LIMIT S. 



In like manner, ihe quantities called by fir Ifaac Newton 

 Taniihing, may never fubfift under that proportion here 

 eftcemed the:r ultimate. 



In the cafe of drawing tangents to curves, wliere the or- 

 dinate bears the fame proportion to the fubtanprent, as that 

 wherewith the diff^-rence <.f the or.linatcs, to the difference 

 of the abfcl(r.e, vanifh ; thefe lines mull not be conceived, 

 by the naine of an evanefcent, or any other appellation, cvi-r 

 to fubfifl under that proportion : for (hould we conceive 

 thefe lines, in any manner, to fubllll under _th!s proportion, 

 though at the iiiilanl of tlieir vaiiiihing, we (hall tall into 

 the unintelhgible notion of "indivilibles, by endeavourir.g 

 to reprefent, to the imagination, fome inconeei\able 

 kind of exillence of thefe lines between tlieir having a real 

 magnitude, and becoming abfolutely nothing. Sir Ifaac 

 Newton was himfelfapprehenfive, tiut tliis midakc might 

 be made ; for as lie thought fit (in compliance with the bad 

 tafte which then prevailed) to continue the uie ot fomeloole 

 and indillind cspreirioiis refembliiig thofe of indivifiblcs, 

 for which he has himfelf apologized, lie exprefsly cautions 

 lis againll railinteipreling him iu this manner, when he fays : 

 ' Si quando di.Kcro qiiantitates quam minimas vel cvanef- 

 centes, vel ultimas, cave intcUigas qnantitates magiiitudine 

 dcterminatas, fed cogita femper diminuendas fine limite.'' 

 Thus exprefsly has he declared to us, that vaniihing quan- 

 tities, or whatever other lefs accurate appellation he names 

 them bv, arc to be confidercd as indeterminate quantities 

 bearing' to each other, under their different magnitudes, dif- 

 • ferent proportions ; which the quantities themfelves can 

 never obtain, and the limit of thefe proportions is that, 

 for the fake of which thefe quantities are confidered: in- 

 fomuch, that fince thefe quantities have different propor- 

 tions, while they obtain the name of vaniiliing quantities, the 

 term ultimate is necefTarily added to denote that proportion, 

 which is the limit of an enJltfs number of varying ,ones. 

 Tne hke remark is necciTary, when thefe quantities are con- 

 fidered in the other light, as ariling before the imagination : 

 for then the proportion intended muil be fpecificd, by call- 

 ing it the firil, or prime proportion of thc(e quantities. 

 And as this additional epithet is neceffary to exprefs the 

 proportion intended, fo it is abfurd to apply it to the quan- 

 tities themfelves ; as fir Ifaac Newton fays, there are '< ra- 

 tjones prim.B quantitatuni nafccntium," but not " qnan- 

 titates primx nafccntes." Philofoph. Tranfadlions, N J42, 

 p. 205. 



So that, according to the author we have been quoting, 

 all the examples given by fir Ilaac in the before mentioned 

 foftion, are to be underftood of fuch limits or ultimate ra- 

 tios, as are never attained to by the quantities and ratios li- 

 tiiited, but to which thefe may approach indefinitely, that is, 

 ib as to differ lefs than by a given quantity. 



On the other hand, a learned gentleman, who affnmed 

 the name of Philalethes Cantabrigienfis, thinks that (ir Itaac 

 means, by the words of the lemma, and proves, in his de- 

 inondration, not that the quantities or ratios are barely to 

 be confidered as ultimately becoming equal, or are to be 

 clU-emed as ultimately equal ; though, in reality, they can 

 never have that proportion to each other ; but that they do 

 at lall become actually, pen'eAly, and abfolutely equal. 

 PreL State of the Republic of Letters for November 1733, 



He alfo diitinguifhes, as above, between quantities and 

 ratios which arrive at their limits, and thofe which do not. 

 And it is innfted on, that every one of the examples given 

 in the lemmata of this firft feftion of the fw-il book ot fir 

 ifaac's Principles, are of fuch quantities and ratios as ac- 

 tually arrive at their refpcftive limits ; nor is there an in- 



8 



fiance there given of a quantity, or rati^ which never 

 arrives at its limit, except one at the latter end of the 

 fcholiupi of this fection (and that by way of illuHration of 

 a particular objection only) of two quantities, liaving a 

 given difFerenee, and being equally increaled, aJ hifiiiiiupi, 

 and whole ratio, it is admitted, never arrives at its limit. 

 But decreafing quantities may really and in faft be di- 

 mini(hed flc/ infnitum ; for they may vanifli and come to 

 nothing. The ratio, therefore, of thefe, fays he, may arrive 

 at its limit ; though that of the others cannot. 



Neither are thefe learned gentlemen agreed ^s to the fcnfc 

 of the word I'ani/ling or evuru-futit, in the feholium of tiiis 

 firll tVclion of fir Ifa.:c's Principles. 



The quelUon is, whether the quantities that vanilh are 

 underilood to fpcnd fom.e finite time in vaniihing, or to vaniHi 

 in an inftant, or point of time ; and confequently, whether 

 they bear one to another an infinite number of difierent fuc- 

 cefiive ratios during the vanidiing, or one ratio only, at the 

 point or inllance of their evauefcence. 



This lall is the fenfe in which Philalethes takes the word 

 evanefcent, or vanifiiing ; and the difpute, on this head, as 

 he obferves, is of no otiier confequeiice than to determine, 

 wliether the fenfe in which he nfes the word be agreeable to 

 fir Ifaac Newton's, For, if the quantities vanilh in an in- 

 llaiit, I take the only ratio with which they vanidi ; or they 

 fpend a finite time in vanifliing, and I take the Inft of the 

 ratios, wiiieh they fucccliively bear to one another during 

 that time ; ftill the ratio, taken in either of thefe cafes, will 

 be one and the fame. Prefent State of the Republic of Let- 

 ters for November 1735, p. 3S3, 384. 



We cannot pretend to give the whole detail of this con- 

 troverfy, but muft refer the curious to the Prefent State of 

 the Republic of Letters for 1735. We (hall only obferve, 

 that this difquifition is partly critical and partly fcientifical. 

 Tiie critical inquiry is into the fenfe of fir Ifaac, fo far as 

 it may be determined from his own words ; and here we can- 

 not help thinking that this is fomewhat doubtful. The 

 other inquiry is about the true or fcientifical notion, upon 

 which this doftrine ought to be founded. With relpett to 

 which we (hall only a(k two queftions, which every reader 

 may refolve for himfelf, to wit, whether the conception or 

 notion he has cf the ratio or proportion of evanefcent quan- 

 tities, at the point or inftance of their cvariefeence, be more 

 clear and diftincl than the notion of infinitefimals .' And 

 wliether the notion of infcribed or circumfcribed polygons 

 to any curve, attaining their lall form, and thereby coin- 

 ciding with their curvilinear limit, be more clear and dilliiic\ 

 than the notion of polygons of an infinite number of fides in 

 the method of iufiniteiimalb ? 



Before we leave this fubjeft, it may be proper to give 

 the fentiments of an eminent mathematician about the doc- 

 trine of limits, or of prime and ultimate ratios, and to Ihe^v 

 the conncftiou of this dodrine with that of fluxicns, Mr. 

 Maclaurin, in his Treat, of Flux., art. 502. 



Sir Ifaac Newton coiifiders the fimulianeous increments 

 of flowing quantities as finite, and then invelligates the 

 ratio which is the limit of the various proportions wh:ch 

 thofe increments bear to each other, while he fuppofes thetn 

 to decreafe together ti;l they vanifii ; which ratio is the laiic 

 with the ratio of the fluxions. In order to difcover this 

 limit, he firil determines the ratio of the increments in gene- 

 ral, and reduces it to the moll fimple terms, fo as that 

 (generally fpcaking) a part at leall of each term may be in- 

 dependent of the value of the increments themfelves ; then, 

 by fuppofing the increments to decreafe, till they vanifii, 

 the limit readily appears. 



For example, let u be an invariable quantity, # a flowing 



quantity, 



