiiis" it is no less true than 

 singular that in this instance a difficulty more peculiarly af- 

 fecting reals receives light from the consideration of iniagi- 

 naries. To exhibit x therefore in its most general form ad- 

 mitted in algebra, including imaginaries as well as reals, let 



a; = j/ + i/^ z (5.) 



y and s, which I call the " constituents" of x, being inde- 

 pendent quantities, positive, negative, or + or — 0. Then 

 we shall have 



Here it is important to remark that the second constituent 

 of ^' is always of a different sign from the corresponding con- 

 stituent of — , for if 2 be positive, T~~Z'^ '^'^^ t)e negative, 



and vice versa. 



I must now lay down some necessary definitions of the no- 

 tation employed, preparatory to the statement of a proposi- 

 tion which will be found further on, and I must here add 

 that, in resorting to a new specificatory or individualizing no- 

 tation, I have unwillingly yielded to necessity, from finding 

 that the indeterminateness of ordinary exponential and in- 

 verse trigonometrical expressions almost always occasioned 

 perplexity and frequently led into material errors of rea- 

 soning ; not that I was unaware of the repugnance which any 

 new notation has to encounter, and the increased difficulties 

 it opposes to the reception of any new theory. Let the sym- 

 bol \/, placed before a positive quantity, be appropriated to 



denote the positive square root ; then ■■ - , — will denote 1 or 



— 1, according as s: is positive or negative. When x is real, 

 and therefore s = 0, we have no more right to consider z 

 positive than negative, unless it be known absolutely that the 

 variable x is in such a varying state as to be about to become 

 x + dy+ \^ — 1 dz, the sign of the real infinitesimal </ 2 being 

 known ; but still, when x is real, and nothing else is known 

 with reference to its varying state, this at least we know al- 

 ternatively, that if we consider — -^= = 1, there would be a 



metaphysical impropriety, a wanton violation of analogy and 

 oontinuity, in not considering j^- = —1, ViXidi vice versd. 



