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THE

METHOD of FLUXIONS AND

INFINITE SERIES; WITH ITS of CURVE-LINES. Application to the Geometry the

By

Sir

I

NEWTON,^

AA C

S

INVENTOR

Late Prefident of the Royal Society.

^ranjlated from the

AUTHOR'* LATIN ORIGINAL

not yet

made

To which

is

publick.

fubjoin'd,

A PERPETUAL COMMENT

upon the whole Work,

Confiding of

ANN OTATIONS,

ILLU STRATION In order to

make

Acomplcat Inftitution for By Mafter of

JOHN

and

SUPPLEMENTS,

this Treatife

the ufe o/'

CO L SON,

Sir Jofeph fFilliamfon's free

s,

M.

LEARNERS.

A. andF.R.S.

Mathematical-School at Rochejter.

LONDON: Printed by

And

Sold by

HENRY WOODFALLJ

JOHN NOURSE,

at the

Lamb

M.DCC.XXXVI.

without Temple-Bar.


'


T O

William Jones Efq; F.R

S.

SIR, [T was a laudable cuftom among the ancient Geometers, and very worthy to be imitated by their SuccefTors, to addrefs their of labours, not fo much to

Men

Mathematical eminent rank

the world, as to Perfons of diftinguidi'd For they knew merit and proficience in the fame Studies. fuch only could be competent Judges of very well, that

and

{ration

in

Works, and would receive them with ''the efteem. So far at leaft I can copy after thofe they might deferve. their

as to chufe a Patron for thefe Speculations, great Originals, whofe known skill and abilities in fuch matters will enable

and whofe known candor will incline him I have had in the prefent judge favourably, of the fhare

him to

to judge,

performance.

For

Work, of which

as

to

am

I

the fundamental

only the Interpreter,

part I

of the

know

it

it will need no cannot but pleafe you protection, nor ean it receive a greater recommendation, than to bear the name of its illuftrious Author. However, it very naturally ;

I am fure you, who had the honour (for you think it fo) of the Author's friendship and familiarity in his life-time ; who had his own confent to publifli nil

applies

itfelf to

of fome of his pieces, of a nature not very elegant edition

and who have

an efteem for, as well as knowledge of, his other moft fublime, moil admirable, andjuftly celebrated Works. But A 2

different

from

this

;

fo

juft


DEDICATION.

iv

\

But befides thefe motives of a publick nature, I had The many perothers that more nearly concern myfelf. fonal obligations I have received from you, and your generous manner of conferring them, require all the teftimonies of gratitude in my power. Among the reft, give me leave to mention one, (tho' it be a privilege I have enjoy 'd in common with many others, who have the hapof your acquaintance,) which is, the free accefs you pinefs have always allow'd me, to -your copious Collection of whatever is choice and excellent in the Mathernaticks. Your judgment and induftry, .in collecting -thofe. valuable

more conspicuous, than the freedom and readinefs with which you communicate them, to all fuch who you know will apply them to their proper ufe, are not

?tg{^tfcu.,

that

is,

improvement of Science. leave, permit me, good Sir,

to the general

Before

I

take

my

to join

my

wiOies to thofe of the publick, that your own ufeful Lucubrations may fee the light, with all convenie-nt ipeed ;

which, if I rightly conceive of them, will be an excellent methodical Introduction, not only to the mathematical Sciences in general, but alfo to thefe, as well as to the other curious and abftrufe Speculations of our great Author. You are very well apprized, as all other good Judges muft be, that to illuftrate him is to cultivate real Science, and to make his Difcoveries eafy and familiar, will be no fmall

improvement in Mathernaticks and Philofophy. That you will receive this addrefs with your ufual candor, and with that favour and friendship I have fo long ind often experienced,

S

I

is

the earneil requeft of,

R, Your moft obedient humble Servant^ J.

C

OLSON.


(*)

THE

PREFACE. Cannot but very much congratulate with my Mathematical Readers, and think it one of the moft forLife, that I have it in to the prefent publick with a moft valuable power fter in Mathematical and Anecdote, of the greatefl

tunate ciicumftances of

my

my

Ma

the World. And Philofophical Knowledge, of this Anecdote is an element becaufe the much fo more, ry nature, to his other moft arduous and fubh'me and introductory preparatory for the instruction of Novices Speculations, and intended by himfelf and Learners. I therefore gladly embraced the opportunity that was put into my hands, of publishing this pofthumous Work, bethat ever appear 'd in

had been compofed with that view and defign. And that my own Country-men might firft enjoy the benefit of this publication, I refolved upon giving it in an Englijh Translation, I thought it highly with fome additional Remarks of my own. and of the great Author, as reputation injurious to the memory well as invidious to the glory of our own Nation, that fo curious and uleful a piece fhould be any longer fupprels'd, and confined to a few private hands, which ought to be communicated to all the caufe

I

found

it

World for general Inftruction. And more efpecially at a when the Principles of the Method here taught have been

learned

time

fcrupuloufly fifted and examin'd, have been vigorouily .oppofed and

(we may fay) ignominioufly rejected as infufficient, by fome Mathematical Gentlemen, who feem not to have derived their knowledge of them from their only true Source, that is, from cur Author's

And on the other Treatife wrote exprefsly to explain them. of Method have this been hand, the Principles zealouily and commendably defended by other Mathematical Gentlemen, who yet own

a

feem


x

lie

fern

have been as

PREFACE.

acquainted with this Work, (or at leaft it,) the only genuine and original Fountain of this kind of knowledge. For what has been elfewhere deliver'd by our Author, concerning this Method, was only accidental and octo

little

to have over-look'd

and far from that copioufnefs with which he treats of it and illuftrates it with a here, great variety of choice Examples. The learned and ingenious Dr. Pemberton, as he acquaints us in his View of Sir Tfaac Newton's Philofophy, had once a defign of this with the confent" and under the Work, publishing infpectkm of the Author himfelf; which if he had then accomplim'd, he would certainly have deferved and received the thanks of all lovers of Science, calional,

The Work would

have then appear'd with a double advantage, as Emendations of its great Author, and likewife in And among the faffing through the hands of fo able an Editor. other good effects of this publication, it poffibly might have prevented all or a great part of thofe Difputes, which have fince been raifed, and which have been fo ftrenuoufly and warmly pnrfued on both fides, concerning the validity of the Principles of this Method. They would doubtlefs have been placed in fo good a light, as would have cleared them from any imputation of being in any wife defective, or not fufficiently demonstrated. But fince the Author's Death, as the Doctor informs us, prevented the execution of that defign, and fince he has not thought fit to refume it hitherto, it became needful that this publication fhould be undertook by another, tho' a much inferior hand. receiving the

la ft

was now become highly necefTary, that at laft the great himfelf fhould interpofe, fhould produce his genuine MeIjaac thod of Fluxions, and bring it to the teft of all impartial and confiderate Mathematicians ; to mew its evidence and Simplicity, to maintain and defend it in his own way, to convince his Opponents, and to teach his Difciples and Followers upon what grounds they mould proceed in vindication of the Truth and Himfelf. And that this might be done the more eafily and readily, I refolved to accomit with an pany ample Commentary, according to the beft of For

it

Sir

my

and (I believe) according to the mind and intention of the Author, wherever I thought it needful ; and particularly with an Eye fkill,

to the fore-mention'd In which I have endeavoui'd to Controverfy. obviate the difficulties that have been raifed, and to explain every thing in fo full a manner, as to remove all the objections of any force, that have been any where made, at leaft fuch as have occtu'd to

my

obfervation.

If

what

is

here advanced, as there

is

good

rea-

fon


PREFACE.

xi

fon to hope, fhall prove to the fatisfadtion of thofe Gentlemen, who ikfl darted thefe objections, and who (I am willing to fuppofe) had of Truth at heart; I fhall be very glad to have cononly the caufe But if tributed any thing, towards the removing of their Scruples. what is here offer'd and fhould not it fhall happen otherwife, appear

and demonflration to them moil other thinking Readers, with unprejudiced and impartial

to be furricient evidence, conviction, I perfuaded it will be fuch to

yet

am

who

fhall

;

apply themfelves to it minds; and then I mall not think my labour ill beflow'd. It fhould however be well confider'd by thofe Gentlemen, that the great number of Examples they will find here, to which the Method of Fluxions is fuccefsfuUy apply'd, are fo many vouchers for the truth of the on which that Method is founded. For the Deductions Principles, are always conformable to what has been derived from other uncontroverted Principles, and therefore mufl be acknowledg'd us true. This argument mould have its due weight, even with fuch as cannot, as well as with fuch as will not, enter into the proof of the And the hypothefn that has been advanced to Principles themfelves. one error in reafoning being ilill corrected of this evade conclufion, equal and contrary to

and

that fo regularly, conftantly, and frequently, as it mufl be fiippos'd to do here ; this bvpothe/is, I not to be ferioufly refuted, becaufe I can hardly think it fay, ought

by another

is

it,

ferioufly propofed.

chief Principle, upon which the Method of Fluxions is here built, is this very fimple one, taken from the Rational Mechanicks ; which is, That Mathematical Quantity, particularly Extenlion, may be conceived as generated by continued local Motion; and that all Quantities whatever, at leaflby analogy and accommodation, may be con-

The

Confequently there mufl be fuch generations, comparativeVelocitiesofincreafeanddecreafe, during ceived as generated after a like manner.

and determinable, and may therefore /proThis Problem our Author blematically) be propofed to be found. here folves by the hjip of another Principle, not lefs evident ; which is infinitely divifible, or that it may (menfuppofes that Qnimity

whole Relations

are fixt

fo far continually diminifh, as at lafl, before it tally at leaft) be call'd to arrive at Quantities that

may

is

totally

vanilhing than any afTignform a Notion, not and of relative but indeed of abioiute, comparative infinity. 'Tis a to the Method of Indivifibles, as aifo to the very jufl exception infiniteiimal Method, that they have rccourfe at once to foreign a 2 infinitely

extinguifh'd,

whk.li are infinitely little, and Quantities, or Or it funnolcs that we may able Quantity.

lefs


The

PREFACE.

little Quantities, and infinite orders and gradations of thefe, thefe Quantities affume not relatively but absolutely fuch. They without as that Quantities finnd any ceremony, Jewel, actually and

infinitely

&

and make Computations with them accordingly ; tlie refult of which muft needs be as precarious, as the abfblute exiftence of the Quantities they afiume. And fome late Geometricians have carry 'd thefe Speculations, about real and abfolute Infinity, ftill much farther, and have raifed imaginary Syftems of infinitely great and infinitely little Quantities, and their feveral orders and properties j which, to all fober Inquirers into mathematical Truths, muft certainly appear very notional and vifionary. Thefe will be the inconveniencies that will arife, if we do not Abfolute rightly diftinguifh between abfolute and relative Infinity. can be as the either of our fuch, Infinity, hardly object Conceptions or Calculations, but relative Infinity may, under a proper regulation. Our Author obferves this diftinction very ftrictly, and introduces none but infinitely little Quantities that are relatively fo ; which he arrives at by beginning with finite Quantities, and proceeding by a His Computations gradual and neceffary progrefs of diminution. finite and intelligible commence by Quantities ; and then at always laft he inquires what will be the refult in certain circumftances, when fuch or fuch Quantities are diminim'd in infinitum. This is a conftant practice even in common Algebra and Geometry, and is no more than defcending from a general Propofition, to a particular Cafe which is certainly included in it. And from thefe eafy Principles, managed with a vaft deal of fkill and fagacity, he deduces his Method of Fluxions j which if we confider only fo far as he himfelf has carry'd it, together with the application he has made of it, either obvioufly exift,

here or elfewhere, directly or indiredly, exprefly or tacitely, to the moft curious Difcoveries in Art and Nature, and to the fublimeft Theories may defervedly efteem it as the greateft Work of nobleft Effort that ever was made by the Hun an and as the Genius, Mind. Indeed it muft be own'd, that many uftful Improvement?, and new Applications, have been fince made by others, and probaFor it is no mean excellence of bly will be ftill made every day.

We

:

Method, that it is doubtlefs ftill capable of a greater degree of and will always afford an inexhauftible fund of curious perfection matter, to reward the pains of the ingenious and iuduftrious Analyft. As I am defirous to make this as fatisfactory as poffible, efptcially to the very learned and ingenious Author of the Difcourle call'd The Analyjl, whofe eminent Talents I acknowledge myfelf to have a

this

;

J

great


The

PREFACE.

xlii

for ; I fhall here endeavour to obviate fome of his great veneration to the Method of Fluxions, particularly fuch as principal Objections I have not touch'd upon in Comment, which is foon to follow.

my

He

thinks cur Author has not proceeded in a demonftrative and fcientifical matter, in his Princip. lib. 2. km. 2. where he deduces the Moment of a Rectangle, whole Sides are fuppofed to be variable I fhall reprefent the matter Analytically thus, Lines. agreeably (I

mind of the Author. Y be two variable Lines,

think) to the

X

and

or Quantities, which at different periods of time acquire different values, by flowing or increaor alike inequably. For inflance, let fing continually, either equably

Let

becomes A fa, A, b and B -+f3, B, f fuccefiively b, are any quantities that may be the fame periods of time the variable

there be three periods of time, at

and and

A -+- 7 a

;

reflectively

aiTumed

;

Y

which

X

becomes B where A, a, B,

and

at pleafure.

Then

XY

Produ<ft or Rectangle +- f * x B -+- h, that

A

AB -+- f^B -f- 7$ A -f- ^ab.

at

will

become A"

AB Now

T <?B

is,

in

fa x B

fM.

f4,

-f-

ab,

the interval from the

AB, and AB, and

firft

period

of time to the fecond, in which X from being A fa is become A, the is become Product XY and in which Y from being B B, 7^ AB that becomes from being AB -f- ^ab is, f^B by Subis f#B -+traction, its whole Increment during that interval of time to the in the fecond interval from the And ^ab. period in which Y from and X A in becomes which A-f-ftZ, third, being frcm being B becomes B -hf^, the Product XY from being AB becomes AB-f- ffiB -f f 4A -+- -^ab that is, by Subtraction, its whole Increment during that interval is 7,76 7^A -+- ^ab. _ Add thefe two Increirents together, and we fhall have <?B -+- bA. for the compleat Increment of the Product XY, during the whole interval of time, while X fk w'd from the value A \a to A -f- ftf or Y flow'd from the value B f to B +7''. Or U might have been found thus: While X f.ows from A tne \a to A, and by Operation, to therce to A -f- ft?, or Y flows f-om B B, and thence to f3 -ithe will Product flow fiom XY AB B f<?B f3A -f- ab f A, to AB, ?nd thence to AB -+- f^B -J'k -f- ^ab > therefore by Subtraction the whole Increment during that interval of time will be tfB-4-M. Q^E. D. This may eafily be illuftrated by Numbers thus: Make A,rf,B,/, equal to 9, 4, i 5, 6, refpeclively; (or any other Numbers to be affumed at pleafure.) Then the three fucceffive values of X will be the three fucceffive values of Y will be 12, 15, 18, 7, 9, ii, and

iA

-,

fA

;

+

,

+

reipcciivcly.


PREFACE.

The

xiv

Produd XY = 4xic-f- 6x9= =

the three fucceflive values of the

Alfo refpeftively. will be 84, 135, 198. 19

8_8 4

Thus

But rtB-f-M

114

Q.E. O.

.

the

Lemma

will be

true of any conceivable finite Increments whatever; and therefore by way of Corollary, it will be true of infinitely little Increments, which are call'd Moments, and which was the thing the Author principally intended here to demonflrate. nitely

Moments

the cafe of

15ut in

A

A, and

ftf,

to be confider'd, that X, or defito be taken indifferently for and definitely B f/;, B, B -+- ~b.

it is

A -+-

a, are

the fame Quantity ; as alfo Y, the want of this Confutation has occafion'd not a few per-

And

plexities.

Now from hence the reft of our Author's Conclufions, in the fame Lemma, may be thus derived fomething more explicitely. The Moment of the Reclangle AB being found to be Ab -+- ^B, when the contemporary Moments of A and B are reprelented by a and b A, and therefore b a, and then the refpedtively ; make B A .or of x will be Moment Aa -+- aA, or 2aA. Again, A, A*, A a and therefore b-=. zaA, and then the Moment of make B

=

=

=

=

,

AxA*, or A', will be 2rfA 4 -f- aA 1 , or 3^A*. Again, make B s 5 and therefore l> , ^aA -, and then the Moment of xA*, or

A A

=

A

3 3 3 3<?A -4-rfA , or 4#A Again, make 3 therefore ^ and of then the Moment , 4^A be 4<?A 4 -i-tfA 4 , or 5<zA 4 And fo on in infinitum.

4

,

will be

=

.

Ax A

B==A-, 4

.

m to

general, afluming reprefent any integer affirmative 1 Moment of A* will be .

Number,

maA"

Now

and

or A', will , Therefore in

the

=

i, (where m is any integer affirmative of Unity, or any other conftant and the becaufe Moment Number,) A* we (hall have x Mom. A~m -f- A~m x Mom. p; quantity, is 110 A- x Mom. A" But Mom. A" o, or Mom. maA m ~*, as found before ; therefore Mom. A"* A~ iw x maA-"-' ma A"-' Therefore the Moment of Am will be ~ m maA , when m is any integer Number, whether affirmative or

becaufe

A*

=

A"=

=

x

A^

ra

A~"=

=

=

.

.

I

negative.

n

And may

or A"=. B" we put A" where m and be any integer Numbers, affirmative or negative ; then we

mall have

is

the

=B,

univerfally, if

ma A"-*

Moment

=

;.^B"^'

of B, or of

,

A"

or

.

b=

mgA<

So that the

,

=

-aA

Moment

i,

of

which

A"

will

be


P E E F A C

The be

wtfA"*"

rtill

1

whether

,

;;/

fraction.

The Moment being </C +- cD

of

AB

fuppofe

;

be affirmative or negative, integer or

= MAB,

aB, and the Moment of CD and therefore d-=. b& +- aB,

-+-

being

D

xv

E.

Moment of ABC will be bA +- aB xC r AB. And likewife the Moment of ~ m n B maA. And fo of any others. connexion between the Method of Mo-

and then by Subftitution the -f- c

AB

=

MC

-+-

rfBC -h

l />B"-'A" -fthere is fo near a ments and the Method of Fluxions, that it will be very eafy to pafs from the one to the other. For the Fluxions or Velocities of increafe, are always proportional to the contemporary Moments. Thus if for A, B, C, &c. we write x, y, z, &c. for a, b, c, &c. we may Then the Fluxion of xy will be xy -f- xy, the write x, y, z, &c. m whether m be integer or fraction, will be rnxx*-* Fluxion of x affiimative or negative; the Fluxion of xyz will be xyz -f- xyz -f-

A*B"

will be

.

Now

,

x my n

xjz, and the Fluxion of

will be

mxx m -*y

-J-

fo of the reft. Or the former Inquiry

nx myy"~ s

.

And

may be placed in another view, thus A-f- a be two fucceflive values of the variable Quantity X, as alfo B and B -+- b be two fucceflive and contemporary values of Y ; then will AB and AB -f- aB-\~ bA+ab be two fucceflive and And while X, contemporary values of the variable Product XY. from A -f- a, or Y flows value to its A by increafing perpetually, flows from B to B -f- b ; XY at the fame time will flow from AB to AB +- aB -+- bA. -f- ab t during which time its whole Increment, Or in as appears by Subtraction, will become aB -h bh. -+- ab. Numbers thus: Let A, a, B, b, be equal to 7, 4, 12, 6, refpectively ; then will the two fucceflive values of X be 7, 1 1 , and the two fucLet

:

A and

ceflive values

of

the Product

XY

ah-

42

48

And

-f-

thus

it

Y

Alib the two fucceflive values of But the Increment aB -+- t>A -J-

will be 12, 18. will be 84, 198.

24= 14=

1 -+198 84, as before. will be as to all finite Increments : But when the In-

crements become Moments, that

is,

when a and b are fo far dirniA and B at the fame time aB or ^A, (for aB. ab ::

nifh'd, as to become infinitely lefs than ab will become infinitely lefs than either

and bA. ab

A. a

and therefore

;

will vanifh in refpect of them. In which cafe the Moment of the Product or Rectangle This perhaps is the more obvious and will be aB -+- bA, as before. in the t relent of direct way proceeding, Inquiry but, as there was room for choice, our Author thought fit to chufe the former way,,

B.

b,

::

y

)

it

;

as


The

xvi

PREFACE.

elegant, and in which he was under no neceflity of having recourfe to that Principle, that quantities arifing in an Equation, which are infinitely lefs than the others, may be neglected or exas the

more

companion of

in

punged

Now

thofe others.

avoid the ufe of

to

tho' otherwife a true one, was all the Artifice ufed which certainly was a very fair and justifiable one.

this Principle, this occaiion,

on

I fhall conclude my Obfervations with confidering and obviating the Objections that have been made, to the ufual Method of finding the Increment, Moment, or Fluxion of any indefinite power x of

the variable quantity x, by giving that Inveftigation in fuch a manner, as to leave (I think) no room for any juft exceptions to it. And the rather becaufe this is a leading point, and has been ftrangely perverted and mifreprefented. In order to find the Increment of the variable quantity or power x, (or rather its relation to the Increment of x } confider'd as given ; becaufe Increments and Moments can be known only by comparifon

with other Increments and Moments, as alfo Fluxions by comparifon with other Fluxions ;) let us make x"=y, and let X and Y be any fynchronous Augments of x and y. Then by the hypothefis we have the Equation x-fc-X\*

fhall

=y

-+-

Y

;

for in

any Equation

the variable Quantities may always be increafed by their fynchronous Augments, and yet the Equation will flill hold good. Then by

our Author's famous Binomial Theorejn -+-

nx"~'X

^=-^*X

n x

-+-

*

we

fhall

have y

+ n x *~ x '-^-V^X

moving the equal Quantities y and

x",

it

-f-

3 ,

=

Y

=

&c. or

nx n ~

xn re -

X +that when X deT

Y

will be

l

So n x ?-^- x ^^x'-'^X 3 , &c. Y will here denote notes the given Increment of the variable quantity the fynchronous Increment of the indefinite power y or x" ; whofe value therefore, in all cafes, may be had from this Series. Now that we may be fure we proceed regularly, we will verify this thus far, by a particular .and familiar instance or two. 2, Suppofe n then Y 2xX -+- X l That is, while x flows or increafes to x +- X, .v* in the fame time, 2xX -+-X 1 will increafe by its Increment Y ny.

*

^-x"--X

-+-

A,-,

to

.v

1

4-

fuppofe

=

= 2xX = fl

creafes to

= =

.

X

,

which we otherwife know to be true. Again, -ja 1 3 Or while x in*. 3, then Y 3* X -+- 3*X H- X J a x r+- X, x"> by its Increment Y X X3 -h 3^ 3^X

will increafe

1

,

to x* -f-

=

3*

1

X -+- ^xX

1

-+- X

3 .

.

+

And

whereby we may plainly perceive, Conclufion mud be certain and indubitable.

particular cafes,

fo

in all ,other

that

this general

This


Tie

PREFACE.

xvii

X

and Series therefore will be always true, let the Augments docs not at all defo little for the truth or ever fo ; ever be great, of their when circumftance the on magnitude. Nay, they are

This

Y

pend

they become Moments, it muft be true alfo, But when and Y are diby virtue of the general Conclufion. minifh'd in infinitum, fo as to become at laft infinitely little, the muft needs vanifli firft, as being relatively of an of or infinitely little,

when

X

X

greater powers

So that when they are all expunged, we ihall neceflarily obtain the Equation Y=znx*~'X ; where the remaining Terms are likewife infinitely little, and confeif there were other Terms in the Equation, quently would vanifh, which were (relatively) infinitely greater than themfelves. But as .there are not, we may fecurely retain this Equation, as having an undoubted right fo to do; and efpecially as it gives us anufeful piece of information, that X and Y, tho' themfelves infinitely little, or vanifli in proportion to each other as vanifhing quantities, yet they f have therefore learn 'd at laft, that the Moment by j to nx"~ vali e than the fmaller powers. infinitely lefs

.

We

which x increafes, or X, is to the contemporary Moment by which And their Fluxions, or Velox a increafes, or Y, as i is to nx"~ in the fame cities of increafe, being proportion as their fynchronous s

.

have nx*-'x for the Fluxion of X", when the Fluxion of x is denoted by x. I cannot conceive there can be any pretence to infinuate here, that any unfair artifices, any leger-de-main tricks, or any Ihifting of the hypothefis, that have been fo feverely complain'd of, are at all have legitimately derived made ufe of in this Inveftigation. this general Conclufion in finite Quantities, that in all cafes the re-

Moments, we

fhall

We

lation

of the Increments will be

of which one particular cafe nually to decreafe, till they

Y

is,

=

nx"~

when

X

l

X+

and

Y

x ~~x*'-1X*, &c. are fuppofed conti-

But by finally terminate in nothing. thus continually decreafing, they approach nearer and nearer to the Ratio of i to nx"~\ which they attain to at ihe very inftant of the'r This therefore is their ultimate Ratio, vanifhing, and not before. the Ratio of their

and x n continually general tine

Moments, Fluxions, increafe

or decreafe.

or Velocities, by which x to argue from a

Now

Theorem

of the moft

to a particular cafe contain'd under it, is certainly legitimate and logical, as well as one of the mofl ufual

in the whole compafs of the Mathemcto ftand we have made and after that object here, for fome quantity, we are not at liberty to make them nothing, or no is not an Objection againft the quantity, or vanishing quantities,

and

ufeful

ticks.

ways of arguing,

X

To

b

Y

Method


Tte

XVlll

PREFACE.

Method of Fluxions, but againft the common Analyticks. This Method only adopts this way of arguing, as a conftant practice in the vulgar Algebra, and refers us thither for the proof of it. If we have an Equation any how compos'd of the general Numbers a, b, c, &c. it has always been taught, that we may interpret thefe by any particular Numbers at pleafure, or even by o, provided that the Equation, or the Conditions of the Queftion, do not exprefsly require the contrary. definite

any

For general Numbers, as fuch, may ftand for in the whole Numerical Scale which Scale

Numbers

;

be thus commodioufly 2> reprefented, &c. 3, &c. where all i, o, i, 2, 3,4, poffible fractional Numbers, intermediate to thefe here exprefs'd, are to be conceived as interpolated. But in this Scale the Term o is as much a Term or Number as any other, and has its analogous properties in common with the refK (I

think)

We

may

are likewife told, that

we may

not give fuch values to general as they could not receive at firft ; which if adafterwards, Symbols mitted is, I think, nothing to the prefent purpofe. It is always moft eafy and natural, as well as moll regular, inftruclive, and elegant, to

make our

Inquiries as

much

and

to defcend to particular cafes nearly brought to a conclufion.

by degrees, But this is a point of convenience

only, and not a point of neceffity. flead of defcending

from

finite

ments, or vanifhing Quantities,

Terms as may be, when the Problem is

in general

Thus

Increments

in the prefent cafe, into infinitely little

Mo-

we might

begin our Computation with thofe Moments themfelves, and yet we mould arrive at the As a proof of which we may confult our Aufame Conclufions. thor's ownDemonftration of hisMethod, in oag. 24. of this Treatife. x In fhort, to require this is jufl the fame thing as to infift, that a

Problem, which naturally belongs to Algebra, mould be folved by common Arithmetick ; which tho' poflible to be done, by purluing backwards all the fleps of the general procefs, yet would be very troubkfome and operofe, and not fo inflrudtive, or according to the true Rules of Art But I am apt to fufpedr, that all our doubts and fcruples about Mathematical Inferences and Argumentations, especially when we are fatisfied that they have been juftly and legitimately conducted, may be ultimately refolved into a fpecies of infidelity and diftruft. Not in refpecl of any implicite faith we ought to repofe on meer human authority, tho' ever fo great, (for that, in Mathematicks, we mould are hardly utterly difclaim,) but in refpedl of the Science itfelf. to that fo Science is the believe, brought perfectly regular and uni-

We

form,


72*

PREFACE.

xix

form, fo infinitely confident, conftant, and accurate, as we mall re&lly find it to be, when after long experience and reflexion we (hall have overcome this prejudice, and {hall learn to purfue it rightly. do not readily admit, or eafily comprehend, that Quantities have an infinite number of curious and fubtile properties, fome near and obvious, others remote and abftrufe, which are all link'd together by a neceffary connexion, or by a perpetual chain, and are then only difcoverable when regularly and clofely purfued ; and require our truft and confidence in the Science, as well as our induftry, application, and obftinate perfeverance, our fagacity and penetration, in That Nature is ever order to their being brought into full light.

We

.

with

confiftent

herfelf,

and never proceeds

faltum, or at random, but

is

in thefe Speculations

infinitely fcrupulous

and

per

felicitous,

as

adhering to Rule and Analogy. That whenever we regular Portions, and purfue them through ever fo great a variety of Operations, according to the ftricT: Rules of Art ; we fhall always proceed through a feries of regular and well- connected tranlmutations, (if we would but attend to 'em,) till at laft we arrive That no properties of Quantity at regular and juft Conclufions. are intirely deftructible, or are totally loft and abolim'd, even tho' profecuted to infinity itfelf j for if we fuppofe fome Quantities to become infinitely great, or infinitely little, or nothing, or lefs than nothing, yet other Quantities that have a certain relation to them

we may

fay,

in

make any

and often finite alterations, will fymand with conform to 'em in all their changes ; and them, pathize their will always preferve analogical nature, form, or magnitude, which will be faithfully exhibited and difcover'd by the refult. This we may colledl from a great variety of Mathematical Speculations, and more particularly when we adapt Geometry to Analyticks, and will only undergo proportional,

Curve-lines to Algebraical Equations. ral

That when we purfue gene-

Nature is infinitely prolifick in particulars that will from them, whether in a direct rubordination, or whether they

Inquiries,

refult

collaterally ; or even in particular Problems, we may often that thefe are only certain cafes of fomething more perceive general, and may afford good hints and afiiftances to a fagacious Analyft, for

branch out

afcending gradually to higher and higher Difquilitions, which may be profecuted more univerfally than was at firft expe<5ted or intended. Thefe are fome of thofe Mathematical Principles, of a higher order,

which we

and which we {hall never be or know the whole ufe of, but from much pracof, attentive confideration ; but more efpecially by a diligent b 2 peruial, find a difficulty to admit,

fully convinced tice

and


xx

The

P

R E

F

A C

E.

and clofe examination, of this and the other Works of our He abounded in thefe fublime views and inAuthor. had acquired an accurate and habitual knowledge of all thefe, quiries, and of many more general Laws, or Mathematical Principles of a not improperly be call'd The Philofophy of fuperior kind, which may aflifted and which, Quantity ; by his great Genius and Sagacity, together with his great natural application, enabled him to become fo compleat a Matter in the higher Geometry, and particularly in the Art of Invention. This Art, which he poflefl in the greateft perfection imaginable, is indeed the fublimeft, as well as the moft diffiperuial,

illuftrious

cult of all Arts, if it properly may be call'd fuch ; as not being reducible to any certain Rules, nor can be deliver'd by any Precepts, but

wholly owing to a happy fagacity, or rather to a kind of divine Enthufiafm. To improve Inventions already made, to carry them

is

on, when begun, to farther perfection, is certainly a very ufeful and excellent Talent ; but however is far inferior to the Art of Difcovery, as haying a TIV e^u, or certain data to proceed upon, and where juft method, clofe reasoning, ftrict attention, and the Rules of Analogy, may do very much. But to ftrike out new lights, to adventure where

no

footfteps

the nobleft

had ever been

Endowment

fet before,

that a

nullius ante trita folo

human Mind

is

capable

of,

is

this

;

is

referved

and was the peculiar and diftinguifhing Character of our great Mathematical Philofopher. He had acquired a compleat knowledge of the Philofophy of Quanor of its moft eflential and moft general Laws ; had confider'd it tity, in all views, had purfued it through all its difguifes, and had traced it through all its Labyrinths and Recefles j in a word, it may be faid of him not improperly, that he tortured and tormented Quantities make them confefs their Secrets, and difcover all poflible ways, to for the chofen

few

quos Jupiter tequus amavit,

their Properties.

The Method

of Fluxions, as it is here deliver'd in this Treatife, is a very pregnant and remarkable inftance of all thefe particulars. To take a cuifory view of which, we may conveniently enough divide The firft will be the Introduction, into thefe three parts. it or the Method of infinite Series. The fecond is the Method of The third is the application of both Fluxions, properly fo culi'd. thefe

Methods

to

fome very general and curious Speculations,

Geometry of Curve-lines. As to the firft, which is the Method of infinite the Author opens a new kind of Arithrnetick, (new

chiefly

in the

time of his writing

this,)

Series, in this at leaft at the

or rather he vaftly improves the old.

For he


The

PREFACE.

xxi

he extends the received Notation, making it compleatly universal, and fhews, that as our common Arithmetick of Integers received a great

Improvement by the introduction of decimal Fractions

;

fo the

common Algebra or Analyticks, as an univerfal Arithmetick, will receive a like Improvement by the admiffion of his Doctrine of infinite Series,

by which the fame analogy

farther advanced towards perfection.

will be

ftill

carry'd on,

Then he fhews how

all

and

com-

be reduced to fuch Series, as will plicate Algebraical Expreffions may continually converge to the true values of thofe complex quantities, or their Roots, and may therefore be ufed in their ftead : whether thofe quantities are Fractions having multinomial Denominators, which are therefore to be refolved into fimple Terms by a perpetual Divi-

whether they are Roots of pure Powers, or of affected Equawhich are therefore to be refolved by a perpetual Extraction. And by the way, he teaches us a very general and commodious Method for extracting the Roots of affected Equations in Numbers. fion

;

or

tions,

And this is chiefly the fubftance of The Method of Fluxions comes

his

Method of

infinite Series.

next to be deliver'd, which indeed is principally intended, and to which the other is only preparatory and fubfervient. Here the Author difplays his whole fkill, and fhews the great extent of his Genius. The chief difficulties of this he reduces to the Solution of two Problems, belonging to the abftract or Rational Mechanicks. For the direct Method of Fluxions, as it is now call'd, amounts to this Mechanical Problem, tte length of the ibed being continually given, to find the Velocity of the Modefer Aifo the inverfe Method of Fluxions has, tion at any time propofcd. for a foundation, the Reverfe of this Problem, which is, The Velocity of the Motion being continually given, to find the Space defer ibed at any

Space

So that upon the compleat Analytical or Geometritime propofcd. cal Solution of thefe two Problems, in all their varieties, he builds

whole Method. His firft Problem, which

his

being given,

to

f

The relation 6J the owing Quantities is, the determine relation of their Fhixiom, he difpatches

A

He does not propofe this, as is ufualiy done, flowvery generally. ing Quantity being given, to find its Fluxion ; for this gives us too lax and vague an Idea of the thing, and does not fhew fufficiently

that Comparifon, which is here always to be understood. Fluents and Fluxions are things of a relative n.iture, and two at leafr,

fuppofe

whofe

mould always be exprefs'd bv Equations. He all fhould be reduced to Equations, by which

relation or relations

requires therefore that the relation of the flowing Quantities will be

exhibited, and their

comparative


f/jg

xxii

PREFACE.

comparative magnitudes will be more eafily eftimated ; as alfo the And befides, by this comparative magnitudes of their Fluxions. means he has an opportunity of refolving the Problem much more For in the ufual way of generally than is commonly done. taking we are confined to. the Indices of the Powers, which are Fluxions,to be made Coefficients ; whereas the Problem in its full extent will allow us to take any Arithmetical Progreflions whatever. By this means we may have an infinite variety of Solutions, which tho' different in form, will yet all agree in the main ; and we may always chufe the fimpleft, or that which will beft ferve the prefent purpofe. the given Equation may comprehend feveral variable Quantities, and by that' means the Fluxional Equation maybe found, notwithstanding any furd quantities that may occur, or even any other quantities that are irreducible, or Geometrically irrational.

He

(hews

And

how

alfo

derived and demonitrated from the properties of Modoes not here proceed to fecond, or higher Orders of Fluxions, for a reafon which will be affign'd in another place. His next Problem is, An Equation being propofed exhibiting the relation of the Fluxions of Quantities, to the relation of find thofe Quantities, or Fluents, to one another ; which is the diredt Converfe of the This indeed is an operofe and difficult foregoing Problem. all this is

He

ments.

Problem,

taking dreis

;

it

in

its full

which

extent, and, requires all our Author's fkill and adyet hefolyes very generally, chiefly by the affiftance of his

Series. He firfl teaches how we may return from the Fluxional Equation given, to its correfponding finite Fluential or when be that can done. But when it cannot be Algebraical Equation, or when there is no finiie fuch .done, Algebraical Equation, as is moft commonly the cafe, yet however he finds the Root of that

Method of infinite

Equation

by an

infinite

converging

Series,

which anfwers the fame purpofe. the Root, or Fluent required, by

often he mews how to find an infinite number of fuch Series. His proceffes for extracting thefe Roots are peculiar to himfelf, and always contrived with much fubtilty and ingenuity. The reft of his Problems are an application or an exemplification of the foregoing. As when he determines the Maxima and Minima of quantities in all cafes. When he mews the Method of drawing to whether Geometrical or Mechanical ; or howCurves, Tangents ever the nature of the Curve may be defined, or refer'd to right Lines or other Curves. Then he {hews how to find the Center or Radius of Curvature, of any Curve whatever, and that in a fimple but general manner ; which he illuftrates by many curious Examples,

And

and


fbe

PREFACE.

xxiii

and purfues many other ingenious Problems, that offer themfelves by After which he difcufTes another very fubtile and intirely the way. new Problem about Curves, which is, to determine the quality of the Curvity of any Curve, or how its Curvature varies in its progrefs or inequability. through the different parts, in refpect of equability He then applies himfelf to confider the Areas of Curves, and fhews

how we may find as many Quadrable Curves as we pleafe, or fuch whole Areas may be compared with thofe of right-lined Figures. Then he teaches us to find as many Curves as we pleafe, whofe Areas may be compared with that of the Circle, or of the Hyperus

bola, or of any other Curve that (hall be affign'd to Mechanical as well as Geometrical Curves.

;

which he extends

He

then determines

the Area in general of any Curve that may be propofed, chiefly by the help of infinite Series ; and gives many ufeful Rules for afcerAnd by the way he fquares the Areas. taining the Limits of fuch the and and Circle Quadrature of this to the conapplies Hyperbola, of Logarithms. But chiefly he collects veryftructing of a Canon of Quadratures, for readily finding the general and ufeful Tables Areas of Curves, or for comparing them with the Areas of the Conic Sections; which Tables are the fame as. thofe he has publifh'd himThe ufe and application of thefe felf, in his Treatife of Quadratures. he (hews in an ample manner, and derives from them many curious Geometrical Conftructions, with their Demonftrations. Laftly, he applies himfelf to the Rectification of Curves, and mews us how we may find as many Curves as we pleafe,. whofe Curvelines are capable of Rectification ; or whofe Curve-lines, as to length, may be compared with the Curve-lines of any Curves that fha.ll be And concludes in general, with rectifying any Curve-lines affign'd. that may be propofed, either by the aflifbncc of his Tables of QuadraAnd tures, when that can be done, or however. by infinite Series. this

is

chiefly the fubflance of the prefent

that perhaps"

may

be expected, of what

I

Work.

As

to ,the account

have added in

my Anno-

the inquifitive Reader to the PrefacCj tations ; will go before that part of the Work. I

{hall

refer

which

THE


-

;

THE

CONTENTS. CT^HE

Introduction, or the

Method of

refolding complex Quantities

into infinite Series of Jimple Terms.

pag.

i

Prob.

i.

From

the given Fluents to find the Fluxions.

p.

21

Prob.

2.

From

the given Fluxions to find the Fluents.

p.

25

p.

44

p.

46

Maxima and Minima

Prob. 3.

To determine the

Prob. 4.

To draw Tangents

Prob.

5.

To find the Quantity of Curvature in any Curve.

P-

59

Prob.

6.

To find the Quality cf Curvature in any Curve.

p.

75

Prob. 7.

To find any number of Quadrable Curves.

p.

80

Prob.

To find Curves whofe Areas may be compared

8.

of Quantities,

to Curves.

Conic SecJions.

Prob. 9.

Prob.

1 1.

thofe

of the p. 8

To find the Quadrature of any Curve

Prob. 10. To find any number of

to

ajjigrid.

rettifiable Curves.

p. p.

1

86

124

To find Curves whofe Lines may be compared with any Curvelines ajfigrid.

Prob. 12. To rectify any Curve-lines ajpgn'd.

p.

129

p.

134


THE

METHOD

of

FLUXIONS,

AND

INFINITE SERIES. INTRODUCTION

:

Or, the Refolution of Equations

by Infinite Series.

IAVING

obferved that moft of our modern Geome-neglecting the Synthetical Method of the Ancients; have apply'd themfelves chiefly to the the affiftance cultivating of the Analytical Art ; by

tricians,

of which they have been able to overcome fo many and fo great difficulties, that they feem to have exhaufted all the of Curves, and Speculations of Geometry, excepting the Quadrature Ibme other matters of a like nature, not yet intirely difcufs'd I thought it not amifs, for the fake of young Students in this Science, to compofe the following Treatife, in which I have endeavour'd to enlarge the Boundaries of Analyticks, and to improve the Doctrine of Curve-lines. 2. Since there is a great conformity between the Operations in and the fame Operations in common Numbers; nor do they Species, :

feem to

differ,

except in the Characters by which they

B

are re-

prefented,.


'The

Method of FLUXIONS,

firft being general and indefinite, prefented, the I cannot but wonder that nite and particular

and the other defino body has thought of accommodating the lately-difcover'd Doctrine of Decimal Fractions in like manner to Species, (unlels you will except the Quadrature of the Hyberbola by Mr. Nicolas Mercator ;) efpecially fince it might have open'd a way to more abftrufe Discoveries. But iince this Doctrine of Species, has the fame relation to Algebra, as the Doctrine of Decimal Numbers has to common Arithmetick ; the Operations of Addition, Subtraction, Multiplication, Divifion, and Extraction of Roots, may eafily be learned from thence,, and the if the Learner be but fk.ill'd in Decimal Arithmetick, and obferves the that obtains beVulgar Algebra, correfpondence and Decimal Fractions Terms continued. tween Algebraick infinitely For as in Numbers, the Places towards the right-hand continually decreafe in a Decimal or Subdecuple Proportion ; fo it is in Species :

when

the Terms are difpofed, (as is often enjoin 'd in in an uniform Progreflion infinitely continued, acwhat follows,) cording to the Order of the Dimenfions of any Numerator or De-

refpedtively,

And as the convenience of Decimals is this, that all and Radicals, being reduced to them, in fome meaFractions vulgar fure acquire the nature of Integers, and may be managed as fuch ; nominator.

it is a convenience attending infinite Series in Species, that all kinds of complicate Terms, ( fuch as Fractions whofe Denominators are compound Quantities, the Roots of compound Quantities, or of affected Equations, and the like,) may be reduced to the Clafs

fo

of fimple Quantities ; that is, to an infinite Series of Fractions, whofe Numerators and Denominators are fimple Terms ; which will no under thofe difficulties, that in the other form feem'd longer labour

almoft infuperable. Firft therefore I mail fhew how thefe Reductions are to be perform'd, or how any compound Quantities may be reduced to fuch fimple Terms, efpecially when the Methods of computing are not obvious. Then I fhall apply this Analyfis to the Solution of Problems. Divifion and Extraction of Roots will be 3. Reduction by plain from the following Examples, when you compare like Methods of Operation in Decimal and in Specious Arithmetick.

Examples


and INFINITE SERIES,

3 .

..ift

Examples of Reduttion by Dhifwn.

The

.4.

Fraction

being propofed, divide

^

following manner

aa by b

+

IjfM/l^^ x

in the

:

aax 1

aax

faa

aax*

a a x*

.

" .

aax aax

O

--7

-f-O

aax*

o

-+-

-

o

~

+o **

*

flt

Jf*

;. v *-\ The Quotient "

-rr^i_

which

i r therefore

being

Series,

Or making x

j^.

infinitely

the

tf*^ a* x* T _-JT-+ -T_

^^

*

is

- - ?4 **

+ toaa + o e

(the Quotient will be * r~ _ _ , % found as by the foregoing Procefs. ,

#

I

6.

+

manner the Fraction

In like

5.

-{-

x4

And

'

A:*

H- x

-j-

34x

T ,

~-

x-*

will

r

i+x*

s

13**

Here

or to , &c. 9 v " 2 *

the Fraction

i1

yx

8

be

equivalent

of the Divifor, in

x

4. n

this

~

will

be

reduced

to

manner, ~

1^ *

#-* _f. ^-

be

a* X+

.

rr + T7-, &c.

.

will

continued,

Term

firft

a* x*

1

V &c AV reduced

to

^-8

to

2x^

2x

3*

&c.

will be

proper to obferve, that I make ufe of x-', - &c. of for &c. i, ;r 7,' x-', x-', x-*, xs, xi, x^, xl, A4, &c. for v/x, v/*S \/ x *> vx , ^x l , &c. and of x'^, x-f. x i &c for **** 1Ui ' * i 7.

it

-

,

^x

j_^

^ ? >' y-^.'

&c.

And

this

by the Rule of Analogy,

as

apprehended from fuch Geometrical Progreflions as thefe x> (or i,) a"*,*-',*'*, x, *, &c.

B

2

may

be

x,

x*,

;

8.

'* /

Av


ffie

Method of FLUXIONS,

er for -In the fame manner

q.

And

1^ + 1^!,

8.

.and

aa

xv|*

inftead

may

be wrote

&c.

',

thus inftead

&c.

xx may be wrote aa of the Square of aa xx; and

>

of^/aa

xxl^

3

inftead of v/

So that we may not improperly diftinguim Powers into Affirmative and Negative, Integral and Fractional. 10.

11.

tract

Examples of Reduction by Extraction of Roots. The Quantity aa -+- xx being propofed, you may thus exits

Square-Root.

- _i_ XX V v (a aa-+^"

-4-

4r

Sfl3

2a

5

i6*

x -

1287

4- J

-

'

c*

2560*

aa

xx 4.

a*

x*

~*

a 4

64 64 a

X* sT*

64^8

~ i;

z$6a'^

_

x

64^ 5* 64 a

"

6

+ 2^1, &c 7^ _ n-i7R/3 7' + I

_-1

256 *

z8rt 8

.

/7lt>

1

,__i!_lll, &c.'

found to be

^

4a~\--^^T,&C. Where it may be obferved, that towards the end of the Operation I neglect all thofe Terms, whofe Dimenfions would exceed the Dimenfions of the laft Term, to which I intend only to continue the Root,

Jo that the Root

fuppofe to

is

*'

,2.


and INFINITE SERIES.

Order of the Terms may be inverted in this manbe in which cafe the Root will be found to

iz. Alfo the

xx

ner

5

+- aa,

aa 10 A*

Thus

13.

the

15.

. i

1

of

xx

.

<z

*

A-

.

AT

*

'-

*

A.'

A'

g

i a

*

A-

4

+

,'_

n

3 x- 6

+ |^^4

T^

-f-

T^

^frx H- rV^ x -+-

6 ,

But thefe Operations, by due preparation,

17.

Sec.

,

.

&c-

.c

j

and more -

.

becomes

it

-|- -i/^r

-4-

**, 8cc.

4-.v*

.Ii*--.4-. ;,,.

over by adually dividing, i

4-

i

^ -Jj --^7 Tr --b*X* ^ '

i**

.

f

+

is

xx is #'" ## is a -f-

x

And v/r^rr,

6.

A-

Root of aa

The Root Of -+-

14.

iz

&c.

may

very often

the foregoing

Example to find \/;_***' if the Form of the Numerator and Denominator had not been the bxx, which would fame, I might have multiply'd each by </ 1 be abbreviated;

as

in

1

-

y^i -f-rt*

have produced

ab x

b

I

*

and the

&

of the work might

reft

xx

have been performed by extracting the Root of the Numerator only, and then dividing by the Denominator. 1 8. From hence I imagine it will fufficiently appear, by what means any other Roots may be extracted, and how any compound Quantities, however entangled with Radicals or Denominators, (fuch

Vx as

x">

\fi

V x i! 2x t

xx

-}-

;

^/axx

-\-

A-

*

3

infinite Series confifting

Of 19.

As

xi

" x-{-xx

v

_. 2X

j

x.1

may

be reduced to

'

of iimple Terms.

the ReduStion of offered Equations.

to aftedled Equations, we mufl be fomething more parhow their Roots are to be reduced to fuch Se-

ticular in explaining ries as thefe ; becaufe

their Doctrine

in

Numbers, as hitherto devery perplexed, and incumber'd with fuperfluous Operations, fo as not to afford proper Specimens for performing the Work in Species. I fhall therefore firfl (hew how the Refoluliver'd

by Mathematicians,

is


Method of FLUXIONS, Refolutidn of affected Equations may be compendioufly perform'd Numbers, and then I fhall apply the fame to Species. 20. Let this Equation _y l be propofed to be rezy 5 and let 2 be a Number how folved, (any found) which differs from the true Root lefs than by a tenth part of itfelf. Then I make and fubftitute 2 4-/> for y in the given Equation, by 2 -\-p which is produced a new Equation p> 4- 6p l 4- iop i whofe Root is to be fought for, that it may be added to the Quote. in

=

=y,

=o,

Thus

Equation the

4-

1

=

o,

Therefore

I

rejecting io/>

/>> i

6//

becaufe

of

fmallnefs, the remaining will approach very near to

its

or/>=o,i,

Quote, and fuppofe 4- ^ =/>, and fubftitute this fictitious Value of p as before, which produces q* 4- 6,3^ 4- 1 1,23? 4- 0,06 1 =o. And fince 1 1,23^ 4- 0,06 1 is near the truth, or 0,0054 nearly, 1 * (that is, dividing 0,06 by 11,23, many Figures arife as there are places between the firft Figures of this, and of the principal QmDte exclufively, as here there are two places between 2 and o,

truth.

write

this

in

the

i

=o

I

write

0,0054 negative; and fuppofing 0,005) before.

manner

y~'

And

^

in the

^=

lower part of the Quote, as being

0,0054 4- r=sg,

I

fubftitute this

thus I continue the Operation as far as of the following Diagram :

zy

5

=o

I

as

in the pleafe,


and INFINITE SERIES. 21. But the

Work may

this Method, efpecially in far firft determin'd

be

much

7

abbreviated towards the end by

Equations of

many

Dimenfions.

Having

how

you intend to extract the Root, count fo after the firft Figure of the Coefficient of the laft Term many places but one, of the Equations that refult on the right fide of the Diagram, as there remain places to be fill'd up in the Quote, and reject But in the laft Term the Decimals may the Decimals that follow. be neglected, after that are reject

fill'd

all

up

many more places as are the decimal places Quote. And in the antepenultimate Term And fo on, by proafter fo many fewer places. fo

in the

that are

ceeding Arithmetically, according to that Interval of places: Or, is the fame thing, you may cut off every where fo many in as the penultimate Term, fo that their loweft places may Figures be in Arithmetical Progreffion, according to the Series of the Terms, or are to be fuppos'd to be fupply'd with Cyphers, when it happens otherwife. Thus in the prefent Example, if I defired to continue the Quote no farther than to the eighth place of Decimals, when I fubftituted 0,0054 -f- r for q, where four decimal places are compleated in the Quote, and as many remain to be compleated, I might have omitted the Figures in the five inferior places, which therefore I have mark'd or cancell'd by little Lines drawn through them ; and indeed I might alfo have omitted the firft Term r J , Thofe Figures therefore although its Coefficient be 0,99999, for the being expunged, following Operation there arifes the Sum 1 which by Divifion, continued as far as 0,0005416 -f- 1,1 62?%

which

the Term prefcribed, gives 0,00004852 for r, which compleats Then fubtracting the negative the Quote to the Period required. from of the the affirmative Quote part, there arifes 2,09455148 part for the Root of the propofed Equation. 22. It may likewife be obferved, that at the beginning of the

had doubted whether o, i -f-/> was a fufficient Apto the Root, inftead of i o, I might have iof> proximation i o, and fo have wrote the firft fuppos'd that o/** -f- i op of its Root in the as nearer to And Quote, Figure

Work,

if I

=

=

being nothing. be convenient to find the fecond, or even the third Figure of the Quote, when in the fecondarjr Equation, about which you are converfant, the Square of the Coefficient of the penultimate Term is not ten times greater than the Product of the laft Term multiply'd into the Coefficient of the antepenultimate Term. And indeed you will often fave fome in this

manner

in Equations of

it

may

many

Dimensions,

if

you feek

pains, efpecially for all the Figures

to-


Tie Method of FLUXION'S,

8

to be added to the

this

manner

lefier

lafl

Terms of

Quote in Root out of the three

that

;

its

is,

if

you

extract the

fecondary Equation

:

For thus you will obtain, at every time, as many Figures again in the Quote. 23. And now from the Refolution of numeral Equations, I mall proceed to explain the like Operations in Species; concerning which, neceflary to obferve what follows. 24. Firft, that fome one of the fpecious or literal Coefficients, if there are more than one, fliould be diftinguifh'd from the reft, which it is

much the leaft or greateft of The reafon of which is, that

is, or may be fuppos'd to be, or neareft to a given Quantity.

either all,

becaufe of its Dimeniions continually increafing in the Numerators, or the Denominators of the Terms of the Quote, thofe Terms may grow lefs and lefs, and therefore the Qtipte may conftantly approach to the Root required ; as may appear from what is faid before of the Species x, in the Examples of Reduction by Divifion and Ex-

And

traction of Roots.

make

for this Species, in what follows, I z ; as alfo I fliall ufe y, p, q, r, s,

mall

&c. generally extracted. Radical to be for the Species or furd Quantities, 25. Secondly, when any complex Fractions, or to in to arife afterwards occur the happen propofed Equation, in

are

ufe of

or

A:

the Procefs, they ought to be removed by known to Analyfts. As if fufficiently

y* -+- j

1

duct by*

=

x"=

1 >'

o,.

Kyi'-l-fry*

fuppofe y x b have i; J -+- &*v* whence extracting the Root v r in order to obtain y.

have

x, and from the Promultiply by b Or bx^ -+ x*-= o extract the Root y.

x=v,

we might we mould

fuch Methods as

we mould

and then writing

^~x

for

=y = =

^hx' -+. x we might divide the Quote by b Affo if the Equation j 3 xy* -f- x$

fax*

6

-\- 3/5***

xj

=

were propofed, we might put v, and will arife v 6 for and z* vv for there x, ting y, which Equation being refolved, y and x may be

y?=

t

o,. x,,

o

and fo wrio; -f- z* reftored. For the Root will befound^=2-f-s3_|_5~s 5cc.andrei1:onngjyandA;, we have x^ -f- x -+- 6x^ &c. dien fquaring, y =x^-+- 2X J ~f- 13*", &c.. y* 26. After the fame manner if there mould be found negative Dimenfions ofx and jy, they may be removed by multiplying by the fame '2.x~ x andjy. As if we had the Equation x*-}-T x*-y~ i6y- =o, 1 3 5 and and x arife there would x*y* -+- 3# jy multiply by j , 2_v

=

z,

z=v

5

I

I

3

>

A

O.

J

-r

And U

1

tjie

-r-v

Equation were

x

=

3

aa

~

2ai

a + ?r y i

4

1

\.

by;


and INFINITE SERIES. by multiplying

And

fo

into

}

there

jy

would

arife

i

xy*-=.a' y*

of others.

when

the Equation is thus prepared, the work be^ of the Quote ; concerning which, as gins by finding the firfr. Term alfo for finding the following Terms, we have this general Rule, when the indefinite Species (x or 2) is fuppofed to be fmall ; to 27. Thirdly,

which Caie the other two Cafes are reducible. 28. Of all the Terms, in which the Radical

Species

(y,/>, q,

or

not found, chufe the loweft in refpect of the Dimenlions &c.) of the indefinite Species (x or z, &c.) then chufe another Term in which that Radical Species is found, fuch as that the Progreflion of the Dimenfions of each of the fore-mentioned Species, being continued from the Term fir ft afTumed to this Term, may defcend as is

r,

much

as

may

or

be,

afcend

as

little

as

may

be.

And

if

there

any other Terms, whofe Dimenfions may fall in with this muft be taken in 1 ikeProgreflion continued at pleafure, they thus felected, and made equal to wife. Laftly, from thefe Terms are

and write nothing, find the Value of the faid Radical Species, the Quote.

may be more

29. But that this Rule

explain right

it

farther

Angle BAC,

its

fides

in

clearly apprehended, I fhall

by help of the following Diagram. divide

it

AB, AC,

into

Making

equal parts,

a

and

Angular Space into equal Squares or Parallelograms, which you may conceive to be denominated from the Dimenfions of the Species x and y, raifing Perpendiculars, diftribute the

Then, when mark fuch of propofed,

as they are here infcribed.

any Equation

is

B

A4

the Parallelograms as correfpond to all and let a Ruler be apply'd its Terms, to two, or perhaps more, of the Parallelet one lograms fo mark'd, of which be the loweft in the left-hand Column at AB, the other touching the Ruler towards the right-hand ; and let all the reft, not touching Then felecl: thofe Terms of the Equation the Ruler, lie above it. which are reprefented by the Parallelograms that touch the Ruler, and from them find the Quantity to be put in the Quote. s out of the Equation y 6 30. Thus to extract the Root y 5xy -+1

)'*

ja*x y

1

+6a

i

x*-\-&

1

x4=o, C

I

mark

the Parallelograms belong-


The Method of

10

ing to the Terms of this Equation with the Mark #, as you fee here Then I apply the Ruler done. to the lower of the Parallelo-

FLUXIONS,

B *

DE

mark'd in the left-hand Column, and I make it turn round towards the right-hand from the grams

C

A

lower to the upper, till it begins in like manner to touch another, or perhaps more, of the Parallelograms that are mark'd

;

and

I fee

5 that the places fo touch'd belong to x 3 , x*-y* y Therefore and_y z 6 as if from the Terms y to equal 7a x*-y -}-6a*x*, nothing, (and .

<L

moreover,

if

you

6=

reduced to v 6

o, by making 7^*4of and find it to be four- fold, $=rv'\fitxt ) y, -\-</ax, </ax, -+-</2ax, and ^/2ax, of which I may take any one for the initial Term of the Quote, according as I defign to extract this or that Root of the given Equation. x =o, I chufe 31. Thus having the Equation y* 6y*-i-()&x* the Terms thence and I obtain -\-gbx*-, by4-3* for the initial Term of the Quote. x* 2rt =o, I make choice of 32. And having y">-i-axy-{-aay 2<2 3 and its Root -\-a I write in the Quote. y'-i-a^y I

pleafe,

feek the Value

3

3

,

33. Alfo having x*y

1

s

^c^xy

c

which like

^/

gives

for

the

firft

c I .v a 4-

Term

7

=o,

I felect

vV

i

f

y 4-<r

And

of the Quote.

7 J

the

of others.

Term

is found, if its Power fhould happen the to be negative, I deprefs Equation by the fame Power of the indefinite Species, that there may be no need of depreffing it in the Refolution ; and befides, that the Rule hereafter delivei'd, for the

But when

34.

this

fuppreffion of fuperfluous Terms, Thus the Equation 8z; 6_)i3 4-^2 5>' a

Root

is

to begin

by the Term

come Sz+yt-^azy

2ja

!>

z~

1

^ =o,

may

be

27^5=0 I deprefs

before

conveniently apply'd. being propofed, whofe

by s% I

that

attempt

it

may

be-

the Refolu-

tion. 3 5.

The

fubfequent

Terms

of the Quotes are derived by the fame

Work, from their feveral fecondary For the whole affair but lefs trouble. with commonly Equations, the of the loweft Terms affected with the is perform'd by dividing 1 3 Radical Speindefinitely fmall Species, (x, x , x , &c.) without the the radical r with which that } &c.) Species by Quantity (/>, q, Method,

in the Progrefs of the

i

of


n

and INFINITE SERIES,

of one Dimenfion only is affected, without the other indefinite SpeSo in the following cies, and by writing the Refult in the Quote. ->

~

~>

Example, the Terms a l x, TrW", TTT-v &c. by ^aa. 36. Thefe things being }

-

&c. are

produced

by dividing

3

,

premifed,

it

remains

now

to exhibit the

Praxis of Refolution. za* xz be

Therefore let the Equation y*-{-a zy-\-axy And from its Terms propofed to be refolved. 2 3 =o, being a fictitious Equation, by the third of the

=o

y=-\-a*y

a=o, and jtherefore I write -{-a in foregoing Premifes, I obtain y the Quote. Then becaufe -\~a is not the compleat Value ofy, I put a+p=y, and inftead of y, in the Terms of the Equation written in the Margin, I fubftitute Terms refulting (/> -{a-\-p, and the 1 from which again, I in the write ; 3rf/ -f-,?,v/>, &c.) Margin again Terms -+-^p the to felect the third of the I Premifes, according 3

p=

-H2 l .v=o

for a fictitious ^x, I Equation, which giving in the Quote. is not the becaufe Then accurate ^.v of p, I put in Terms for and the marginal x-\-q=p, p

~x

write

Value

3 -^x^+^a^, &c.) ^x-t-q, and the refulting Terms (j I again write in the Margin, out of which, according to the fore_I3-drx*=o for a fictigoing Rule, I again feledl the Terms

I fubftitute

tious Equation,

Again, fince

^

and inftead of a thus

I

continue

exhibits to view.

which giving is

I

4^ =^> I write

not the accurate Value of fubftitute

g,

-^ I

in

the Quote.

make -^--{-r=q

~--\-r in the marginal Terms. '

&4 the Procefs at pleafare,

as the following

t

And

Diagram


Method of FLUXIONS,

12

X

-

3

X*

2a'

axp

T

'

a*-x

;

643

*

*

axq *-

'31**

509*4

were required to continue the Quote only to a certain 37. If it that x, for inilance, in the laft Term {hould not afcend Period, a beyond given Dimenfion ; as I fubftitute the Terms, I omit fuch as For which this is the Rule, that after I forefee will be of no ufe. the firft Term refulting in the collateral Margin from every QuanTerms are to be added to the right-hand, as the Intity, fo many dex of the higheft Power required in the Quote exceeds the Index of that

firft

refulting

Term.

38. As in the prefent Example, if I defired that the Quote, (or the Species .v in the Quote,) mould afcend no higher than to four Dimenfions, I omit all the Terms after A-*, and put only one after x=.

Therefore


and INFINITE SERIES. Therefore the Terms after the

And

expunged. to the

Terms

Work being

thus the

-^

Mark

13

are to be conceived to

*

continued

till

axr,'m which

-^--H-rfV

at laft />, q,

r,

be

we come or

the Supplement of the Root to be extracted, are only of one Dimenfion ; we may find fo many Terms by Divifion, reprefenting

509*4

131*3 _, 5121.

So that

\

as

16384(13 /

at laft

we

we

fl^n

... {hall have

e

want n g

y=a

to compleat the Quote.

j

XX

1.*' 'SI*' 13

kuyAT 509*4

7*-f"6^-t-^l~*- r^I;

_

icc -

farther Illustration, I mail propofe another 39. For the fake of From the Equation -L_y< .Ly4_f_iy3 iy=. to be refolved.

Example _^_y

z=o,

let

the

Quote be found only

and the fuperfluous Terms be

to

rejected after the

the fifth Dimenfion,

Mark,

_!_

+^

5 ,

-L;S 4

5j

&c.

&c. Z'p, &C. 6cc.

2; s

,

&c.

% &c.

+

40. And thus if we propofe the Equation T 4-rjrJ' '+TT|-T )'' 7 J 3 to be refolved only to the ninth Di-rT T ;' -t-TW' -i-r.)' menfion of the Quote ; before the Work begins we may reject the Term -^^y" ; then as we operate we may reject all the Terms 7 beyond 2', beyond s we may admit but one, and two only after

+y

=o,


The Method of

Y4 z

becaufe

f

;

we may

obferve,

FLUXIONS,

that the

to afcerrd

Quote ought always

two Units, in this manner, z, .s , z &c. Then 9 s have ;'=c T_5__ 2; ^_ _^_'T ^_. 3 & C . fs3_j__|_. is difcover'd, by which Equations, 41. And hence an Artifice tho' affected hi injinitum, and confiding of an infinite number of Terms, may however be refolved. And that is, before the Work of begins all the Terms are to be rejected, in which the Dimenfion not the radical affected by the indefinitely fmall Species, Species, Interval of

by the

we

at laft

_

fliall

,

J

Dimenfion required

the greateft

exceeds

s

j

in the

)

Quote

;

or from,

which, by fubftituting inftead of the radical Species, the firfl Term, of the Quote found by the Parallelogram as before, none but fuch Terms can arife. Thus in the laft Example I mould have exceeding omitted all the Terms beyond y>, though they went on ad And fo in this Equation tum. 8 -f-3 1 1 )

4S 4 -f-92

lS

s 4 -}- z

6

in z*

j'

8

l6

injini-

&C.

,

z*y &c.

Root may be extracted only to four Dimenfions of z, Terms in infinitum beyond -f-j 5 in z, 1 J.-4_|_.L 2 > 1 a 4 -(-.c 6 and all beyond -+-y in .c 1 2z 4 and all beyond y- in z 4 And therefore I aflurr.e this Equation and beyond S-}-;s 42 6 s 6^ 1 -}-^ 4^ 1 be to refolved, -^z y* 2z*y z^y* z*y* -{-?* ;> only

that the Cubick I omit all the

,

,

tt

.

'

4s4_j_ s

-i-z'-y

8=0.

i

of y

being fubflituted inflead

by

',(*''-

in the reft

What

I

Qi\adraticks.

have

As

faid

if I

of the

1

A*

A4

h-r-f--;

y 2"

as the

-y+ &

in \

Equation deprefs'd

of higher Equations may alib be apply'd to the Root of this Equation

.r

far

of the Quote,)

defired

r

as

Term

~^ {

every where more than four Dimenfions.

z^y gives

42.

Becaufe?.

Period x f , I omit

<?_[-*+

'

all

the

and affume only

Terms

this

-

&c.

in infinititm.,

Equation, j*

ay

beyond xy

4

4-*-*

=0.

This

I

refolve either in the ufual

manner, by making


and IN FINITE SERIES. j-^; or more expedition fly by the

Method of

have _}'=

affected Equations deliver'd before, #>

3

where the

Term

laft

by which we

required

vanifhes,

fhall

or

becomes equal to nothing. 43. Now after that Roots are extracted to a convenient Period, they may fometimes be continued at pleafure, only by oblerving the Analogy of the Series. So you may for ever continue this z-t-i-z* ^_^.25_j__'_ 2; 4_{_ T i_2;s &c. (which is the Root of the infinite Equaj

the laft Term by thefe 5r==)'-f-^ _j_^5_|_y4 j foe.) by dividing Numbers in order 2, 3, 4, 5, 6, &c. And this, z f^-H-rlo-^' yj TB.27-f_ TTy2;9 j &c. may be continued by dividing by thefe Numi

tion

'

l

'

TrT

bers

2x3, 4x5, 6x7, 8x9, &c.

"-'g

,

&c.

refpectively fo of others.

may by

Again, the Series

be continued at pleafure, by multiplying the

Terms

TV, &c>

And

thefe Fractions,

f}

7,

,

-,

44. But in difcovering the firft Term of the Quote, and fometimes of the fecond or third, there may ftill remain a difficulty For its Value, fought for as before, may happen to be overcome. to be furd, or the inextricable Root of an high affected Equation. Which when it happens, provided it be not alfo impoffible, you

and then proceed as if it were known. As in the Example y*-\-axy-{-ii*-y x 3 2a>=o If the Root of this Equation y^^-a'-y 2 =o, had been furd, or unknown, I mould have put any Letter b for it, and then have perform'd the Refolution as follows, fuppofe the Quote found only to the third Dimenfion.

may

reprefent

it

by fome

Letter,

:

5


fbe Method of

i6 s

y -\-aay-\rtxy ,

tf^A-

2a3 4jC ft

;

FLUXIONS,


and INFINITE SERIES,

17

47. Hitherto I have fuppos'd the indefinite Species to be little. But if it be fuppos'd to approach nearly to a given Quantity, for that indefinitely fmall difference I put fome Species, and that being

Equation as before. Thus in the Equation x o, it being known or fupf}-' y* -t-y -\-a ^y* -+- ^y pos'd that x is nearly of the fame Quantity as a, I fuppofe z to be their difference; and then writing a-\-z or a z for x, there will arife y* -fwhich is to be folved y* -{-y jj fubftituted, I folve the

=

l

y

+ z=o,

5

as before.

48. But if that Species be fuppos'd to be indefinitely great, for Reciprocal, which will therefore be indefinitely little, I put fome Species, which being fubflituted, I proceed in the Refolution as

its

Thus having

before.

y* -+-\

or fuppos'd to be very -

I

and

put z,

being reflored.

&c

great,

if

is

.y

=

=o,

x>

-f-jv

for the

- for fobflituting

~ =o, whofe Root x

l

.v,

reciprocally

+

^z

you pleafe, J

-f-

known

Quantity 1

-f-.)'

+

y

^2', &c. where - H-

y=:x *

will be

it

z*

is

little

there will arife y>

^

where x

3

9*

H

8 i**

'

49. If it fhould happen that none of thefe Expedients mould fucceed to your defire, you may have recourfe to another. Thus 1 1 in the Equation y* i whereas -+-fx^y o, xy* Z) 2y -+-

=

Term

be obtain'd from the Suppofition that ought 2 1 which 0, jy-4_j_2yt y yet admits of no poffible Root; you may try what can be done another way. As you may fuppofe that x is but little different from 2, or that 2-{-z-=x. Then inftead of there will arife 2-{-z A*, fubftituting y* the

firft

+

=

to

+

2y

-f-

1

y

and the Quote

-+-2y*

2y H-

i

=

o,

and

the Quote. 50. And thus by you may extract and 51. If

from

will begin l

1

--{--

-\zy*

z'-y*

0,

to be indefinitely great, or -

fuppole x >*

=

-f-

=

z

z,

-j- i.

if

you

will

have ^ 4

initial

Term

you

for the

Or

of

,

proceeding according to feveral Suppofitions, exprefs Roots after various ways.

you mould

delire

to

find

after

how many ways

this

may be done, you mufl try what Quantities, when fubfHtuted for the indefinite Species in the propofed Equation, will make it divifible fome Quantity, or by^ alone. Which, for Example by_y, -f-or l x> 20 fake, will happen in the Equation y* -}-axy-+-a y o, 3

D

4

=

by


1

The Method of

8

by of

fubftituting -f-rf, or

And

.v.

from

differ little

you may

thus

-j-tf,

extract the

or

T

&c. inftead , the conveniently fuppofe Quantity x to 2a, or za*l^, and thence a, or a,

you may

FLUXIONS,

or

za,

or

2

}

|

Root of the Equation propofed

after

fo

And perhaps many other ways, by fupbe differences to thofe Befides, if you take indefinitely great. poling for the indefinite Quantity this or that of the Species which exprefs many

ways.

your defire after other ways. farther ftill., by fubftituting any fictitious Values for the indebz 1 , -> ~n^> &c. and then proceeding Species, fuch as az

the Root, you

And finite

alfo after fo

may

perhaps obtain

+

as before in the Equations that will refult. 52. But now that the truth of thefe Conclufions

may be maniand feft ; is, that the Quotes thus extracted, produced ad libi-* turn, approach fb near to the Root of the Equation, as at laft to differ from it by lefs than any afilgnable Quantity, and therefore when infinitely continued, do not at all differ from it You are to confider, that the Quantities in the left-hand Column of the righthand fide of the Diagrams, are the laft Terms of the Equations whofe Roots are p, y, r, s, &c. and that as they vanifh, the Roots p, q, r, s, &c. that is, the differences between the Quote and the Root fought, vanifh at the fame time. So that the Quote will not then differ from the true Root. Wherefore at the beginning of the that

:

Work,

if

you

fee

that the

Terms

in the faid

Column

will

all

de-

you may conclude^ that the Quote fo far exBut if it be otherthe perfect Root of the Equation. will fee however, that the Terms in which the indefi-

one' another,

ftroy tracted

wife,

is

you

of few Dimenfions, that is, the greate ft Terms, out of that Column, and that at laft none unlefs fuch as are lefs than any given Quantity, will remain there, and therefore not greater than nothing when the Work is continued ad infinitum. So that the Quote, when infinitely extracted, will at laft be the true Root. fake of perfpieuity I 53. Laftly, altho' the Species, which for the have hitherto fuppos'd to be indefinitely little, fhould however be fuppos'd to be as great as you pleafe, yet the Quotes will ftill be This true, though they may not converge fo faft to the true Root. here the Limits But is manifeft from the Anal'ogy of the thing. of the Roots, or the greateft and leaft Quantities, come to be For thefe Properties are in common both to finite and confider'd. The Root in thefe is then greateft or leaft,. infinite Equations. is

nitely fhiall Species are continually taken

when


and INF INITE SERIES.

19

the greateft or leaft difference between the Sums of the affirmative Terms, and of the negative Terms ; and is limited when the indefinite Quantity, (which therefore not improperly I but that the Magfuppos'd to be fmall,) cannot be taken greater, nitude of the Root will immediately become infinite, that is, will

when

there

become 54.

Is

impoffible.

To

Diameter

MakeAB

illuftrate this,

and

BC

AC D

be a Semicircle defcribed on the be an Ordinate.

let

=AD,^,BC=7,AD =

^.

Then

xx before.

as

Therefore BC, or y, then becomes greateft

iax moft exceeds all the Terms " f- V S^x 4- i6a> VS ax &c that la Sax -f- ga*

when

-

>

it

will

when x

be terminated

a.

For

S ax

if

is >

when *

we

=

**

i

but

take x greater than

V

ax ax> TbTs *S s7 &c. will be infinite. There is another Limit alfo, when x o, ax Radical to which the of reafon of the ; by impoffibility Terms or Limits, the Limits of the Semicircle A, B, and D, are cor^ a t the

Sum

of

all

the

Terms

^

=

S

refpondent. Tranfttion to the

55.

And

thus

much

METHOD

for the

OF FLUXIONS.

Methods of Computation, of which

Now it remains, that mall make frequent ufe in what follows. I mould of an Illuftration the for give fome SpeciAnalytick Art, of the nature Curves will fupmens of Problems, efpecially fuch as But firft it may be obferved, that all the difficulties of thefe ply. be reduced to thefe two Problems only, which I mall propofe I

may

concerning a Space defcribed by local Motion, any

how

,

x

'

accelerated

~

or retarded.

The Length of the Space defcribed being continually ( that -*"* 56. at fill ?V, Times) given; to find the Velocity of the Motion at any ffo^ Tune propofed. JbotA.*** if* 57. II. The Velocity of the Motion being continually given ; to find I.

Length of the Space defcribed at any Time propofed. in the Equation xx=y, if y reprefents the Length of 58. Thus the Space t any time defcribed, which (time) another Space x, the

by increafing with an uniform Celerity #, mea/ures and

D

2

exhibits as

defcribed

:

/

SJLJ tt


?%e Method of

20 defcribed y,

:

Then zxx will reprefent moment of Time,

the fame

at

by which the Space and ;

the Celerity

to be defcribed

proceeds

And

hence it is, that in what follows, I confider if were generated by continual Increafe, after the they

contrary-wife.

Quantities as

FLUXIONS,

manner of a Space, which a Body or Thing in Motion defcribes. 59. But whereas we need not confider the Time here, any expounded and meafured by an equable local whereas only Quantities of the fame kind can be compared together, and alfo their Velocities of Increafe and Therefore in what follows I fhall have no regard to Time Decreafe formally conficter'd,, but I fhall fiippofe fome one of the Quantities propofed, being of the fame kind, to be increafed by an equable Fluxion, to which the reft may be referr'd, as it were to Time j and therefore, by way of Analogy, it may not improperly receive Whenever therefore the word Time occurs in the name of Time. what follows, (which for the fake of perfpicuity and diftindlion I have fometimes ufed,) by that Word I would not have it underftood as if I meant Time in its formal Acceptation, but only that farther than

Motion

as

and

;

it

is

befides,

:

other Quantity, by the equable Increafe or Fluxion whereof, is expounded and meafured.

'60.

f

T H _' .i

(

>

%f& f'df**

which

thofe Quantities

I

confider as gradually and

hereafter call Fluents, or indefinitely increafing, Flowing Quantities, and fhall reprefent them by the final Letters of the Alphabet v, x, y, and z ; that I may diftinguifh them from other Quantities, which in Equations are to be confider'd as known and. the initial determinate, and which therefore are reprefented I

2

U

Now

Time

fhall

by

And the Velocities by which every Fluent Letters a, b, c, &c. is increafed by its generating Motion, (which I may call Fluxions, oi V* ffm***4t*'Qr fimply Velocities or Celerities,) I fhall reprefent by the fame V*>

i*i~-

Letters pointed thus

K

J

id

tti '(/

t4

JO the Quantity v Quantities x, y,

-y,

I fhall

and

x,

y.,

and

z, I fhall put x, y,

1.

two Problems

juft

now

is,

for the

Celerity

of

put v, and fo for the Celerities of the other

Thefe things being premifed, to the matter in hand } and firft I 6

That

z.

I

and z

mall

fhall

now

refpeftively.

forthwith proceed

give the Solution of

the:

propofed.

PROF,


and INFINITE SERIES.

R O

P

B.

21

I.

The Relation of the Flowing Quantities to one another being given, to determine the Relation of their Fluxions.

SOLUTION. Difpofe the Equation, by which the given Relation is exprefs'd, according to the Dimenftons of fome one of its flowing Quantities, fuppofe x, and multiply its Terms by any Arithmetical 1.

and then by And perform this Operation feparately for every one of the flowing Quantities. Then make the Sum of all the Products equal to nothing,, aad you will have the Equation .

Progreflion,

required.

EXAMPLE

2.

y be X'

ax*--{-

If the Relation of the flowing Quantities A; and 3 firft axy ^ difpofe the Terms according i.

=o;

and then according to manner.

to x, ing,

y,

and multiply them in the follow-

*

Mult.

by makes

%xx*

zaxx

-{-

axy

*

zyy* -f- ayx

* *

The Sum of

zaxx -k- axy W*-f- ayx=zo, the Relation between the Fluxions x and y. ax 1 -{-axy pleafure, the Equation .v yt

the Produdls

-jx**

is

i

which Equation gives For if you take x at

=

o will give y.

7v* 3..

ax Ex.

:

2.

.

3

Which

being determined,

it

will

be x

:

y

::

yx^zax -{- ay. of the Quantities

If the Relation

the Equation preis'd by

2j

3

-f-

x*y

zcyz +-

x,

y,.

yz*

and z r be exz''

=

QJ


22

Method of FLUXIONS,

*The

Wherefore the Relation of the Fluxions

,v, v,

and

is

z,

Celerities

+

tyy* -\-

or of the

of Flowing,

$zz

2xxy

l

-f-

6zzy

zczy

.

4.

But

there are here three

fince

flowing Quantities,

y,

.v,

and

z, another Equation ought alfo to be given, by which the Relation among them, as alfo among their Fluxions, may be intirely deter2 0. From whence mined. As if it were fuppofed that x -\-y

=

=

o would be z another Relation among the Fluxions AT-HV thefe with the found by this Rule. Now compare foregoing Equaand alfo any of three one the Quantities, tions, by expunging any one of the Fluxions, and then you will obtain an Equation which will intirely determine the Relation of the reft. In the Equation propos'd, whenever there are complex Frac5.

I put fo many Letters for each, and fuptions, or furd Quantities, Afto reprefent flowing Quantities, I work as before. them pofing as the afTumed and exterminate terwards I Letters, you fee

fupprefs

done

here.

aa If the Relation of the Quantities .v and y be yy xx I write z, and thence I ## o; for x</aa x\/aa 3 1 2* x4 aa two have the %,=.&., and a -* Equations^' z will give zyy o, as before, for the o, of which the firfl and z, and the latter will give 2<j*xx Relation of the Celerities Ex.

6.

=

3.

=

.

i

y

a * xx

o, or

Celerities

=

x and

-./** 4-

^ aa

g*.>*

XX

=

z, for the Relation of the

Now z being expunged, it will be zyy xx for z, we fhall reftoring x^aa

__ 0>

the Relation between

for

x and

quired. 7.

Ex.

4.

If

.v

3

that

the Relation

ay* is

+&

The

firft

Zy^-yz

-u

=

gives

2W=

o,

between

3^

&

=

zayy o,

and

AT

v

:

I

-+-

xx

y, as

=

make

have zyy

o,

^^

was

re-

expreffes

=

5;,

and

Lave the three Equations x1^=0. and ax*y + x 6 -0=0, the fecond gives az +

fhall

^ =o, 3

az-\-yz

3**'

XX \fay

4- j4r

^x \/~ay-+-xx=v, from whence I ay*

-

z.

and then

o,

~ ****

+

z

and the third

gives

4.axx>y-+-6xx'-i-a}>x*

o, for the Relations of the Velocities

-y,

.v,

y,

and

.

But the


and INF NIT E SERIES.

23

i

the Values of & and

? iSj /.

/.

.

v-.

nrft Equation,

=

found by the fecond and third Equations,

z

and

11

.

7n

,.

and there anies %xx*

Then

o.

for

i',

inflead of

vz

2a)y-^-~^T.

z and v

their Values

refloring

a f

and >

.

XX \/ ay

-+-

xx, there will

arife

awMf

6*- A- 3

aa -f-

+ yy

2^

Velocities

x and y

=

...

o.

.

_

^xx*-2ayy .

.

which the Relation by >

r

,

or the

will be exprefs'd.

what manner the Operation

in other is to be performed manifefl believe is from hence j as when in the Equation there are found furd Denominators, Cubick Radicals, Ra-

After

8.

2

.

the Equation fought

Cafes, I

propos'd xx} or any other comdicals within Radicals, as v ax -+- \/'aa of Terms the like kind. plicate fhould 9. Furthermore, altho' in the Equation propofed there be Quantities involved, which cannot be determined or exprefs'd by any Geometrical Method, fuch as Curvilinear Areas or the Lengths of Curve-lines ; yet the Relations of their Fluxions may be found, as will appear

from the following Example.

EXAMPLE

Preparation for 5* 10. Suppofe BD to be an Ordinate at right Angles to AB, ancL be any Curve, which is defined by that

ADH

between AB and BD exhibited Let AB be called A;, and an Equation. by of the Curve the Area ADB, apply 'd to Unity, Then erect the Perpendicular AC be call'd z. the Relation

C

equal to Unity, and thro' to

AB, and meeting

BD

in

E.

draw

CE

Then

parallel

conceiving

two Superficies ADB and ACEB to be generated by the Motion of the right Line BED it is manifeft that their Fluxions, (that is,, the Fluxions of the Quantities i x z and i x v, or of the

thefe

;

t

Quantities

s

and BE.

Therefore

z

= * AndBD.

and

.

x,) are to each other as the generating Lines :

x

::

BD

:

BE

or

i,

and

BD

therefore

x

hence it is, that z may be involved in any Equation, the Relation between .v and expre fling any other flowing'Quantityjv ; and yet the Relation of the Fluxions x and however be dif1

1.

y may

cover'd,

12

.


<

fhe Method

24 As

FLUXION s,

<J/"

the Equation zz -\-axz _y*=r=o were propos'd to exprefs the Relation between x and; as alfo BD, for determining a Curve, which therefore will be a Circle.

Ex.

12.

5.

if

=

\/axxx

1

,

The Equation zz-^-axz azX

-f-

and

z.

axz

=

4_y_y

And

o,

j^=o,

iubftitute this

the Relation

for

therefore fince

z

it is

Value inftead of

it,

xx 4- axz Celerities x and

2xz -t- axx \/ax-r Relation of the

before, will give

as

= x BD = x

and there

or

-x

\/ax

will arife the

o,

qyy*

2zz-i-

of the Celerities x,y,

xx

t

Equation

which determines the

y.

DEMONSTRATION

of the Solution.

The Moments

of flowing Quantities, (that is, their indefithe acceffjon of which, in by indefinitely fmall are of Time, they continually portions increafed,) are as the Velocities of their Flowing or Increafing. 13.

nitely fmall Parts,

14. t>y the

Wherefore

if

Product of

its

Moment of any Celerity x into an

the

one, as x, be reprefented indefinitely fmall Quantity

by xo,} the Moments of the others <y, y, z, vot yo, zo ; becaufe vo xo, yo, and zo, are reprefented by o (that

is,

y

other as v, x, ,.

,

p. /fc, // natti** cttA

Now

to each

and x.

y,

fince the

15. u tt i e ^cceflions

will be

Moments,

as

xo and yo, are the indefinitely

of the flowing Quantities

.v

and

y,

by which thofe

any therefore the Equation, which at all times indifferently exprefles the Relation of the flowing Quantities, will as well exprefs the

And

x -3- xo and

y-+-yo, as between x and y: So that x -+- xo and y -f- yo may be fubftituted in the fame Equation for thofe Quantities, inftead of x and y. 1 6. Therefore let any Equation #' ax* -+- axy be ^' Relation between

=

given, and

fubftitute

x~\-xo

for

x and y }

-j-

yo for y,

will arife

+ ax 1

2axox

axy +- axoy y:

lyoy-

$x*oox

-f- x*o''

ax*oo

-h ayox

~ yfooy

-h axyoo

and there


and INFINITE SERIES.

3 ax--3raxy =o, which thereby Suppofition x the Terms and remaining being divided by o, expunged, remain ^xx* -f- ^ox -+- x>oo zaxx ax 1 o -f- axy -f-

Now

17. fore being

there will

ayx

3

_}'

axyo

_f_

3_vy*

to be infinitely tities

the

;

of the

zaxx

little,

Terms

axy

-+-

y*oo

3y*oy that it

may

=

ayx

reject

3_yj*=

o,

But whereas

o.

reprefent the

that are multiply'd by

Therefore I

reft.

-f-

25

it

o is

fuppofed Qiian-

Moments of

will be nothing in relbedl

them, and there remains $xx* as above in Examp. i.

Here we may

obferve, that the Terms that are not multiply'd by o will always vaniih, as alfo thole Terms that are multiply'd by o And that the reft of the Terms of more than one Dimenfion. 1

8.

being divided by o, will always acquire the form that they ought Which was the thing to be proved. to have by the foregoing Rule now this fhewn, the other things included in the being 19. And Rule will eafily follow. As that in the propos'd Equation feveral flowing Quantities may be involved ; and that the Terms may be of the Dimenlions of the flowmultiply'd, not only by the Number other but alfo Arithmetical Progreilions ; fo by any ing Quantities, that in the Operation there may be the lame difference of the Terms according to any of the flowing Quantities, and the ProgrefTion be difpos'd according to the fame order of the Dimenlions of each of And thele things being allow'd, what is taught belides in them. Examp. 3, 4, and 5, will be plain enough of itfelf. :

P R O

B.

II.

An

Equation being propofed, including the Fluxions of O^uantitieS) to find the Relations of tbofe Quantities to one another.

A PARTICULAR SOLUTION. i.

As

folved

this

by

Problem

is

the Converfe of the foregoing,

proceeding in a contrary

multiply'd by

x

manner.

That

is,

it

the

muft be

Terms

being difpofed according to the Dimenfions of

they muft be divided by

* x

,

and then by the number of

their

x;

Di-

menfions, or perhaps by fome other Arithmetical Progreffion. Then the fame work muft be repeated with the Terms multiply'd by v, y,

E

or


The Method of

26

FLUXIONS,

or z, and the Sum refulting muft be the Terms that are redundant.

made

equal to nothing, re-

jeding 2.

EXAMPLE. Let 4- ayx

Divide

=

3 ATA?*

Quot. by Divide by

5

3A: .

3

The

Operation will be after this

2axx-i-axy

Divide

2ax* -\-ayx

by

2

i.

ax 1

A; 5

Quote

the Equation propofed be ^xx*

o.

Sum #

-{-ayx

2axx 4- axy manner :

* -f-

Quot. Divide by ^.

*

3

2

3 5

Quote

_y

*

ayx

4- axy .

i.

4- axy

=

o, will be the required y* axy Relation of the Quantities x and y. Where it is to be obferved, that tho' the Term axy occurs twice, yet I do not put it twice in ax* -+- axy the Sum x'> y* =. o, but I rejed the redundant Term. And fo whenever any Term recurs twice, (or oftener when there are feveral flowing Quantities concern'd,) it muft be wrote only once in the Sum of the Terms. are other Circumftances to be obferved, which I mall/ 3. There leave to the Sagacity of the Artift for it would be needlefs to dwell too long upon this matter, becaufe the Problem cannot always be folved by this Artifice. I mail add however, that after the Rela-

Therefore the

3

ax*

-f-

-,

of the Fluents is obtain'd by this Method, if we can return, by Prob. i. to the propofed Equation involving the Fluxions, then the work is right, otherwife not. Thus in the Example propofed, after I have found the Equation x> ax 1 -{- axy o, if from y*

tion

-

thence

I

feek the Relation

of the Fluxions

=

x and y by

the

firft

mall arrive at the propofed Equation ^xx* 2axx 4o. Whence it is plain, that the Equation i,yy* -faxy 3 o is But if the Equation AT 3 -ax*-+-axy _y rightly found.

Problem,

I

ayx=

=

=

o were propofed, by the prefcribed Method I xy -\- ay fhould obtain this ^x* o, for the Relation between ay xy be Conclufion would erroneous: Since by Prob. i. x and y ; which

xx

the Equation xx is

different

4. lhall

+

xy

yx -+- ay

= =

o would be produced, which

from the former Equation.

.Having therefore premiled this in a perfundory manner, I now undertake the general Solution.

A


and IN FINITE SERIES.

27

A PREPARATION FOR THE GENERAL SOLUTION. mufl be

the propofed Equation the Symbols of the Fluxions, (fince they are Quantities of a different kind from the Quantities of which they are the Fluxions,) to afcend in every Term to the fame number of Dimenfions : Firft

5.

it

that in

obferved,

ought

And when

it happens otherwife, another Fluxion of fome flowing be underflood to be Unity, by which the lower mufl Quantity Terms are fo often to be multiply'd, till the Symbols of the Fluxions As if arife to the fame number of Dimenfions in all the Terms. o were the Fluxion axx z the Equation x -+ x'yx propofed, of fome third flowing Quantity z mufl be underilood to be Unity, by which the firfl Term x mufl be multiply'd once, and the lafl axx twice, that the Fluxions in them may afcend to as many Di-

=

Term xyx

menfions as in the fecond

had been derived from

z

=

gine

xz -{-xyx-

the propofed Equation azzx*o, by putting if

=

And

thus in the Equation yx =}')'-, you ought to imato be Unity, by which the Term yy is multiply'd. Equations, in which there are only two flowing Quan-

i.

x

Now

6.

which every where

tities,

may

this

As

:

the fame

arife to

always be reduced to fuch a form,

had the Ratio of the Fluxions,

(as \

4 x

, .

number of Dimenfions,

as that

or y

,

other fide the Value of that Ratio, exprefs'd *

Terms

;

as

you may

fee here,

4-

=

2

-h 2X

on one

fide

may be

or ~ and on the x ,&c.)

by fimple

Algebraic

And when

y.

the

foregoing particular Solution will not take place, it is required that you fhould bring the Equations to this form. of that Ratio any Term is de7. Wherefore when in the Value nominated -by a Compound quantity, or is Radical, or if that Ratio be the Root of an affected the Reduction mufl be per;

Equation form'd either by Divifion, or by Extraction of Roots, or by the Refolution of an affected Equation, as has been before fhewn. 8.

pofed

As j

av+y'

if the firfl

And

Equation ya

by Reduction in

the

firfl

xa

yx this

-+-

xx

xy

=

o were pro-

x

Cafe, if I reduce

minated by the compound Quantity a

E

-==

becomes T-=i-f--^-, ax or

2

the

Term ^^.,

y

deno-

x, to an infinite Series of

fimple


The Method of

28 Terms j

fimple

^ y

~ ^

- -f-

&c. by dividing the Numerator

-+-

Denominator a

the

y by

-f-

have -

mall

I

x,

FLUXIONS,

i

-

+-

-f-

x and

7; &c. by the help of which the Relation between to be determined.

-f-

is

So the Equation

9.

=

_y_y

xy

being given, or ^-

-j- .XVY.V A:

A-*

and by a

i- xx,

4=4 +V/T

Reduction

farther

-+- A-*

^ -f.

=

4, x

I extract

:

AT

the fquare Root out of the Series f -{-x* x* -f- 2X 6

Terms

tute for

have -

&c. is

\/t H- xx, ~

or.

either

=

2x*a>=o anfes ~ x

-I,

or

14*', &c. which i

8

'

-f-

-j-

^__ ^-_|_ 640 5i2 a

as

-f-

a1-

2rf 3

>v 3

x

and

Equation,

COQi'4 ^

&c

as

.

16384^3

~

x*x">

-f- a'-x^y

axx*y

Cubick

4.

2x 6

it.

ax

111*5

if I fubfti-

-f-

&c. according

,

A:

affected

x*

x*

-f-

from

A:5

XX

4

5*

Equation y*

Root of the X

-+-

or fubtracled

were propofed,

=a

V

2X 6

thus if the

I extract the /-

to

=

X

x^-ir-x*

added

And

10.

I (hall

-f-

5*"

xr, and obtain the infinite

-f-

-J

may^

=

o

there.

be feen

before.

But here

look upon thofc Terms only as compounded, which are compounded in refpect of flowing For I efteem thofe as fimple Quantities which are comQuantities. pounded only in refpect of given Quantities. For they may be reduced to fimple Quantities by luppofing them equal to other givea 11.

^

Quantities. -

1

it

Thus

may

be obferved, that

I

" I

eonfider

the

->

Quantities ^^'

c

"-TT, rra*4b' ax-\~bx

>

4

~^,

may

L

,

xi all

>

v/tfA-

H- bx, &c.

as

fimple Quantities, becaufe they

be reduced to the fimple Quantities

^

x*} &cc. by fuppofing a -f- b =r= e. 12. Moreover, that the flowing Quantities

i,

,

-^-,

may

the

may

\/ex (or

more

eafily

be diflinguifh'd from one another, the Fluxion that is put in the Numerator of the Ratio, or the Antecedent of the Ratio, may not improperly be call'd the Relate Quantify, and the other in the Denominator, to which it is compared, the Correlate : Alfo the flowing


and INFINITE SERIES.

29

be diftinguifli'd by the fame Names refpecflowing Quantities may And for the better understanding of what follows, you may tively. is Time, or rather any other conceive, that the Correlate Quantity

which Time is expounded and Quantity that flows equably, by or the Relate Quantity, is Space, that the And other, meafured. which the moving Thing, or Point, any how accelerated or retarded, And that it is the Intention of the Problem, defcribes in that Time. the Motion, being given at every Inftant of the that from Velocity of Time, the Space defcribed in the whole Time may be determined. in refpedt of this 13. But Orders. into three

Problem Equations may be

and only one

In which two Fluxions of Quantities,

of

diftinguifli'd

14. Firft: their flowing Quantities are involved.

Second: In which the two flowing Quantities are involved, their Fluxions. together with Third: In which the Fluxions of more than two Quantities 1 6. 15.

are involved. 17.

With

Problem,

Premifes

thefe

according to thefe

I

{hall

the

attempt

Solution of the

three Cafes.

SOLUTION OF CASE

I.

in which alone is contain Suppofe the flowing Quantity, accordand the be the the Equation, to Correlate, Equation being on one fide to be only the is, making by ingly difpos'd, (that Ratio of the Fluxion of the other to the Fluxion of this, and on the other fide to be the Value of this Ratio in fimple Terms,) mulValue of the Ratio of the Fluxions by the Correlate Quantiply the then divide each of its Terms by the number of Dimenfions tity, with which that Quantity is there afTeded, and what arifes will be 1

'd

8.

other flowing Quantity. equivalent to the the Equation yy xy -+- xxxx ; I fuppofe x 19. So propofing and the Correlate be the to Quantity, Equation being accordingly

=

reduced,

tiply

we

mall have -

the Value of

&c. which Terms

and the Refult x

=

into x, I

i

-f-

x1

.v 4

-f-

fv'

fv'-f-fv

,

'

and there

arifes

divide feverally by their

+-

Now I mul-

2X & &c.

1 ,

&c.

.v-f-AT

3

xf

-{-

2X\

number cf Dimenfions, I

put

=y. And

by this


Method ^/"FLUXIONS,

77je

30

Equation will be defined the Relation between

this

required.

20. Let the Equation be will arife

=

y

ax

Relation between

And

21.

y

gives

x

and

'

^

Value of -

into

-f-

^

.

and

A;,

2

or

A:-

the

x' 1 -\-

-f-

=

&c.

^

|.

x

*+

**

there

the

-tI

#*,

multiply

the

x*

v* X ,

For

.

-

ax^

X -f.

-

.

-*-

Terms being divided by of y will arife as be-

which

x^-i-x^,

number of Dimenfions,

x*

_i_ -, I

,

*.!

v-J

Jf^

ax*

was

'* 3

for determining o

.

OJ.oi.t~

becomes

it

3

5i2*

6-}<z

-^ &c

*;

1

-4-

y, as

'

j-

y.

+

'

-

4

y ZM

thus the Equation

=

a

~

y -+-

A;

=

--

x and

Value

the

fore.

22. After the fame \- cy,

Value of

gives

-

A-

being or

-{-n'3

-the Value of

x

=

manner the Equation

^_

H- -

-}-

multiply'd

by

2^^-y* -h -~i refults,

3

;'

1.

3

v/^)'

there

j,

-^ 4- S7=5-7=== \/ f

-.

-i-

~

arifes

+ v/^ +

cy~>

c %y*.

A -+-

For the

.

*

-^

_j_

And

thence

by dividing by the number of the Dimen-

of each Term.

lions

23.

And

fo

=?

~

=z\ gives y = $z*.

=

, 3

And

=

-1

=-

*

47

,

gives

r=

But the Equation . For f multiply'd ; gives 7 f ^ A: makes a, which being divided by the number of Dimen~ an infinite fions, which is o, there arifes Quantity for the Value

3f^L into

.

,

,

_

24. Wherefore, whenever a like

of

-.

,

whofe Denominator

Term

mail occur in the Value

involves the Correlate

Quantity of one

Dimenfion only

inftead of the Correlate ; Quantity, fubftitute the or the Difference between the fame and fome other given Quantity to be affumed at pleafure. For there will be the fame Relation of Flowing, of the Fluents in the Equation fo. produced, as of the Equation at firft and the infinite Relate j

Sum

propofed

Quantity


and INFINITE SERIES.

means will be diminifh'd by an infinite part of itfelf, become finite, but yet confifting of Terms infinite in

this

by

tity

and will number.

= =

-

the Equation 4 25. Therefore

v

-

=

:

,

-

now the Rule aforegoing will give_}'= j x and

the Relation between

So

26.

_f

.

4*

= ^

you write

if

->

xx.

2X

2Ar 3

ax 4- fx

And

tranfmute x, .

_{_

j,

&c

At

c

-rr b+

~p

~j^

3

And

-

&c. for

xx; becaufe

4-3 there

will

Term ~-^

arife

4

into an in-

will have 4

,

X

then according to the Rule for the Relation of x

6cc.

,

thus if the Equation -.-=x'^-i-x-

Term x

I here obferve the pofed j becaufe

i

I

y.

27.

I

ax^

4- 2x% &c. you

s

|x 4- ^x

X

for x,

And

4

3

=

reducing the

,

1

4.x

Then

x

-f-

2x 3 4- 2x 4 &c.

x*

_{_

i

2x4- 2x l

4-2

finite Series

y and

~

x

2

(_

X

you have the Equation

if

Term

of the

y.

^ 4-

x

there will arife

O

v

u-^r~X

if for

being propofed,

x, affuming the Quantity b at pleafure, a^ ax^ fl 11 T^ /* v and by Divifion 4 T rr 4- 77 * &

^4-

write

31

_'

x

_L

_|_

.i#

__

by

-

1

i

4-

are fubftituted,

I

-i-x

it,

and there

Term

&c. and the

Term

\/i

x

1

1

4- 4-x 4-

fhall

then by the Rule y

or

an(^ therefore |-x

have 4 X

=. x 4-

3 ,

=

&c. i ~f-

1

x 4-

4-x*

to be found,

(or ~j

the

V^S

4-x

the fame as

,

l

Now

v/

3

for

were pro-

AT*

A;".

-

x 1 4- x

x

I

fubftituting

1

So that

4

-

x produces

is

i

arifes

l

equivalent

_ _ v

when

-

a

^

is

thefe Values

2x 4- 4x i 4-4-^-x

4- ri* 4 , &c

.

JL;(

to

3

,6cc.

An<i

And

^

oi

others.

28. Alfo in other Cafes the Equation may fometimes be conveniently reduced, by fuch a Tranfmutation of the flowing Quantity.

As

if this

Equation were propofed 4

= -^

^^. ^_ xi c

inflead


^

O52

of

.v

write c

I

Method of FLUXIONS, i/

and then

AT,

and then by the Rule y

=

J

-

4=

mall have

I

L.

-f,.

^

or

^

7

~i>

5

But the ufe of fuch Tranf-

mutations will appear more plainly in what follows.

SOLUTION OF CASE

II.

And fo much for Equations that involve But when each of them are found in the only one Fluent. Equation, fiift it muft be reduced to the Form prefcribed, by making, that on one fide may be had the Ratio of the Fluxions, PREPARATION.

29".

fide. aggregate of fimple Terms on the other be any fo reduced there 30. And befides, if in the Equations the flowing Quantity, they muft be freed Fractions denominated

equal to an

by from thofe Denominators, by the above-mentioned Tranfmutation of the flowing Quantity. o being propofed, or aax xxy 31. So the Equation yax f becaufe of the Term -, I afiume b at pleafure, and _{_

=

i_l

.

for

x

*

x

a

x

write b -+- x,

it

will

become 4

=

-

-f-

rrr.

As

- b.

x

x, or

or b

either write b -+- x,

I

And

.

we

mall have

&C.

manner the Equation

the fame

after

Term

=

X

2x

37 X

I

<

,

And

72. J

-

-

,

fhould

then the

into an infinite Series, being converted byDivifion

-1-1

if I

..

being propofed;

write

i

for

y

_ o V -4- 2 3/

x

-f-

y

=-^

^1

infinite Divjfion gives

and the

^x fore

Term

_f-

6 x *-y

r-=

X

-

t

and

y

~.-\- X*r ZX

I

3

a '

-f->'

a

.

there will

x,

xy

But the

8x

^

1

y

^y

5 ,

&c.

2 -i-

2_y

IOAT*, -i-

= by J

xy* t &c.

xy* -J-_y

-f-_y

y3

'-

3

+ iox*y

xy* -{-

and^.,

arife

Term

like Divifion gives 5

2v

-

X

a

x -+-y

.

for

2

2- v

-4-

_^2~+ xx by 6x a 4- S* ^

3^-i- 3^J

x

i

y i

by reafon of the Terms

if,

+

&c.

6^^

^xy

There6x*

33-


and INFINITE SERIES.

33

The Equation being thus prepared, when need rethe Terms according to the Dimenfions of the flowquires, difpofe ing Quantities, by fetting down fir ft thofe that are not affected by the Relate Quantity, then thofe that are affected by its lead DimenRULE.

33.

fion,

and

In like manner alfo diipofe the

fo on.

Terms

in

each of

thefe Clafies according to the Dimenfions of the other Correlate Quantity, and thofe in the firft Clafs, (or fuch as are not affected

by the Relate Quantity,) write in a collateral order, proceeding towards the right hand, and the reft in a defcending Series in the lefthand Column, as the following Diagrams indicate. The work being thus prepared, multiply the firft or the loweft of the Terms in the firft Clafs by the Correlate Quantity, and divide by the number of Dimenfions, and put this in the Quote for the initial Term of the Value of the Relate Quantity. Then fubftitute this into the the Terms of Equation that are difpofed in the left-hand Column, inftead of the Relate Quantity, and from the next loweft Terms you will obtain the fecond Term of the Quote, after the fame manner as you obtain'd the firft. And by repeating the Operation you the as continue far as you But this will appear Quote may pleafe. plainer

34.

by an Example or two.

EXAMP.

i.

Let the Equation 4

be propofed, whofe Terms i by the Relate Quantity _v, you

T.V -+-

fee

= A'

1 ,

i

^x-\-y-\- x*-{-.vy

which

are not affected

difpos'd collaterally in the

up-


The Method of

3-4

FLUXIONS,

xx for the by the number of Dimenfions 2, gives Then this Term fecund Term of the Value of y in the Quote. afiumed to likewifc the Value of the compleat being Marginals -{-y xx and x and -+- xv, there will arife alfo to be added to the Terms -j-x and -{-xx that were before inferted. Which being done, I again a flume the next loweil Terms -f-xx, xx, and -{-xx, which I collect into one Sum xx, and thence I derive (as before) the third Term -|-.ix ; to be put in the Value of y. Again, taking this Term -i-x 3 into the Values of the marginal Terms, from the 4 x 3 added together, I obtain for next loweft -f-y# 3 and ^-x And fo on in infinitum. the fourth Term of the Value of y. In like manner if it were required to determine 35. Ex AMP. 2.

beino; divid'-d

5

,

,

y the Relation of x and y in this Equation, - -=. < ^

-

I

-f-

- -f- --v -fa &*

r'-f-

&*

fuppofed to proceed ad infinitum ; I put I in the beginning, and the other Terms in the left-hand Column, and then purfue the work according to the following Diagram. ,

&c. which

Series

is


and INFINITE SERIES. EXAMP.

37.

=

6..Y

3,v -+1

-f- SA-

extract the

Terms

J

3.

In like

Xj* -t-j

i*y -4-;* -

8.v

_V

3

manner

as

3

.vy

A^

-4-;-

IOA-*, &c. and far as feven Dimensions

4- \oxy*

Value ot y

Equation were propofed

if this 3

35

it

is

of x.

intended to I place the

order, according to the following Diagram, and I work as before, only with this exception, that iince in the left-hand Column y is not only of one, but alfo of two and three Dimensions; (or of

beyond

in

more than

Value of y, ted

three,

the degree of fo

by degrees

far

x~*

,)

intended to produce the Value of y fubjoin the fecond and third Powers of the

if I I

gradually produced, that when they are fubftituin the Values of the Marginals

to the right-hand,


The Method of

36

FLUXIONS,

it appears Equation required. But whereas this Value is negative, that one of the Quantities x or y decreafes, while the other inAnd the fame thing is allb to be concluded, when one of creafes. the Fluxions is affirmative, and the other negative.

EXAMP. 4. You may proceed in like manner Equation, when the Relate Quantity is affected with 38.

menfions. this

As

Equation,

if it

-

=

to refolve the

Diwere propofed to extract the Value of x from iy

fractional

1

^y-

-+-

zyx*

-J.v

-f-

77*

-f-

;

2_y

,

in


and INFINITE SERIES.

37


tte Method of

38 And

41.

and of

Quantities,

laft

moft

I

have compleated this moft troublefo'me Problem, when only two flowing

difficult

together with their Fluxions,

are

comprehended

in

an

But

befides this general Method, in which I have taken the Difficulties, there are others which are generally fhorter,

Equation. in all

thus at others

all

FLUXIONS,

which the

Work may often be eafed;

by

to

givefome Specimens of which,

ex abundantly perhaps will not be diiagreeable to the Reader. 42. I. If it happen that the Quantity to be refolved has in fome places negative Dimenfions, it is not of ablblute necefllty that therefore the Equation mould be reduced to another form. For thus

=

-

xx being

propofed, where y is of one negative Dimenfion, I might indeed reduce it to another Form, as by writing i -f- y for y ; but the Refolution will be more expedite as you have it in the following Diagram.

the Equation y


and INFINITE SERIES,

3.9


The Method

^FLUXIONS,


and INFINITE SERIES, 50. And tion proceeded, I

here

the

Terms

it

be obferved by the way, that as the Opera-

may

might have -

and

4**

41

inferted

any given Quantity between

for the intermediate

,

Term

that

is

deficient,

and fo the Value of y might have been exhibited an infinite variety of ways. 51. V. If there are befides any fractional Indices of the Dimenfions of the Relate Quantity, they may be reduced to Integers by

fuppofing that Quantity, which is affected by its fractional Dand then by ftibftitutii g menfion, to be equal to any third Fluent ;

Fluxion, ariling from that fictitious Quantity, inftead of the Relate Quantity and its Fluxion. Equation, 52. As if the Equation 3*7* -\- y were propofed, where the Relate Quantity is affected with the fractional Index .1 of its Dimenthat

as

alfo

its

y=

fion; a

= y= y

Fluent the

z'> ;

32Z

1

=

z

z, or being afTumed at pleafure, fuppofe y^ be will Relation of the Fluxions, by Prob. i.

Therefore fubftituting ^zz* for

.

=

alfo

as

v,

z*

for

=

y,

3 and z* for y$, there will arife yzz 1 or z x -\-^z, ^xz*- -+- z where z performs the office of the Relate Quantity. But after the Value of z is extracted, as z &c. inx* -f-J-f-^-Q ,

ftead of

z

reftore

tween x and

Cubing each 53. In

by Prob.

=

i

;

y\ and you

that

is,

=

will have the defired Relation i.v 1

+ -V^ H- T-nr* 3

y =.^x -\- T'_.v -+- T Y T X manner if the Equation y

xM

2^^ -J-

2zz

i.

y?

^

6

fide,

like

given, or_y

z

v

= =

,

-+- {-x^.

=

=

;

7

I

S

^c

=

make z =)'^

>

4 ;

&c

-

an(^

be-

^7

-

</xy were or zz=y, and thence </^y

-+-

=

and by confequence 2zz 2z -f- x*z, or Therefore by the firft Cafe of this 'tis z x -fy,

=

then by fquaring each fide, v=y>; -+- -|Jf^ Ar-f", , 5 But if you mould defire to have the Value of y exhibited -i- -i-x infinite number of ways, make z =. c -f- x -f- -ytf , aiTuming any an initial Term c, and it will be ss, that is y, c* -{- zcx ^cx*

-i-v

1

or y'1

-i.v

1

.

=

+

-v

1

1

-+-

-i-x

*

-t-

^v

3 .

ing of fuch things as

But perhaps I may feem too minute, will but feldom come into practice.

SOLUTION OF CASE

+

in treat-

III.

54. The Refolution of the Problem will foon be difpatch'd, the Equation involves three or more Fluxions of Quantities.

G

when

For between


Method of FLUXIONS,

?$

42

between any two of thofe Quantities any Relation may be afiumed, when it is not determined by the State of the Queftion, and the Relation of their Fluxions may be found from thence ; fo that either of them, together with its Fluxion, may be exterminated. For which reafon if there are found the Fluxions of three Quantities, only one need be two to if there be and fo on j affumedj four, Equation be transform'd into another And then Equation, in which only two Fluxions may be found. this Equation being refolved as before, the Relations of the other Quantities may be difcover'd. that the Equation propos'd

may

finally

=

z -f- yx o ; that I 55. Let the Equation propofed be zx of the obtain Relation the and Quantities x, y, z, whofe Fluxions may and z are in the contained form I a Relation at x, y, Equation ; pleafure between any a -+- z, or or 2y

=

and thence x

=

two of them,

x=yy,

&c.

as

x and y,

fuppofing that

But fuppofe

at prefent

x=y,

x=yy,

Therefore writing zyy for x, and yy for x, the Equation propofed will be transform'd into this q.yy z-^-yy* o. And thence the Relation between y and z will arife, 2yy-{2yy.

=

:

In which if x be written for yy, and x* for y~>, we mall z. ~x^ So that among the infinite ways in which have 2X -fx, y, and z, may be related to each other, one of them is here found, which is reprefented by thefe Equations, .v =yy, 2y* +- y* z. z, and 2X -+- ^x*

^y=

=.z.

=

=

=

DEMONSTRATION. thus we have folved the Problem, but the DemonftraAnd in fo great a variety of matters, that we behind. and with too great perplexity, from it not derive fynthetically, foundations, it may be fufficient to point it out thus in

56. tion is

may its

And ftill

genuine

by way of Analyfis. That is, when any Equation is propos'd, after you have finifh'd the work, you may try whether from the derived Equation you can return back to the Equation propos'd, by fhort,

And therefore, the Relation of the Quantities in the deI. rived Equation requires the Relation of the Fluxions in the propofed and contrary-wife : which was to be fhewn.

Prob.

Equation, 57. So

=

the Equation propofed were y x, the derived Equal on and the be ; contrary, by Prob. i. we have

if

y={x

tion

will

y

xx, that

is,

y=.x,

becaufe

x

is

fuppofed Unity.

And

thus

from


and INFINITE SERIES.

=

=

4.3

x* -f- Lx 1 tf -+-y -f- xx -+- xy is derived _y ! vS > &c i And thence by Prob. i. y 2x -4T' ^v+ -+- ^o x 1 x! Values of Which two &c. ^-x -V y agree %x> -+- ^-x*

from y

3*

I

-

=

)

with each other, as appears by fubftituting x -^x* xx+^x> ->-J-x , <5cc. inftead of^ in the firft Value. But in the Reduction of Equations I made ufe of an Opera.,8. of which alfo it will be convenient to give fome account. And tion, that is, the Tranfmutation of a flowing Quantity by its connexion Let AE and ae be two Lines indefinitely with a given Quantity. s

extended each way, along which two moving Things or Points may pafs from afar, and at the fame time E c p E may reach the places A and a, B and and let &c. and B C and d, c, b, ^ ? -4 be the Point, by its diftance from which, or of the the Motion moving thing

A

D

'

:

i

AE

is eftimated ; fo that BA, BC, BD, BE, fucceffively, point in the moving thing is in the when be the Quantities, flowing may E. Likewife b be a let like point in the other Line. places A, C, D, BA and ba be will Then contemporaneous Fluents, as alfo if inftead of the BC and be, BD andZv/, BE and be, 6cc. points B and b, be fubftituted A and c, to which, as at reft, the Motions and are refer'd ; then o and ca, AB and cb, AC and o, and will be Therece, cd, contemporaneous flowing Quantities.

Now

AD

AE

fore the flowing Quantities are changed by the Addition and Suband ac ; but they are not changed traclion of the given Quantities as to the Celerity of their Motions, and the mutual refpect of their

AB

For the contemporaneous parts AB and ab, BC and be, CD and cd, DE and de, are of the fame length in both cafes. And thus in Equations in which thefe Quantities are reprefented, the

Fluxion.

contemporaneous parts of Quantities are not therefore changed, notwithftanding their ablblute magnitude maybe increafed or diminimed by fome given Quantity. Hence the thing propofed is manifeft For the only Scope of this Problem is, to determine the contemporaneous Parts, or the contemporary Differences of the abfolute QuanAnd tities f, x, _>', or z, defcribed with a given Rate of Flowing. of what one thofe abfolute magnitude it is all Quantities are, fo that their contemporary or correfpondent Differences may agree with the of the Fluxions. prcpofed Relation reaibn of this matter may alfo be thus explain'd AlThe 59. :

gebraically.

Let the

Equation

G

y=xxy 2

be propofed,

and fuppole


Method of FLUXIONS,

77je

44.

x=

pofe

i

Then by

-+-Z-

Prob.

x

i.

=

So that for y =-. xxy

z.

,

fince ,v=s, it is plain,, that may be wrote y =. xy -h xzy. and z be the not of the fame length, yet that x Quantities though in of y, and that they have equal contemrefpecl: they flow alike therefore may I not reprefent by the fame poraneous parts. in their Rate of Flowing,; and to dethat agree Symbols Quantities

Now

Why

their

termine,

+

=== xy

v

not

why may

contemporaneous Differences, of y initead xxy xxy ?

=

uie

I

appears plainly in what manner the contemporary Lartly be found, from an Equation involving flowing Quantities. parts may ~ -+- x be the Equation, when # Thus if y 2, then _y 24. it

60..

=

But when x 2 to 3,

y

6

3,

2

3

then

=

=.

y

from

3.1.

2-i to

and

i,

3.1.

proceed to

more

2-i

ltijt'1

^

determine the 1.

When

at that

=f

.

thus laid for

what

follows, I fhall

particular Problems.

PROB. A

=

Therefore while x flows from So that the parts defcribed in

3-^

This Foundation being

1.

now

=

will flow

time are

this

=

m.

Maxima and Minima of

H^

is the greateft or the leaft that neither flows backwards or forwards.

a Quantity

moment

it

it

can be,

For

if it

flows forwards, or increafes, that proves it was lefs, and will prethan it is. And the contrary if it flows backwards, fently be greater Wherefore find its Fluxion, by Prob. i. and fuppofe or decreafes. it

to be nothing. 2.

Ex AMP.

greatefl

Value

i.

of,

If in the

Equation x>

x be required

;

ax 1

+ axy

3

jy

=

o the

find the Relation of the Fluxions

l 2axx -f- axy of x and y, and you will have 3X.v a %yy -i-ayx 1 Then making x o. o, there will remain yyy -\- ayx=o, the ax. or 3j* help of this you may exterminate either x By

= = may ax = or

=

y out of the primary Equation, and by the determine the other,

refulting

Equation you

and then both of them by

3^* -f-

o.

you had multiply 'd the Terms of the propofed Equation by the number of the Dimenfions of the other flowing Quantity.^. From whence we may .derive the famous 3.

This Operation

is

the fame,

2.

as

if


and INFINITE SERIES.

45

famous Rule of Huddenius,

that, in order to obtain the greateft or Relate Quantity, the Equation mufl be difpofed according to the Dimenfions of the Correlate Quantity, and then the Terms are to be multiply 'd by any Arithmetical ProgrelTion. But fince neither leaft

Rule, nor any other that

this

tions affected

with

iiird

I

know

yet publiihed, extends to

Equa-

Quantities, without a previous Reduction

j

the following Example for that purpofe. If 2. the greatest Quantity 4. EXAMP. y in the Equation x* xx ay ~+" xx ay~ be to be determin'd, feek the I fhall give

+ 7+--

^

_

= =

.Fluxions of xand^y, and there will ^^v)

1

+ 2^n5__ Aaxxy-\-6x\* + I

a1

-\-

zay

-

2

+j*

^

ay -\- xx

atx 2 -

the Equation 3^^* zayy-{A j r r r And fince by fuppofition y o,

arife

=

\

0.

,

omit the Terms multiply'd by y, (which, to fhorten the labour, might have been done before, in the Operation,) and divide the reft

by xx, and there will remain %x

= %xx =

"*"-'**

^-a"xx

o.

When

Re-

the

made, there will arife ^ay-\o, by help of which of either the or y out of the exterminate x quantities you may prothen from and the refulting Equation, which will, pos'd Equation, duction

is

be Cubical, you 5.

From

this

extract the Value of the other. Problem may be had the Solution of thefe

may

lowing. In a given .Triangle, or in a Segment of any I. Reftangle. ir.fcribe the greatejl

fol-

given Curve,

ft>

II. To draw the greatejl or the leafl right Line, 'which can lie: between a given Point, and a Curve given in pofition. Or, to draw. a Perpendicular to a Curve from a given Point. III. To draw the greatejl or the leajl right Lines, which

pajjin?.-

through a given Point, can lie bet-ween two others, either right Lines or Curves. IV. From a given Point within a Parabola, to draw a rivbt Line, which Jhall cut the Parabola more obliquely than any other. And to do the fame in other Curves.

V. To determine the Vertices of Curves, their greatejl or lealT Breadths, the Points in which revolving parts cut each other, 6cc. VI. To find the Points in Curves, where they hcrce the great ejT or leajl Curvature. VII. To find the Icaft Angle in a given EHi/is, in which the. Ordinates can cut their Diameters. VIII..


The Method of

4.6

FLUXIONS,

that pafs through four given Points, to determine the greateft, or that which approaches neareft to a Circle. IX. 70 determine fuch a part of a Spherical Superficies, which

Of EHipfes

VIII.

in its farther part, by Light coming from a which is refracted by the nearer Hemijphere. great dijlance, and And many other Problems of a like nature may more eafily be propofed than refolved, becaufe of the labour of Computation.

can be illuminated,

R O

P

B.

draw Tangents

To

Firft

IV.

Curves.

to

Manner.

be varioufly drawn, according to the various And firft let BD be a right Relations of Curves to right Lines. Line, or Ordinate, in a given Angle to another right Line AB, as a Bafe or Abat the Curve ED. fcifs, and terminated 1.

Let

Ordinate

this

move through an

inde-

Space to the place bd, fo be increafed by the Moment

finall

finitely

that

may

Tangents

may

it

AB

by the Moment ^ A equal and parallel. Let Da be produced till it meets with AB in T, and touch the Curve in D or d ; and the Triangles dcD, cd,

while

increafed

is

which DC

Bb, to

is

1

So that

fimilar.

TB

it is

BD

:

:

DC

:

(or

B)

:

this

Line will will be

DBT

cd.

of BD to AB is exhibited by the of the Curve is determined feek for the nature which Equation, by Then take TB to BD of the Prob. i. the Relation Fluxions, by of AB the Fluxion to of the Fluxion in the Ratio of BD, and will touch the Curve in the Point D. Ex. i. Calling AB x, and BD =jy, let their Relation be 3. o. And the Relation of the Fluxions will ax* -h axy x _y 2. Since therefore the Relation

;

TD

3

-,

be 3xx-i

2ax ...

w* ~~

^yy* -+-

2axx-i-axy -4-

f!X

and AB, gent

==

TD

ay

:

^

ax

::

BD

ayx-=. (;-)

or v and x, the length is

BT

D

So that y x Therefore BT.

o.

:

::

BT

^xx

=

DB

being given, and thence will be given, by which the Tan-

Therefore the Point

~

:

determined. 4-


and INFINITE SERIES. 4.

But

this

47 Make

Method of Operation may be thusconcinnated.

the Terms of the propofed Equation equal to nothing multiply by the proper number of the Dimenfions of the Ordinate, and put the Then multiply the Terms of the fame Refult in the Numerator Equation by the proper number of the Dimenfions of the Abfcifs, and the Produdl divided by the Abfcifs, in the Denominator of the :

:

put Value of BT.

Then way

BT

take

but the contrary

towards A, if its Value be affirmative, Value be negative.

if that o

5.

Thus

the

Equation*

-f-

Numbers,

being multi-

axy

z

3

ply'd by the upper

13y*=o, 10

o

ax*

3

for the

3

gives axy

3_y

Numerator

j

and multiply 'd by the lower Numbers, and then divided by x, gives 1 zax -+- ay for the Denominator of the Value of BT. 3-x-

=

Thus

the Equation jy 3 o, (which by* cdy -f- bed -\-dxy denotes a Parabola of the fecond kind, by help of which Des Cartes 6.

confirufted Equations of fix Dimenfions

fee his

;

7.

And

r

thus a 1

whofe Center

is

-x*

y

A,) gives -

1

^X

= ,

o,

or

Geometry,

^--"fr+'^v

Amfterd. Ed. An. 1659.) by Infpeftion gives

=

42. Qr

^

(which denotes an

^

p.

BT.

And

Ellipfis

fo in others.

1

And you may

take notice, that

it matters not of what quantity be. the Angle of Ordination may 9. But as this Rule does not extend to Equations afFefted by furd Quantities, or to mechanical Curves ; in thefe Cafes we mufl have

8.

ABD

recourfe to the fundamental 10.

Ex.

2.

Let

A;

Method. 1

S

ay

xx

-+- j-

\/'ay -+-

xx

Equation exprefling the Relation between

AB and BD

the Relation of the Fluxions will be

3***

=0.

o be the

and by Prob. i.

::

(y

:

+ ~

*"*"*

zayy

-f. */,.,,,

Therefore

T^Tp ^^ fay

;

=

it

x

will be <ixx

::)

BD

:

4

v

2

V

BT.

II.


48

Method of FLUXIONS,

TJoe

ED

ii. Ex. 3. Let be the Conchoid of Nicomedes, defcribed with the Pole G, the Afymptote AT, and the Diftance and let ;

LD

lar

=

=

LD And becaufe of fimic, AB=.v, andBD=;>. BD DM MG Triangles DEL and DMG, it will be LB

'GA

,

: :

:

that

is,

=yx. thus

I

v/'cc

x

:

:

:

:

;

b -+- y, and therefore b-\-y ^/cc

y cc Having got this Equation, I fuppofe yy fliall have two bz andzz Equations ~\-yz =yx, yy

:

==

V

yy and z,

cc

yy.

By the help of thefe I find the Fluxions of the Quantities x, y, and From the firft arifes bz -+-yz -\- yz =y'x -+- xy, z, by Prob. i. and from the fecond 2zz o. Out of 2yy, or zz -j- yy

=

thefe if

-+ xy, (y

:

x

we

exterminate z, there will

which being ::)

BD

But

BD AL -f-

as

=

BT

will be

it

-^

arife

is

y

y,

-

-i-yz

z

:

=yx

-

therefore

~

x

BT=

BT

is,

!-

J_l

:

:

.3-

BL -; where the Sign denotes, that the Point T mufl be taken contrary

iff

prefixt

refolved

BT.

:

That to

=

(_J

to

the Point A. 12.

SCHOLIUM.

point of the

And hence it appears by the bye, how that Conchoid may be found, which Separates the concave

from the convex

then v

x the work, for is

before,

part.

x

and

make

=

fubftitute it

will

AT AT

For when

Therefore

be that point.

-

be

z

-+-

^l!5 >

-

?

-f.

=

2K

is

v

;

and fmce

by -\- yv

-+-

Value -

Fluxions v, y, and z being found by Prob.

-

i.

is

BT

Here

-+-

w hich z

the lea ft poffible,

derived

=

v.

to

D -

will

z

morten

from what

Whence

and fuppofing

the

^=0


and INFINITE SERIES. )K --Azy-4-zvy --- ~-t-z --=i; + iy-l-zyy z za '

iy,

...

.,,

-3. there will anfe bvProb. J J

jy

y

:

fubftituting in this

Laflly,

z and zz

values of

for z,

-

and

cc

had from what goes

are

49

=

o.

yy for zz, (which

=

before,)

and making a

o. -2.be* due Reduction, you will have y'- -+- ^by* By the ConThen thro' ftrudlion of which Equation y or AM, will be given. of fall the Point to it will AB, drawing parallel upon

MD

M

D

contrary Flexure. if the Curve be Mechanical whofe Tangent is to be 13. of the Quantities are to be found, as in Examp.5. Fluxions the drawn, and then i. the reft is to be perform'd as before. of Prob. and Ex. 4. Let be two Curves, which are cut in

Now

AD

AC

14.

C

the Points

BCD,

D

and

by the right Line in a to the Abfcifs

apply 'd Let given Angle.

-=

and

Preparat. to

AB

z.

Examp.

=

x,

Then 5.)

it

AB BD

= z = x y,

(by Prob.

will be

i.

~T>

^ ^

B~

xBC.

Now

15.

AC

let

be a Circle, or any

mine the other Curve AD, z is involved, as zz +- axz

known Curve

and

;

to deter-

any Equation be propofed, in which

let

Then by

2zz +- axz And writing x x BC for z, it will be zxz x BC -+- axz 4X7*. Therefore 2z x BC -+- ax x BC -{-+- axx x BC H- axz 4)7'. BD BT. So that if the nature of the x ::) :: az (y 4jy Curve AC be given, the Ordinate BC, and the Area ACB or z ; the Point T will be given, through which the Tangent DT will

=

pafs. 1 6.

AD

(y

:

x

Prob.

i.

:

:

After the fame manner, if

Curve

.

=

J

:

=_y 4

::)

be (3.3) 3^ x

'twill

;

BD

BT.

:

And

= BC = 32

zy be the Equation to the zy.

So that

3BC

:

2

::

fo in others.

Ex. 5. Let AB=,v, BD =y, as before, and let the length 17. of any Curve AC be z. And drawing a Tangent to it, as Cl, 'twill be Bt

::

x

z, or

:

z

=

x x C/

^-

Now

for determining the other Curve AD, whofe Tangent to be drawn, let there be given any Equation in which z is in-

18. is

Ct

:

volved, fuppofe

(y : gent

x

:

DT

:}

:

z

BD

==)'. :

BT.

may be drawn.

Then

it

will be

But the Point

H

z=y, fo that Ct T being found, the

:

Bf

''

Tan19-


The Method of Thus fuppofmg xzsssyy,

19.

^

z writing

for

y->

*

fore

-I-

Ex

20.

:

f~-'

BD

: :

AC

xz

KZ

be

'twill

there will arife

27

Let

6.

.

FLUXIONS,

-f-

+ zx = zyj

^-^ O/

=

and

>,

ayy. ''

There-

known

Curve,

DT.

:

be a Circle,

or

other

any whofe Tangent is Ct, and let AD be any other Curve whofe Tangent DT is to be drawn, and let it be defin'd by afTuming AB to the Arch AC ; and (CE, BD

=

being Ordinates to AB in a given Angle,) let the Relation of BD to CE or be

AE

exprels'd

by any Equation. 21. Therefore call AB or

= And AC, AE, = ^= Now = Then = z

CE

v

is

it

.

and

C7

i>,

22.

as

are^to

and

let

y

.

BD

:

23.

BT. Or

let

=

BD =y, AE=z,

x,

z.

any Equation be given to define the Curve AD, and therefore Et ; y Ct :: (v x "') K :

the Equation be r~>T? .

I

TT-

.

yz+vx,

:

Or

24.

::

(y

:

x

finally,

:

::)

BD

:

BT.

the Equation

let

= (3^ =)

zayy

BD

Ct

3*1;' x

.

and

it

will

be

y-.

And Ct

and

plain that v, x, and z, the Fluxions of CE, each other as CE, Therefore *x Ct, and Et.

x

.v

AC

T

t.

be ayy

So that

CE

therefore

=

31;*

v* y and

x

CE

it

-4-

Et

will be

2 ay x Ct

:

::

BT.

FC

be a Circle, which 25. Ex. 7. Let be a and let Curve, which is defined by affuming any Relation of the to the Arch FC, which is Ordinate

FD

is

touched by

CS

in

C;

DB

intercepted by

DA

Then

fall

letting

the Circle,

=

CF

/;

call

and

AE

is

AC

or

AF=i, AB

tz=(t^=) tv = (/x -^ =) z. Here I and .

v,

drawn to the Center. CE, the Ordinate in

it

.

T

KB

will be

,S

^..

dirninifh'd while

EC

is

increafed.

put z negatively, becaufe

And

befides

AE

:

EC AB

:: :


and INFINITE SERIES. AB

vx

tx*

xv.

-f-

= Now

zy

vx,

xy.

26.

let

pofe

=

/

x

::

(

ty*

be defined by any Equation, from

y

may be

t

that

fo

y,

_x

:

:

yx

yx*

yy*

BD(;')

= xx+yy =

AT

x; and

-

yx

-f-

derived, to be fubftituted here. Supfirft to the and i. Prob. (an Equation Quadratrix,) by

^=_y, be

will

DF

the Curve

let

which the Value of it

=

and thence by Prob. i. zy Then exterminating v, z, and v, 'tis yx

fo that

BD,

:

51

:)

if 27. After the fame manner, 6r, and thence

=

AT= x~ /

=

//

is

Whence y

xy.

Therefore

ADa ^/.

it

z/

=

BT.

:

:

xx

=

x*

there will arife

ly,

And

.

BT

of others.

fo

r

be taken equal to the Arch FC, the the fame names Curve being then the Spiral of Archimedes ; of the Lines ftill remaining as were put Becaufe of the right Angle afore 'tis xx -{-yy=tf ) and therefore (by Prob. i.)

Ex.

28.

Now

8.

AD

if

ADH

ABD

:

=

xx +yy DB CE,

Tis

//.

AD

alfo

AC

:

:

:

tv=y znd thence (by Prob. i.) tv -4- vf =y. Laftly, the Fluxion to the Fluxion of the of the Arch FC to AE, or as AD as Line AC CE, right :

fo that

t

is

to

that

AB,

ix

=

'tis

tv

fore

QP_ take

vf.

t

is,

v

:

:

thence

compleating the

(BD

AP

29.

=

:

BT !

::

>

x,

and thence

xx -\-yy

now

=

Parallelogram

y

PD

;

And from

:

the Equations

Compare

-+-ix=y, and

::

t

:

hence

x

:

::)

X

(tt

found, and you will fee

=)

ABDQ^_, :

y

^^ if

^

.

And there-

you make

;

that

is,

QD if

:

you

will be perpendicular to the Spiral. (I

imagine)

it

will

be fufficiently manifeft,

of all fcrts of Curves are to be by what methods the Tangents be not it However drawn. foreign from the purpofe, if I alfo may be fliew how the Problem may perform'd, when the Curves are reSo that havfer'd to right Lines, after any other manner whatever of feveral Methods, the eafieft and moil fimple may ing the choice :

always be ufed.

H

2

Second


The Method of

$2

FLUXIONS,

Second Manner.

D

be a point in the Curve, from which the Subtenfe be anOrdinate in any given is drawn to a given Point G, and let AB. the let the to Abfcifs Angle flow for an infinitely fmall fpace Point let Gk be D^/ in the Curve, and in taken equal to Gd, and let the Parallelogram dcBl> be compleated. Then Dk and DC will be the contemporary Mo30. Let

DG

DB

Now

D

GD

---

ments of

GD

and BD, by which they

are diminifh'd while

D

~Dd be produced,

till it

meets with

the Subtenfe

let fall

Dcdk and

GD

DBTF will be

Now

transfer'd to d.

the right Line to T, and from the Point the perpendicular TF, and then the Trapezia is

like;

AB

let

T

in

and therefore

BD

DB

:

DF

::

DC Dk. :

GD

the Relation of to is exhibited 31. Since then by the find the Curve the for of Relation the ; determining Fluxions, Equation in the Ratio of the Fluxion of to and take to the Fluxion of BD. Then from F raife the perpendicular FT, which in T, and being drawn will touch the may meet with muft be taken towards G, if it be affirmative, Curve in D. But

GD

DB

FD

DT

AB

DT

and the contrary way 32. Ex. i. Call ax 1 -f- axy be x

if negative.

==

and

BD

o.

Then

+-

-f-

GD

x,

~,

will be

^xx zax -h ay ~

.'

V

axy .

and

From whence

alfo.

concourfe

T

::

ayx

(y

Then any

,

BD

axy

ax

^yy

:

1

and thence

y"=

2axx

1

GD if

with the

=_>', and let their Relation the Relation of the Fluxions

:

x

^yy:

Point

:)

=

DB

D in

Therefore

o. :

DF.

^xx

So that

(y) the Curve being given,

the Point F will be given be raifed, from its Perpendicular Abfcifs AB, the Tangent may be or

y and

x,

the

FT

DT

drawn.

And might appears, 33 For having difpofed all the as in the former Cafe. that a Rule

be derived here, as well Terms of the given of the Ordinatejy, the one Dimensions on fide, multiply by Equation of Numerator a Fraction. Then multiply in the refult the and place of Subtenfe the its Terms feverally by the Dimenfions x, and dividing the refult by that Subtenfe x, place the Quotient in the DenotoAnd take the fame Line minator of the Value of DF. Where otherwile the if it be wards affirmative, contrary way..

hence

.

it

DF

G

you


and IN FINITE SERIES, you may obferve, that it is from the Abfcifs AB, or

is

no matter how if

it

be at

53

all diftant,

Angle of Ordination ABD. be as before x* 34. Let the Equation

G

Point nor what is the

far diftant the

ax*

=

3

J axy 3 for the Numerator, and 3** it axy 3>' gives immediately of DF. -+- ay for the Denominator of the Value 35. Let alfo a -+it

Sedtion,)

-x~y=o, for the

y

gives

-f-

(which Equation

is

;

Conick

to a

for the

Numerator, and

o

2ax

Denomi-

fly

DF, which

nator of the Value of

And

therefore will be

7

thefe things will be

thus in the

Conchoid, (wherein 36. perform'd more expeditioufly than before,) putting

= GA =

(5)

cb is,

GD=x,

c, :

GL

(x

<:).

BD=^,

will be

it

Therefore xy

cy

BD =

:

<r=DF.

Therefore prolong

GD

DL

(;) cb, or xy

This Equation according to the Rule gives

o.

x

and

=

GA

to F,

fo

^-^

(c)

::

cy

that

-

,

that

b,

DF

=

LG, and at F raife the perpendicular FT meeting the Alymptote AB in T, and DT being drawn will touch the Conchoid. or furd Quantities are found in the 37. But when compound to the general Method, mufl recourfe have except you Equation, you fliould chufe rather to reduce the Equation.

38. Ex. 2. If the Equation

yy =zyx, were gven the ; (fee foregoing Figure, find the Relation of the Fluxions by Prob. i. As fuppoiing 52.) p. will z have the bz v/ff ) -+- yz yx, and )')' you Equations

for the Relation

between

GD

xv/cr

and

BD

=

cc

=

yy=.zz, and

=

yx

-f-

yx,

thence the Relation of the Fluxions bz-\-yx

and i

2yy=2Z,z.

And now

z,

and z being exter-


T&e Method of

FLUXIONS,

=

v

exterminated, there will arife

Therefore y

^/cc

:

\/ cc

v

J2^

yy

'Jjl U

yy .v

::

:

(y

,v

\x

BD

::)

1

(ji

)

:

xy.

DF.

Third Manner.

two Subtenfes

39. Moreover, if the Curve be refer'd to

A

BD, which being drawn from two meet

D

given Points the Curve: Conceive that Point

at

flow on through an infinitely Space Del in the Curve ; and in to

BD

Ak

take

=

Ad, and Bc

= AD

the Ratio

DF

therefore

BD

to

Moment D&

of the

Moment

and B, may

and and

Bc/;

AD

Take

and

little

then kD and cD will be contemporaneous Moments of the Lines and

BD.

AD

-

in

/r

the

to

Ratio of the Fluxion of the Line DC, (that is, to the Fluxion of the LineBD,) and draw BT, FT perpendicular to BD, AD, meeting in T. Then the Trapezia DFTB and DM: will be fimilar, and therefore the will touch the Diagonal the

in

AD

DT

Curve. 40. Therefore from the Equation, by which the Relation is defined between and BD, find the Relation of the Fluxions by Prob. i. and take in the fame Ratio. to

AD

FD

41.

Ex AMP.

AD

Suppofing

=

e

a

tion be

BD

=

andBD=;',

x,

let

their Rela-

o. This Equation is to the Ellipfes of y the fecond Order, whofe Properties for Refracting of Light are fhewn by Des Cartes, in the fecond Book of his Geometry. Then the -f-

j

e Relation of the Fluxions will be -

d ::(>:# 42. e

And

_d

:

BD

::)

for

BD

: :

:

'Tis therefore e

DF.

the fame reafon

DF.

:

y ==o.

In the

if

a

^

Cafe take

firft

y

=

DF

o,

'twill

:

be

towards A, and

contrary-wife in the other cafe. 43. COROL.

comes

DF

a

i.

Hence

if

Conick Section,)

= DB.

DFT

angles the Angle

And and

FDB

the Tangent.

d-=.e,

(in

which

cafe

the Curve be-

be therefore the Tri'twill

DBT will

being equal, be bifected by v

-K

A 44.


and INFINITE SERIES.

55

And hence alfo thofe things will be manifeft of 44. COROL. 2. themfelves, which are demonstrated, in a very prolix manner, by Des Cartes concerning the Refraction of thcfe Curves. For as much and DB, (which are in the given Ratio of d to e,) in refpect as

DF

DTF

and DTB, of the Radius DT, are the Sines of the Angles that is, of the Ray of Incidence upon the Surface of the Curve, and of its Reflexion or Refraction DB. And there is a like reafon-

AD

of the Conick Sections, fuppofing ing concerning the Refractions or B be conceived to be at an infinite of the Points that either

A

diftance.

of the 45. It would be eafy to modify this Rule in the manner Curves of it when and more alfo to As Examples foregoing, give are refer'd to Right lines after any other manner, and cannot commodioufly be reduced to the foregoing, it will be very eafy to find out other Methods in imitation of thefe, as occafion mall require. :

Fourth Manner.

As

46.

if the right

Line

B, and one of its Points Point C fhould be the interfection of the right Line BCD, with another right

Line

pofition. lation of

AC

BCD

D

mould revolve about a given Point mould defcribe a Curve, and another

given in

Then the Re-

BC

and

BD be-

ing exprefs'd by any Equation ; draw BF parallel to

meet DF, perpendicular perpendicular to DF; and take FT

AC,

fo as to

erect

FT

BC,

that the Fluxion of

BD

to

BD,

in F.

Alfo

fame Ratio to has to the Fluxion of BC. Then DT in the

being drawn will touch the Curve. Fifth Manner.

A AC

47. But if the Point being given, the Equation ihould exprefs the Relation between and } draw parallel to DF, and in the fame Ratio to BG, that the Fluxion of take has to

BD

CG

FT

BD

the Fluxion of AC. Sixth Manner.

Or

AC

again, if the Equation exprefles the Relation between let and meet in and take in the fune ; Ratio to BG, that the Fluxion of has to the Fluxion of AC. A. id

48.

and

CD;

AC

H

FT

HT

CD

the like in others.

Seventh


Method of FLUXION

*fhe

Manner

Seventh

The Problem

For

:

Spirals.

not otherwise perform'd, when the Curves 49. are refer'd, not to right Lines, but to other Curve-lines, as is ufiial in Mechanick Curves. Let BG be the Circumference of a Circle, in whole Semidiameter AG, while it revolves be conabout the Center A, let the Point is

D

ceived to

move any how,

ADE.

fo as to defcribe the

And

fuppofe ~Dd to be an inof the Curve thro' which finitely part take Ac flows, and in Ad, then cD and Gg will be contemporaneous Moments of the right Line and of the Periphery BG. Therefore draw Af parallel to cd, that is, perpendicular to AD, and let the Tangent meet it in cd then it will be cD ; AT. Alfo let Gt be parallel to the Tangent DT, and it will be cd At. ::) Gg :: (Ad or Therefore 50. any Equation being propofed, by which the Relation is exprefs'd between BG and find the Relation of their ; in the fame Fluxions by Prob. i. and takeAi? Ratio to AD: And then Gt will be parallel to the Tangent. let their Relation be x, and 51. Ex. i. Calling EG 5 A: 3 ax 1 -f- axy and i. Prob. zax-\- ay : 3^* o, by 3^* jy

Spiral

=

little

D

AD

AD

DT AD

T

: :

:

:

AD

:

AG

:

AT

:

AD

==

ax

: :

Gt, and

(y

:

DT

x

:

:)

AD

to parallel

52. Ex. 2. If

'tis

AD=^,

At.

:

The

which

it,

y =y>

Point

/

being thus found, draw

will touch the Curve.

(which

is

the Equation to the Spiral

=

: of Archimedes,} 'twill be j y, and therefore a : b (y : x : :) be produced to P, At. Wherefore by the way, if be be AB :: a : b y PD will that it may perpendicular to the Curve.

AD

TA

:

AP

53.

A.

:

Ex.

And

3.

If

xx

:

=

=

AD

then 2XX b :: by, and 2x be to drawn eafily may any Spirals what-

by,

thus Tangents

:

:

ever.

Eighth


and INFINITE SERIES.

57

Eighth Manner : For Quad ratr ices. CA. Now if the Curve be fuch, that any Line AGD, being drawn from the Center A, may meet the Circular Arch inG, and the Curve in D; and if the Relation between the Arch BG, and the right Line DH, which is an Ordinate to the Bafe

AH

in a given Angle, be determin'd by any Equation to whatever Conceive the Point infinitean move in the Curve for

or Abfcifs

D

:

to d, ly {mail Interval

dhHk

rallelogram ed,

=ArchADBG produce

and the Pa-

Jf

being compleat-

Ad

to

c,

fo

that

and D/' will be contemporaneous Moments of and of the Ordinate DH. Now produce Dd ftrait the on to T, where it may meet with AB, and from thence let fall Then the Trapezia Dkdc and DHTF the Perpendicular TF on DcF. therefore and D/fc DF. And befides DC :: will be fimilar; if Gf be raifed perpendicular to AG, and meets AF in f; becaufe of the Parallels DF and Gf, it will be DC Gg :: DF Gf. There-

Ac

then

;

Gg

DH

:

:

:

fore ex aquo,

D

'tis

:

G^

: :

DH

DH

Gf, that

:

:

as the

is,

Moments

or

and BG. Fluxions of the Lines the Equation which exprefies the Relation of 55. Therefore by BG to DH, find the Relation of the Fluxions (by Prob. i.) and inDraw that Ratio take Gf, the Tangent of the Circle BG, to DH.

DF

parallel creel the

which may meet A/* produced

to Gf,

And

F.

in

at

perpendicular FT, meeting AB in T; and the right being drawn, will touch the Quadratrix. i. Making Ex. x, and DH=;', let it be xx fy; 56. b :: Therefore x 2.x : then (by Prob. i.)2xx (y by. ::) and the Pointy being found, the reft will be determin'd as above.

F

DT

Line

EG

=

=

=

:

DH

:

GJ; But perhaps this Rule may be thus made fomething neater Make x :y :: AB AL. Then AL AD :: AT, and then the For Curve. touch becaufe of DT will equal Triangles AFD and AT x DH, and therefore AT AD (DF or ATD, 'tis AD x :

:

AD

:

DF=

JB

x 57.

Gf

:

DH

Ex.2. Let

or 1 G/::)

:

:

AD

:

f-

AG

or)

:

:

AL.

x=y, (which the Equation to the Quadratrix #=v. Therefore AB AD AD AT. is

of the Ancients,) then

:

I

::

:

8.


*fhe

58 58. 3;-*

:

Ex.

zax

:

3.

Let

(x

:

:

y

Method

axx=y*, :

:)

^FLUXIONS,

then

zaxx=sMy*.

AB AL. Then AL AD :

:

: :

Therefore make

AD

:

AT.

And

thus you may determine expeditioufly the Tangents of any other howfoever Quadratrices, compounded.

Ninth Manner.

ABF be any given Curve, which Line and Bt a part BD of ; right the right Line BC, (being an Or59. Laftly, if

is

touch'd by the

dinate in any given Angle to the Abfcifs AC,) intercepted between

and another Curve DE, has a Relation to the portion of the Curve AB, which is exprefs'd by any Equation: You may draw a this

Tangent DT to the other Curve, by taking (in the Tangent of this Curve,) BT in the fame Ratio to BD, as the Fluxion of the Curve right Line 60. Ex.

therefore

^ AB

^

<f-

hath to the Fluxion of the

BD. i.

ax

Calling

=

zyy.

6j. Ex.2. Let

be a Circle,) then

AB ==x, and BD =y-

t

Then a

:

zy

:

(y

:

x

:

:)

BD

be ax==yy, and :

BT.

(the Equation to the Trochoid, if

^#==7,

fX=y

:

let it

t

and a

:

b

::

BD

:

ABF

BT.

62. And with the fame eafe may Tangents be drawn, when the Relation of to AC, or toBC, is exprefs'd by any Equation; or when the Curves are refer 'd to right Lines, or to any other Curves, after any other manner whatever.

BD

63. There are alfo many other Problems, whofe Solutions are to be derived from the fame Principles ; fuch as thefe following. I. To find a Point of a Curve, where the Tangent is parallel to the or to any other Abfcife, right Line given in pofition ; or is perpendicular to it, or inclined to it in any given

Angle.

To find the Point where the Tangent is moft or leajl inclined to the Abfcifs, or to any other right Line given in That is, to find 'pofition. the confine of contrary Flexure. Of this I have already given a Specimen, in the Conchoid. III. From any given Point without the Perimeter of a Curve, to draw a right Line, which with the Perimeter may make an Angle of II.

Contact.


and IN FINITE SERIES.

59

that is, from Contaft, or a right Angle, or any other given Angle, or to draw or a given Point, right Lines 'Tangents, Perpendiculars^ that Jhall have any other Inclination to a Curve-line. IV. From any given Point within a Parabola, to draw a right Line, which may make with the Perimeter the greateji or leaft Angle And Jb of all Curves whatever. poj/ible. V. To draw a right Line which may touch two Curves given in or the fame Curve in two Points, when that can be done. pojition,

VI. To draw any Curve with given Conditions, which may touch another Curve given in pojition, in a given Point. VII. To determine the RefraSlion of any Ray of Light, that falls upon any Curve Superficies. The Refolution of thefe, or of any other the like Problems, will not be fo difficult, abating the tedioufnefs of Computation, as that And I imagine if there is any occalion to dwell upon them here have mention 'd to Geometricians more to be barely may agreeable :

them. ;

P At any given

R O

B.

:

V.

Point of a given Curve^ Quantity of Curvature.

to

find the

There are few Problems concerning Curves more elegant than c that give a greater Infight into their nature. In order to its or this, Refolution, I mufl: premife thefe following general Confederations. 2. L The fame Circle has every where trie fame Curvature, and 1.

it is reciprocally proportional to their Diameters. If the Diameter of any Circle is as little again as the Diameter of If another, the Curvature of its Periphery will be as great again. be of the will be the one-third Curvature thrice the Diameter other,

in different Circles

as

much, &c. 3.

II.

If a Circle touches

any Curve on

its

concave

fide,

in

any

be of fuch magnitude, that no other tangent be interleribed in the Angles of Contact near that Point ; can Circle that Circle will be of the lame Curvature as the Curve is of, in that given Point, and

if

it

For the Circle that conies between the Curve Point of Contact. and another Circle at the Point of Contact, varies lefs from the Curve, and makes a nearer approach to its Curvature, than that other Circle does.

And

therefore that Circle approaches nea'-eil to its I 2 Curvature,


60

*fbe

Method of FLUXIONS,

which and the Curve no other

Curvature, between

Circle can

in-

tervene. 4. III.

Curve, dius

is

or

Therefore the Center of Curvature to any Point of a the Center of a Circle equally curved. And thus the Raof the Perpendicular Semidiameter of Curvature is part

to the Curve,

which

is terminated at that Center. the 5. proportion of Curvature at different Points will be known from the proportion of Curvature of aequi-curve Circles, or from the reciprocal proportion of the Radii of Curvature. 6. Therefore the Problem is reduced to this, that the Radius, or

IV.

And

Center of Curvature

may

be found.

the Curve 7. Imagine therefore that at three Points of Peipendkulars are drawn, of which thofe that are

D and ^ meet in H, and thofe that are at D and d meet in h And the Point D being in the

<f ,

D, and

d,

at

:

/

middle, Curyity at the part Dj^ But than at DJ, then will be lefs than db. are and how much the dh /H by Perpendiculars nearer the intermediate Perpendicular, fo much the lefs will the diftance be of the Points and h : And at laft when the Perpendiculars meet, thofe Points will coincide. Let them coincide in the Point C, then will C be the Center of Curvature, at the if there is a greater

DH

H

D

of the Curve, on which the Perpendiculars ftand ; which is manifeft of itfelf. 8. But there are feveral Symptoms or Properties of this Point C', which may be of ufe to its determination. 9. I. That it is the Concourfe of Perpendiculars that are on each lide at an infinitely little diftance from DC. 10. II. That the Interfeftions of Perpendiculars, at any little finite diftance on each fide, are feparated and divided by it ; fo that thofe which are on the more curved fide D,f fooner meet at H, and thofe which are on the other iefs curved fide -Dd meet more remotely Point

at h.

DC

be conceived to move, while it infifts perpendicularly on the Curve, that point of it C, (if you except the motion of approaching to or receding from the Point of Influence C,) will be leaft moved, but will be as it were the Center of Motion. 12. IV. If a Circle be defcribed with the Center C, and the diftance DC, no other Circle can be defcribed, that can lie between 11. III. If

at the Contact.


and INFINITE SERIES. n. V.

61

the Center II or b of any other touching Circle degrees to C the Center of this, till at la it it coif

Laftly,

approaches by

then any of the points in which that Circle mall with the point of Contact D. cut the Curve, of each thefe Properties may fupply the means of folving 14. And But we fliall here make choice of the the Problem different ways incides with

'it

;

will coincide

:

being the moit fimple. At any Point D of the Curve let DT be a Tangent, DC a 15. and C the Center of Curvature, as before. And let Perpendicular, AB be the Abfcifs, to which let DB be apply 'd at right Angles, and which DC meets in P. Draw DG parallel to AB, and CG perin which take pendicular to it, Cg of any given Magnitude, and draw gb perpendicular to it, which as

firlt,

meets

DC

Cg gf :

: :

in

Then

<T.

(TB BD :

:

it

will

be

the Fluxion :)

of the Ablcifs, to the Fluxion of Likewife imagine the Ordinate. to move in the Curve the Point an infinitely little diftance Dd, and

D

drawing de perpendicular to DG, and Cd perpendicular to the Curve, let Cd meet DG in F, and $g in/ Then will De be the Momentum of the Abfcifs, de the Momentum of the Ordinate, and J/ the contemporaneous Momentum of the right Line g. Therefore DF

-De^.^t

Having therefore the Ratio's of thefe Moments, or, which is the fame thing, of their generating Fluxions, you will have the Ratio of CG to the given Line C^, (which is the fame as that of DF to Sf,) and thence the Point C will be determined. z 16. Therefore let AB x, BD =y, Cgi, and g .

'

LJC

=

then

it

will

be

i

:

z

:

x

:

:

y, or

mentum

S-f of z be zxo, (that of an and infinitely fmall Quantity

Dt'==xxo, de=yx.o, and 'tisQ-(r) xx

:

CG

::

(Jf

:

DF

z

::)

=

o,}

zo

r-

=

Now

.

let

the

;

Mo-

X

the Product of the Velocity and therefore the Momenta

is,

thence

=

*

DF :

=

xo

.\o

-f-

+^

.

.

Therefore

X

That

is,

CG=

\y

J

7-


7%e Method of FLUXIONS, And whereas we are at liberty to afcribe whatever

62

Velocity of the Abfcifs x, (to which, as to an Fluxion the pleafe the reft may be referr'd j) make x i, and equable Fluxion, z -^. '} and And thence then J z,' and y 17.

we

to

=

=

CG

'-^

=

DG

.

=

any Equation being propofed, in which the Relation of BD to AB is exprefs'd for denning the Curve ; firft find the Relation betwixt x and y t by Prob. r. and at the fame time fubThen from the Equation that arifes, ftitute i for ,v, and z for y. 18. Therefore

by the fame Prob.

between

find the Relation

i.

and z, and

#, y,

at

for x, and z for y, as before. And thus the will obtain Value former of z, and by operation you by the the latter you will have the Value of z ; which being obtain'd, pro-

the fame time fubftitute

duce

may

DB

-

-

make 19.

Ex.

TM-T

DH==

a,

DC in C D of the

PT

is

o,

DP

=;o

;

)

there will arife (by Prob.

for x,

l

and z

for

y

zb

hence again there arifes wrote for ,v and y.) By the

z

=

2zz firft

2zy

we have

^^

Therefore any Point

a +. zbx

i.)

in the refulting Equation,

which otherwife would have been ax -+ 2&xx

latter

AB, and meet-

parallel to

1 the Equation ax^-hx* y being proan Equation to the Hyperbola whofe Latus redtum

(writing

i

it

z

and Tranfverfum 2

2zy

that

then will C be the Center of CurOr fince it is i -|- r.y. -7 Curve.

Tk/->

or

'

j

HC

of the Curve,

Thus

i.

pofed, (which is

and draw

,

ing the Perpendicular vature at the Point

PT

the concave part

H, towards

to

DH =

be

i

=

= z z= L^L zyy

o, (i

and

o

;)

and

being again

C

}

an( j

by tne

D of the Curve being given,

and confequently xand y, from thence z and z will be given, which 7 GC or DH, and draw HC. being known, make Z

20.

As

if

definitely

=

you make

=

3,

b=i,

and

be the condition of the Hyperbola.

xx=yy may aliume x=i, ^11^

=

2,

z=, z=

li being found, raife the Perpendicular

HC

9

T T , and

fo that

And

3#-fif

DH=

you gL.

meeting the Perpendicular


and IN FINITE SERIES.

DC

cular

HC

::

before

drawn

z

i

:

(i

::)

63

or, which is the fame thing, make HD Then draw DC the Radius of Curva:

;

:

.

ture.

When you think the

21.

may

z and z

fabfHtute the indefinite Values of

Thus

CG.

be too perplex, you

will not

Computation

-

into

the

,

Example, by a due Reduction Yet the Value of DH by -jyou will have Calculation conies out negative, as may be feen in the numeral Exmufl be taken towards B ample. But this only fhews, that for if it had come out affirmative, it ought to have been drawn the Value of

DH

in the prefent 4

=y

'

S

^

r*

.

DH

contrary way. 22. COROL.

changed, that Ellipfis,)

--

thence

o,

let

be

may

the Sign prefixt to the Symbol -\-b be -bxx yy=zo, (an Equation to the

ax

DH=;--f- ilLll^: fuppofing b=. o, that the

then

23. But

yy

Hence it

;

.

may become ax

Equation (an Equation to the Parabola,) then

DG

=

DH

\a

24. From that the Radius

=

y

~

-f-

;

and

2X.

-f-

thefe feveral Exprefilons

may

it

eafily

be concluded,

Conick Seftion

of Curvature of any

is

always

aa 2. If xy- be propofed, (which is the Equa25. Ex. l tion to the CiiToid of Diodes,") by Prob. i. it will be firft T> x =.2azy 2xzz 6x zxzy 2azy-+-2azz -2zy y-t and then

x*=ay*

zxzy 2Z\ J So :

1

that

z=

=

-3*x

-4- yy

z=

,

3-^. and 2.vy'

zay

a%z

T.X

-

^ 2cv+ *~~ -4-

ay

n-.!

There-

.

xj

thence .v and any Point of the Ciflbid being given, and there will be given alfo & and z, ; which being known, make

fore

26. Ex.

_

_

= CG.

3.

If b-jf-y^/cc

yy

K

=.vy

were given, (which

make \/cc Equation to the Conchoid, inpag.48;) there will arife

=

hi) -+-

=

yv

Now

xy.

Prob. i.) vv,) will latter and the give l>v -+-yv will give (by

+

2yz

zv

v and z

Equations rightly difpofed may alfo be found; out of the i>,

by

fubilituting

^

,

y,

laft

the

=

=y

firft

is

the

y\=zv, and

of thele, (cc

z

for

_vv v

2vv, (writing ;) thefe And from xz. -{-

will be determined.

But

that

z

Equation exterminate the Fluxion

and there will

arife

7

-I- ~"^


Method of FLUXIONS,

=

y -f- xz, an Equation that comprehends the flowing Quantities, without any of their Fluxions, as the Refolution of the firft Problem requires. Hence therefore by Prob. i. we mall have

^2*

Ijzv

byz

2)zs

"\vzv

)?

+-

This Equation being reduced, and difpofed zz But when z and z are known, make + '

27. If

by Prob. 2

i

.

divided the

=

in order,

=

2Z +- XZ.

will give z.

CG.

Equation but one by z,

laft

-

we mould have had

which would have been

;

^,

we had

ZV

a

-f-

^

-f-

---

then -i;

-f.

=

more fimple Equation than the

former, for determining z. 28. I have given this Example, that it may appear, how the operation is to be perform'd in furd Equations: But the Curvature of the Conchoid may be thus found a fhorter of the The

Equation b -\-y ^/cc there arifes

~

v\'

=

*"

-f.

^

way.

parts

xy being fquared, and divided by yy, x*, and thence by Prob. i. 2by y*

=

x

or

And the

hence again by J Prob. *^

z

i.

^^ y4

-f-

~ y/9

...

z

1 z,

zz

m

By

determined, and z by the latter. be a Trochoid [or Cycloid] belonging to 4. Let the Circle ALE, whofe Diameter is and making the Ordinate j BD to cut the firft

refult

is

ADF

29. Ex.

AE

Circle

in

L,

AB=x, BD and the Arch AL=/, and the Fluxion of the fame Arch

=

firfl

the

/.

And

(drawing Semidia-

meterPL,)the Fluxion ofthe Bafe or Abfcifs

AB

will be to the

Fluxion of the Arch AL,

as

BL to


and INFINITE SERIES. to

PL

;

that

or

is, A*

I

/

:

v

: :

:

from the nature of the Circle ax Prob.

i.

2X

a

=

2-yy, or

And

~a.

-~~*

=

xx

65

therefore

=

-y-y,

=

^

and therefore by

v.

LD= Arch

nature of the Trochoid, 'tis 30. Moreover from the And thence (by Prob. i AL, and therefore -y

-M =y.

of the Fluxions Laftly, inftead tuted,

rived

and there

-

-f-

*ut/

v and

/ let

a

will arife

-

w

*v

=

-^ =z. z.

And

Then

/.

v -h / =z.

)

be lubfti-

their Values

Whence

(by Prob.

thefe being found,

i.)

is

de-

make z,

and raife the perpendicular HC. i. Now it follows from hence, that COR. 2BL, and 31. CH 2BE, or that EF bifeds the radius of Curvature CO in N. And this will appear by fubftituting the values of z and z now

==

DH,

DH

'

found, in the Equation the

.

**= DH,

=

and by a proper reduction of

refult.

32. COR, 2.

Hence the Curve FCK,

defcribed indefinitely

by the

another Trochoid equal to this, ADF, whofe Vertices at I and F adjoin to the Cufpids of this. For let the Circle FA, equal and alike pofited to ALE, be defcribed, and

Center of Curvature of

C/3 be

let

drawn

FA = The

parallel

is

to

EF, meeting the

(Arch EL= NF =) CA.

Circle in

A

:

Then

Arch right Line CD, which is at right Angles to the 33. COR. 3. Trochoid IAF, will touch the Trochoid IKF in the point C. in verted Trochoids,) if at theCufpid K 34. COR. 4. Hence (in the be a a Thread the at the diof Weight hung by upper Trochoid, the while ilance KA or 2EA, and Weight vibrates, the Thread be to apply itfelf to the parts of the Trcchoid KF and KI, fuppos'd which refift it on each fide, that it may not be extended into a right Line, but compel it (as it departs from the Perpendicular) to will

be by degrees inflected above, into the Figure of the Trochoid, while the lower part CD, from the loweft Point of Contact, ftill remains a right Line The Weight will move in the Perimeter of the lower Trochoid, becaufe the Thread CD will always be perpen:

dicular to

it.

35. COR.

5.

Therefore the whole Length of the Thread

equal to the Perimeter of the Trochoid equal to the part of the Perimeter CF.

K

KCF, and

its

part

KA

is

CD

is

36.


The Method of

66

FLUXIONS,

COR. 6. Since the Thread by its ofcillating Motion revolves moveable Point C, as a Center ; the Superficies through the about which the whole Line continually pafles, will be to the Superabove the right Line IF pafles at ficies through whichjthe part Therefore the the fame time, as CD* to CN*, that is, as 4 to i. and the Area Area of the is a fourth Area ; part 36.

CD

CN

CFD

CFN

KCNE

of the Area AKCD. fubtenfe EL is equal and parallel to 37. COR. 7. Alfo fince the CN, and is converted about the immoveable Center E, juft as moves about the moveable Center C ; the Superficies will be equal through which they pafs in the fame time, that is, the Area CFN, and the Segment of the Circle EL. And thence the Area will will be the triple of that Segment, and the whole area be the triple of the Semicircle. arrives at the point F, the 38. COR. 8. When the Weight whole Thread will be wound about the Perimeter of the Trochoid KCF, and the Radius of Curvature will there be nothing. Wherefore the Trochoid IAF is more curved, at its Cufpid F, than any Circle ; and makes an Angle of Contact, with the Tangent /3F produa right Line. ced, infinitely greater than a Circle can make with that are of Contact are there But infinitely greater Angles 39. than Trochoidal ones, and others infinitely greater than thefe, and fo on in infinitum ; and yet the greateft of them all are infinitely 3 Thus xx lefs than right-lined Angles. y, x* ==ry 5 , ay, x &cc. denote a Series of Curves, of which every fucceeding x* is

a fourth part

CN

NFD

EADF

D

=

=

=

dy+,

one makes an Angle of Contact with its Abfciis, which is infinitely And the the preceding can make with the fame Abfcifs. greater than firft xx=ay makes, is of the fame kind Angle of Contact which the which the fecond x*-=by z makes, is of that and Circular with ones; And tho' the Angles of the fucceedthe fame kind with Trochoidals. in Curves do always infinitely exceed the Angles of the preceding, yet of a right-lined Angle. they can never arrive at the magnitude 1 x fame manner the After ==y, xx=ay, x*=l> y, x4 c*y, 40. &c. denote a Series of Lines, of which the Angles of the fubfequents,

=

their Abfcifs's at the Vertices, are always infinitely lefs than the Angles of the preceding. Moreover, between the Angles of Contact of any two of thefe kinds, other Angles of Contact may be found ad infwitum, that mall infinitely exceed each other. that Angles of Contact of one kind are in41. Now it appears, of another kind ; fince a Curve of one than thofe finitely greater it may be, cannot, at the Point of Contact, kind, however great he I

made with


and INFINITE SERIES.

67

between the Tangent and a Curve of another kind, however fmall Or an Angle of Contacl of one kind cannot that Curve may be. an Angle of Contact of another kind, as the whole necefTarily contain Thus the Angle of Contaft of the Curve x* contains a part. cy*, or the Angle which it makes with its Abfcifs, neceflarfly includes the i and can never be contain'd Angle of Contacl of the Curve x~' that it. For can exceed each other are of the by Angles mutually fame kind, as it happens with the aforefaid Angles of the Trochoid, and of this Curve x> by*. And hence it 42. appears, that Curves, in fome Points, may be more infinitely ftraight, or infinitely more curved, than any Circle, and on that But all account, lofe the form of Curve-lines. yet not, lie

=

=^y

,

=

by the way only. 43. Ex. 5. Let ED be the Quadratrix from Center A; and letting fall DB this

AE, make AB and AE Then i.

perpendicular to

BD =y, be yx

Then

=

=xy,

yx*

yy*

i for x, and z for l becomes zx Equation zy y ; and thence, by Prob.

zx*

again writing

&

But z and

for y,

make

being found,

zx zzyy

and z

x

for

i

i.

zzxx

zx

-f-

x,

'twill

y, the

=

zx*

the Circle, defcribed

as before.

writing

zy*

=

to

'

=

y

Then arifes z

m

there

reducing, and

xxxjy

J, **

T

DH,

=

and draw

HC

as

above.

44. If you defire a Conftrudtion of the Problem, you will find it Thus draw in P, very mort. perpendicular to DT, meeting

and make

=

and zy into "PT* r 1

aAP

AE

:

=

g.

zy-\-x-=.

::

+x=

AP=2.

AB

it is

TATI a

=

i-4-zz

\

I

j

. 1

i

I

-j:

nrfr

=

T3T r 1

** U r BlJq := i-{- rrTT =-T-:T " and tnereiore Bl? BI? DH CH==^^. BT BD Laftly, 2-

it

is

the negative Value only as

AP, and -_^_..

Moreover

1

,)

= DH.

=r

BTy

f.

BT

way

For *

CH.

:

into x

T> P\

/i

AT

-BP; and;ey

z -

AE

DP PT

from

::

:

mews,

that

:

CH

Here

mufl be taken the fame

DH.

other 45. In the fame manner the Curvature of Spirals, or of any Curves whatever, may be determined by a very mort Calculation. 2 46.

K


7&e Method of

68

FLUXIONS,

to determine the Curvature without any pre46. Furthermore, vious reduction, when the Curves are refer'd to right Lines in any other manner, this Method might have been apply'd, as has beer* done already for drawing Tangents. But as all Geometrical Curves, as alfo Mechanical, (efpecially when the defining conditions are reduced to infinite Equations, as I mail mew hereafter,) may be refer'd to rectangular Ordinates, I think I have done enough in this matter. He that defires more, may eafily fupply it by his own inilluflration I mall add the Method duftry ; efpecially if for a farther for Spirals.

BK

be a Circle, 47. Let its Circumference. Let

A

its

Center, and

B

a given Point in

ADd

be a

DC

Spiral,

and

dicular,

C

its

Perpen-

the Center of the Point D.

Curvature at

Then drawing the right Line

ADK,

and

equal to

CG

AK,

pendicular

parallel and as alfo the Per-

GF

meeting

inF: Make AB

AK

or

CD

=

i=CG, BK=#, AD==y, and

GF

=

Then conD to move in

z.

.

the Spiral for an infinitely little Spree and then through rfdraw the Semidiameter A/, and Cg parallel to gC, fo that G/ cuts gf equal to it, draw gf perpendicular to <p, fo that in P; produce and /, and draw

ceive the Point Drf',

and

in/

GF

GF

de perpendicular to

AK,

G p= CD <

and produce

it till it

meets

and

Then De

at I.

be Kk,

Moments of BK, AD, and Fa, which therefore may be call'd xo, yo, and zo. Ae (AD) :: kK Je=yo, where I 48. Now it is AK x=i, as above. Alfo CG GF :: de eD oyz, and

the contemporaneous

:

:

G<p, will

aflurne

= dD CF dD = = LDAd, =(=Z-GG?) ADJ, CP

there-

:

:

CF de Befides CG oy x y becaufe x d\ Z_PC<p CF?. Moreover, oy the Triangles L. - eSQ -f- Red.) LCdl and /.CPp :: thence and AD Dd (CF) CP<p and AD</ are fimilar, take F<p and there will remain PF From whence o x CFq. fore

= (= = = oxCF^ yz

: :

:

f

for

CFa,

:

dl

::

'twill

:

:

t

ex z.

PF

:

=

P<p

'tis

:

: :

Laftly, letting

=

CG eH or DH :

be

DH

=

y

-

ya!g

fall

CH

LlHf CFyx

.

Here

perpendicular

Or it

to

fubftituting

may

AD

}

i+zz

be obferved, that


and IN FINITE SERIES.

69

and that in this kind of Computations, I take thofe Quantities (AD from little differs but the Ratio of which infinitely Ae) for equal, of Ratio the Equality. from hence arifes the following Rule. The Relation 49. exhibited of x and any Equation, find the Relation of

Now

by y being and fubftitute i for x, and yz x and Prob. the Fluxions i.) y, (by Then from the refulting Equation find again, (by Prob. i.) for y.

the Relation between x, y, and z, and again fubftitute i for x. The firft refult by due reduction will give y and z, and the latter

z

will eive

which being known, make

;

=

1

-f-

Z.X.Z.

= DH,

and

raife

DC

the Perpendicular HC, meeting the Perpendicular to the Spiral Or before drawn in C, and C will be the Center of Curvature. :: z and : i, which comes to the fame thing, take

HD

CH

:

draw CD. the Equation be ax=y, (which will belong to 50. Ex. i. If or (writing i the Spiral at Archimedes,) then (by Prob. i.) y

ax=y

7

=

and yz for_y,) ^ =yz. And hence again (by Prob i.) o Point D of the Spiral being given,, and yz+y'z. Wherefore any or y, there will be given z and thence the length , for x,

=

AD

(

3-

H-iz

Which

.

or)

DA

:: (y) hence you will :

make i-t-zz-z z :: DH CH.

known,

being

DH.

z=

And

i

:

:

:

deduce the following Conftrucftion. AB that fo Arch BK :: Arch BK to AB Produce Q, BC^, :: a AB HC. and make -+- AQ^: AQj: 1 2. If ax =_)" be the Equation that determines the Re51. Ex. lation between BK and AD; (by Prob. i.) you will have 2axx=.

And

eafily

:

DA

or

3Jy,-*,

therefore

i-\-zz

2ax= 3y.

= ^7

z

K

:

,

DH

:

Thence again 2a'x= ^zy -+- gsiyy*. 'Tis ' a~ 9 ~ z and z Thefe being known, make s

=

'-

.

DA DH.

i-t-zz

to a better form,

:

:

:

make gxx

1

-f-

10

:

gxx

Or, the -f-

4

::

work being reduced

DA

DH.

:

Ex. 3. After the lame manner, if ax* determines bxy=yi 52. ~ to there will arife I"* and the Relation of ; z,,

AD

BK

g .

*~

9 *'-

;

*7~^;~ Point C,

is

8

=

'

bxy

g.

From which

DH/

-f- $)*.

=

and thence

the.

determined as before. 5qi-


I'he

yo

Method of FLUXIONS,

thus you will eafily determine the Curvature of any53. And other Spirals ; or invent Rules for any other kinds of Curves, in imitation of thefe already given. 4. And now I have finim'd the Problem ; but having made ufe

Method which

of a

is

pretty different

from the common ways of

of the number of thofe operation, which are not very frequent among Geometricians : For the illuflration and confirmation of the Solution here given, I mall not think

and

Problem

the

as

itfelf is

much

to give a hint of another, which is more obvious, and has a nearer relation to the ufual Methods of drawing Tangents. Thus if

from any Center, and with any Radius, a be defcribed, which may cut any Curve in

Circle be conceived to feveral Points

;

if that

Circle be fuppos'd to be contracted, or enlarged, till two of the And Points of interfeclion coincide, it will there touch the Curve. to towards, or recede befides, if its Center be

approach fuppos'd from, the Point of Contadt, till the third Point of interfedtion fhall meet with the former in the Point of Contadt ; then will that Circle be cequicurved with the Curve in that Point of Contadt In like manner as I infmuated before, in the laft of the five Properties of the Center of Curvature, by the help of each of which I affirm'd the Problem might be folved in a different manner. a Circle be 55. Therefore with Center C, and Radius CD, let :

defcribed, that cuts the Curve in the Points d, D, and <f ;

and letting fall the Perpendiculars DB, db, <T/3, and CF,

AB AB = AF = = BD Then to the Abfcifs x,

;

call

v,

y,

FC=/,andDC=J.

BF=vx,

and DB-f-FC The fum of the

=_>>-{-/.

the Squares of thefe is equal to 1 that is, -D ; Square of

DC

2VX

-+-

=ss. ate this,

X* -f- )" -h 2yt -+- / If you would abbrevi-

make 1

v* -f-/ 1

s 1

1

=f,

(any Symbol at pleafure,) and it o. After you have found q 1

=

2vx -f-jy -f- zfy -+becomes x r 1 /, y, and q*, you will have s-=\/ v -+- 1* q*. for defining the Curve, 56. Now let any Equation be propofed the quantity of whofe Curvature is to be found. By the help of the of this Equation you may exterminate either Quantities x or y, and


and INFINITE SERIES.

71

an Equation, the Roots of which, (db, DB, <f/g, &c. if y u exterminate x ; or A/>, AB, A/3, &c. if you exterminate Wherefore fince are "at the Points of interfedtion d, D, J\ &c. _y,) the Circle become both them the Curve, touches of "three equal, of be fame of the Curvature as the Curve, in and will alfo degree But they will become equal by comparing the point of Contact the Equation with another fictitious Equation of the fame number of Dimenfions, which has three equal Roots ; as Des Cartes has Or more expeditioufly by multiplying its Terms twice by fhew'd. an Arithmetical Progreflion. Let the Equation be ax =yy, (which is an 57. EXAMPLE. the Parabola,) and exterminating x, (that is, fubftituEquation to

and there

ting

its

will arife

Value

--

in the forego-

ing Equation,) there will arife Three of whofe Roots ^ are to be made equal. And for this purpofe I multiply the Terms twice by an

*

^~y*_

_j_

*

u

=

+ \a.

Whence

\a, as before.

i

-J

it is

=

o.

= 2x

-{-

zty -f- ?

a

yi

4*2

Arithmetical Progrellion, as you fee done here j and there arifes

Or

-+

1

+

eafily infer'd,

I

o

o

i

2J X

that

= BF

D

-

of the Parabola being given, draw the to the 2AB, Curve, and in the Axis take Perpendicular in then will to and erect C; FA, meeting Perpendicular 58.

Wherefore any Point

PF

DP

=

DP

FC

C

be the Center of Curvity defired. in the Ellipfis and Hyperbola, 59. The fame may be perform'd but the Calculation will be troublefome enough, and in other Curves generally very tedious.

Of

^uefiions that have fome Affinity to the preceding Problem.

60. From the Refolution of the preceding Problem fome others may be perform'd ; fuch are, I. To find the Point where the Curve has a given degree of Curvature. if the Point be required 6 1. Thus in the Parabola, whofe Radius of Curvature is of a given length f: From the Cen-

ax=yy,

ter

of Curvature,

found

as before,

you

will

determine die Radius


7%e Method of

72 to be

which muft be made equal

-~^ \/aa -+- ^.ax,

by reduction there

arifes

FLUXIONS,

x

=

^a

Then

to f.

-f-

1/^aff. To find the Point of ReElitude. 62. I call that the Point of ReEiitude, in which Flexure becomes infinite, or its Center at an infinite it is at the Vertex of the Parabola a*x=y*. And of Limit is the contrary Flexure, whole commonly II.

the Radius of diftance

Such

:

fame Point Determination But another Determination, and that not I have exhibited before. from this Problem. derived be Which is, the inelegant, may fo of Flexure much the lefs the Angle DCJ is, longer the Radius (Fig.pag.6i.) becomes, Fluxion of the Quantity

and

z

Moment

the

alfo

this

<F/j

fo that

the

diminim'd along with it, and by the of Therefore find the Infinitude that Radius, altogether vanimes. Fluxion z, and fuppofe it to become nothing. 63. As if we would determine the Limit of contrary Flexure in the Parabola of the fecond kind, by the help of which Cartefius conftructed Equations of fix Dimenfions ; the Equation to that Curve is AT o. And hence bx* cdx -+- bed 4- dxy (by Prob. i .) arifes is

=

3

2bxx

3***

and z

for y,

writing

i

cdx

dxy

-4-

-f-

a

becomes

it

again (by Prob.

=

-

3-v

dxy zbx

+

&

o.

Now

cd-{- dy

=

x

t

=

whence ; Here again zb -+- zdz dz in the bx arife

o.

(>x

3* for exterminating z, by dxz there will cd -+- dy -fzbx o, Equation 3^,v this being fubftituted in the o) cd-$-dy o.

for

-f- dxz=.o

putting b

And

i

writing

zbx -+ dy dxz +- dxz for y, and o for z, it becomes

i,) 6xx

for x,

=

=

ory=c-{-^;

of y in the Equation of the Curve, we fhall have x* which will determine the Confine of contrary Flexure. 64. By a like Method you may determine the Points of Rectitude, which do not come between parts of contrary Flexure. As if the

room

+- bcd-=z. Q

}

=

o exhave firfl, prefs'd the nature of a Curve ; you i2a*x faz=o, i2ax*-+(byProb. i.)4^3

Equation x*

4<w 3

-}-

ba^x*

and hence again 12X* =o. Here fuppofe z tion there will

arife

ABi=fl, and

erect

Curve

x

in the Point of

24^7^ -f- 12^'

==

the

b>y

o,

a.

b*z

and by ReducWherefore take

perpendicular

BDj

this

will

meet

Re&itude D, as was required. III.


and IN FINITE SERIES. III.

73

To find the Point of infinite Flexure.

65. Find the Radius of Curvature, and fuppofe it to be nothing. Thus to the Parabola of the fecond kind, whole Equation is A;* a

<7y

,

comes nothing when x

=

CD

that Radius will be

=

=

q*

4"

\/q.ax-\- gxx

6a

,

which be-

o.

IV. To determine the Point of the greatefl or leaft Flexure. 66. At thefe Points the Radius of Curvature becomes either the Wherefore the Center of Curvature, at that moor leaft. greateft ment of Time, neither moves towards the point of Contact, nor Therefore let the Fluxion the contrary way, but is intirely at reft. be found; or more exof the Radius let the Fluxion of either of the peditioufly, be found, and let it be or Lines

CD

AK

BH

made

equal to nothing.

if the Queftion were propofed con67. As the Parabola of the fecond kind cerning firft to determine the Center of xl o*y ;

=

Curvature

you

and therefore

BH

=

?AV

A-

"'

=

ftitute

1 rf .y,

^

BH,

;

6 ^'

Hence (by Prob. i.) Fluxion of

DH =

find

will

"-

-

_j_

^y

aa

make

==

9X ,

->

ox

BH

t}.

But

now

fuppofe

be nothing ; and belides, lince by and thence (by Prob. i.) yxx 1 =<?*.}', putting

for

v,

to

-y,

Hypothecs

x=

and there will

arife

4.5x4=0+.

or the

Take

i,

fub-

therefore

^

AB ==a

=<7 x45|

and raifrng the perpendicular BD, it will" meet the Curve in the Point of the greateft Curvature. Or, which is the fame BD I. thing, make AB 3^/5 68. After the fame manner the Hyperbola of the lecond kind l 3 will be reprefented by the Equation xy , moft inflected in the points and d, which you may determine by taking in the Abfcifs AQ== r, and erecting the Perpendicular QP_=z= v/5, and it on the other fide. to Then drawQ^/> equal and AP will meet the Curve in the ing A/>, they D and d required. points y'^j-

,

: :

:

D

:

=

V,


The Method of

74

FLUXIONS,

V. To determine the Locus of the Center of Curvature, or to defcribe the Curve, in which tbaf* Center is always found, have already {hewn, that the Center of Curvature of the 69. Trochoid is always found in another Trochoid. And thus the Center of Curvature of the Parabola is found in another Parabola of the fecond kind, reprefented by the Equation axx=y*, as will

We

eafily

appear from Calculation.

VI. Light falling upon any Curve, to find its Focus, or the Conrafted at any of its Points. courje of the Rays that are ref 70. Find the Curvature at that Point of the Curve, and defcribe a Circle from the Center, and with-the Radius of Curvature. Then find the Concourfe of the Rays, when they are refracted by a Circle about that Point For the fame is the Concourfe of the refrac:

ted

Rays

in the propofed Curve.

To

be added a particular Invention of the Curvawhere they cut their Abfcifles at right For in which the Perpendicular to the Curve, the Point Angles. with the it cuts Abfcifs, ultimately, is the Center of its meeting So that having the relation between the Abfcifs x, Curvature. and the rectangular Ordinate y, and thence (by Prob. i.) the relation between the Fluxions x and y ; the Value yy, if you fubftitute 71.

thefe

may

ture at the Vertices of Curves,

r

for

x

into

it,

and make y

=

o,

will be the

Radius of Curva-

ture.

72.

Thus

in the Ellipfis

ax

xX=yy,

we fuppofe^=o, and

it is

-*

"

= =

yy ;

/>, confequently x And a of Curvature. fo Radius for the i for x, becomes ^writing at the Vertices of the Hyperbola and Parabola, the Radius of Curvature will be always half of the Latus rectum.

which Value of yy,

if

73-


and INFINITE SERIES. 73.

And zicc

in like

+

~T

Prob.

i.)

EG

^

=

for the

c or

of Curvature Conchoids.

C

=

xx

zbx

""*

and

:

c,

eG at

by the Equation the Value of yy (found by t

Now

^ "~~ *

IT

we

f,

yy,

eG

zb

Therefore

and you the Vertices E and

::

:

o,

AE EG

::

c,

make

or :

have the Centers e of the Conjugate

will

ec,

PROB.

=

fuppofing y

mail have

Radius of Curvature.

EC, and he

:

for the Conchoid, defined

bb

will be

and thence # 2(5 -f- f,

manner

cc

75

VI.

To determine the Quality of the Curvature, at a given Point of any Curve. I.

or

the Quality of Curvature I mean inequable, or as it is varied more or

By

lefs

So

if

L

2

its

Form,

lefs,

in

its

as

it

is

more

progrefs thro'

were demanded, what is the Quality of the Curvature of the Circle ? it might be anfwer'd, that different parts of the Curve. it

is

it

uniform, or invariable.

And thus what

if it

were demand-

the Quality of the ed, Curvature of the Spiral, which is described by the motion of is

the point D, proceeding from

A

AD

with an accelerated while the right velocity, Line AK moves with an uniform rotation about the Center A ; the acceleration of in

which


7&? Method of FLUXIONS,

76

which Velocity is fuch, that the to the Arch BK, defcribed from

AD

has the fame ratio right Line a given point B, as a Number has

be afk'd, What is the Quality of the Curvature of this Spiral 1 It may be anfwer'd, that it is uniformly And thus other Curves, in varied, or that it is equably inequable. their feveral Points, may be denominated inequably inequable, according to the variation of their Curvature. 2. Therefore the Inequability or Variation of Curvature is reConcerning which it may be obquired at any Point of a Curve. to

its

Logarithm

:

I

fay,

if

it

ferved, 3.

I.

That

Points placed alike in like Curves, there

at

is

a like

Inequability or Variation of Curvature. 4. II. And that the Moments of the Radii of Curvature, at thofe Points, are proportional to the contemporaneous Moments of the

Curves, and the Fluxions to the Fluxions. III. And therefore, that where thofe Fluxions are not propor5. For tional, the Inequability of the Curvature will be unlike. there will be a greater Inequability, where the Ratio of the Fluxion of the Radius of Curvature to the Fluxion of the Curve is And therefore that ratio of the Fluxions may not improgreater.

perly be call'd

the Index of the Inequability

or of the Variation

of Curvature.

D

and the points be let there Curve AD^, 6.

At

d,

infinitely near

to each other,

in the

drawn the Radii of Curvature DC and dc , and D</ of the Curve, Cc being the Moment will be the

contemporaneous Radius of Curvature,

of the

will be the

Moment and

-^

Index of the Inequability of

>

For the Inequability may Curvature. be call'd fuch and fo great, as the quanof that ratio

tity j

mews 7^ Ja

it

to be

:

the Curvature may be faid to be fo much the more unlike to the uniform

Or

II

Curvature of a Circle. 7.

to

Now

any

line

letting fall

AB

the

perpendicular

meeting DC in P and thence B&

= j

-1

= DBBD = = = =

make AB

xo,

and dbt

Ordinates

it

T^

will be ,

#,

y\

Cc

vo; and

making x

i.

Wherefore


and IN FINITE SERIES.

77

Wherefore the relation between x and y being exhibited by any and 5.) the PerpendicuEquation, and thence, (according to Prob. 4. of Curvature i and the lar DP or /, being found, and the Radius 1

,

of that Radius, (by Prob. i.) the Index '^ of the Inequabiof Curvature will be given alfo. lity 8. Ex. i. Let the Equation to the Parabola tax vy be given therefore then (by Prob. 4.) BP and a -\-\y=^t. a, Alfo (by Prob. 5.) BF a -+- 2X, and BP DP BF "i)C Fluxion

<y

=

= =

at

.v

Now

=1;.

-

t-~

=v,

+

+ 2fx __ ^

Z fx

=

And

y

2ax

give

i.)

Which

there will arife

i,

:

the Equations

(by Prob.

and zyy

and

=

2jvy,

=

and putting

being reduced to order,

= = r^ = -, /

- > an<^ )

f

ztt,

v=

-

v

thus y,

x= 4

j^=

and v being found, there will be had ^ the Index

t,

But

v/5,

that^=j

2#==n

or a

>

,

then y (==

;

which

3,

10.

=

=

:

:

-and

+ 7 "=

+

/

:

2ax =}'}', aa-\-yy=tt,

of the Inequability of Curvature. 9. As if in Numbers it were determin'd,

and

^

DP=

;

the Index of Inequability.

is

were determin'd, that

if it f

therefore

= </,

and

3v /2.

=

-17

A:

=2,

So that

3^/5.

then y

=

2,

So that

^'=T>

^-=) 6 will be here

the Index of Inequability. 11. Wherefore the Inequability of Curvature at the Point of the Curve, from whence an Ordinate, equal to the Latus reftum of the Parabola, being drawn perpendicular to the Axis, will-be double to the fo drawn is half Inequability at that Point, from whence the Ordinate the Latus rectum ; that is, the Curvature at the firft Point is as unlike again to the Curvature of the Circle, as the Curvature at the fecond Point. 12. Ex. 2. Let the Equation be zax bxx-=.yy, and (by Prob. 4.) it

will be

=)

na

where,

&v=BP,

a

--

byy if for

Tis

yy

alfo

yy.

byy

you ::

fubftitute

DH

:

f.ll

the Equations

zaxbxx^yy,

2a6x-lrb

tf=(aa

Alfo (by Prob.

BD DP :

and thence

5.)

it

is

DH =}'

aa, there ariies

//

DC= - =v. U1

aa

-{

DH

=

Now (hv Prob.i.)

byy-\-y\-=^t!,

and give


78

Method of FLUXIONS,

77je

bx =}')', and yy

give a

being found, the Index

^ of

=

the

/'/,

and

~=

13.

Thus

thus

Inequability of Curvature,

we make

it

is

a

x=-, *

v

and therefore ;

*

=

this

it

will

" L a 3

then Sr-

=|,

which

is

v

x

~~

;

~"

the In-

of Curvature.

A

b

P

appears, that the Curvature of

Ellipfis,

fign'd,

D

at the Point

here af-

by two times left inequable, two times more like to the Curis

(or 'by vature of the Circle,) than the Curvature of the Parabola, at that Point of

Curve, from whence an Ordinate

o

V

Jl

let fall

upon the Axis

is

to half the Latus rectum. 14. If

we have

a

mind

to

=

-

-

x

BP

j

and

fo in the

Hyperbola 2ax

-+-

in

> v for the )^ a

Examples, in the Parabola 2ax=yy arifes (~ V ' s Index of Inequability j and in the Ellipfis zax bxx=yy,

=J

equaj

compare the Conclufions derived

thefe

(^7-

v

3 ATA:

and b=-.^

r,

dex of the Inequability

Hence

2X

the Ellipfis

in

=}'}', where

its

And

v.

be known.

aJib

if

byy

bxx

arifes

=yy,

y+3b the analogy being obferved, there arifes the Index ("2^ && J \. t x BP. Whence it is evident, that at the different Points of

any Conic Section conn'der'd apart, the Inequability of Curvature is as the Rectangle BD x BP. And that, at the feveral Points of the Paraboh, it is as the Ordinate BD. 15. Now as the Parabola is the moft fimple Figure of thofe that are curved with inequable Curvature, and as the Inequability of its Curvature

is

fo

eafily

determined, (for

its

Index

is

t h er e6x^ll^i,) .. .

fore the Curvatures to the Curvature

As if Ellipfis 2X 1

6.

it

of other Curves

of this. were inquired,

$xx=yy,

at

determined by affuming x it might be anfwer'd, that

may

not improperly be compared

what may be the Curvature of the of the Perimeter which is

that Point

= it

:

is

Becaufe its Index is 4., as before, like the Curvature of the Parabola 6.v


and IN FINITE SERIES. of the Curve, between =)')', at that Point the perpendicular Odinate is equal to |. Thus, as the Fluxion of the Spiral 6.v

d

fuppofe as

=y

AP

and P

to

the Fluxion

AD,

AD

x dd

is

the Axis

in a certain given Ratio, on its concave fide erect

to e;

-

AP

which and

ADE

17.

of the Subtenfe

79

ee

perpendicular to

AD,

will be the Center of Curvature,

and

t

r=?

or

1J

y

a.i

will be the

ee

Index of Inequa-

So that this Spiral has every where its Curvature alike inequable, as the Parabola 6x yy has in that Point of its Curve, from whence to its Abfcifs a perpendicular Ordibility.

=

nate

1

let

is

And

8.

Trochoid, fore

its

which

fall,

is

to

equal

thus the Index of Inequability at any Point

(fee Fig.

in

Art. 29. pag. 64.)

Curvature at the fame Point

D

to that of a Circle, as the Curvature of

the Point where the Ordinate 19.

the

And from

is

is is

AB

found to be

.

as inequable,

any Parabola AB

D

of the

Where-

or as unlike

ax

-

is

at

of the Problem,

as

yy

^a x -^

thefe Confiderations the Senfe

conceive, mufl be plain enough; which being well underftood, it will not be difficult for any one, who obferves the Series of the to furnifh himfelf with more Examples, and things above deliver'd, of other Methods as occafion to contrive many operation, may rethat he will be able to manage Problems of a like nature, So quire. (where he is not difcouraged by tedious and perplex Calculations,) Such are thefe following ; with little or no difficulty. I

To find the Point of any Curve, where there is either no Inequabior infinite, or tie grcatej?, or the leajl. lity of Curvature, the Vertices of the Conic Sections, there is no Inat 20. Thus of Curvature; at the Cuf] id of the 1 rcchoid it is infiequability nite ; and it is greatefl at thofe Points of the Ellif.fis, where the I.

Rectangle BD x BP is greatefl, that is, where the Diagor.al-Lines of the circumfcribed Parallelogram cut the Elliriis, whofe Sides touch

it

in their principal Vertices.

1o determine a Curve of fame definite Species, l'nfp rje a Section, liioje Curvature at any Point may be cqiu:l and Jiitiilar Curvature of any other Curve, at a given P./:./ of it. II.

C.n:c r

to the


8o

"The

III.

ri?//i7V

to the

Method of FLUXIONS,

To iL-termine a Conk Sctfion, at any Point of which, the Curand Pojition of the tangent, (in refpeSt of the AxisJ) may be like Curvature and Pofition of the Tangent, at a Point ajfigrid of

any other Curir. 21. The ufe of which Problem is this, that inftead of Ellipfes of the fecond kind, whofe Properties of refradling Light are explain'd by Des Cartes in his Geometry, Conic Sections may be fubftituted, which mall perform the fame thing, very nearly, as to their ReAnd the fame may be underfhood of other Curves. fractions.

R O

P To find as

many Curves

as

B.

VII.

you

pleafe y

Areas may

ivbofe

be exhibited by finite Equations.

AB

be the Abfcifs of a Curve, at whofe Vertex be raifed, and let CE be i perpendicular drawn parallel to AB. Let alfo DB be a rectanI.

Let

AC

=

CE

gular Ordinate, meeting the right Line the Curve in D. And conceive

AD ACEB and ADB

and

Areas

in

A

let D

E,

thefe

be generated by the right Lines BE and BD, as they move along the Line AB, Then their Increments or Fluxions will be always as the defcribing Lines BE and BD. Wherefore the Parallelogram

Curve

ADB

to

ACEB, And

call z.

the

make

or AB x i, =.v, and the Area of the the Fluxions x and z will be as BE and

= =

=

BD. BE, then z be a/Turned at any Equation pleafure, for determining the relation of z and x, from thence, (by Prob. i.) may z be deAnd thus there will be two Equations, the 'latter of which rived. will determine the Curve, and the former its Area.

BD;

2.

fo that

Now

making x

i

if

EXAMPLES. 3.

Aflume ##:=:, and thence (by Prob.

becaufe 4.

x=,

Aflame

i.)

2xx=s

}

or

2x=c:,

i.

^=z,

and thence will

arife

=;s

?

an Equation

to the Parabola. 5.

A flume ax* =zz, or

or ^(?x

=

a'f

x*=z, and there

will arife \a^x'

=^^,

zz, an Equation again to the Parabola. i

6.


and INFINITE SERIES.

81

=

a*x * Affume a 6 x~ 1 =zz,or a*x-' =z, and there arifes z, Here the negative Value of z only infinuates, o. or a'' -j-2xx that BD is to be taken the contrary way from BE. 1 z 1 , you will have zc*x if you affume c'-a -+- c^x* f

6.

=

Again 2zz and

=

7.

=

;

8.

=

Or

if

and

v,

z

you affume

it

will be

the Equation aa if

being eliminated, there will

-f-

you exterminate 9.

firflieek

\/aa --+- xx

=

'Z. aa

z,,

-J-.VA-

make

^ =s,and then (by Prob.i.) ^p

xx <u,

= it

011;

gives

2X

become

will

= 3

-j^-

<z

^

-}- ATA-

Alfo

zvv, by the help of which

= = z

j-

\/ aa-^-xx.

you affume 8 3^2 -f- ^&=. zz, you will obtain Wherefore 2Z&. 3x2; -f- $z by the affumed Equation the Area z, and then the Ordinate z by the reiulting Equa-

Laftly,

32;

aa-^-xx -

arife

if

=

tion.

may

And

thus from the Areas, however they may be feign'd, you always determine the Ordinates to which they belong.

10.

P To fad as

R O

many Curves

have a relation

to the

B.

VIII.

you pleafe, -wbofe Areas fiall Area of any given Curve, a/fignas

able by finite Equations.

FDH

i. Let be a given Curve, and GEI the Curve required, and conceive their Ordinatss and EC to move at right Angles upon

DB

A .11

C G,

/V

their Abfciffes or Bafes

AB and

AC.

of the Areas which they defcribe,

M

Then

the Increments or Fluxions thofe Ordinates drawn

will be as

into


fhe Method of

82

into their Velocities of

Therefore

Abfcifles.

CE =y,

the Area

moving, that

= =

AFDB

zy

= 2.

we

if

fuppofe x

and thence

t,

Therefore

let

-

=

AB

make

the Fluxions of the Areas be

Therefore

FLUXIONS,

BD

x,

=

and

t

:

and

i,

And

= =

AC AGEC

and the Area

j,

s

into the Fluxions of their

is,

v,

will be

it

v=s,

/,

xv zy :

as before;

and and let

z,

s

: :

t.

:

will be

it

=y.

any two Equations be affumed

;

one of which

may exprefs the relation of the Areas s and t, and the other the relation of their Abfciffes x and z, and thence, (by Prob. let the i.) Fluxions 3.

Ex.

xx

= w,

and

s=:f, and thence

to determine z, x and z. 4.

As

if

xx

s

FDH

:

=)

[

let

=

ax=zz;

for

-

=^-.

= =

i.)

a

\/ aa

s.s,

therefore

=. 2zz: it

zz

aa

aa \/

a

But

.

'

-

remains

It

between the Abfciffes

relation

then (by Prob. "

then y

z,

=

and y

f,

fome

by afluming

you fuppofe

that fubflituting

(\/ax

=>'.

be a Circle, exprefs'd by the other Curves be fought, whofe be equal to that of the Circle. Therefore by the Hy-

may

pothefis

-

make

be found, and then

Let the given Curve

i.

Equation ax Areas

z

and

t

=

So

is

v

=

*y

is

the

Equation to the Curve, whofe Area is equal to that of the Circle. After the fame manner if you fuppofe xx =. z, there will 5. ariie

2x

=s,

(==] ~

and thence _)'= ;

exterminated, 6.

Or

if

it

will be

7.

cc

you fuppofe

-- = T-V

thence

y=-

2

= --

(5

y

y

ax

Ex.

pofe alfo

2.

ax

=

ZZ.

-j

and x being

I

7"-' 2Z 2

-

xz,

v az +-

'-

?>

Let the Circle ax Curves be fought, whofe Areas As to the Area of the Circle. 8.

whence

there arifes o

=z +

xz, and

/

2^3

Again, fuppofing

-^and thence

,

=

;

cc.

=

z, (by Prob. i.) \t'isa

which denotes

a

+ s=:z,

mechanical Curve.

=w

be given again, and let xx have any other aflumed relation may if

(By Prob.

you i.)

afliime

'tis

c

+

cx s

+

=

t,

s

=

and fup-

t,

and a

=

Therefore


and INFINITE SERIES.

= = ~~; 2

~

Therefore y 5

for x,

and f

But

9.

you affume

if

^! =/,

s

= =

i;

(by Prob.

Where

xx and

\/ ax 10.

if

But

for

you expunge v and there will arile

2;,

i

=

ss

2zz; and

=

A;

therefore

_y

j,

=j

2!2L

Oi '

fl

K

ax

xx

'tis

y=.

fubftituting their values

\/tf;s

= = V = 4^-

and x

f,

-

for

you will have

and therefore

by

_)=-"

z,

xx

the Equation

2x= 2vv,

^

gives

i.)

y-

Therefore

z.

exterminating v,

you affume

if

2w=r^, and

=

=

and x

=/,

j

i

Now

.

iJ'u, .

and

^ ax

and fubftituting

^ v'^

4-

'tis;'=

83

will

there

zz,

arife

Anc

=

xx and &z, it will become y and x fubftituting \/ ax \Sa-zz;, which is an Equation to a mechanical Curve. be found, which 1 1. Ex. 3. After the fame manner Figures may have an aflumed relation to any other given Figure. Let the Hyperbola cc -{- xx and wu be given ; then if you affume s /, 5

=

xx=cz,

.r=

you

will have

Then

-.

12.

=: 2Z

And

will have

= = vx

-u,

13.

i/cz

-{-

Therefore

y= -

i.

%

x=^-ui;, and

and c*z* Ex.

4.

-

xx

for x,

'tis

=

cz;

for

j,

= =^

But

~.

if

now

***

Then

y

/'

-,

xx

+-

- s

=

therefore h +- -

s

M

t

it

(by Prob.

=1; and -

;

=

i.)

cc-\-xx

^

^c

,

^-^^_-

2

for x,

fubftituting \/i<,-t-xx

~

-

the Ciffoid

=

=

related Figures are to be found,

Fluxion

and C-z^

_)'

s=zf, and xx cz, you But and thence cz. v=.s,

becomes y =.

it

Moreover

^/ax

and thence

zz.

v-^-vx

you affume its

and

xv s=t, and 2X

which other

and

f

thus if you affume

1?^ gives

for

=

fubftituting v/cc -+-

s;

will be y *

s

=

2X =

fuppofe

were given,

for that purpofe

*/ ax

xx

=

*** /.

to

But the Equation

h,

-

=M


84 3

=/j/j gives

*A

7%e Method of

FLUXIONS,

.V.

if

==2.^, where

^

And

= ax

\/ aa

=z

bcfides

fuice

Now

t.

i

is

- s

-

3

3

a

.)

=

/&,

z,

2zz, or z a

it

will be

xx ,

z and

determine

Jaxxx

V..

And

it

to

then (by Prob.

;

you exterminate

afTume

=

-

'

A;

Equation belongs to the Circle, we mall have the relation of the Areas of the Circle and of the Ciflbid. ~.

v/tftf

And

14.

and x

=

Equation

thus

as this

if

V

you had aflumed

z, there would have been again to the Circle.

">/

ax

derived

xx -h ~

y-=.\/as>

s

=

fy

an

.22-,

like manner if 15. In any mechanical Curve were given, other mechanical Curves related to it might be found. But to derive it will be of that convenient, geometrical Curves, right Lines deon each other, fome one may be taken for pending Geometrically and that the Area which compleats the Parallethe Bafe or Abfcifs logram be fought, by fuppofing its Fluxion to be equivalent to the Abfcifs, drawn into the Fluxion of the Ordinate. ;

1

Ex.

6.

5.

Thus

the Trochoid

ADF

being propofed, I refer

it

to the Abfcifs

ABj

and the

Parallelogram being

ABDG

compleated, I leek for the

complemental

Superficies

ADG,byfuppofing

it

to

be

defcribed

the

by Motion of

the right Line

GD,

and therefore

drawn as

AL

as the

into the is

GD

Fluxion to be equivalent to the Line Now whereVelocity of the Motion ; that is, x*v.

parallel

its

to the

Tangent DT,

Fluxion of the fame

AB

to the

therefore

AB

will be to

Fluxion of the Ordinate

BL BD, that


and INFINITE SERIES. that

as

is,

to

r

So that

-j.

ADG

Therefore the Area

<u

== BL. BL fince

and therefore xv

Ah

described by the Fluxion is defcribed by the fame Fluxion,

is

therefore the circular Area they will be equal.

=

85

;

ALB

ADF

manner if you conceive to be a Figure of or of verfed is that Ordinate whole Arches, Sines, is, equal to Arch the lince the Fluxion of the Arch is to the Fluxion ; In like

17.

BD

AL

AL

of the Abfcifs then

-y

=

AB,

as

--/

v ax

2

.

xx

to

BL,

Then

vx,

Wherefore

7=^=. xx

will be 2V

PL

<,*

if

that

is,

v

:

i

::

a

:

\/ ax

the Fluxion of the Area

.v.v,

ADG,

--.' Line equal to ly .x

a right

ATV

.

be conceived to be apply 'd as a rectangular Ordinate at B, a point of the Line AB, it will be terminated at a certain geometrical Curve, whole Area, adjoining to the Abfcifs AB, is equal to the Area

ADG. And

thus geometrical Figures may be found equal to other Figures, made by the application (in any Angle) of Arches of a Circle, of an Hyperbola, or of any other Curve, to the Sines 1

8.

right

or verfed of thole Arches, or to any other right Lines that Geometrically determin'd.

As

may

be

the matter will be very fliort For from the the Arch being defcribed, with any Radius AG, cutting the right Line in G, and the Spiral in D ; fince that Arch, as a Line moving upon the Abfcifs AG, delcribes the Area of the Spiral fo that the Fluxion of that Area is 19.

to Spirals,

DG

Center of Rotation A,

AF

AHDG,

to the Fluxion of the Rectangle i x AG, as to i ; if you raife the perpenthe Arch dicular right Line equal to that Arch,

GD

GL

in

by moving Line AC, it

like

manner upon the fame

will defcribe

the Area

The And

A/LG

AHDG

the Area of the Spiral Curve A/L being a geometrical Curve.

equal to

if

the Subtenfe

= |AGx GD =

fartlirr,

xGL

ADH

:

=

AL be drawn, then A ALG | AGDj therefore the complernental

Sector

will alfo be equal. AL/ and And this not only agrees to the Spiral of Archimedes^ (in which cafe A/L becomes the Parabola of Apoliomus,) but to any other whatever; fo that all of them

Segments

may

be converted into equal geometrical Curves with the fame

eale.

20.


tte Method of

86

FLUXIONS,

20. I might have produced more Specimens of the Conftruction of this Problem, but thefe may fuffice; as being fo general, that whatever as yet has been found out concerning the Areas of Curves, or (I believe) can be found out, is in fome manner contain'd herein, and is here determined for the moil part with lefs trouble, and with-

out the ufual perplexities. 21. But the chief ufe of this and the foregoing Problem is, that nffuming the Conic Sections, or any other Curves of a known magnitude, other Curves may be found out that may be compared with

Equations may be difpofed orderly in a Catalogue or Table. And when fuch a Table is contracted, when the Area of any Curve is to be found, if its defining Equation be either immediately found in the Table, or may be transformed into another that is contain'd in the Table, then its Area may be known. Moreover fuch a Catalogue or Table may be apply'd to the determining of the Lengths of Curves, to the finding of their Centers of Gravity, their Solids generated by their rotation, the Suand to the finding of any other flowing perficies of thofe Solids, Fluxion a analogous to it. quantity produced by thefe,

and that

their defining

P R O

B.

IX.

To determine the Area of any Curve propofed. 1.

The

refolution of the

Problem depends upon

this,

that

from

the relation of the Fluxions being given, the relation of the Fluents may be found, (as in Prob. 2.) And firft, if the right Line BD, by the motion of which the Area required is defcribed, move upright upon an Abfcifs the Paralgiven in pofition, conceive (as before) mean time to be defcribed in the

AFDB AB

lelogram

ABEC

on the other

a line equal to unity. Fluxion of the Pathe being fuppos'd will be the Fluxion of the Area rallelogram, fide

AB, by

And BE

BD

required. 2.

and

Therefore

BE

=

BD=z,

make

=

Call alfo

x.

as alfo

tion expreffing

AB

=~, X

BD,

and then alfo the Area AFDB

becaufe

at the

ABEC=i \x=x,

=

x,

x=i.

fame time the

z,

and

it

will be

Therefore by the Equaratio

of the Fluions

-

IS


and INFINITE SERIES. is exprefs'd, and thence (by Prob. 2. Cafe relation of the flowing quantities x and z.

3.

Ex.

or z,

and (Prob.

rabola,)

=

AB

Jc.

Let there be ^ eiven

x

BD,

=

r

z,

=

z,

Area

=

a

Therefore ^>

z.

AFDB.

to a Equation *

fan

the fecond kind,) and there will arife

=

(the Equation to the Pa-

,

Area of the Parabola aa

be found the

may

i.)

equal to fome fimple quantity.

there will arife

2.)

-L

is

~

Let there be given

4.

or

When BD,

i.

87

-^

=

z, that

is,

Parabola of

~ AB

x

BD

AFDB.

Let there be given z' XX ~ 1 or a^x x-:, (an Equation to an Hyperbola of the fecond kind,) and there will arife a3 x 1 z 6.

=

or

7

=

Area

=

HDBH,

the Ordinate

And

j.

is,

of an

BD,

thus

AB x BD

infinite

were given

if there

=

ax

let

Moreover,

=

BD 9

.

Let

10. 1 1.

And

fide

of

=

z, there

would

arife

Area

AFDB.

~=zz-

t

Let ax*

=

z~>

= ^= fV =

z,

~a^x^

arife

za*x

then

s, or 2

then ;

then

z, or

(an Equation again z,, that is, i-AB

=

= AFDH. AB BD = HDBH. i AB xBD = AFDH. AB x BD

s, or 2

f

=zz',

Let

will

=

x

fo in others.

12. 13.

Ex.

2.

Where z

14.

15. 6.

equ.il to

is

an Aggregate of fuch Quantities.

LetAT-H^ij then^-h ~ then ax Let -{- ^ ,

1

^

zz, or a*x*

to the Parabola,) and there

x

on the other

length, lying

as its negative value insinuates.

Z.

2XX 8.

That

z.

<z

=

*-

J

.

\

= =

z>

&>

+-

=

2r Let 3*i 4** z ; then 2x^ x ^ Ex. 3. Where a previous reduction by Divifion is required. -

Let there be given j~, =.& (an Equation to the Apollonian in injinittun, it will be Hyperbola,) and the divifion being performed 17.


FLUXION s

l%e Method of

_ ^ 4.

x __

5

^l

?f

And

&c.

,

Let there be given ^

8.

~=i

=

thence (by Prob. 2.) 2

2

=

H- ^-

-i

X

=

2X

2x^

=

divifion

will be

it

3

L

l

-f-^

r

1#

+ 7^

1

I3AT

=z,

And

&c.

=AFDBi

&c.

,

\/ aa

divifion

And + V A<^ &c

x' -fProb. 2.) z T3x3 $x* yx* 20. Ex. 4. Where a previous reduction of Roots.

=

and by

34*%

-f-

'

z

7

A

~!Li X

,

-

' X4 -f- -., A.

X-^1

=HDBH.

&c.

5,

s=

elfe

^

SA.''

19. Let there be given

be z

--^

~J

x

-

3X*

==*, and by

^ XX

1

x 6 , &c. or

x*-{-x*

"

or

a*x

*

A/4

5^/3 1

^^

z= -y

1.

will obtain

you

U~A ^

"x a

/.

as (by Prob. 2.)

thence, 11

the fecond Set of Examples,

in

,

6cc.

will

it

thence (by

8

-

required by Extraction

is

xx, (an Equation to the an infinite multitude Hyperbola,) and the Root being extracted to 21. Let there be

of terms,

it

*

.

will be

given

.

r

22. In the fame given, b

23.

z

(which

And

=

= ax+ manner

,

-h

Circle,)

x'f

x*

4-**

^x

_

-r'-g-x^,

=

-

xx were

z=ax

And

6cc.

therefore

bx x

And

H-irf

6<

=

-h 43^4, &c.

-f^ T

z=a-\-

.ix*

24^

I

-

-

.\

gjs

j

occ.

-

/'*v3

-,

4

thus v^~ZT7~

z=i-+- T^-V*

gives

-- ---

=

=

z

xx, (an Equation again to the Cir-

by extraction of the Root it ^* Jf3 whence 2; <7Ar -f25.

&c

TT^

77^

the Equation z \/aa would be produced

TT ___ Thus s === v//z<* -^- AV .

,

&c. whence

r, fit*

were given z-=\/x xx, (an Equation there would arife Root the by extracting

Vz

cle,)

-7

x

X

~

I I

to the Circle,) there

1

24.

X 6

if

-

7-

~\ Q f,9.*if. f ,l

a

fo if there

the

to

ft

ii2a s

^Oi.J

alfo

is

*

z=i a 4-

as in the foregoing ss

-\-

,

&c.

^, by a due reduction gives

then 2

=

AT -f-

^ + V^ V^ 3

-f-

_l_^

T

S &c.

T

'

- Vo^

26.


and INFINITE SERIES. Thus

26.

z=a

Root, gives

+

gfr C

-t-A'

Ex.

x

:

:

''

1

Where

5.

5

But

-fA

5

~

-

if z~'

by

this

will be

2x*z

cz* f?

.v

V

2;

+

A

**

is

by the

required,

Equation z>

+

c *z -f-

2x ?

-+- c*

=

c

=

-

32'

=

z

2"

2rc

r^

-

At T

&c.

correfponding Areas, i.v 1

.

z-=ax

o were the

.v-f- !i'

c

x

a

Root;

-,

<-

--(_

f

=

refo-

a*z

obtain'd as before

&c. or S Jl, 1

-

three

tb,e

&c.

-1

=7

a previous reduction

whence

zaa

i

X4 8.

*

extradl the Root, and there will arife

o,

c

of

T^t,

6c

=

= + s=

z

or

1

thence

to the Curve, the refolution will afford a three-fold

Equation either

values

3

2.)

elfe

A And

!4^-. &c.

_j_

64.2

29.

by the extraction of the Cubic

~ ~ w &c. and then (by Prob. T> &c. = AFDB. Or *=

lution of an affected Equation. 28. If a Curve be defined

j_

,

= HDBH.

&c. 567*

2a"'

5

-+-

-+-

*=** + -~,

27.

z=l/a*

finally

89

+^

^0,

And

z

hence will

=

ex

&e. and

+

arife

x

the

x*

=

ex

.,S

!

30. I add

24^'

flrr CCC>

nothing here concerning mechanical Carves, becaufe form of geometrical Curves will be taught af-

their reduction to the

terwards.

But whereas the values of z thus found belong to Areas are fituate, fometimes to a finite part AB of the Abfcifs, fometimes to a part BH produced infinitely towards H, and fometimes to both parts, according to their different terms: That the due value of the Area may be alTign'd, adjacent to any portion of the Abfcifs, that Area is always to be made equal to the difference of the values of z, which belong to the parts of the Abfci/s, that are terminated at the beginning and end of the Area. 31.

which

32.

For Inflance

;

to the

Curve exnrefs'd bv the Equation i-^-'xx '

m^^ JTC-


fhe Method of ~,

z=x

found that Now that &c. is

it

_l_ 4_,vS

I

^x

may

FLUXIONS,

}

de-

termine the quantity of the Area MDll, adjacent to the part of the Abfcifs /'B; from the value of z,

which

=

by putting AB z, which

arifes

I take the value of

remains x that Area

WDB.

be jqere will

To

33.

+

-Lx*

Whence

had the whole

Whence

Ab=x,

by putting

+

x>

and there

&c. the value of

-J-x',

if A*, or x, be put equal to nothing, x' -+- -^x', &c. x Area

=

AFDB

the fame Curve

L, &c.

x

&c.

^-x',

x,

arifes

there

found

alfo

is

what

again, according to

z,

is

- -+

==.

before, the Area

5** 1

I 1

__

]V\T\

1

_.

t

I

,_

--

^

oCC

'-

"

*

I

I

T

'

^""1

'

-)

-

1

&/~f* OCC.

*"T

J.

ppr^TOt'f ijCl CIUI C

1

AB, or x, be fuppofed infinite, the adjoining Area bdH H, which is alfo infinitely long, will be equivalent to if

-f-

&c.

.

For the

vanifh, becaufe of

34.

To

i

-4-

-

=

its

infinite

^-35 -f-

~, &c.

denominators.

the Curve reprefented by the Equation a-\-

found, that

:s

Series

latter

CA

-

z=.ax

Area &/DB.

-. X

But

this

Whence becomes

it

is

that

infinite,

toward

x

= -

^ will

Z,^

it

ax

X

whether x be fup-

and therefore each Area AFDB and ; and the intermediate parts alone, fuch as

pofed nothing, or x infinite

&/H

is

infinitely great,

And

always happens when the Abwell in the numerators of fome of the terms, as in the denominators of others, of the value of z. But when x is in the numerators, as in the firft found the value of only

&/DB, can be fcifs x is found

exhibited.

this

as

Example,

And belongs to the Area fituate at AB, on this fide the Ordinate. when it is only in the denominators, as in the fecond Example, that value, when the figns of all the terms are changed, belongs to the whole Area infinitely produced beyond the Ordinate. 35. If at any time the Curve-line cuts the Abfcifs, between the

z,

b and B, fuppofe in E, inftead points of the Area will be had the difference of the Areas at the diffe&/E*

BDE

rent parts of the Abfcifs ; to which if here be added the Rectangle

t

he Area t

dEDG

will be obtain'd.

G


and INFINITE SERIES. 36. But

that when x of only one dimension

chiefly to be regarded,

it is

divided by

in the value

of

&

the Area corre; term belongs to the Conical Hyperbola ; and thereAs is done in fore is to be exhibited by it felf, in an infinite Series

any term fponding

is

to that

:

what

follows.

~ glA

fl3

'= z, be an

77. Let ax J it

becomes

Z

=

xx

-f-

=

z

-

Equation to a Curve

2a +- 2X

h^

_

aa

;

and by divifion J

and thence

&c. y

X*

2X> l '

l^

2ax

j

-f-

x1

To*

^T

And

&c.

the Area

2*5

aa

Terms

Now

38.

and

1-1 aa

aa

-

to the

Marks

the

xx

zax

,&*.

Where by

I

&/DB

denote the

little

,

Areas belonging

1

and that |^

and

may

|j|

make Kb,

be found, I

x

or

y

to

be definite, and bE indefinite, or a flowing Line, which therefore I call

;' ;

ing to

fo that

will be

it

B, that

-^;

j

=

'

.

'

-{-

y

=

-

x

x

.

and then

if Z>B

of a definite length,

and

\

&c.

-,

40. Moreover,

i>,

-

~

X

definite Line,

=

-

will be x

it

and therefore the whole Area required

the fame manner, 39. After

a

Area adjoin-

or -

therefore,

-*

21 3 2A3

xx H

to that Hyperbolical

Divifion But by J

is,

A4

WDB

=

C

or

_i- ^-^-'

CB

it

AB,

or x,

might have been ufed

would have been

be bifefted in C, and

and Cb and =_)',

for

'twill

CB be

AC

indefinite

;

be affumed to be then making AC

bd= -^ s=-\-

'

-)-

~

&c. and therefore the Hyperbolical Area adjacent

N

2

to


'A Mt&od to the Part of the Abfcifs

Twill be

.&c.

&c.

And

=

2f

Thus

in the

nature of a Curve,

&c.

Whence

6cc.

And

4

r

I

1

4'

~r,

Root

there arifes

,

And

&c.

&c. will be

-f- z,* ~$-

will be

z,

1

=

z

z

Sum

of thefe

equivalent to

x~=

,v

Lxx

=-.

=o,

y

_

-x

-|

denoting the 8

Y TA

the Area

the

5'

3 Equation a

its

'

-f-

*

CB

1

Si"*-

'

-

Jf

-\~

7-

41.

V

= -~ = ? - ~ + ~ - ^ + st^

+ ~r

l ,

f

Areas

DB

a

will be

therefore the Area adjacent to the other part of the Abfcifs 11 7*

1

"

alfo

&C

FLUXIONS,

of

J

A

X

8

_

_1

Six'

7

T, &c. &c.

_ ^

&c.

-

-

that

is,=:|.v

-

i

A

i

_!_

'

ox

I

KC

/^r^ '

^'

.'X TA

Six<