Royal Society. 37 



shire, which I have recently examined : but I must reserve 

 the communication for a future Number of your Annals. 

 Hampstead, Dec. 14, 1830. ROBERT BAKEWELL. 



P.S. I omitted to mention, that the bone found in the Not- 

 tingham sand-rock appeared partially mineralized, and much 

 resembled bones from some of the tertiary beds. 



XL Proceedings of Learned Societies. 



ROYAL SOCIETY. 



Nov. 18, A PAPER was read, entitled, " On the nature of ne- 



1830. gative and imaginary quantities." By Davies Gil- 



bert, Esq. President of the Royal Society. 



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

 given rise to much controversy, and has been involved in much un- 

 necessary mystery. Paradoxes and apparent solecisms, when in- 

 volved with facts and indubitable truths, will always be found, upon 

 accurate examination, to be near the surface, and to owe their ex- 

 istence either to ambiguities of expression, or to the unperceived 

 adoption of some extraneous additions or limitations into the com- 

 pound terms employed 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, 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 multiplication, 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 belong- 

 ing to the assumed unity, or universal antecedent of the ratios ; 

 and unlike signs, the contrary. 



Since either the one or the other of the arithmetical scales de- 

 rived from the two unities is in itself equally affirmative, but nega- 

 tive 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 apparent discrimination of the two scales is removed ; so that 

 the properties belonging to the two scales are interchangeable, and 

 all formulae become universally applicable to both, by changing the 

 signs according to the side in which the universal antecedent is 

 taken. Imaginary quantities, then, are merely creations of arbi- 

 trary 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 



