356 . - PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



inches long, like those already erected: and, 6. That these copper terminations 

 be very finely tapered, and as acutely pointed as possible. 



We give these directions, being persuaded, that elevated rods are preferable to 

 low conductors terminated in rounded ends, knobs, or balls of metal ; and con- 

 ceiving that the experiments and reasons, made and alleged to the contrary by 

 Mr. Wilson, are inconclusive. 



March 12, 1778. J. Pringle, p.r.s.; W. Watson; H. Cavendish; W.Henly; 



S. Horsley; T. Lane; Mahon; E. Nairne; J. Priestley. 



Xf^J. On the Arithmetic of Impossible Quantities. By the Rev. John Play/air * 



A.M. p. 318. 



The paradoxes which have been introduced into algebra, and remain unknown 

 in geometry, point out a very remarkable difference in the nature of those 

 sciences. The propositions of geometry have never given rise to controversy, 

 nor needed the support of metaphysical discussion. In algebra, on the other 

 hand, the doctrine of negative quantities and its consequences have often per- 

 plexed the analyst, and involved him in the most intricate disputations. The 

 cause of this diversity, in sciences which have the same object, must no doubt 

 be sought for in the different modes which they employ to express our ideas. 

 In geometry every magnitude is represented by one of the same kind; lines are 

 represented by a line, and angles by an angle. The genus is always signified by 

 the individual, and a general idea by one of the particulars which fall under it. 

 By this means all contradiction is avoided, and the geometer is never permitted 

 to reason about the relations of things which do not exist, or cannot be exhibited. 

 In algebra again every magnitude being denoted by an artificial symbol, to which 

 it has no resemblance, is liable, on some occasions, to be neglected, while the 

 symbol may become the sole object of attention. It is not perhaps observed 

 where the connection between them ceases to exist, and the analyst continues to 

 reason about the characters after nothing is left which they can possibly express: 

 if then, in the end, the conclusions which hold only of the characters be trans- 

 ferred to the quantities themselves, obscurity and paradox must of necessity ensue. 

 The truth of these observations will be rendered evident by considering the na- 

 ture of imaginary expressions, and the different uses to which they have been 

 applied. 



2. Those expressions, as is well known, owe their origin to a contradiction 

 taking place in that combination of ideas which they were intended to denote. 

 Thus, if it be required to divide the given line ab (fig. 2, pi. 4) = a in c, so 

 that AC X CB may be equal to a given space b-, and if ac = x, then x =^ ^a + 

 i\/ {^a- — b'-); which value of x is imaginary when b'^ is greater than -\d'; now 



* Now professor of natural philosophy in the uiii\eisily of Edinburgh. 



