LIMITS. 



ratios, wliea if is greater than the one, and lefs than the 

 otlier. 



But limit is often ufed in a more reflricled fenfe j thus, 

 when a variable quantity approaches continually to feme de- 

 terminate quantity, and may come nearer to it than to have 

 any given difference, but can never go beyond it ; then is 

 tTie determinate quantity faid to be the limit of the variable 

 quantity. 



Hence, the circle may be faid to be the limit of its cir- 

 cumfcribed and infcribed polygons; becaufe thefe, by in- 

 creafing the number of their fides, can be made to differ 

 from the circle lefs than by any fpace that can be propofcd, 

 how fmall foever. 



The limit of a variable ratio, is fome determinate ratio, to 

 which the variable ratio may continually approach, and 

 come nearer to it than to have any given difference, b\it can 

 never go beyond it. Hence, the ratio of the ordinate to 

 the fub-tangent of a curve, is faid to be the limit of the 

 variable ratio of the differences of the ordinates, to the dif- 

 ferences of the abfciffi. 



The word limit, in this fenfe, fignifies the fame as what 

 fir Ifaac Newton calls a firll or prime, and a laft or ultimate 

 ratio. 



There arc two cafes of a variable quantity, or variable 

 ratio, tending to fuch a limit, ?.3 we have been defcribing. 

 In the firfl cafe, the variable quantity, or ratio, will not 

 only aj)proach to its limit within lefs than any given differ- 

 ences, but will aftually arrive at its limit. 



In the fecond cafe, the variable quantity, or ratio, will 

 only approach its limit within lefs than any given difference, 

 but will never actually arrive at it. 



Sir Ifaac Newton, to avoid theharihnefs of the hypothefis 

 of indivifibles, and the tedioufnefs of demonftrations, ac- 

 cording to the method of the ancients, by dedutlions ad 

 abfurdum, has premifed leveral lemmata, in the iirll feclion 

 of the tiril book of his Princ^jles, relating to the firlt and 

 lali fums, and ratios of nafcent and evanefcent quantities ; 

 that is, to the limits af fums and ratios. This doftrine 

 chiefly depends on the firft of thofe lemmata ; the words 

 of which are, " Quantitates ut & quantitatum rationes, quae 

 ad iqualltatem tempore quovis finito conllantcr tendunt, & 

 'ante finetn temporis illius propius ad invicem accedunt 

 quam pro data quavis differentia, Hunt ultimo squales.'' 



The learned gentlemen, who have written in defence of 

 fir Ifaac, againft the author of the Analyll, are not agreed 

 among themfclves as to the precile meaning of this lemma. 

 One of thefe gentlemen fays, that the genuine meaning of 

 tliis propofition is, that thofe quantities are to be elleemed 

 ultimately equal, and thofe ratios ultimately the fame, which 

 are perpetually to each other, in fuch a manner, that any 

 difference, bow minute foever, being given, a finite time 

 may be adigned, before the end of which, the difference of 

 thofe quantities, or ratios, fhall become lefs than that given 

 difference. See Pref. State of the Rep. of Letters for Ott. 

 l-^j, and for Oct. 17J6. 



What fir Ifaac Newton intends we fliould underfland by 

 the ultimate equality of magnitudes, and the ultimate iden- 

 tity of the ratios propofed in this lemma, will be belt known 

 from the demonftration annexed to it. By that it appears, 

 fir Ifaac Newton did not mean that any point of time was 

 affignable, wherein thefe varying magnitudes would become 

 artually equal, or the ratios really the fame ; but only that 

 no difference whatever could be named, which they ftiould 

 not pafs. The ordinate of any diameter of an hyperbola, 

 is always lefs than the fame continued to the afymptote ; yet 

 the demonftration of this lemma can be applied, without 

 thanging a fingle word, to prove their ultimate equality. 



Vol.. XXI. 



The fame is evident from the lemma immediately following', 

 where parallelograms are infcribed, and others cirrtimfcribed 

 to a curvilinear fpace. Here the iirft lemma is applied ta 

 prove, that by multiplying the number, and diminifhiiig the 

 breadth of thefe parallelograms in tiifimtum, that is, per- 

 petually and without end, the infcribed and circurnfcribed 

 figures become ultimately equal to the curvilinear fpace, 

 and to each other ; whereas, it is evident, that no point of 

 time can be affigncd, wherein they are aftually equal ; to 

 fuppofe this were to affcrt, that the variation afcribcd to the 

 figures, though endlefs, could be brought to a period, and 

 be perfeflly accompliihed ; and thus we fhould return to 

 the unintelhgible language of indivifibks. The excellence 

 of this method confifts in making the fame advantage of 

 this endlefs approximation towards equality, as by the ufe 

 of indivifibles, without being involved in the abfurdities of 

 that doftrine. In fhort, the difference between thefe two 

 may be thus explained. 



There are but three ways in nature of comparing fpaces : 

 one is by (hewing them to conlilt of fuch, as by impofition 

 on each other will appear to occupy the fame place : an- 

 other is, by (hewing their proportion to fome third ; and 

 this method can only be direCtly applied to the like fpaces 

 as the former ; for tliis proportion muff be finally deter- 

 mined by {hewing when the multiples of fuch fpaces are 

 equal, and when they differ : the third method to be ufed, 

 where thefe other two fail, is by defciibing upon the fpaces 

 in queftion fuch figures as may be compared by the former 

 methods ; and thence deducing the relation between thofe 

 fpaces, by that indireft manner of proof, commonly called 

 dcdudio ad ahfurdum; and this is as conclufive a demonitra- 

 tion as any other, it being indubitable, that thoil- things arc 

 equal which have no difference. Thus Euclid and .Vrchi- 

 medes demonllrate all they have written concerning the 

 comparifon and menfuration of curvilinear fpaces. The 

 method advanced by fir Ifaac Newton for the fame purpuie 

 differs from their's, only by applying this indirect form of 

 proof to fome general propofitions, and from thence de- 

 ducing the rclf by a direCt form of reafoning. Whoever 

 compares the fourth of fir Ifaac Newton's lemmas with 

 the firfl:, will fee, that the proof of the curvilinear fpaces, 

 there confidered, having the proportion named, depends 

 wholly upon this, that if otherwife the figure infcribed 

 within one of them, could not approach, by fome certain 

 diftance, to the magnitude of that fpace : and this is pre- 

 ciffly the form of reafoning, whereby Euclid proves the 

 proportion between the different circles. As this method of 

 reafoning is very diffufively fet out in the writings of the 

 ancients ; and fir Ifaac Newton has here expreffed himfelf 

 with that brevity, that the turn of his argument n^ay pof- 

 fibly eicape the unwary, the reading 01 the ancients mull 

 be the belt introduAion to the knowledge of his method. 

 The impoffible attempt of comparing curvilinear fpaces, 

 without having any recourfe to the foremenlioned indirect 

 method of arguing, produced the abfurdity of indivifibles. 



As the magnitudes, called in this lemma ultimately equal, 

 may never abfolutcly cxiff umier that equality ; fo the va- 

 rying magnitudes holding to each other the variable ratios, 

 here confidered, may never exilt under that, which is here 

 called the ultimate ratio. Of this fir Ifaac Newton gives 

 an inftanee, from lines i-.icreafing tOi^elher by equal addi- 

 tions, and having from the firit a given difference. For 

 the ultimate ratio of thefe lires in the fenfe of this lem.ma, 

 as fir Ifaac Newton himfelf obferves, will bo the ratio of 

 equality, though thefe lines can never have this ratio ; fince 

 no po-nt of time can be afligned, when one does not exceed 

 the other. 



