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XXXIX. Το others it may possibiy seem, that we mouid sorm a justeridea of fluxions, by assuming the finite unequat isochronat increments KL, L M, MN, M. and considering them in statu nascenti, also their
increments in satu nascenti, and the nascent increments of those incremenis, and so on, supposing the sirst nascent increments to be proportional to the sirst fluxions or velocities, the nascent increments of those increments to be proportional to the second fluxions, the third nascent increments to be proportional to the third fluxions, and so onwards. And, as the first fluxions are the velocities of the sirst nascent incremenis,so the second fluxions may be conceived to he tho velocities of the second nascent incremenis, rather than the velocities of velocities. Bywhicli means the analogy of fluxions may seem better preserved, and thenotion rendered more intelligibie.
XL. And indeed ii mould stem, that in the way of obtaining the second or third fluxion os an equation, the gi ven fluxions were consideredrather as increments than Velocities. But the considering them some- times in one sense, semetimes in another, One while in themselves, another in their exponenis, stems to have occasioned no smalI mare of that confusion and obscuri ty, whicli is soland in the doctrine os fluxions. Itmay stem there re, that the notion might be stili mended, and that instead os fluxions os fluxions, or fluxions os fluxions os fluxions, and in flead os secones, third, or seurth, Sc. fluxioris of a gixen quantity, it might he more consistent and lesi liable to exception to say, the fluxiones the first nascent increment, i. e. the second fluxion ; the fluxion of the second nascent increment, i. e. the third fluxion; the fluxion of the third nascent increment, i. e. the fourth fluxion, whicli fluxions are conccived
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respectively proportional, each to the nascent principie of the incrementsuccceding that whereos it is the fluxion. XLI. For the more distinct conception os ali whicli it may be consider-cd, that is the finite increment L M be divided into the isochronalpariS Lm, mn, n o, o με, and the increment MN into the paris MAρ ρ, g ri r N isochronal to the former; as the whole jncrements L M, M N are proportional to the sums of their describing velocities, evenso the homologous particles L m, up are also proportional to the respective accelerated velocities with whicli they a re described. And asthe velocity with which Μρ is genera ted, exceeds that with whicli L mmas generaled, even so the particle up exceeds the particie L m. Andiu generat, as the i chronat velocities describing the particles of MN exceed the i chronat velocities describing the particles of L M, even sothe particles of the former exceed the correspondent particles of the lalter. And this will hold, he the said particles ever se smali. MN there re will exceed L M is they are both taken in their nascent states: and that excess will bc proportional to the excess of the velocity b aho vethe velocity a. Hence we may see that this last account of fluxions Comes, in the ut mot, to the fame thing with the first '. XLII. But notis illistanding what hath been Rid it must stili be a linowledged, that the finite particles L m or mi, though taken ever sos mali are not proportional to the velocities a and b; but eachito a series of velocities changing every moment, or Whicli is the fame thing, to an accelerated velocity, by which it is generaled, during a certain minute particle of time: that the nascent beginnings or evanescent end-ings of finite quantities, whicli are produced in moments or infinitelysmali paris of time, are alone proportional to gi ven velocities: that there- fore, in order to conceive the first fluxions, we must conceive time di-
See the soreming scheme in Sect. 36. et Sech. 36.
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vied into momenis, increments generaled in those momenis, and velo cities proportional to those incremenis: that in order to conceive secondand third fluxions, we must suppose that the nascent principies or momentaneous increments have themselves also other momentaneous incremenis, whicli are proportional to their respective generating velocities: that the velocities of these second momentaneous increments are second fluxionsalliose of their nastent momentaneous increments third suxions. Aiad se on ad infinitum.
XLIII. By subducting the increment generaled in the first momentsrom that generaled in the second, we get the increment of an increment. And by subduci ing the velocity generating in the first moment rom that generating in the second, we get the fluxion os a fluxion. In like manner, by subducting the disserenoe of the velocities generat ing iathe two first momenis, stom the excess of the velocity in the third above that in the second moment, we obtain the third fluxion. And aster the fame analogy we may proceed to Gurth, fifth, si xth fluxions, M. Andis we cali the velocities of the first, second, third, fourth moments ara, c, d, the series of fluxions will be as above, a. b - a. c a b - a. d - 3 c ε 3b - a. ad in itum, i. e. x x x. x. ad infriitum.
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XLV. One would thinli that men could not speah too examy on sonice a subjeci. And yet, as was fore hinted, we may osten observethat the exponents of fluxions or notes representing fluxions are con- unded with the fluxions them selves. Is not this the case. when justaster the fluxions of fio ing quantities were seid to he the celerities of their increasing, and the second fluxions to the mutations of the sirst
suxions or celerities, we are told that z. z. z. z. d. z. represenis a se-ites of quantities, whereos each subsequent quantity is the fluxion of the preceding, and each sorinoing is a fluent quantity having the sollowing one sor iis fluxion tXLVI. Divers series of quantities and expressions, geometrical and algebraical, may be easily conceived, in lines, in sursaces, in species, to hecontinued without end or limit. But it will not be Mund se east toconceive a series, ei ther of mere velocities or os mere nascent incremenis, distinet there om and corresponding thereunto. Some perhaps may be led to think the author intended a series of ordinates, where- in each ordinate was the fluxion of the preceding and fluent of the following, i. e. that the fluxion of one ordinate was itself the ordinate of another curve; and the fluxion of this last ordinate was the ordinato of yet another curve; and se on ad in itum. But who canconceive how the fluxion whether veloci ty or nascent increment) of an ordinate 8 Or more than that cach preceding quantity or fluent is related to iis subsequent or fluxion, as the area of curvilinear figureto iis ordinate , agreeably to what the author rema rhs, that each pr ceding quantity in such series is as the area of a curvilinear figure, whereos the abscisi is et, and the ordinate is the solio ing quantity. XLVIL Upon the whole it appears that the celerities are disinissed, and instead thereos areas and ordinates a re introduced. But however
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expedient such analogies or such expressions may be found for facilitating the modern quadratures, yet we mali not find any light gi ven usthere by in to the original reat nature os fluxions; or that we are enabledio frame frona the nce just ideas of fluxions considered in them lues. Inali this the generat ultimato drise of the aut hor is very clear, but his principies a re obscure. But perhaps those theories of the great author arenot minutely considered or canuassed by his disciples: who seem ea ger,as was besore hinted, rather to operate than to know, rather to apply hisrules and his forms, than to understand his principies and enter into his notions. It is neverthelest certain, that in order to solio is him in his quadratures, they must find fluents fio m fluxions; and in order to this, they must know to find fluxions fio m fluenis; and in order to find suxioris, they must first know what fluxions are. Other i se they proceed without clearnesis and without science. Illius the direct method precedegthe inverse, and the knowledge of the principies is supposed in both. But as for opera ting according to rules, and by the hel p of generat fornis,uhereos the original principies and reasons are not understood, this is lobo esteemed meret y technicat. Be the principies there re e ver so abstruse and metaphysical, they must be studi ed by whoe ver would comprehend the doctrine os fluxions. Nor can any geometrician have aright to apply the rules of the great author, Without first considerint his metaphysical notions whence they Were derived. These how necesiarysoever in ordor to science, whicli can ne ver be atta ined without a preci se, clear, and accurate conception of the principies, a re neverthelesia by se verat caretesty passed over; while the expressions alone are dweli on and considered and trealed with great skill and management, the nce to obtain other expressions by methods, suspicious and indirect to say the least) is considered in them selves, ho e ver recommended by inductionand authori ty ; two motives Whicli are aclino ledged sufficient to hegeta rational faith and morat persuasion, but nothing highen
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heen seid, and to screen false principies and inconsistent rea nings, by a generat preten ce that these objections and remarlis are metaphysical. But this is a va in pretence. For the plain sense and truth of what is ad-vanced in the foregoing rema rhs, I appeal to the undersianding of e very unprejudiced intelligent reader. To the fame Ι appeal, whether thepoinis rema rhed upon a re not mosi incomprehensi ble metaphysios. Andmetaphysics not of mine, but yoUr o n. I Would not be understood to 1nser, that your notions are false or Vaci because they are metaphysicat. Nothing is either triae or false sor that reason. Whether a potnt be call-ed metaphysical or no, avalis litile. The question is, whether it be clearor Obscure, right or wrong, weli or ill- deduced t
XLIX. Although momentaneous incremenis, nascent and evanescent
quantities, suxions and infinitesiimals of ali degrees, are in truth lachmadowy entities, se disticuli to imagine or conceive distinctly, that stosay the least) they cannot be admitted as principies or objects of clear and accurate science: and although this obscurity and i ncom prehen sibili ty ἡyour metaphysios had been alone lassicient to allay your pretensions toe viden ce; yet it hath, is I mistahe not, hecn further mewn, that your inferences are no more just than yOur conceptions are clear, and that yourlogies are as eXceptionable as your metaphysios. It mould Qem there- fore u pon the whole, that your conclusions are not attained by just rea- Toning from clear principies; consequently, that the employment of modern analys g, ho ver useful in mathematical calculations and constructions, doth not habituale and qualisy the mind to apprehend clearly and infer justly; and consequently, that you have no right in virtve of such habiis, to dictato out of your proper sphere, beyond which your judν
ment is to past for no more than that os other men.
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L. Os a long time I have suspected, that these modern analytics werenot scientifical, and gave seme hinis thereos to the public about twenty-five years ago. Since whicli time, I have been diverted by other occu pations, and imagined Ι might employ myself better than in deducingand laying together my thoughis on se nice a subjeci. And though oflate I have been called upon to mahe good my suggestions ; yet as the
thematics which he would patronige, I mould have spared myself the troubie of writing for his conviction. Nor mould I now have troubledyou or myself with this addrest, after se long an intermission of these studies, were it not to prevent, se far as I am able, your imposing onyourself and others in matters of much higher moment and concern. And to the end that you may more clearly comprehend the force and
design of the foregoing remarlis, and pursue them stili farther in yourown meditations, Ι mali su oin the solio ing eries. lues r. Whether the object of geometry be not the proportions os assignable extension si And whether there be any need os considering quantities ei ther infinitely great or infinitely smalit Qu. z. Whether the end of geometry be not to measure assignablefinite extension t And whether this practical vlew did not first put menon the study of geometrytav. a. Whether the mistaking the objeci and end of geometry hathnot created needlest dissiculties, and wrong pursu iis in that science tau. ψ. Whether men may properly be seid to procred in a scientisiemethod, without clearly conceiving the object they are conversant a boui.
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Qu. s. Whether it doth not sussice, that every assignable number os paris may be contained in seme assignabie magnitude t And whether ube not un necessary, as weli as absurd, to suppose that finite extension is
sti. 6. Whether the diagrams in a geometrical demonstration are notio he considered as signs os ali possibie finite figures, of ali sensibie and imaginable extensions or magnitudes of the same hind t
culties and absurdities, so long as either the abstract generat idea os e tension, or absolute externat extension be supposed iis true object isu. 8. Whether the notions os absolute time, absolute place, and absolute motion he not most abstractedly metaphysical Θ Whether it be posisi bie sor us to measure, compute, or lino them taeu. 9. Whether mathematicians do not engage themselves m disputes and paradoxes, concerning What they neither do nor can conceive t Andwhether the doctrine of forces be not a lassicient proos of this t Qu. Io. Whether in geometry it may not sussice to consider assigna-hle finite magnitude, without concerning ourselves with infinityi And.whether it would not be righter to measure large polygons having finites des, instead of curves, than to suppost curves are polygons os infinite-fimal sides, a supposition neither true nor conceivablat u. II. Whether many potnis, which are not readily assented to, arenot neverthelesi true t And whether those in the two sollowing eries may not be of that number tu See the Latin Treatist De Diti.
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notion os extension prior to motion t Or whether is a man had neverperceived motion, he Would ever have known or conceived one thing tobe distant stom another t Qu. I 3. Whether geometrical quantity hath coexistent parist AndWhether ali quantity be not in a flux as weli as time and motion tetu. I . Whether extension can be supposed an attribute os a bella gimmutabie and eternat tau. Is . Whether to decline examining the principies, and unravetiling the methods used in mathematics, would not me a bigotry in mathematicians t2n. I 6. Whether certain maxims do not pasi current among analysts, whicli are mocking to good sense t And whether the common assumption that a finite quantity divided by nothing is infinite be not os
this numbertetu. 17. Whether the considering geometrices diagrams absolutetyor in themselves, rather than as representatives of ali assignabie magnitudes or figures of the fame hind, be not a principat cause of the supposing finite extension infinitely divisibie; and of ali the difficulties and
absurdities consequent thereupon teu. 18. Whether from geometrical propositions being generat, and the lines in diagrams being there re generat substitutes or representatives, it doth not follow that we may not limit or consider the numberos paris, into which lach particular lines are divisibie t
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198 The Ana s. Qu. 19. When it is seid or implied, that such a certain line delinealed On paper contains more than any assignable number of paris, whether any more in truth ought to be understood, than that it is a signindisserently represen ting ali sinite lines, he they e ver se great. In whichrelative capaci ty it contains, i. e. stands for more than any assignablenum ber of paris i And whether it he not altogether absurd to suppose a finite line, considered in iiself or in iis own possitive nature, stould comtain an infinite number of paris taeu. 2O. Whether ali arguments for the infinite divisibili ty of sinite extention do not suppose and imply, et ther generat abstract ideas or absolute externat extension to be the object of geometry t And theresere, whether, along with those suppositions, such arguments also do not cease iand vanim tDv. 21. Whether the supposed infinite divisibili ty of sinite extensionliath not been a mare to mathematicians, and a thorn in their fides tAnd whether a quantity infinitely dimini med and a quantity infinitelysmali, a re not the fame thingtetu. 22. Whether it be necessary to consider velocities of nascent oreVanescent quantities, or momenis, or infinitesimals t And whether the introducing of things Q inconceivable be not a reproach to mathematiciis tetu. 23. Whether inconsistencies can be truthst Whether potnis repugnant and absurd are to be admitted upon any subject, or in any science ' And whether the use of infinites ought to be allo ed, as a su D
sicient pretexi and apology for the admitting of such potnis in geometry t