3-M:/
M
University of California • Berkeley
THE
ANALYST;
O R, A
DISCOURSE
AddrcfTcd to an
Infidel Mathematician.
WHEREIN
It IS examined whether the Objedl, Princi- ples, and Inferences of the modern Analy- fis are more diftindly conceived, or more evidently deduced,than Religious Myfteries and Points of Faith.
By the A u T H o R of ne Minute Phllofipher.
Firjl cajl out the beam out of thine own Eye ; and then Jhalt thou fee clearly to cajl out the mote out of thy bro- ther's eye, S. Matt. c. vii. v. 5".
LO N D ON: Printed for J. Tonson in the Strand, 17J4.
(P
.1^''
.#■
THE
CONTENTS.
SEC T. I. Mathematipiam frefumed to be the great Mafier^ cf Reafon, Hence an nmche Reference ^o their decifions where they have no right to.dedde. This .one Cmfe fif .Infidelity.
II. Their Prirwiplejs and Methods to be exa-^ mined with the fame Jreedom^ which they -ajfume witi regard to the Principles and Myjleries of Religion, In what Senje and how far Geometry is to be allowed an Jfnproipement of :the Mind.
III. Fltixions'the •greatObjedi and Employment of the profound Geometricians in the pre-
feut ^e. What tkeje pluxioiu are.
IV". "Moments or nafoent Increments cfflowirig ^antities difficiilt 4o conceive. Fluxions <f different -Orders. Second and third Fluxions obfcure Myjleries.
A 2 V. Differ
4340iG
The CONTENTS.
V. Differences^ i. e. Increments or Decre-^ ments infinitely fmall^ ufed by foreign Ma^ thematicians infiead of Fluxions or Velo^ cities of nafcent and evanefcent Incre-- ments.
VI. Differences of various Orders^ i. e. ^an-- tiiies infinitely lefs than ^antities infi* nitely little ; and infinitefimal Farts, of infinitefimals of infinitefimalsy ix.c, without end or limit,
VII. Myfieries in faith unjufily objeSiedagainJi by thofe who admit them in Science.
VIII. Modern Analyfis fuppofed by themfelves - to extend their views even beyond infinity :
Deluded by their own Species or Symbols.
IX. Method for finding the Fluxion of a ReSi-- 'ungle. of two indeterminate ^antities^ .Jhewed to be illegitimate andfalfe.
X. Implicit Deference of Mathematicalmen
for the great Author of Fli^xions. 7'heir eamefinefs rather to go onfafi and far ^ than tofet out warily and fee their way difiin^lly,
XL Momen^
TSe CONTENTS.
XI. Momenfums difficult io comprehend, Nd mddk ^antity to be admitted between a finite ^antifj and nothings 'without admitting Infiniiefimals.
XII. ^he Fluxion of any Power of afiowing ^antity. Lemma premifed in order to examine the method for finding fucb Fluxion.
XIII. The rule for the Fluxions of Powers attained by unfair reafoning,
XIV. The aforefaid re afoning farther unfold-^ ed andjloew'd to be illogical
XV. No true Conclufion tobe jufily drawn by dire6l confequence from incon/ifient Sup^ pojitiom. The fame Rules of right rea-^ fon to be obfer^ved^ whether Men argue
in Symbols or in Words,
XVI. An Hypothefis being defiroyed^noconfe^ quence offuch Hypothefis to be retained,
XVII. Hard todifiinguifij between evanefcent Increments and infinitefimal Differences, Fluxions placed in various Lights, The great Author^ it feemsy not fatisfied with his own Notions.
XVIII. ^^/f-
The CONTENTS.
XVIU. ^4mt^iesinjituielyfmallfid^fiqfed and rejeBed by Leiboitz and Ms Polbwers. 3S[o ^antityy according ^o them^ greater orfmaUerfar .the Addkim or Subdue^ tion of its InfiniteJimaL
XIX. CmcJyfions io he ^Dvtd byibe Princi^ ples^and not Principles by. the Gonclu/ions.
XX. The Geometrical Analyft confidered as a Logician'^ and his Di/coverieSy mot in themfehes^ iut 4is derived fram fucb Principles and byfuch Inferences.
XXI. A I'angent drmsDnto, the Parabola ac-^ cording to the calculus difFerentialis. "Truth Jhewn to be the refult oferror^ and how,
XXII. 3y vlrjiut of a (twofold miftake Ana* lyjls arrive at Truthjxutmt at Science : ignorant how they come at their own Concliifions,
XXIII. The Conclupon never evident or accu-^ rate^ in viitue cf ibfcure or inaccurate Premifes. Pinite '^antities mi^ht be rejeUed as well as Infiniiefimals.
XXIV. The foregoing DjsBrit^ farther illu* ftraied. XXV. Sundr^f
Tte CONTENTS.
jtXV. Sundry Obfirvathns thereupon.
XXVI. Ordinate found fram the Area by means of evanefcent Increments.
XXVII. In the foregoing Cafe the fuppofed evanefcent Increment is realty a finite ^antity, defiroyed by ojj equal ^aniity mth an oppofite Sign.
XXVIII. ^he foregoing Cafe fut generally. Algebraical ExpreJ/ions compared with Geometric at ^anfitiiS.
XXIX. Correfpondent ^antities Algebraical and Geometrical equated. The Analyfis fhe*wed not to obtain in Infintefnvals^ hut
it muji alfo obtain infinite ^antities.
XXX. The getting rid of ^antities by the received Principles, whether of Fluxions or of Differ ences, neither good Geometry nor good Logic. Fluxions or Velocities, i»hy introduced,
XXXI. Velocities not to be abfiraEted from Ttme and Space: Nor their Proportions to be invefiigated or confidered exclufively ofJime and Space.
XXXII. Dijicult
The CONTENTS.
XXXII. Difficult andobfcure Points conjlitufe the Principles of the modern Analyjis^ and are the Foundation on which it is built.
XXXIIL The rational Faculties whether im^ proved byfuch objcure Analytics.
XXXIV. By what inconceivable Steps finite Lines are found proportional to Fluxions, Mathematical Infidels firain at a Gnat andfwallow a Camel,
XXXV. Fluxions or Infinitefimalsmt to bea- voided on the received Principles. Nice Ab" firaBions and Geometrical Metaphyfics. -
XXX VI. Velocities of nafcent or evanefcent ^antities^ whether in reality underfiood andfignified by finite Lines and Species.
XXXVII. Signs or Exponents obvious-^ but' Fluxions the mf elves not fi.
XXXVIII. Fluxions^ whether the Velocities with which infinitefimal Differences are generated ?
XXXIX. Fluxions of Fluxions or fecond Fluxions^^ whether to be conceived as Velo- cities of VclocitieSy or rather as Velocities of the fecond nafcent Increments?
XL, Fluxions
The CO NT E NTS.
XL. Fluxions confidered, fometimes in oni Senfe^ fometimes in another : One while in themfehes^ another in their Exponents : Hence Confujion and Obfcurity,
XLI. Ifochronal Increments^ whether finite or ndfcenty proportional to their refpeEiive Velocities.
XLII. T'ime fuppofed to be divided into Mo^ ments: Increments generated in thofe Moments : And Velocities proportional to thofe IncrementSi
XLIII. Fluxions^ fecond, thirds fourthy &c. *wh^t they are; how obtained, and how re'- prefented. What Idea of Velocity in a Mo^ ment of Time and Point of Space.
XLIV. Fluxions of all Orders inconceivable;^
XLV. Signs or Exponents confounded with the Fluxidnii
XLVI. Series of Expffjions or of Notes eafily contrived. Whether a Series, of mere Ve- locities, or of mere nafcent Increments^ cerrefponding thereunto, be as eafily con^ teivtdf
B j^y. Cekriliei.
The CONTENT!
XL VII. Celerities difmijfed, and injiead there* of Ordinates and Areas introduced, Ana^ logies and Exprejfiom ufeful in the modern ^adratiireSy may yet be ufelefs for ena- bling us to conceive Fluxions, No right to apply the Rules without knowledge of the Principles,
XLVIII. Metaphyjics of modern Analyjis mojl incomprehenfble.
XLIX. Analyjis employ* d about notional Jl:>a'* dowy Entities, Their Logics as exception nable as their Metaphyjics.
L. Occafion of this Addrefs, Conclujiont Queries.
THE
t H E
ANALYST.
I. ^5SSS^^23SHOtJGli i ama Stranger to your Perfon, yet I am not. Sir, a Stranger to the Repu- tation you have acquired, in that branch of Learning which hath beeli your peculiar Study ; nor to the Authority that you therefore aflume in things foreign to your Profeffion, nor to the Abufe that you, and too many more of the like Cha- rader, are known to make of fuch tindu6: Authority, to the mifleading of tinwary Perfons in matters of the higheft Con- cernment, and whereof your mathemati- cal Knowledge can by no means qualify you to be a competent Judge. Equity in- deed and good Senfe would incline one to* difiregar d the Judgment of Men, in Points* B 2 whielif
Thb Analyst;
III. The Method of Fluxions is the ge- neral Key, by help whereof the modern Mathematicians unlock the fecrets of Geo- metry, and confequcntly of Nature. And as it is that which hath enabled them fq remarkably to outgo the Ancients in difr covering Theorems and folving Problems, the exercifc and application thereof is be- come the main, if not fole, employment of all thofe who in this Age pafs for pro- found Geometers. But whether this Me- thod be clear or obfcure, confiftent or I'epugnant, demonftrative or precarious, as I {hall inquire with the utmoft impar- tiality, fo I fubmit my inquiry to your own Judgment, and that of every candid Reader; Lines arc fuppofed to be gene- rated * by the motion of Points, Plains by the motion of Lines, and Solids by the motion of plains. And whereas Quan- tities generated in equal times are greater or leffer, according to the greater or leffer Veltcity, wherewith they increafe and are generated, a Method hath been found to determine Quantities from the Velocities of their generating Motions.
* Introd. ad Quadraturam Curvarum.
And
The a n a t y s tJ f,
And fuch Velocities are called Fluxions: and the Quantities generated are called flowing Quantities. Thefe Fluxions are faid to be nearly as the Increments of the flowing Quantities, generated in the leaft equal Particles of time ; and to be accurately in the firft Proportion of the nafcent, or in the laft of the evanefcenr, Increments. Sometimes, inftead of Velo- cities, the momentaneous Increments or Decrements of undetermined flowing Quantities are confidered, under the Ap* pellation of Moments.
IV. By Moments we are not to under- fl:and finite Particles. Thefe are faid not to be Moments, but Quantities genera- ted from Moments, which laft are only the nafcent Principles of finite Quanti- ties. It is faid, that the minuteft Errors are not to be negled:ed in Mathematics : that the Fluxions are Celerities, not pro- portional to the finite Increments though ever fo fmall ; but only to the Moments or nafcent Increments, whereof the Pro- portion alone, and not the Magnitude, is confidered. And of the aforefaid Fluxions ^ 4 there
P T H E A N A L y S T.
there be other Fluxions, which Fluxions pf Fluxions are called fecond Fluxions. And the Fluxions of thefe fecond Fluxions are called third Fluxions : and foon, fourth, fifth, fixth, &c. ad infinitum. Now as our Senfe is ftrained and puzzled with the perception of Objedls extremely minute, even fo the Imagination, which Faculty derives from Senfe, is very much ftrained ^nd puzzled to frame clear Ideas of the leaft Particles of time, or the leaft Incre- nients generated therein : and much more fo to comprehend the Moments, oi: thofe Increments of the flowing Quanti- ties in Jiatu nafientiy in their very^ firft origin or beginning to exift, before they become finite Particles. And it feems ftill more difficult, to conceive the abftradled Velocitie? of fuch nafcent impeffed En- tities. But the Velocities of the Velocities, the fecond, third, fourth and fifth Velo- cities, ^c, exceed, if I miftake not, all Humane Underftanding. The further the Mind analyfeth aqd purfueth thefe fugi- tive Ideas, the niore it is loft and be- yv^ildered; the Objeds, at firft fleeting and piinute, foon vanifbing put of fight. Cer- tainly
The Analy s t,
tainly In any Senfe a fecond or third Fluxion feems an obfcure Myflery. The incipient Celerity of an incipient Celerity, the nafcent Augment of a nafcent Aug- ment, /. e, of a thing which hath no Magnitude: Take it in which light you pleafe, the clear Conception of it will, if I miftake not, be found impofiible, whe- ther it be fo or no I appeal to the trial of every thinking Reader. And if a fecond Fluxion be inconceivable, what are we to think of third, fourth, fifth Fluxions, and fo onward without end ?
V. The foreign Mathematicians are fuppofed by fome, even of our own, to proceed in a manner, lefs accurate per- haps and geometrical, yet more intelligi- ble. Inftead of flowing Qu^antities and their Fluxions, they confider the variable finite Quantities, as increafing or dimi- nifhing by the continual Addition or Sub- dudion of infinitely fmall Quantities. In- ftead of the Velocities wherewith Incrc- nients are generated, they confider the In- crements or Decrements themfelves, which ;hey call Differences, and which are fup- pofed
to THEANALYSrr
pofed to be infinitely fmall. The Diffe-* rence of a Line is an infinitely little Line ; of a Plain an infinitely little Plain. They fuppofe finite Quantities to confift of Parts infinitely little, and Curves to be Poly* gones, whereof the Sides are infinitely lit- tle, which by the Angles they make one with another determine the Curvity of the Line. Now to conceive a Quantity in* finitely fmall, that is, infinitely lefs than any fenfible or imaginable Quantity, or than any the leaft finite Magnitude, is, J confefs, above my Capacity. But to con-» ceive a Part of fuch infinitely fmall Quan- tity, that fhall be ftill infinitely lefs than it, and confequently though multiply*d infinitely fliall never equal the ipinuteft finite Quantity, is, I fufpedl, an infinite Difficulty to any Man whatfoever; and will be allowed fuch by thofe who can- didly fay what they think ; provided they really think and refle<ft, and do not take things upon truft.
VI. And yet In the calculus differentialit^ which Method ferves to all the fame In- tents and Ends v^ith that qf Fluxions,
our
T H E A N A L y S T. II
pur modern Analyfts are not content to confider only the Differences of finite Quantities: they alfo confider the Differ rences of thofe Differences, and the Diffe- rences of the Differences of the firfl Diffe- rences. And fo on ad infinitum. That is, they confider Quantities infinitely lefs than the leafl difccrnible Quantity ; and others infinitely lefs than thofe infinitely fmall oncsj and flill others infinitely lefs than the prece- ding Infinitefimals, and fo on without end pr linaic. Infomuch that we are to ad- pit an infinite fucceflion of Infinitefimals, each infinitely lefs than the foregoing, $ind infinitely greater than the following* As there are firfl, fecond, third, fourth, iifth, Csfr, Fluxions, fo there are Diffe- rences, firft, fecond, third, fourth, &fr. in $in infinite ProgrefHon towards nothing, which you ftill approach and never arrive at. And (which is nKjfl ftrange) although you (hould take a Million of Millions of thefe Infinitefinaals, each whereof is fup- pofed infinitely greater than fome other real Magnitude, and add them to the leaft given Quantity, itfhall be never the bigger. For this is one gf the modefl pojtulata of
pur
rt T H E A N A L Y S T?
our modern Mathematicians, and is a Cor- ner-ftone or Ground-work of their Specu- lations.
• VII. All thefe Points, I fay, are fup- pofed and believed by certain rigorous Ex- a<ftors of Evidence in Religion, Men who pretend to believe no further than they can fee. That Men, who have been con- verfant only about clear Points, fhould with difficulty admit obfcure ones might not feem altogether unaccountable. But he who can digeft a fecond or third Fluxi- on, a fecond or third Difference, need not, methinks, be fqueamifli about any Point in Divinity. There is a natural Prefump- lion that Mens Faculties are made alike. It is on this Suppofition that they attempt to argue and convince one another. What, therefore, Ihall appear evidently impoffi- ble and repugnant to one, may be pre- fumed the fame to another. But with what appearance of Reafon {hall any Man prefume to fay, that Myfteries may not be Objedts of Faith, at the fame time that he himfelf admits fuch obfcure Myfteries to be the Obje(^ of Science ?
VIII. It
Irviii
The a k a l V s t. "i'f
VIIL It mu ft indeed be acknowledged, the modern Mathematicians do not confi- der thefe Points as Myfteries, but as clear- ly conceived and maftered by their com- prehenfive Minds. They fcruple not to fay^ that by the help of thefe new Analy-^ tics they can penetrate into Infinity it felf i That they can even extend their Views be- yond Infinity : that their Art comprehends not only Infinite, but Infinite of Infinite (as they exprefs it) or an Infinity of Infinites^ But, notwithftanding all thefe Aflfertions and Pretenfions, it may be juftly queftibn- ed whether, as other Men in other Inqui>- ries are often deceived by Words or Terms-, fo they like wife are not wonderfully de- ceived and deluded by their own peculiar Signs, Symbols, or Species. Nothing iseafier than to devife Expreflions or Notations for Fluxions and Infinitefimals of the firft, fc-^ cond, third, fourth and fubfequent Orders^ proceeding in the fame regular form with-* out end or limit x\ ^^ ^' ^;, Gfc. or dx, ddx, dddx, ddddx &c, Thefe Expreflions in- deed arc clear and diftindl, and the Mind finds no difficulty in conceiving them to be continued beyond any aflignable Bounds;
But
14 The ANALVst.
fiiit if we remove the Veil and look uric(ef-. neath, if laying afide the Expreflions wc fct our felves attentively to confider the things themfelves, which are fuppofed to be expreffed or marked thereby, we fhall difcover much Emptinefs, Darknefs, and Confufion j nay^ if I miftake not> direft Impoffibilities and Contradictions. Whe- ther this be the cafe or no, every think- ing Reader is intreated to examine and judge for himfelfi
iX. Having cotifidered the ObjcdV, I proceed to confider the Principles of this pew Analyfis by Momentums, Fluxions^ or Infinitefimals ; wherein if it (hall appear that your capital Points, upon which the reft are fuppofed to depend, include Er- ror and falfe Rcafoning j it will then fol- low that you, who are at a lofs to eon- dud your felves, cannot with any decen- cy fet up for guides toother Mcri. The main Point in the method of Fluxions is to obtain the Fluxion or Momentum of the Redtangle or Produdt of two indetef- fninate Quantities. Inafmuch as from thence arc derived Rules for obtaining thd
Fluxiofitr
The ANALYst. ij^
Fluxions of all other ProduiSts and Pow- ers; be the Coefficients or the Indexes what they will, integers or fradtions, rational or furd. Now this fundamental Point one would think fhould be very clearly made out, confidering how much is built upon it, and that its Influence extends throughout the whole Analyfis. But let the Reader judge. This is given for De- monftration. * Suppofe the Produd: or Redangle AB increafed by continual Mo- tion: and that the momentaneous Incre- ments of the Sides A and B are a and i^ When the Sides A and B were deficient, or lelTer by one half of their Moments, the Rect- angle was ji^—^a xi^. — i^ i.e. AB—^^aB ^^ibji-^-iab. And as foon as the Sides A and B are increafed by the other twa halves of their Moments, the Rectangle becomes ^TT^ ^ FTT^or AB-^iaB^ ibA-^r-ab, From the latter Redangle fubdudl the former, and the remaining difFc* fence will be aB-^bA. Therefore the Increment of the Redangle generated by the intire Increments a and b\s aB'\-bA,
r * Naturalis Philofophix principia mathematica, I. z. km. 2.
1 tf T H E A N A L Y S t.
^jE. jD. But it is plain that the direct: and true Method to obtain the Moment or Increment of the Rectangle jiB^ is to take the Sides as increafed by their whole In- crements, and (o multiply them together*, A-^-a by 5 + ^, the Produdt whereof jiB + aB + 6A'\'ab is the augmented Redlangle- whence if we fubdudl AB^ tht Rerriainder aB + iA + aif will be the trub Increment of the Reftangle, exceeding that which was obtained by the former illegitimate and indiredl Method by the Quantity ab. And thfS holds univerfally be the Quantities a and i what they will, big or little, Finite or Infinitefimal, Incre- ments, Moments, or Velocities. Nor will it avail to fay that a^ is a Quantity ex- ceeding fmall t Sihct we are told that in re-^ bus mathematicis errores quam minimi nOn funt contemnendi, * Such reafoning as this,, for Demonftration, nothing but the obfcurity of the Subject could have encouraged or indu- ced the great Author of the Fluxionary Me- thod to put upon his Followers, and nothing but an implicit deference to Authority couH move them to admit. The Cafe indeed is
* Imrod. ad Quadraturam Curvarum.
difBcufc
T H E A N A L V S t. 17.
difficult. There can be nothing done till you have got rid of the Quantity a b. In order to this the Notion of Fluxions is fhifted: It is placed in various Lights: Points which fhould be clear as firft Prin-^ ciples are puzzled 3 and Terms which fhould be fteadily ufed are ambiguous. But notwithftanding all this addrefs and skill the point of getting rid oi ab can- not be obtained by legitimate reafoning. ^ If a Man by Methods, not geometrical or demonftrative, (hall have fatisfied himfelf of the ufefulnefs of certain Rules; which he afterwards (hall propofe to his Difciples for undoubted Truths; which he under- takes to demonftrate in a fubtile man- ner, and by the help of nice and in- tricate Notions 5 it is not hard to conceive that fuch his Difciples may, to fave them- felves the trouble of thinking, be inclined to confound the ufefulnefs of a Rule with the certainty of a Truth, and accept the one for the other; efpecially if they are Men accuftomed rather to compute than to think; earneft rather to go on faft and far, than folicitous to fee out warily and fee
their way diftindtly.
C XI. The
I 8 T H E A N A t r S T.
XL The Points or meer Limits of nal- cent Lines are undoubtedly equals as hav- ing no more magnitude one than ano- ther, a Limit as fuch being no Quantity. If by a Momentum you mean more than the very initial Limit, it muft be either a finite Quantity or an Infinitefimal. But all finite Quantities are exprefly excluded from the Notion of a Momentum. There* ^ fore the Momentum muft be an Infini- tefimal. And indeed, though much Ar- tifice hath been employed to efcape or a- void the admiffion of Quantities infinitely fmall, yet it feems ineifedtual. For ought I fee, you can admit no Quantity as a Medium between a finite Quantity and nothing, without admitting Infinitefimals. An Increment generated in a finite Parti- cle of Time, is it felf a finite Particle ; and cannot therefore be a Momentum^ You muft therefore take an Infinitefimal Part of Time wherein to generate your Momentum. It is faid, the Magnitude of Moments is not confidered : And yet thefe- fame Moments are fuppofed to be divided into Parts. This is not eafy to conceive,, no more than it is why we fhould take
Quantities'
T H E A N A L Y S ir; ffll
Qoantities lefs than A and 5 in order to obtain the Increment of A By of which proceeding it muft be owned the final Caufe or Motive is very obvious; but it is not fo obvious or eafy to explain a juft and legitimate Reafon for it, or (hew it to be Geometrical.
XII. From the foregoing Principle fo demonftrated, the general Rule for find- ing the Fluxion of any Power of a flow- ing Qujantity is derived *. But, as there feems to have been fome inward Scruple or Confcioufnefs of defed: in the forego- ing Deraonftration, and as this finding the Fluxion of a given Power is a Point of primary Importance, it hath therefore been judged proper to demonftrate the fame in a different manner independent of the foregoing Demonftration. But whe- ther this other Method be more legitimate and conclufive than the former, I pro- ceed now to examine j and in order there- to (hall premife the following Lemma. " If with a V^iew to demonftrate any
* Philofophije naturalis principia Mathematica, lib. 2. lem. 2.
B 2 ^ Propo-
;3L0 T HE A N A L Y S f,
*^ Propofition, a certain Point is fuppofcd, " by virtue of which certain other Points *^ ^re attained; and fuch fuppofed Point " be it feif afterwards dcftroyed or rejec- " ted by a contrary Suppofition ; in that " cafe, all the other Points, attained thereby " and confequent thereupon, muft alfo " be deftroyed and rejeded, fo as from " thence forward to be no more fuppo- " fed or applied in the Dcmonftration." This is fo plain as to need no Proot
XIII. Now the other Method of ob- taining a Rule to find the Fluxion of any Power is as follows. Let the Quantity x flow uniformly, and be it propofed to find the Fluxion of x"» In the fame time that X by flowing becomes x + o^y the Power a:» becomes x-\-o\ » , i. e. by the Method of infinite Series x^ + nox^ — ^
+ '^''^ -oox" ^ + £f^. and the Incre- ments 0 and nox^ — ^ -f lUZf ^o^v^—z
2
4- &c, are one to another as i to «;c»~i ^ IJLlLJLoxn—^ + 0*c. Let now the In- crements vanifli, and their laft Proportion will be I to «x» — ^ Butitfliould feem
that
I
T H E A N A L Y S T- 11
that this reafoning is not fair or conclufive. For when it is faid, let the Increments vanifh, /. e, let the Increments be nothing, or let there be no Increments, the former Suppofition that the Increments were fomething, or that there were Increments, is deftroyed, and yet a Confequence of that Suppofition, i, e, an Expreflion got by virtue thereof, is retained. Which, by the foregoing Lemma, is a falfe way of reafoning. Certainly when we fuppofe the Increments to vanifh, we muft fup- pofe their Proportions, their Expreflions, and every thing elfe derived from the Sup- pofition of their Exiftcnce to vanifh wittt theoi.
XIV. To make this Point plainer,- -J lift fliall unfold the reafoning, and propofe Jt in a fuller light to your View. It amounts therefore to this, or may in other Words be thus expreflfed. . I fuppofe that the Quantity x flows, and by flowing is in- creafed, and its Increment I call o, fo that by flowing it becomes x-^-o. And as X increafcth, it follows that every Power of X is likewife increafed in a due Pro- C 3 portion.
tjf, The AaN^^lyst*
portion. Therefore as x becomes x-irJ?, A?« will become x -^ o\": that is, .accord- ing to the Method of infinite Series, ^«
+ nox""—^ -V^-^^-^oox^—^ + &c. And
2
if from the two augmented Quantities we fubdud the Root and the Power refpec^ lively, we fhall have remaining the two Increments, to wit, o and nox^—'^ + ^'—-^oox^ — '^ 4- G'r. which Increments, being both divided by the common Divi- for Oy yield the Quotients i and nx^ — ^
■4- l^'^^ox^ — ^ 4- &c. which are there- fore Exponents of the Ratio of the Incre- ments. Hitherto I have fuppofed that x flows, that X hath a real Increment, that
0 is fomething. And I have proceeded all 1 along on that Suppofition, without which
1 fhould not have been able to have made fo much as one fingle Step. From that Suppofition it is that I get at the Incre- ment of a:», that I am able to compare it with the Increment of Xy and that I find the Proportion between the two In- crements. I now beg leave to make a new Suppofition contrary to the firft, /. e, I will fuppofc that there i? no Increment
of
T H E A N A L Y S T. I5
of AT, or that 0 IS nothing ; which fecond Suppofition deftroys my firft, and is in- confiftent with it, and therefore with eve- ry thing that fuppofeth it. I do never- thelefs beg leave to retain ^a:» — », which is an Expreffion obtained in virtue of my firft Suppofition, which neceffarily pre- fuppofeth fuch Suppofition, and which could not be obtained without it : All which feems a moft inconfifl:ent way of arguing, and fuch as would not be allow- ed of in Divinity.
XV. Nothing is plainer than that ih jufl: Conclufion can be diredly drawn from two inconfiftent Suppofitions. You may indeed fuppofe any thing poflible : But af- terwards you may not fuppofe any thing ^^that deftroys what you firft fuppofed. Or ' if you do, you muft begin de novo. If therefore you fuppofe that the Augments vanifli, i,e, that there arc no Augments, you are to begin again, and fee what fol- lows from fuch Suppofition. But nothing will follow to your purpofe. You cannot by that means ever arrive at your Con- clufion, or fugceed In, what is called by
B 4 the
5t4 .The Anal y s^t.
the celebrated Author, the Inveftigation of the firft or laft Proportions of nafcent and evanefcent Quantities, by inftituting the Analyfis in finite ones. I repeat it again: You are at hberty to make any poflible Suppofition: And you may de- ftroy one Suppofition by another: But then you may not retain the Confequences, or any part of the Confequences of your firft Suppofition fo deftroyed. I admit that Signs may be made to denote either any thing or nothing : And confequently that in the original Notation x-\- Oy o might have fignified either an Increment or no- thing. But then which of thefe foever you make it fignify, you muft argue con- fiftently with fuch its Signification, and not proceed upon a double Meaning : Which to do were a manifeft Sophifm* Whether you argue in Symbols or in Words, the Rules of right Reafon are ftill the fame. Nor can it be fuppofed, you will plead a Privilege in Mathematics to be exempt from them.
XVI. If you affume at firft a Quantity increaf^c} by nothing, and in the Expref-
fion
The Analyst. 15
lion x + 0, 0 ftands for nothing, upon this Suppofition as there is no Increment of the Root, fo there will be no Increment of the Power; and confequently there will be none except the firft, of all thofe Mem- bers of the Series conftituting the Power of the Binomial ; you will therefore never come at your Expreflion of a Fluxion le- gitimately by fuch Method. Hence you are driven into the fallacious way of pro- ceeding to a certain Point on the Suppo- fition of an Increment, and then at once {hifcing your Suppofition to that of no Increment. There may feem great Skill in doing this at a certain Point or Period. Since if this fecond Suppofition had been made before the common Divifion by c?, all had vanifhed at once, and you muft have got nothing by your Suppofition. Whereas by this Artifice of firft dividing, and then changing your Suppofition, you retain i znd nx^—K Bli€, notvvithftand- ing all this addrefs to cover ir, the fal- lacy is ftill the fame. For whether it be done fooner or later, when once the fe- cond Suppofition or Aflumption is made, in the fame inftant the former Affumpti-
on
^4 The Analy^Y.
Oft and all that you got by it is deftroyed, and goes out together. And this is univer- ifally true, be the Subjedt what it will, throughout all the Branches of humane Knowledge ; in any other of which, I believe. Men would hardly admit fuch k reafoning as this, which in Mathematics is accepted for Demonftration.
XVII. It may not be amifs to obferve, ■that the Method for finding the Fluxion of a Rectangle of two flowing Quantities, as it is fet forth in the Treatife of Qua- dratures, differs from the abovementioned taken from the fecond Book of the Prin- ciples, and is in effedl the fame with that ufed in the calculus different talis *. For the fuppofing a Quantity infinitely dimi- niflied and therefore rejed:ing it, is in ef- fed: the rejed:ing an Infinitefimal; and indeed it requires a marvellous fharpnefs of Difcernment, to be able to diftinguifh between evanefcent Increments and infinir tefimal Differences. It may perhaps be faid that the Quantity being infinitely di- minished becomes nothing, and fo no- thing is rejedled. But according to the
* Analyfe des infiniment petits, part. i. prop. 2.
received
ff The a k a l y s t. 17
received Principles it is evident, that no Geometrical Quantity, can by any divifion or fubdivifion whatfoever be exhaufted, or reduced to nothing. Confidering the var%» ous Arts and Devices ufed by the great Author of the Fluxionary Method: in how many Lights he placeth his Fluxions : and in v^hat different ways he attempts to demonftrate the fame Point : one would be inclined to think, he was himfclf fufpici- Gus of the juftnefs of his own demonftra- tions 5 and that he was not enough pleafed with any one notion fteadily to adhere to it. Thus much at leafl: is plain, that he owned himfelf fatisfied concerning certain Points, which neverthelcfs he coqld not undertake to demonftrate to others *. Whe- ther this fatisfaftion arofe from tentative Methods or Inductions ; which have often been admitted by Mathematicians, (for inftance by Dr. fFallis in his A- rithmetic of Infinites) is what I fliall not pretend to determine. But, whatever the Cafe might have been with rcfped: to the Author, it appears that his Followers {i^ve {hewn themfeives more eager in ap-
* See Letter to Collins, Nov. 8, 1676.
plying
The Analyst.^
plying his Method, than accurate in exa- mining his Principles.
♦ XVIII. It is curious to obferve, what fubtilty and skill this great Genius em- ploys to ftruggle with an infuperable Dif- ficulty; and through what Labyrinths he endeavours to efcape the Dodrine of Infiniteiimals ; which as it intrudes up- on him whether he will or no, fo it is admitted and embraced by others without the leaft repugnance. Leibnitz and his Followers in their calculus differentialis making no manner of fcruple, firft to fup- pofe, and fecondly to reje(5l Quantities infinitely fmall: with what clearnefs in the Apprehenfion and juftnefs in the reafoning, any thinking Man, who is not prejudiced in favour of thofe things, may eafily difcern. The Notion or Idea of an infinitefimal Quantity, as it is an Objedl fimply apprehended by the Mind, hath been already confidered *. I fhall now only obferve as to the method of getting rid of fuch Quantities, that it is done without the leaft Ceremony. As in
* Se^. 5 and 6.
Fluxions
The a n a l y s Tr a.>
Fluxions the Point of firft importance, and which paves the way to the reft, is to find the Fluxion of a Product of two in- determinate Quantities, fo in the calculus differentialis (which Method is fuppofed to have been borrowed from the former with fome fmall Alterations) the main Point is to obtain the difference of fuch Produdl. Now the Rule for this is got by rejecting the Produdl or Redlangle of the Differen- ces. And in general it is fuppofed, that no Quantity is bigger or leffer for the Addi-« tion or Subdudlion of its Infinitefimal : and that confequently no error can arife from fuch rejection of Infinitefimals.
XIX. And yet it fliould feem that, whatever errors are admitted in the Pre- mifes, proportional errors ought to be ap- prehended in the Conclufion, be they finite or infinitefimal: and that therefore the a'jc^/jSgia of Geometry requires nothing fliould be negledted or rejected. In anfwer to this you will perhaps fay, that the Conclufions are accurately true, and that therefore the Principles and Methods from whence they are derived muft be fo too.
But
50 The An a: l y* s t.
But this inverted way of demonftrating^ your Principles by your Concluiions, as it would be peculiar to you Gentlemen, fo it is contrary to the Rules of Logic. The truth of the Conclufion will not prove either the Form or the Matter of a Syl- logifm to be true : inafmuch as the Illation might have been wrong or the Premifcs falfe, and the Conclufion neverthelcfs true, though not in virtue of fuch Illation or of fuch Premifes. I fay that in every other Science Men prove their Conclufions by their Principles,and not their Principles by the Conclufions. But if in yours you fhould allow your felves this unnatural way of proceeding, the Confequence would be that you muft take up with the Inducftion, and bid adieu to Demonftration. And if you fubmit to this, your Authority will no longer lead the way in Points of Reafon and Science.
XX. I have no Controverfy about your Conclufions, but only about your Logic and Method. . How you demonftrate ? What Objeds you are converfant with, and whether you conceive them clearly?
What
The Analyst.
What Principles you proceed upon; how fpqnd they may be ; and how you apply them? It muft be remembred that I am not concerned about the truth of your Theorems, but only about the way of conling at them ; whether it be legitimate or illegitimate, clear or obfcure,fcientificor tentative. To prevent all poflibility of your miftaking me, I beg leave to repeat and infift, that I confider the Geometrical A- nalyft as a Logician, /. e. fo far forth as he reafons and argues ; and his Mathematical Conclufions, not in themfelves, but in their Premifes ; not as true or falfe, ufe- ful or infignificant, but as derived from fuch Principles, and by fuch Inferences. And forafmuch as it may perhaps fcera an unaccountable Paradox, that Mathe- maticians fliould deduce true Propofitions from falfe Principles, be right in the Con- clufion, and yet err in the Premifes ; I ihall endeavour particularly to explain why . this may come to pafs, and fliew how Er- ror may bring forth Trutb> though it cannot bring forth Science.
XXI. la
V
i^
The Analyst;
XXI. In order therefore to clear up this Point, we will fuppofe for inftance that a Tangent is to be drawn to a Parabola, and examine the progrefs of this Affair, as it is performed by infinitcfimal Differences,
Let AB be a Curve, the Abfciffe AP — Xi the ordinate PB==y, the Difference of the Abfciffe PM=dxy the Difference of the Ordinate RN=Jy, Now by fuppofing the Curve to be a Polygon, and confequenc- ly BNy the Increment or Difference of the Curve, to be a ftraight Line coincident
with
T H E A N A L Y S T- 3 5
With the Tangent, and the differential Triangle B RN lo be fimiliar to the tri- angle TP-B the Subtangent PT is found a fourth Proportional to RN: RB:PBi that is to Jy : dx: y. Hence the Subtangent
will be ^~, But herein there is an error
arifing from the forementioned falfe fup- pofition, whence, the value of PT comes out greater than the Truth : for in reality it is not the Triangle RNB but RLB, which isfimilar to P B T", and therefore (in- ftead o£RN)RL fhould have been the firft term of the Proportion, /. e. RN -{- NL, i, e. dy -Vzi whence the true expreflion
for the Subtangent fhould have been ^^^«
There was therefore an error of defedt in making dy the divifor : which error was equal to z, / . e. NL the Line comprehend* ed between the Curve and the Tangent. Now by the nature of the Cnrwc yy=px, fuppofing p to be the Parameter, whence by the rule of Differences 2ydy^^pdx and dy^==^^—^. But if you multiply^ -V dy
by it felf, and retain the whole Produdt
without rejedling the Square of the Diffe-
D rence,
34 The Analyst.
rence, it will then come out, by fubftitu- ting the augmented Quantities in the E-
quation of the Curve, that dy=-^-^^^^
truly. There was therefore an error of
excefs in making ^;' = ^-~, which followed
from the erroneous Rule of Differences. And
the meafure of this fecond error is -^^ = z.
Therefore the two errors being equal and contrary deftroy each other ; the firft er- ror of defe(5l being corrected by a fecond error of excefs.
XXlI. If you had committed only one error, you would not have come at a true Solution of the Problem. But by virtue of a twofold miftake you arrive, though not at Science, yet at Truth. For Science It cannot be called, when you proceed blindfold, and arrive at the Truth not knowing how or by what means. To de-
monftrate that z is equal to -^^, let BR
or dx be m and RN or Jy be n. By the thirty third Propofition of the firft Book of the Conies oi jlpolloniuSy^ and from fimilar
Triangles,
T H E A N A L Y S t. 5 5
Triangles, as 2^ to;r fo is m to «+ ^ =— . Likewife from the Nature of tHp
2X
Parabola 7 ;' + 2 yn+nn= xp^r mp, and 2yn-\-nn=^mp\ wherefore ^-^^ ^'^ '^ = /;7 :
and becaufe;'j=/>^, y will be equil to X, Therefore fubfiitutirig thefe values inftead of m and x we ftiall have
^yUAH: which being reduced gives
2y zy ^^
' XXIII. Now I obfervc In the firft place, that the Conclufion comes out right, not becaufe the rejefted Square of dy was in- finitely fmall ; but becaufe this error was compenfated by another contrary and e^ qual error. I obferve in the fecond places that whatever is rejedled, be it ever {o fmall, if it be real and confequently makes a real error in the Premifes, it will pro- duce a proportional real error in the Con- clufioni Your Theorems therefore cannot be accurately true, nor your Problems accurately folved, in virtue of Premifesi D 2 trhieh
3,^ Th E A N A L Y S T.
which thcmfelves are not accurate, it be- ing a rule in Logic that Conclufio fequitur partem debiliorem. Therefore I obferve in the third place, that when the Conclufion is evident and the Premifes obfcure, or the Conclufion accurate and the Premifes in- accurate, we may fafely pronounce that fuch Conclufion is neither evident nor accurate, in virtue of thofe obfcure inaccurate Pre- mifes or Principles; but in virtue of fome other Principles which perhaps the De- monftrator' himfelf never knew or thought of I obferve in the laft place, that in cafe the Differences arc fuppofed finite Quantities ever fo great, the Conclufion will neverthelefs come out the fame : in- afmuch as the rejed:ed Quantities are le- gitimately thrown out, not for their fmallnefs, but for another reafon, to wir, becaufe of contrary errors, which deftroy- ing each other do upon the whole caufe that nothing is really, though fomething is apparently thrown out. And this Rea- fon holds equally, with refpedt to Quan- tities finite as well as infinitefimal, great as well as fmall, a Foot or a Yard long as well as the minutefl: Increment.
XXIV. For
Ipxxr
Tbte Analyst.
XXIV. For the fuller illuftration of thisf Point, I (hall confider it in another lighr,> amd proceeding in finite Quantities to the Conclufion, I (hall only then make ufe
df one Infinitefimali Suppofe the (Iraight Line M^ cuts the Curve ^ T* in the Points 2? and »S. Suppofe L2? a Tangent at the Point R, AN the Abfcifle, NR and OS Ordinates. Let ^A'' be produced to O, and RP he drawn parallel to NO. Suppofe AN=x, NR=y, N O = v, PS = %y the fubfecant MN=S, Let the Equation ^=Ar^ exprefs the nature of the Curve: and fuppofing y and iV increafed by their finite Increments, we get y -\- z ^^^x-^ 2y;v\-vvi whence the former P 3 Eq^ua-.
?7
The Analyst.
Equation being fubdudbed there remains Z=2xv+vv. And by reafon of fimilar Triangles PS: PR:: NR: NM, i.e.
z :v :: y: j = — > wherein if for y and z
we fubflitute their values, we get —'"\^ — ==j=— ^. And fuppofins; NO to be
infinitely diminiflied, the fubfecant NM will in that cafe coincide with the fubtan- gent NL, and v as an Infinitefimal may be rejedted, whence it follows that
S = NL = — = - 5 which is the true va-
2 AT .2
lue of the Subtangent. And fince this was obtained by one only error, /. e. by once rejecting one only Infinitefimal, it fhould feem, contrary to what hath been faid,that an infinitefimal Quantity or Difference may be negledted or thrown away, and the Conclufion neverthelefs be accurately true, although there was no double miftake or ledifying of one error by another, as in the firft Cafe. But if this Point be through- ly confidered, we {hall find there is even here a double miftake, and that one com- penfates or reftifics the other. For in the
firft
T H E A N A L Y S T. 5 P
iirfl: plaee, it was fuppofed, that when NO is infinitely diminifhed or becomes an, Infinitefimal, then the Subfecant NM be- comes equal to the Subtangent NL. But this is a plain miftake, for it is evident, that as a Secant cannot be a Tangent, fo a Subfecant cannot be a Subtangent. Be the Difference ever fo fmall, yet ftill there is a Difference. And if NO be infinitely fmall, there will even then be an infinitely fmall Diflference between NM and NL, There- fore NM or S was too little for your fup- pofition, (when you fuppofed it equal to. N L) and this error was compen fated by a fecond error in throwing out i;, which lafl error made s bigger than its true va- lue, and in lieu thereof gave the value of the Subtangent. This is the true State of the Cafe, however it may be difguifcd. And to this in reality it amounts, and is at bottom the fame thing, if we fliould pretend to find the Subtangent by hav- ing firft found, from the Equation of the Curve and fimilar Triangles, a gcr neral Expreffion for all Subfecants, and , then reducing the Subtangent under this general Rule, by confidering it as the P 4 Subfe-
40 T H B A Ijl A L y $ T.
Subfecant when v vanijthcs or becomes nothing.
XXV. Upon the whole I obferve, Firjf^ that V can never be nothing fo long a& there is a fecant. Secondly^ That the fame Line cannot be both tangent and fecant. thirdly, that when v or NO * vanifheth, PS and iS-R do alfo vanifh, and with them the proportionality of the fimilar Triangles. Confequently the whole Expref- fion, which was obtained by means thereof; and grounded thereupon, vanifheth whea V vanifheth. Fourthly^ that the Method, for finding Secants or the Expreffion of Se- cants, be it ever fo general, cannot in com- mon fenfe extend any further than to alls Secants whatfoever: and, as it neceffarily fuppofeth fimilar Triangles, it cannot be fuppofed to take place where there are not fimilar Triangles. Fifthly ^ that the Subfe- cant will always be lefs than the Subtaa^ gent, and can never coincide with it; which Coincidence to fuppofe would be abfurd J for it would be fuppofing, the fame Line at the fame time to cut and.
* S(e the foregoing Figure,
not
T HE A N A L Y S T. 41
not to cut another given Line, which is a manifeft Contradicftion, fuch as fubverts the Hypothefis and gives a Demonftration of its Falfhood. Sixthly, If this be not admitted, I demand a Reafon why any other apagogical Demonftration, or De- monflxation ad abfurdum fhould be ad- mitted in Geometry rather than this : Or that fome real Difference be affigned be- tween this and others as fuch. Seventhly, I obferve that it is fophiftical to fuppofe NO or RF, PS, znd SR to be finite real Lines in order to form the Triangle UPS, in order to obtain Proportions by fimilar Triangles ; and afterwards to fup- pofe there are no fuch Lines, nor confe- quently fimilar Triangles, and neverthe- lefs to retain the Confequence of the firft Suppofition, after fuch Suppofition hath been deftroyed by a contrary one. Eighthly, That although, in the prefent cafe, by in- confiftent Suppofitions Truth may be ob- tained, yet that fuch Truth is not demon- ftrated: That fuch Method is not conform- able to the Rules of Logic and right Rea- fon : That, however ufeful it may be, it muft be confidered only as a Prefumptior,
as
43L T H E A N A L Y S T.
as a Knack, an Arc or rather an Artifice, but not a fcientific Demonftration.
XXVI. The Dodrine premifed may be farther illuftrated by the following fimple and eafy Cafe, wherein I (hall proceed by evanefcent Increments. Suppofe jiB = Xj
FH
BC=yy BD = Oy and that xx is equal ta the Area ABC : It is propofed to find the Ordinate ^ or BC, When x by flowing becomes x + o, then x x becomes xx-{- 2XO'\-oo: And the Area ABC becomes ADH, and the Increment of xx will be equal to BDHC the Incremem of the
Area.
-T H E A N A L Y S T. 4.J
Area, I e. to BCFD+CFH. And If we fuppofe the curvilinear Space C FH to he go 0, then 2x0 -i-oo =y o-\- qoo which xlividedby 0 gives 2^:4-0 =^4- §'(?. And, fuppofing 0 to vanifli, 2x=y, in which Cafe ACH will be a ftraight Line, and the Areas ^BC, CFH, Triangles. Now with regard to this Reafoning, it hath been already remarked *, that it is not le- gitimate or logical to fuppofe 0 to vanifh, /. e, to be nothing, /. e. that there is no Increment, unlefs we rejedt at the fame time with the Increment it felf every Con- fequence of fuch Increment, /. e. what- foever could not be obtained but by fup- pofing fuch Increment. It mufl never- thelefs be acknowledged, that the Problem is rightly folved, and the Conclufion true, to which we are led by this Method. It will therefore be asked, how comes it to pafs that the throwing out 0 is attended with no Error in the Conclufion } I an- fwer, the true reafon hereof is plainly thi5: Becaufe q being Unite, qo \s equal to o\ And therefore zx-\-o — qo=:y^=2Xy
* Se^. 12 and i^, fupra.
the
44 T H E A N A L Y S t.
the equal Quantities qo and o being dci* ftroyed by contrary Signs.
XXVIL As on the one hand' it were abfurd to get rid of o by faying, let mc contradidt my fclf : Let me fubvert my own Hypothefis : Let mc take it for grant* ed that there is no Increment, at the fame time that I retain a Quantity, which I could never have got at but by afluming an Increment: So on the other hand it Would be equally wrong to imagine, that in a geometrical Dcmonftration we may be allowed to admit any Error, though ever fo fmall, or that it is poffible, in the nature of Things, an accurate Conclufion fliould be derived from inaccurate Prin^ ciples. Therefore a cannot be thrown out as an Infinitefimal, or upon the Principle that Infinitefimals may be fafely negleded. But only becaufe it is deftroyed by an equal Quantity with a negative Sign, whence o^^qo is equal to nothing. And as it is illegitimate to reduce an Equation, by fubduding from one Side a Quantity when it is not to be deftroyed, or when an equal Quantity is not fubduilcd from
the
The A N A L Y s T. 45
i;he other Side of the Equation : So it mufl: be allowed a very logical and juft Method of arguing, to conclude that if from E- qual« either nothing or equal Quantities are fubdu<fted, they fhall ftill remain equal. And this is a true Reafon why no Error js at laft produced by the rejecting of o. Which therefore muft not be afcribed to the Dodtrine of Differences, or Infinitefi- mals, or evancfcent Quantities, or Mo- mentums, or Fluxions.
XXVni. Suppofe the Cafe to be gene- ral, and that ;c« is equal to the Area ABC^ whence by the Method of Fluxi- ons the Ordinate is found nx^-^^ which we admit for true, and (hall inquire how it is arrived at. Now if we are content to come at the Conclufion in a fummary way, by fuppofing that the Ratio of the Fluxions of x and x» are found * to be I and «a:»— ', and that the Ordinate of the Area is coniidcred as its Fluxion ; wc fliall not fo clearly fee our way, or per- ceive how the truth comes out, that Me- thod as we have fticwed before being ob-
* s,a. 13.
fcure
4(5 /.T^H E Analyst.
fcure and illogical. But if we fairly de^ lineate the Area and its Increment, and divide the latter into two Parts BCFD and C FH^y and proceed regularly by E- quatlons between the algebraical and geo- metrical Quantities, the rcafon of the thing will plainly appear. For as a: « is equal to the Area AB C^ fo is the In- crement of ^« equal to the Increment pf the Area^ /. e. to BDHG-, that is^ to fay, «o:v« — I +/-^-^ o o at » - ^ 4- ^c.
==BDFC 4- CFH. And only the firft Members, on each Side of the Equation being retained, nox*"--^ =BDFC: And dividing both Sides by <? or B D, we fliall get nx» — ^=BC. Admitting, therefore, that the curvilinear Space CFH is equal to the rejedaneous Quantity
VLTU^oox^-^ + &c. and that when this
2
is rejected on one Side^ that is rcjedled on the other, the Reafoning becomes juft and the Conclufion true. And it is all one whatever Magnitude you allow to B D^ whether that of an infinitefimal Difference or a finite Increment ever fo great. It is there- fore plain, that the fuppofing the rejectaneous
* See tie Figure in Sea. z6. alge-
T H E A K A L Y S T. 47
algebraical Quantity to be an infinitely fmall or evanefcent Quantity, and there- fore to be negleded, muft have produced an Error, had it not been for the curvi- linear Spaces being equal thereto, and at the fame time fubduded from the other Part or Side of the Equation agreeably to the Axiom, If from Equals you fubdudl Equals^ the Remainders ivill be equal For thofe Quantities which by the Analyfts are faid to be neglected, or made to vanifli, are in reality fubduded. If therefore the Conclufion be true, it is abfolutely necef- iary that the finite Space C F H he equal to the Remainder of the Increment
expreffed by ^"'"•"•oox^ — ^ &c. equal I fay
to the finite Remainder of a finite Incre- ment.
XXIX. Therefore, be the Power what you pleafe, there will arife on one Side an algebraical Exprefllon, on the other a geometrical Quantity, each of which na- turally divides it felf into three Members: The algebraical or fluxionary Exprefllon, into one which includes neither the Ex- prefllon
4> T H E A N A L Y S T.
preflion of the Increment of the Abfcifs nor of any Power thereof, another which includes the Expreflion of the Increment it felf, and a third including the Expref- fion of the Powers of the Increment. The geometrical Quantity alfo or whole in- creafed Area confifls of three Parts or Members, the firft of which is the given Area, the fecond a Redtangle under the Ordinate and the Increment of the Ab- fcifs, and the third a curvilinear Space, And, comparing the homologous or cor- refpondent Members on both Sides, we find that as the firft Member of the Ex-r preflion is the Expreflion of the given Area, fo the fecond Member of the Ex- preflion will exprefs the Re(flangle or fe- cond Member of the geometrical Quanti- ty ; and the third, containing the Powers of the Increment, will exprefs the curvi- linear Space, or third Member of the geo- metrical Quantity. This hint may, per- haps, be further extended and applied to good purpofe, by thofe who have leifurc and curiofiry for fuch Matters. The ufe I make of it is to fliew, that the Analyfis cannot obtain in Augments or DiiFerences,
but
T H E A N A L Y S t. 45>
but it muft alfo obtain in finite Quantities, be they ever fo great, as was before ob- ferved.
XXX. It fecms therefore upon the whole that we may fafely pronounce, the Conclufion cannot be right, if in order thereto any Quantity be made to vanifli, or be negledled, except that either one Error is redrefled by another ; or that fe- condlyj on the fame Side of an Equa- tion equal Quantities are deftroycd by contrary Signs, fo that the Quantity wc mean to rejcd: is firfl annihilated 3 or laftly, that from the oppofite Sides equal Qu^antities are fubducled. And therefore to get rid of Quantities by the received Principles of Fluxions or of Differences is neither good Geometry nor good Logic. When the Augments vanifli, the Veloci- ties alfo vaniCh. The Velocities or Fluxi- ons are faid to h^ primo and ulfimo, as the Augments nafcent and evanefcent. Take therefore the Ratio of the evanefcent Quantities, it is the Tame with that of the Fluxions. It will therefore anfwer all Intents as well. Why then are Fluxions E intro*
5© The Analyst.
introduced? Is it not to fliun or rather to palliate the Ufe of Quantities infinitely fmall ? But we have no Notion whereby to conceive and meafure various ' Degrees of Velocity, befide Space and Time, or when the Times are given, befide Space alone. We have even no Notion of Ve- locity prefcinded from Time and Space. , When therefore a Point is fuppofed to move in given Times, we have no Notion of greater or lefler Velocities or of Pro- portions between Velocities, but only of longer or (horter Lines, and of Proporti- ons between fuch Lines generated in equal Parts of Time.
XXXL A Point maybe the limit of a Line : A Line may be the limit of a Sur- face: A Moment may terminate Time. But how can we conceive a Velocity by the help of fuch Limits ? It neceflarily im- plies both Time and Space, and cannot be conceived without them. And if the Velocities of nafcent and evanefcent Quan- tities, /. e, abftraded from Time and Space, may not be comprehended, how can we comprehend and demonflrate their
Propor-
The Analyst. fi
Proportions ? Or confider their rationes frimce and ultima. For to confider the Proportion or Ratio of Things implies that fuch Things have Magnitude : That fuch their Magnitudes may be meafijred, and their Relations to each other known. But, as there is no meafurc of Velocity except Time and Space, the Proportion of Velo- cities being only compounded of the di- red Proportion of the Spaces, and the reciprocal Proportion of the Times ; doth it not follow that to talk of inveftigating, obtaining, and confidering the Proportions of Velocities, exclufively of Time and Space, is to talk unintelligibly ?
XXXII. But you will fay that, in the ufe and application of Fluxions, Men do not overftrain their Faculties to a precife Conception of the abovementioned Velo- cities, Increments, Infinitefimals, or any other fuch like Ideas of a Nature fo nice, fubtile, and evanefcent. And therefore you will perhaps maintain, that Problems may be folved without thofe inconceiva- ble Suppofitions: and that, confequently, the Dodlrin^ of Fluxions, as to the prac-
E 2 tical
The Analyst.
tical Part, ftands clear of all fuch Diffi- culties. I anfwer, that if in the ufe or application of this Method, thofe difficult and obfcure Points are not attended to^ they are neverthelefs fuppofed. They are the Foundations on which the Moderns build, the Principles on which they pro- ceed, in folving Problems and difeover- ing Theorems. It is with the Method of Fluxions as with all other Methods, which prefuppofe their refpedlive Principles and are grounded thereon. Although the Rules may be pradifed by Men who nei- ther attend to, nor perhaps know the Principles. In like manner, therefore, as a Sailor may practically apply certain •Rules derived from Aflronomy and Geo- metry, the Principles whereof he doth not underftand : And as any ordinary Man m:ay folve divers numerical Queftions, by the vulgar Rules -and Operations of Arith- metic, which he performs and applies without knowing the Reafons of them: Even fo it cannot be denied that you may apply the Rules of the fluxionary Me- thod : You may compare and reduce par- ticular Cafes to general Forms : You may
operate
w
TheAnalyst. 5}
operate and compute and folve Problems thereby, not only without an adual At- tention to, or an adlual Knowledge of, the Grounds of that Method, and the Prin- ciples whereon it depends, and whence it is deduced, but even without having ever confidered or comprehended them.
XXXIII. But then it muft be remembred, that in fuch Cafe although you may paf^ for an Artift, Computift, or Analyft, yet you may not be juftly efteemed a Man of Science and Demonflration. Nor fliould any Man, in virtue of being converfanc in fuch obfcure Analytics, imagine his rational Faculties to be more improved than thofe of other Men, which have been exercifed in a difterent manner, and on different Subjefts ; much lefs ered: him- felf into a Judge and an Oracle, concern- ing Matters that have no fort of conne- xion with, or dependence on thofe Species, Symbols or Signs, in. the Management whereof he is fo converfant and expert. As you, who are a skilful Computift or Analyft, may not therefore be deemed skilful in Anatomy : or* vice verja, as a E 3 Mm
54 TheAnalyst.
Man who can diffedl with Art, may, nc- verthclefs, be ignorant in yoqr Art of com- puting : Even fo you may both, notwith- ^ {landing your peculiar Skill in your re-
fpedtive Arts, be alike unqualified to de- cide upon Logic, or Metaphyfics, or E- thics, or Religion. And this would be true, even admitting that you underftood your own Principles and could demon- ftrate them.
XXXIV. If it IS faid, that Fluxions may be expounded or expreffed by finite Lines proportional to them : Which finite Lines, as they may be diftindtly conceiv- ed and known and reafoned upon, fo they may be fubftituted for the Fluxions, and their mutual Relations or Proportions be confidered as the Proportions of Fluxions : By which means the Doctrine becomes clear and ufeful. I anfwer that if, in or- der to arrive at thefe finite Lines propor- tional to the Fluxions, there be certain Steps made ufe of which are obfcure and inconceivable, be thofe finite Lines them- felves ever fo clearly conceived, it muft neverthelefs be "acknowledged, that your
proceed-
IHP The Analyst.
proceeding is not clear nor your method fcientific. For inftance, it is fuppofed that 4B being the Abfcifs, B C the Ordinate,
55
and VCH 2i Tangent of the Curve AC, Bb ov CE the Increment of the Abfcifs, Ec the Increment of the Ordinate, which produced meets V H \n the Point T*, and Cc the Increment qf the Curve. The right Line C c being produced to K, there are formed three fmall Triangles, the Redilinear CEc, the Mixtilinear CEc, and the ReftiUnear Triangle GET. It is evident thefe three Triangles are dif- ferent from each other, the Redilinear C E c being lefs than the Mixtilinear CEc, whofe Sides are the three Incre- ments abovementioned, and this ftill lefs ^han the Triangle GET. It is fuppofed that the Ordinate b c moves into the place BC, fo that the Point c is coincident with the' Point C^ and the right Line C K E 4. an4
5(> Thi Analyst.
and confequently the Curve Cr, is coin-' cident with the Tangent C H. In which cafe the mixtilinear evanefcent Triangle CE c will, in its laft form, be fimilar to the Triangle GET: And its evanefcent Sides C E, E f , and C c will be porpor- tional to CE^ ET, and CtT the Sides of the Triangle C E T, And therefore it is concluded, that the Fluxions of the Lines j4B, BC, and AC, being in the » laft Ratio of their evanefcent Increments, are proportional to the Sides of the Tri- angle GET, or, which is all one, of the Triangle V B G fimilar thereunto, * It it particularly remarked and infifted on by the great Author, that the Points C and c muft not be diftant one from ano- ther, by any the leaft Interval whatfoever; But that, in order to find the ultimate Proportions of the Lines C E, E c, and C c (/. e, the Proportions of the Fluxi- ons or Velocities) expreffed by the finite Sides of the Triangle FBG, the Points C and c muft be accurately coincident, i, e. one and the fame. A Point therefore is confidered as a Triangle, or a Triangle is iuppofed to be formed in a Point, Whicjn
^ Introdud. ad Quad. Cur v. tO
^ TheAnalyst. 57
to conceive feems quite impoffible. Yet fome there are, who, though they (hrink at all other Myfteries, make no difficulty of their own, who ilrain at a Gnat and fwal- low a Camel.
XXXV. I know not whether it be worth while to obferve, that poffibly fome Men may hope to operate by Symbols and Suppofitions, in fuch fort as to avoid the ufe of Fluxions, Momentums, andln- finitefimals after the following manner. Suppofe X to be one Abfcifs of a Curve, and z another Abfcifs of the fame Curve. Suppofe alfo that the refpe6tive Areas are XXX 2indizzz: and that z* — x is the In- crement of the Abfcifs, and zzz — xx x the Increment of the Area, without confi- dering how great, or how fmall thofe In- crements may be. Divide nov^zzz — xx x by 2J — AT and the Quotient will be zz '\- z x-]r X X : and, fuppofing that z and X are equal, this fame Quotient will be 3 ;^ X which in that cafe is the Ordinate, which therefore may be thus obtained in- dependently of Fluxions and Infinitefi- •mals. But herein is a diredt Fallacy: for
in
5g The Analyst.
in the firft place, it is fuppofed that the Abfcifles z and x are unequal, without which fuppofition no one ftep could have been made ; and in the fecond place, it is fuppofed they are equal ^ which is a mani- feft Inconfiftency, and amounts to the fame thing that hath been before confi- dered * And there is indeed reafon to ap- prehend, that all Attempts for fetting the abftrufe and fine Geometry on a right Foundation, and avoiding the Dodrine of Velocities, Momentums, &c, will be found impradlicable, till fuch time as the Objed: and End of Geometry are better un- derftood, than hitherto they feem to have been. The great Author of the Method of Fluxions felt this Difficulty, and there- fore he gave into thofe nice Abftradions and Geometrical Metaphyfics, without which he faw nothing could be done on the received Principles ; and what in the way of Demonftration he hath done with them the Reader will judge. It muft, in- deed, be acknowledged, that he ufed Fluxions, like the Scaffold of a building, as things to be laid afide or got rid of, as ' foon as finite Lines were found proportion
• Sea. 15. ' ' ^^^
p The Analyst. 59
nal to them. But then thefe finite Expo- nents are found by the help of Fluxions. Whatever therefore is got by fuch Expo- nents and Proportions is to be afcribed to Fluxions: which mull therefore be previ- oufly underftood. And what are thefe Fluxions? The Velocities of cvanefcent Increments ? And what are thefe fame cva- nefcent Increments ? They are neither fi- nite Quantities, nor Quantities infinitely fmall, nor yet nothing. May we not call them the Ghofts of departed Quanti*-
lies ?
«•
XXXVI. Men too often impofc on themfelves and others, as if they conceived and underftood things expreffed by Signs, when in truth they have no Idea, fave only of the very Signs themfelves. And there are fome grounds to apprehend that this may be the prefent Cafe. The Velo- cities of evanefcent or nafcent Quantities are fuppofed to be expreffed, both by fi- nite Lines of a determinate Magnitude, and by Algebraical Notes or Signs : but I fufpedt that many who, perhaps never having examined the matter, take it for
granted.
Co TheAnalyst.
granted, would upon a narrow fcrutiny find it impoffible, to frame any Idea or Notion whatfoever of thofe Velocities, ex- clufive of fuch finite Quantities and Signs.
a / c B e
} 1 I I I ■, ! t I i \ (
K X'^^^"l/[/^7^]Sr o P
Suppofe the Line K P defcribed by the Motion of a Point continually accelerated, and that in equal Particles of time the unequal Parts KL, LM, MN, NO &e. are generated. Suppofe alfo that a, b^ c^ J, <f, Gfc. denote the Velocities of the genera- ting Point, at the feveral Periods of the Parts or Increments io generated. It is eafy to obferve that thefe Increments are each pro- portional to the fum of the Velocities with which it is defcribed : That, confcquently, the feveral Sums of the Velocities, generated in equal Parts of Time, may be fet forth by the refpedlive Lines KL^ LM, MJV, &c. generated in the fame times : It is likewife an eafy matter to fay, that the laft Velocity generated in the firft Parti- cle of Time, may be' exprefTed by the Symbol ^, the laft in the fecond by i, the laft generated in the third by f, and fo
on :
P Th E A N A L Y S T. ^1
on : that a is the Velocity of L M in fiatu nafcentiy and b, c, d, f, &?r. are the Velocities of the Increments MAT, A^O, O P, Gfr. in their refpedive nafcent eftates. You may proceed, and confider thefe Ve- locities themfelves as flowing or increafing Quantities, taking the Velocities of the Velocities, and the Velocities of the Ve- locities of the Velocities, /. e, the firft, fecond, third, S*c. Velocities ad infinitum : which fucceeding Series of Velocities may be thus exprefTed. a, b — a, c — ib'\- a, d — 3^'i"3^ — ^ ^c, which you may call by the names of firft, fecond, third, fourth Fluxions. And for an apter Expreffion you may denote the variable flowing Line KL, KM, KN, &c, by the Letter Xi and the firft Fluxions by x, the fecond by X, the third by x, and fo on ad infini^ turn,
XXX Vn. Nothing is cafier than toaflign Names, Signs, or Expreflions to thefe Fluxions, and it is not difficult to compute and operate by means of fuch Signs. But it will be found much more difficult, to omit the Signs and yet retain in our
Mipds
CZ T H E A N A L Y S T.^
Minds the things, which we fuppofe to be fignlfied by them. To confider the Ex- ponents, whether Geometrical, or Alge- braical, or Fluxionary,is no difficult Mat- ter. But to form a precife Idea of a third Velocity for inftance, in it felf and by it felf. Hoc opus^ hie labor. Nor indeed is it an eafy point, to form a clear and diftind Idea of any Velocity at all, exclufive of and prefcinding from all length of time and fpace ; as alfo from all Notes, Signs or Symbols whatfoever* This, if I may be allowed to judge of others by my felf, is impoffible. To me it feems evident, that Meafures and Signs are abfolutely neceffa- ry, in order to conceive or reafon about Velocities ; and that, confequently, when we think to conceive the Velocities, Am- ply and in themfelves^ we are deluded by vain Abftraftions.
XXXVIII. It may perhaps be thought by fome an eafier Method of conceiving Fluxions, to fuppofe them the Velocities wherewith the infinitefimal Differences are generated. So that the firft Fluxions fhall be the Velocities of the firft Differences,
the
11^ TheAnalyst. ^3
the fecond the Velocities of the fecond Differences, the third Fluxions the Veloci- ties of the third Differences,and fo on adin^ Jinitum, But not to mention the infurmoun- table difficulty of admitting or conceiving Infinitefimals, and Infinitefimals of Infinite- fimals, &c, it is evident that this notion of Fluxions would not confift with the great Author's view ; who held that the minuteft Quantity ought not to be negled:ed, that therefore the Do<flrine of Infinitefimal Diffe- rences was not to be admitted in Geome- try, and who plainly appears to have in- troduced the ufe of Velocities or Fluxions, onpurpofc to exclude or do without them.
XXXIX. To others it may poffibly feem, that we fhould form a jufter Idea of Fluxions, by affuming the finite unequal ifochronal Increments KL, LM, MN, &c. and confidering them mjiatu nafcenti^ alfo their Increments in Jlatu nafcenti, and the nafcent Increments of thofe Increments, and fo on, fuppofing the firft nafcent In- crements to be proportional to the firft Fluxions or Velocities, the nafcent Incre- ments of thofe Increments to be propor- tional
4^4 T H E A N A L Y S t.
tional to the fecond Fluxions, the third nafcent Increments to be proportional to the third Fluxions, and fo onwards. And, as the firft Fluxions are the Velocities of the firft nafcent Increments, fo the fe- cond Fluxions may be conceived to be the Velocities of the fecond nafcent Incre- ments, rather than the Velocities of Ve- locities. By which means the Analogy of Fluxions may feem better preferved, and the notion rendered more intelligible.
XL. And indeed it fhould feem, that in the way of obtaining the fecond or third Fluxion of an Equation, the given Fluxions were confidered rather as Incre- ments than Velocities. But the confider- ing them fometimes in one Senfc, fome- times in another, one while in themfelves, another in their Exponents, feems to have occafioned no fmall fhare of that Confu- fion and Obfcurity, which is found in the Dodlrine of Fluxions. It may feem there-^ fore, that the Notion might be ftill mend* ed, and that inftead of Fluxions of Fluxi- ons, or Fluxions of Fluxions of Fluxions, and inftead of fecond, third, or fourth^G^r.
Fluxions
T H E A N A L Y S T. ^5
Fluxions of a given Quantity, it might be. more confiftent and lefs liable tocxception* to fay, the Fluxion of vthe firft nafcenti Increment, i, e. the fecond Fluxion ; the Fluxion of the fecond nafcent Increment,^ i. e, the third Fluxion ; the Fluxion ofi the third nafcent Increment, /. f- the. fourth Fluxion, which Fluxions are con- ceived refped:ively proportional, each to. the nafcent Principle of the Increment fuccecding that whereof it is the Fluxion. ^
XLI. For the more diftindt Conception* of all which it may be confidered, that if the finite Increment LM*be divided into* the Ifochronal Parts L niy ?n n, no, o M-, and the Increment MN into th^, Parts Mp, pqy qr, rN Ifochronal to the for-, mer -, as the whole Increments L M, MN* are proportional to the Sums of their dc-' fcribing Velocities, even fo the homolo-' gous Particles L m, Mp are alfo propor- tional to the refpedtive accelerated Veloci- * ties with which they are defer ibed. And' as the Velocity with which Mp is gene-' rated, exceeds that with which Lj?2 was generated, even fo the Particle Mp ex-
* See the foregiing Scheme in Se£i. 36.
F ceeds
6d The Analyst,
cceds the Particle Lm. And in general, as the Ifoehronal Velocities dcfcribing the Particles of MN exceed the Ifoehronal Velocities defcribing the Particles of L Af> even fo the Particles of the farmer e^^ceed the correfpondent Particles of the latter. And this will hold, be the faid Particles ever fo fmall. MN therefore will exceed J/ M if they are both taken in their naf- cent States : and that excefs will be pro- portional to the excefs of the Velocity b above the Velocity a. Hence we may fee that this laft account of Fluxions comes, in the upfliot, to the f^me thing with the firft *.
XLII. But notwithftanding what hath been faid it muft ftill be acknowledged, that the finite Particles L m or Af /», though taken ever fo fmall, are not pro- portional to the Velocities a and i ; but each to a Series of Velocities changing every Moment, or which is the fame thing, to an accelerated Velocity, by which it is generated, during a certain minute Parti-, cle of time : That the nafcent beginnings or evanefcent endings of finite Quantities^
♦ Sf£i. 36 which
TheAnalyst: cy
which are produced in Moments or .inft- nitely fmall Parts of Tin>e, are alone proportional to given Velocities: That, therefore, in order to conceive the firft Fluxions, we muft conceive Time divi- ded into Moments, Increments generated in thofe Moments, and Velocities propor- tional to thofe Increments : That in order to conc«ive fecond and third Fluxions, v^^e muft fuppofe that the nafcent Principles or momentaneous Increments have themfelves alfo other momentaneous Increments, w^hich are proportional to their refped:ive genera- ting Velocities: That the Velocities of thefe fecond momentaneous Increments are fecond Fluxions: thofe of their nafcent momentaneous Increments third Fluxions, And fo on ad infinitum,
XLIII. By fubdudling the Increment generated in the firft Moment from that generated in the fecond, we get the Incre- ment of an Increment. And by fubdudt- ing the Velocity generating in the firft Mo- ment from that generating in the fecond, we get the Fluxion of a Fluxion. In like manner, by fubduding the Difference of
F 2 the
6^ The Analyst.
the Velocities generating in the two firft Moments, from the excefs of the Velocity in the third above that in the fecond Mo- ment, we obtain the third Fluxion. And after the fame Analogy we may proceed to fourth, fifth, fixth Fluxions, &c: And if we call the Velocities of the firft, fe- cond, third, fourth Moments a, b^ c^ d^ the Series of Fluxions will be as above, a. b — a. c — 2^-4-^. d — 3^+ 3^5 — a.
ad infnitumy /. e, x, x, x. x, ad infi- nitum,
XLIV. Thus Fluxions may be confider- ed in fundry Lights and Shapes, which feem all equally difficult to conceive. And indeed, as it is impoffible to conceive Ve- locity without time or fpace, without either finite length or finite Duration |, it muft feem above the powers of Men to comprehend even the firft Fluxions. And if the firft are incomprehenfible, what ftiall we fay of the fecond and third Fluxions, G?^? He who can conceive the beginning of a beginning, or the end of an end, fomewhat before the firft or after
I Sea. 31. * -
ihe
T H E A N A L Y S t, ^9
the laft, may be perhaps fliarpfighted enough to conceive thefe things. But moft Men will, I believe> find it impoffible to underftand them in any fenfe whatever.
XLV. One would think that Men could notfpeak too exactly onfo nice aSubjeA. And yet, as was before hinted, we may often obferve that the Exponents of Fluxions or Notes reprefenting Fluxions arc con- founded with the Fluxions themfelves. Is not this the Cafe, when juft after the Fluxions of flowing Quantities were faid to be the Celerities of their increafingi and the fecond Fluxions to be the muta- tions of the firft Fluxions or Celerities^
we are told that z. z. z, z, z, z. * re- prefents a Series of Quantities, whereof each fubfequent Quantity is the Fluxion of the preceding ; and each foregoing is a fluent Quantity having the following one for it^ Fluxion ?
XLVL Divers Series of Quantities and Expreflions, Geometrical and Algebraical^
* De Quadratura Curvarum.
F 3 may
70 TheAnalyst.
may be eafily conceived, in Lines, in Sur- faces, in Species, to be continued without end or limit. But it will not be found fo eafy to conceive a Series, either of mere Velocities or of mere nafcent Increments, diftinft therefrom and correfponding there- unto. Some perhaps may be led to think the Author intended a Series of Ordinates, wherein each Ordinate was the Fluxion of the preceding and Fluent of the following, /• e. that the Fluxion of one Ordinate was it felf the Ordinate of another Curve; and the Fluxion of this laft Ordinate was the Ordinate of yet another Curve > and fo on ad infinitum. But who can conceive how the Fluxion (whether Velocity or nafcent Increment) of an Ordinate fhould be it felf an Ordinate? Or more than that each preceding Quantity or Fluent is related to its Subfequent or Fluxion, as the Area of a curvilinear Figure to its Ordi- nate 5 agreeably to what the Author re- marks, that each preceding Quantity in fuch Series is as the Area of a curvili- near Figure, whereof the Abfcifs is z^ and the Ordinate is the following Quan- tity.
XLVII. Upon
'The AnaIysj.
XLVIL Upon chc whole it appears that the Celerities are difmiiled, and iaftead thereof Areas and Ordinates are introduced. But however expedient fuch Analogies pr foch Expreffions may be found for facili- tating the modern Quadratures, yet we fhall not find any light given us thereby into the original real nature of Fluxions; or that we are enabled to frame from thence juft Ideas of Fluxions confidered in them- felves. In all this -the general ukimaie drift of the Author is v^ry clear, but hi^ Principles are obfcure. But perhaps thoft Theories of the great Author are not mi- nutely confidered or canvajQTed by his Dif- dples J who ieem eager, as was before hinted, rather to operate than to know, rather to apply his Rules and his Forms^ than to underftand his Principles and en- ter into his Notions. It is neverthelefs cer- tain, that in order to follow him in his Quadratures, they muft find Fl^uents from Fluxioms ; and in order to this, they muft know to find Fluxions from Fluents ; and in order to find Fluxions, they muft firft know vdiat Fluxions are. Otherwife they proceed without Clearnefs and without F 4 Seienee.
y% TheAnalyst.
Science. Thus the direct Method precedes the' inverfe, and the knowledge of the Principles is fuppofed in both. But as for operating according to Rules, and by the help of general Forms, whereof the ori- ginal Principles and Reafons are not on- derftood, this is to be efteemed merely technical. Be the Principles therefore ever fo abflrufe and metaphyfical, they muft be ftudicd by whoever would comprehend the Doctrine of Fluxions. Nor can any Geometrician have a right to apply the Rules of the great Author, without firfl * confidering his metaphyfical Notions whence they were derived. Thefe how neceffary foever in order to Science, which can never be attained without a precife, clear, and accurate Conception of the Principles, are neverthelefs by feveral carelefly paffedover; while the Expref- fions alone are dwelt on and coniidered and treated with great Skill and Manage- ment, thence to obtain other Expreffions by Methods, fufpicious and indirect (to fay the lead) if confidered in themfelves, however recommended by Induction and
Authorityj
The Analyst*
Authority; two Motives which are ac- knowledged fufficient to beget a rational Faith and moral Perfuafion, but nothing higher.
XL VIII. You may poffibly hope to e- vade the Force of all that hath been faid, and to fcreen falfe Principles and incon- fiftent Reafonings, by a general Pretence that thefe Objedions and Remarks are Metaphyfical. But this is a vain Pretence. For the plain Senfe and Truth of what is advanced in the foregoing Remarks, I ap- peal to the Underiftanding of every un- prejudiced intelligent Reader. To the fame I appeal, whether the Points re- marked upon are not moft incomprehen- fiblc Metaphyfics. And Metaphyfics not of mine, but your own. I would not be un- derftood to infer, that your Notions are falfe or vain becaufe they are Metaphyfi- cal. Nothing is either true or falfe for that Reafon. Whether a Point be called Metaphyfical or no avails little. The Queftion is whether it be clear or obfcure, right or wrong, well or ill-deduced ?
XLIX. Al-
7J
^^ T H I A N A L Y S t.
XLIX. Although momentaneous Incre- ments, nafccnt and evancfcent Quantities, Fluxions and Infimtefimals of all Dcgwes, are in truth fuch fhadowy Entities, fo difficult to imagine or conceive diftin^tly, that (to fay the leaft) they cannot be ad- mitted as Principles or Ol^ds of clear and accurate Science : and although this ob- fcurity and incomprehenfibility of your Metaphyfics had been alone fufEcient, to allay your Prctenfions to Evidence j yet it hath> if I miftake not, been further fliewn, that your Inferences are no more juft than your Conceptions are clear, and that your Logics are as exceptionable as your Meta- phyiics. It fhould feem therefore upon the whole, that your Conclufions are not attained by juft Reafoning from clear Prin- ciples; confequently that the Employ- ment of modern Analyfts, however ufeful in mathematical Calculations, and Con- ftrudtions, doth not habituate and qualify the Mind to apprehend clearly and infer juftly ', and confequently, that you have no right in Virtue of fuch Habits, to did:ate out of your proper Sphere, beyond which
your
Iv T H E A K A L ir s t. 71
I your Judgment is to pafs for no mxjrc
than that of other Men.
li
L. Of a long time I have fufpcdkd, that thefe modern Analytics were not fcientifi- cal, and gavefomc Hints thereof to the Pub- lic about twenty five Years ago. Since which time, I have been diverted by other Occupations, and imagined I might em- ploy my felf better than in deducing and laying together my Thoughts on fo nice a Subje<a. And though of late I have been called upon to make good my Suggefti- ons; yet as the Perfon, who made this Call, doth not appear to think maturely enough to underftand, either thofe Meta- phyfics which he would refute, or Ma- thematics which he would patronize I fhould have fpared my felf the trouble of writing for his Convidion. Nor /hould I now have troubled you or my felf with this Addrcfs, after fo long an Intermiflion of thefe Studies ; were it not to prevent, fo far as I am able, your impofing on your felf and others in Matters of much higher Moment and Concern. And to the end that you may more clearly comprehend
the
jr^ The Ana l y s r:
the Force and Defign of the foregoing Remarks, and purfuc them ftill further in your own Meditations, I fhall fubjoin the following Queries.
^ery I. Whether the Objedl of Geome- try be not the Proportions of affignable Extenfions? And whether, there be any need of confidering Quantities either in- finitely great or infinitely fmall ?
^. 2. Whether the end of Geometry be not to mcafure affignable finite Ex- tenfion ? And whether this practical View did not firft put Men on the ftudy of Geometry ?
^. 3. Whether the miftaking the Ob- jedt and End of Geometry hath not crea- ted needlefs Difficulties, and wrong Puf- fuits in that Science ?
^. 4. Whether Men may properly be faid to proceed in a fcientific Method, without clearly conceiving the Object they are converfant about, the End propofed, and the Method by which it is purfued ?
^. 5. Wiie-
ThV Analyst. 7;^
^. 5. Whether it doth not fuffice, that every affignable number of Parts may be contained in fome affignable Magnitude ? And whether it be not unneceflary, as well as abfurd, to fuppofe that finite Extenfioii is infinitely divifible ?
^. 6. V/hether the Diagran^s in a Geo- metrical Demonftration are not to be confi- dered, as Signs of all poffible finite Fi- gures, of all fenfible and imaginable Ex- tenfions or Magnitudes of the fame kind ?
^. 7. Whether it be poffible -to free Geometry from infuperable Difficylties and Abfurdities, fp long as either, the abftracl general Idea of Extenfion, or abfolute ex- ternal Extenfion be fuppofed its true Ob-r jed?
^. 8. Whether the Notions of abfolute Time, abfolute Place, and abfolute Mo- tion be not moft abfl.rad:edly Metaphyfi- cal ? Whether it be poffible for us to mea- fure, compute, or know them ?
^. 9. Whether Mathematicians do not engage themfelves in Difpute3 and Para- doxes,
^8 T HE A N AL Y S T.
doxes, concerning what they neither do nor can conceive ? And whether the Doc- trine of Forces be not a fufficient Proof of this? *
^. 10. Whether in Geometry ir may not fuffice to confider affignable finite Mag- nitude, without concerning our felves with Infinity? And whether it would not be Tighter to meafure large Polygons having finite Sides, inftead of Curves, than to fuppofc Curves are Polygons of infinitefi- mal Sides, a Suppofition neither true nor conceivable ?
Slu, II, Whether many Points, which are not readily affented to, are not never- thelefs true? And whether thofe in the two following Queries may not be of that Number ?
%. 13. Whether it be poffible, that we fhould have l^ad an Idea or Notion of Extenfion prior to Motion? Or whether if a Man had never perceived Motion, he would ever have known or conceived one thing to be diftant from another ?
* See a Latin Trc^tife De Motu, publifhed at London, in the Year 1721.
^. 13. Whe-
H TheAnalysT. ^9
Qui. 13. Whether Geometrical Quantity hath coexiftcnt Parts ? And whether all Quantity be not in a flux as well as Time ^nd Motion ?
^. 14. Whether Extenfion can be fup- pofed an Attribute of a Being inrmiutable and eternal ?
^. 15. Whether to decline examining the Principles, and unravelling the Me- thods ufed in Mathematics, would not fhew a bigotry in Mathematicians ?
%. 16. Whether certain Maxims do not pafs current among Analyfls, which are fhocking to good Senfe ? And whether the common Affumption that a finite Quantity divided by nothing is infinite be not of this Number ?
^. 17. Whethpr the confidering Geo- n^trical Diagrams abfolutely or in them- fejves, rather than as Reprefentatives of mi aflignable Magnitudes or Figures of the fame kind, be not a principal Caufe of ;hc fuppofing finite Extenfion infinite-
3o . TheAnalyst:
ly divifible ; and of all the Difficulties and Abfurdities confequent thereupon ?
^i. 1 8. Whether from Geometrical Propofitions being general, and the Lines in Diagrams being therefore general Sub- flitutes or Reprefentatives, it doth not fol^ low that we may not limit or confider the number of Parts, into which fuch parti- ticular Lines are divifible?
^. 19. When it is faid or implied, that fuch a certain Line delineated on Paper contains more than any affignable number of Parts, whether any more in truth ought to be underftood, than that it is a Sign indifferently reprefenting all- finite Lines, be they ever fo great. In' which relative Capacity it contains, /. ej ftands for more than any effignable num- ber of Parts ? And whether it be not alto-' gether abfurd to fuppofe a finite Line, confidered in it felf or in its own pofitive Nature, flibuld contain an infinite num-r- ber of Parts ?
^. 20. Whether all Arguments for. the infinite Divifibility of finite Extenfion
do
The Analyst. S i
do not fuppofe and imply, either general abftrad: Ideas or abfolute external Exten- fion to be the Objedt of Geometry ? And, therefore, whether, along with thofe Sup- pofitions, fuch Arguments alfo do not ceafe and vanifh ?
^. 21. Whether the fuppofed infinite Divifibility of finite Extenfion hath not been a Snare to Mathematicians, and a TKorn in their Sides ? And whether a Quantity infinitely diminiflied and a Quan* tity infinitely fmall are not the fame thing ?
^. 22. Whether it be neceflary to confider Velocities of nafcent or eva- nefcent Quantities, or Moments, or Infi- nitefimals? And whether the introducing of Things fo inconceivable be not a re- proach to Mathematics ?
^. 23. Whether Inconfiftencies can be Truths ? Whether Points repugnant and abfurd are to be admitted upon any Sub- je<5l, or in any Science? And whether. the ufe of Infinites ought to be allowed, as a G fufficient;
8i The Analyst.
fufficicnt Pretext and Apology, for the ad- mitting of fuch Points in Geometry ?
^. 24. Whether a Quantity be not properly faid to be known, when we know its Proportion to given Quantities ? And whether this Proportion can be known, but by Expreffions or Exponents, either Geometrical, Algebraical, or Arith- metical ? And whether Expreffions in Lines or Species can be ufeful but fo far forth as they are reducible to Numbers ?
25. Whether the finding out proper Expreffions or Notations of Quantity be not the moft general Character and Ten- dency of the Mathematics ? And Arithme- tical Operation that which limits and defines their Ufe ?
^. 26. Whether Mathematicians have fufficiently confidered the Analogy and Ufe of Signs? And how far the fpecific limit- ed Nature of things correfponds thereto?
^. 27. Whether becaufe, in ftating a general Cafe of pure Algebra, we are at
full
I
The Analyst. 8j
full liberty to make a Charaifter denote, either a pofitive or a negative Quantity, or nothing at all, we may therefore in a geometrical Cafe, limited by Hypothefes and Reafonings from particular Proper- ties and Relations of Figures, claim the fame Licence ?
^, 28. Whether the Shifting of the Hypothefis, or (as we may call it) the fal^ lacia Suppojitionis be not a Sophifm, that far and wide infedts the modern Rea- fonings, both in the mechanical Philo- fophy and in the abftrufe and fine Geo- metry ?
^«. 29. Whether we can form an Idea or Notion of Velocity diftind: from and exclufive of its Meafures, as we can of Heat diftin(5t from and exclufive of the Degrees on the Thermometer, by which it is meafured ? And whether this be not fuppofed in the Reafonings of modern Analyfts ?
^. 30. Whether Motion can be con- eeived in a Point of Space ? And if Mo-
G 2 tion
§4 The Analyst.
tion cannot, whether Velocity can ? And if not, whether a firft or laft Velocity can be conceived in a mere Limit, ei- ther initial or final, of the defcribed Space ?
^. 31. Where there are no Incre- ments, whether there can be any Ratio of Increments ? Whether Nothings can be confidered as proportional to real Quan- tities ? Or whether to talk of their Pro- portions be not to talk Nonfenfe ? Alfo in what Senfe we are to underftand the Proportion of a Surface to a Line, of' an Area to an Ordinate ? And whether Species or Numbers, though properly ex- preffing Quantities which are not homo- geneous, may yet be faid to exprefs their Proportion to each other?
^. 32. Whether if all aflignable Cir- cles may be fquared, the Circle is not, to all intents and purpofes, fquared as well as the Parabola? Or whether a pa- rabolical Area can in fadl be meafured more accurately than a Circular ?
% 33. Whe-
r
T H E A N A L Y S T. 85
^. 33. Whether it would not be righter to approximate fairly, than to endeavour at Accuracy by Sophifms ?
^^ 34. Whether it would not be more decent to proceed by Trials and Induc- tions, than to pretend to demonflrate by falfe Principles ?
^. 35. Whether there be not a way of arriving at Truth, although the Prin- ciples are not fcientific, nor the Reafon- ing juft ? And whether fuch a way ought to be called a Knack or a Science ?
^. 36. Whether there can be Science of the Conclufion, where there is not Science of the Principles ? And whether a Man can have Science of the Princi- ples, without underftanding them ? And therefore whether the Mathematicians of the prefent Age adl like Men of Science, in taking fo much more pains to apply their Principles, than to under- ftand them ?
G 3 ^«. 37, Whe-
9^ T H E A N A L Y S T.
^. 37. Whether the greateft Genius wreftling with falfe Principles may not be foiled? And whether accurate Quadratures can be obtained without new Foftulata or Affumptions ? And if not, whether thofe which are intelligible and confiftent ought not to be preferred to the contrary ? See Sed. XXVIII and XXIX.
^. 38. Whether tedious Calculations in Algebra and Fluxions be the liklieft Method to improve the Mind ? And whe- ther Mens being accuftomed to reafon altogether about Mathematical Signs and Figures, doth not make them at a lofs how to reafon without them?
%/. 39. Whether, whatever readinefs Analyfts acquire in ftating a Problem, or finding apt Expreffions for Mathematical Quantities, the fame doth neceflarily in- fer a proportionable ability in conceiving and expreffing other Matters ?
i^/. 40. Whether it be not a general Cafe or Rule, that one and the fame Co- efficient dividing equal Produd:s gives e-
qual
b
T H E A N A L Y S T. 87
qual Quotients ? And yet whether fuch Coefficient can be interpreted by o or; nothing ? Or whether any one will fay,' that if the Equation 2 x 0 = 5 x c?, be di- vided by (?, the Quotients on both Sides are equal? Whether therefore a Cafe may not be general with refpedl to all Quantities, and yet not extend to No- things, or include the Cafe of Nothing? And whether the bringing Nothing un- der the Notion of Quantity may not have betrayed Men into falfc Reafoning ?
^. 41. Whether in the mofl general Reafonings about Equalities and Propor- tions, Men may not demonflrate as well as in Geometry? Whether in fuch De- monflrations, they arc not obliged to the fame flrid; Reafoning as in Geometry ? And whether fuch their Reafonings are not deduced from the fame Axioms with thofe in Geometry ? Whether therefore Alge- bra be not as truly a Science as Geo- metry ?
^. 42. Whether Men may not reafon in Species as well as in Words? Whether . G 4 the
8S T HE A N A L Y S T.
the fame Rules of Lpglc do not obtain in both Cafes ? And whether we have not a right to expcd and demand the fanie Evi- dence in both ?
^L 43. Whether an Algebraift, Fluxio- nift, Geometrician or Demonftrator of any YmA can expedl indulgence for obfcure Principles or incorred Reafonings? And v^hether an Algebraical Note or Species » can at the end of a Procefs be interpreted in a Senfe, which could not have been fub- ftituted for it at the beginning ? Or whe- ther any particular Suppofition can come under a general Cafe which doth not con- fift with the reafoning thereof ?
%f. 44. Whether the Difference be- tween a mere Computer and a Man of Science be nor, that the one computes on Principles clearly conceived, and by Rules evidently demonftrated, whereas the other doth not ?
^. 45. Whether, although Geometry be a Science, and Algebra allowed to be a Science, and the Analytical a moil excel- lent
The Analyst. 8^
licnt Method, in the Application neverthe- Icfs of the Analyfis to Geometry, Men may not have admitted falfe Principles and wrong Methods of Reafoning ?
^. 46. Whether although Algebraical Reafonings are admitted to be ever fo juft, when confined to Signs or Species as gene- ral Reprefentatives of Quantity, you may not neverthelefs fall into Error, if, when you limit them to ftand for particular things, you do not limit your felf to rca- fon confiftently with the Nature of fuch particular things ? And whether fuch Er- ror ought to be imputed to pure Algebra ?
%. 47. Whether the View of modern Mathematicians doth not rather feem to be the coming at an Expreflion by Artifice, than the coming at Science by Demonftra- tion ?
I.i ^. 48. Whether there may not be found Metaphyfics as well as unfound? Sound as well as unfound Logic? And I ivhether the modern Analytics may not be brought under one of tjiefe Denominations, and which?
^. 49. Whe-
jS T H E A N A L Y S T.
^. 49. Whether there be not really a Philofophm prima, a certain tranfcenden- tal Science fuperior to and more exteniive than Mathematics, which it might behove our modern Analyfts rather to learn than defpife ?
^. 50. Whether ever fince the recovery of Mathertiatical Learning, there have not been perpetual Difputes and Controverfies among the Mathematicians ? And whether this doth not difparage the Evidence of their Methods ?
^. 51. Whether any thing but Meta- phyfics and Logic can open the Eyes of Mathematicians and extricate them out of their Difficulties ?
^. ^2. Whether upon the received Principles a Quantity can by any Divifion or Subdivifion, though carried ever fo far, be reduced to nothing ?
%/. 53. Whether if the end of Geo- metry be Pradtice, and this Practice be Meafuring, and \y& meafure only affigna-
ble
The Analy s t.
ble Extenfions, it will not follow that un- limited Approximations compleatly an- fwcr the Intention of Geometry ?
^. 54. Whether the fame things which are now done by Infinites may not be done by finite Quantities? And whether this would not be a great Relief to the Imagi- nations and Underilandings of Mathema- tical Men ?
^. 55, Whether thofe Philomathema- tical Phyficians, Anatomifts, and Dealers in the Animal Oeconomy, who admit the Dodbrine of Fluxions with an implicit Faith, can with a good grace infult other Men for believing what they do not com- prehend ?
^. 56. Wlicther the Corpufcularian, Experimental, and Mathematical Philo- fophy fo much cultivated in the laft Age, hath not too much cngroffed Mens At- tention; fome part whereof it might have ufefuUy employed ?
^. ^j, Whe-
51
!t -
SI T HE A N A L Y S T.
^. 57. Whether from this, and other concurring Caufes, the Minds of fpecula- tive Men have not been born downward, to the debafing and ftupifying of the higher Faculties ? And whether we may not hence account for that prevailing Narrow- nefs and Bigotry among many who pafs for Men of Science, their Incapacity for things Moral, Intelledtual, or Theological, their Pronenefs to meafure all Truths by Senfe and Experience of animal Life ?
^. 58. Whether it be really an EfFe(ft of Thinking, that the fame Men admire the great Author for his Fluxions, and de- ride him for his Religion ?
^/. 59. If certain Philofophical Vir- tuofi of the prefent Age have no Religion, whether it can be faid to be for want of Faith >
^, 60. Whether it be not ajufter way of reafoning, to recommend Points of Faith from their Effeds, than to demon- ftrate Mathematical Principles by their Conclufions ?
^.61. Whe-
IPI^^ The Analyst. 5>5
^^. 6 1. Whether it be not lefs exccp- I tionable to admit Points above Reafon than contrary to Reafon ?
^u. 62. Whether Myfteries may not with better right be allowed of in Divine Faith, than in Humane Science ?
^. 63. Whether fuch Mathematicians as cry out againft Myfteries, have ever examined their own Principles ?
^. 64. Whether Mathematicians, who are fo delicate in religious Points, are flrid:- ly fcrupulous in their own Science ? Whe- ther they do not fubmit to Authority, take things upon Truft, believe Points incon- ceivable ? Whether they have not their Myfteries, and what is more, their Re- pugnancies and Contradidions?
^. 65. Whether it might not become Men, who are puzzled and perplexed a- bout their own Principles, to judge wari- ly, candidly, and modeftly concerning o- ther Matters?
^r/.66. Whe-
J4 T HE A N AL Y S T.
^. 66. Whether the modern Analytics do not furnifli a ftrong argumentum ad ho- minem^ againft the Philomathematical In- fidels of thefc Times ?
^. 67. Whether it follows from the abovementioned Remarks, that accurate and juft Reafoning is the peculiar Cha- radler of the prefent Age? And whether the modern Growth of Infidelity can be afcribed to a Diftindlion fo truly valuable >
FINIS.
fe
S?, 36, It