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being destroyed by contrary signf. XXVII. As on the one hand it were absurd to get rid of O by saying, let me contradidi mystis; let me subvert my own hypothesis; let me
alie it for granted that there is no increment, at the fame time that Ireta in a quantity, Whicli I could never have got at but by assuming an increment: se on the other hand it would be equalty wrong to imagine, that in a geometrical demonstration we may be allo ed to admit any error, though e ver se smali, or that it is possibie, in the nature of things,
an accurate conclusion should be derived Dom in accurate principies. There re o cannot he thrown out as an infinitesimal, or Upon the principie that infinitesimais may be lasely neglected; but only because it is destroyed by an equat quantity with a negative sigra, Whence o - po isequat to nothing. And as it is illegitimate to reduce an equation, bysubducting from one sit de a quantity when it is not to be destroyed, orWhen an equat quantity is not subducted stom the other side of the equation: so it must be allowed a very logical and just method os argu- in g, to conclude that is fio m equais ei ther nothing or equat quantities a re subducteis, they mali stili re main equat. And this is a true rea n whyno error is at last produced by the rejecting of o. Whicli there re must
not be ascribed to the doctrine os differences, Or i irinitesimais, or evanescent quantities, or momentum S, or fluxions.
XX Ui II. Suppose the case to be generat, and that x' is equat to the area ABC, when e by the method os fluxions the ordinate is se undn xv I, which we admit for true, and thali inquire how it is arrived at. Now is me a re content to come at the conclusion in a summary Way, hysupposing that the ratio of the fluxions of x and x' is found o he i
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and n x-I, and that the ordinate of the area is considered as iis fluxion ;we shali not so clearly see our Way, or perceive how the truth comes out, that method as we have me ed besore being obscure and illogical. Butis we Dirly delinea te the area and iis increment, and divide the lalter into two paris B CFD and CF Η ', and proceed regularly by equations het cen the algebraical and geometrical quantities, the rea n of thething will plainly appear. For as xu is equat to the area ABC, se is the increment of xv equat to the increment of the area, i. e. to B D ΗC ,
is equat to the rejectaneous quantity-o o κη ' λ ε ροα and that when
this is rejected on one side, that is rejected on the other, the rea ning hecomes just and the conclusion true. And it is ali one Whatever magnitude you allow to B D, whether that os an infinitesimal disserence ora finite increment e ver so great. It is there re plain, that the supposing the rejectaneous algebraical quantity to be an infinitely smali or evanes.cent quantity, and there re to be neglected, must have produced an error, had it not been sor the curvilinear spaces heing equat thereio, andat the same time subducted Dom the other pari or si de of the equation, agreeably to the axiom ; Is fram equati Iou subduct equais, the remalaira Sisi be equat. For those quantities which by the analysts are seid to beneglected, or made to Vanim, are in reality subducted. Is there re me conclusion be true, it is absolutely necessary that the finite space C FΗ
he equat to the re mainder of the increment expressed by -ooxn-2 Uc.
equat I say to the finite re mainder of a finite increment.
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one sit de an algebraical eXpression, On the other a geometrical quantity, each of whicli naturalty divides itself into three members: the algebraicalor fluxionary expression into one, whicli includes neither the expressionos the increment of the abscisi nor of any power thereos, another Whichincludes the expression of the increment itself, and a third including the expression of the powers of the increment. The geometrical quantityalso or whole increased area consists of three paris or members, the firstof whicli is the gi ven areas the second a rectangle under the ordinate and the increment of the abscisi, and the third a curvilinear space. An i, comparing the homologous or correspondent members on both ssides, wefind that as the first member of the expression is the expression of thegi ven area, so the second member of the expression will express therectan e or second member of the geometrical quantity; and the third, containing the powers of the increment, Will express the curvilinearspace, or third member of the geometrical quantity. This hint may perhaps be further extended, and applied io good purpose, by those whohave leisure and curiosity sor such matters. The use Ι malae of it is tomeis, that the analysis cannot obtain in augments or differences, but it must also obta in in sinite quantities, be they e ver so great, as was fore observed. XXX. It seems there re u pon the whole, that We may sesely pronounce the conclusion cannot be right, is in order thereto any quantitybe made to vanish, or be neglected, eXcept that Cither one error is re
dresied by another; or that secondiy, on the same fide os an equationequat quantities are destroyed by contrary signs, so that the quantity wemean to reject is first annihilaled , or lastly, that froni the opposite fides equat quantities are subducted. And there re to get rid of quantities
by the received principies of fluxions or os disserenoes is nei ther good geometry nor good logic. When the augments vanim, the velocities also vanis h.
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vanim. The velocities or suxions are seid to be primo and ultimo, as theaugments nascent and evanescent. I ake there re the ratio of the evanescent quantities, it is the same with that of the fluxions: it will therea fore an wer ali intenis as well. Why then a re fluxions introduced i Isit not to naun or rather to palliate the use of quantities infinitely smali tBut we have no notion whereby to conceive and mea re various degrees of veloci ty, besides space and time, or when the times are given, hesides space alone. We have even no notion of veloci ty prescinded Dom timeand space. Whcn there re a pol ni is supposed to move in gi ven times, we have no notion os greater or lesier velocities or os proportions he- tween velocities, but only os longer or morter lines, and of proportions bet ween such lines generaled in equat paris of time.
XXXI. A potnt may be the limit of a line: a line may be the limit ofa furface: a moment may terminate time. But how can we conceive a
space, and cannot be conceived without them. And is the velocities of nascent and evanescent quantities, i. e. abstracted from time and space, may not he comprehended, how can we comprehend and demonstratetheir proportion S; or Consider their rationes primae and ultimae J For toconsider the proportion or ratio of things implies that such things have magnitude ; that such their magnitudes may be measu red, and their relations to eacti other known. But, as there is no measure of veloci tyexcepi time and si ace, the proportion os velocities being only compo unded of the direct proportiori of the spaces and the reciprocat proportion of the times; doth it not sol low that to talli of investigatin g, obtaining, and considering the proportioris of velocities, eXclusi vely of timeand space, is to talli unintelligibly tXXXII. But you will say that, in the use and application os fluxions
meri do not overstra in their faculties to a preci se conception of the a bove- mentioned
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mentioned velocities, incremenis, infinitesimais, or any other lach like ideas of a nature so nice, subtil, and evanescent. And there re you vili perhaps maintain, that problems may be solved without those inconceivabie suppositions; and that, consequently, the doctrine os fluxi-ons, as to the praetical pari, stands clear of ali such difficulties. I an-Mer, that is in the u se or application of this method those dissiculi and obscure potnis are not attended to, they are nevertheleia supposed. Theyare the foundations on whicli the moderns bulld the principies on whichthey proceed, in solving problems and disco vering theorems. It is withthe method os fluxions as with ali other methods, whicli presuppo se their respective principies and are grounded thereon ; although the rules maybe practised by men who neither attend to, nor perhaps know the principies. In like manner, there re, as a stilor may practicatly apply certain rules derived from astronomy and geometry, the principies Whereos he doth not understand; and as any ordinary man may solve divers numerical questions, by the vulgar rules and operations os arithmetic, whichhe performs and applies without lino ing the reasons of them : even soli cannot be dented that you may apply the rules of the fluxionary method: you may compare and reduce particular cases to generat fornas; yOu may operate and compute and solve problems thereby, not onlywithout an actuat attention to, or an actual knowledge OG tho groundsos that method, and the principies whereon it depends, and wherace it is deduced, but even without having ever considered or comprehended
XXXIII. But then it must be remembered, that in lach case althoughyOu may Pasi sor an artist, compulist, or analyst, yet you may not bejustly esteemed a man os science and demonstration. Nor mould any man, in virtve of heing conversant in such obscure analytics, imagine his rational sacrilites to he more improved than those of other men, Which have been exerci sed in a disserent manner, and on different sub
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jects ; much lese erect himself into a judge and an oracle, concerning matters that have no sort of connexion with, or dependence on those species, symbolf or signs, in the management whereos lie is Q conversant and expert. As you, who are a sicilial compulist or analyst, maynot there re be deemed skilful in anatomy ; or vice Tersa, as a man whocan dissect with ari, may, neverthelest, be ignorant in your ari os computing : even Q you may both, not illistanding your peculiar skill in your respective aris, be alike unqualified to decide Upon logic, or meta- physios, or ethics, or religion. And this Would be true, even admittingillat you understood your o n principies and could demonstrate them. XXXIV. Is it is seid, that fluxions may be expounded or expressed by finite lines proportional to them ; Whicli finite lines, as they may hedistinctly conceived and known and rea ned upon, so they may be substituted for the fluxions, and their mutuat relations or Proportions be considered as the proportions os fluxions; by Whicli means the doctrine hs comes clear and useful: I ans er that is, in order to arrive at these finite lines proportional to the fluxions, there be certain Reps made use of whicli are obscure and inconceivable, be those finite lines themselves everso clearly conceived, it must neverthelest be aclino ledged, that your proceeding is not clear nor your method scientific. For instance, it is supposed that A B heing the abscisi, B C the ordinate, and VC Η a tangent of the curve AC, B b or CE the increment of the abscisi, Eo theiocrement of the ordinate, whicli produced meets VH in the poliat T,
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and C e the increment of the curve. The right line C c heing producedio Κ. there a re formed three smali triangles, the rectilinear C E c themixtilinear CEc, and the rectilinear triangle C E T. It is evident thesethree triangles a re different Dom each other, the rectilinear GEc be ingle se than the mixti linear CEO, whose sit des are the three increments above mentioned, and this si ill lese than the triangle C E T. It is suppo sed that the ordinate b c moves in to the place B G so that the potui e is coincident with the potnt C; and the right line CK, and con quently the curve C c, is coincident with the tangent C H. In whicli casu the mixti linear evanescent triande C Eo will, in iis last form, he similar to the triangle C ET: and iis evanescent sides C E, E c, and C e Will be proportional to C E, E T, and CT the sidcs of the triangle C E T. And there re it is concluded, that the fluxions os ille lines A B, B C, and A C, being in the last ratio oftheir evanescent incremenis, a re proportional to the fides of the triangle C E T, or whicli is ali one, of the triangle UB C similar thereunto. Itis particularly rema rhed and insisted on by the great author, that thopolitis Cand c must not be distant one from another, by any the least intervat what ever: hut that, in order to find the ultimate proportions of the lines C E, E c, and C c i. e. the proportions of the fluxions or velocities) expressed by the finite fides of the triangle V B C, the potnis C and emust he accurately coincident, i. e. one and the fame. A potnt there reis considered as a triangle, or a triangle is suppo sed to be formed in a potnt. Which to concei ve stems qui te impossibie. Yet some there a re, who, though they mrin k at ali other mysteries, malae no disii culty of their own, vilio stra in at a gnat and Mallow a camel. XXXV. I know not whether it be worth While to observe, that possibiyso me men may hope to operate by symbols and suppositions, ita sicli sortas to avo id the use of fluxions, momentums, and infinitesimais after thesollo ing manner. Suppo se x to be one abscisi os a curve, and Z another
Introd. ad quadraturam curvarum.
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abscisis of the same curve. Suppose also that the respective areas are xxx and zzz: and that z-x is the increment of the abscisi, and zzz - xxx the increment of the area, Without considering how great, or ho smallthose increments may be. Divide no zzz-xxx by Σ - X and the
quotient will be arid, supposing that z and x are equat, this sanae quotient Will be s xx whicli in that case is the ordinate, whichthere re may be thus obtained independently of fluxions and infinitesimais. But herein is a direct fallacy : sor in the first place, it is supposed
that the abscisses et: and x are u nequat, Without whicli supposition no onestep could havc hcen made ; and in the second place, it is supposed theyare equat; whicli is a manifest inconsistency, and amounts to the famething that hath been besere considered '. And there is indeed rea n to apprehend, that ali attempis sorsetting the abstruse and fine geometryon a right foundation, and avoiding the doctrine os velocities, momentums, tac. will be found impracticabie, tili such time as the object andend of geometry are betier understood, than hitherio they seem to have been. The great author of the method os fluxions seli this difficulty, and thercsore he gavo into those nice abstractions and geometrical metaphysics, without which he Ois nothing could be done on the received principies ; and What in the way of demonstration he hath done with themthe re ader will judge. It mus , indeed, be aclino ledged, that he used fluxions, like thc scassold os a buit ling, as things to be laid aside or gotrid os as soon as finite lines isere found proportional to them. But thenthese finite exponents are found by the help oi fluxions. Whateverthere re is Di by fuch exponents and proportions is to be ascribed tofluxions : whicli must there re he previora sty undercto . And what arethese fluxionsi The velocities of evanescent incrementst And what arethese sanae evanescent increments Z They are netther finite quantities,
nor quantities infinitely mali, nor yet nothing. May We not cali themthe ghosts of departed quantities t
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XXXVI. Μen too osten impose on theniselves and others, as is thcyconceived and understood things expressed by signs, when in truth theyliave no idea, fave only of the Very signf themselves. And there are me grounds to apprehend that this may be the present case. The velocities of evanescent or nascent quantities are supposed to he expressed, both by sinite lines of a determinate magnitude, and by algebraical notesor sigris: but I suspect that many who, perhaps never having examinedihe matter, ta ke it sor granted, would upon a narrow scruti ny sind it impossibie, to Danae any idea or notion what e ver of those velocities, egelu- sive of suci, sinite quantities and signf. ab edet l-l -l-l-l - - - l l
Suppose the line R P described by the motion os a poliat continuatly accelerated, and that in equat particles of time the uiaequat paris XL, L M, MN, NO, Fc. are generaled. Supposie also that a, b, c, d, e, Sc. denote the velocities of the generating potnt, at the severat periods of the parisor increments se generaled. it is easy to observe that these increments are each proportional to the sum of the velocities with whicli it is describ-ed : that, consequently, the severat sums of the velocities, generaled in
equat paris of time, may be set sortii by the respective lines X L, L M, MN, G. generaled in the fame times: it is lihewise an easy matter to. say, that the last veloci ty generaled in the firsi particle of time, may beexpressed by the symbol a, the last in the second by b, the last generaledin the third by c, and so on: that a is the velocity of L M in statu nascenti,
and b, c, d, e, tac. are the velocities of the increments MN, NO, OP, Uc. in their respective nascent estates. You may proceed, and considerthese velocities themselves as flowing or increasing quantities, taliing the Velocities of the velocities, and the velocities of the velocities of the Velocities, i e. the first, second, third, tac. velocities His itum : Whicli succeed ing series of velocities may be thus expressed, a. b a. C - 2 b Φ a.
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XXXVII. Nothing is easier than to assign names, signs, or expressions to these fluxions, and it is not difficult to compute and operate by means os suci, signf. But it will be sound much more difficuli, to omit thesignS, and yet retain in our minds the things, which we suppost to be sisenified by them. Τo consider the exponenis, Whether geometricat, or algebraicat, or fluxionary, is no difficult matter. But to form a precise idea os a third veloci ty for instance, in iiself and by itself, Hoc opus, hic labor. Nor indeed is it an east potnt, to sorm a clear and distinct idea of any velocity at ali exclusive of and prescinding from est tength of time and space ; as also stom ali notes, signS or symbols what ever. This,
it stems evident, that measures and signs are absolutely necessary, in o der to conceive or reason about velocities; and that, consequently whenwe thin k to conceive the velocities, simply and in them selves, we are deluded by va in abstractions. XXXVIII. It may perhaps bethought by some an easter method of conceiving fluxions, to suppost them the velocities wherewith the infinitesima, disserentes a re generaled. So that the firsi fluxions mali be the velocities of the fissi disserences, the second the velocities of the second disse- rences, the third suxions the velocities of the third differences, and se onad in itum. But not to mention the insurmolantable difficulty of admitting or conee ving infinitesimais, and infinites mals of infinitesimal it k.cvident that this notion os fluxions would not con sist with thegreat author's vlew ; Who held that the minutest quantity ought not to beneglected, that theresere the doctrine os infinitesimal differences was notis