336 Car not on the Theory of 



equation contains affigned quantities only, and, confequentW, 

 it cannot be what I call an imperfect equation, q. e. d. 



34. Theorem III. — Every imperj eel equation which hath 

 undergone fuch transformations as are indicated in the f.rjl 

 theorem, and from which all unajjigned quantities have been 

 eliminated, by thofe transformations, will be neceffarily and 

 rigoroufly exacl. 



For, by the fir ft theorem, the equation cannot be abfb- 

 lutely falfe ; and, by the fecond, it cannot be imperfect ; 

 therefore it is neceiiarily and rigoroufly exa<S. q.e.d.* 



35. Corollary. — All that hath been faid on the fubject 

 of imperfect equations, ought to be underftood of all the pro- 

 portions, propofitions, and reafonings whatsoever, which can 

 be exprefTed and delivered by fuch equations. 



The leading Principle of this Analyfis. 



36. Scholium. — Such are the general principles into 

 which the theory of the Infinitefimal Calculus is rcfolvable. 

 From thefe principles it appears, that if, after exprefling the 

 conditions of a problem in imperfect equations, we arrive, by 

 means of fuch transformations as are indicated in the firrt 

 theorem, at the elimination of all auxiliary or unaffigned 

 quantities, a compenfation of errors muft neceffarily have 

 taken place, in the courfe of the procefs. It further apptars, 

 that the advantage of the Infinitefimal Calculus confifts in 

 this, That, the conditions of a queftion being often very diffi- 

 cult to be exprefl'ed accurately by rigorous equations, it i* 

 eafv to do it by imperfect equations, from which as certain 

 Tefults can be derived, as if the original equations had been 

 perfectly accurate ; and this by the fimple expedient of eli- 

 minating the quantities whofe prefence occafioned the errors. 



* The word falfe, as before intimated (in the Note on § 9), feems by 

 far too ftrong a term to be applied, in any allowable fenfe, to kich equa- 

 tions as the author is conhdering. He lhould have defined it, and adhered 

 to his definition ; for, in art. 31, he uksfa/jl and impoffd as convertible 

 terms, and, in art. 3Z and 33, he takes them in oppofite fenfes. This 

 ijnfteadincfs renders his ireaning fomewhat ambiguous. But if, by falfe 

 or imperfeft equations, he uniformly mean fuch whofe fides differ infinitely 

 little from equalin , then his whole meaning becomes clear, and his three 

 SUiorems almoft fdf-cvident.— \V. D. 



The 



