IIATIO. 



\V<i may farther obferve, that though the terms of this ratio, 

 ■viz. X + dt.o X, never approximate nearer to each oilier, 

 their conllaiit difference being </, yet the ratio of the terms 

 approximates to the ratio of equably ; that is, though the 

 terms get no nearer in their difference, yet, if we may be 

 allowed the expreflion, they get nearer in tlieir ratio. The 

 difference of the terms, and the ratio of tin- terms, are ideas 

 very dillinft from each other, and in no wife to be fo con- 

 netted, but that one may vary whilft the other is conftant. 

 Although the ratio of equality may ftriftly be called the 

 limh of the varying ratio of the quantities x -r d and x, 

 yet the terms of' this ratio can never be ftriftly faid to be 

 equal, nor ultimately equal, as that fuppofes an ultimate Itate 

 in which they are equal ; nor equal when they vani/h into 

 infinity, or, when they pafs out of finity into infinity. 

 There is no finite quantity next to infinity, no number, for 

 inftance, which is the next number to infinity. Nor is there 

 any flep from a ftate of nothingnefs into finite exiftence ; 

 there is no fraftion fo fmall as to be the very next fraftion 

 to nothing; no fraftion can be affigned fo fmall, but another 

 fraftion may be affigned that is fmaller. Neither can we 

 fay, in (Iridlnefs, that two infinitely great numbers with a 

 finite difference are equal, it being a propofition obvioufly 

 abfurd and contradiftory. There is no fucli thing in 

 nature as an infinitely great number ; and it is contradiftory 

 to fay of any two numbers, both that they have a difference, 

 and that they are equal. Whoever confiders that the idea of 

 infinity is a general or abftraft idea, that the idea of 

 number is always particular, that infinity is a property of 

 numbers, a property of extenfion, &c. itfelf, will readily 

 perceive that thefe and fuch like expreffions have no literal 

 meaning. (See Locke, b. ii. ch. i6. and ch. 17.) As to 

 the metaphorical ufe of them, to avoid circumlocution, or 

 the inlroduftion of new terms, it may be allowed, when 

 once the literal meaning has been explained, in this, as 

 well as on various other occafions both in fcience and in 

 common life. 



When the difference between any two quantities decreafes, 

 fo as to become a lefs fraftional part of the one of them than 

 any afiigned fractional part whatever ; or when the difference 

 between the terms of a ratio becomes lefs in refpeft of one 

 of them (the greater for example) than any affig;ned ratio, 

 this may be expreffed by faying, that fuch difference ^)a/2^/2lfJ 

 in reipeft of that greater quantity. 



But if the difference between two terms vanifh in refpeft 

 to one of them, it will alfo vanifii in refpeft of the other. 

 It is true that at any affigned lime, when the terms have a 

 particular magnitude, the difference between the terms will 

 always be a lefs fraftional part of the greater term than 

 it is of the lefs term ; but as this difference continually 

 decreafes, it will become the fame fraftional part of the 

 lefs term that it was before of the greater, however fmall 

 that fraftional part of the greater term may be ; therefore, if 

 the difference vanifh with regard to one of them, it will vanifli 

 alfo with regard to the other. We may, therefore, inftead of 

 the third condition of our definition, fay, that " the difference 

 muff vanifh in refpeft either of the fixed or of the varying 

 quantity, fince one of thefe imphes the other." 



The above is true as well for the cafe of two variables, as 

 for one fixed and one variable, though both of fuch variables 

 mcreafe or decreafe without limit. Thus, in the example 

 above given, in whicli the terms were x ^- d and x, and 

 where x is continually increafing, the terms themfelves both 

 continually increafe ; for in this initance it is obvious, that if 

 the difference vanifh with refpeft to one of the terms, it will 

 alfo vanifh with rofpeft to the other. Let .-cbe to d, at any 

 affigned point of time, as n is to I, and i/ at that inftant of 



time will be the th part of the greater term, and the 



n H- 1 



- th part of the lefa term : now, thotigh the contemporary 



value of thefe fraftions can never be equal, yet in fucceffion 

 the value of the latter Iraftion will become whatever the for- 



mer has been ; therefore, if - 



n + I 



become lefs than any af- 



figned fraftion, fo will - hkewife ; and thus, if the difference 



Tl 



vanifhes with refpeft to one of the terms, x + d, to will it 

 alfo vanifli with refpeft of the other, x. And the fame would 

 be true if the terms were *; and x — d, all things elfe being 

 as before. 



Again, if x by decreafing vanifh with refpeft to fome fixed 

 quantity n, then will *, multiplied by a given number n, or 

 n X, vunifh in refpeft of a ; or, which is the fame, if* vanilh 

 in rcfpift of a, fo likewife will the quantity n « ; bearing to 

 X the affigned ratio of n to i . For though at any particular 

 point of time x is a fmalh-r fraftional part of a than n x, 

 yet X can be no affigned fraftional part of a whatever ; but 

 by farther diminifiiing x, the quantity n * may become the 

 fame fraftional part of a that x was before. If, then, x 

 may become equal to, or k-fs than, any affigned fraftional 

 part of a, fo likewife may n x ; that is, if * vanilh in refpeft 

 of a, fo likewife will n x. 



For alike reafon, if x vanifli in refpeft of any quantity, 

 it will likewife vanifli in refpeft to that quantity multiplied 

 or divided by any number ; or it will vanifli in reipeft of a 

 quantity, bearing to a any affigned ratio, as that of ;n to l. 

 Thus, if X vanifli in refpeft of a, it will alfo vanifli in re- 

 fpeft of 3 a, 2 a, 5 a. Sec. Or, if any fraftion vanifli in 

 refpeft to the diameter of a circle, it will likewife vanifli in 

 refpeft of the radius. For whatever part of the diameter 

 the line x may be at any affigned point of time, let .-c farther 

 decreafe, till it be half what it was at that affigned time, and 

 it will now be the fame part of the radius that it was before 

 of the diameter ; and the fame with various other lines and 

 quantities. We have fhewn, in the preceding part of this 

 article, that the ratio which two quantities bear to each 

 other may have a limit, although the terms themfelves may 

 increafe or decreafe perpetually without limit. I. If the 

 terir.3 approximate tcwaids each other ; 2, if the lefs never 

 pafs the greater; and, laftly, if their difference vanifli in re- 

 fpeft of either term, then the limit of their varying ratio is 

 that of equality : and this, whether the terms themfelves are 

 fuch, as by increafing they both become greater than any 

 affigned quantity, or, which is more common, fuch as by 

 decreafing become lefs than any affigned quantity, or as it is 

 called, infinitely fmall. Becaufe the idea of the terms of a 

 ratio is lefs abttraft than that of the ratio itfelf, it is more 

 ufual to fay that the terms themfelves in this cafe are ulti- 

 mately equal, though, ftriftly, it is the ratio only of the 

 terms that comes to a limit, for the terms themfelves are fup- 

 pofed to increafe or decreafe without limit, or as we com- 

 monly fay, ad infinitum. 



Prep. — In a circle whofe centre is C, {Plate XIII. 

 Analyfis, fig. 9.I radius C A, diameter A a, let A B be the 

 chord, F B the fine, and A D the tangent of the arc A B. 

 I fay, that while the arc A B continually decreafes without 

 limit ; ift, the fine continually approximates to the tan- 

 gent ; zdly, the fine never exceeds the tangent ; 3dly, their 

 difference will vanifli in refpeft of either fine or tangent. 



Firft ; B F : D A :: C F : C A, but while the arc de- 

 creafes, C F approximates to C A ; B F approximates to 



DA. 



