I N F 



breadth, it have a finite thicknefs; all fuch folids fliallbe as 

 the given finite diftances one to another. 



But thefe quantities, though infinitely greater than the 

 «pther, are yet infinitely lefs than nny of tliole wherein all tlie 

 three ditnenfions are infinite — Such are the fpaces intercepted 

 between two inclined, planes infinitely extended : the fpace 

 intercepted by the furface of a cone, or the fides of a py- 

 ramid, likewife infinitely continued, &c. all which, not- 

 withftanding the proportion of one to another, and to the 

 TO ■srav, or vaft abyfs of infinite fpace (wherein is the locus 

 of all things that are, or can be) or to the folid of infinite 

 length, breadth, and thicknefs (taken all manner of ways) 

 are eafily aflignable — For the fpace between two planes is to 

 the whole, as the angle of thofe planes to the 360 degrees of 

 the circle. As for cones and pyramids, they are as the 

 fpherical furface intercepted by ihem, is to the furfsce of the 

 fphere; and therefore cones are as the verfed fines of half 

 their angles to the diameter of the circle ! thefe three forts 

 of infinite quantity are analogous to a line, furface, and folid ; 

 and like them, cannot be compared or have any proportion 

 one to another. 



Infinites, Arithmetic of. See Akithmetic. 



Infinites, Charaaers in ylrithmelic of. See Charac- 

 ttER. 



Infinite Decimals. See Repetend. 



Infinite Dif.refs, in Lanv. Sec Distress. 



Infinite Proprjition, in Logic. See Proposition. 



Infinite Series. See Series. 



INFINITELY fmall quantity, called alfo an infmitefi- 

 tr.al, is that which is fo very minute, as to be incomparable to 

 a-.y finite quantity ; or it is that which is lefs than any af- 

 fignable quantity. 



An infinite quantity carnot be either augmented or lef- 

 fened, by adding or taking from it any finite quantity ; 

 neither can a finite quantity be either augmented or leCened, 

 by adding to, or taking from it an infinitely fmall quan- 

 tity. 



If there be four proportionals, and the firfl be infinitely 

 greater than the fecond ; the third will be infinitely greater 

 than the fourth. 



If a finite quantity be divided by an infinitely fmall one, 

 the quotient will be an infinitely great one ; and if a finite 

 quantity be multiplied by an infinitely fmall one, the pro- 

 duct will be an infinitely fmall one ; if by an infinitely great 

 one, the produft will be a finite quantity. 



If an infinitely fmall quantity be multiplied into an infi- 

 nitely great one, the produ£^ will be a finite quantity. 



In the method of infinitefimals, or of infinitely fmall 

 quantities, .the clement by which any quantity increafes or 

 decrcafes, is fuppofed to bcinfisiitcly fmall, and is generally 

 exprefled hf two or more terms ; fome of which arc infinitely 

 lofs than the reft, which being neglected as of no importance, 

 the renvjiining terms form what is called the difference of the 

 propofed quantity. The terms that are neglefted in this 

 manner, as infinitely lefs than the other terms of the element, 

 ai'c the verj' fame which r.rife in confequence of the accelera- 

 tion, or retardation of the generating motion, during the 

 infinitely fmall time in which the element is gcEerated ; fo 

 that the remaining terms cxprefs the element that would 

 have been produced in that time, if the generating mo- 

 tion had continued uniform, at is farther explained under 

 Flu.xion. 



Therefore thofe differences are accurately in the fame 

 ratio to each other as the generating motions or fluxions. 

 And hence, though infinitefimal parts of the elements are 

 neglefted, the conclufions are accurately true, without even 

 ^1 infinitely fmall error, and agree prccifely with thofe that 



X3 



I.N.F 



are dediiccd by the methods of fluxion?. In order to rercr-: 

 the application of this rfiethod eafy, fome analogous princi- 

 ples are admitted, as that the infinitely fmall elements of a 

 curve are right lines, or that a circk is a polygon of an in- 

 finite number of fides, which being produced, give the tan» 

 gents of a curve, and by their inclination to each other mea- 

 fure the curvature. This is as if we (hould fuppofe that 

 when the bale flows uniformly, the ordinate flows with a 

 motion which is uniform for every infinitely^ fmall part (^ 

 time ; and increafes, or dccrcafes, by infinitely fmall differ- 

 ences at the end of every fuch time. 



But however convenient this principle may be, it muft be 

 applied with caution and art, on various occafions. It is 

 ufual, therefore, in many cafes, to refolve the element of the 

 curve into two or more infinitely fmall right fines ; and fome- 

 times it is neceffary (if we would avoid error) to refolve it 

 into an infinite number of fuch right fines, which are infinite- 

 fimals of the fecond order. In general it is zfofulatum in this 

 method, that we may defcend 10 infinitefimals of any order 

 whatever, as we find it neceffary ; by which means any error 

 that might arife in the application cf it may be difcovercd 

 and correfted by a proper ufe of tl.is method itfelf. 



It is alfo to be obferved, that wlien the value of a quan- 

 tity that is required in a philofcphical problem becomes, -n 

 certain particular cafes, infiniteiy great cr irfii.iely little, 

 the folution wouid not be always juft, though fuih magni- 

 tudes were admitted. As when it is required, to find by 

 what centripetal force a curve would be defcribed about a 

 fixed point that is either in a curve, or is fo fituated '.hat a 

 tangent m.ay be drawn from it to the curve. The value of 

 tlie force is found infinite at tlie centre of the forces in the 

 former cafe, and at the point of contaft in the latter ; yet 

 it is obvious, that an infinite force could not inflcft the 

 line defcribed by a body that (hould proceed from either of 

 thefe points into a curve ; becaufe the direftion of its 

 motion in either cafe pafics through the centre of tiie forces, 

 and no force, how great fbever, that tends towards the cen- 

 tre, could caufeit to change that direftion. But it is to be 

 obferved, that the geometrical magnitude by which the force 

 is meafured, is no more imaginary in this than in other cafes, 

 where it becomes infinite ; and pliilofophical problems have 

 limitations that enter not always into the general folution 

 given by geometry. 



But although by proper care errors may be avoided ' 

 the method of infinitefimals, yet it muft be owned that ; > 

 fuch >vho have been accuftomed to a more ftrift and rigid 

 kind of demon ftration in the elementary parts of geometry, 

 it may not focm to be confiftent with perfeft accuracy, that, 

 in determining the firft differences, any part of the element 

 of the variable quantity ftiould be rejcAed, merely becaufe 

 it is infinitely lefs than the reft ; and that the fame part 

 fhould be afterwards employed for determining the fecond 

 and higiier differences, and refolving fome of the moft im- 

 portant problems. Nor can we fuppofe that their fcruples will 

 be removed, but rather confirmed, when they come to confi- 

 der what has been advanced by fome of the moft celebrated 

 writers on this method, who have expreffed their fentiments 

 concerning infinitely fmall quantities in the precifeft terms ; 

 while fome of them deny their reality, and confider them 

 only as incomparably lefs than infinite quantities, in the fame 

 manner as a grain of fand is incomparably lefs than the "whole 

 earth ; and others rcprefent them in all their orders, as no 

 lefs real than finite quantities And although it appears, 

 from what has been faid in this article, that a fatisfaftory ac- 

 count may be given for the brief way of reafoning that is 

 ufed in the method of infiniielinials ; while nothing is neg- 

 ledtcd without accounting for it ; and then the harmony 



between 



