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