The analyst, or, A discourse addressed to an infidel mathematician : wherein it is examined whether the object, principles, and inferences of the modern analysis are more distinctly conceived, or more evidently deduced, than religious mysteries and points / Berkeley, George, 1685-1753

kd i art 




Library 

of the 

University of Toronto 



THE 

ANALYST; 

O R, A 

DISCOURSE 

Addrefled to an 

Infidel MATHEMATICIAN. 

WHEREIN 

It is examined whether the Objec~l, Princi 
ples, and Inferences of the modern Analy- 
fis are more diftindtly conceived, or more 
evidently deduced,than Religious Myfterie 
and Points of Faith. 



By the A u T H o R of fbe- Minute Philofopher. 



irft caft out the beam out of thine own Eye ; arid then 
Jhalt thoujee clearly to caft out the Jnote out of thy bro 
thels eye. S. Matt. c. vii. v. f. 



L O N D O N: 
Printed for J. TONS ON m the Strand. 1754* 






3 ; 



? Jf 



A 



THE 

CONTENTS. 

SECT. I. Mathematicians prefumed to 
be the great Mafters of Reafon. Hence 
an undue deference to their decifions 
where they have no right to decide. This 
one Caufe of Infidelity. 

II. Their Principles and Methods to be exa 
mined with the fame freedom, which 
they affume with regard to the Principles 

and Myfteries of Religion. In what Senfe 
and how far Geometry is to be allowed ajt 
Improvement of the Mind. 

III. Fluxions the great ObjeSi and Employment 
of the profound Geometricians in the pre- 

fent Age. What thefe Fluxions are. 

IV. Moments or nafcent increments of flowing 
Quantities difficult to conceive. Fluxions 
of different Orders. Second and third 
Fluxions obfcure Myfteries. 

A 2 V. Dip- 



The CONTENTS. 

V. Differences, i. e, r Internments or Deer & 
ments infinitely fmall, ujed by foreign Ma 
thematicians injlead of Fluxions or Velo 
cities of nafcent and evanefcent Incre 
ments. 



I. Diff efjnc-es, of various Ordcfs^ i. e. 
Cities Infinitely Icfs than* Quantities i nJL- 
? Jffidfa ]>it tie ^ - y and Infinite fimal farts of 

infinikejinifils of mfinitefimalS) &c. without - 

end or limit. 



^Vf^^V^^^A objeStedagainJt 
by thoje who admit them in Science. 

VIII, Madern Analyfis fuppdfedby tbemfefoes 
to.extend their views even beyond infinity : 
Deluded by their own Species or Symbols. 



\% f Method for finding the Fluxion of a Reff- 
angle of two indeterminate Quantities^ 
fuvwedto be illegitimate andfalfe. 

X. Implicit Deference of Mathematicalmcn 

for the great Author of Fluxions. Their 

ecrnejlnefe rather to g& on f a/I a?2dfar y 

than to jet out warily andje? their way 



XL Momen- 



The CONTENTS. 

XI. fyoinentums difficult to comprehend. 
middle Quantity to be admitted between. 
a finite Quantity drid*nothing y without 
admitting Infimtejimah, 

XII. The Fluxion of any Power .of a flowing 
Quantity. Lemma premifed in order to 
examine the "method for finding fucb 

p Fluxion. 

XIII. The rule for ile PtutioM . o/ Powers 

attained by unfair reajbnihg. 

XIV. The aforefaid reafoning farther unfold* 
ed anclfiewd tb Ve 



XV; N&^true Cvnclufiontobejuftly dnwn by 

diretf confequence from inconfift-ent Sup~ 

fofitions. The fame Rules of right rea- 

fon to be obferved y whether Me ft argue 

in Symbols or in Words. 
< 

XVI. An Hypothecs being dejiroyed.no confe- 
quence offuch Hypothecs to be retained. 

XVII. Hard todijiinguijh between evanefcent 
Increments and infinitefimal Differences. 
Fluxions placed in ^various Lights. The 
great Author, it feems, not fatisfied with 
bis Q*ivn~ Notions. 



The GONTEN,TS; K 

^P?^ 

rejected by Leibnitz and bis Fdllow&rir; 

No Quantity, according to them y greater 
or /mailer for the Addition or Subdue* 



^ ^ 

XIX. Conclujions to be proved by the Princi 
ples >and not Principles by the Conclujions. 

XX. The Geometrical Analyflconfidered as a 
Logician ; and his Difcoveries, not , in 
ihemfelves, but as derived from fucb 
Principles and by fuch Inferences.^ 

XXI. A tangent, drawn to the Parabola ac 
cording to the calculus differentialis. 
Truth fifwn to be therefult of error, and 

- Sic ^^ iut^ 

.XXII. By- virtue of a twofold mi/take Ana- 
lyfts arrive at Truth ^but not at Science : 
ignorant how they come tit thdroivn 
Conclufions. 

XXIII. The Conclufion never evident vr -accu 
rate ^ in virtue of chfcure or inaccurate 
Premifes. Finite Quantities 
reje&ed as well a 



XXIV. The foregoing Dotfrme far f her illu- 
^ : J?rat& X5CV. Sundry 



The CO NT E N T 

XXV. Sundry Obfervations thereupon. 

-am t*s *& ^mcbj <$ V^^A\ 

XXVL Ordinate found, from ibe Area by 
^ means of evanefcent Increments*. 

^ . V 

XXVII. J //6^ foregoing Cafe the Jtippofed 
^*&vanefcent Increment is really a finite 
x\^^uantity^ dejtroyed by an equal Quantity 
^ with an oppoflte Sign. 

XXVIIL The foregoing Cafe put generally. 
Algebraical Exprejfions compared with 
Geometrical Quantities. 

XXIX. Correfyondent Quantities Algebraical 
and Geometrical equated. The Analyfa 
Jhewed not to obtain in Infintejimals, but 

it mujl alfo obtain infinite Quantities. 

XXX, The getting rid of Quantities by the 
received Principles, whether of Fluxions 
or of Differences, neither good Geometry 
nor good Logic. Fluxions or Velocities, 



XXXI. Felocities not to be abftrafted from 

fime and Space: Nor their Proportions 

tobe inixftigated or confdered exclufi^ely 



^vvA-KTv .VlXX 
XXXII. Difficult 



XXXII. Difficult 

the. Principles of the modern Analyfu y and 
are the Foundation w whisk it is. builf* 



XXXIII. the rational Faculties whether Im 
proved byfucb obwc$ Analytic, &.\;\ 

XXXIV. By wk4f inconceivable Steps fyjte 
Lines are found proportional to Fluxions. 
Mathematical Injideh Jlrain at a Gnat 
andfwallow a Camel 

XXXV. Fluxions or Infmitefiwqls not 1o be a- 
voided on the received Principles. Nice Ab- 
ftra&ions and Geometrical $getaphyjics. 

* ^ - Vi V A. Vl\ 

XXX VI . F,chcities of nafient or evanefcent 
* Quantities .whether in reality underjlood 
and Jignijied by finite Lines and Specie^ 

XXXVII. Signs or Exponents obvious i Ut 

Fluxions themfehes not fo. 

i^^T - - J d ^< itvg^- >-JX 

XXXVUI- Fluxions, ivhetlitr tfy .Vehtfties 
with which infriitefmal Differences arj 

7 <) 

gentrafea * 

$r^$$l 

XXXKX./F/^w.?/* Fluxions or Mond 
^F^ions, vbetber to be conceded, as Velo 
* : Velocities, or " 



tftbe Jecond najcent Increment 
^v^. & XL. Fluxions 



Tfie CO NT E NTl 

XL. Fluxions conjidered, fometimes in one 
Senfe, fimetimes in another : One while in 
themfefoeS) another in their Exponents : 
Hence Confujion and Qbfeurity. 

XLI. Ifochronal Increments, whether finite or 
nafcent, proportional to their refyeStwe 
Velocities. 

XLII. Time fuppofed to be divided into Mo 
ments: Increments generated in thofe 
Moments : And Velocities proportional to 

ihofe Increments. 

, . ... 

XLII I. Fluxions, fecohd, third \ fourth, &c 

what they are, bow obtained, and how re- 
prefented. What Idea of Velocity in a Mo 
ment of Time and Point of Space. 

JCLIV. Fluxions of all Orders inconceivable, 

XLV. Signs or Exponents confounded with 
the Fluxiohs; 

XLV1. Series ofExprcffions or of Notes eafily 
contrived. Whether a Series, of mere Ve 
locities, or of mere nafcent Increments^ 
terrefponding therewith, be as eafily cfoi- 
ceiled ? 

B 47. Ctttfifftk 



: The CONTENTS 

XL VII. Celerities difmi/ed, and injltad them. 
ofOrdinates and Anas introduced. Ana 
logies and Expreffions-iifeful in the modern 
Quadratures^, may yet be ttfelefs fof ena 
bling us to conceive Fluxions. No right, 
to apply the Rules without knowledge of 
the Principles. 

XL VIII. Metapbyfies of modern Analyjis mojl 
incomprehenfible. 

XLIX. Analyjis employed about notional Jb a- 
dowy Entities* The.ir Logics as except io- 
tbeirMetaphyfics. 



tr. Qccajion of this Addrefs. Conclufion. 

3$ 

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ajmrij msmulk;^- v\: ; 




vhirn 003 ba t ,m.. t 

- s ^ Du ^*^i |^iii^^ 
v u io 

HS* - : 

^t - gbb 

r ^i s/3 df o:? no 

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saMo 1 ! IK ^o^M lo 

. .-iw. 





*)P H E 

ANALYST. 

HOUGH I am a 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 been 
your peculiar Study ; nor to the Authority 
that you therefore affume in things foreign 
to your Profeffion, nor to the Abufe that 
you, and too many more of the like Cha- 
radler, are known to make of fuch undue 
Authority, to the mifleading of Unwary 
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* 
difregard the Judgment of Men, in Points 
B 2 which 



THE ANA L Y s T. 

which they have not conlidered or exarrri 5 * 
ned. But feveral who make the loudeft 
Claim to thofe Qualities, do, neverthelefs^ 
the very thing they would feem to defpife, 
clothing themfelves in the Livery of other 
Mens Opinions, and putting on a general 
deference for the Judgment of you, Gen 
tlemen, who are prefumed to be of ll 
Men the greateft Matters of Reafon, to be 
moft converfant about diftmdt Ideas, and 
never to take things upon truft, but aU 
ways clearly to fee your way, as Men 
whofe conftant Employment is the de 
ducing Truth by the jufteft inference from 
the moft evident Principles. With this" 
bias on their Minds, they fubmit to your 
Decifions v>here you have no right to de 
cide. And that this is one (hort way of 
making Infidels I am credibly informed. 






II. XVhereas then it is fuppofed, that 
you apprehend more diftindly, confider 
more clofely, infer more juftly, conclude 
morp accurately than ether Men, and that 
you are therefore lefs religious becaufe 
more judicious, I fhall claim the privilege 
of a Free-Thinker j and take the Liberty 

to 



THE ANA L Y s r7 

roinquire into the Object, Principles, and 
Method of Demonftration admitted by the 
Mathematicians of the prefent Age, with 
the fame freedom that you prefume to 
treat the Principles and Myfteries of Reli 
gion ; to the end, that all Men may fee 
what right you have to lead, or what En 
couragement others have to follow you. 
It hath been an old remark that Geome 
try is an excellent Logic. And it muft be 
ewned, that when the Definitions are clear; 
when the Poftulata cannot be refufed, nor 
the Axioms denied ; when from the dif- 
tindt Contemplation and Comparifon of 
Figures, their Properties are derived, by a 
perpetual well-connected chain of Confe- 
quences, the Objects being ftill kept in 
view, and the attention ever fixed upon 
them ; there is. acquired an habit of rea- 
foning, clofe and exact and methodical : 
which habit ftrengthens and Sharpens the 
Mind, and .being transferred to other 
Subjects, is of general ufe in the inquiry 
after Truth. But how. far this is the cafe 
of our Geometrical Analyfts, it jnay hp 
while to confides 



B 3 III. The 



THE ANALYST; 

III. The Method of Fluxions is the 
neral Key, by help whereof the modern 
Mathematicians unlock the fecrets of Geo 
metry, and confequently of Nature. And 
2S it is that which hath enabled them fa 
remarkably to outgo the Ancients in dif- 
covering Theorems and folving Problems, 
the exercife 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, confident or 
repugnant, demonftrative or precarious, as 
I ihall inquire with the utmoft impar 
tiality, fo I fubmit my inquiry to your 
pwn Judgment, and that of every candid 
Reader. Lines are 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 
<yr leffer, according to the greater or 
leffer Velocity, wherewith they increafe 
and are generated, a Method hath been 
found to determine Quantities from the 
Velocities of their generating Motions. 

rit ^trod. ad Quadraturam Curvaram, 

And 



T HE .ANA t y s f; 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 
Jeaft 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 considered, under the Ap 
pellation of Moments. 

IV. By Moments we are not to under- 
fland finite Particles. Thefe are laid 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 negle<3ed 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 
onfidered. And of, the ^forefaid Fluxions 
B 4 there 



TrB E A T: A L Y S T. 

.there be other Fluxions, which Fluxions 
of Fluxions are called fecond Fluxions.. 
And the Fluxions of thefe fecond Fluxions 
are called third Fluxions : and fo on, fourth, 
fifth, fixth, &-c. ad infinitum. Now as our 
Senfe is {trained and puzzled with the 
perception of Objeds extremely minute, 
<even fo the Imagination, which Faculty 
derives from Senfe, is very much ftrained 
and puzzled to frame clear Ideas of the 
Jeaft Particles of time, or the leaft Incre 
ments generated therein : and much more 
fo to comprehend the Moments, of 
thofe Increments of the flowing Quanti 
ties in ftatu nafcenti^ in their very firft 
origin or beginning to exift, before they 
become finite Particles. And it feems ftill 
more difficult, to conceive the abftrafted 
Velocities of fuch nafcent imperfed: En-> 
tities. But. thq Velocities of the Velocities, 
the fecond,^ third, fourth and fifth Velo 
cities, ^. exceed, ^jfj^rpiftake not, all 
Humane Underftanding. The further the 
Mind analyfeth. and purfueth thefe fugi- 
tive Ideas, the more it is Jpft and be 
wildered; the Objects, at firft fleeting and 
minute, fpon varjifhing:OUt of fight. Cert ^ 

tainly 



T H 1 A^H^X 1 * &T& 

tarnfy in any Senfe " a fecond or third 
Fluxion feems an obfcure Myftery. The 
incipient Celerity of an incipient Celerityi 
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 inipofiible, 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 fo 
think of third, fourth, fifth Fluxions, and 
fo on ward without end? 



V. The foreign Mathematicians arc 
fuppofed by fome, even of our own, to 
proceed in a manner, lefs accurate per 
haps and geometrical, yet more intelligi 
ble. Inflead of flowing Quantities and 
their Fluxions, they confider the variable 
finite Quantities, as increafing or dimi- 
nifliing by the continual Addition or Sub^ 
dud:ion of infinitely fmall Quantities. In^ 
ftead of the Velocities wherewith Incre 
ments are generated, they confider the In 
crements or Decrements themfetves, which 
* hey call Differences, and which are 



to T HE A N A L Y s tC 

pofcd 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 eonfift 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 iit* 
finitely fmall, that is, infinitely lefs than 
any fenfible or imaginable Quantity, or 
than any the leaft finite Magnitude, is, I 
confefs, above my Capacity. But to coa^ 
ceive a Part of fuch infinitely fmall Quan 
tity, that (hall be ftill infinitely lefs than 
it, and confequently though multiply d 
infinitely {hall never equal the minuteft 
finite Quantity, is, I fufpe<3:, -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 refleft, and do not take 
things upon truft. 

" 2 U./ / i ) i i il i 1 ) t w : : = ">" KOQ 

VI. And yet in the calculm differentially 
which Method ferves to all the fame In 
tents and Ends with that of Fluxions, 

our 



T H E A N X L-yj-T. ** 

our modern Analyfts are n6t content to 
confider only the Differences of finite 
Quantities : they alfo confider the Diffe 
rences of thofe Differences, and the Diffe 
rences of the Differences of the firft Diffe 
rences. And fo on ad infinitum. That- is, 
they confider Quantities infinitely lefs than 
the leaft difcernible Quantity ; and others 
infinitely lefs than thofe infinitely fmall ones j 
and {till others infinitely lefs than the prece^ 
ding Infinitefimals, and fo on without end 
or limit. Infomuch that we are to ad^ 
jnit an infinite fucceffion of Infinitefimals, 
each infinitely lefs than the foregoing, 
and infinitely greater than the following. 
As there are firft, fecond, third, fourth, 
fifth, &c, Fluxions, fo there are Diffe 
rences, firft, fecond, third, fourth, &c. in 
^n infinite Progreffion towards nothing^ 
which you ftill approach and never arrive 
at. And (which is moft ftrange) although 
you fhould take a Million of Millions of 
thefe Infinitefimals, each whereof is fup- 
pofed infinitely greater than fome ether 
real Magnitude, and add them to the leaft 
given Quantity, it {hall be never the bigger, 
for this is one of the modeft fojlulafa of 

our 



|t T H E A N A L Y S f * 

our modern Mathematicians, and is a Gbr- 
ner-ftone or Ground-work of their Specu* 
lations. 

>q> ibdb yd tmafkm kia JwvfoM&j^i 
VII All thefe Points, I fay, are 
pofed and believed by certain rigorous 
afters 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. JBut 
he who can digeft a fecond or third Fluxi 
on, a fecond or third Difference, need nor, 
rnethinks, be fqueamifh about any Point 
in Divinity. There is a natural Prefump* 
tion that Mens Facnkies are made alike. 
It is on this Suppofuion that they attempt 
to argue and convince one another. What, 
therefore, fhall appear evidently impoffi* 
ble and repugnant to one, may be pre- 
f umed the fa mfe to another. But with 
what appearance of Reafon (hall any Man 
prefume to fay, that Myileries may not 
be Obje<3:s of Faith, at the fame time that 
he hirnfelf admits fuch obfcure lylyfteries 
to be -the Object of Science ? 

VIII. It 



T HE A N A LT8 ti 1$ 

It muft indeed be acknowledged, 
the modern Mathematicians do not confi* 
der thefe Points as Myfteries, but as elear^ 
ly conceived and mattered by their com- 
prehenfive Minds. They Scruple not to 
fay, that by the help of thefe new Analy 
tics they can penetrate into Infinity it felf 2 
That they can even extend their Views be 7 
yond Infinity : that their Art comprehends 
not only Infinite, but Infinite of Infinite (as 
they exprefs it) or an Infinity of Infinite^ 
But, notwithftanding all thefe AfTertions 
and Pretenfions, it may be juftly queftion* 
cd 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 iseafietf 
than to devife Expreffions or Notations for 
Fluxions and Infinitefimals of the firft, fe- 
cond, third, fourth and fubfequent Orders^ 
proceeding in the fame regular form with- 
out end or limit # m #\# %. &c. or dx. ddx+ 
dddx* ddddx &c. Thefe Expreffions in 
deed are clear and diftinfl, and the Mind 
finds no difficulty in conceiving them tec 
be continued beyond any affignable Bounds^ 

But 



THE ANALYST. 

But if we remove the Veil and look under 
neath, if laying afide the Expreffions we 
fet our felves attentively to eonfider the 
things themfelves, which are fuppofed to 
be exprefled or marked thereby, we fhall 
difcpver much Emptinefs, Darknefs, and 
Cpnfufion 5 nay, if I miftake not, direct 
Impoffibilities and Contradictions. Whe 
ther this be the cafe or no, every think 
ing Reader is intreated to examine and 
judge for himfelfc 



IX. Having considered the Object, I 
proceed to eonfider the Principles of this 
pew Analyfis by Momentums, Fluxions, or 
Infinitefimals ; wherein if it fhall appear 
that your capital Points, upon which the 
reft are fuppofed to depend, include Er- 
for and falfe Rcafoning; it will then fol 
low that you, who are at a lofs to con 
duct your felves, cannot with any decent 
cy fet up for guides to other Men. The 
main Point in the method of Fluxions is 
to obtain: the Fluxion or Momentum of 
the Re&angle or Product of two indeter 
minate Quantities. Inafmuch as from 
thence ar* derived Rules for obtaining the 

Fluxions 



ANA ty^st. 

Fluxions of all other Products and Pow 
ers j be the Coefficients or the Indexes what 
they will, integers or fractions, rational 
or funj. Now this fundamental Point 
one would think fliould 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- 
inonftration. * Suppofe the Produd: or 
Redtangle AB increafed by continual Mo 
tion: and that the momentaneous Incre 
ments of the Sides A and B are a and b* 
When the Sides A and B were deficient, or 
JefTer by one half of their Moments, the Reft- 
angle was 4~^ a * ti^b i. e, AB~~aB 
r^bA + ^ab. And as foon as the Sides 
A and B are increafed by the other two r 
halves of their Moments, the Redangle 
becomes A -\- a x B --f ^b or AB + aB + 
ib A -\-\ab. From the latter Redtangle 
fubdud: the former, and the remaining diffe 
rence will be &B + bA. Therefore the 
Increment of the Redangle generated by 
the intire Increments a and b is 



* Natiif alls Philofcph i^ principia mathematica, I. z- 

Isife %ninl^da iai . :.TOO &"\* ^pr^a 



THE ANALYST. 

. But it is plain that the difeffe 
and true Method to obtain the Moment of 
Increment of the Reftangle AB is to tak6 
the Sides as increafed by their whole In 
crements, and fo multiply them together, 

by + , the Produft whereof 
B^rbA-^ ab is the augmented 
Reftangle ; whence if we fubduft AB, the 
Remainder aB-t bA + ab will be the trufc 
Increment of the Reftahgle, exceeding 
that which was obtained by the former 
illegitimate and indirect Method by thfe 
Quantity ab. And this holds univerfally 
be the Quantities a and b what they will, 
big or little, Finite or Infinitefimal, Incre 
ments, Moments, or Velocities. Nor will 
it avail to fay that a b is a Quantity ex 
ceeding fmall : Since we are told that in re 
bus mathematicis errores quam minimi ncn 
funt contemnendi. * Such reafoning as this, 
for Demonftration, nothing but the obfcuricy 
of the Subjedl 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 t Authority could 
move them to admit. The Cafe indeed k 

* Jntrod. ad Quadraturam 



THE ANALV s t 

difficult. There can be nothing done till 
you have got rid of the Quantity ab. 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; and Terms which 
ihould be fteadily ufed are ambiguous. 
But notwithstanding all this addrefs and 
skill the point of getting rid of a b 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 ; 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 
Menaccuftomed rather to compute than to 
think; earned rather to go on faft and far, 
than felicitous to fet out warily and fee 

their way diftinftly. #*?*&# *+ t& 
ofer C XL The 



i8 THEANALYS T. 

XI. The Points or meer Limits of nal- 
cent Lines are undoubtedly equal, 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 Infiniteftmal. 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 employ *d to efcape or a- 
void the admiffion of Quantities infinitely 
fmall, yet it feems ineffectual. 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, 
mo more than it is why we fhould take 

Quantities 



THE A N A L t s t. 

Quantities lefs than A and B in order to 
obtain the Increment of AB> of which 
proceeding it muft be owned the final 
Caufe or Motive is very obvious - y but it 
is not fo obvious or eafy to explain a juft 
and legitimate Reafon for it, or fhew 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 Quantity is derived *. But, as there 
feems to have been fome inward Scruple 
or Confcioufnefs of defeat in the forego 
ing Demonftration, 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 $ and in order there 
to fhall premife the following Lemma. 
< c If with a View to demonftrate any 

* Philofophi* naturalis principia Mathematica, lib. 2. 
1cm. 2, 

B a " Propo- 



" 



THE ANA i y s T. 

Propofition, a certain Point is fuppofed, 
" by virtue of which certain other Points 
" are attained ; and fuch fuppofed Point 
" be it felf afterwards destroyed or rejec- 
* { ted by a contrary Suppofition \ in that 
" cafe, all the other Points, attained thereby 
" and confequent thereupon, muft alfo 
" be deftroyed and rejected, fo as from 
<c thence forward to be no more fuppo- 
cc fed or applied in the Demonftration." 
This is fo plain as to need no Proof. 

XIII. Now the other Method of ob 
taining a Rule to find the Fluxion of any 
Power is as follows. Let the Quantity A? 
flow uniformly, and be it propofed to find 
the Fluxion of x n . In the fame time 
that x by flowing becomes x + a, the 
Power x n becomes x -\-o\ n , i. e. by the 
Method of infinite Series x n + nox* \ 

. n n n n 2 i > 11 

-\ oox + &c. and the Incre^ 

J^Q. ^ (t -\% -riif 

ments o and nox n ~~ l 4- ^ oox n ? 
4- &c. are one to another as i to nx^i 
4. "JLl^L xn-~2 + fc Let now the, In 
crements vaniih, and their laft Proportion 
will be i to nx n ~ l . But it fhould feem 

that 






T H E A N A L Y S T. 

that this reafoning is not fair or conclufive. 
For when it is faid, let the Increments 
vaniih, i.- e. let the Increments be nothing, 
or let there be no Increments, the former 
Suppofition that the Increments were 
fpmething, or that there were Increments, is 
deftroyed, and yet a Confequence of that 
Suppofition, i. e. an Expreffion 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 Expreffions, 
and every thing elfe derived from the Sup 
pofition of their Exiftence to vanifli with 
them. 

XIV. To make this Point plainer, I 
fliall unfold the reafoning, and propofe it 
in a fuller light to your View. It amounts 
therefore to this, or may in other Words 
be thus exprefled. I fuppofe that the 
Quantity x flows, and by flowing is in- 
creafed, and its Increment I call 0, fo 
that by flowing it becomes x + o. And 
as # increafeth, it follows that every Power 
of x is likewife increafed in a due Pro- 
C 3 portion, 



THE ANALYST. 

portion. Therefore as x becomes #4-0, 
x n will become x + o\ n i that is, accord 
ing to the Method of infinite Series, x* 

,4- nox* 1 + *"~~~-oox n *- 2 + &c. And 
if from the two augmented Quantities we 
fubduft the Root and the Power refpec- 
tively, we lhall have remaining the two 
Increments, to wit, o and n-ox n -~ l + 
^ %0tf*--2 Jf & Ct which Increments, 
being both divided by the common flivi- 
for o, yield the Quotients I and nx n i 

*$- ~ 0#*-~ 2 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 

is fomething. And I have proceeded all 
along on that Suppofition, without which 

1 Ihould not have been able to have made 
fo much as one fmgk Step. From that 
Suppofition it is that I get at the Incre 
ment of x* , that I am able to compare 
it with the Increment of #, 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 fuppofe that there is no Increment 

Of 



THE ANALYST. 

of AT, or that o 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 n x i, which 
is an Expreflion obtained in virtue of my 
firft Suppofition, which neceflarily pr e - 
fuppofeth fuch Suppofition, and which 
could not be obtained without it : All 
which feems a moft inconfiftent way of 
arguing, and fuch as would not be allow 
ed of in Divinity. 

XV. Nothing is plainer than that no 
juft Conclufion can be diredtly drawn from 
two inconfiftent Suppofitions. You may 
indeed fuppofe any thing poffible : But af 
terwards you may not fuppofe any thing 
that deftroys what you firft fuppofed. Or 
if you do, you muft begin de now. If 
therefore you fuppofe that the Augments 
vanifli, i.e. that there are 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 fuccced in, what is called by 
to B 4 the 



14 THE ANAL "Ys T. 

the celebrated Author, the Inveftigation 
of the firft or lafl Proportions of nafcent 
and evanefcent Quantities, by inftituting 
the Analyfis in finite ones. I repeat it 
again: You are at liberty to make any 
poffible Suppofition: And you may de 
ft roy 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 + o, o might 
have fignified either an Increment or no 
thing. But then which of thefe foever 
you -make it fignify, you muft argue con- 
fiftendy.wrth. fuch its Signification, and 
not proceed upon a double Meaning : 
Which :tb do- were a manifeft Sophifm. 
Whether, you argue in Symbols or in 
Word$j 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 affame ,at firft a Quantity 
by .nothing, and in the.Expref- 

fion 



TH E A-N A L Y S T. Z5 

fion x 4- o, o 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 tKe firft, of all thofe Mem 
bers of the Series conftituting the Power 
of the Binomial ; you will therefore never 
come at your Expreffion 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 
Shifting 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 0, 
all had vanished at once, and youi muft 
have got nothing by your Suppofition. 
Whereas by this Artifice of firft dividing, 
and then changing your Suppofition, you 
retain iand#* J . But, notwithftand- 
ing all this addrefs to cover it, the fal 
lacy is ftill the fame. For whether it be 
done fooner or later, when once the fe 
cond Suppofition or Affumption is made, 
in the fame inftant the former Aflumpti- 

on 



T HE A N AL Y S T. 

on and all that you got by it is deftroyed, 
and goes out together. And this is univer- 
fally true, be the Subjeft what it will, 
throughout all the Branches of humane 
Knowledge ; in any other of which, I 
believe, Men would hardly admit fuch a 
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 Redlangle 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 effeft the fame with that 
ufed in the calculus differentials *. For 
the fuppofing a Quantity infinitely dimi- 
niftied and therefore rejecting it, is in ef- 
fe<5t the rejedting an Infinitefimal ; and 
indeed it requires a marvellous fharpnefs 
of Difcernment, to be able to diflinguifli 
between evanefcent Increments and infini- 
tefimal Differences. It may perhaps be 
faid that the Quantity being infinitely di~ 
rniniihed becomes nothing, and fo no 
thing is rejeded. But according to the 

* Analyfe des infiniment petits, part, I. prop. 2. 

received 



THE. ANALYST. 

received Principles it is evident, that no 
Geometrical Quantity, can by any divifion 
or fubdivilion whatfoever be exhausted, or 
reduced to nothing. Confidering the vari 
ous Arts and Devices ufed by the great 
Author of the Fluxionary Method : in 
how many Lights he placethihis Fluxions: 
and in what different ways he attempts to 
demonftrate the fame Point : one would be 
inclined to think, he was himfelf fufpici-^ 
ous of the juftnefs of his own demonftra- 
tions ; and that he was not enough pleafed 
with any one notion fleadily to adhere to 
it. Thus much at leaft is plain, that he 
owned himfelf fatisfied concerning certain 
Points, which neverthelefs he could not 
undertake to demonftrate to others *. Whe 
ther this fatisfa&ion arofe from tentative 
Methods or Inductions ; which have 
often been admitted by Mathematicians* 
(for inftance by Dr. Wallh in his A- 
rithmetic of Infinites) is what I fhall not 
pretend to determine. But, whatever the 
Cafe might have been with refped: to the 
Author, it appears that his Followers 
have (hewn themfelves more eager in ap- 

* $(( Letter to Collins, Nov. 8, 1676. 

plying 



T H X A K A t Y S T\ 

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 Do&rine of 
Infinitefimals ; which as it intrudes up 
on him whether he will or no, fo it is 
admitted and embraced by others without 
the lea ft repugnance. Leibnitz and his 
Followers in their calculus differentialis 
making no manner of fcruple, firft to fup- 
pofe, and fecondly to rejeft 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 Objeft 
limply apprehended by the Mind, hath 
been already confidered *. I {hall now 
only obferve as to the method of getting 
rid of fuch Quantities, that it is done 
without the leaft Ceremony. As in 

* Sea. 5 and W^l 

Fluxions 



T HE A N A L Y S T. 

Fluxions the Point of firft importance, 
and which paves the way to the reft, is to 
find the Fluxion of a Produft of two in 
determinate Quantities, fo in the calculus 
differenti&lis (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 Produdt or Redlangle of the Differen 
ces. And in general it is fuppofed, that no 
Quantity is bigger or lefler 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 fhould 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 
cfagfietct of Geometry requires nothing 
fhould be negle&ed 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 



TH $ ANALYST. 

But this inverted way of demonstrating 
your Principles by your Conclufions, 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 Premifes 
falfe, and the Conclufion neverthelefs true, 
though not in virtue of fach Illation of 
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 Induction, 
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. 
iyriw fr ii lpir3 &* -^hriMirt^^ 11 ^^! 1115 

XX. I have no Controverfy about your* 
Conclufions, but only about your Logic 
and Method. How you demanftrate ? 
What Objects you are cortverfant with,- 
and whether you conceive them deafly? 

What 



T H E A N A L Y S T. 
What Principles you proceed upon; how 
found they may be j 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 
coming at them ; whether it be legitimate 
or illegitimate, clear or obfcure,fcientific or 
tentative. To prevent all poffibilityofyour 
miftaking me, I beg leave to repeat and 
infift, *hat I conlider 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 feem 
an unaccountable Paradox, that Mathe 
maticians mould deduce true Propofitions 
from falfe Principles, be right in the Con- 
cluiion, and yet err in the Premifes ; I mall 
endeavour particularly to explain why 
this may come to pafs, and mew how Er 
ror may bring forth Truth, though it 
cannot bring forth Science. 



XXI. In 



THE ANALYST: 

XXL 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 infinitefimal Differences, 



p 







Let AE be a Curve, the Abfciffe 
the ordinate PBy y the Difference of 
the Abfciffe PMdx, the Difference of 
the Ordinate RN=dy. Now by fuppofing 
the Curve to be a Polygon, and confequent- 
ly BN y the Increment or Difference of 
the Curve, to be a ftraight Line coincident 

with 



T H E A K A L Y S T. 3 J 

with the tangent, and the differential 
Triangle B RN to be finilliar to the tri 
angle TPB the Subtangent PT is found 
a fourth Proportional to RN: RB:PB< 
that is to dy : dx\ y. Hence the Subtangent 

will be y --. But herein there is an error 
d y 

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 RLE, 
\#hich is fimilar to P B *T, and therefore (in- 
ftead of R N)RL fhould have been the firft 
term of the Proportion, /. c. RN + N L, 
i> e. dy 4- z : whence the true expreffion 

for the Subtangent fhould have been *- 

There was therefore an error of defeci in 
making dy the divifor : which error was 
equal to 2?, /. e. NL the Line comprehend 
ed between the Curve and the Tangent. 
Now by the nature of the Curve yy=zpx, 
fuppofing p to be the Parameter, whence 
fey the rule of Differences 2ydy 
and dy p ~. But if you multiply^ 

by it felf, arid retain the whole Produdl 

without rejecting the Square of the DifFe- 
fi;ci <-:-_ ^ 

D; rence, 



j4 THE AHA L Y s T. 

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 */y= -- -, which followed 

& . J 2 y> 

from the erroneous Rule of Differences. And 
the meafure of this fecond error is -^- = z t 

Therefore the two errors being equal and 
contrary deftroy each other ; the firft er 
ror of defect being corrected by a fecond 
error of excefs. 

%~* 
XXII. 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 
riot 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 dy be ~n. By the 
thirty third Propofition of the firft Book of 
the Conies of ^po/lonius, and from fimilar 

Triangles, 
r^ ; Q 



T H E A N A L Y s f. 3 j 

Triangles, as 2x to y fo is m to h + z 

*=~- x . Likewife from the Nature of the 

farabola yy + 2 yn+n n = xp+ mp y and 

2 y n + n n = mp : wherefore ^ **** = m : 

and becaufe^^ = p x, y will be equal 
to x. Therefore fubftituting thefe valued 
inftead of m and AT we (hall have 



2 yy p 






which being reduced 



XXIII. Now I obferve in the firft place, 
that the Conelufion comes out right, noe 
becaufe the reje&ed 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 place, 
that whatever is rejeded, be it ever fa 
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- 
clufion. Your Theorems therefore cannot 
be accurately true, nor your Problems 
accurately folved, in virtue of Premifes, 
D 2 whieh 



THE ANALYST. 

which themfelves 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 are fuppofed finite 
Quantities ever fo great, the Conclufion 
will nererthelefs come out the fame : in- 
afmtfch as the rejected Quantities are le 
gitimately thrown out, not for their 
fmallnefs, but for another reafon, to wit, 
becauie 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 refpecl: to Quan 
tities finite as well as infinitefimal, great 
as well as fmall, a Foot or a Yard long as 
well as the minuteft Increment. 

XXIV. For 



T H I r A N A L Yr S V T. 

A 

XXIV. For the fuller illustration of this 
Point, I (hall confider it in another light, 
and proceeding in finite Quantities to the 
Conclufiou, I ihall oply then make uft 



37 




of one Infinitefimal. Suppofe the ftraight 
Line Mt^ cuts the Curve AT? in the 
Points R and S. Suppofe LR a Tangent 
at the Point R, A N the Abfcifle, NR 
and OS Ordinates. Let AN be produced 
to O, and R P be drawn parallel to N O. 
Suppofe AN=x, NR=y y N O = v, 
PS = z, the fubfecant MN=S. Let the 
Equation y z=x x exprefs the nature of the 
Curve ; and fuppofing y and x increafed 
by their finite Increments, we get y +z 
zxVtk V ui whence the former 
P 3 Equa- 



5 8 THE ANALYST. 

Equation being fubdudled there remains 
%=2xv+i)v. And by reafon of fimilar 
Triangles PS: PR:: NR: NM y i.e. 

z : v : : y: s = > wherein if for y and z 

we fubftitute their values, we get fj^^jj 
==j== - ~*L. And fuppofing NO to be 

infinitely diminished, the fubfecant NM 
will in that cafe coincide with the fubtan- 
gent NL 9 and v as an Infinitefimal may 
be rejected, whence it follows that 

S = NL ===-; which is the true va- 

f y * f,,.^ 2, X 2 . i V v .*<..._ 

lue of the Subtangent. And fince this was 
obtained by one only error, /. e. by once 
rejecting one only Infinitefimal, it fliould 
feem, contrary to what hath been faid,that 
an infinitefimal Quantity or Difference 
may be neglected or thrown away, and the 
Conclufion neverthelefs be accurately true, 
although there was no double miftake or 
rectifying 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 rectifies the other. For in the 

firfl 



THE ANA L Y s T. 

firfl place, it was fuppofed, that when 
NO is infinitely diminiftied 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 
Difference between NM and NL. There 
fore NM or S was too little for your fup- 
pofition, (when you fuppofed it equal to. 
NL) and this error was compenfated by a 
fecond error in throwing out i;, which 
laft 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 diiguifed. 
And to this in reality it amounts, and is 
at bottom the fame thing, if we mould 
pretend to find the Subtangent by hav 
ing firft found, from the Equation of 
the Curve and fimilar Triangles, a ge 
neral Expreffion for all Subfecants, and 
then reducing the Subtangent under this, 
general Rule, by confidering it as the 
D 4 Subfe- 



40 T H2 A N A L^ * T. 

Subfecant when v vanishes or becomes 
nothing. 

^ooioQ fi esvig bos <i^ o< 

XXV. Upon the whole I obferve, Firft, 
that ij can never be nothing fo long as 
there is a fecant. Secondly, That the fame 
Line cannot be both tangent and fecant. 
thirdly, that when v or NO % vanifheth, 
PS and SR do alfo variifh, and with 
them the proportionality of the limilar 
Triangles. Confequently the whole Expref- 
fion, which was obtained by means thereof 
and grounded thereupon, vaniiheth when 
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 all 
Secants whatfoever: and, as it neceflarily 
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 t hap the Subtan-, 
gent, and can never coincide with it$ 
which Coincidence to luppoie would be 
abfurd 5 for it would be fuppofing, the 
fame Line at the fame time to cut and 

* See tie foregoing Figure, 

not 



T RE A N ALTS T. 41 

not to-, cut another given Line, which is a 
manueft Contradiction, fuch as fubverts 
the Hypothecs and gives a Demonftration 
of its FaMboo.l. Sixthly, If this be, not 
admitted, I demand a Reafon why any 
other apagogical Demonftration, . or De- 
monftratKm ad abfurdum\ fhould he 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 RP, PS, and SR to be finite 
real Lines in order to form the Triangle 
R PS, in order to obtain Proportions by 
limilar 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 Suppoiition hath 
been deflroyed by a contrary one. Eighthly, 
That although, in the prefentcafe, 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 ccnfidered only as a Prefumption, 

/ -- & : ?: "o" " v o - - > 

as 



4 1 T H E A N A L^Y S T. 

as a Knack, an Art or rather an Artifice^ 
but nqt a fcientific Demonstration. 



XXVI. The Doctrine premifed may be 
farther illuflrated by the following fimple 
and eafy Cafe, wherein I fhall proceed by 
eyanefcent Increments. Suppofe 

fcVs 

A - 




. 
= o y and that ATA: is equal to 

the Area ^5C : It is propofed to find the 
Ordinate y or B C. When x by flowing 
becomes x -\- o, then x x becomes x x ~t" 
zxo-\-oo: And the Area ABC becomes 
ADH, and the Increment of xx will be 
equal to BDHC the Increment of the 

Area, 



THE ANALYST. 4j 

Area, / . e. to BCFD+CFH. And if 
we fuppofe the curvilinear Space C F H to 
be goo, then 2x0 +o o=yo -\-qoo which 
divided by o gives zx-\-oy-\- qo. And, 
fuppofing o to vanifti, 2xy, in which 
Cafe ACH will be a ftraight Line, and 
the Areas ABC, CFH, Triangles. Now 
with regard to this Reafoning, it hath 
been already remarked *, that it is not le 
gitimate or logical to fuppofe o to vanifh, 
*. e. to be nothing, /. e. that there is no 
Increment, unlefs we reject 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 muft 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 o is attended 
with no Error in the Conclufion ? I an- 
fvver, the true reafon hereof is plainly 
this: Becaufe q being Unite, qo is equal 
to o: And therefore 2x-\-o j0=^ = 2.y, 

* Sefi. 12 and 13. fupra. 

the 



ANALYST. 

the equal Quantities qo and o being de- 
ftroyed by contrary Signs. ^ 



As on the one hand it were 
abfurd to get rid of o by faying, let me 
contradict my felf : Let me fubvert my 
own Hypothefis : Let me 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 aflbming 
an Increment : So on the other hand it 
Would be equally wrong to imagine, that 
in , a geometrical Demonftration 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 
fhould be derived from inaccurate Prin 
ciples. Therefore o cannot be thrown out 
as an Infinltefimal, or upon the Principle 
that Infinitefirnds may be fafely neglected. 
But only becaufe it is dtftroyed 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 fubdufting from one Side a Quantity 
when it is not to be deftroyed, or when 
an equal Quantity is not fubdu&ed from 

the 



THE AN AL vs T. 4$ 

the other Side of the Equation: So it muft 
be allowed a very logical and juft Method 
of arguing, to conclude that if from E- 
quals either nothing or equal Quantities 
are fubdufted, they {hall ftill remain equal. 
And this is a true Reafon why no Error 
fe at laft produced by the rejecting of o. 
Which therefore muft not be afcribed to 
the Dodrine of Differences, or Infiniteli- 
mals, or evanefcent Quantities, or Mo- 
mentums, or Fluxions. 

_ f . v.ft ;-\J>~ - f -i w =(, ,. ! 

XXVIII. Suppofe the Cafe to be gene 
ral, and that x n is equal to the Area 
ABC y whence by the Method of Fluxi 
ons the Ordinate is found nx n "~ l 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 n are found * to be 
I and nx n r , and that the Ordinate of 
the Area is confidered as its Fluxion ; we 
{hall riot fo clearly fee our way, or per 
ceive how the truth comes out, that Me 
thod as we have fhewed before being ob- 

13. - 

fcure 



THE ANALYST. 

fcure and illogical. But if we fairly de 
lineate the Area and its Increment, and 
divide the latter into two Parts BCFD 
andC-Ffl"*, and proceed regularly by E- 
quations between the algebraical and geo 
metrical Quantities, the reafon of the 
thing will plainly appear. For as x n is 
equal to the Area AEC^ fo is the In 
crement of x n equal to the Increment 
of the Area, i. e. to J3DHC; that is, 
to fay, nox"-* + n ^-~ oo x*~ 2 + &c> 

= BDFC + CFH. And only the firft 
Members, on each Side of the Equation 
being retained, nox* 1 =BDFC: And 
dividing both Sides by o or B D y we 
fliall get n-x* 1 = B C. Admitting, 
therefore, that the curvilinear Space CFH 
is equal to the rejeftaneous Quantity 
n JLrf00#*-2 + (g Cu and that when this 

2 

is rejedled on one Side, that is rejected on 
the other, the Reafoning becomes juft and 
the Conclufion true. And it is all one 
whatever Magnitude you allow to B D y 
whether that of an infinitefimal Difference 
or a finite Increment ever fo great. It is there 
fore plain, that the fuppofing the rejectaneous 

* % the Figure in Sett. 26. 



THE ANALYST. 47 

algebraical Quantity to be an infinitely 
fmall or evanefcent Quantity, and there 
fore to be negle&ed, muft have produced 
an Error, had it not been for the curvi 
linear Spaces being equal thereto, and at 
the fame time fubdufted from the other 
Part or Side of the Equation agreeably to 
the Axiom, If from "Equals you JubduEi 
Equals, the Remainders will be equal. For 
thofe Quantities which by the x\nalyfts are 
faid to be negleded, or made to vanifh,, 
are in reality fubdu&ed. If therefore the 
Conclufion be true, it is abfolutely necef- 
fary that the finite Space C F H be equal 
to the Remainder of the Increment 

exprefied by nn ~"- ox n ~ 2 &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 Expreffion, on the other a 
geometrical Quantity, each of which na 
turally divides it felf into three Members: 
The algebraical or fluxionary Expreffion, 

into on,e whicbt includes neither the Ex- 
?*" ~ 

preffion 



48 THE ANALYST. 

preffion of the Increment of the Abfcift 
nor of any Power thereof, another which 
includes the Expreffion 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 confifts of three Parts or 
Members, the firft of which is the given 
Area, the fecond a Rectangle 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 
preffion is the Expreffion of the given 
Area, fo the fecond Member of the Ex 
preffion will exprefs the Re&angle 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 leifure 
and curiofity for fuch Matters. The u(e 
i make of it is to fhew, that the Analyfis 
cannot obtain in Augments or Differences, 

but 



T H E A N A L Y S T. 4? 

but it muft alfo obtain in finite Quantities, 
be they ever fo great, as was before ob- 
ferved. 

XXX. It feems therefore upon the 
whole that we may fafely pronounce, the 
Conclufion cannot be right, if in order 
thereto any Quantity be made to vaniflh, 
or be neglected, except that either one 
Error is redreffed by another ; or that fe- 
condly, on the fame Side of an Equa 
tion equal Quantities are deftroyed by 
contrary Signs, fo that the Quantity we 
mean to reject is firfl annihilated ; or 
laftly, that from the oppofite Sides equal 
Quantities are fubducted. Aad 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 vanifh. The Velocities or Fluxi 
ons are faid to beprimb and ultimo^ as the 
Augments nafcent and evanefcenr. Take 
therefore the Ratio of the evanefcent 
Quantities, it is the fame with that of 
the Fluxions. It will therefore anfwer all 
Incents as, well. Why then are Fluxions 

E intro- 



50 THE A. N A \L Y I T. 

introduced? Is it nat to fhun or rather- 
tP palliate the Ufe of Quantities infinitely 
finall ? 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 fupppfed to 
move in given Times, we have no Notion 
of greater or leffer Velocities or of Pro 
portions between Velocities, but only of 
longer or fhorter Lines, and of Proporti 
ons between fuch Lines generated in equal 
Parts qf Time, 

XXXI. A Point may be 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 neceffarily 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 

Proper- 



THE A N A L y s T, 51 

Proportions ? Or confider their rationed 
frimtz and idtimce. For to confider the 
Proportion or Ratio of Things implies that 
fuch Things have Magnitude : That fuch 
their Magnitudes may be meafured, and 
their Relations to ea^ch other known. But, 
as there is no rneafure of Velocity except 
Time and Space, the Proportion of Velo^ 
cities being only compounded of the di- 
reft 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, exclusively of Time and 
Space, is to talk unintelligibly ? 

XXXII. But you \yUl fay that, in the 
ufe and application of Fluxions, Men dp 
^ipt Qverfcrain their faculties to a precife 
Conception of the ^bovementioned Velpr 
cities, Incrempnts, Infinitefimals, or any 
other fuch like Ideas of a Nature fo nice, 
fubtile, and evanef^nt. And therefore 
you \y,ill perhaps maintain, that Problems 
may b folved without thofe inconceiva 
ble Suppofcions : and that, confequently, 
the Doftriae of Fluxions, as to the prac- 

E 2 tical 



52, T H E A N A L Y S T. 

tical Parr, 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 difcover- 
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 pradtifed 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 Aftronomy and Geo 
metry, the Principles whereof he doth 
not underftand : And as any ordinary Man 
may 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 



THE ANALYST. 55 

operate and compute and folve Problems 
thereby, not only without an aftual 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 pafs 
for an Artift, Computift, or Analyft, yet 
you may not be juilly efteemed a Man of 
Science and Demonftration. Nor fhould 
any Man, in virtue of being converfant 
in fuch obfcure Analytics, imagine his 
rational Faculties to be more improved 
than thofe of other Men, which have 
been exenpifed in a different manner, and 
on different Subjedls ; much lefs ered him- 
felf into a Judge and an Oracle, concern 
ing Matters that have no fort pf 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 
^kilful in Anatomy : or vice verfa, as a 
E 3 Man 



J4 T H A N A t f S T. 

Man who can difledl with Art, ftiay, ne- 
verthelefs, be ignorant in your Art of com 
puting : Even fo you may both, notwith- 
ftanding your peculiar Skill in your re- 
fpe&ive 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 

fDur own Principles and could demon- 
rate them. 

XXXIV. If it is faid, that Fluxions 
may be expounded or exprefled by finite 
Lines proportional to them : Which finite 
Lines, as they may be diflindtly conceiv 
ed and known and reafoned upon, fo they 
may be fubflituted for the Fluxions, and 
their mutual Relations or Proportions be 
confidered as the Proportions of Fluxions: 
By which means the Do6lrine becomes 
clear and ufeful. I anfwer that if, in or 
der to arrive at thefe finite Lines propor- 
, Honal to the Fluxions, there be certain 
Steps made ufe of which are obfcure and 
inconceivable, be thofe finite Lines them- 
/elves ever fo clearly conceived, it muft 
r-cverthelefs be acknowledged^ that your 

proceed- 



THE ANA L y s T. 

proceeding is not clear nor your method 
fcientific. For inftancejt is fuppofed that 
AB being the Abfcifs, B C the Ordinate, 

*& :rfi5^ ft? J/r-4? indb: 



55 




and FCU a Tangent of the Curve 
/; or CE the Increment of the Abfcifs, 
EC the Increment of the Ordinate, which 
produced meets V H in the Point 
and C c the Increment of the Curve. 
right Line C c being produced to K, there 
are formed three fmall Triangles, the 
Rectilinear CEc, the Mixtilinear CEc, 
and the Redlilinear Triangle GET. 
is evident thefe three Triangles are dif 
ferent from each other, the Redtilinear 
CEc being lefs than the Mixplinear 
CEc y whofe Sides are the three Incre 
ments abovementioned, and this ftill 
than the Triangle CE <T. It is fuppofed 
that thfe Ordinate b c moves into the place 
B C, fo that the Point c is coincident with 
the Point C; and the right Line C K, 

E 4 



T H* A N A L Y S T. 

and confequently the Curve C c, is coin 
cident with the Tangent C H. In which 
cafe the mixtilinear evanefcent Triangle 
CEc will, in its laft form, be fimilar to 
the Triangle GET: And its evanefcent 
Sides C E, EC, and Cc will be porpor- 
tionalto C, E T, and C^ the Sides of 
the Triangle GET. And therefore it 
is concluded, that the Fluxions of the 
Lines AB, BC, and AC, being in the 
laft Ratio of their evanefcent Increments, 
are proportional to the Sides of the Tri 
angle C E T y or, which is all one, of the 
Triangle V E C 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 y and 
C c (i. e. the Proportions of the Fluxi 
ons or Velocities) expreiTed by the finite 
Sides of the Triangle VEC, the Points C 
and c muft be accurately coincident, /. e. 
one and the fame. A Point therefore is 
confidered as a Triangle, or a Triangle is 
fuppofed to be formed in a Point. Which 

*- Introdadl, ad Qaad. Curv. to 



T HE A N A L Y S T. 57 

to conceive feems quite impoffible. Yet 
fome there are, who, though they (brink at 
all other Myfteries, make no difficulty of 
their own, who drain at a Gnat and fwal- 
low a Camel. 

.Kxno [ .VK? iir>.< j o I fjii , > A ,&.0 ^3bic 
XXXV. I know not whether it be 
worth while to obferve, that poflibly 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 and zzz: and that z< x is the In 
crement of the Abfcifs, and zzz xxx 
the Increment of the Area, without confi- 
dering how great, or how fmall thofe In 
crements may be. Divide now zz z x x x 
by z x and the Quotient will be 
zz 4- z x + x x : and, fuppofing that 
z and x are equal, this fame Quotient will 
be 3 x x which in that cafe is theOrdinate, 
which therefore may be thus obtained in 
dependently of Fluxions and Infinitefi- 
inals. But herein is a- direct Fallacy : for 

in 



58 T H A tt A L Y S T. 

in the firft place, it is fuppofed that the 
Abfciffes z and x are unequal, without 
which fuppofition no one flep could haVe 
been made j and in the fecond place, it is 
fuppofed they are equals 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 Doftrine of 
Velocities, Morhentums, &c. will be 
Found impracticable, till fuch time as the 
Objed: and End of Geometry are better un* 
derftood, than hitherto they feeni to have 
been, "the great Author of the Method 
of Fluxions felt this Difficulty, and there 
fore he gave iritd thoib nice Abftradtions 
and Geometrical Metaphyiics, without 
which he faw nothing could be done on 
the received Principles ; 2nd 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 proportio- 

* Sea-. 15. nal 



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 underflood. And what are thefe 
Fluxions? The Velocities of evanefcent 
Increments? And what are thefe fame eva 
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 
ties? 

XXXVI. Men too often impofe 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 
fufpecT: that many who, perhaps never 
having examined the matter, take it for 

granted, 



60 T H E A N *A L Y S T. 

granted, would upon a narrow fcrutiny 
find it impoilible, to frame any Idea ;><i^ 
Notion whatibever of thofe Velocities, ex- 
clufive of fuch finite Quantities and Signs* 

/SHJ Vi- 
c o e 

\ ! 1-4- \~\ I- ir hH 1 J 

K. -jjnu o vjy? 7 7- jj n p 

-^ AI/ / ^ ^ -^ijiuu rf 



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 y LM, M N, NO &c. 
are generated. Suppofe alfo that a, b y r, d, e y 
&c. denote the Velocities of the genera 
ting Point, at the feveral Periods of the 
Parts or Increments fo 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, confequently, 
the feveral Sums of the Velocities, generated 
in equal Parts of Time, may be fet forth 
by the refpeftive Lines KL, LM y MN y 
&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 expreffed by the 
Symbol a, the laft in the fecond by ^, the 
laft generated in the third by c, and fo 

on : 



THE ANALYST. 

on : that a is the Velocity of L M*\n 
ftatu nafcenti^ and ^, c, </, e, &c. are the 
Velocities of the Increments MN, NO, 
O P, Gfc. in their refpeftive 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, 
iecond, third, &c . Velocities adinfinitum : 
which fucceeding Series of Velocities may 
be thus expreffed. a. b a. c- 2^+ a. 
d T>c-\-^b a. &c. which you may call 
by the names of firft, fecond, third, fourth 
Fluxions. And for an apter Exprcffion 
you may denote the variable flowing Line 
KL, KM, KN y &c. by the Letter x ; 
and the firft Fluxions by .v, the fecond 

by x, the third by x, and fo on adinfini- 
tum. 

XXXVII. Nothing is eafier than toaffiga 
Names, Signs, or Expreflions to thefe 
Fluxions, and it is not difficult to compute 
and operate by means of fuch Signs. But 
,ifc Jbaafound much more difficult, to 
ipar, and" yet retain in our 

Minds 



6Z THE ANALYST. 

.1 \l A 

Minds the things, \yhich we fuppofe to 
be fignified 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, {im 
ply and in themfelves, we are deluded by 
vain Abftradtions. 

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 (hall 
be the Velocities of the firft Differences, 

the 



THE. ANALYST. 

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- 
firnals, &c. it is evident that this notion of 
Fluxions would not confift with the great 
Author s view j who held that the minuteft 
Quantity ought not to be neglected, that 
therefore the Doctrine 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, 
onpurpofe to exclude or d without them. 

XXXIX. To others it may poflibly 
feem, that we fhould form a jufler Idea of 
Fluxions, by afluming the finite unequal 
ifochronal Increments KL, L M y MN y &c. 
and conlidering them injlatu nafcenti^ alfo 
tl^eir Increments in ftatu nafcentij and the 
nafcent Increments of thpfe Increments, 
and fo on, fuppofing the ftrft nafcent In 
crements to be propqrtional to the firft 
Fluxions or Velocities, the nafcent Incre 
ments of thofe Increments to be prppori- 

tional 



4 THE ANALYST. 

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 coniidered rather as Incre 
ments than Velocities. But the confider- 
ing them fometimes in one Senfe, fome- 
times in another, one while in themfelves, 
another in their Exponents, feems to have 
occafioned no fmall {hare of that Confu- 
fion and Obfcurity, which is found in the 
Dodtrine 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,<Sfr. 

Fluxions 



THE ANALYST. || 

Fluxions of a given Quantity, it.migbr te 
more confiftent and lefs liable to 
to lay, the Fluxion of the firft 
Increment, t. e. the iecond Fluxion j tr^ 
Fluxion of the fecond nafcent Increiiv 
i. e. the third Fluxion $ the F.JXI 
the third nafcent Increment, ;. t\ 
fourth Fluxion, which Fluxions /are icoifrn 
ceived refpedtively proportional,} each .to. 
the nafcent Principle of the 
fucceeding that whereof it is the 

ry -i/ > / * li*/^ 

XLI. For the more dntiricl Concepi on 
of all which it may be confidered, that if 
the finite Increment LM*be divided into* 
the Ifochronal Parts Lm, mri, no y oM{ 
and the Increment M N into the Parts 
Mp, pq, qr, rN Ifochr onal to the for 
mer 5 as the whole Increments L M } MN 
are proportional to the Sums of their de- 
fcribing Velocities, even fo the homolo 
gous Particles L m, Mp are alfo propor 
tional to the refpedive accelerated Veloci 
ties with which they are defcribed. And* 
as the Velocity with which Mp is gene 
rated, exceeds that with which Lm was 

generated, even fo the Particle Mp ex- 

* c , ,- . o , -v^MiU 

* See tbt foregwig Scheme in Se8. 36 

F ceeds 



T H E A N A L Y 6 f J 

cceds the Particle Lm. And in general, 
as the Ifochronal Velocities defcribiag the 
Particles of M N exceed the Ifochronal 
Velocities defcribing the Particles of Z* Af, 
even fo the Particles of the former exceed 
the correfpondent Particles of the latter. 
And this will hold, be the faid Particles 
ever fo fmall. MN therefore will exceed 
fj 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 upfiiot, to the fame thing with 
the firft *, 
j. ci3Clfn . ii jxi j -K 

XLII. But, notwithftariding what hath 
been faid it mufl ftill be acknowledged, 
that the finite Particles Z/ m or Mp 9 
though taken ever fo fmall, are not pro 
portional to the Velocities a and^; 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, 
s.<8. 36 which 



THE ANAL Y ST. 

which are produced in Moments or infi 
nitely fmall Parts of Time, are alone 
proportional to given Velocities: Thar, 
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 conceive fecond and third Fluxions, we 
muft fuppofe that thenafcent Principles or 
momentaneous Increments have themfelves 
alfo other momentaneous Increments, which 
are proportional to their refpedtive 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 fubdufting the Increment 
generated in the .firft Moment from that 
generated in the fecond, we get the Incre 
ment of an Increment. And by fubduft- 
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 fubdudling the Difference of 

F 2 the 



8 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- 
merit, we: obtain the third Fluxion. And 
after the fame Analogy we may proceed to 
fourth, fifth, fixth Fluxions, fr. And if 
We call the Velocities of the firft, fe 
cond, thirdj. fourth Moments a y b, c, d, 
the Series of Fluxions will be as above, 
a. b a. cib -\-a. d^c + ^b a. 

ad infinltum^ /. e. x x. x. x. ad infi- 

nitum. 

fcr/Jr :/; . ^ 

XLIV. Thus Fluxions may be confider- 
ed in fundry Lights and Shapes, which 
feem alj equally difficult to conceive. And 
indeed, as it is impoffible to conceive Ver- 
locity without time or fpace, without 
either finite length or finite Duration "jr, 
it muft feem above ;he powers of Men 
..to comprehend even the firft Fluxions. 
And if the firft "are incomprehenfible, 
what fhall we fay of the fecond and third 
: Fluxions, &c? He who can conceive the 
beginning of a beginning, or the end of 
an end, fomewhat before the firft or after 

> aiojii^ijP ^ 
f Sea. 31. 

the 



T H E . A N A L V S*ff 

i 

the feft, may be perhaps lharpfighted 
enough to conceive thefe things. Butmoft 
Men will, I believe, find it impoffible to 
tmderftand them in any fenfe whatever* 
oj barxnq vsm sw vgoknA am*) wb isft* 
XLV. One would think that Men could 
nocfpeak too deadly onfo nice a Subject 
And yet, as Was before hinted^ we may 
often obferve that the Exponents of Fluxions 
or Notes reprefenting Fluxions are cori* 
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 increafing, 
and the fecond Fluxions to be the muta 
tions of the firft Fluxions or Celerities* 

we are told that is. z. ar. 2:. z. z. * fe- 
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 its Fluxion ? 

fov.iU bnfe bno:^! wir to v>. **^ iU(ij ^^iiw 

XL VI. Divers Series of Quantities and 

Expreffions, Geometrical and Algebraical, 
?3H.s KO .ij?rt t?n? ^lol^c? jJB.ny^ixlp* -^bn^ n^ 

* De Qaadratura Curvarum. 

F 3 may 



70 THE ANALYST. 

* 

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, 
diftincT: 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, 
i. 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 j and 
fo on ad infinitum. But who can conceive 
how the Fluxion (whether Velocity Of 
nafcent Increment) of an Ordinate fliould 
be it feif 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 ; 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 Qua ri- 
. an^^CTjyQfiriw *m;^> 

< 

XLVII. Upon 



THE ANALYST. 71 

XL VII. Upon the whole it appears that 
the Celerities are difmifled, and inftead 
thereof Areas and Ordinates are introduced, 
But however expedient fuch Analogies or 
fuch 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 5 
or that we are enabled to frame from thence 
juft Ideas of Fluxions confidered in them- 
felves. In all this the general ultimate 
drift of the Author is very clear, but his 
Principles are obfcure. But perhaps thofe 
Theories of the great Author are not mi 
nutely confidered or canvafled by his Dif- 
ciples ; who feem 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 Fluents from 
Fluxions; and in order to this, they muft 
know to find Fluxions from Fluents; and 
in order to .find Fluxions, they muft firft 
know what fluxions are. Otherwife they 
proceed without Clearnefs and without 

F 4 Science, 



T HE A N A L Y 8 T. 

Science. Thus the diredl Method precedes 
the inverfe, - and the knowledge of the 
Principles* is fuppofed in -both. But as for 
operating according to Rules, and by the 
3 of general Forms, whereof the ori- 
! Principles and Reafons are not un- 
oei:lood, this is to be efteemed merely 
t c .nical. Be the Principles therefore ever 
, r :;uie and metaphyficat, they muft 
ftii "led by whoever would comprehend 
[Jodtrine of Fluxions. Nor can any 
,/:jLtrician have a right to apply the 
tides of the great Author, without firft 
.:^:ilidering his metaphyfical Notions 
whencjp they were derived. Thefe how < 
neceftary foever in order to Science, which 
can never be at t lined without a precife, 
clear, and accurate Conception of the 
Principle^ "are neverthelefs by feveral 
carelcfly pafled over ; while the Expref- 
fions alone are dwelt on and confidered 
and treated with great Skill and Manage 
ment, thence to obtain other Expreffions 
; by Methods, fufpicious and indirect fto 
fay the leaft) if confidered in themfelves, 
however recommended by Induction and 

Authorit- 



THE ANALYST. 

Authority $ two Motives which are ac 
knowledged fufficient to beget a rational 
Faith and moral Perfuafion, but nothing 
higher.^* 

^ITO $1(1* : teoh rttw ! ^rnvvi f*4ry* ^ 
. XLVIII. You may poffibly hope to e- 
yade the Force of all that hath been faid, 
and to fcreen falfe Principles and incon- 
fiftent Reafonings, by a general Pretence 
that thefe Objections 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 Underftanding of every un 
prejudiced intelligent Reader. To the 
fame I appeal, whether the Points re 
marked upon are not moft incomprehen- 
fible 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- 



TH I A K A t Y;ST. 

XLIX. Although momentaneous Incre 
ments, nafcent and evanefcent Quantities, 
Fluxions and Infinitefimals of all Degrees, 
are in truth fuch fhadowy Entities, fo 
difficult to imagine or conceive diftindtly, 
that (to fay the leaft) they cannot be ad 
mitted as Principles or Obje&s of clear and 
accurate Science : and although this ob- 
fcurity and incomprehenfibility of your 
Metaphyfics had been alone fufficient, to 
allay your Pretcnfions to Evidence j yet it 
hath, if I miflake not, been further fhewn, 
that your Inferences are no more juft than 
your Conceptions are clear, and that your 
Logics are as exceptionable as your Meta 
phyfics. It fhould feem therefore upon 
the whole, that your Conclusions 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 didate 
out of your proper Sphere, beyond which 



T H E A N A L S T. 

your Judgment is to pafe for rto -more 

than that of Other Men. 

\$%yi*p cltatQ Jbrf& r3aia$.uS : ! 

L. Of a long time I have fufpeaed, that 
thefe modern Analytics were not feientifi- 
cal, and gave fome Hints thereof to the Pub* 
lie 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 Subjedh 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 
fliould have fpared my felf the trouble of 
writing for his Conviction. Nor fhould I 
now have troubled you or my felf with 
this Addrefs, after fo long an Intermiflion 
of thefe Studies; were it not to prevent, 
fo far as I am able, your imposing on your 
felf and others in Matters of much higher 
Moment and Concern. And to the end 
that you may more clearly comprehend 



ANA I*YS T: 

the Force and Pefign of the foregoing 
Remarks, and purfue them ftill further 
in your own Meditations, I (hall fubjoin 
the following Queries. ~> j ibdisrtw.b nA 

. ji j . . 

%^ry i. Whether the Objea 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 ? 

i~* tk- i iff 

<gu. 2. Whether the end of Geometry 
be not to meafure affignable finite Ex- 
tenfion ? And whether this practical View 
did not firft put Men on the ftudy of 
Geometry ? 



-X3S31 t - rfJlO&y .,... , ^, 

g>u. 3. Whether the millaking the Ob- 
jeft and End of Geometry hath not crea 
ted needlefs Difficulties, and wrong Pur- 
fuits in that Science ? 

*~4^^ 

nrvWE * ; hn ^ftW aiu.lokfp .amiT 
Qu. 4. Whether Men may properly be 

faid to proceed in a fcientific Method, 
without clearly conceiving the Objeft they 
are converfant about, the End propofed, 
and the Method by which it is purfued ? 
^it^l jwijfc.. j&t&oihCI m .,vfcimsdj ^g^gns 

M. . Whe- 



THE A N A L Y s l r. 77 

<gu. 5. Whether it doth not fuffice, that 
evfy 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 Extenfion 

is infinitely divifible ? 

$w jQft -m YIJ 

$>u. 6. Whether the Diagrams 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? 

-^v^t ,HI? it &tfi\ s^ Jfa & -i^^.^iii^of 1* -3(1 

Qu. j. Whether it be poffible to free 
Geometry from iofuperable Difficulties and 
Abfurdities, fo long as either the abftract 
general Idea of Extenfion, or abfolute ex 
ternal Extenfion be fuppofed its true Ob- 
Jed? 

<$u. 8. Whether the Notions of abfolute 
Time, abfolute Place, and abfolute Mo 
tion be not moft abflradtedly Metaphyfi- 
cal ? Whether it be poffible for us to mea- 

fure, compute, or know them ? 
Dsloqoiq T) n3 *rfjf ^uodfi *njsV^ ^fis* >TH 

>u. 9. Whether Mathematicians do not 
engage themfelves in Difputes and Para 
doxes, 



-? & TH-E A N AL Y s ii 

doxes, concerning what they neither do 
nor can conceive ? And whether the Doc 
trine of Forces be not a fufficient Proof of 

this? * 

i_ ^ij .,i . . / * 

>u. 10. Whether in Geometry it may 
not fuffice to confider affignable finite Mag 
nitude, without concerning our felves withi 
Infinity? And whether it would not be 
righter to meafure large Polygons having 
finite Sides, inftcad of Curves, than to 
fuppofe Curves are Polygons of infinitefi- 
mal Sides, a Suppofition neither true nor 
conceivable ? 

%y. n. Whether many Points, which 
are not readily aflented to, are not never* 
thelefs true ? And whether thofe in the 
two following Queries may not be of that 
Number ? 

^u. 12. Whether it be poflible, that 
we (hould have had 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 ona 
thing to be diftant from another ? 

* See a La fin Treatife DC Motu, publiflied at lyndm, , 
in the Year 1731. 

4&. 13. Whe- 



THB-ANXLYSIT. 

^hf* 13. Whether Geometrical Quantity 
hath coexiilent Parts ? And whether all 
Quantity be not in a flux as well as Time 
and Motion ? 

*21Iwi. *U*> On** <**- - v . 

<$u. 14. Whether Extenfion can be fup- 
pofed an Attribute of a Being immutable 

and eternal ? 

;riA < vjfciital 



>u. 15. Whether to decline examining 
the Principles, and unravelling the Me 
thods ufed iu Mathematics, would not 
hew a bigotry in Mathematicians ? 

>u. 1 6. Whether certain Maxims do 
not pafs current among Analyfts, which 
are Shocking to good Senfe ? And whether 
the common Affumption that a finite 
Quantity divided by nothing is infinite be 
not of this Number ? 



Qu. 17. Whether the confidering 
metrical Diagrams abfolutely or in them- 
felves, rather than as Reprefentatives of 
all affignable Magnitudes or Figures of 
the fame kind, be not a principal Caufe 
of the fuppofing finite Extenfion infinite- 

iy 



8o THE A N A L Y s T. 

ly divifible ; and of all the Difficulties and 
Abfurdities confequent thereupon ? 

. 

%. 1 8. Whether from Geometrical 
Propofitions being general, and the Lines 
in Diagrams being therefore general Sub- 
ftitutes or Reprefentatives, it doth not fol 
low that we may not limit or confider the 
number of Parts, N into which fuch parti- 
ticular Lines are divifible ? 

<j)u. 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 representing all 
finite Lines, be they ever fo great. In 
which relative Capacity it contains, /. e. 
ftands for more than any affignable 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, fhould contain an infinite num 
ber of Parts ? 

^u. 20. Whether all Arguments for 
the infinite Diviiibility of finite Extenfion 

:>*fe do 



THE ANALYST. 

do not fuppofe and imply, either general 
abftrad: Ideas or abfoiute external Exten- 
iion to be the Object of Geometry ? And, 
therefore, whether, along with thofe Sup- 
pofitions, fuch Arguments alfo do not 
ceafe and vanifli ? 

* -M^r-VSl 

<j)u. 21. Whether the fuppofed infinite 
Divifibility of finite Extenfion hath not 
been a Snare to Mathematicians, and a 
Thorn in their Sides? And whether a 
Quantity infinitely diminifhed and a Quan 
tity infinitely fmall are not the fame 
thing ? 



22. Whether it be neceffary to 
coniider 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 Inconfiilencles can 
be Truths ? Whether Points repugnant and 
abfurd are to be admitted upon any Sub 
ject, or in any Science? And whether the 
ufe of Infinites ought to be allowed, as a 
G Sufficient 



THE ANALYST. 

fufficicnt Pretext and Apology, for the ad 
mitting of fuch Points in Geometry? 

f a ^W f lb JAi^fi. . 

<$u. 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 ? 

-olicML iMinsffawn -W? m. & #(i** 
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 ? 

XV . : -* ,;v ? . . Jit \ .,( 

sdi io sv.au bxs bue mcii j&rrifixb js^H 
..-{.s^. 26. Whether Mathematicians have 
fufficiently confidered the Analogy and Ufe 
of Signs? And how far the fpeci^c limit 
ed Nature of things correfponds thereto? 

<%u. 27. Whether becaufe, in ftating a 
general Cafe of pure Algebra, we are at 

full 



THE A N A L Y s 4. 85 

full liberty to make a Character denote, 
cither a pofitive or *& L 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 ? 
efcjfifjnoqjjua so anoru3Tqx3 yd ?^ j ..jwoM^i 

%. 28. Whether the Shifting of the 
Hypothefis, or (as we may call it) thefa/- 
lacia Suppofitionis be not a Sophifm, that 
far and wide infecfls the modern Rea 
fonings, both in the mechanical Philo- 
fophy and in the abftrufe and fine Geo 
metry ? 



>u. 29. Whether we can form an Idea 
or Notion of Velocity diftindl from and 
exciufive of its Meafures, as we can of 
Heat diftindt from and exciufive of the 
Degrees on the Thermometer, by which 
it is meafured ? And whether this be not 
fuppofed in the Reafonings of modern 
Analyfts? 

$u. 30. Whether Motion can be con 
ceived in a Point of Space ? And if Mo- 

G 2 tion 



84 TH E A N A L Y S T. 

lion cannot, whether Velocity can ? And 
if not, whether a firil or laft Velocity 
can be conceived in a mere Limit, ei 
ther initial or final, of the defcribed 
Space ? 
,;<;K K al&iiT ^4 fmoo*?q:Oi ttm^b 

>u. 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 laid to exprefs their 
Proportion to each- other ? 
BnA ^ rmrif gatfaaufnabnu luoiijiw ^[q 

Qu. 32. Whether if all affignable 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 fact be meafured 
more accurately than a Circular ? 



. 33. Whe- 



THE ANALYST. 

Qu. 33. Whether it would not be 
righter to approximate fairly, than to 
endeavour at Accuracy by Sophifms ? 

Qu. 34. Whether it would not be more 
decent to proceed by Trials and Induc 
tions, than to pretend to demon/Irate by 
falfe Principles ? 

Qy* 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 luch a way ought 
to be called a Knack or a Science? 

$u. 36. Whether there can be Science 
of the Conclufion, where there is not 

CLidencz of the Principles? And whether 

<">. i 

a Man can have^wence of the Princi 
ples, without underftanding them ? And 
therefore whether the Mathematicians 
of the prefent Age aft like Men of 
Science, in taking fo much more pains 
to apply their Principles, than to under- 
ftand them ? 

G 3 % 37 Whe- 

*MiW . .^ 



T H E ANAL Y s Y. 

$u. 37. Whether the greateft Genius 
wreftling with falfe Principles may not bs 
foiled ? And whether accurate Quadratures 
can be obtained without new Poflulata or 
Aflumptions? And if not, whether thofe 
which are intelligible and confident ought 
n6t to be preferred to the contrary? See 
Sed. XXVIII and XXIX. 

>u. 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? 

Sty. 39. Whether, whatever readiuefs 
Analyfts acquire ira&siting a Problem, or 
finding apt Exprefiions for Mathematical 
Quantities, the fame doth neceffarily in 
fer a proportionable ability in conceiving 
and expreifmg other Matters ? 

** 4 

%. 40. Whether it be not a general 
r Rule, that one and the fame Co- 
dividing equal Products gives e- 

qual 




THE ANA 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 x0, be di 
vided by o, the Quotients on both Sides 
are equal? Whether, therefore a Cafe 
may not be general with refpecl 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 falfe Reafonirsg ? 

Qi;: .. &^">J :>.:: f i-jf r 

%. 41. Whether in the moft general 
Reafonings about Equalities and Propor 
tions, Men may not demonflrate as well 
as in Geometry? Whether in fuch De- 
monftrations, they are not obliged to the 
fame ftricl: Reafoning as in Geometry ? 
And whether fuch their Reafonings are not 
deduced from the fame Axioms with thofc 
in Geometry ? Whether therefore Alge 
bra be not as truly a Science as Geo 
metry ? 



Qu. 42. Whether Men may not reafon 

in Species as well as in Words ? Whether 

G 4 the 



88 T H E A N A L Y 5 "?. 

the fame Rules of Logic do not obtain in 
both Cafes ? And whether we have not a 
right to expcft and demand the fame Evi 
dence in both ? 

%. 43. Whether an Algebraift, Fluxio- 
nift, Geometrician or Demonftrator of any 
kind can expect indulgence for obfcure 
Principles or incorredt Reafonings? And 
whether 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- 
the-r any particular Suppofition can come 
under a general Cafe which doth not con-" 

fift with the reafoning thereof ? 
TTOuom J.O- wt>tv aS rj^p^vv .^ 

%/. 44. Whether the Difference be 
tween a mere Computer and a Man of 
Science be not, that the one computes on 
Principles clearly conceived, and by Rules 
evidently demonftrated, whereas the other 
doth not ? 

Qu. 45. Whether, although Geometry 
be a Science, and Algebra allowed to be a 
Science, and the Analytical a moft excel 
lent 



THE A N A L Y s 

lent Method, in the Application neverthe- 
lefs of the Analyfis to Geometry, Men may 
not have admitted felfe Principles and 
wrong Methods of Reafoning ? 

$u. 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 Jimit your felf to rea- 
fon confiftently with the Nature of fuch 
particular things ? And whether fuch Er 
ror ought to be imputed to pure Algebra ? 

>u. 47. Whether the View of modern 
Mathematicians doth not rather feem to be 
the coming at an Expreffion by Artifice, 
than the coming at Science by Demonftra- 
tion ? 



>u. 48. Whether there may not be 
found Metaphyiics as well as unfound? 
Sound as well as unfound Logic? And 
whether the modern Analytics may not be 
brought under one of thefe Denominations, 
and which ? 

>u. 49. Whe- 



T H 1 A N A L Y S T. 

y. 49. Whether there be not really a 
Philofophia prlma^ a certain tranfcenden- 
tal Science fuperior to and more extenfive 
than Mathematics, which it might behove 
our modern Analyfts rather to learn than 

defpife ? 

5?-. rfj T3ii Jor {77 bn-A T&JI j| Jfl sfip $?ifdd yrf 

^. 50. Whether ever fince the recovery 
of Mathematical Learning, there have not 
been perpetual Difputes and Controverfies 
among the Mathematicians ? And whether 
this doth not difparage the Evidence of 

their Methods? 
snt !i/m>js onw /y 

^r. 51. Whether any thing but Meta* 
phyfics and Logic can open the Eyes of 
Mathematicians and extricate them out of 
their Difficulties ? 



. 52. Whether upon the received 
Principles a Quantity can by any Divifion 
or Subdivifion, though carried ever fo far, 
be reduced to nothing ? 
ft Klgrtf! n looiariv^ *i&q[ tsniol ^nom^s 

%u. 53. Whether if the end of Geo 
metry be Practice, and this Practice be 
Meafuring, and we -meafure only affigna- 

ble 



THE A N A L Y s T. 

t>I$iExtenfions y it will not follow that un 
limited Approximations compleatly 
fwer the Intention of Geometry ? 



an- 



<^w. 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 Underftandings of Mathema 
tical Men ? 

<$u. 55. Whether thofe Philomathema- 
tical Phyficians, Anatomifts, and Dealers 
in the Animal Oeconomy, who admit the 
Dodrine of Fluxions with an implicit 
Faith, can with a good grace in fu It other 
Men for believing what they do not com 
prehend ? 



56. Whether the Gorpufcularian, 
Experimental, and Mathematical Philo- 
fophy fo much cultivated in the laft Age, 
hath not too much engrofTed Me-ns At 
tention; fome part whereof it might have 
ufefully employed ? 



u. 57. Whc- 



THE ANALYST. 

%<?. 57. Whether from this, and other 
concurring Caufes, the Minds of fpecula- 
tive Men have not been born downward, 
to the debating 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, Intellectual, or Theological, their 
Pronenefs to meafure all Truths by Senfe 
and Experience of animal Life? 

Qu. 58. Whether it be really an Effect 
of Thinking, that the fame Men admire 
the great Author for his Fluxions, and de 
ride him for his Religion? 

.>u. 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 wa.y 

of reafoning, to recommend Points of 
Faith from their Effects, than to demon- 
ftrate Mathematical Principles by their 
Conclufions ? 

%tf.6i. Whe- 



THE ANA L Y s T. 

%^. 6 1. Whether it be not lefs excep 
tionable to admit Points above Reafon 
than contrary to Reafon? 

-Dfh Tto 5rnvliqu.fi: bm$ snikcfab s/b o* 

t . sif . - v 

<^/. 62. Whether Myfteries may not 
with better right be allowed of in Divine 
Faith, than in Humane Science ? 



>u. 63. Whether fuch Mathematicians 
as cry out againft Myfteries, have ever 
examined their own Principles ? 

>u. 64. Whether Mathematicians, who 
are fo delicate in religious Points, are fidd 
ly fcrupulous in their own Sgience ? Whe 
ther they do not lubmit to Authority, take 
things upon Trull, believe Points incon 
ceivable ? Whether they have not their 
Myfteries, and what is more, their Re 
pugnancies and Contradictions? 

<2>u. 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? 

$tf. 66. Whe- 



THEANALYST. 

Qu. 66. Whether the modern Analytics 
do not furnifh a ftrong argumentum ad hd- 
minem^ againft the Philomathematical In 
fidels of thefe Times ? 

>u. 67. Whether it follows from the 
abovementioned Remarks, that accurate 
and juft Reafoning is the peculiar Cha- 
radter of the prefent Age? And whether 
the modern Growth of Infidelity can be 
afcribed to a Diftinftion fo truly valuable ? 



FINIS. 




-ril 



A SB T- 

-dd , 



garni T a 



rno ewooi 






ERRATA 

Page 16. 1. 20. r. contemnendi * 
page ^o. 1. 17- r - with In^ u ^ ion -