LIMITS. 



•riuantity, and u any increment of a? : then the fimultaneoiis out them both; andhence the ofculator}' circle is fuppofej 

 increments of .r.r and rt A' will be 2«o+ooandao, which to have an equal curvature with the curve at that point. See 

 ■ "' "" •■- ' ' --' ■ -:-.-- '1^1-;- Mr. Maclaiirin's Flux. art. ^g.y. 



■ara in the fame ratio to each other as 2 x + o is to a. This 

 ratio o( 2 X- + to a continually decreafes while o dccrcafcs, 

 and is always greater than the ratio of 2 x to a, while o is 

 any real increment ; but it is manifeft, that it continually 

 approaches to the ratio of 2 x to a as its limit ; whence it 

 ■follows, that the fluxion of xx is to the fluxion of tin, as 

 2 .r is to a. If x be fuppofed to flow uniformly, ax will 

 likewife flow uniformly, but xx with a motion continually 

 accelerated ; the motion with which a x flows, may be 

 •mj-afured br a o ; but the motion with which 2 .v flows is 

 not to be meafured by its increment 2x0 + 00, but by the 

 •part 2. f 9 only, which is generated in confequence of that 

 motion; and the part 00 is to be rejefted, becaufe it is 

 generated in confequence only of the acceleration of the 

 motion with which the variable fquare flows, wliile n, the 

 increment of its fide, is generated; and the ratio of 2.1-0 

 to a <; is that of 2 >• to a, which was found to be the limit 

 of the ratio of the increment 2x0 + 00 and ao. See 

 Fluxion-. 



It is cbjeflcd aijainft fir Ifaac Newton's method of in- 

 vcftigating this Imiit, that he firft fuppofes tliat there are 

 inorement'ri ; that when it is faid iel the increment I'aniji, the 

 former fuppofition is deilroyed, and yet a confequence of 

 this fuppofition, J. e. an expreffion got by virtue thereof, is 

 retained. But the fuppofitions that are made in this method 

 of inveftigating the limit arc not fo contradicfory as this ob- 



fuppofes that 



jeftion feems to imoort. He firft luppoles that tliere are ■ j • u 1 n- - 



"increment^ generated, and reprefents their ratios by that of C"'"^-'d« with the elhplis ; but this fcems a confeq 

 " " ■:h is given fo as not to vary with the 'anguagc of nnhnitchmals. It would be more 



two quantities, one of which is gi' 



the increments. If he had afterwards fuppofed that no in- 

 crements had been geiicr.ited, this indeed had been a fup- 

 pofition dirtflly contradiAory to the former But when he 

 Xuppofrs thole increments to be diminilhed till they vanifli, 

 this fuppofition lurely cannot be faid to be fo contradictory 

 to the former as to hinder us from knowing what was the 

 ratio of thofe increments, at any term of the time, n hile 

 they lud ,i real exiftence ; how this ratio varied, and to what 

 limit it approached while the increments were continually 

 diminifhed: on the contrary, this is a very concife and jull 

 method of .hfcov -ring t!ie limit which is required. 



Jt is to be obfjrved, that the limiting, prime, or ultimate 

 ratio of increments, (trittly fpeaking, is" not tlie ratio ot any that one fliall be greater and one lefs than the root required 



Now if we conceive the ofculatory circle at the end of the 

 great a.\is of an ellipfis, it will fall entirely within the ellipfis ; 

 and the curvature of the ellipfis and ofculatory circle may 

 both be faid to be limits of the curvatures of all the circles 

 falling wholly within, and touching the ellipfis at the end of 

 its great axis. But the term limit will not in both ca.Q-i 

 have prccifely the fame meaning; for the ofculatory circle 

 is a limit mcluji-ve, being the lait of the circles limited ; and 

 the ellipfis is a limit exchfive, none of the circles limited 

 ever coinciding with it. As to the circles which fall wholly 

 without the ellipfis, and touch it at the end of its great axis, 

 they have no limit inclttfi-ve, no circle touching the cllipfii 

 fo clofely, that no other can pafs between ; the or.Iy huiit 

 here is exclu/ive, the ellipfis ilfelf. 



The contrary of tiiis happens at the end of the lefl^er axi.'!. 

 At any other point of the ellipfis, one half of every ofculatory 

 circle is a limit incluftvs of the femirircles that fall within, 

 and the other half is a limit e.xclufivs of thofe that fall 

 without. 



_ May we not aflv, if a curve is the limit of its infcribed or 

 circumfcribed polygons in any other fenfe, than the curva- 

 ture of the elliplis is the limit of the curvatures of the circles 

 before defcribed, which approach nearer and nearer to the 

 curve, but never coincide with it .' It is true we hear it 

 often faid, that the ofculatory circle is equicurval, and fo 



uence of 

 accurate 

 to fay. that the curvature of the eHipfis is the hmit exclu/ive 

 of all the before mentioned circles, and that the ofculatory 

 circle is their limit indiifive. That excellent geometer, Mr. 

 Simfon, in his Conic Seftions, lib. v. prop. 36. cor. fays 

 only, after demonltrating the chief property of the ofcula- 

 tory circle, that eanciem habere cum Jl-f.ior.e conica curvaluram 

 Jicitiir, giving this only as an appellation, but not as a pro- 

 pofiion. See on the fubjeft of this article, Robins's Difc. 

 on Fluxions, in his Trafts, vol. li. 



LiMlT.s of the Roots of an Equation.— 'V'c have already 

 obferved, that fey finding the limits of the roots of an ecua- 

 tion, is to be underfiood the finding of tuo fuch numbers, ■ 



real increments whatfoever. But as the tangent of an arch 

 is the right line that limits the pofition of all the fecants that 

 tan pafs through the point of contatt, though, llrictly 

 fpeaking, it be no fecant ; fo a ratio may hmit the variable 

 ratios of the increments, though it cannot be faid to be the 

 ratio of any real increments. The ra.io of the generating 

 motion may be likewife faid to be the lail or ultimate ratio 

 of the increments, while they are fuppofed to be diminiilied 

 till they vanilh, for a like reafon. It may jull be added, 

 that there being two cafes of variable quantities and ratios 

 tending to a limit, it might have conduced to pcrfpicuity, 

 and preventing difputei, to have dilfinguifhed thcfe different 

 limits by fome addition. As m the firll cafe to have called 

 It a limit or ultimate ratio i.irlufve ; becaufe the limit is the 

 lad of the quantities or ratios limited : and in the fecond to 

 have called it a limit or ultimate ratio exclufve ; becaufe the 

 quantities limited never attain to the limit, though they .ap- 

 proach t'l it indefinitely. 



This dillinftion may perhaps receive fome farther ilhiftra- 

 tion from the following example. It is known that the 

 olculatory circle is a circle that touches a curve fo clofily 

 that no other circle can be drawn through the point of con- 



iad between them, all other circles paflitng within or with- refults with contrary lign 



by which means an approximation is evidently made to- 

 wards the true root, and the nearer thefe limits approach 

 towards each other, fo much the more accurate will be the 

 approximation. I.a Grange, in his " Traite de la Rcfolution 

 iiumerique des Equation,<!," has carried the method of 

 limits to itj utmoit poihble perfoftion, by {hewing, in all 

 equations, how the limits of each of its roots may be afcer- 

 tained, and has fliewn, tliat the method of approximation 

 employed by Newton, and in fact every method, except 

 that ot his own, is defective in this refpeif, iiir.. that be- 

 tween the hmits afcertained in their operation, there may 

 be one, two, or more roots, and confequently, that tliey are 

 not neceflfarily the limits of one root, but merely the limits 

 between whicli one at leall of the real roots of the equation 

 muft lie. The nature of this article will not admit of our 

 entering into an explanation of the procel's of this celebrated 

 analyll ; we can, therefore, only refer the reader to the 

 work itlelf, and mult content ourfelvcs in this place with 

 giving a few of the moil remarkable cafes relating to the 

 limits of the roots of an equation. 



I. If we can find two quantities, which, being fnbftitutcd 

 for the unknown quantity in any equation, give two 



, then will thcfe two quantitie» 

 >i- 2 be 



