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It may appear flrange after all that has been written upon the method of prime and ulti- 

 mate ratios (or perhaps more p'Opoily fpeaking of limits) that with refpeifl to a de- 

 monflration in which that method ;s ufed, more fliould be deemed neceflary than merely a 

 reference to it, as already eftabli(hed upon logical principles. However of late, eminent ma- 

 thematicians both at home and abroad have again called in quellion the principles of the 

 doflrine, and rather flian ufe a method fo admirably adapted to the purpofes intended by 

 the illuftrious inventor, compendium of invention and demonftration, have had recourle 

 to methods which fall little (hort of the tedious demonftrations of the ancients. From 

 hence it Ihould feem that its principles have not been put by its defenders, and com- 

 mentators in that clear light in which they ought to be placed ; at leaft, judging from 

 the eminence of its prefent opponents. Without acceding to fuch an opinion, there can, I 

 think, be no impropriety in endeavouring to fliew that a demonftration in which that method 

 is ufed, is unimpaired thereby. Principally with this intention the following fummary of the 

 doflrine is given, in which is more particularly noted that part of it introduced into the above 

 demonftration. The whole is contained in effefl in the fiift fection of the Principia. 

 But the great author, preffing forward to more important matters, did not flop to give that pre- 

 cifion to the doflrine which has been Gnce required by its opponents. He was fatisfied 

 with anticipating fome objeiflions, and giving fome cautions, which, if fully attended to, 

 wilienable any one to place the method on the moft folid foundations. 



It being granted that the doftrine is founded upon logical principles, no objeiflion can 

 be made to its ufe in demonftrations either purely geometrical or analytical. 



The fuperiority it poffeffes over other methods whether ancient or modern, with ref- 

 pea to facility of invention and demonftration, appears moft fatisfadorily by comparing 

 the proceffes inftituted according to the refpeflive methods. 



This fuperiority I confider as a fufficient apology for introducing the controverfy 

 refpefting its principles. It if not convenient here to enquire particularly how the 

 opponents of the doftrine have underftood thofe points, which they conceive are un- 

 anfwerable objeflions. If I do not deceive myfelf, the method as deduced from what 

 the illuftrious author has left us, is capable of the moft logical proof. What is here 

 given is by no means intended as a complete fummary. If it ferves to (hew that the 

 above application of the method is logical, and alfo in any degree to (hew, that the 

 general principles of the method are neither obfcure nor inaccurate, the purpofe is an- 

 fwered. Not that 1 can flatter myfelf with the hope of advancing much new illuftration, 

 but as the old objeftlons have again been brought forward in another iliape, I thmk it 

 incumbent upon thofe, who imagine a method fo important to be logically founded, to 

 endeavour, if they cannot advance new defences, to bring forward the old ones, in a 

 manner likely to be moft effeilual. 



The method of limits may be confidered as reducible to the following heads. 



I. The 



