35° Car not on the Theory of 



we have it entirely in our power to confider them either a:s 

 real quantities, or as abfolute nullities. The difference be- 

 tween thefe two ways of confidering this queftion, confifis in 

 this, that, by regarding evanefcent quantities as nullities, the 

 propofiiions, equations and refults, whatever they may be, 

 are always accurate and rigorous ; but hive a reference to 

 quantities which are creatures of the underftanding, and ex- 

 prefs the relations which exift between quantities which do 

 not themfelves exift *. On the other hand, by confidering 

 infinitely fmall quantities as having feme reality, the propo- 

 finons, equations and refults, whatever they may he, have 

 for their fuhjeet real quantities. But thefe laft proportions, 

 equations and refills are falfe, or rather imperfect, and be- 

 come exact in the end only in confequence of the compen- 

 fation of their errors, a compenfation, however, which is the 

 neceffary and infallible refult of the operations of the calculus. 



* Thus the ratios of the ordinal numbers (one, two, three, &c.) to 

 each other, while thofc numbers remain floating, fo to fpeak, in indeter- 

 minate abftra&ion, atid unapplied to any particular objefts of fenfe, may 

 be faid to " exprefs the relation-, which exift between quantities which do 

 not themfelves exift.'' Thus alio, if a body be fuppofid to fall from any 

 moderate height, its velocities at any two points (refiftance apart) will 

 ha\e to each other the ratio of the'fquare roots of the fpaces fuppofed to 

 bedefcribedj although no body ever aclaally fell, or perhaps ever will 

 actually fair, fiom that precJfe height. Thefe examples, it is hoped, will 

 prevent readers who arc not much accuftomed to fuch fpeculations, from 

 rafhly elwgin - Mir author with abferdity, in talking of the relations be- 

 tween quantities which do not themfelves exift; that is, which have no 

 eitfterice in external nature. For the truth is, and a furprifihg, unac- 

 countable truth it appears to many beginners, that the objedte of Pure 

 Mathematics, though originally abftiadied. or copied, from external ob- 

 jects, have no exiftence out of the minds which conceive them ; and hence 

 proceeds all that accuracy for which thofe fciences are jufrly valued. The 

 inaccuracy of the figures* motions* feci of external objects induces a corrc. 

 fponding inaccuracy into Mixed Mathematics. — The inaccuracy of lan- 

 guage has an analogous effect in metaphyfics ; for metaphyfical relations 

 and deductions may be perfectly accurate in the mind, yet few of them, 

 can be adequately and unexceptionably cxprcflcd, for want of an accurate, 

 unambiguous language. Hence the endlefs difputes with which men 

 unhappily difpofed to cavil, and who affect to doubt of every thing, never 

 ceafe to embroil that important, and otherwile not unpleafant, region of 

 philofophy. — \V. D. 



47, The 



