■'I 



A Paper was read, entitled, " On the nature of negative and ima- 

 ginary quantities." By Davies Gilbert, Esq. President of the Royal 

 Society. 



The object of this paper, the author observes, is one that has given 

 rise to much controversy, and has been involved in much unnecessary 

 mystery. Paradoxes and apparent solecisms, when involved with 

 facts and indubitable truths, will always be found, upon accurate 

 examination, to be near the surface, and to owe their existence either 

 to ambiguities of expression, or to the unperceived adoption of some 

 extraneous additions or limitations into the compound terms em- 

 ployed for definition, and which are subsequently taken as constituent 

 parts of their essence. 



The first misapprehension pointed out, is that of considering any 

 quantity whatever as negative per se y and without reference to an- 

 other opposed to it, which has previously been established as positive. 

 In order to avoid previously formed associations of ideas, the author 

 prefers employing in his reasonings on this subject, the symbols (a) 

 and (b) to express this quality of opposition, rather than the usual 

 ones of plus and minus. By the aid of this notation he is enabled to 

 present, in its full generalization, the law of the signs in multiplica- 

 tion, — a process which, it is well known, is founded solely upon the 

 principle of ratios ; and to show that like signs invariably give the 

 sign belonging to the assumed unity, or universal antecedent of the 

 ratios j and unlike signs, the contrary. 



Since either the one or the other of the arithmetical scales derived 

 from the two unities is in itself equally affirmative, but negative with 

 relation to the other, it follows, that by using the scale of (6), all even 

 roots in the scale of (a) will become imaginary, and thus the appa- 

 rent discrimination of the two scales is removed ; so that the proper- 

 ties belonging to the two scales are interchangeable, and all formulae 

 become universally applicable to both, by changing the signs accord- 

 ing to the side in which the universal antecedent is taken. Imaginary 

 quantities, then, are merely creations of arbitrary definitions, endowed 

 with properties at the pleasure of him who defines them ; and the whole 

 dispute respecting their essence turns upon the very point that has 

 been contested from the earliest times, between the hostile sects of 

 realists and nominalists. 



It is now, however, universally agreed, that all abstractions and 

 generalizations are mere creatures of the reasoning faculty, existing 

 nowhere but in the mind contemplating them. Such, in algebra, 

 are the supposed even roots of a real quantity, taken in the scale 

 opposite to that which has given the universal antecedent : the 

 sign indicating the extraction impossible to be performed, veils 

 the real quantity, and renders it of no actual value until the sign 

 is taken away by an involution, the reverse of the supposed opera- 

 tion which the sign represents; although the quantity itself is, in 

 the mean time, by its arbitrary essence, made applicable to all the 

 purposes for which real quantities are used, in every kind of for- 

 mula. 



Several illustrations of these views of the nature of imaginary 



