i 3 4 On the USE of NEGATIVE 



the other is taken in an oppofite fituation j and that the fecond 

 and fourth equation ought to be confidered as one, and as be- 

 longing to the fame ftraight line referred to the fame axis. 



EVEN long after this time, we find, that Dr WALLIS, al- 

 though, in his Algebra, he confiders at fome length the meaning 

 and ufe of negative qviantities, yet, in his Arithmetic of Itifi- 

 mtes, falls into a ftrange miftake, which leads us to fufpeft, 

 that his notions on this fubjecl: were not perfectly clear. For, 

 obferving that his general expreffion for hyperbolic areas would, 

 in certain cafes, be a fraction with a negative denominator, he 

 did not perceive, that this exprefled the area on the other fide 

 of the ordinate, inftead of the area fought ; but he contents 

 himfelf with faying, that the denominator of the fraction be- 

 ing lefs than nothing, denoted the area to be more than infinite. 



THAT even fome time afterwards, the ufe of negative quan- 

 tities had not become familiar, appears from the aftonifhment 

 which Dr HALLEY exprefles at his own difcovery of a formula, 

 which, by the mere change of the figns, gave the focus both of 

 converging and diverging rays, whether reflected or refradled 

 by convex or concave fpecula or lenfes. And Mr MOLYNEUX 

 fpeaks of the univerfality of HALLEY'S formula as fomewhat 

 that refembled magic. t 



THUS it appears to have been long, before mathematicians 

 ventured to employ negative quantities fb freely, as we now per- 

 ceive them to be employed. The reafon of which probably 

 was, that no fatisfadlory account had been given of the 

 grounds upon which the conclufions drawn from them are 

 founded. But the confiftency of thefe conclufions, with all the 

 mod indifputable truths of the mathematical fciences, and the 

 great beauty and advantage, derived from the very general fo- 

 lutions which are thus obtained, gradually eftablifhed their ufe. 



STILL, however, a complaint remains, which appears to be 

 too well founded, that the Method of negative quantities, as 

 has been the cafe with fome other rules of the art, is fupport- 



ed, 



