Page 1


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

-

for the moft pnrt, may be very 42. But this Hyperbolical term, the avoided, by altering commodioufly beginning of the Abfcifs, it or that is, by increafing diminiihing by fome gi\ en quantity. As

in the

=

*

?

former Example, where

+

z was

the Equation to the Curve, if I fhould make beginning of the Ablcifs* determinate of to be Al> and fuppofmg any length 4/7, for the renow write fliall x I Abfcifs Thst is, if I dimiB, mainder of the x a -finftead of it will nifti the Abfcifs a, by writing v

vv

b to be the

:

by

^~^,.

-become

&

_!! 273

=

whence

Ci

x,

~> and

arifes

= \ax

s

j

Area

'

4

^ z 4--^'bia-' &c.

=

.

thus by affuming another and another point for the beArea of any Curve may be exr-ivib'd an ginning of the Abfcifs, the

43.

And

infinite variety

of ways.

44. Alfo the Equation into the

two

rj-p

=z --

-2

infinite Series z,

.V

**--}-;

&c. where there

is

might have been refolved "

X

1

found no 2

-+- "-^ X }

Term

&c.

a -f

divided b} the

.v

fir ft

Power


and INFINITE SERIES.

93

But fuch kind of Series, where the Powers of A* afcend in the numerators of the one, and in the denominators infinitely of the other, are not fo proper to derive the value of z from, by

Power of

x.

Arithmetical computation,

when

the Species are to be changed in-

Numbers.

to

difficult can occur to any one, who is to un45. Hardly any thing in fuch a after value of the Area the dertake Numbers, computation Yet for in the more is obtain'd Species. compleat illufhation of the I mall add an Example or two. foregoing Doctrine, be propofed, the Let Hyperbola 46. whofe Equation is \/x-+-xx=z; its Vertex be-

AD

ing at A, and each of

its

Axes

equal to Unity.

is

From what goes before, its Area ADB=-i.v> A*' -+- T'T* ? -+T^P*'"' &c that j'^ "

1

'

+

s x* into Lx -+- x* T T .v T4T V y T .v &c. which Series may be infinitely produced by '

*

is

'

4

-

>

multiplying thelaft term continually by the fucceeding terms of this ~~'" q J 5 That is &c. Proereffion i- #. .v x Xi A 2 S 8.-n 10.15*. 47 6-9 1 the firft term I_3 x makes the fecond term -L.v* : Which x

^^r

^..v

"

'

2 -5

-~ x makes the l

multiply 'd by

^x

tuin.

Now let AB Number

y

x T>

for

~

This

.v,

being

&c

-^3333333> '

v

into

term

TV-vl

'

by

this

third

:

makes T T .v ? the fourth term; and

tiply'd

or

^^'v

-

-

Which mulfo

ad

hifini-

be affumed of any length, fuppofe ^, and writing and its Root 4 for x*, and the firft term ^x^ reduced to a decimal Fraction, it becomes

This into

makes

'- ^

makeso.oo625 the fecond term.

0.0002790178, &c. the

third term.

And

4-7 4

But the term?, which I thus deduce by degrees, I in one, and the negadifpole in two Tables; the affirmative terms tive in another, and I add them up as you fee here. fo

on

for ever.

-i-o.


"The

94

+

Method of FLUXIONS, 00002790178571429

0.0833333333333333 62500000000000 271267361111 5135169396 144628917 4954581 190948 7963

34679066051 834^65027 26285354 961296 3 86 7 6

1663

75

_

35 2 1

0.0002825719389575 0.0896109885646518

1

-f-

4- 0.0896109885640518

"0^3284166257043

fum of the Affirmatives I take the fum of the neand there remains 0.0893284166257043 for the quantity gatives, which was to be found. of the Hyperbolic Area ADB

Then from

the

;

47.

which

Now

let

isexpreffed by the equation

whofe Diameter is its Area AdB goes before

that

AdF

the Circle

is,

be

\/x

propofed,

xx

=z

>

unity, and from what will

be -!#*

..#*

&

I n which Series, fince c> T T xi -fT^i from differ the terms of the Sethe terms do not which above exprefs'd the Hyperbolical Area, unlefs in the ries, ; nothing elfe remains to be done, than to Signs -4- and terms with other fignsj that is, by numeral fame conned: the of both the afore -mention'd tables, fubtracting the connected fumsfrom the firft term doubled 1 6 0.1666666666666, 93 0.08989 3 560 503 &c. and the remainder 0.0767731061630473 will be the portion A^B of the ciicular Area, fuppoiing AB to be a fourth part of the And hence we may obferve, that tho' the Areas of the diameter. Circle and Hyperbola are not compared in a Geometrical confideration, yet each of them is dilcover'd by the fame Arithmetical com'

putation.

48.

The

portion of the circle A^/B being found, from thence the may be derived. For the Radius dC being drawn,

whole Area

multiply Ed, or -^v/S? s-Vs/3'

or

Ulto

-^C, 2 75

-54 12 ^5 8 773^5

cWB;

which added

Sector

ACd

=

to

the

or i,

w '"

^e

Area AdB,

0.1308996938995747, whole Area.

and half of the product

^e

va ^ ue

there

f the Triangle be had the

will

the fextuple

of

which

49.

And


and INFINITE SERIES.

95

49- And hence by the way the length of the Circumference will be 3.1415926535897928, by dividing the Area by a fourth part of the Diameter. 50. To thefe we mail add the calculation of the Area comprehended between the Hyperbola dfD and its Afymptote CA. Let C be the Center of the Hyperbola, and putting

be -^

a+x =BD,

'twill

the Area

=

AFDB

and -^

-=.bd; whence - -

bx

4-

"

4

-*,

&c. and the Area

4-

,

&c. and the fum 0aL>&=. 2ox-\

~ 4- ^ AB = Cb

?

?

4-

,

Now

&c.

let

CA

us fuppofe

=

c

AB

/'/.<

= AF=i,

and

Kb

i.i ; and fubftituting thefe being 0.9, and CB for a, b, and x, the firft term of the Series becomes 0.2, the fecond 0.0006666666, &c. the third 0.000004 ; and fo on, as

or

TL.,

numbers

you

fee in this Table.

O.2OOOOOOOOOOOOOOO 6666666666666 40000000000 285714286 2222222 l8l82

The fum

0.200670695462151

1= Area bdDB. AD

be defired feparately, 51. If the parts of this Area Ad and fubtract the lefler from the greater dA, and there will remain

BA

3-+ -^4and

-jig.

h --

-

for x,

&c.

Where

if

i

be wrote for a and

the terms being reduced to decimals will

b,

iland

thus; O.O

IOOOOOOOOOOOOOO

500000000000 3333333333 25000000 2OOOOO 1667

The fum

o.

= A^

AD, 52-


The Method of

96

Now

FLUXIO N

s,

of the Areas be added to, and fubtracted from,their fum before found, half the aggregate o. 1053605156578263 be the greater Area hd, will and half or the remainder will be the lefler Area AD. 0.0953101798043248 52.

53.

as

By the fame tables thofe Areas AD and hd will be obtain'd when AB and Ab are fuppos'd T ~, or CB=i.oi, and

=

alfo,

d>

if this difference

o.gg, if the numbers are but duly transferr'd to lower places, may be here feen. O O2OOOOOOOOOOOOOC0 O.O30ICOOOOOOOO3OO 66666666666 50020000 4000000

3^

28

Sum

o 020000(5667066(195

Sum

0.0001000050003333

=

AJ

AD.

==AD. 0^

fo putting 0.999, there will be obtain'd

= AD = =

and

A or

or

AB andA=-~o-> orCB=i.oor,

And

54.

and'

Ad= 0.0010005003335835,

o. 0009995003330835.

AF=

CA

AB

and i) putting 55. In the fame manner (if fe Areas will arife, o.2, or 0.02, or 0.002, the

and

A^=o.223 1435513 142097, and ADz=o. 1823215567939546, 0.0 19802 627296 1797, A</= 0.0202027073 175194, and AD o.ooi AW=o.oo2oo2 andAp

= =

From

56.

thefe Areas thus

found

it

will be eafy to derive others, f'

I

by addition and fubtradtion alone. the

fum of

the Areas 0.693

I

For

as

it

2.

\

into

is

-^

=

2,

to the Ratio's

47 I ^5599453 belonging

2

o 8 upon the parts of the Abfcifs 1.2 and 1.2 2, as is known. o.9,)will be the Area AFcPjS, C/3 being the fum 1.0986122886681097 of the Again, fince ^ into 2 3,

^|and

^-

(that

,

is,

infifting

=

=

Area's belonging to and 2, ^-|

Again, as Areas

will

c /3=5;

And

it

thus,

x 10 xo.98 -

is

be

and

~=

and 2

x5=

obtain'd

Area AFcT/3, C/3 being

10,

10x10=100,

7,

=499

=AF

;

and

lox

it is

plain,

i.i

and

=

<

10x100=1000,

= n,

x

and

that the Area

the compofition of the Areas found before,

3..

by a due addition of

AF^/3, 1.6093379124341004 when C/3 T/3, 2.3025850929940457

fince

=

5,

will be the

'

=when 10.

and

^5

'

I

.'

^)

and

AF^/3 may be found by when C/3 i oo i ooo

=

j

i

7>


and IN FIN ITE SERIES,

97

=

of the above-mention'd numbers, AB BF being 7; or any other I to method was a This that llill infinuate, willing might unity. be derived from hence, very proper for the conftrudtion of a Canon

which determines the Hyperbolical Areas, (from which the Logarithms may ealily be derived,) correfponding to fo many Prime numbers, as it were by two operations only, which are But whereas that Canon feems to be derivanot very troublefome. ble from this fountain more commodioufly than from any other, what if I mould point out its contraction here, to compleat the whole ? of Logarithms,

of the 57. Firfl therefore having affumed o for the Logarithm is as for the of and i number the 10, genei, Logarithm

number

done, the Logarithms of the Prime numbers 2, 3, 5, 7, 1 1, the Hyperbolical 13, 17, 37, are to be inveftigated, by dividing which is the Area corAreas now found rally

by 2.3025850929940457, Or which is the fame thing, by mulThus for Inftance, its reciprocal 0.4342944819032518. tiplying by if 0.69314718, &c. the Area correfponding to the number 2, were multiply'd by 0.43429, &c. it makes 0.3010299956639812 the Lorefponding to the number 10:

garithm of the number 2. 58. Then the Logarithms of

all

the numbers in the Canon,

are made by the multiplication of thefe, are to be found And the void the addition of their Logarithms, as is ufual. by be the are to help of this interpolated afterwards, by places

which

Theorem. be a Number to which a Logarithm is to be adapted, A59. Let the difference between that and the two neareft numbers equally diflant

on each

fide,

whofe Logarithms

are already found,

be half the difference of the Logarithms.

Then

and

let

d

the required Loga-

Number

n will be obtain'd by adding d-\+- gr^, &c. to the Logarithm of the leffer number. For if the numbers

rithm of the

expounded by C/>, C/3, and CP, the rectangle CBD or C,&T=i, if n be wrote as before, and the Ordinates pq and PQ^being raifed for C/3, and x for p or /3P, the Area pgQP or ~, -+are

;

^

~ + }

^

&c. will be to the Area pq}$ or +-f&c. as the diffe^, rence between the Logarithms of the extream numbers or 2(i, to the difference between the Logarithms of the leffer and of the middle *-

O

one;


Tie Method of

g8

dx*

dx

-

FLUXIONS,

one: which therefore will be x

-

divifion

is

perform'd, d-\-

*

The two

60.

firft

2n

-4-

dx*

-+-

-f. A' 3

,

A"*

+i

Zfj s

&C.

-4-

that

is,

when the

a

&c.

&c.

terms of this Series d-\-

I 2n

think to be accu-

enough for the construction of a Canon of Logarithms, even tho' they were to be produced to fourteen or fifteen figures; provided the number, whofe Logarithm is to be found, be not lefs rate

And

can give

little trouble in the calculation, beor the number 2. Yet it is not necefunit, all the the of this For the Rule. places by help fary to interpolate are of numbers which the produced by Logarithms multiplication or divifion of the number laft found, may be obtain'd the numbers

than 1000.

caufe

x

is

this

generally an

by whofe Logarithms were had before, by the addition or fubtraction Moreover by the differences of the Logaof their Logarithms. rithms, and by their fecond and third differences, if there be occalion, the void places may be more expeditioufly fupply'd ; the foregoing Rule being to be apply'd only, when the continuation of fome wanted, in order to obtain thofe differences. 6 1. By the fame method rules may be found for the intercalation of Logarithms, when of three numbers the Logarithms of the leffer and of the middle number are given, or of the middle number and of the greater; and this although the numbers mould not be in full places is

Arithmetical progreffion. 62. Alfo by purfuing the fteps of this method, rules might be for the conftruction of the tables of artificial Sines eafily difcover'd, and Tangents, without the affiftance of the natural Tables. But of thefe things only by the bye. 63. Hitherto we have treated of the Quadrature of Curves, which

by Equations confirming of complicate terms ; and that by means of their reduction to Equations, which confift of an infiBut whereas fuch Curves may fomenite number of fimple terms. times be fquared by finite Equations alfo, or however may be comother Curves, whofe Areas in a manner may be confipared with der'd as known ; of which kind are the Conic Sections For this

are exprefs'd

:

to adjoin the two following catalogues or tables thought of Theorems, according to of promife, conflructed by the

reafon

I

fit

my

help

the jtb and Bth aforegoing Propofitions.

64.


and IN FINITE SERIES.

99

firft of thefe exhibits the Areas of fuch Curves as can be the fecond contains fuch Curves, whole Areas may be and fquared compared with the Areas of the Conic Sections. In each of thefe, the letters d, e, f, g, and h, denote any given quantities, x and z the Abfcifles of Curves, v and y parallel Ordinares, and s and t The letters and 6, annex'd to the quantity z, Areas, as before. denote the number of the dimenfions of the fame z, whether it be

64.

The ;

integer or fractional,

JZ1ZZZ2 3 , z

is

l

"=z

s ,

affirmative or negative.

z-=z-~>

or-' 3

,

&+'

=

As

if

=3,

z*, and z*-'

then

=z*.

in the values of the Areas, for the fake of brevity, 65. Moreover inftead of this Radical \Se-{-f& t or </e-t-fzi-\-gz**>, written

and/

R

inflead of </b-t-iz* t

affected.

by which the value of the Ordinate^

is


10O

"fhe

Method of FLUXIONS,

,

t CO rt

I

n

ii

'5

en

e*

I

i

CO

U 3

H

N

s

1 I

* u

N v, S 1 1

1

a

Curve

+ T 1

-

1

N

V CO

'

*~ *J-

^ ~


and INFINITE SERIES. T-

T*

*

II

101

II

II

II

II

II

a

CO

o G

s bo

*

**

c

X

t

II

T

U

1

a f o

^

u

O iM

OJ

01

^

\ ol

X"

i

\O

""* I

I

v

1

I

cno

N

*?>

X X s

*

CO

t

H-

oa'j j?

M M

cr>

-f'

i

N

f

?r M

x

x"

s:


e

IO2

jff>e

Method

-o^

FLUXIONS,

67. Other things of the fame kind might have been added ; but I now pafs on to another fort .of Curves, which be com-

fhall

may

And in this Table or Catalogue pared with the Conic Sections. you have the propofed Curve reprefented by the Line QE^R, the beginning of whole Abfcifs is A, the Abfcifs AC, the Ordinate CE, the beginning of the Area and the Area a^, But the beginning of this or the initial term, (which com-

defcribed

Area,

a^EC.

commences at the beginning of the Abfcifs A, or recedes to an infinite diftance,) is found by feeking the length of the Abfcifs Aa, when the value of the Area is nothing, and by eredling the per-

monly

either

pendicular a^/. 68. After the fame manner you have the Conic Sedlion reprefented by the Line PDG, whofe Center is A, Vertex a, rectangular

Aa and AP,

the beginning of the Abfcifs A, or a, or a, the Abfcifs AB, or aB, or aB, the Ordinate BD, the Tangent meeting AB in T, the Subtenfe aD, and the Re&angle infcribed or adfcribed ABDO. the letters before defined, it will be 69. Therefore retaining or aB and t, x, i;, z, a.%EC

Semidiameters

DT

=

AC ABDP

CE=y,

or

=

aGDB=j. And

AB

befides,

=

BD

when two Conic

=

Sections are

of any Area, the Area of the latter required, for the determination mall be call'd <r, the Abfcifs |, and the Ordinate T. Put p for


and INFINITE

103 Tl

.V3 .

+ BL,

>*,

i?

Q O rt

O

2

S

S ea

u

Q O 14

o

o

c

a

en

O Q

_2

3

Q O

Q O

M OH Q pa

rt

>s| = CO

O eg Q

Q O

oa

rt

-y

CO

5

U

**

V

V

I 4-

o U.

--V-


Method of FLUXIONS,

104

.<v fa.

a

Q Oa

o Q O

c

O

c h

O OJ

.5

o

Pi

Q O

cj

O Q

o on

rt

O

C

(LI

o

O

Q O

m 3

rt

B

2

^1^

X 15

^

"T

K+ I

dina

fr

SJ

4 I

ed

C

t 3|

s

% II

II

3 +

v

a -

ji

3

U

-

V

+

^

I o

4-

4-

H

fe

-V"


and INFINITE SERIES. < Q CO

Q

O

+

u

1

+

5

13

)

$N

+

1

I.Hf 1

4-

+

H

Mj Tf-

1

P

s-

O

^ ^

U S)

V

i

u O It

*-

tt

<5

+ b<j

4 I?

U

'

-V

E

<J-.

o 4-

H-| ^i t.

1=

o

.

X


Method of FLUXIONS,

io6

s

J*

U

i

s

J

to

_u

o

I 3

+

U

+ v.

t/>

o fa

X

x


and INFINITE SERIES, 71. Before

go on

I

107

by Examples the Theorems that

to illuftrate

of Curves, I think it proper to obferve, in the whereas Equations reprefenting Curves, I have 72. I. That all along fuppofed all the figns of the quantities d, e, f\ g, />, and i to be affirmative ; whenever it fhall happen that they are negative, Abfcifs and Orthey muft be changed in the fubfequent values of the alfo of and the Area ninate of the Conic Section, required. numeral II. Alfo the of the Symbols and 0, when they figns 73. Moreare negative, muft be changed in the values of the Areas. are deliver'd in thefe claffes

Theorems themfclvcs may

over their Signs being changed, the Thus in the 4th quire a new form.

ac-

Form of Table 2, the Sign ot d ~~~' becomes -;_ iv,.-j~~^ ,-^ '

being changed, the 3d

x &c.

that

,

Theorem

-}>

I

=}', *"==*, 7=^= cz -f-/a

is,

'

2.w

into

And

3^===^.

The

the fame

is

to be obferved in others.

of each order, excepting the 2d of the ift Taeach way ad infinitum. For in the Series of ble, may be continued the numeral cf co-efficients of the Table Order and the -;d i, 4th are fonn'd by multiinitial terms, (2, 96, 768, Sec.) 4, 16, 6, 8, ro, &c. continually 2, 4, plying the numbers and the co-efficients of the fubfequcnt terms are deinto each other III.

74.

feries

;

rived

1

from the

initials in

A, *

>

,

,

i,

4-, plying by the denominators

the 3d Order, by multiplying gradually by -Li, &rc. or in the 4th Order by multiBut the co-efficients -rV. &C. T> f,

&c. a rife by multiplying the i, 3, 15, 105, numbers i, 3, 5, 7, 9, &c. gradually into each other. d ft h and the ad Table, the Series of the i , 2 3'', 4 c; ", 75. But in Thus having io th Orders are produced in infinitum by diviiion alone.

of'

1

,

.4--1-'

t/x

=

v,

there will arifo

venient period,

==.)'.

Table

i,

Whence __

_il

s

The

firft

it

appears,

putting

s

x=r

,

^

three terms belong to the

that

Area

the

to the ift Species d Jc 3n -

is

7^-

~

1^f

'j

7

^

ift

Order of

cf

this Order. ~

z

--4

+

<:

?r

~

n

r

for the Area of the Conic Section, whofe Abfcifi

*'

is

diviiion to a con-

~z

j~

and the fourth term belongs

.

}

you perform the

in the ift Order, if

,

and Ordinate v

=

d - r--. g

:

P

2


7&e Method of

io8

FLUXIONS,

the Series of the ^th and 6th Orders may be infinitely 76. But the continued, by help of the two Theorems in the 5th Order of As alib the 7th and 8th Table i. by a due addition or fubtraction means of the Theorems in the 6th Order of Table i. and Scries, by the Series of the nth, by the Theorem in the roth Order of Table i. For inftance, if the Series of the 3d Order of Table 2. beto be farther continued, fuppofe 6 and the ift Theorem of the 4>j, :

=

jth Order

of Table

=. this Series

i.

-^-=^f.

to be

8fts~

wll become

4l|

produced, writing

for

^

</,

v/^-h/ 2

=/>

~

1

into

I2e

~

it is

f%>

and 'Qfr'-'S/*' ize

So that fubtrafting the former values of / and RS a J- 1 4J Ii;/?} 10/1/3 / qnez

3>1

5/b~

.

But according to the 4th Theorem of

<x=v, '

~~ 1

/,

ft

__

t

there will remain _,,

-

,

.

Thefe being mul-

ij j

tiplied

a 5th

by

;

and,

Theorem of

=

(if you

pleafe) for

writing xv*, there will arife

the Series to be produced,' ,

v,

and

1

-r-

-

s

!

Some of thefe Orders may As in the 2d Table, the 5th,

77. IV. others.

-~

alfo

=

f.

be otherwife derived from

6th, 7th, and nth, from the So that I might have omitted them,

8th; and the 9th from the loth but that they may be of fome ufe, tho' not altogether necefftry. Yet I have omitted fome Orders, which I might have derived from the ifr, and 2d, as alfo from the 9th and loth, becaufe they were affected by Denominators that were more complicate, and therefore can hardly be :

of any ufe. the defining Equation of any Curve is compounded of 78. V. If feveral Equations of different Orders, or of different Species of the Afame Order, its Area mufl be compounded of the

correlponding taking care however, that they may be rightly connected with For we mufl not always add or fubtra<fl at the their proper Signs. fame time Ordinates to or -from Ordinates, or correfponding Areas to or from correfponding Areas ; but fometimes the fum of thefe, and the difference of thofe, is to be taken for a new Ordinate, or to And this muft be done, when the conftitute a correfponding Area. constituent Areas are pofited on the contrary fide of the Ordinate. reas

;

Huf that the cautious Geometrician

may

the

more

readily avoid this in-


and INFINITE SERIES.

109

d their proper Signs to the feveral Vainconveniency, I have prefix' lues of the Areas, tho' ibmetimes negative, as is done in the jth and yth Order of Table 2. It is farther to be obferved, about the Signs of the Areas, 70. VI. that -f- * denotes, either that the Area of the Conic Section, adjoinis to be added to the other quantities in the value ing to the Abfcifs, or that the Area on the other the ifl fee of t , ( Example following ;) be s is to fubtracled. on the contrary, And fide of the Ordinate denotes ambiguoufly, either that the Area adjacent to the Abfcifs is to be fubtradled, or that the Area on the other fide of the Ordinate Alfo the Value of f, if is to be added, as it may feem convenient. it

comes out

affirmative, denotes the

Area of the Curve propoled ad-

: And contrariwife, if it be negative, it reprejoining to its Abfcifs fents the Area on the other fide of the Ordinate. 80. VII. But that this Area may be more certainly defined, we mull enquire after its Limits. And as to its Limit at the Abfcifs, at the Ordinate, and at the Perimeter of the Curve, there can be no unBut its initial Limit, or the beginning from whence its de-

certainty:

commences, may obtain various pofitions. In the following Examples it is either at the beginning of the Abfcifs, or at an infinite But it diftance, or in the concourfe of the Curve with its Abfcifs. fcription

And wherever it is, it may be found, by be placed elfewhere. of the Abfcifs, at which the value of f becomes ieeking that length an Ordinate. For the Ordinate fo raifed nothing, and there erecting will be the Limit required. 8 1. VIII. If any part of the Area is pofited below the Abfcifs, / will denote the difference of that, and of the part above the Abmay

fcifs.

the dimenfions of the terms in the values of too high, or defcend too low, they may be afcend .v, i;, and /, .reduced to a juft degree, by dividing or multiplying fo often by any be fuppos'd to perform the office of Unigiven quantity, which may as often as thole dimenfions mail be either too high or too low. 82. IX.

Whenever (hall

ty,

83. X. Befides the foregoing Catalogues, or Tables, we might allb conftrucT: Tables of Curves related_tp_ other Curves, which may be the

moftfimple intheirkind; as to <Ja-\-fx* =v, ortox</e-t-fx* =v, So that we might at all times derive the or to ^/e-\-Jx* y, &c. Area of any propoled Curve from the fimpleft original, and know But now let us illuitrute by Exto what Curves it llands related. delivered. been already what has amples.

=

<

84-


no

The Method

^FLUXIONS,

QER

be a 84. EXAMPLE I. Let Conchoidal of fuch a kind, that the Q Semicircle

AC

QH A being defcribed, and

being creeled perpendicular to A Q^_ if the Parallelobe gram QACI compleated, the Diabe AI drawn, meeting the Segonal

R

the Diameter

micircle in

H the'per-

H, and from

HE

be

1C

let fall to

pendicular Curve, whole Area

ACEQJs

E

then the Point

;

will defcribe

a

fought.

make AQ^==a, AC=z, ^.Therefore

CE=y,

and becaufe of the

continual Proportionals AI, 'twill be ECor_>'= AQ^, AH, EC, 86. that this may acquire the Form of the Equations in the make and for z~- in the denominator write z*, and Tables,

--^

Now

=2,

*

for

a*z~-*

flf a -\-x,

or

;]-'

in the numerator,

an Equation of the

>

f=

I

.fo

j

that 4/ v

.J'' ii

T-<

x,

i

will arife_y

=

Species of the ad Order of Table 2,

ift

and the Terms being compared, /

and there

it

will 3

be^ tf

1

1 .*;

= =

rf

3 ,

-u,

e

=

a*,

and xv

and 2s

t.

Now that the values found of x and v may be reduced to a number of dimen lions, choofe any given quantity, as a, by

87.

as unity,

which,

in the value

a*

of v, a>

may be may be

multiplied once in the value of x, and 1 divided once, and And by twice.

^x

1 .v =1', and xv s/"^niTr =^,^/a t: of which the conllradion is thus. 2s, 88. Center A, and Radius AQ^_ defcribe the Qigadrahtal Arch raiie the perpendicular BD meeting ; QDP ; in AC take AB Then the double of the Scclof that Arch in D, and draw AD.

this

means you

l

will obtain

= AH

ADP or

=

will be equal to the '

on

AB.?=) BD, or-y and .vj ADB'-f- aBDP, that is, either ;

2*A

Of which

ACEQ^

Area fought

values the affirmative

aDAP

2s= 2 A ADB 2 aOAD, or=2DAP:

=

belongs to the Area ACEQ, aC^AD belongs to the Area

fide EC, and the negative extended ad infi.ritum beyond EC. 'of Problems thus found 89. The folutions this

For

RE R

made more

elegant.

Thus

in the prefent cafe,

may fometimes

drawing RH

be

the le-

midiameter


in

and INFINITE SERIES. the Sector

QRH

of the Surface

QH A,

is

ACEQ^

EXAMPLE Angular point E

QH

becaufe of equal Arches and DP, half the Sector DAP, and therefore a fourth part

midiameter of the Circle

AGE be a

Let

Curve, which

is defcribed by the whilft one of of the Norma AEF, the Legs AE, the interminate, continually through given point A, being paffes and the other CE, of a given length,

90.

flides

upon the

II.

AF giEH per-

r,

right Line

ven in pofition.

Let

fall

and compleat pendicular and the Parallelogram AHEC ; CE AC z, calling =_y, and EF rf, becaufe of HF, HE, to

AF,

=

=

HA

continual Proportionals,

y=

HAor 91. or 2

will be

it

Now

that the

Area

AGEC may

=

be known, fuppofe

*,

tr-i

=

,

and thence

it

j== ~z^

will be

'

numerator

is

Here

=}'

fl

"ce

z

in the

1

of a fraded dimenfion, deprefs the value of/ by di~ V)

viding by

a

z&, and

it

will be

~

=

I

7=7= ~

ya

*

S> an

of the

Equation

i

ad Species of the ;th Order of Table 2. And the terms being comSo that z 1 a*. i, and pared, it is </= i, /-

'

__ N A

\*~

.i

)

jV/^i

_

e= .v

1

=

/=

_=

-u,

and

xv

5

=

/.

Therefore fince

;

^a-x*

an Equation to a : with the Center A, and diftancq a or Circle EF let the Circle PDQ^be defcribed, which CE meets in D, and let then will AC ^, the' Parallelogram ACDI be compleated ; ACDP xv ^ CD=<u, and the Area fought AGEC *

and z

and fince is a Diameter whofe are equal,

v

is

=

=

=

92.

Ex-


The Method of

112 EXAMPLE

92. Circle

Let AGE be the Ciflbid belonging to with Let DCE

III.

the

defcribed

ADQj

the diameter

FLUXIONS,

AQ..

be drawn perpendicular to the and meeting the diameter, and E. And naCurves in

D =

ming AC

AQj== CA, CE

a

y

;

of

and

CD,

=

and dividing by z,

'tis

X

~~

/

az

'

Therefore zr~ l I

== ^, or

=V

y

CE =.y, y

:,

=

t

continual Proportiowill be CE or

it

nals,

z

becaufe

=

i

,and thence

an Equation or

aai-i

The the 3d Species of the 4th Order of Table 2. 'tis d-=. I, e and i, being compared,

= *AC =

% it

x,

is

xx

</ax

CD 4AADC

3ACDH

x,

==

=

=

v,

and

v,

f=a.

and 3^ thence

2xv

3*

Terms

=

= = =ACDH 2x1;

Wherefore

/.

s

;

fo

that

Area of the Ciflbid Area Segments ADHA

t

ACEGA. Or, which is the fame thing, 3 Area AHDEGA. ADEGA, or 4 Segments ADHA IV. Let PE EXAMPLE 93.

=

therefore

Therefore

=

Conchoid of the Ancients, defcribed from Center G, with the Afymptote AL,. and diftance LE. Draw its Axis GAP, and let fall the OrThen calling AC dinate EC. CE a, and =: z, =.y, GA

be the

firft

=

Ap

c

.

;

becaufe of the Pro-

AL

A C CE :

portionals

GC

CE,

:

it

will be

CE

* 04.

Now

that

its

Area

:

or

:

y

PEC may

be found from hence,

the

And if the Ordinate CE are to be confider'd feparately. paits'of , divided in D, that it is v/^ the Ordinate CE is fo

CD

=

and


and INFINITE SERIES. and

= *\/V

DE

^ CD ;

will be

the Ordinate of a Circle de-

from Center A, and with the Radius AP. Therefore the the Area PDC is known, and there will remain the other of part DPED to be found. Therefore fince DE, the part of the Orfcribcd

part

by which

dinate

=

pofe 2

the

ift

it

and

w,

Species

is

it

defcribed,

becomes

.

j

1

=

x,

\/

i

= =

t

c

1 -+- c* x

,

;

= DE,

of the 3d Order of Table

being compared, itisd=t>, 1 Z

z*

-^/e* f

z*

equivalent to -\/e*

is

an Equation of The terms therefore

2.

i; and therefore

and/=

v, and

fup-

--

zbc l s

-

=

t.

Z.

reduce them to a juft number of 95. Thefe things being found, and dimenfions, by multiplying the terms that are too deprefs'd, fome given Quantity. If this dividing thofe that are too high, by

be done by

=

--

ex

c

c,

there will arife

t

The

:

~

=

x,

x%

c * -t-

</

Conflruclion of which

in this

is

=

v,

and

manner.

96. With the Center A, principal Vertex P, and Parameter aAP, defcnbe the Hyperbola PK. Then from the point C draw the right And it will be, as Line CK, that may touch the Parabola in

K

AP

to

2AG,

fo

EXAMPLE

97.

the Area

is

5.

CKPC

Let the

to the

Norma

:

Area required

GFE

fo revolve

F may

DPED.

about the Pole

upon the right continually angular point Curve PE to be dein conceive the then given pofition ; fcribed by any Point E in the Now that the other Leg EF. Area of this Curve may be as that

G,

Line

its

AF

found,

let fall

GA

and

flide

EH per-

pendicular to the right Line AF, and compleating the Pa-

=

rallelogram

AHEC,

AC

call

=

and 2, CE=j, AG EF=; and becaufe of the AG Proportionals HF EH :

AF,

we

bz .

,

Va

mall

have

Therefore

: :

,

=

:

AF CE or

y

zz b

But whereas </cc

of a Circle defcribed with the Semidiameter

c

zz ;

is

the Ordinate

about the Center

A let


Method of FLUXIONS, PDQ_be defcribed, which CE produced

*fhe

fuch a Circle

let

D

then

;

it

will be

DE

= ^=rS

there remains the Area

tion

Suppofe therefore

=:2, and

:

PDEP

G=^

3

B? or

the hel P of

it

will be

which EqUa ~

be determin'd.

DERQ^to

and

DE= V

sn Equation of the ift Species of the 4th Order of Table the Terms being compared, it will be b-= d, cc =e, and fo that

=

bV cc zz Now as the value

meets ia

i^~ ft

>

^

And

i. j

==/;

l>R=f.

of t is negative, and therefore the Area 98. that its initial Limit / lies ; beyond the Line reprefented by t becomes nowhich for of at feek that z, length may be found, to Therefore continue thing, and you will find it to be c. Q^> that it may be AQ==c, and erect the Ordinate QR.; and

DE

AC

DQRED

will be the

Area whofe value

now found is to know the

b\/cc

zz.

fhould define quantity of the Area 99. If you the at co-extended Abfcifs with it, without and AC, PDE, pofited determine thus it. the Limit QR, you may knowing 100. From the Value which / obtains at the length of the Abfcifs AC, fubtract its value at the beginning of the Abfcifs ; that is,

from

PAGK,

M

;

fubtract

&, and

there will arife the defired

Therefore compleat the Parallelogram fall and let perpendicular to AP, which meets will be equal to the Area and the Parallelogram

quantity A: in

zz

b\/ cc

b\/ LC

zz.

DM

GK

PKML

PDE. Whenever

the Equation defining the nature of the Curve cannot be found in the Tables, nor can be reduced to limpler terms be transform'd into by divifion, nor by any other means ; it muft of Curves related to it, in the manner fhewn in other 101.

Equations

Prob. 8. till at laft one is produced, whofe Area may be known by And when all endeavours are ufed, and yet no fuch the Tables. can be found, it may be certainly concluded, that the Curve probe compared, either with rectilinear Figures, or with pofed cannot Sedions. Conic the 102. In the fame manner when mechanical Curves are concern'd, they muft fir ft be transform'd into equal Geometrical Figures, as is fhewn in the fame Prob. 8. and then the Areas of fuch Geometrical

Curves are to be found from the Tables.

Of

this

matter take

the following Example. 103.


and IN FINITE SERIES. EXAMPLE

115

Let it be propofed to determine the Area of Arches of any Conic Section, when they aie the Figure of the made Ordinates on their Right Sines. As let A be the Center of the Conic Section, " 103.

AQ_and AR

6.

^

the

V

^\

.'

CD

the Semiaxes, Ordinate to the Axis AR, and PD a Perpendicular at the Alfo let point D. AE be the fa id

Curve

mechanical

meeting CD in E; and from its nature

CE

before defined, will be equal to the

ThereArea A EC

Arch QD. fore the

fought, or comthe parallelogram ACEF, the excefs is To required. pleating which purpole let a be the Latus rectum of the Conic Section, and b its Latus tranfverfum, or and CD=_>'; 2AQ^_ Alfo let

is

AEF

AC=z,

then as

is

it

will be

known.

Alfo

Now

i,

-f-

-zz

=y,

an Equation to a Conic Section,

PC= -z, and thence PD = v/^H ~- zz. QD

is to the fluxion of Arch the fluxion of the Abfcifs be fupthe Fluxion of the Arch QD, or of the Ordinate CE,

104. the Abfcifs pos'd

V ^bb

fince the fluxion of the

AC,

as

PD

to

CD

;

if

**+"-*~~ will be i/

will

If

arife

4 .

Draw

z /

therefore

for

in

the

Ordinate

into

this

FE,

or z, and there

fluxion of the

the

CD

take

you

Area

AEF.

CG

-

-zz

V

,

the

Area

AGC, which

is

defcribed

-zz

moving upon AC,

will be equal to the

Area

AEF,

by

CG

and the Curve

AG


n6 AG

Method of FLUXIONS,

77je will be

fought.

AGC

Geometrical Curve. Therefore the Area let z* be fubflituted for z* in the purpofe

a

To

this

= CG,

<

Equation, and

becomes &*-*

it

j-^

\/^-j-,

i;

is

laft

an Equa.-

M

tion of the

ad Species of the comparifon of terms it is d

$= ~s

=

a

=

fo that

*:

That

t.

i

\/ ^bb * is,

CD

the Conftruction of

~]

=

i,

is

'

\/

=

now

=,/=

e-=.i-bb

zz=x. x, DP

a

what

And from

ith Order of Table 2.

v,

Afl

and

-f-

Jj

=

a

-

xx

~ r,

,

and

i>.

and

/

And

/.

a

this is

found.

QK

perpendicular and equal to QA, and thro* draw HI parallel to it, but equal to DP. And the the point Line KI, at which HI is terminated, will be a Conic Section, and the comprehended Area HIKQ^will be to the Area fought

At Q^ erect

105.

D

PC

as b to a, or as

to

AEF,

AC.

Here obferve, that if you change the fign of b, the Conic Section, to whofe Arch the right Line CE is equal, will become an if you make b the Ellipfis becomesEllipfis; and befides, And in this cafe the line KI becomes a right line parallel a Circle. 106.

=

,

the Area of any Curve has been thus found and con107. After confider about the demonftration of the conwe fhould ftrucled, ftruction

that laying afide

;

be, the

may

become

fit

for publick view.

monftrating, which I mail

Algebraical calculation, as much as adorn'd, and made elegant,, fo as to And there is a general method of de--

all

Theorem may be

endeavour to

iiluftrate

by the follow-

ing Examples.

Demonftration of the Conjlruflion in Example 5. 1

08. In

Arch PQ^take a point d indefinitely near 113.) and draw de and dm parallel to DE and

the

(Figure p. and meeting of the Area ment

DM

Area

LMKP.

AP

p and

in

Draw

But

it

therefore Dj>

:

::

pd

: :

will

will be

DE^/ be the momoment of the

the

AD, and conceive the indewere a right line, and the triand therefore D/> pd:: AL LD.

and ALD will be like, HF EH AG AF that

angles -D/^/

:

Then

D,

DM,

the femidiameter

fmall arch ~Dd to be as finitely it is

/.

LM//

PDEP, and

to

ML

:

:

;

DE.

:

is,

AL LD

Wherefore

:

::

:

ML

Dp DE x

= pdDE; ML

and

:

x

That


and IN FINITE SERIES, That

is,

the

moment DEed

is

equal to the

moment

LM;;//.

117 And

demonflrated indeterminately of any contemporaneous it is plain, that all the moments of the Area all the to are PDEP contemporaneous moments of the Area equal whole Areas compofed of thofe moments the therefore PLMK, and are equal to each other. C^JE. D. fince this

is

moments whatever,

Demonftration of the ConftruSfion in Example

be the momentum of the fuperficies 109. Let DEed A</DA be the contemporary moment of the Segment ADH. Draw the femidiameter DK,

and it

Cc

is

Befides

AC

:

Cc

:

AC

:

zDd-x.

AK

meet

let de :

it is

in c

-,

3.

AHDE,

and

and

Dd :: CD DK. DC QA (aDK) :

: :

:

DE. And therefore 2Dd :: DC aDK :: DE, and Cc x DE AC. Now to the mo:

=

periphery Dd that is, to the tanproduced, the of Circle, let fall the gent and AI will perpendicular AI,

ment of the

AC.

So that AI x zDd 4 moment that 4 Triangles So AD</. Triangles is DE^/. Therefore every moment of the fpace quadruple and the therefore moment of of the contemporary Segment ADH, whole that whole fpace is quadruple of the Segment. Q^E. D.

be equal to zDd x AC

=

=

AD^/=C^xDE= AHDE

Bemvnftratwn


iiS

"The

Method of FLUXIONS,

Demonftration of the ConftruRion in Example 4.

no. Draw from

ce

CE, and

parallel to

at

an indefinitely fmall diflance

and the tangent of the Hyperbola ck t and let fall it,

KM

AP. Now perpendicular from the nature of the Hyperbola it will be AC A? :: to

:

AP AM, and therefore AC? GLq AC?: LE? (or AP V :: :

:

::

')

AP?

AM?

:

and

;

divlfim*

AG/

:

AL? (DE?) ::.AP?: AM? AP?(MK?) And invent, AG: ;

AP

DE

::

little

Area

:

MK.

DEed

is

But the

to the Tri-

DE

angle CKr, as the altitude as to -LAP. Wherefore to are all the contemporaneous

AG

And

to 4-AP.

is

to half the altitude

KM

in. Draw c*/ parallel and meeting the Curve

Then by

q.

litude

of the

Eq

::

(

Dp

x

P

HI

:

that

is,

AG

Demonjlration of the Conjlruftion in Example

and

;

all the moments of the Space PDE moments of the Space PKC, as therefore thofe whole Spaces are in the fame ratio.

AE

in

e,

infinitely near to and draw hi and

the Hypothefis

~Dd=

Triangles Ddp and DCP, (PD) HI, fo that Dp x HI

moment

CD,

6.

(Fig. in p. 115-)

fe meeting DCJ in p Eg, and from the fimiit

will

be

= Eg xCPj AC moment

D/>

:

(Dd)

and thence

(the Eg x AC. Wherefore

EF/e) :: fince PC and AC EyxAC :: CP are in the given ratio of the latus tranlverfum to the Jatus rectum of the Conic Section QD, and fince the moments HI//) and EFfe of the Areas HIKQ^and AEF are in that ratio, the Areas themfelves will be in the fame ratio. Q-^E. D.

E ? xCP

(the

HI/'/.)):

:

:

112. In this kind of demonilrations it is to be obferved, that I affume fuch quantities for equal, whofe ratio is that of equality And that is to be efteem'd a ratio of equality, which differs lefs from equality than by any unequal ratio that can be Thus :

affign'd.

in the laft

demon ftration

fuppos'd the rectangle E^xAC, or FE?/, to be equal to the fpace FEt/j becaufe (by realon of the difference lefs than them, or nothing in comparifon of Eqe infinitely them,) I

they


and INFINITE SERIES.

119

of inequality. And for the fame reafon I they have not a ratio HI//6 ; and fo in others. DP x HI here made ufe of this method of proving the Areas 1 13. I have of Curves to be equal, or to have a given ratio, by the equality, or by the given ratio, of their moments ; becaufe it has an affinity to

=

made

But that feems more natural the ufual methods in thefe matters. which depends upon the generation of Superficies, by Motion or Fluxion. Thus if the Confbuclion in Example 2. was to be demonftrated

ID

line

ID

AGE

From

(Fig. p.i

and

;

:

it is

1

AI

the nature of the Circle, the fluxion of the right is to the fluxion of the 1.) right line IP, as AI to ID ID from the nature of the Curve CE, :

:

:

:

=

=

ID x IP. But CE x ID to and therefore CE x ID And therefore thofe Areas, being gethe fluxion of the Area PDI. nerated by equal fluxion, muft be equal. Q^E. D. fake of farther the 1 For illustration, I fliall add the demon14. flration of the Confrruc~r.ion, by which the Area of the Ciffoid is Let the lines mark'd with points in the determin'd, in Example 3. and the Afymptote fcheme be expunged; draw the Chord from of the Ciffoid. the nature of the Circle, it Is Then, QR and AQ_x CQ^, DQj;

DQ^

=

thence (by Prob.

Fluxion of

And

i.)

DQj= AQjcCQ.

therefore

AQ_:

Alfo from the it is of the Ciffoid nature

2DQj CX^

AD

::

ED

DQ^

AQ^:

:

There-

fore

ED AD

and

EDxCC^=ADx2DQ^,

or

:

:

:

4xiADxDQ^ Nowfmce

DQ__

perpendicular at the end of AD, revolving about

A

;

is

and i AD x QD

=

to the fluxion

generating the Area

ED x CQ^== fluxion generating the Ciffoidal Area Wherefore that Area QREDO QREDO. infinitely long, is geneof the other rated quadruple ADOQ^ Q^E. D. its

quadruple alfo

SCHOLIUM.


The Method of

120

FLUXIONS,

SCHOLIUM. the foregoing Tables not only the Areas of Curves, but of any other kind, that are generated by an analogous quantities of flowing, may be derived from their Fluxions, and that by way

By

115.

the affiftance of this Theorem That a quantity of any kind is to an unit of the lame kind, as the Area of a Curve is to a fuperficial unity ; if fo be that the fluxion generating that quantity be to an unit of its kind, as the fluxion generating the Area is to an unit of its kind alfo ; that is, as the right Line moving perpendicularly upon the Abfcifs (or the Ordinate) by which the Area is defcribed, to a :

Wherefore if any fluxion whatever is expounded by fuch a moving Ordinate, the quantity generated by that fluxion will be expounded by the Area defcribed by fuch Ordinate ; or if the Fluxion be expounded by the fame Algebraic terms as the Ordinate, the generated quantity will be expounded by the fame as the deTherefore the Equation, which exhibits a Fluxion of fcribed Area. any kind, is to be fought for in the firft Column of the Tables, and the value of t in the laft Column will mow the generated Quan-

linear Unit.

tity. 1 1 6.

As

if

\/

_ 1

exhibited a Fluxion of any kind, make it may be reduced to the form of the Equations

-h

and that equal to y,

it

in the Tables, fubftitute z* for z,

7y, And

an Equation of the

comparing the 8a

and thence

+-i8z

^~

,~

it

\S

i

gz -

+-

quantity

1/1

-id

which

-4-

be

will

==

a

Z

will be

it

Species of the

firft

terms,

and

-p.

R> is

'

</

1

-+-

z 43

3d Order of Table

</=

=/.

z~

i,

e=i,f=2.

_,

>

_

Therefore

generated

i.

it

is

the

by the Fluxion

4" 3 17,

And

thus if v'l -f- J^l- reprefents a Fluxion, by a due re7 9

&

duftion, (or by extracting for

2~^) there will be had

-or,

z

out of the radical, and writing

*/s&--! ga*

=7,

the ad Species of the 5th Order of Table 2.

an

_

Equation

of

Then comparing

the

terms,


and INFINITE SERIES. it is

terms,

'

^7

_j_

d=.

=

e

i,

-u,

=

and/=

,

the fluxion v/ quantity generated by

making

it

to be to an Unit of

fuperficial unity

;

its

k

1 1

8.

Thus fuppofing \/i 4-

l

9

no longer inftance, which

imagine for

t

w

found,

the

being ill

be known,

kind, as the Area j*

to the fame,

_~

to fuperficial unity.

+ L^Z

**,

by

by

is

to

fuppofing the

Superficies, but a quantity of anits own kind, as that fuperficies

no longer to reprefent a quantity other kind, which is to an unit of t

j

own

which comes

or

Which

*

= = -

So that x 7

i.

=- =A

and 4 J

121

to reprefent a linear Fluxion,

T

I

to fignify a Superficies, but a Line ; that Line, is to a linear unit, as the Area: which (accord-

is to a fuperficial unit, or ing to the Tables) is reprefented by t, On that which is produced by applying that Area to a linear unit. which account, if that linear unit be made e, the length generated

by the foregoing

fluxion will be

~

.

And upon

this

foundation

be apply'd to the determining the Lengths of Curve-lines, the Contents of their Solids, and any other quantities whatever, as well as the Areas of Curves.

thofe Tables

may

Of I.

^uejlions that are related hereto.

To approximate

method

Areas of Curves mechanically, that the values of two or more

to the

this, right119. The lined Figures may be fo compounded together, that they may very the value of the Curvilinear Area required. nearly conftitute

120.

Thus xx

is

for the Circle

= zz

AFD

which

is

denoted by the Equa-

tion } having found the value of ** #* the Area AFDB, viz. /,** of values fome the &c. Rectangles are to J-x*, .v

be fought, fuch

r

the value

x\/x

xx, or x* of the z* T#* rectangle the value of AD x and or BD x AB, x^/x, #', be are to thefe values AB. Then multiply'd by is

TV#% & c

any

different letters,

that

-

ftand for

R

numbers

indefinitely,

and then to


2^2 Method of

122

FLUXIONS,

added together, and the terms of the fum are to be compared with the correfponding terms of the value of the Area AFDB, that As if thofe Paralleloas far as is poffible they may become equal. \ex^ grams were multiply'd by e and f, the fum would be ex* to be

{$$, &c. the terms of which being compared with

^x

*

or

e

,^x

= = AB

*

/=

to .!#*

equivalent

from the Area

tracted

i^=

+/=-!, and 4., 4 that So x AB x AD -fTr TT ^-BD * For AB is x AD AB x -f. T T nearly. ^-BD &c. which _^.v* _L.,v*, 4.** being fub-

TV*% &c. there arifes

and % Area AFDB very ,

thefe terms

e

=

AFDB,

*

leaves the error only

T'-#

-j-

TV#*,

&c.

Thus

121.

AB x DE -2-#* 128

if

will be

--

were bifected

x\/x

x*, &c.

1024

the rectangle

AB

AB

=

J-x*

AD

Area

--

-\

E, the value of the rectangle

-^x* %xx, or x* with this And compared r 8DE + zAD

x AB, gives

AFDB,

the

&c. which

x*

in

into

error is

being only always lefs than

5760

560

AFDB

even tho' part of the whole Area, But this Theorem may be thus of a Circle. were a quadrant prois the fo into DE, added to a rectangle pounded. As 3 to 2, and DE, to the Area AFDB, fifth part of the difference between TJ^JTJ.

AB

AD

very nearly. and 122. And thus by compounding two rectangles AB x BD, or all the three rectangles together, or by taking in ftill other Rules may be invented, which will be fo more

ABxED

rectangles,

much the more exacT:, as there are more Rectangles made ufe of. And the fame is to be understood of the Area of the Hyperbola, or of any other Curves. Nay, by one only rectangle the Area may often be very

as in the foregoing Circle, to the as v/io x will be 5, rectangle as 3 to 2, the error being only T TAT* -fr-

commodioufly exhibited,

by taking BE to AB to the Area AFDB,

AB

ED

f

The Area being g hen, to determine the Abfcifs and Ordinate. 123. When the Area is exprefs'd by a finite Equation, there can But when it is exprefs'd by an infinite Series, the be no difficulty affected root is to be extracted, which denotes the Abfcifs. So for II.

:

the


W^^

w

**

and INFINITE SERIES. the Hyperbola, defined by the Equation

*

=

^

123

=

we

z, after

have

*

bx

&c. that from the given Area , -^ -+the Abfcifs x may be known, extract the affedled Root, and there - 4will arife x &c. And ,

found

=

that

by a

is,

Thus

124.

-xx

ax 1 i

=

} -+

as

-^

-f-

zz,

after the

were required, divide ab by /z 4- AT, &c. and there will arife z=l>> , s

= is

t*

,

^

equal to x.

the Equation arifes

*

which

Area

I

ax

And a

-xx 3

5<:

is

exprefs'd

found z

=

by the Equation a%x *

^a?x* ,

,

and

/

for x*,

and

becomes

it

&c. and extracting the root

this value

=

is

x

for

-i-j,

= **. ^L

z, the given Area,

i;'

-JjjL

~

.5

to the Ellipfis

Hf_, &c. write

^!^

&c.

-f-

+

^

the Ordinate

if

moreover,

+

/=

being fubflituted inftead of

x

in

zz, and the root being extracted, there

38*' __ '7Sf*

4

Q7^

7

5c c>

So that from

225018

and thence v or ./"I, the Abfcifs # f za*

will be

All which things may be accommogiven, and the Ordinate z. dated to the Hyperbola, if the flgn of the quantity c be changed, only wherever it is found of odd dimenfions.

RO

B.


*The

124-

Method of FLUXIONS,

P R O

B.

X.

many Curves as we pleafe, vohofe Lengths may be exprcfsd by finite Equations.

1o find as

The

1.

this

following pofitions prepare the

way

for

the

foltirion

of

Problem.

If the right Line DC, ftanding perpendicularly Curve AD, be conceived thus to move, 2.

I.

upon

any.

its points G, g, r, &c. will defcribe other Curves, which are equidiftant, and perpendicular to that line : As GK, gk,

all

rs,

&c.

Line is continued each its extremities will indefinitely way, move contrary ways, and therefore there will be a Point between, which will have no motion, but may therefore be call'd the Center of Motion. This Point will be the fame as the Center of Curvature, hath at the point D, which the Curve Let that point as is mention'd before. 3.

II.

If that right

AD

beC.

we

AD

not fuppofe the line but circular, unequably curved, fuppofe more curved towards <T, and lefs toward A; that Center will continually change its place, approaching nearer to the parts more curved, as in K, and going farther off at the parts lefs curved, as in. 4. III. If

to

be

kt and by that means will defcribe fome line, as KG. IV. The right Line DC will continually touch the line de5. fcribed by the Center of Curvature. For if the Point of this line moves towards ^, its point G, which in the mean time pafTes to K, and is fituate on the fame fide of the Center C, will move the fame way, by pofition 2. moves Again, if the fame point towards A, the point g, which in the mean time paffes to k, and k fituate on the contrary fide of the Center C, will move the conmoved in the former cafe, trary way, that is, the fame way that Wherefore K and k lie on the fame fide of while it pafs'd to K. the right Line DC. But as K and k are taken indefinitely f :>r any

D

D

G

points,


and INFINITE SERIES.

125

the whole Curve lies on the fame fide of the plain that is not cut, but therefore and only touch'd by it. DC, the line <rDA is continually more 6. Here it is fuppos'd, that for if its greateft or leaft ; curved towards <T, and lefs towards it is

points, right line

A

Curvature D, then the right line DC will cut the Curve KC ; but yet in an angle that is lefs than any right-lined angle, which is the fame thing as if it were faid to touch it. Nay, the point C in or which the two parts of the at the is cafe Limit, this Cufpid, moft in the oblique concourfe, touch each other ; Curve, finishing and therefore may more juftly be faid to be touch'd, than to be cut, which divides the Angle of contact. by the right line DC, V. The right Line CG is equal to the Curve CK. For conis

7. ceive

in

the points r, 2r, 3;-, ^.r, &c. of that right Line to defcribe the arches of Curves rs, 2r2s, 3^3;, &c. in the mean time that they the motion of that right line ; and approach to the Curve CK, by fmce thofe arches, (by polition i.) are perpendicular to the right all

lines that

touch the Curve

CK, (by

follows that they Wherefore the parts of

pofition 4.)

it

will be alfo perpendicular to that Curve. the line CK, intercepted between thofe arches, which by reafon of their infinite fmallnefs may be confider'd as right lines, are equal to the intervals of the fame arches ; that is, (by polition i.) are equal And equals being added to fo many parts of the right line CG. will be equal to the whole Line to equals, the whole Line

CK

CG. thing would appear by conceiving, that every part of the right Line CG, as it moves along, will apply itfelf fuccefof the Curve CK, and thereby will meafure fively to every part them ; juft as the Circumference of a wheel, as it moves forward by revolving upon a Plain, will meafure the diflance that the point of 8.

The fame

ContacT; continually defcribes.

And

hence it appears, that the Problem may be refolved, by afiuming any Curve at pleaflue A/'DA, and thence by determining the other Curve KC, in which the Center of Curvature of the aftumed Curve is always found. Therefore letting fall the perpendiculars DB and CL, to a right Line AB given in pofition, and in .v and BD v to AB taking any point A, and calling AB be affumed AD let relation between x and v, define the Curve any and then by Prob 5. the point C may be found, by which may be determined both the Curve KC, and its Length GC. 9.

=

=

;

10.


Method of FLUXIONS,

126

=yy

EXAMPLE. Let ax

10.

be the Equation to the Curve, And, by Prob. 5.

which

therefore will be the Apollonian Parabola.

will be

found

^

and DC

* ,

AL=|

= 2if

-+.

a

KC

Which is

ax.

being obtain'd, the Curve determin'd by AL and LC, and

Length by DC.

For

as

its

we

are at

K

C

and liberty to aflume the points let us in where the Curve KC, anf

K

Curvature of the Parabola at its Vertex ; and putting therefore AB and BD, or x and y, to be nothing, it will be fuppofe

=

DC

AK,

to be the Center of

And

-irf.

or

from the former

DC,

and

2

u*

is

the Length fubtracted

indefinite value of

=

GC or KC -^- V aa +. ax Now if you defire to know what Curve

leaves

11. its

this

DG, which being

\a. this

is,

and what

=

is

zt Length, without any relation to the Parabola ; call KL and or AL and it will be &==. LC AT, v, \a 3 x, ^z ' S! == or Therefore CL ax =yy. v, 4v

=

=

/-

'''

which fhews the Curve

j

And

for

writing

its

~z

KC

Length there

for

>r

t

27 #

aa

to be a Parabola of the fecond kind.

arifes

in the value

The Problem

27

= = = = =

llil

^/^aa

-f-

az

a,

by

of CG.

be refolved by taking an Equation, which fhall exprefs the relation between AP and PD, fuppofing P to be the interfeclion of the Abfcifs and 12.

Perpendicular.

For

alfo

may

calling

AP=,v,

CPD to move and PD =/, an infinitely fmall fpace, fuppofe to the place Cpd and in CD and Cd taking CA and CeT both of the fame r, and to given length, fuppofe CL let fall the perpendiculars A^ and fy of which Ag, (which call may meet Cd inf. Then compleat the Parallelogram gyfe, and making conceive

}

=

=z)

y

x,y, and

z

the fluxions of the quantities

,v,

y,

and x,

as

before it


and IN FINITE SERIES. it

Ae

will be

And A/:

P/>

:

::

A/

CA

A?P

:t

:

C

'

All*

Then

P.

" ?

Q"P

:

127

CA]

1

Ae:Pp

a>quo t

TT

::

::

'

^11

:

CP.

But P/> is the moment of the Abfcifs AP, by the acceiTion of which it becomes Ap ; and Ae is the contemporaneous moment of the perthe decreafe of which it becomes fy. pendicular Ag-, by fore Ae and Pp are as the fluxions of the lines Ag (z) and that

as

is,

is

z and x. Wherefore a i CAI AgT

=~*

*

Cgl

*

CP_=

:

=

z

::

&&, and

:

CP.

CA

=

And

fmce

i ;

will be

it

it

an uniform fluxion, to which the reft are to be be that fluxion, and its value is unity, then CP

three x,y, and

x

z

13. Befides

zz )

x

(x),

Moreover fmce we may aflume any one of the

.

referr'd, if

2,

~-

There-

AP

it

: :

~z

for

CA

is

(i)

CP CL :

Z z.

j

=

Ag

:

(z}

therefore

;

Laftly,

CP

::

it is

PL; alfo 2Z PL :

=

drawing /^parallel to the

CA ,

(i)

and

:

CL

Cg

=

infinitely fmall

X

Arch D</, or perpendicular to DC, P^- will be the momentum of DP, by the acceflion of which it becomes dp, at the fame time that AP becomes A/>. Therefore Pp and Pg are as the fluxions of AP (x)

PD

and

triangles

fame

that

(;'),

as

is,

ratio,

it

will be

y

=

and

i

Ppq and CAg, fmce

CA .

y.

Therefore becaufe of fimilar

and Ag,

or

Whence we

i

and z,

have

are

in

this folution

the

of

the Problem.

the propofed Equation, which exprefles the relation between x and^x, find the relation of the fluxions x and y, (by Prob. i.) be had the value of _)-, to which z and putting x i, there will 14.

From

=

Then fubftituting z for/, by the help of the lafl Equaequal. tion find the relation of the Fluxions x,y, and z, (by Prob. i.) and Thefe fubftituting i for x, there will be had the value of z. is

again

being found

= CL;

make

^21= CP, Z

C

zx

CP

=

PL, and

CP

x v/

1

yy

be a Point in the Curve, any part of which Line CG, which is the difference of the to the Curve \)d from the points tangents, drawn perpendicularly

KG

is

and

will

equal to the right

C and K,


7%e Method of

I2 8 Ex. Let

15.

between

tion

Trob.

a

= y

I

it

i.)

=

be the Equation which exprefles the and PD ; and (by

ax=yy

AP

will be

ax= 2yy,

firft

Then zyz -f- zyz

2yz.

z.

Thence

yy

FLUXIONS,

it

=

is

rela-

or

o, or

CP

=

l_J

4-vv

And from CP and PL

taking

x.

=

and

AL

=

away y and

away y

CD ~ Now

there remains

and

aa.

?a

.

x, becaufe

aa I

,

take

c

when CP and

have affirmative values, they fall on the fide of the point P toand A, and they ought to be diminiihed, by taking away wards the affirmative quantities PD and AP. But when they have negative will the fide fall on of the point P, and then values, they contrary they muft be encreafed, which is alfo done by taking away the affirmative quantities PD and AP. in which the point 1 6. Now to know the Length of the Curve, of between two its and C ; we rauft ieek C is found, any points the length of the Tangent at the point K, and fubtradt it from CD.

PL

D

K

K were the point, at which the Tangent is terminated, when and CA Ag, or i and z, are made equal, which therefore is fituate in the Abicifs itfelf AP ; write i for z in the Equation 2yz,

As

if

a=

whence that

is

Therefore for y write ^a in the value of CD, and it comes out And this is the length a.

a=2y. in

,

DG

the difference between of the Tangent at the point K, or of ; which and the foregoing indefinite value of CD, is -i#> that

--

is

GC,

to

which the

part of the Curve

KC

is

equal.

may appear what Curve this is, from AL (hav17. firft its ing changed fign, that it may become affirmative,) take AK, which will be ^a, and there will remain KL %a, which

Now

that

it

=

call /,

and in the value of the

aa anc

^

->

is

l

^ere

l

a "fe

line

CL, which

\/^at

=

v

;

call

or

v, write

=

vv,

for

which

an Equation to a Parabola of the fecond kind, as was found before. i a,


and INFINITE SERIES.

129

the relation between t and v cannot conveniently be reduced to an Equation, it may be fufficient only to find the lengths PC and PL. As if for the relation between AP and PD the Equa3 were affumed; from hence (by Prob. i.) tion 8.

1

When

^x-^-^y

firft

there arifes a

and therefore given

PC

=

it

=o

_}'

1

4-^*2

is

z

""',

=

and

y*z yy

PL

aa

=

then

o,

z=

and

, '

=

aaz aa

y*z=o,

zyyz

Whence

.

are

yy

2rxPC, by which

the point

C

is

And the length of the Curve, determined, which is in the Curve. be known will fuch between two points, by the difference of the

two correfponding Tangents, DC or PC y. i, and in order to determine 19. For Example, if we make fome point C of the Curve, we take y 2 then AP or x becomes

a=

=

.y

3"'.v_ _

T'

Zaa

Then

.

-_. T> zz

*

T>

determine another point,

to

AP=6,

=i, z

=

;

PC"1V if

we

and rLl PI ana

2>

take

PC=

^'

=

3,

-

it

? will be

andPL=

ioi. 84, >ir, Which being had, if y be taken from PC, there will remain 4 in the firil cafe, and the j 87 in the fecond, for the lengths difference of which 83 is the length of the Curve, between the two

DC

points found C and c. 20. Thefe are to be thus underftood, when the Curve is contiand C, withnued between the two points C and c, or between For when out that Term or Limit, which we call'd its Cufpid.

K

one or more fuch terms come between thofe points, (which terms are found by the determination of the greateft or leaft PC or DC,) the lengths of each of the parts of the Curve, between them and the muft be feparately found, and then added together. points C or K,

PROB.

XI.

To find as many Curves as you pie ofe, whofe Lengths may be compared with the Length of any Curve propofed, or with its Area applied to a given Line y by the help of finite Equations.

performed by involving the Length, or the Area of the in the Equation which is affumed in the foregoing propofed Curve, the relation between AP and PD (Figure to determine Problem, Eut that z, and z may be thence derived, (by Art. 12. pjg. 126.) i.

It

is

S

Prob.


7%4 Method of

130

FtuxioNS,

Prob. i.) the fluxion of the Length, or of the Area, muft be firft difcowr'd. 2. The fluxion of the Length is determin'd by putting it" equal to the fquare-root of the fum of the fquares of the fluxion of the Abbe the perpendicular OrdiFor let fcifs and of the Ordinate. the Abfcifs and MN, nate, moving upon be the propofed Curve, at which let

RN

QR RN terminated. NR=/, and

=

s,

Fluxions ceive the

nr

s,

/,

Line

and

MN

Then calling QR='i>, and

is

their

refpeclively ; coninto the place

<u

NR to move

and letting to nr, then RJ, sr,

infinitely near the former,

fall

RJ perpendicular

_ M"

v

^

N"

and Rr will be the contemporaneous moments of the lines MN, NR, and QR, by the accetfion of which they become nr, and And as thefe are to each other as the fluxions of the fame

M,

lines,

=

and becaufe of the right Angle Rsr,

Rr, or

\/V

-f- f-

=

it

>/R/

will be

-f-Tr*

<v.

But to determine the fluxions s and t there are two Equationsand NR,. required; one of which is to define the relation between or s and /, from whence the relation between the fluxions s and tis to be derived ; and another which may define the relation beor in the given Figure, and of AP or x in that retween 3.

MN

MN

quired, x or i

NR

from whence the

may Then

relation of the fluxion s or t to the fluxion

be difcover'd.

being found, the fluxions y and z are to be fought by a third aflumed Equation, by which the length PD or y may be 4.

defined.

DC

<u

Then we

= PC

y,

are to take

as in the

PC

= '-^,

PL

=y x PC,

and

foregoing Problem.

ss=tt be an Equation to the given Curve Ex. i. Let as 5. be a Circle; xx as the relation between the which will QR, and the relation between the length of lines AP and MN, Lv=.y, the Curve given QR, and the right Line PD. By the firft it will

=

be as

By

2ss

=

a

the fecond

third

u=y,

~

2tt, or it

is

that

zt

2X is,

2

's=i.

= ^=

as,

And

thence

and therefore -t

z } and hence

^

zt

=v

=. v.

s*-i-t* ==: v.

And by

^'=2;.

the

Which being


and INFINITE SERIES. PC

muft take being found, you

== PC

PC

or

y,

= -^. 1

Where

131

PL=/x PC,

,

DC

and

of appears, that the length cannot be found, but at the fame time 'the

QR-

QR

it

the given Curve muft be known, and from thence the length of the right Line and fo on the is found ; length of the Curve, in which the point C

DC

contrary.

Ex.2. The Equation

6.

an d

irv

And by

^ax-=.^ay.

2iw

third

4^

=

"

3L

Then from hence

ss

the

But by the fecond

as above.

by the

as

i

=

ff

= ^=

make #

remaining,

there will be found

firft

407,

== z

=

=

and therefore ^

s,

or (eliminating

-y,

And

v.

=

i

^

-y)

j,

z.

=

,

st, a + Let there be fuppos'd three Equations, aa denotes an which the Then firft, *s by x, and A: -f- v =}' and therefore '-V" 4- " ', Hyperbola, it is o=rf+/i, or 7

Ex.

=

j.

3.

=

V/M - v/w

= =

-f- tf

-+-

v.

v.

By the fecond it And by the third

is

3*

it

is

=

+

i

and therefore

i,

== y

-u

or

t

+

i

3'

</ss-4-tt=:z; then

it is

from hence

w =s,

that

w

putting

is,

3'

Fluxion of the radical

for the

equal to

^=

iv,

or

|

And

2W7i;.

~

-f-

=

firft

7C'i;,

-^

</"

-t-

^, which

fubftituting

dividing by aw, there will arife

P^

3

then

/',

= = iv

be

it

from thence

there will arife ~ for

if

1.

for

Now

z.

_>'

made

^ s,

and

and

z

in the fivft Example. perform'd as is 8. Now if from any point Q_of a Curve, a perpendicular let fall on MN, and a Curve is to be found whofe length may be

being found, the

known from

reft

is

QV

the length

to any given Line

;

let that

arifes

produced by fuch application be fince the fluxion of the Area

which

is

call'd

<y,

and

its

QRNV reiTtangular parallelogram made upon VN, as the Ordinate or moving line NR by which

And

Area of a E,

QRNV

by applying the Area given Line be call'd E, the length

which

fcribed, to the

is

=

to the

t,

moving Line E, by which the other S 2

fluxion v.

Fluxion of the with the height is

this

is

deicribcd

dcin

the


FLUXION

tte Method of

132

s,

the fame time ; and the fluxions v and } of the lines v and or of the lengths which arife (or s,) by applying thofe Areas to the

MN,

this

by

Rule the value of v as in the

perform'd 9.

Ex.

or

= =v

Then

t.

and y

the

;

Examples aforegoing. be an Hyperbola which

=

Equation, aa -+

firft

being found,

and thence other

will give

=

or ct

;

if for the

the latter will give y

and fubftituting

//

ft

make -r~

v,

or

for

t,

arifes

= z =

j,

reft to

I.)

=tf,

aflumed

x=s

(by Prob. are

= =

whence v

z

becomes

=== CP, and_y x

}

^

then from hence

-g,

be

defined by this

is

two Equations

i

it

Therefore

.

be inquired, and the

QR

Let

4.

will be

it

;

to

is

v= ~ s

given Line E, are in the fame ratio

=~

z= ^

Now

.

hit

CP =n PL,

as

and

y

,

and z

beforehand

thence the Point C will be determin'd, and the Curve in which all fuch points are fituated The length of which Curve will be known from the length DC, which is equivalent to CP v, as is fuffifliewn before. ciently 10. There is alfo another method, by which the Problem may be refolved ; and that is by finding Curves whofe fluxions are either equal to the fluxion of the propofed Curve, or are compounded of the fluxion of that, and of other Lines. And this may fometimes be of ufe, in converting mechanical Curves into equable Geometri:

Curves

cal

;

of which thing there

is

a remarkable

Example

in fpiral

lines. 1

1.

Let

ing upon taining

which

A

AB be AB as

Line given in pofition, an Abfcifs, and yet re-

a right

as its Center,

that arch

AD^

a

Spiral,

BD

an Arch mov-<

at

is

continually terminated, bd an arch indefinitely near it, or the place into which the arch by its motion next

BD

DC

a perpendicular to the arch bdt the difference of the arches, another Curve equal to the Spiral AD, a arrives,

dG

AH

BH

right Line moving perpendicularly upon AB, and terminated at the Curve AH, bh the

next place into which that right lane moves,

^ andHK

~B~<T perpendicular to bb.


And

bb.

and INFINITE SERIES. and HK, infinitely little triangles DG/

in the

133 lince

DC

HK

are equal to the fame third Line Bb, and therefore equal and to each other, and Dd and Hh (by hypothecs) are correfpondent and therefore equal, as alfo the angles at G parts of equal Curves, third fides dC and hK will be equal and K are angles ; the

right

Moreover

alfb.

it

is

AB BD

(CG)

j

-

therefore - A B

=

Ab bC CG. If :

&

fince

dG, and /jK

B,

AB

hb

this

and

AB=*, BD=-y, andBH=>',

therefore

::

(Qb)

be taken

= dC =

*

dG

remain

there will

from dG,

::

:

*

BD

bC

fince

away Call

/6K.

their

:

fluxions

are the

contemporaneous moments of the fame, by the acceflion pf which they become to each other as the fluxions. A, bd and bb, and therefore are lafl in the moments Equation let the fluxions be Therefore for the and there will arife-y as alfo the letters for the Lines,

and

z, v,

y refpedtively,

t

.

fubftituted,

^~ == "

Now

-.y. *'

z be

fuppos'd equable, or the

the Equation will be

reft are refer'd,

which the

unit to

if

of thefe fluxions,

i;

^=)'-

z Wherefore the relation between AB the Spiral is defined, which and v,) being given by any Equation, by the fluxion v will be given, (by Prob. i.) and thence alfo the fluxion and BD, (or between

12.

;',

it

by putting

the line

Ex.

i?.

If the

i.

v.

And

.

^

of which

BH,

or

y,

equal to

(by Prob. 2.) this will give

the fluxion.

it is

Equation jzrzr-u were given, which 2

thence (by Prob.

of Archimedes,

Spiral

take -

,

or -

2?_-r.

,

and there

Which fhews

will

-

i.) -^

=

to the

is

From hence

v.

remain -

=y, and thence (by Prob. 2.) AH, to which the Spiral AD

the Curve

i is

2U

to be the Parabola of Apollonius, whofe Latus reclum equal, is or whole Ordinate always equal to half the Arch BD. If 2. the Ex. Spiral be propofed which is defined 14.

is

2??;

by

the

BH

5

3 Equation a

=a'v

1 ,

or

v

}_

=;^T

there arifes (by Prob. i.)

,

_l_

from which

if

you take ^, or

~,

fl

there will remain i

thence (by Prob.

EU,

AH

2.)

will be

,

2i T

x

produced ^l

=

v.

That

2^ T

=

I

;

i.

;

=-r,

v,

ano

-BD

nrr

3^ being a Parabola of the fecond kind,

t

>


tte Method of

134 15.

Ex.

Prob. (by '

^/- ?,

If the Equation to the Spiral be

3.

-

i.)

FLUXIONS,

2

a ,

.

?.~-

V ac -\- cz

=

v

from whence

;

there will remain

, ~..

=

-

z</"^ =-y,

if

you take away

Now

y.

thence ""-

K

or

fince the quantity

generated by this fluxion y cannot be found by Prob. 2. unlefs it be refolved into an infinite Series; according to the tenor of the Scholium to Prob. 9. I reduce it to the form of the Equations in the firft

column of the Tables, by

fubftituting z*

=.y, which Equation belongs Orderof Table

andf=c,

that

fo

-~

2

-

^

ac

belongs to a Geometrical Curve Spiral

-f-

cz

z,

then

;

it

becomes

to the 26. Species of the 4th

And by comparing the

i.

for

terms,

it is

== f=y.

AH, which

is

d=,e=:ac,

Which

Equation

equal in length to the

AD.

PROB.

XII.

To determine the Lengths of Curves. In the foregoing Problem we have fhewn, that the Fluxion of is equal to the fquare-root of the fum of the fquares of of the Abfcifs and of the perpendicular Ordinate. the Fluxions Wherefore if we take the Fluxion of the Abfcifs for an uniform and 1.

a Curve-line

determinate meafure, or for an Unit to which the other Fluxions are to be refer'd, and alfo if from the Equation which defines the we mall have the Curve, we find the Fluxion of the Ordinate, Fluxion of the Curve-line, from whence (by Problem 2.) its Length may be deduced. be propofed, which is defined by 2. Ex. i. Let the Curve

FDH

--

the Equation

-f-

-

'-

moving Ordinate DB from the Equation (by \ s

Prob.

fluxion of

=_y

3

i.) '

aa

z being

12Z.S.

i,

making the

be had,

=

y,

-/ '

the

be

-h |-f-

AB

=

^

-^ ==

it,

s, and the Jr

J

v-

and y being

Then adding the the fluxion of y. of the fluxions, the fum fquares v/ill

Abfcifs

Then

=y. will

;

X~

and extracting the

root,


and INFINITE SERIES.

=

^

135

= t and thence (by Prob. 2.) ^ Here / ftands for the fluxion of the Curve, and / for its Length. if the length </D of any portion of this Curve were 3. Therefore d and D let fall the perpendiculars db and required, from the points DB to AB, and in the value of t fubftitute the quantities Ab and AB feverally for z, and the difference of the refults will be JD the a, writing La for #, Length required. As if Ab === ?a, and AB t,

becomes

it

t

^

.

=

=

from whence

:

=

then writing a for #, it becomes / the firfl value be taken away, there will remain ;

if

Or

for the length </D.

if

only h.b be determin'd to be ^a, and

AB

be look'd upon as indefinite, there will remain aa for the value of

1

2ft

_i_ -1 24

you would know the portion of the Curve which is reprefented by /, fuppofe the value of / to be equal to nothing, and there If

4.

z*

arifes

and

eredT:

or

t

=

z= --

Therefore if you take AB=-^- > V*z y, 2 the perpendicular bdt the length of the Arch ^D will be And the fame is to be underflood of all Curves ,

or

11%

aa

in general. 5.

..

12

manner by which we have determin'd

After the fame

-^L

^

the

=y

if the Equation be propofed, -flength of this Curve, for defining the nature of another Curve ; there will be deduced

^

.

_lL -=.t\ or 1

if this

3"

La*y?~*. "*

Equation be propofed, *

2_

there will arife *

_

.-

5 ^ -f-i^ '=

=_>', where

or Fraction, Integer o

6

we

t.

Or

in

if

general,

it

is

cz* -{-

u fed for reprefenting any number, either ,-8" (hall have cz* /. is

4&Qi

od<r

=

Ex.2. Let the Curve be propofed which is defined by this "" + \/ #a -t=t^,V; then (by Prob. i.) will be had Equation 6.

_y

= ^^-r

To the

^+

^f*-*

fquare of

1

4* ^

or exterminating y t

which add

i.

y=

and the fum will be

i

'-'</~aa-{- zz. -J-

~ aa

4 4

4- a*

.

and


ttt Method of

136 and

Root

its

2

tain'd

7.

+

Ex.

Equation

i

-f-

aa

FLUXIONS,

Hence (by Prob. * *

t.

be ob-

will 2.) /

^

A

Let a Parabola of the fecond kind be propofed, whofe

3.

z*

is

=

1

ay

Therefore

r==y.

=

*

~ =_y, and

or

,

<

1

=

-+- 2f:

2a a

thence

~

i -+-

.by Prob.

ss

yy

derived

i. is

Now fmce the

.

4<*

length of the Curve generated by the Fluxion / cannot be found by Prob. 2. without a reduction to an infinite Series of fimple Terms, I confult the Tables in Prob. 9. and according to the Scholium belonging to

have

I

it,

/

=

'

v/

1

Z

the lengths of thefe Parabolas

1

-t-

=

And

.

ay*, 2?

thus you

r=

may

z>

ay*,

=

find ay*,

&c. 8.

=

Ex.

Let the Parabola be propofed, whofe Equation 4

4.

*

3

rfy

,

and thence (by Prob.

^=:^;

or

"

Therefore v/

1

-f-

i^7

=

-+- i

</yy

=

will arife

i.)

t.

1^

*

is

=

_y.

This being found,

I

ga

confult the Tables according to the aforefaid Scholium, and by comthe 5th Order of Table 2, I have paring with the 2d Theorem of

sF

=

1

x,

v/i

-f9

^ = v,

and

7

|j=?.

Where x

denotes the

Ab-

the Area of the Hyperbola, and / the by applying the Area %s to linear unity. length which the fame manner the lengths of the Parabolas z 6 =ay', 9. After 7 z* :z=y , z' =ay', &c. may alfo be reduced to the Area of the fcifs,

y

the Ordinate,

and

s

arifes

Hyperbola. jo.

Ex.

Equation

v/ az

5. is

Let the CuToid of the Ancients be propofed, whole __ ; an d thence (by Prob. i.) ' 22,* V az. 2.Z.

^T^jL"

zz=y,

and

.

therefore

-^

^/"^

=^

yy

-f-

'

which by writing 2?

for

^

or

z~\ becomes

^

v/ az"

i

= =

-f- 3

t

;

/,

an Equation of the ift Species of the 3d Order of Table 2 ; then fo that is d, 3 =. e, and ^ comparing the Terms, it ^

=

=

=^5

i

'u,

and

/:

6;

20;'

4</c

i3

AT

My

2iA?

___.l_into

s=f. And


and INFINITE SERIES. And

a for Unity,

taking

by

the

=

f

:

1 1.

to a juft

=

<

Which are thus conftructed. The Ciflbid being VD, AV

AF

Multiplication

be reduced

which, thefe Quantities may xx, menfions, it becomes az

it

is

its

DB

D

= AG = jAV,

Of

or Divifion

number of Diand

v,

a

ax

the Diameter of the Circle to

adapted, Afymptotc, and in Curve with the the ; AV, cutting and the AF Semiaxis AV, Semiparalet the meter Hyperbola

which

37

perpendicular to

and taking AC a mean AB and AV, at C between Proportional drawn perpendiVK and let CA and V

YkK be defcribed

;

the Hyperbola in , right Lines kt and in touch it in thofe points, and cut the let at and /and T; Rectangle

AV,

to

cular

and K, and

<:ut

KT AV

let

AV

AVNM

be defcribed, equal to the Space TK&. Then the length of the Ciflbid will be fextuple of the Altitude VN. 12. Ex. 6. Suppofing Ad to be an Ellipfis, which the Equation

VD

2zz

i/az

=y be

AD

reprefents

;

the mechani-

let

v ''

cal Curve propofed of fuch a nature, that if B</, or_)', be produced till it meets this Curve at D, let BD be equal to the Elliptical Arch &d. that the length of this may be deter-

d

Now

2.zz=. y will give min'd, the Equation \/ az to the fquare of which if i be added, there ariies zy az aa 4 of the fluxion of the arch A.J. To which , the fquare

=y,

Szz

02

if

-^

^ ^-

the fluxion of the Curve-line

AD.

i

be added again, there will

=.*

__ 2ZZ

/ 2y az

is

and

tracted out of the radical, "

--

'

Table e

=

and

,

a

arife

for

Fluxion of the

z

~

ift

,

whofe fquare-root

Where

2,

=rt;

-1 + ,=

fo

that

into,

z=

z be ex-

be written c", there will be Species of the 4th Order of

Therefore the terms being collated, there will

2.

if

=

x,

\/ ux

arife

d=.^a,

_.v.v

=

<


Method of FLUXION

J3 8

s.

is thus; that the right line </G be the center of the Ellipfis, a parallelogram may double of the and its equal to the fedlor AC/,

The Conftruaion of which

i-i.

being drawn

to

made upon AC,

the Curve height will be the length of

AD.

Ex. 7. Making A/3= tp, (Fig, a -+ whofe bola, Equation is v/ let the Curve being drawn be propofed, whofe

%=

14.

i.)

being an Hyper$&, and its tangent <TT

and

CL

;

WD

Abfcifs

is

and

,

its

Ordinate

pendicular

BD, which

length

applying the Area

the

arifes

by

a^To.

to

Now

linear unity. length of this

per-

is

that the

VD

Curve

may be determin'd, I feek the fluxion of the Areaa<rTa, when AB and

flows uniformly,

find

I

'

'tis

ax, fluxion unity.

its

AT

= = t>p

into the altitude

defcribed

, '

-p

v/ b

Ordinate

</a*b

o

-^

=, For

</z, and

/3<^,

its

or v/

whofe half drawn

fluxion

is

-rV za v z

+-

is

the fluxion of the Area

a

by the Tangent

,

,

Therefore that fluxion

<TT.

is

az, and this apply'd to unity becomes the fluxion of the

To

BD.

the fluxion

is

be

putting AB

v/ b

and

to

it

add ~^~ ^ ^~

the fquare of this

BD, and

a>z-\- ibfrz*-

l6

the fquare of

ta

whofe root -^ is the fluxion of the Curve VD. But this 2 and Table of of the Species 7th Order

there

a fluxion of the ift

fl

i,

arifes

^+

,

a

:

the terms being collated, there will be

=

-

=

</,

aab=e,

a*=f,

l and therefore z a*x -fx, and \/a b to one Conic Section, fuppofe HG, (Fig. 2.) whofe (an" Equation v ;) alfo Area EFGH is j, where EF ==%, #, and FG

=g,

=

and */i6bb

-

a*%

+

a&t-

i

=Y

)

=

*-

(an Equation to another Conic Section,


and INFINITE SERIES. ML (Fig. 3.) whofe Area IKLM

Section, (hppofe T/"T

i

*w*

T

\

and Kl_/= TiJ

g

/XT

fl5^Y*tf4y

2aftbb^f

is

where IK

<r,

" Aaabb?

T.2abbs 2

L,aitiy

DJ

that the length of

Wherefore

139 /\

of the Curve

any portion db perpendicular to AB, and make Kb may z ; and thence, by what is now found, feek the value of t. Then make AB=,s, and thence alfo feek for /. And the difference of thefe two values of / will be the length Dd required. 16. Ex. 8. Let the Hyperbola be propos'd, whofe Equation is 15.

VD

be

=

known,

let fall

=)', and

To

\/aa 4- tzz

aa -\-

will be ^/

thence, (by Prob.

the fquare of this add

bz.z.

+

=

bits.

in the Tables, I 'reduce y

it

;= </ 1 t ==

becomes

the root,

t

hence (by Prob.

2.)

will be

Now as this fluxion

/.

i

/ 3

y

4 5

'

7*

z

is

and

;

/ 1

^2:4 H-r 2

1

-f-jaS

Or

(

* >

not to be found firft

&c

-

divifion

by

a d

z

a-

may

=-

had^

and the root of the fum

i,

to an infinite Series

it

"

i.)

extracting ,

c

& ,*

&c.

A And

be had the length of the Hyperbolical Arch

the Ellipfis \/aa bz,z=.y were propofed, the Sign of 17. If b ought to be every where changed, and there will be had z 41 i ^t_s 7 , &c. for the length of its *-z' -^ _f-

^

&

And

Arch.

likewife putting Unity for

b,

it

will be

z

-+-

-^

-f-

Now the of the Circular Arch. &c. for the length 3ii_4_ IJil > O 2V.'' 104 numeral coefficients of this feries may be found adinfinitum t by mul,

I

the terms of this Progreflion tiplying continually S x 9

'

10 x

j

,

,

^-

>

' i

i

Ex. 9. Vertex is V, 18.

VDE AV

Laftly, let the Quadratrix being the Center, and

A

be propofed, whole

the femidiameter of the interior Circle, to which it is adapted, and the Angle

VAE

Now

any right Line

being

a right

Circle

being drawn through A, cutting the in K, and the Quadratrix in D, and

AKD

Angle.

the perpendiculars KG, call AV =.a, to }

AE

DB

AG

=

being c;,

T

let

VK 2

fall

=

x,

and

BD

=

y,

and

it

will


The Method of will be as in the foregoing

Extract the root

&c.

*7 ,

s

,

4 "

js,

and there will

and there

AG GK

it is

:

i.)

which add

i

,

::

= y=- ^

will arife y

(by Prob.

j; 4-

-^ 9Aa9

a

AB

:

BD

^ ^.

AB VR x, AB x GK by AG,

'tis

-^--,

^

and the root of the fum

,

_6o4^

6 4 v7

&c __ ^

1Z /S~S U

,

&c.

And thence,

&c. to the fquare of

will be

\vhence (by Prob.

Arch of the Quadratrix

;

viz.

YD

=

GK.

= =

,

(y), divide ^

&c. will be

, *7?r\/j9; *

i

4-

^ '-J

or the

-

4

4-

whereas by the nature of the Quadratrix

fince

4-

-r~

z= x ^ 4-

arife

a

# 2^j

and

=.z 4-

Example, x

&c. whofe Square fubtract from AKq. or a l s and the

root of the remainder

Now

FLUXIONS.

2.)

/

x 4-

may

^j

il il

-f-

be obtain'd, -f

'

&c.

895025

THE


THE

METHOD

FLUXIONS

of

AND

INFINITE SERIES; O

A PERPETUAL the foregoing

R,

COMMENT TREATISE,

upon


u

.

;


THE

METHOD

of

FLUXIONS

AND

INFINITE SERIES. ANNOTATIONS

on the Introduction

:

OR,

The

SEc

T.

Refolution of Equations by

I.

Of

the

Nature and

INFINITE SERIES.

ConftruElion of Infinite

or Converging Series. great Author of the foregoing Work begins with a fhort Preface, in which he lays down his main defign very concifely. He is not to be here underftood, as if he would reproach the modern Geometricians with deferting the Ancients, or with abandoning their Synthetical Method of Demonftration, much lefs that he intended to difparage the Analytical Art ; for on the contrary he has very nauch improved both and this he in Treatife Methods, particularly wholly applies himfelf in he which has fucceeded to univerial apto cultivate Analyticks, Not but and admiration. that we mail find here fome explaufe of the Method which are very mafterly likewife, Synthetical amples and elegant. Almoft all that remains of the ancient Geometry is indeed Synthetical, and proceeds by way of demonftrating truths it

already

known, by mewing

their

dependence upon the Axioms, and other


:

-tb e

144 other

But the mediately or immediately. fuch Mathematical Truths as be fuppos'd at leaft to be unknown. It afiumes

fir ft Principles, hiiinefs of Analyticks

really are, or

may

Method of FLUXIONS, either

is

to invcftiga'te

thofe Truths as granted, and argues from them in a general ner, till after a .fcries of argumentation, in which the -feveral

manfteps

connexion wjth each other, it arrives at the knowof the ledge propofition required, by comparing it with fomething This therefore being the Art of Invention, really known or given. be it deferves to cultivated with the utmoft certainly induftry. Many of our modern Geometricians have been perfuaded, by confidering the intricate and labour'd Demonftrations of the Ancients, that they .were Mailers of an Analyfis purely Geometrical, which they ftudiouily conceal'd, and by the help of which they deduced, in a direct and fcientifical manner, thofe abftrufe Proportions we fo much admire in tome of their writings, and which they afterwards demonftrated Synthetically. But however this may be, the lofs of that have

a.

neceftary.

Analyfis, if any fuch there were, is amply compenfated, I think, by our prefent Arithmetical or Algebraical Analyfis, especially as it is now improved, I might fay perfected, by our fagacious Author in the Method before us. It is not only render 'd vaftly more univerfal,

and

exterriive

than that other in

probability could ever be, but is Analyiis for the more abftrufe Geome-

likewife a moft

all

compendious Speculations, and for deriving Conftructions and Synthetical Demonftrations from thence ; as may abundantly appear from the trical

enfuing Treatife. 2. The conformity or correfpondence, which our Author takes notice of here, between his new-invented Doctrine of infinite Series, and the commonly received Decimal Arithmetick, is a matter of confiderable importance, and well deferves, I think, to be let in 3. fuller

Light, for the mutual illuftration of both ; which therefore I fhall For Novices in .this Doctrine, tJho' they here attempt to perform. be well inay already acquainted with the Vulgar Arithmetick, and with the Rudiments of the common Algebra, yet are apt to appre-

hend fomething

abftrufe

and

difficult in infinite Series

deed they have the fame general foundation

;

whereas in-

Decimal Arithmetick, Decimal and the fame or Notation is only Notion Fractions, efpecially ftill farther, and rendered more univerfal. But to mew this tarry'd in fome kind of order, I muft inquire into thefe following particulars. Firft I muft (hew what is the true Nature, and what are the genuine Principles, of our common Scale of Decimal Arithmetick. Secondly what is the nature of other particular Scales, which have been, or as

may


and INFINITE SERIES.

145

Thirdly, what is the nature of a which lays the foundation for the Doctrine of infinite general Scale, Scale ot Series. Laftly, I ihall add a word or two concerning that the Root is in which and thcrefoi-e Arithmetick unknown, propofcd to be found ; which gives occafion to the Doctrine of Affected Equa-

may

be, occasionally introduced.

tions.

Firft then as to the

common

Scale of

Decimal Arithmetick,

it is

that ingenious Artifice of expreffing, in a regular manner, all conceivable Numbers, whether Integers or Fractions, Rational or Surd, the feveral Powers of the number Ttv/, and their Reciprocals;

by with the affiftance of other fmall Integer Numbers, not exceeding Nine, which are the Coefficients of thofe Powers. So that Ten is here the Root of the Scale, which if we denote by the Character X, as in the Roman Notation and its feveral Powers by the help of this 3 Root and Numeral Indexes, (X 1000, ico, X 10, X X4 10000, &c.) as is ufual then by ailuming the Coefficients o, i, 2, 3, 4, 5, 6, 7, 8, 9, as occafion (hall require, we may form or Thus for inflance 5X 4 -f- jX 3 -fexprefs any Number in this Scale. 1

=

4X

1

1

=

=

;

+ 8X

Scale,

=

and

1

-rf-

is

3X

will be a particular Number exprefs'd by this as 57483 in the common way of Notation.

the fame

Where we may

from the other way of Powers of X (or Ten) are fupprefs'd, together with the Sign of Addition -f-, and are left For as thofe Powers afcend to be fupply'd by the Underftanding. of Units, (in which is always X, or i, regularly from the place muhiply'd by its Coefficient, which here is 3,) the feveral Powers will ealily be understood, and may therefore be omitted, and the

Notation only

obferve, that this laft differs that here the feveral

in this,

Thus Coefficients only need to be fet down in their proper order. 6 5 3 will oX* (land for the Number 7906538 -+-f-+-6X -f-

gX

yX

^X* -f-3X' -f-3X, when you fupply all that is underftood. And the Number 1736 (by fuppreffing what may be ealiiy -underftood,) will be equivalent to

Integer

Root X,

X

3

-+-

Numbers whatever,

1

6

and the

of all other exprefs'd by this Scale, or with this

7X

3X

-f-

-f-

;

like

or Ten.

The fame

Artifice

is

uniformly carry'd on, for the expreffing of

all Decimal Fractions, by means of the Reciprocals of the ll-vcral c.:c. Powers of Ten, fuch as ^ o, i 0,0 1 ; 0,001 5^1

=

;

=

^

=

which Reciprocals may be intimated by negative Indices. Decimal Fraction 0,3172 (lands for 3X~'-j- iX~~ -f-7X and the mixt Number 526,384 (by {applying what is

;

Thus

:

U

-{-

the

2\~~4 i

underfl

;

becomes


Method becomes

X + 2X> 4

5

-f-

6X

<?/*

FLUXIONS,

-f- 3 X~' -f-

infinite or interminate Decimal Fraction

9 X^'

-f-

1

gX-

nite Series

is

-4-

9X~

3

H-

4

-f-

equivalent to Unity.

Ten and

by the feveral Powers of their Coefficients,

conceivable

9X~

which

are

all

Numbers may be

fracled, rational or irrational

9 X~

5

8X"1

4 X-

and the ; &c. ftands for 0,9999999, & &c. which infi-+- yX~ -f-

,

So that by

this

Decimal

their Reciprocals,

the whole

Scale, (or

together with

Numbers below Ten,)

all

exprefs'd, whether they are integer or at leaft by admitting of a continual ;

progrefs or approximation ad infinitum, And the like may be done by any other Scale, as well as the Decimal Scale, or by admitting any other Number, befides Ten, to be

Root of our Arithmetick. For the Root Ten was an arbitrary Number, and was at firft aflumed by chance, without any previous

the

Other Numbers perhaps be may affign'd, which would have been more convenient, and which have a better elaim for being the Root of the Vulgar Scale of ArithBut however this may prevail in common affairs, Mathemetick. maticians make frequent life of other Scales ; and therefore in the fecond place I (hall mention fome other particular Scales, which confideration of the nature of the thing.

have been occafionally introduced into Computations. The moft remarkable of thefe is the Sexagenary or Sexagefimal Scale of Arithmetick, of frequent ufe among Aflronomers, which expreffes poffible Numbers, Integers or Fractions, Rational or Surd, by the Powers of Sixty, and certain numeral Coefficients not exceeding fiftyThefe Coefficients, for want of peculiar Characters to reprenine. fent them, muit be exprefs'd in the ordinary Decimal Scale. Thus all

if

ftands for 60, as in the Greek Notation, then one of the/e Num53^ -f- 9^' -+- 34!, or in the Sexagenary Scale 53", 9*,

bers will be

Again, equivalent to 191374 in the Decimal Scale. the Sexagefimal Fraclion 53, 9', 34", will be the fame as 53^= -f-

34, which

is

z

Decimal Numbers will be 53,159444, &c. infinitum. appears by the way, that fome Numbers may be exprefs'd by a finite number of Terms in one Scale, which in another cannot be exprefs'd but by approximation, or by a pro-

9|f+ 34~

greffion of

Another

which

,

aa

Whence

in

it

Terms

in infinitum. particular Scale that

has been confider'd, and in fome meafure has been admitted into practice, is the Duodecimal Scale,

which

mon

exprefles

affairs

we

all

Numbers by the Powers of Twelve. So Dozen, a Dozen of Dozens or a Grofs,

fay a

in a

comDozen

of GrofTes or a great Grofs, Off. And this perhaps would have been the mod convenient Root of all otherSj by the Powers of which to


and IN FINITE SERIES. to conftruct the popular Scale of Arithmetick

but that

its

to

as

;

not being fo lig

Multiples, and all below it, might be eafily committed and it admits of a greater variety of Divifors than any

memory Number not much ;

147

greater than

itfelf.

Befides,

it is

not fo fmall,

Numbers exprefs'd hereby would fufficiently converge, or by a few figures would arrive near enough to the Number required; the contrary of which is an inconvenience, that muft neceflarily

'but that

attend the taking too fmall a

Number

for the

Root.

this Scale into practice, only two fingle Characters to denote the Coefficients Ten and Eleven.

Some have which

to admit

would be wanting,

confider'd the Binary Arithmetick, or that Scale in

the Root, and have pretended to and to find considerable advantages in it. TIDO

And

is

make Computations

But this can never it, be a convenient Scale to manage and exprefs large Numbers by, becaufe the Root, and confequently its Powers, are fo very fmall, that they make no difpatch in Computations, or converge exceeding flowly. The only Coefficients that are here necelTary are o and i. Thus i x 2 5 -f- i x 2* -h o x2 3 i x2* -f- i x 2' -f- 0x2 is one of thefe

by

+

Numbers, tation

is

(or compendioufly 110110,) which in the common Nono more than 54. Mr. Leibnits imngin'd he had found

great Myfteries in this Scale.

of Paris, Anno 1703. In common affairs

See the

Memoirs of

the

Royal Academy

we

have frequent recourfe, though tacitly, to and other Scales, whofe Roots are certain Millenary Arithmetick, Powers of Ten. As when a large Number, for the convenience of reading, is diftinguifli'd into Periods of three figures: As 382,735,628,490. Here 382, and 735, &c. may be confider'd as Coefficients, and the Root of the Scale is 1000. So when we reckon by Millions, Billions, Trillions, &c. a Million may be conceived as the Root of our Arith-

Alfo when we divide a Number into pairs of figures, for metick. the Extraction of the Square-root ; into ternaries of figures for the Extraction of the Cube-root ; &c. we take new Scales in effect, whofe

Roots are 100, 1000, &c.

Any Number

whatever, whether Integer or Fraction, may be made a particular Scale, and all conceivable Numbers may be exprefs'd or computed by that Scale, admitting only of integral and affirmative Coefficients, whofe number (including the Cypher c) need not be greater than the Root. Thus in

the

Root of

(Quinary Arithmetick,

which the

compofed of the Powers of the Root 5, the need be Coefficients only the five Numbers o, i, 2, 3, 4, and yet all Numbers whatever are expreffible by this Scale, at leaft by approxiin

Scale

is

U

2

mation,


Method of FLUXIONS, Thus the common Number accu-racy we pleafe.

j^B

77oe

mation, to v/hat would be 4 x 5 4 2827,92 in this Arithmetick I

ox5 H-2x5-f-4x5~ H-3x 5~ I

s ;

as

Powers of 5 by the Imagination only,

common

Scale, this

Number

will be

-+- 2

x 5'

-|- 3

x 5*

we may fupply the we do thofe of Ten

or if

in

42302,43

-\~

feveral in the

Quinary Arithme-

tick.

All vulgar Fractions and mixt Numbers are, in fome meafure, the a particular Scale, or making the Denoexpreffing of Numbers by be the Root of a new Scale. to Thus Fraction is of the minator 1 is the fame as 8 and 8-fx 5 '-f- 3 x j-'j 2 in effect o x ;

+

3

and

25-5-

ther

x^"

reduced to this Notation will be

2x9'

-4-

7x9

1

-4-4X9""

.

And

fo

25x9 of

+ 4x 9'

or ra-

,

other Fractions and

all

mixt Numbers.

A Number to

computed by any one of thefe Scales is eafily reduced by fubftituting inftead of the Root in one what is equivalent to it exprefs'd by the Root of the other

any other Scale affign'd,

Scale, Scale.

=

Thus 6x10,

reduce Sexagenary

to

Numbers

=

to Decimals,

becaufe

60 3 6X ^=2i6X 3 , or|=6X, and therefore | &c. by the fubilitution of thefe you will eafily find the equivalent Decimal Number. And the like in all other Scales.

The

s

1

,

not neceflarily confin'd to be affirmative integer Numbers lefs than the Root, (tho' they mould be fuch if we would have the Scale to be regular,) but as occafion may Coefficients in thefe Scales are

be any Numbers whatever, affirmative or negative, require they may And indeed they generally come out promiffractions. integers or Nor is it neceflary that the cuoully in the Solution of Problems.

Powers mould be always

Numbers, but may be any regular Arithmetical Progreffion whatever, and the Powers

Indices of the

themielves either rational or irrational. come by degrees to the Notion of what

integral

And is

thus (thirdly)

call'd

an univerfal

we

are

Series,

For fuppofing the Root of the or an indefinite or infinite Series. or a to be Scale indefinite, general Number, which may therefore be reprefcnted by x, or y, &c. and affuming the general Coefficients are Integers or Fractions, affirmative or negaa, b, c, d, &c. which as it may happen ; we may form fuch a Series as this, ax* -ftive,

ex* -f- dx -f- ex, which will reprefent fome certain Number, If fuch a Number prothe Scale whofe Root is x. exprefs'd by ceeds in hfif.itum, then it is truly and properly call'd an Infinite x being then fuppos'd greater than Series, or a Converging Series,

lx*

l

_j_

+

Such for example is x \x~ '-\-^.x'--+ ^*~3 , &c. where Unity. the reft of the Terms are underftood ad in/initum, and are iniinuated


and INFINITE SERIES. And

bv, oV. for

its

as

Indices,

And

it

may have ~any dcfcending Arithmetical m xm -+- ^v* -+-"*.. \s, Gfc. \x

we

thus

149 Progreffion

1

l

have been led by proper gradations, (that

arguing from what

well

is

known and commonly

received,

by what

is,

to

difficult and obfcure,) to the knowledge of the Learner will find of which infinite Series, frequent Examples And from hence it will be eafy to in the lequel of this Treatife. make the following general Inferences, and others of a like nature, which will be of good ufe in the farther knowledge and practice of t-hefe Series ; viz. That the firft Term of every regular Series is al-

before appear'd to be

ways the

mo ft

coniiderable,

which approaches nearer

or that

to the

Number

intended, (denoted by the Aggregate of the Series,) than other That the fecond is next in value, and fo on lingle Term any That therefore the Terms of the Series ought always to be difpoled :

:

in this regular defcending order, as is often inculcated by our Author : there is a Progreflion of fuch Terms-/;? infinitum, a few of

That when

the firft Terms, or thofe at the beginning of the Series, are or fhould be a fufficient Approximation to the whole ; and that thefe may come as near to the truth as you pleafe, by taking in ftill more Terms That the fame Number in which one Scale may be exprefs'd by a finite number of Terms, in another cannot be exprefs'd but by :

an

infinite Series,

or by approximation only,

and vice

versei

:

That

the bigger the Root of the Scale is, by fo much the fafter, cafen'.i for then the Reciprocals of the paribus, the Series will converge Powers will be fo much the lefs, and therefore may the more fafely ;

That

T os

by increafing Powers, fuch bx* -+- ex* -|-</.v , &c. the Root x of the Scale mull be underftood to be a proper Fraction, the lefler the better. Yet whenever a Series can be made to conveige by the Reciprocals of Ten, or its Compounds, it will be more convenient than a Series that converges fafter j becaufe it will more eafily acquire the form of the

be neglected as

:

if a Series

ax -^

Decimal

coir

e

4

Scale, to

which,

in particular Cafes,

all Series

are to be ul-

timately reduced. LafHy, from fuch general Series as thefe, which refill are commonly the t in the higher Problems, we muft pafs (by fubftitution) to particular Scales c; Series, and thofe are finally to be And the Art of finding fuch general reduced to the Decimal Scale. their and then Reduction to -particular Scales, and laft all Series, to the

common

Scale of

of the prefent TrcuUic. j

abrtiull-r pares

Decimal Numbers,

Amly

ticks, as

may

ulmoll the whole of be fecn in a good meaiiire'by is

I


Method of FLUXIONS, took notice in the fourth place, that this Doctrine of Scales, and Series, gives us an eafy notion of the nature of affected Equations, or fhews us how they ftand related to fuch Scales of Numbers. In Inflances other of and even of general ones, the particular Scales, the Root of the Scale, the Coefficients, and the Indices, are all fiippos'd to be given, or known, in order to find the Aggregate of the But in affected Equations, on Series, which is here the thing required. I

the contrary, the Aggregate and the reft are known, and the Re ot of the Scale, by which the Number is computed, is unknown and reThus in the affected Equation $x* -j- 3*2 -f- ox* -+- 7*quired. the Aggregate of the Series is given, viz. the Number 53070,

53070, to find x the Root of the be 10, or to be a

Number

Scale.

This

exprefs'd by the

is

eafily difcern'd to

common

Decimal

Scale,

we

efpecially fupply the feveral Powers of 10, where they are underftood in the Aggregate, thus 5X 4 -+- 3X 3 -f-oX 1 -4-oX if

=

53070. will not be

Whence by companion

'tis

+7X' = X=io. As ^x~ ^x~ = x

x

fo eafily perceived in other instances.

3 Equation 4^+4- ax

-f-

3** -f-ox"

-f-

2x

-f-

f

if I

But this had the

1

-f-

2827,92

Ihould not fo eafily perceive that the Root was 5, or that this is a Number exprefs'd by Quinary Arithmetick, except I could reduce 3 * it to this form, 4x5* -+- 2x $ 3*5* 0x5' -f- 2 x 5 H- 4x5

I

;

=

+

+

2827,92, when by comparifon

would preiently apthe be that Root muft So that fought finding the Root of 5. pear, an affected Equation is nothing elfe, but finding what Scale in Arithjnetick that Number is computed by, whofe Refult or Aggregate is -+- 3

x

5~~

it

which is a Problem of great ufe and given in the common Scale this is to be done, extent in all parts of the Mathematicks. either in Numeral, Algebraical, or Fluxional Equations, our Author ;

How

its due place. difmiis this copious and ufeful Subject of Arithmetical Scales, I fhall here make this farther Observation ; that as all conceivable Numbers whatever may be exprefs'd by any one of theie

will inflruct us in

Before

I

or by help of an Aggregate or Scries of Powers derived frcm fo likewife Root any any Number whatever may be exprefs'd by fome fingle Power of the fame Root, by affuming a proper Index,

Scales,

;

integer or fracted, affirmative or negative, as occafion fhall require. Thus in the Decimal Scale, the Root of which is 10, or X, not

Numbers i, 10, 100, 1000, &c. or i, o.i, o.oi, o.ooi, &c. the feveral integral Powers of 10 and their Reciprocals, is, may s the fingle Powers of X' , 1 or 10, viz. be exprefs'd , , only the

that

X

by

or

X, X- X~% 1

,

X--% &c.

X

but alfo refpectively,

X X ,

all

the inter-

mediate


and INFINITE SERIES.

151

mediate Numbers, as 2, 3, 4, Gff. u, 12, 13, Gfr. may be exprefs'd or 10, if we aflame Powers of proper Indices. by fuch fingle '477",&c. 4 =__ Xo/o-.o, &e. g^ Qr jj JOI03> &CX X' Thus 2 3

X

=

=

,

=

X*.s89s,&c. And the like of 2=== X'>7i" 8 &e> 456 Thefe Indices are ufually call'd the Logarithms of all other Numbers. Numbers the (or Powers) to which they belong, and are fo many Ordinal Numbers, declaring what Power (in order or fucceflion) any And different Scales of Loof any Root aflign'd given Number is,

_.X''4'3!>.

&C

-

'

i

:

different Roots of thofe Scales. garithms will be form'd, by afluming thefe how But Indices, Logarithms, or Ordinal Numbers may be conveniently found, our Author will likewife inform us hereafter. All that I intended here was to give a general Notion of them, and to mew their dependance on, and connexion with, the feveral Arith-

metical Scales before defcribed. It is eafy to obferve from the Arenariiu of Archimedes, that he confider'd and difcufs'd this Subject of Arithmetical Scales, had fully in a particular Treatife

or Principles a'^^tl,

which he there quotes, by the name of his which (as it there appears) he had laid the

in

;

foundation of an Arithmetick of a like nature, and of as large an It extent, as any of the Scales now in ufe, even the moft univerlal. a he had that notion of the acquired very general appears likewife, But how far he had accommoDodtrine and Ufe of Indices alfo. dated an Algorithm, or Method of Operation, to thofe his Princimuft remain uncertain till that Book can be recover'd, which ples, than expedled. However it may be is a thing more to be wim'd his great Genius and Capacity, that fince he from concluded fairly thought fit to treat on this Subject, the progrefs he had made in it

was very confiderable. But before we proceed ration with

to explain cur Author's methods of it may be expedient to enlarge a

infinite Series,

Opelittle

and formation, and to make fome general upon Reflexions on their Convergency, and other circumftances. Now their formation will be beft explain'd by continual Multiplication after the following manner. Let the quantity a -+- bx -{-ex 1 -+- <A' -+- ex 4 6cc. be aflumed as a Multiplier, confming either of a finite or an infinite number of their nature

farther

3

,

Terms

;

and

let alfo - -+-

x=

the

Root

will

produce *

3

-

+

.

x

=

If thefe

2Xf?* a

+

o be fuch a Multiplier, as will give

two

are multiply'd

2f_V + "1^5^ + a

a

together,

*i n

V,

they

&c.


152 o

.

wi

tp

and

FLUXIONS,

we

its

=

+ bg

x

fy+"<!

o p

;

dj>

we

+

eg

if

tp+bq -f-

divide

by

/*

ep

^

give us a

-,

x

and

+ Jq x -

f-

dp

+

,

&c.

,

the Series

'

/3 -. -f-

cq -*-

x 11*

tranfpofe,

t*

-

value

=

it

ef+t/f -^-^-?

p* -

x

9

will be

>

j* "*"

aq .

....

,7

which

:

Series,

infight into the nature of infinite plain that this Series, (even though it continued to infinity,) mufl always be equal to a, whatever be fuppofed to be the values of p, q, a, b y c, d} &c. For

may

For

Series in general.

were

may ,

here fubflitute

q

x

j

thus derived,

f x -

q

or if

+

-

TTT

q

y

-

of x

if inftead

ap become -

1

&c.

;

The Method of

the

firft

good

is

it

part of the firflTerm, will always be

removed or deflroy'd

by its equal with a contrary Sign, in the fecond part of the feeond Term. And x- the firfl part of the fecond Term, will be re,

i

i

moved by its equal with third Term, and fo on Aggregate of the whole

we may

a contrary Sign, in the fecond part :

So

Series.

as

-finally to leave

And

here

it is

,

of -the

or a, for the

likewile to be obferv'd,

flop whenever we

pleafe, and yet the Equation will be good, provided we take in the Supplement, or a due part of the next Term. And this will always obtain, whatever the nature of the Series may be, or whether it be converging or diverging. If the

that

be diverging, or if the Terms continually increafe in value, then there is a neceflity of taking in that Supplement, to preferve But if the Series be converging, or .the integrity of the Equation. if the Terms continually decreafe in any compound Ratio, and therefore finally vanifh or approach to nothing ; the Supplement may be Series

as vanishing alfb, and any number of Terms maybe taken, the more the better, as an Approximation to the QiumAnd thus from a due confederation of this fictitious Series, a. tity the nature of all converging or diverging Series may eafily be apprefafely neglected,

Diverging Series indeed, unlefs when the afore-mention'd increafing Supplement can be affign'd and taken in, will be of no

hended.

And

Supplement, in Series that commonly occur, will be generally fo entangled and complicated with the Coefficients of the Terms of the Scries, that altho* it is always to be understood., neverthelef?, ii is often impoffible to be extricated and affign'd. But however, converging Series will always be of excellent ufe, as Affording a convenient Approximation to the quantity required, when In thefe the Supplement aforefaid, it cannot be othei wile exhibited. feivice.

this

tho'


and INFINIT

SERIES.

E

153

and unnflignable, yet continually decreafes the of the with Terms Series, and finally becomes lefs than any along tho' generally inextricable

aflignable Quantity.

The. lame Quantity may often be exhibited or exprefs'd by feveral converging Scries but that Series is to be mod edeem'd that has the Rate of Convergency. The foregoing Series will converge greateft ;

fo

much

the fader, cteteris paribus, as - is lefs

Fraction

For

than Unity.

if

p

is

than q

lefs

be equal

it

or as the

y

to, or greater

than

Unity, it may become a diverging Series, and will diverge fo much the fader, as p is greater than q. The Coefficients will contribute or

little

nothing to

Convergency or Divergency,

this

if

they are

fuppos'd to increafe or decreafe (as is generally the cafe) rather in a fimple and Arithmetical, than a compound and Geometrical Proportion. To make fome Edimate of the Rate of Convergency in this Series, and by analogy in any other of this kind, let k and / reprefent two Terms indefinitely, which immediately fucceed each other in the progrefTion of the Coefficients of the Multiplier a -+bx -if ex* -f-^x 3 , &c. and let the number n reprefent the order or

place of

fented

Then any Term of the

k.

by

Series indefinitely

mud

l'-Jf-~*- where the Sign

-f-

be

may

be repre-

-+- or

accor-

,

?"

ding

as

n

k

a,

1

=

then & fo

=

1',

and the

= = l

/>,

of the red.

grefTion after

Term

m

if

the

be

Terms

l

then

in the

Term will next Teim

-f-

-^~lp"~

-f-

fame

^

7 /."

==

if

will be -f- *_LlL^Z

and the fecond

c,

Alib

/,

firft

?"

cefiive

Thus

an odd or an even Number.

is

]f

.

then

i,

==2j

c

be

And

^~p.

in the aforefaid

will

be any

pro-

two

fuc-

?"

Now

Series.

in

order to a due Conver-

gency, the former Term abfolutely confider'd, that is fetting afide the Signs, mould be as much greater than the fucceeding Term, as conveniently

may

Let us fuppoie therefore that

be.

JL^Jp-i

j

s

i"

than

greater

'

" r

that ^

+

/f?

is

that Ipq -f-

mon

' ~^ ^p",

IpqJ

that - x

(

dividing

all

by the

common

factor c"

\ }

'

t"

than

,

^

or

(

multiplying both by pq,

)

than nip* +- Ipq, or (taking away the comkrf is greater 1 that kf is greater than //.y, , or (by a farther Diviiion,) is

fl

greater

or

greater than unity

;

X

and

as

much

greater as

may

be.

This


7%e Method of

FLUXIONS,

on a double account

; firft, the greater k is in the is in refpect of p\ of ;;;, and fecondly, greater 5* refpecl: in the Multiplier a -\-bx -f- ex* -\-dx>, &c. if the Coefficients a, b, in any decreafing ProgreiTion, then k will be greater than r, &c. are

This

will take effeft

Now

m

will be greater than therefore m. Alfo if q be greater than p, and (in a duplicate ratio) is So that be than will (cater faribus) the degree of /*. greater j* from, the Rate according to is here to be eftimated, Convergency

which

/,

is

greater than

fo that a fortiori k

;

w hich

&c. continually decreafe, compounded its duplicate,) according to which q fhall be fuppos'd to be greater than />.

the Coefficients a, b, with the Ratio, (or rather

The fame

c,

n

A /

Term

the things obtaining as before,

.j_

be

will

i

For if the Series be call'd the Supplement of the Series. continued to a number of Terms denominated by n, then inftead of all the reft of the Terms in itifinitutn, we may introduce this Supvalue of a, inftead of plement, and then we fhall have the accurate Here the firft Sign is to be taken an approximation to that value. Thus if if n is an odd number, and the other when it is even.

what was

n=

i,

*

and confequently

== a.

Or .

c\i

L---a.

Or

f

x

f

if

n

=

and

/=

j

7

3,

And

=.$.

^

== 2,

if

q

k=a, a,

fo on.

ment always compleats the

/=

and

/=

we

<,

ill +

lX

b

c,

.i

then

then

1

Here the taking

in

of

and

a,

tll

e

xt

ff->r f-a p _L_L_I x - -4-

J-If

bb-^-a-j i

value

have

fhall

<{$ -4-

-

i

cq

q

of the Supple-

makes

it

perfect,

whether the Series be converging or diverging ; which will always be the beft way of proceeding, when that Supplement can readily happens, in fuch infinite Series as gewe muft have recourfe to infinite converging Series, nerally occur, wherein this Supplement, as well as the Terms of the Series, are

But

be known.

as this rarely

diminifh'd ; and therefore after a competent number of them are collected, the reft may be all neglected in infinitum. From this general Series, the better to aflift the Imagination, we will defcend to a few particular Inftances of converging Series in infinitely

Let the Coefficients

pure Numbers. <

,,

|

,

^ '

;

,

to, refpectively

^c! (XC<

J

r

.

/>,

c-,

** _

then

d,

^

&c. be expounded by

*

x ^

+

L^_fl^ x ^ H_-2^ x ^_^-+5ix 4, f /

'

or

5 '.

;

a,

That the

27

Series

r.x; ?

7

hence arifmg

3

x 4?

may

4x55.

converge,

5

3'

make/

x

&C. lefs

than


FINITE SERIES.

a?:d IN than q in any given

A

ratio,

TV x

|.x|H-4^x^

Fractions, which

= =

-

fuppofe

&c.

-J.,

~, or

155

=

/>

That

i.

q

i,

=

Series

this

is,

then

2,

of

or by the

computed by Binary Arithmetick, Reciprocals of the Powers of Two, if infinitely continued will Or if we defire to flop at thefe four finally be equal to Unity. and reft inftead of the ad infinitum if we would introduce Terms, the Supplement which is equivalent to them, and which is here

known

to be

T'o-

i,

=

eafy to

is

pounded by

|,

i,

1

3

=

1f

X 47

4f

&c. then

-f,

i

or

infinitely,

-

3

may

it

will be -

by

by introducing the Supplement

;?,

-+-

be fum'd after any number of

~ i,

exprefs'd

-

Thu Series m

&

/

4 X 5?

J*

be continued

i,

^

Hull have 4 -fT | -+Or let the fame Coemdents be ex-

prove.

-i,

4iz^

f

we

x T or TV, y,

j

as

is

_

.

ehhei

Terms

n

infteadof

all

then -2

_f.

;

H-IXJ*

the

Or more

reft.

we make (jr=

particularly, if

=

$p,

^ &c. i, v/hich is a Number 30X;;!' 20X^4 And this is eafily reduced to the exprefs'd by Quinary Arithmetick. Decimal Scale, by writing for -f, and reducing the Coefficients ; for then it will become if we take thefe i. 0,99999, &c. five Terms, mall have exadly we together with the Supplement, --

7-^-. -+-

6x5!

-

liXjS

-f

.

(

~

-f-

2x5

11-

r 6x,i

we make

^^-

6

-

>c

x 3

-12-

+

12x5}

-iiccoo

-

-f-

20x54

77= ioo/^,

here JJ

+

40

~9

X 3

which converges very

=

we

"

30x5'

i

<

co

oooooo -f-

And

4- ^-, 6x;

=

i.

Again,

if

have the Series

fhall

9

x

4 fa ft.

+ -~

Now

-':

4X 5

27

x

locoocooo

we would

reduce this to the regular Decimal Scale of Arithmetick, (which is always fuppos'd to be done, before any particular Problem can be faid to be coinplcatly folved,)

we muit

let

if

when

the Terms,

decimally reduced, orderly

under one another, that their Amount or Aggregate may be tlifcover'd and then they will ftand as in the Margin. Here the Aggregate of the firfc five Terms is 0,99999999595, 0,985 which is a near Approximation to the Amount of the ;

whole lake,

=

infinite Series, or to Unity.

we add

And

for proof-

if,

=

+/

'

to

this

the Supplement

0,00000000405, the wh<

be

.

X

2

_

+

1L 5 '

,

,/'

,-

'" |OJ

Unity exaclly.

There


T

The Method of

3f6

FLUXIONS,

*f

Methods of forming converging

are alfo other

There

whe-

Series,

ther general or particular, which fhall approximate to a known quanand therefore will be very proper to explain the nature of Contity, and to how the Supplement is to be introduced, when

mew vergency, be done, in order to make the Series finite ; which of late it can has been call'd the Summing of a Series. Let A, B, C, D, E, &c. and a, />, c, d, e, &c. be any two Progrcffions of Terms, of which is to be exprefs'd by a Series, either finite or infinite, compos'd of itfelf and the other Terms. Suppofe therefore the firft Term of

A

the Series to be a, and that

Then as

a

or

-}-/>,

the fupplement to the value of a.

is

~a

=

p

That

plement. /A

a

b

xi

f

q

a A x ~~B~

\

R=7

value of the Supplement

nominated by the Fradion

> .

Cc Cc x x

B as

/;

x

(^

I

we

r

7 i

A

==

where

That

is

A

a

-\-

=

s

_,.

r

x

A

11

b

7,

TV-

x

orr

x

Now

B

x

as this

=

Cc -jj-

r.

= (--

a

/A

is

x

the whole

a r-

at

B

x

lafr.

=

-

'>

r-

- x Cc, -rr-a U -+-

That

s.

/;

^

x

Cc

or

s,

T>d x

77

we have

s

A j And

.

= /-

lo

~

is,

A

a x -77Ij

on

c

as far

the value of A.'=a-\-p,

~l)-\-q, where the fecond Supple-

b

B

AA

a

D

TT-C -}- r,

x '^D

finally

x

B

a

g

.,

next Supplement put

for the

x

)

where the Supplement p inent q

....

'

-g-

rr-

'

A

So that

A

..

b

-\-q, or

-XTJ

only afTume fuch a part of it as is denominated

TJ

pleafe.

-

and for the next Supplement put

,

= =

Bl>

x -7^

I

E

= (p=)

.

x value of the Supplement r,

-

and

= (?=)

by the Fraction

,

1S 1S the whole Again, as only aflume fuch a part of it as is de-

I fhall

q>

whole Supple-

the

is

only take fuch a part of it and put q for the fecond Sup-

I will afiimie

is,

this

I fhall

denominated by the Fraction -

is

As

.

order to form a Series,

in

ment,

=

A

is

p

where r

=

A

E

a

g

x

C c D d -7- x -rr U x H,r-e-\- 1. And A a. B A a a -+b .+x ^-c -\

Cc

l>

^~ x -]y

,

4-

s,

b

fo

on ad

(*.

/;

x

D

d

-J7-e,

c

\

-a

r^

A

a

tnfinitum.

Eb x -7

TT\

-O Of* ere.

Kc. where A, B, C, D, E,

C

c

,

x -jj-

and a y

be any two Progreffions of Numbers whatever, whether regular or defultory, afcending or defcending. And when

b, r, d, e, 6cc.

may

it


it

happens

and INFINITE SERIES. thefe Progreffions, that either A

157

=

in

a,

B=^,

or

or

of itfelf, and exhibits the 5cc. then But in other cafes it apvilue of A in a finite number of Terms the value to of But in the cafe of an A. proximates indefinitely faid Progreffions ought to the infinite Approximation, proceed reHated it will be Law. Here to feme eafy to ob"ularlv, according if 1C and k are put to reprefent any two Terms indefifcrvc," that aforefaid Progreffions, whofe places are denoted by the the in nitely number ;/, and if L and / are the Terms immediately following ; then the Term in the Series denoted by n -f- i will be form'd from ___

the Series terminates

:

(v

the preceding

K

=

A, k /t

i

DC

**

and

=

j

A

',

"

fo

Series will

f-*

at this

fo

Series

l=b,

B,

TC

=

/-

As by -^ /. and the fecond

t"1~ipn

L

17

-

-L

-

k

"R

--

n

if

Term

"

i,

will

u

'

Term

A will be

A

B

a,

much

the a,

B

a

== ~tT jr-^* ~7cT happen that L =/,

x

r

Term, and proceed no

approximates Numbers A, B, C, D, &c. and each other refpedively.

Now

it

And whenever it fhall

reft.

ftop

T *

H

and the third

of the

=

L

a,

r>

1

z c,

I

Term, by multiplying

b, c,

r;

then the

And

farther.

catcris paribus,

fafter,

f>

~Tr~

the the

as

d, &c. approach nearer to

fome Examples in pure Numbers. Let A, B,C, D, we &c. i, i, i, i, &c- then 2, 2, 2, 2, &c. and a, b, c, d, &c. fo when And c i -h 1 Hfhall have 2 always, T +* -V> Series will be a the of are Ranks the given Progreffions equals, this we would have If G<~ metrical Progrefnon. Progieffion ftop at

=

to give

=

the next Term, we to be 2, 2, 2, 2, 2,

one which.

For

=

+

&

-

either fuppofe the or the fecond to be

may i,

in either cafe

we

mall have

firft

i,

given

i, i,

L= TC

/, ^

i,

Progreilion i,

that

2, is p*

'tis

all

F ==/,

=

or i. and therefore the laft Term muft be multiply'dby , I Then the Progreffion or Series becomes 2 Tci +T-r-ir~+"T and 'if A, B, C, D, &c. &c. d, c, a, b, 5, 5, +-TT5, 5, Again, + c & &c. =4, 4> 4, 4, &c then 5 H-H T 4 T *-TT, T TTT &c if Or &c. -ior ^. H- T T -4 T 4, A, B, C, D, T!T> and </, *, f d, &c. then 5, 5, 5, 5, &c. 4, 4, 4, 6cc. - S &c. If A, B, C, D, &c. &c. *- T H- ^Tr, 5, 5, 5, 5, i -4- -fV 6 6, 7, 8, 9, &c. then 5 and tf, ^, c, d, &c. T7-f-4-Xy8 ^-xf x-f-y -h x-fx AX 10, &c. If we would have the Series or if we v/oiud find one more Term, or Supplement, ftop here, which fhculd be equivalent to all the reft ad inftnitum y (which indeed -

=

-

+

'

,

=

-^-

= == + =

+

+

= =

=

4=5

-


Method of FLUXIONS, deed might be deiirable here, and in fuch cafes as this, becaufe of thcllow Convergency, or rather Divergency of the Series,) fuppofe F==/j and therefore ~ T mu ^ be niultiply'd by the la ft -

Term. ^ X

3

n

T9

3, 4, 5,

=

So that the Series becomes 5 '* * ^ 4 * 3 10 T vX T vX> T vx TJf T vX T X T vX T TO &c. and i, 2, b, c, d, &c. *

I

= ""7^ ^ '

'

=

1

i,

-

2,

13

Tf r

1.7 -f-

.1.

x

CD

-^-S

-

A R for **> ^ ^> ^> (XU

fx

=

..

2> -->

1-43, 4, &c. then 2 If A,B,C,D,6cc. &c. 3 +|x^xi4-|-|x^xix^5, and ^, b, c, d, &c. 2, 3,4, 5, &c. then i 3, 4, &c. X HT T XT;<i5 T x|4 *T T X T 6 > &c And from Series may infinite other particular Series be eafily de<--,

T-+^x^ =^

6

'

=

+

=

-

this general

which fliali perpetually converge to given Quantities the chief ufe of which Speculation, I think, will be, to iliew us the nature rived,

;

of Convergency in general.

There are many other fuch form'd, which mall converge confliucl a Series

ther with a

that

Rank

flrali

like general Series that to a given Number.

converge to Unity, I fet

&c --"""""'"' '*

-h

As

be readily

if I

down

i,

would toge-

of Fractions, both negative and affirmative, as

here follows.

I

may

-

-


and INFINITE SERIES, both affirmative and negative, which being added together obliquely

with

iiich as are feen

it,

following Series.

i

4-

159

as before,

here below

;

will produce the


j6o

Method of FLUXIONS,

*fi>e

In order to obtation of the ufual praxis of Divifion in Numbers. of the comaa or relblve the divided b to tain Quotient by -f- x, into a Series of fimple

pound Fraction T|T-

Quotient of aa divided by is

^

which write

,

Term, and

this

whence

Then the

}

the Product aa -h

the

find

next

"~

orderly under the find the

^

This

Divifor.

the Divifor by

under the Dividend, from

will leave the

~

Remainder

.

Term (or Figure) of the Quotient, divide Term of the Divifor, or by b, and put the fecond Term of the Quote. Multiply

firft

for

Term, and

the Divifor by this fecond

tracted, to

Term of the Then multiply

firft

in the Quote.

Remainder by the

the Quotient

fet

the

muft be fubtracted, and

it

to

fet

l>

the

find

firft

Terms,

Remainder

laft

;

the Product

from whence

new Remainder -h

"-^bo

.

^r

^ it

muft be fub-

Then

to find the

of the Quotient, you are to proceed with th-is new Remainder as with the former ; and fo on in infimtum. The Qup-

Term

next

r

? .+-

^

^

K*

a*

.

tient therefore

is

-+-

^-

So that by

6cc.)

,

a*x3

c*x*

c

j

^

this

c

&c.

,

-

,

(or

Operation the

*

-j

into

i

Number

or

1 from that Scale in , (or a x^-t-*!" ) is reduced Quantity Arithmetick whofe Root is b -+ x, to an equivalent Number, the 1

Root of whofe

Scale,

whofe converging quantity)

(or

is

.

And much

found, will converge fo the fafter to the truth, as b is greater than x. To- apply this, by way of illustration, to an inftance or two in common Numbers. Suppofe we had the Fraction |, and would jeduce it from the feptenary Scale, in which it now appears, to an Then mall converge by the Powers of 6. equivalent Series, that and therefore in the foregoing general we (hall have j ;

Number, or

this

infinite Series thus

=^

,

\

Fraction

-^x b

,

^

make a-=.

i,

b

=

6,

and

#==1, and

the Series

-"j~

will Y.

of

~ become f ^ Or if we would reduce

8,

+

becaufe

f=

~

,

^, &c. which it

to

make

will

be equivalent to

by the Powers and .v i,

a Series converging

a=

i,

^=8,

=

then


and IN FINITE SERIES.

=

then ~

T +

Or

than the former.

x=

=

then 7 0,1428, &c. as 3

if

^

-+-

;

=

>

&- c -

which

Series will converge fafter

we would reduce it to

becaufe f

Scale,

Decimal)

&

-+-

~*

-~r-

-rV -4- -4-0- -+-

be

may

Wo-o-

common Denary (or

the

niake

,

j6r

a=

-f- -o-Vo-o-

=

l>

i,

+

and

10,

TO-O-^O-S-J <* c '

And hence we may maybe reduced a great va-

eafily collected.

obferve, that this or any other Fraction of ways to infinite Series ; but that Series will converge iafteft riety But that to the truth, in which b mall be greateft in refpect of x.

mod

reduced to the common Arithmetic^, which converges by the Powers of 10, or its Multiples. If we will be

Series

eafily

2+5,

or i -f- 6, refolve 7 into the parts 3 -f- 4, or or fuch have of we mould inftead Series, &c. converging diverging be to taken in. as require a Supplement And we may here farther obferve, that as in .Divifion of common Numbers, we may flop the procefs of Divifion whenever we and inftead of all the reft of the Figures (or Terms) ad inpleafe, we may write the Remainder as a Numerator, and the

mould here

:

finituniy

'Divifor

Denominator of

a Fraction, Quotient : fo the

as the

which Fraction

will

be

fame will obtain in the the Supplement to the in Thus the of Divifion prefent Example, if we will flop Species. at the

Or

firft

we

if

we mall Term of the Quotient, ^

have

'

will

Or

ft

if

at the

op

we

Term, then

will flop at the third

_ ^-

x

thefe Supplements

fecond

.

may

And

fo

-^b + X --r\.

= =j o

Term, then

in the fucceeding

always be introduced, to

a

"~

bX

^L. |- x

.

/;

"-~r

-f-

^- = ^

Terms,

make

in

which

the Quotient

in fome of compleat. to the following Speculations, when a complicate Fraction is not to be reduced a to or be intirely refolved, but only to be deprefs'd, commodious form. fimpler and more havhence we Or change Divifion into Multiplication. For may its and of the firft Term Supplement, or Quotient, ing found the *' K ta aax aa i i -i /lit the Equation ^ multiplying it by ? , we fhall x

This Obfervation will be found of good ufe

=

have IldVC

-^i

=

-^

T~^3a

*

-,

fo

, '

ant

_ffL_ .

a>A'*-

that fubftituting Ml

in

the

firft

Equation,

where the two

firft

it

will

become

Terms of

Y

this

^= y aa

the Quotient are

aa

value of aaX

-^

-f-

now known. Multiply


162 this

Multiply *

,

L

aa

=.

r

t-^-x

i

6

hi

become

--

I*

and

,

-

r

8

,

-

become 17

become

=

-

a

a*

z

Again,

^7^ = 5 .v4

x

fubftituted a*x*

.v

5T-

,

t9-^-6x fo every

Equation s a1x !

a**4

.<3 1

where eight of the

'

*

-

4

a*x*

in the laft 7

this a*x 6

---JT+ ~ multiply

fl-.v4

i V8

-4

b

will

. '

now known.

which being

,

aa

it

r-

i

rf

^

r- -,

i

-.

iS+i*X

A.4

^-

fyi-

--

the Quotient are

,

-p will

will

it

b*

Equation by

it

and

,

----

.

...

^

by

which being fubfthuted in the laft Equation, it will become 1 ra aav a' x*' fi^.v a*** .1 n c -4 1where the four nrlt rb b*

Terms of ,-,

Method of FLUXIONS*

The.

firft

p- 4-

f

Terms

are

now num-

known. And fucceeding Operation will double the ber of Terms, that were before found in the Quotient. This method of Reduction may be thus very conveniently imitated in Numbers, or we may thus change Divifion into Multipliof the cation. Suppofe (for inftance). I would find the Reciprocal m Decimal Prime Number 29, or the value of the Fraction T T Numbers. I divide 1,0000, Gfc. by 29, in the common way, fo far as to find two or three of the firft Figures, or till the Remainder becomes a fingle Figure, and then I afliime the Supplement to compleat Thus I mail have T ~ =. 0,03448^ for the compleat the Quotient. Quotient, which Equation if I multiply by the Numerator 8, it will '

.

give

^ = of0,275844^., Fraction

or rather

^.==0,27586^.

I fubftitute

the in the firft Equation, and I (hall have this initead Again, I multiply this Equation by 6, ^=1:0,0344827586^. * == and it will give T 7 o, 2068965517^, and then by Subftitution T 7 I multiply this Equation by 7, 0,03448275862068965517^. Again, anditbecomes T7? =o,24i3793io3448275862oi|-,andthenbySubfti-

=

'

where every Operation

will at

leaft

double the number of Figures this will be an eafy Expe-

And

found by the preceding Operation.

dient for converting Divifion into Multiplication in the Reciprocal of the Divifor being thus found, it into the Dividend to produce the Quotient. ply'd

Now

.

as

.

it is

,

c

,

aa

,

here found, that

j

aa

=7

n*x

all

Cafes.

may

For

be multi-

-**

77 -+ ~jr

**S

Z7~>

&c. which Series will converge when b is greater than A* fo when than b, that the it happens to be otherwife, or when x is greater muft have recourfe to Powers of x may be in the Denominators we ;

the


and INFINITE SERIES, the other Cafe of Divifion, a

i

"^

^-

_j_

,

in

which we

fhall

find

and where the Divifion

&c.

163 -^-^ is

=^

perform 'd as

before.

In thefe Examples of our Author, the Procefs of Divifion exercife of the Learner) may be thus exhibited : (for the 6.

5,

o

xi 1

+o .V4

AT

+7 *-

Now we

order to a due Convergency, in each of thefe Examples, muft fuppofe x to be lefs than Unity; and if x be greater than in

Unity,

we muft

=^

+

I

1

^

invert the I

7*

I

Terms, and then c &c

i *

we i

l

fhall

have

XX "^

1

-

ii

*/*

io. This Notation of Powers and 7, 8, 9, fractional, affirmative and negative, general

Roots by integral and and particular Indices, was certainly a .very happy Thought, and an admirable Improvement of Analyticks, by which the practice is render'd eafy, regular, and univeifal. It was chiefly owing to our Author, at leaft he carA Learner fhould .ried on the Analogy, and made it more general. be well acquainted with this Notation, and the Rules of its feveral Operations fhould be very familiar to him, or otherwife he will often I fhall not enter into any farfind himfelf involved in difficulties. ther difcuffion of it here, as not properly belonging to this place,

or fubject, but rather to the vulgar Algebra. 1 1. The Author proceeds to the Extraction of the Roots of pure Equations, which he thus performs, in imitation of the ufual Pro^ To extract the Square-root of aa +- xx ; firft the cefs in Numbers.

which muft be put in the Quote. Then the Square from the given Power, leaves -+-xx Divide this by twice the Root, or 2a, which is a Refolvend.

Root of aa of for

is

a,

this, or aa, being fubtradted

Y

2

th


the

Method' of FL u X r or N s,

?$

164

part of the Divifor,

firft

Term

fecond

of the Root,

and the Quotient as alfo the

fecond

xx

of the Root, and the Produft

from the Refolvend.

Term

of the Divifor.

x

Multiply the Divifor thus compleated, or

Term

muft be made the ~

-za -J-

+

This will leave

by the fecond

,

muft be fubtrafted

new

for a

,

Refolvend,

4-"

which being divided by the

for the third

will give

j

before found, with this

of the double Root, or

Term

be fubtrafted from the

laft

added to

it,

or 2a -+-

^

^-

Root

-^

,

be-

muft 640''

and the Remainder

Refolvend,

2tf,

the

1-

8^4

4a*

-f-

B

new Refolvend, to be next Term of the Root we (hall have \/ aa -+-xx

proceeded with as before,

will be a

finding the

for

Twice

the Root.

Term, the Product

ins multiply 'd by this

.

Term Term of

firft

A

.

'

So that

pleafe. 1

;

and

fo

on

as far as

= a+ _ '-

-' _i_ oa*

T.a

you

~

io*

eafy to obferve from hence, that in the Operation every new Column will give a new Term in the Quote or Root; and therefore It is

no more Columns need be form'd than it is intended there mall be Terms in the Root. Or when any number of Terms are thus exThus havtraded, as many more may be found by Divifion only. ing;

found the three

firft

their double -za

folvend

-\-

in.

c* 8 1

2 Oil

The

; '

-\

,

7^:

4

the

Root a

',-

a

2a

fcu3

, "

by J

,

the three

firft

Terms of

the Quotient ^*

Rel6&^

l

2 COrt*

Series

-f-

dividing the third Remainder or

04*.

7x

H

Terms of v4

v^

will be the three fucceeding fj

^i H

-f-

TT*

>

^ c<

t

Terms of

^ us f untl f

the Root. r

the fquare-

root of the irrational quantity aa -f- xx, is to be understood in the In order to a due convergency a is to be (iippos'd following manner. greater than x, that the

than Unity, and that a root required.

quantity

,

But

Root or converging quantity

may

as this

-

may

be

leis

be a near approximation to the fquaretoo little, it is enereafed by the fmall

is

which now makes

it

too big.

Then by

the next

Operation


and INFINITE SERIES. Operation it is diminim'd by diminution being too much, it quantity

-7--r

which makes

,

the

it

is

165

fmaller quantity

ftill

which

^;

again encreas'd by the very fmall

too great, in order to be farther di-

by the next Term. And thus it proceeds in infinitum, the Augmentations and Diminutions continually correcting one another, till at lalt ihey become inconfiderable, and till the Series (fo far conminifli'd

01

verted, in

Ids than x, the order of the Terms muft be inihe fquare-root of xx -+- aa muft be extracted as before;

Wh-ii a

12.

Approximation to the Root required.

a lufficiemly near

is

tinued)

which

cafe

is

x

will be

it

-+- 2X

-f-.5

&c.

, '

And

in

Series

this

the converging quantity, or the Root of the Scale, will be -. Thefe two Scries are by no means to be understood as the two different Roots of the quantity aa -+- xx for each of the two Series will exhibit thofe two Roots, by only changing the Signs. But they are accommodated to the two Caf s of to according as a or x may -,

Convergency, happen be the greater quantity. I (halt here refclve the foregoing Quantity after another manner, the better to prepare the way lor what is to follow. Suppofe then fi'-d the value of the Root where we the xx, may y

yv=.cni-\f

11

,wir

or zap -+- pp or

--qq;

= = xx =

Proccfy

;.j

;

yy

aa (If

-+-

XX=

p=

(\f)'

+

-

= -.

'-,

6

oi,'

-

(if

r

=

O-|t.

zaq

(if

+ +

r

rr

rf-f-/)

q}

^ -H^=

2rf?-J-

= aa^-zap ~ xx + = _^

1

VU*

by

-\-pp-,

-{-

?===

or

>

&c. which Procefs

j)

may J

in wo-ds.

be thus explain 'd In order to find

V ua --xx, 1

yy-=aa-\-xx,

=

or

Root y of this Equation wheie a is to be undeiftood as

the

^-f-/', iuppofc y a pretty near Approxii: arion to the value of _y, (the nearer the betand p is the lnv.,11 Supplement to that, or the quantity which ter,) makes it compleat. Then by Subftitution is deiivcd the fir It Sunxx, whole Root/; is to bt fou:,d. plementiil Lqu^i'oa zap -+-//;

=

INOW

as 2uJ>

plement/,) exactly

;-

=

n:iich bigger than ff,

is

v;c : ;

fh;.!l

-f-

-',

za

-

have nearly p

fuppofmg q

(lor

,

to

is

bigger than the Sup-

or at leaft

ve

(hall

have

reprefent the fecoiid Supple-

ment


j66

ment of the Root. ^1

Then by

will be the fecond

fecond Supplement.

much

lefs,

=.

q

Method of FLUXIONS,

*ft>e

And

^

fo -that -f- r,

Subftitution zaq -+-

-4-^=

Supplemental Equation, whofe Root q

Therefore

we

if r

^q

q will be a

mall have nearly

little

q=

=

the

is

quantity, and qq g--3

,

or accurately

be made the third Supplement to the Root.

r therefore zar -fU

r

-f- r*

4^

=

will be

fou*r

the

L^,"

whofe Root is r. And thus we may form Refidual or Supplemental Equations, whofe Roots will continually grow lefs and lefs, and therefore will make nearer and nearer Approaches to the Root y, to which they always converge. For y =5= a -{-/>, where p is the Root of this

third Supplemental Equation, go on as far as we pleafe, to

Or y =: a~\-

Equation zap-- pp-=xx.

Root of

this

Equation zaq

* -. -f- r.

oa>

Ztt

~

where r

rr-=. -~ I

~.

is

-\

the

And

-q-\-qq-=z

Root of

fo on.

this

The

-+-g,

--

^

.

where q

Or y

Equation zar "Refolution of

the

is

a

;

-f-

a

-f-

r

any one

of thefe Quadratick Equations, in the ordinary way, will give the which will compleat the value of y. refpeclive Supplement, I took notice before, upon the Article of Divifion, of what may be call'd a Comparifon of Quotients; or that one Quotient be

may

exhibited by the help .of another, together v/ith a Series of known Here we have an Inftance of a like 'Comparifon or iimple Terms. of Roots; or that the Root of one Equation may be exprels'd by the Root of another, together with a Series of known or fimple

Terms, which will hold good in all Equations whatever. And to we mall hereafter find a like Comparifon of carry on the Analogy, one where Fluents ; Fluent, (fuppofe, for inftance, a Curvilinear Area,) will be exprefs'd by another Fluent, together with a Series of fimple Terms. This I thought fit to infinuate here, by way of that I might mew the conftant uniformity and haranticipation, mony of Nature, in thefe Speculations, when they are duly and regularly purfued.

I mall here give, ex abundanti, another Method for this, and of Extractions, tho' perhaps it may more properly bekind fuch Refolution of Affected Equations, which is foon to follong to the low ; however it may ferve as an Introduction to their Solution.

But

j

The


and INFINITE SERIES. The

167

Refidual or Supplemental Equation in the foregoing Procefs was which xx, 2ap -\-pp-=. may be refolved in this manner. firft

Bccaufe **

/>= -^-, za + t

it

y

'

will be

" by 3 Divilion p

=-

-{ Aa*

za

^

-f-

x*tA

!

+ -^

,-^7

,

&c.

Divide

all

the

Terms of

this Series (except the

by p, and then multiply them by the whole value of />, and you will have p fir ft)

=-

ia

^ -g

where the two

6cc.

,

Terms of p

in

are clear'd of />.

-^ -f-

3

8*4

3Z

Divide

all

the

except the two firft, by />, and multiply them or />, by the firft Series, and you will have a Series And by rethe three firft Terms are clear'd of p.

which

peating the Operation, you

So that at

pleafe.

Terms

+

or by the

'

this Series,

by the value of for

firft

8*'

Series,

laft

you

may

clear as

will have

p=

many Terms of p

~

-+-

as

you

7^

,

&c. which will give the fame value of y as before. of thefe Examples, and 13, 14, 15, 16, 17, 18. The feveral Roots of all other pure Powers, whether they are Binomials, Trinomials, or any other Multinomials, may be extracted by purfuing the Method of the foregoing Procefs, or by imitating the like Praxes in Numbers. But they may be perform'd much more readily by gene-+-

^~,

for that purpofe. And as there will be frecertain general Operathe for enfuing Treatiie, quent as Multiplication, fuch be tions to perform'd with infinite Series, Divilion, railing of Powers, and extracting of Roots ; 1 mall here derive fomc Theorems for thofe purpofes. ral

Theorems computed occalion, in

A H- B 4- C + D -+-

E, &c. P-f-Q^-R-f-S-t-T, &c. and a _l__j_^_{_j\.4_ g) &c. reprefent the Terms of three feveral Series and let A-|-B-{-C-f-D-|-E, &c. into P+Q-t-R-f-S+T, refpedlively, a, -\- /B -{&c. y -i- -\- e, &c. Then by the known Rules of be multiMultip'ication, by which every Term of one Factor is to AP, /3 AQ^-jply'd into every Term of the other, it will be I.

Let

=

<f~

=

BP, BS

=

7=AR-i-BQ^-CP, ^z^AS-i-BR-i-LQH-DP, g=AT-f-

+ CR-t-DQ^-4- E'P .

x

1-

;

4-

and

fo on.

"^.-t-K -f- o -t-

Then by

Subftitution

it

will be

i7ov. = AP +BP -i-Cf'+DP-f-E,-,

<3c.

And


1

68

Method of FLUXIONS,

'The

And

be a ready Theorem for the Multiplication of any each other 5 as in the following Example.

this will

infinite Series into (A)

(D) *J

(C)

(B)

afr*+ JL/rv t* A ,

_ ^^TT

(QJ

(P)

A'4

(T)

(S)

(R) 1

.

,,

-

+~, $cc, =^+t^+tf^

&>+X^+i*?,rb =

(E)

+ & + &> &c mto*-fx-f- -x

X*

1

X*

v* a **

-P.

A4 '-

V-1

_

\2a"

9#

+**'+ i! 7a

1

+7|? * 4

_

i

\a '*

.3* ^

-*-

9

And

fo in all

II.

From

other cafes.

the fame

Equations

.-DQ.-CR-BS-AT

we

above

have

fhall

A

=

-.

And then by Subftitution ^

^

i

^(A + B-J-C+D + E, &c. =) S +

a

p

p

commodioufly for the Divifion of one infinite Series by Here for conveniency-fake the Capitals A, B, C, D, &c. another. in the Theorem, to denote the firft, fecond, retained are third, fourth, &c. Terms of the Series refpedively. will ferve

Thus, (>)

ii x * the

for

if

Example,

(/)

(t)

2

_}_ ^-ii^-_{_

Quotient

we would

'

z ,

.

&c. by the Series

be

a

a *"~ -

-f-

,

A, B, C, D, &c. which reprefent the order, the Quotient will become a

And

after the

fame manner in

all

a+^x-i-

~^.

-f-

,

&c.

T*

fx*

-f-

&c.

(0)

W

J

"

.

will

M

divide the Series #* _f. .ax -+. p ( (QJ (S) (ij

Or

feveral

f#

the Values

reftoring

Terms

+

as z

5

7

of

they /land in _i_ .11 *^

ga 3

'

& Ut r ^'

other Examples.

HI.


and INFINITE SERIES. In the

III.

laft

Theorem make .r=r,

.

=

/3

o,

=

o>=o, ^

169 0,60:.

^_

then

V.

DQ+CR+BS+AT

&(

^

~~F~

p

l'

whkh Theorcm win

,

~~p

readl]y find th

Here A, B, C, D, &c. denote the feveral of any infinite Series. Terms of the Series in order, as before. cal

p) (QJ the Reciprocal of the Series a-\- f.v-{(

Thus

we would know

if

(T)

(S)

<R)

^

_|_

And

&c.

^

4-

---12^--h 1-

^- 2. l

fhall

have by Subftitution

reftoring the Values of

-

la

we

&c.

,

^~-

84

it

will be

_

*-

&c. for the Reciprocal required.

>

720*'

,. x + f ..- A< ..{

fa of others. IV. In the

A, B, C, D, &c.

t_i

I

if

=

f.

i

+ + i*+i<, &c.

And

f.v

=

Theorem if we make P=A, Q^==B, R C, if we make both to be the fame Series that we mail have &c. is, S=D, * zAD + 2 AE + zAF + zAG.tff. A+B+C+D+E+F+G7&^ tf= A+ zAB + zAC+ 1 firft

;

I

+ B + zBC + BD + zBE + aBF + L* + zCD+ zCE + D* 2

will be a

which

Theorem

for finding the Square of

infinite

any

Series.

Fv

-_ -'_

i

aa

-

.

Sa'^lba 5

iz8a 7

256^

4^*

"

ga^iea*

1

I

zSafl"

64

-

---

x1

*^c

Ex. J 3.

^a

2 u

H--^3 bx*

o

'

L

t 1 x'

i

&c.

txl

8

25!.,

S

i

.'<>

2a s

A-4

TTTI

i

.7/4 4

64*4

Ex. 4

.

2

_l H_H_l^J

8

Ii

2434"

^-,

I

*_- ii._fil

,

30." >

9<?4

v.


Method of FLUXIONS, V. In this laft Theorem, if we make A*= P, aAB Qv 2 AC 1 1 2 AE 2BC C -+- 2BD -fS, T, &c. we _f- B =R, 2AD -+R O R S BC D fhallhave A P^ B -^- E== ^-, C==-^-

=

T ~ 2BD ~ C

,

Qr

R

B

-4~ zA ^^

-4-

=

z 1

==

=

*

,

T

2BD

-4-

zA

U

C^

,

&c.

2

-4-

2A

,

"*

+ Q + K-hS+TH-U,

2RC

S

2A

.-

Or p

&c.

-

iA

=

BE 2A

^ |

=

^CD '

pi

&c <xu

Theorem

the Square-root of any infinite Series may eafily be B, C, D, &c. will reprelent the feveral Terms of the Series as they are in fucceffion. this

By

Here A,

extracted.

~

^i

^1

Ex 2^1-

P,

a ,

4-2a/3-f-

Q^ R, S, i /

,

is

of any of the

a4

fli

by the fourth Theorem a -{-@-i--y-\-<f<-t-t,&tc.

2a^

+ 2a^H-

1 T, &c. write a

&c. refpedively. X

And this will

_i_

"'~~ o

VI. Becaufeit

=

i^-

be a

,

2ae, &c. in the third

2> + j8S

2a/3,

-f- 2/3y,

2ag-

Then

for finding the Reciprocal of the Square

Here A, B, C, D, &c.

ftill

denote the

R,

S,

Terms

Series in their order.

VII. If in the

Theorem

firft

for P,

Q^

&c.

we

write

+ B 2AD -H 2BC, &c. (that A+B+C+D,&c.| byTheor.4.)wemallhaveA+B+C+D+E+F = A 3A*B + sAB -h sA*D 3AC + 360, &c. 6ABC+ 36^0 3BD 6ACD B' 6ABD-f+ -- 6ABE

A*,

|*

for

A

Theorem

infinite Series.

2a^

Theorem

2AC

2AB,

4

refpedively,

,

is

1

3

3

1

1

s

-j-

-i-

6cc.|

5.

-f-

t

which

will readily " give the

Cube of any r.

*' *"

*

"

^

*'* "T~

2*

" 11 "

13

yjf^

v9' X

infinite Series. A-

11

^

^^

15 3

Ex.


and INFINITE SERIES,

171

Ex.2. t* 1 -i~ VIII. In the laft

if

Theorem,

'+.3A'C = B= C=

we make

A =P, 3

=

--

R, B'-f-6ABC-|-3A D _ R 3 AB* S-6ABC

Q_

p:

1

,

?A ,

Sec.

x

U

,

I

=

&c. then

S,

O

3A*B

Bi

+l + ^ j^

i^K +

,

A=PT, that

fisc.

,

is

Here alfo A, B, C, D, root of any infinite Series may be extracted. &c. will reprefent the Terms as they ftand in order. xs IPX'* T? x' 1 8* 15 ;** 7*"* 7_ ~ _* ~ "' 8 I

- z'

-~"I"

Ex.

2.

8 f* 4 -h T 7 A; H- T | T x '

IX. Becaufe a*

R,

+ 3a S,

i

/3

it is

,

6cc. l^

=t**-t-r T ** H-Trr^ 4

by the feventh Theorem a

T, &c. write

1

-f-

5

/3

,

+

3j8, 3/S -f-Sa

3'fr &c.

1

refpeflively

^ ;

}

/3

,

&c.

&c. y -\Theorem for

-f-

&c. in the third

i

',

243a

'

7

-f- 3 a/3

^

^ + 8i

1

I

-f-

,

J j

P,

6a/3>-f-

then

This Theorem will give the Reciprocal of the Cube of any infinite Series ; where A, B, C, D, &c. ftand for the Terms in order. ; X. Laftly, in the firft Theorem if we make , Q ==3A 1 B, &c. we {hall have >

P=A

A+B-f-C-i-D, &c.

will be a

And

4 I

=A^H-4A

s

4

B-{-6A 1 B 1 -|-4ABs&c. which

Theorem for finding the Biquadrate of any infinite Series. we might proceed to find particular Theorems for any

thus

other Powers or Roots of any infinite Series, or for their Reciprocals, or any fractional Powers compounded of thefe ; all which will

be found very convenient to have at hand, continued to a competent number of Terms, in order to facilitate the following Operations. Or it may be fufticient to lay before you the elegant and general

Theorem, contrived for this purpofe, by that fkilful Mathematician, and my good Friend, the ingenious Mr. A. De Mo'rore, which was firft publifh'd in the Philofophical Tranfa&ions, N 230, and which will readily

perform

all

thefe Operations.

Z

2

Or


The Method of

172

FLUXIONS,

Or we may have

recourfe to a kind of Mechanical Artifice, by the which foregoing Operations may be perform'd in a very eafy and general manner, as here follows. all

When two

infinite Series are

to find a third

which

is

to be

multiply 'd together, in order one of them the

to be their Product, call

and the other the Multiplier. Write dawn upon your Multiplicand, Paper the Terms of the Multiplicand, with their Signs, in a defcending order, fo that the Terms may be at equal diftances, and juft under one another. This you may call your fixt or right-tand Paper. Prepare another Paper, at the right-hand Edge of which write down the Terms of the Multiplier, with their proper Signs, in an afcending Order, fo that the Terms may be at the fame equal diftances from each other as in the Multiplicand, and juft over one another.

This you may call your moveable or left-hand Paper. Apply your movenble Paper to your fixt Paper, fo that the firft. Term of your Multiplier may ftand over-againft the firft Term of your Multiplicand. Multiply thefe together, and write down the Product in its for the firft Term of the Product required. Move your moveplace, abie Paper a ftep lower, fo that two of the firft Terms of the Mulftand over-againft two of the firft Terms of the Multitiplier may Find the two Produces, by multiplying each pair of the plicand. Terms together, that ftand over-againft one another ; abbreviate

may be done, and- fet down the Refult for the fecond of the Product required. Move your moveable Paper a ftep the of firft Terms of the Multiplier may ftand lower, fo that three of the firft Terms of the Multiplicand. three Find the over-againft each of the Terms three Products, by multiplying pair together that one another fet down the abbreviate them, and ftand over-againft j And proceed in the lame Refult for the third Term of the Product. them

if it

Term

manner

to find the fourth,

I ihall iiluftrate this

from the

common

ana

all

the following Terms.

Method by an Example of two

Scale of

Denary

01

taken

Series,

Decimal Arithmetick

;

which

will equally explain the Procefs in all other infinite Series whatever. &c. and Let the Numbers to be multiply 'd be

528,73041, &c. which, by fupplying ftood, will _j_

3X~

-t-

cand, and prefcribed,

J

10 where

it

is

under-

3X -\- jX -+- jX-'-f- aX-'-f- 8X-* &c. and 5 X* -f- aX 4- 8X -j- 7 -+-4X-4-f- iX-s, &c. and call the firft the Multipli-

become

oX-

37,528936,

or

the Series

9 X-44- 3X-5H- 6Xl

X

the fecohd the Multiplier. will ftand as follows.

X- +

Thefe being difpofed

as

is

Multiplier,


and INFINITE SERIES.

Multiplier,

-H4X-+ -f-oX-'

8X

Multiplicand

?X


^

t

174 der one another.

Method of FLUXIONS,

}e

This

is

your

fixt

or right-hand Paper.

Prepare

another Paper, at the right-hand Edge of which write down the Terms of the Divifor in an afcending order, with all their Signs changed except the firft, fo that the Terms may be at the fame equal This will be your distances as before, and jufl over one another.

moveable or left-hand Paper.

Apply your moveable Paper

to

your

Paper, fo that the firft Term of the Divifor may be over-againft the firft Term of the Dividend. Divide the firft Term of the Dividend by the firft Term of the Divifor, and fet down the Quotient over-againft them to the right-hand, for the firft Term of the Quotient required. Move your moveable Paper a ftep lower, fo that two of the firft Terms of the Divifor may be over-againft two of the firft Terms of the Dividend. Colleft the fecond Term of the Dividend, together with the Product of the firft Term of the Quotient now found, multiply'd by the Terms over-againft it in the lefthand Paper ; thefe divided by the firft Term of the Divifor will be the fecond Term of the Quotient required. Move your moveable Paper a ftep lower, fo that three of the firft Terms of the Divifor may ftand over-againft three of the firft Terms of the Dividend. Collecl the third Term of the Dividend, together with the two Produds of the two firft Terms of the Quotient now found, each being multiply'd into the Term over-againft it, in the left-hand Paper. Thefe divided by the firft Term of the Divifor will be the third Term of the Quotient required. Move your moveable Paper a ftep lower, fo that four of the firft Terms of the Divifor may ftand overCollecl: the fourth againft four of the firft Terms of the Dividend. Term of the Dividend, together with the three Products of the three firft Terms of the Quotient now found, each being multiply'd by Thefe divided by the Term over-againft it in the left-hand Paper. the firft Term of the Divifor will be the fourth Term of the QuoAnd fo on to find the fifth, and the fucceeding tient required. fixt

Terms. For an Example .a*

_]_

+ tax 4-j_

L

let

x 1 H-+-

-^

,

it

be propofed to divide the infinite Series

^+ I2IA-5

&c.

28|X4

7^1

.,

,

1C'

,

&c. by the

Thefe being difpofed

as

Series is

a 4- fx 1

prefcribed,

will ftand as here follows.

Divifor,


and INFINITE SERIES. Divifor, ;

fr

175


7^^ Method of

^

FLUXIONS,

c on tne otner 3-4> Paper, when they meet together, will numeral Coefficients. Apply therefore the fecond Term compkat the .of the move-able Paper to the uppertnoft Term of the fixt Paper, >

ij

2

'

-

;ind the Product made by the continual Multiplication of the three Factors thatftand in a lin-e over-againft one another, [which are the fecond Term of the given Series, the numeral Coefficient, (here the given Index,) and the firft Term of the Series already found,] divided by the firft Term of the given Series, will be the fecond Term of the Series required, which is to be let down in its place overMove the moveable Paper a ftep lower, and the two againft I. Produces made by the multiplication of the Factors that ftand overand elfewhere, care muft be had to -againft one another, (in which, Coefficients numeral take the compleat,) divided by twice the firft Term of the given Series, v/ill be the third Term of the Series reMove quired, which is to be fet down in its place over-againft 2. the moveable, Paper a ftep lower, and the three Products made by the multiplication of the Factors that ftand over-againft one another, divided by thrice the firft Term of the given Series, will be the And fo you may proceed to fourth Term of the Series required.

and the fubfcquent Terms. be amifs to give one general Example of this Reducnot

find the next, It

may

If the Series az will comprehend all particular Cafes. c&' -+-dz*, ,&c. be given, of which we are to find any Power, or to extract any Root; let the Index of this Pov>er or Root be m. Then prepare the moveable or left-hand Paper as you fee below, where the Terms of the given Scries are fet over one another tion, _l_

which

b^

_j_

edge of the Paper, and put a full point, as a

Alfo of after every Multiplication, and after every one, (except the firft or loweft) are put the feveral as m, zm, pn, 40;, &c. with the negative Multiples of the Index, Likewife a vinculum may be undei flood to after them. Sign be placed over them, to connect them with the other parts of the numeral Coefficients, which are on the other Paper, and which make them compleat. Alfo the firft Term of the given Series is the reft by a line, to denote its being a Divifor, or feparated from of a Fraction. And thus is the moveable Paper Denominator the in

order, at the

Term

is

at equal diftances.

Mark

prepared. To prepare the fixt or right-hand Paper, write down the natural Numbers o, i, 2, 3, 4, &c. under one another, at the fame equal diftances as the Terms in the other Paper, with a Point after them as a Mark of Multiplication ; and over-againft the firft 1 erm o write


and INFINITE SERIES. Term of the Series required. The reft ot the Terms are to be wrote down orderly under this, as they (hall be To the firft Term o in the found, which will be in this manner. the fecond Term of the moveable Paper, and they fixt Paper apply write a*"z m for the

firft

*-* m ~~ will then exhibit this Fraction

-

to this

Term

aw*

<

of the

and you

will

~ t &s*+ I

" z

which being reduced

,

as,.

I

fet down in its place, for the fecond Move the moveable Paper a ftep lower, a have this Fraction exhibited cz*. 2m o. a z m ,

muft be

Series required.

+

az. 2

which being reduced

become mu m -

will

l

c-{-

mx "LL a m --b* xz m+'~,

Bring to be put down for the third Term of the Series required. the moveable Paper a ftep lower, and you will have the m n dz,*. o. a z

down

Fraction

yn

-f-

.+-

cz*.

bz?.

m

ma

*c -+-

m

L a m- l b 3

x

-

az. 3

for the fourth

ner are

all

the

Term of

Moveable Paper, &c.

And

the Series required.

in the

fame man-

of the Terms to be found.

reft

Fixt Paper o. i. -

2.

*.

m

-3 . J

mx-

a^i-o*

-x7.

1

-f-

ma m~*c x z"

-a m *l>>+mx.'-

'

am

1

6c+Ma m

l

dxz"

T

az.

will produce Mr. De Moivre's Theorem the mentioned before, Inveftigation of which may be feen in the place there quoted, and fhall be exhibited here in due time and And this therefore will fufficiently prove the truth of the

N. B. This Operation

place.

prefent Procefs.

In particular Examples this

very eafy and practicable.

A

a

Method

will be

found

But


The Method of

178 now

FLUXIONS,

mew

fomething of the ufe of thefe Theorems, and jit way for the Solution of Affected and we will here Fluxional Equations; make a kind of retrofpect, and refume our Author's Examples of fimple Extractions, beginning with Divifion itfelf, which we fhall perform after a different and an eafier manner.

But

to

the fame time to prepare the

Thus

aa by b

to divide

into a Series of fimple

Now is

r^=y,

make

;

-

or by -f- xy

,

to find the quantity

manner

this

Terms

or to refolve the Fraction

-f- x,

+

y difpofe the Terms of this Equation after a 1 , and proceed in the Refolution as you fee

=

J

*-J

done here. I

a*x

=**

a*.*

1

a^J

+ xy\

h-r

a *x t -77-

7T +

-T-*--T*

TT

+

-75

&C..

>

,

IT

+

C

-JT-

OCC.

>

Here by the difpofition of the Terms a*- is made the firft Term of the Series belonging (or equivalent) to by, and therefore dividing of the firfl be Term Series will the to equivalent by b, y, as is fet

down

below.

Then

will

a^x

+

be the

firft

Term

of the Series

which is therefore fet down over-againft it; as alfo it is fet down over-againft by, but with a contrary Sign, to be the fecond a Term of that Series. Then will be the fecond Term of y -4- xy,

~

t

to

be

fet

down

in

cond Term of -f- xy for the third

y,

Term

and therefore

its

of

with a contrary Sign fore

'~

and

;

+

will be

place,

by.

~

which

a

will give

-^- for the fe-

with a contrary Sign muit be Then will + be the third

this

~-

down Term of

fet

Term of 4- xy, which mufl be made the fourth Term of by, and therethe fourth Term of y. And fo on for ever. will

be the third

Now

the Rationale of this Procefs, and of all that will here follow of the fame kind, may be manifeft from thefe Confiderations.

The unknown Terms

of the Equation, or thofe wherein y

are (by the Hypothecs) equal to the

known Term

aa.

is

And

found, each of thofe


IN FINITE SERIES.

a?id

unknown Terms

thofe

refolved into

is

170

equivalent Series, the

its

Ag-

gregate of which muft (till be equal to the fame known Term aa Therefore all the fubfidiary and adventitious (or perhaps Terms.) are which introduced into the Equation to aflift the Solution, Terms, the Supplemental (or Terms,) muft mutually deftroy one another. Or we may refolve the fame Equation in the following manner ;

:

y^ Here a 1

made

is

the

= --

firft

a*

la*

A"

.V

-

Term

of

k*a* -X

-4

-+-

-Ha* A;4

. ,

&c.

down for the firft Term of y. This will give + firft Term of by, which with a contrary Sign muft be ~ muft be put down Term of -+- xy, and therefore be put

--

Term

cond

which with

of y.

^

will

be the fecond

a contrary Sign will be the third

+-

therefore

Then

-

will be the third

fore the Fraction propofed

is

Term

refolved

muft

xv, " and therefore x

of y. into

the

for

the fecond for the fe-

Term

of by y

Term of -|- xy, and And fo on. There-

the fame

two

Series as

were found above.

+ * were given to be refolved, make x y=. i, the Refolution of which Equation

If the Fraction

-

:

'

1

or y -+- l rrxpre than writing v

t

y

=

i

x+x

down

the Terms, in the

xx,&cc. ,

ccc..

y

7

+x*y 3

is

manner following

+

-V"

little

:

----(-x- 1 *-4-|_xi x-'-+x- x~ & &c.

=

,

Here in the firft Paradigm, as i is made the firft Term of y, fo 1 x*- will be the be the firft Term of x*y, and therefore will x be and will therefore x* the of fecond Term of fecond Term y, x* will be third Term of y ; &c. Alfo in the x*y, and therefore -+fecond Paradigm, as i is made the firft Term of x*y, fb will -f- x~'be the

Term

firft

Term

of x*y, or

of y, and therefore x~- will be the fecond x~* will be the fecond Term of y ; &c.

A

a 2

To


tte Method of

180

FLUXIONS, i zx

To

refolve the i

=y,

7

I+* quation

compound

1

.

13 =

i *

2.y a

make

Fraction

#v

or 2**

3.

~* -

into fimple

Terms,

i

^xy, which E-

y 4- AT^

3*

be thus refolved

may

=

2A^

:

X^

*

1

3**

+ 34^*

73**, &c. 34* } &c. 3

39**, &c.

Terms of

Place the

the Equation, in

which the unknown quan-

defcending order, and the known Terms here. Then bring down zx^ to be the firfl

in a regular

is

found, tity y above, as you fee is done Term of y, which will give -f-

2x

for

the

firfl

Term of

the Series

which mufl be wrote with a contrary Sign for the fecond Term of y. Then will the fecond Term of 4- x^y be 2x%, and will be which the firfl Term of the Series 6x^, 3*7 together And this with a contrary Sign would have been wrote SAT*. make 4-

x*y,

for the third

Term of y, had J

reduces

it

4- jx

to

*

Term x* been above, which Term of y. Then will 4- yx* and 4- 6x* will be the fecond Term

not the

for the third

Term

of 4- x*y, collected with a contrary Sign, will make of 3fly, which being 1 3** for the fourth Term of y ; and fo on, as in the Paradigm. If we would refolve this Fraction, or this Equation, fo as to accommodate it to the other cafe of convergency, we may invert the Terms, and proceed thus

be the third

:

O

W

3-v

V1 x

f

/

y Bring down will be the firfl

AT*

=f

*-

-i-

1, &c. X'

-f-

to be the

Term

4f *

7

firfl

of y, to be

Term of down in

fet

'

ft*3-vy, its

1

,

6cc.

whence

place.

~\-

Then

^

the

firfl


and INFINITE SERIES. Term

Term

fecond

Then

of y.

fx, which with

a contrary Sign will be the 3*?, and therefore -+- f the fecond Term of -+- x^y will be -f- f#s

of +- x^y will be will be the fecond Term of firft

181

-f-

of y being -+- f x*, thefe two collected with a *.#* for the third Term of have made 3*}', contrary Sign would z had not the Term +- zx' been prefent above. Therefore uniting x* for the third Term of 3*7, which thefe, we fhall have -f-

and the

firft

Term

?j-x~* f of -+- xh be

r

will g lve

Term

the third

Then will the third fecond Term of y being -+- -%, the Sign will make -f- if for

Term

of

y.

if, and the a contrary with thefe two collected and therefore fourth Term of T,xy, Term of y and fo on.

1

TT*""

will be

l

^ e fourth

-,

thus much for Divifion ; now to go on to the Author's pure or fimple Extractions. To find the Square-root of aa -f- xx, or to extract the Root y of

And

=

=

a -+-/>, then we fhall have aa-{- xx ; make y Equation yy xx, of which affected Quadratick Equaby Subftitution zap -f- pp

this

=

we may thus extract the Root p. Difpofe the Terms in this manner zap-^= xx, the unknown Terms in a defcending order oa H-/AJ one fide, and the known Term or Terms on the other fide of the tion

Equation, and proceed in the Extraction as

-

-)

**/==*

*4

+

-

*

here directed. 7 *'

+ H53.

5x8

5i

s74

is

_^i + ^!_^:, +"J-\.__ +^ J 4* 8*4

*

x*

A4

za

8a l

this Difpofition

By

h

izSa 1

is

made

'

7*

10

.,

&C.

-t-f-t

25O' the

Term

firft

of

x

then

zap

Term

as here

of the Series p,

Term

firft

,

\6a !

of the Terms, x 1

the Series belonging to

will be the

CA 9

*6

-I-

f

12Sa 8

640*

;

fet

of the

we

down

Series

have

fhall

for the

underneath.

*,

to

be put

firft

Therefore

down

in

its

4474

place over-againft

be put

which

p

1

down with will

make

.

Then, by what a contrary

the fecond

Sign

Term

is

obferved before,

as the fecond

of/> to

be

-

^

.

Term

it

muft

of zap,

Having

therefore


Method of FLn /

77jt2

fore the

two

firft

of the foregoing

two

the

ries,)

Terms of P Methods for

firft

Terms

=

*-

v

i

o

~, we

!-;

s,-

fhall liave,

(by any

finding the Square of an infinite Se-

of p 1

=~

.

3-

'

#4 4

AfCi

which

la ft

Term

be wrote with a contrary Sign, as the third Term of zap. Therefore the third Term of * is ^ * and the third Term of Lp* irmft

, '

-~

(by the aforefaid Methods) will be a

Term

of p will be

""" 7IsI

which

'

is

Term

fourth

contrary Sign, as the

}

and therefore the

-||_,

to be

is

wrote with

Then the fourth fourth Term of/* is

of zap.

8

to be

which

wrote with a contrary Sign for the

fifth

Term of zap. This will give 2^- for the fifth Term of p and fo 2^O' we may proceed in the Extraction as far as we pleafe. Or we may difpofe the Terms of the Supplemental Equation thus .

:

----

zap

J

-f-

a*

>

zax

za*

-+-

-

~ * x

*

^

'

c &c

AX3

, y

&c.

'

* ,&c.

Term of the Series/ 4 and therefore x, be the firft Term of p. Then zax will be the (or elfe x,) will zax will be the fecond Term of firft Term of zap> and therefore Here * a

is

made

the

So that becaufe /> 1

firft

=

,

#a

by extracting the Square-root of this Series by any of the foregoiug Methods, it will be found a will be the fecond Term of the Root />. x a, &c. or / Therefore the fecond Term of zap will be 2<2% which muft be 1 third for the Term a wrote with contrary Sign of/ , and thence (by

p"-

.

Extraction) the third .third

to be

of/.

Then

Term

2rfx, 6cc.

Term

of zap to be

of ,

/

will be --

and therefore (by Extraction) o This makes the fourth Term of zap ,

will be the fifth

This

will

make

the

which makes the fourth Term of/4

-

^

.

Term I

of/.

will be the fourth

to

be o, as

Then

alfo

the fifth

Term of / z

.

Term of zap


and INFINITE SERIES, zap f

will be

.ll

and therefore o

4*

;

which

. , 5

will

make

the fixth

= =x

aa

-\-

Term

Term of p,

will be the fixth

Here the Terms will be alternately deficient Equation yy

183

xx, the Root will be y

;

is

y

-f-

to

be

fo that in the given

=

a

-f-

&c. which

,

/>*

&c.

^-} -h -^-s fhould change the order of the Terms, or if

&c. that

of

x

a

-f-

"->

the fame as

is

we fhould change a. into x and a into x, If we would extradl the Square-root of aa xx, or find the aa xx make y a -f- p, as beRoot y of the Equation yy x then zap -f-/* fore x*-, which may be refolved as in the following Paradigm : if

we

= =

;

J

f I"

+r

t>

1

X6

.v4

_

z

*4

\

1-

J

;

4

^

J*

.

,^_ __

^.V

8a4

4fl

f

=

;

X'

64,1

8

6

H- ^~; 84

<;*

f

-f-

-

CX

JC .

i

-{-

6^.6

"4 k~^_

8

.

^^^^

-

^

^^

"7

X

J,

c cSCC*

x 1 the firft Term of -J-/ 1 Here if we mould attempt to make we mould have ^/ x 1 or x^/ i, for the fi rfl Term of/ ; which no Series can be form'd from that being rnpoflible, fliews Suppofi,

,

tion.

To

of #

xx, or the Root y in this Equa1 x^ x tion yy x xx, make y p, then x -+- zx^p -f- p x*, which may be refolved after this xx, or zx^p -+- /* find the Square-root

manner

=

=

=

=

+

:

x* the firft being rightly difpofed, make x* be the firft Term of p. Therefore of zx^p; then will a 3 firft Term of be the will which is alfo to be with wrote ~\- px /

The Terms

,

a contrary Sign for the fecoiid Term of 2x'-p, which will give fA * Then (by fquaring) the fecond Term of for the lecond Term of p. -

^

will be

i^ 4 which ,

will give

-

i* 4

for the

fecond

Term

of


zx^p,

Method of FLUXIONS,

ffi?

184

and therefore

-V^

Term

for the third

of

p

and

;

fq op.

Therefore in this Equation it will be y=z x'* f x* f A-'" rV*''"* &c. So to extract the Root y of this Equation yy =.aa-\-bx xxt make y then bx which be -+thus a-{-p p* xx, zap may

=

=

}

refolved.

=fa

+ *fi,

X*

&C

.

1

L'-x'

tx

Make of

p.

firft

l

of zap; then will

Term

firft

be the

~-

firft

be

will

x*

-<

^'

to be

gjr

,

b

of p 1 will be

which

Then by

will

Term

20.

-+-

^

which

,

be wrote with a contrary Sign, fo that the fecond *

p

Ixl

Term

Therefore the

alfo to

zap

bx the

x*

make

the fecond

fquaring, the fecond

is

Term of Term of

Term of/*

which muft be wrote with a contrary Sign 7 *-* -g^ for the third Term of zap. This will give the third Term of p Therefore the Square-root of the as in the Example; and fo on. will be

5

Quantity a^

Alfo

if

-f-

bx

xx

a -+

will be

~

^

we would extract the Square-root

of

a* <

_

-f.

-^ _,

we may

,

ex-

Roots of the Numerator, and likewife of the Denominator, and then divide one Series by the other, as before ; but more tract the

dire.ctly

Make

thus.

=

_**!

= =

-f- p, then ax* Suppofe y which Suppplemental Equation i

yy,

or

i

zp -+-p

may

z

-{-

ax 1 bx*-

=

yy

zbx*-p

b*x*y*. bx*-p*,

be thus refolved.

zp


and INFINITE SERIES,

185

+TT*\ ab

^ab

1 ,

&c.

~a^b &c. t

bx^p* _

,

&c.

Make ax -f- bx* the firfl Term of 2/>, then will frf.v l -f- f&v abx* b*x* will be the firfl Therefore "be the firfl Term of /. and Term of ^a*x* -f- -^abx* -f- -^bx* will be the firfl 2bx*p, Term of/*. Thefe being collected, and their Signs changed, muil be made the fecond Term of 2/, which will give abx* -f- |J*A 2bx*p %a*x* for the fecond Term of/. Then the fecond Term of i and the fecond Term of p* l>*x -f- ^a bx will be -^ab^x 1

6

6

6

>

(by fquaring) the

firfl

will be

Term

l

found f a bx l

of

bx*p

which

Term

of 2p,

far as

you

\ab*x 6 will be ^a^bx

j-a''X

-{-

$frx

6 ,

and

6

4

-^ab'-x

f^'AT

;

pleafe.

thus if

Root y of

this

Subflitution a

=

6

6

+-

and the Signs changed, will make the third half which will be the third Term of p ; and fo on as

being collected

And

6

A: S ,

3

we were

to extract the

Equation 7' 1

=

a 3 41

Cube-root of a* 4- x*, or the make y a -f-/, then by j

=&

3d / -f- ^ap -+- p> which fupplemental Equation -f-

=

tf 3

-+- x*,

may

243

B b

or 3

i

/

-f

be thus refolved.

l

The


Method of FLUXIONS, The Terms <ia*p

which

will be #',

make

will

order, the

being difpos'd in

the

will

Term

make the firft Term ofp

Term of/ 1

firft

firft

to be

^

And

.

of the Series

*.

to be

Thiss

make

this will

the

^

Term

which with a contrary Sign muft be of ^ap* to be , the fecond Term of 3/z*/>, and therefore the fecond Term ofp will

firfl

be .

Then

.

r

(by fquaring) the fecond

and (by cubing) the

ff! Oa*

Term

of

*"=

of ^ap 1 will be

win be

fi 6

Thefe

.

27<:< r

the third .

y9

make

being collected

ill __ _j_

firft

Term

,

which with

a contrary

Sign muft be

Term of ^a^p, and therefore the third Term of p will be Then by fquaring, the third Term of ^ap* will be

and by cubing, the fecond Term of/ collected will

make

^T,

will be

8j<i'

y^-j

>

will be

anc^ therefore the

and the fourth

'

3

Term

^^, which being fourth Term of-^^p

of p will be

--

*

2 43 a

-

11

.

And

fo on;'

Arid thus in a

more

All that

is

may direct

the Roots of

all

pare Equations- be extracted, but

and fimple manner by the foregoing Theorems.

here intended,

is,

to prepare the way for the Refolution in Numbers and Species, as alfo of

affected Equations, both Fluxional Equations, in- which this Method will be found to be of And firfl we mall proceed with our Author to very extenfive ufe. the Solution of numerical affected Equations.

of

SECT.

II L

The Refolution of Nttmeral AffeSted Equations*

W

Refolution of affected Equations, and firft in Numbers ; our Author very juftly complains, that before his time the exegefa numcroja, or the Doctrine of the Solution as to the

of affected Equations in Numbers, was very intricate, defective, and What had been done by Vieta, Harriot, and Oughtred inartificial.

Attempts for the time, yet howSo that he had good reaever operofe. fon to reject their Methods, efpecially as he has fubftituted a much better in their room. They -affected too great accuracy in purfuing in this" matter,

tho' very laudable

was extremely perplex'd and

exact


and INFINITE SERIES. exact Roots,

which

led

them

into tedious perplexities

;

but he

187 knew

very well, that legitimate Approximations would proceed much more and would anfwer the fame intention regularly and expeditioufly,

much

better.

20, 21, 22. His Method may be eafily apprehended from this one Inftance, as it is contain'd in his Diagram, and the Explanation of

Yet for farther

it.

of

When

Illuftration Lfhall venture to give a fhort rationale

Numeral Equation

propos'd to be refolved, he Root as can be readily and takes as near an Approximation And this may obtain'd. always be had, either by the conveniently known Method of Limits, or by a Linear or Mechanical ConitrucIf this be greater or tion, or by a few eafy trials and fuppofitions. lefs than the Root, the Excefs or Defect, indifferently call'd the Supplement, may be reprefented by p, and the affumed Approximation, together with this Supplement, are to be fubftituted in the given. it.

a

is

to the

Equation inftead of the Root. By this means, (expunging what will be fuperfluous,) a Supplemental Equation will be form'd, whole Root is now p, which will confift of the Powers of the affumed Approximation orderly defcending, involved with the Powers of the Supplement which accounts the Terms will be conregularly afcending, on both a in decuple ratio or falter, if the affumed Aptinually decreafmg, proximation be -fuppos'd to be at leaft ten times greater than the Supplement. Therefore to find a new Approximation, which fhall nearly exhauft the Supplement p, it will be fufficient to retain only the two firft Terms of this Equation, and to feek the Value ofp from the refulting fimple Equation. [Or fometimes the three firft Terms be may retain'd, and the Value of p may be more accurately found from the refulting Quadratick Equation; Sec.] This new Approximation, together with a new Supplement g, muft be fuhftituted initead of p in this laft fupplemental Equation, in order to form a And the fame things may be obferved fecond, whofe Root will be q. of this fecond fupplemental Equation as of the firft; and its Root, or an Approximation to it, may be difcover'd after the fame manner. And thus the Root of the given Equation may be profecuted as far as we pleafe, by finding new iiipplemental Equations, the Root of every one of which will be a correction to the preceding Supplement.

So that

in the

y

=

-{-/>,

As p

is

2x2x2

3

= 2x2 = 2y

5

4,

o,

'tis

eaiy to perceive,

which mould make

= = 5.

p be the Supplement of the Root, and it will be y i -fand therefore by fubftitution Q. lop -+- 6p* -\-p= here fuppos'd to be much lefs than the Approximation 2,

Therefore 2.

prefent Example jy 2 fere ; for let

B b

2

ty


The Method of

i88 by

an Equation will be form'd, in which the Terma

this fubftitution

and

will gradually decreafe,

2

is

=

fere,

p

T

'o-

-h ?

a

x

fere

and

o, 6

q=

r for

f^e,

the third

=

1,237 + 6>3?* 4-

which

in

Therefore or q-=.

it

all

0,61

0,0054-)^

=o,

<f

=

much 1

-f-

and therefore

than the

=

1,23.^

o,

ym?,.

by afluming

fubftituted

being

is

are farther

lefs

accurate,

laft

which

o,

Terms

the

q will be

is

This

Supplement.

11,162;-, &c.

0,000541554-

1

-+-

Equation,

former Supplement p. or

1

which the Supplement

in

i -fTerms, io/>=o, Supplement q, 'tis

firft

This being fubftituted for p in the

accurately.

new Supplemental

deprefs'd,

the fafter, cateris parities, as

affuming a fecond

or

;

becomes

it

Equation,

much

Ib

So taking the two

greater than p. or p T_.

=.

FLUXIONS,

will

r-=

=

give

'^^-^

=

&c. or_y 0,00004852, &c. So that at laft/=2 -{-^> &c. 2,09455148, And thus our Author's Method proceeds, for finding the Roots of affedted Equations in Numbers. Long after this was wrote, Mr. Rapb-

Jon publifh'd his Analyfis Mquationum imiverjalis, containing a Method for the Solution of Numeral Equations, not very much different from this of our Author, as may appear by the following Com parifon. To find

Root of the Equation y*

takes as

His

thus.

that he has_y==g-+Ar.

or

if

ment

=

5, Mr. Rapbfon he calls Approximation g, which he near the true Root as he can, and makes the Supplement x, fo

the

would proceed

g=2,

'tis

Then by

Subftitution g 3 -f-3^ 1 Ar+3^x a -f-x 3 2 2 <

iOAr-f-6.v* -4- x*

=

=

i,

=5,

to determine the Supple-

Powers may be rejected, This added tog or 2, nearly.

This being fuppofed fmall,

x.

zy

firft

its

and therefore iox= i, or ,v o, i and makes a new x 2,1, being ftill the Supplement, 'tis y 5 2,1 +x, which being fubftituted in the original Equation _y zy 3 x 0,6 1, to determine the, 5, produces 11,23^-4- 6,3** new Supplement x. He rejects the Powers of x, and thence derives

=

=

+

^___oj __ I

1

,25

=

=

and confequently y becaufe the Powers of x were 0,0.054,

=

2,0946, which rejected, he makes

not being exaft, the Supplement again to be x, fo that 2,0946 -f- x, which be&c. ing fubftituted in the Original Equation, gives 11,162^-+he has 0,00054155. Therefore to find the third Supplement x,

y=

.v

="

54 ,'

1

62

'

5S

=

2,09455148, &c. and

0,00004852,

fo that

=

y =.2,0946

+ *=

To on.

By


and IN FINITE SERIES. we may

this Procefs

By

fee

how

nearly thefe

189

two Methods

agree,

and wherein they differ. For the difference is only this, that our Author conftantly profecutes the Refidual or Supplemental Equations, But to find the firft, fecond, third, &c. Supplements to the Root Mr. Raphjbn continually corrects the Root itfelf from the fame fupwhich are formed by fubftituting the corrected plementaf Equations, And the Rate of Convergency will the Roots in Original Equation. be the fame in both. In imitation of thefe Methods, we may thus profecute this Inmanner. Let the given Equation to be quiry after a very general m m o, in refolved be in this form ay -+- by"-* -4- cy-* -J- dy ~* &c. which fuppofe P to be any near Approximation to the Root y, and P -4-/>. Now from Then is y the little Supplement to be p. Powers and extracof the is (hewn what before, concerning raifing m m P* -f- wP m-'/>, &c. P -h/> ting Roots, it will follow that y m or that thefe will be the two firft Terms of y ; and all the reft, And for being multiply'd into the Powers of />, may be rejected. m~ l m ~l m~ m~ P m ~- -+ iP P the fame reafon y -h m p, &c. y :

1

=

,

=

l

m

2 P"-=p, &c.

=

and

.4- ~niaP

Therefore thefe being fub-

fo of all the reft.

p

,

it

,

=

I

ftituted into the Equation, n- l

a ]>>

I

= =

will be

&c."l

-, &c.

m m

n

2c

~*p, &c.

7 JP "-"</>,

>= o

;

Or

by P"

dividing

&c.

&c. '-

-j-^/P-s

-\-m

= --

^dP~*p, &c.

we mail have/ confequently ^ J

&c.

= ,

o.

ma?- 1

,

+

From whence

+

*P-'

+

cP-

lbV~' -{.m

_

taking the Value of

+ rfp-*

.

ar,.

z^~3 + m

J^P-4

,

/>,

and

Jjff .

r=

,. To reduce

this to a

more commodious form, make Pi= - , whence i B% &c. which being fubftituted, and Numerator and Denominator by A" , it will be

P=A-'B, P- =Aalfo the multiplying ~~ ~ +" I

'B

7

A."-"-B*+

=4rfA"-?Bi.

be a nearer Approach to the Rootjy, than jp

or P,

^c-.

and

will fo

much the

2


Method of FLUXIONS,

77je '

the nearer as

And hence we may

near the Root.

is

derive a very

convenient and general Theorem for the Extraction of the Roots of Numeral Equations, whether pure or affected, which will be this. m ~~m~ s Let th,e general Equation ay m -^- by" &c. -+ cy -f- d) , =: o be propofed to be folved ; if the Fraction - be affumed 1

Root y

the

near

as

iAA m

.

1

31

as

F

3

4.

z

this

inflead of the Fraction puted, may,be,ufed

the

be,

4B4,'feff

3n/A

And

to the .Root.

nearer Approximation

may

conveniently

zcA m -S

+7

1

.

Fraction,

,

Fraction

when com-

by which means

Bearer Approximation may again -be had ; and fo on, we pleafe. proach as near the true Root as

we

till

a

ap-

This general Theorem

Theorems

particular

tion ify

+ by === c,

1

as

may be conveniently refolved into as many we pleafe. Thus in the Quadratick Equa-

will be

it

if

Equation y*

+ ty +

=

cy

=

y *

A1

f

.

2t\

will be

be

^ ==

irt* -+- 2/)AB-4- iB~ x

A

1

rB4

-f-

==

y

In the Cubick

fere.

,

2

A .

3<i 1

In the Biquadratick Equation y*

y^r^.

p D

....

.

it

d,

rB z

-4-

,"7 D X DO J

,

_

111-1 c llke of hl

.

it

-+-dy=ze,

-{- by* -\- cy

i

>/^' And the

l_

S her

Equations. ;

For an 'Example of the Solution of a Quadratick Equation,

let

it.be propofed to extract the Square-root of 12, or let us find the Then by comparing with value of_y in this Equation y 1 1.2. And 12. the general formula, we fliall have b =. o, and <: or making g taking 3 for the firft approach to the Root,

#=

=

/I

=T>

that

Az=3

is,

^~

and

B;=

i,

we

fliall

have by Substitution y ^==.

=

=4-, fora nearer Approximation. Again, making A 7 12l == || for a nearer Approxiand B 2, we fliall have y 14 X 2 mation. A 28, we fliall have _y= Again, making 97 and B

=

97j

+

x 28i

i:!

'94*

= =

.

2

__

lil7

fo r

7o8ic8o77 /ior a nearer 1|^

fame method,

nearer

a

S45 2

making A= 18817 and B =543

==

=

we may

2

>

we ^a11 have

A

i

Approximation.

find as near an

_ y=

Approximation.

And

Aeain,

__

'

-r if

we go on

Approximation

to the

i

in

the

Root

as

y/e pleafe,

This


and INFINITE SERIES.

191

This Approximation will be exhibited in a vulgar Fraction, which, if it be always kept to its loweft Terms, will give the Root of the That is, it will alEquation in the fhorteft and fimpleft manner. the true Root than any other Fraction whatever^ nearer be ways whofe Numerator and Denominator are not much larger Numbers If by Divifion we reduce this laft Fraction to a Dethan its own.

we mall have 3,46410161513775459 for the Square-root of 12, which exceeds the truth by lefs than an Unit in the lall place.For an Example of a Cubick Equation, we will take that of our cimal,

Author

=.

j

_y

2,

*

2?

= =

and d==.

Root, or making

=

and therefore by Companion b o, the to for the firft 2 Approach taking we mall 2 and that is, A

5,

5.

^-

And 4.,

have by Subftitution y ==the Root. Again, make mall have y

~

=

3x11761

1

1

6615

1000

^561 5

And

= 44 A= __ 6= =

2x5615

f

B=i,

r

and B

21

+ 2500

A= 11761

make

Again, y

=

9-

=

a nearer

=

10,

Approach to and then we

Hj-L for a nearer Approximation. 5615

and

J

3 1

5615,

and

we

mall

have .

9759573 16 495

we might proceed to find as near an Approxi' proximation. mation as we think fit. And when we have computed the Root near enough in a Vulgar Fraction, we may then (if we pleafe) reThus in the prefent Example we duce it to a Decimal by Divifion. &c. And after the fame manner fhall have ^ 2,094551481701, we may find the Roots of all other numeral affected Equations, of whatever degree they may be. fo

=

SECT. IV. The Refolution of Specious Equations by infinite Series ; and firft for determining the forms of the Series^

and

their initial Approximations.

TTT^ROM

the Refolution of numeral affected Equations, our Author J/ proceeds to find the Roots of Literal, Specious, or Algebraical Equations alfo, which Roots are to be exhibited by an infinite converging Series, confiding of fimple Terms. Or they are to be exprefs'd by Numbers belonging to a general Arithmetical Scale, as has been explain'd before, of which the Root is denoted by .v or z. The affigning or chufing this Root is what he means here, by diftinguiming one of the literal Coefficients from the 23, 24.

reft,

if

there are feverul.

And

this

is

done by ordering or difpofing the


Method of FLUXIONS, the given Equation, according to the Dimenfions of It is therefore convenient to chufe fuch a that Letter or Coefficient. Series may is choice of the Root Scale, (when allow'd,) as that the Fraction a lefs or If it be the leaft, converge as faft as may be. the

Terms of

than Unity, its afcending Powers muft be in the Numerators of the Terms. If it be the greateft quantity, then its afcending Powers muft be in the Denominators, to make the Series duly converge. If

it

be very near a given quantity, then that quantity

made

may

be con-

Approximation, and that fmall difference, veniently or Supplement, may be made the Root of the Scale, or the conThe Examples will make this plain. verging quantity. 26. The 25, Equation to be refolved, for conveniency-fake, iliould be reduced to the fimpleft form it can be, before its Refoalways But Jution be attempted ; for this will always give the leaft trouble. all the Reductions mention'd by the Author, and of which he gives the

firft

us Examples, are not always neceflary, tho' they may be often conThe Method is general, and will find the Roots of Equavenient. tions involving fractional

or negative

Powers,

as

well as cf other

Equations, as will plainly appear hereafter. be refolved, in diftin27, 28. When a literal Equation is given to its Root is to conwhich or a guifhing proper quantity, by affigning three cafes or varieties ; all which, verge, the Author before has made For becaufe for the fake of uniformity, he here reduces to one. the Series mull neceffarily converge, that quantity muft be as fmall

of the other -quantities, that its afcending Powers may continually diminim. If it be thought proper to chufe the greateil quantity, inftead of that its Reciprocal muft be introAnd if it approach duced, which will bring it to the foregoing cafe.

-,as

poffible,

in

refpect

near to a given quantity, then their fmall difference may be introduced into the Equation, which again will bring it to the firft cafe. So that we need only purfue that cale, becaufe the Equation is always fuppos'd to be reduced to it. But before we can conveniently explain our Author's Rule, for finding the firft Term of the Series in any Equation, we muft confider the .nature of thofe Numbers, or Expreffions, to which thefe

Equations are reduced, whofe Roots are required ; and in this Inquiry we ihall be much aiTifted by what has been already difcourfed of Arithmetical Scales. In affected Equations that were purely numethe feveral Powers ral, the Solution of which was juft now taught, of the Root were orderly difpoied, according to a fingle or limple

literal

Arithmetical Scale, which proceeded only in longum, and was there fufficient


#nd INFINITE SERIES. But we muft enlarge our views in thefe Equations, in which are found, not only the Powers of the Root to be extracted, but alfo the Powers of the Root of the Scale, or of the converging quantity, by which the Series for the Root of the Equation is to be form'd ; on account of each of which circumftances the Terms of the Equation are to be fafficient for their Solution.

literal affected

regularly difpofed,

and therefore are to conftitute a double or combined Arithmetical Scale, which muft proceed both ways, in latum as well as in longum, as it were in a Table. For the Powers of the Root to be

extraded, fuppofe y, are to be difpofed in longum, fo as that their Indices may conftitute an Arithmetical Progreffion, and the vacancies, if any, may be fupply'd by the Mark #. Alfo the Indices of the Powers pf the Root, by which the Series is to converge, fuppofe x, are to be difpofed in latum, fo as to conftitute an Arithmetical Progreffion, and the vacancies may likewife be fill'd up by the fame Mark *, when it hall be thought neceffary. And both thefe together will make a combined or double Arithmetical Scale. Thus if the

Equa-6a* x* fax* 44were i!y4 4y $xy 7* #/ =a=-o, given, find the Root y, the Terms be thus may difpofed 6 V4 y* y* y yS yl yo

tion

to

s

:

x

= =o fhould be thus

Alfo the Equation v f x* by* 4- gbx\ in order to its Solution : jpofed, * * * y' by*

0;

dif-


Method of FLUXIONS,

When is

Terms of

the Equation are thus regularly difpos'd, ft then ready for Solution ; to which the following Speculation will the

be a farther preparation. 29. This ingenious contrivance of out' Author, (which we may call Tabulating the Equation,) for finding the firft Term of the indeed be to the finding all the Terms, extended Root, (which may or the form of the Series, or of all the Series that may be derived from the given Equation,) cannot be too much admired, or too careThe reafon and foundation of which may be fully inquired into thus generally explained from the following Table, of which the :

Construction

is

thus.

ba+bb

a^-bt,

ja-\-bb

2-+J*

+4*

40+4^ 711+3^

70+ zJ

\a-\-zb

za-\-zb

za+b

za

b

b

a

za

zb

zl

azb

b

b

b

3a 3a

50

b

bab bazb

zb

;

3

73*

In a Pfor.e draw any number of Lines, parallel and equidiftant, and cthers_at; right Angles to them, fo as to divide the far as is neceffary, into little equal Parallelograms.

whole Space, as Aflume any one

of thefe,- in which write the Term o, and the Terms a, za, 30, 4.a, &c. in-the fuceeeding Parallelograms to the right hand, as alfo the the left hand. Over the Term Terms -*-^ 2a, 3^7, &c. to the write Terms fame ^, zb, 3^, 4^, &c. fucColumn, o, in. the And and the Terms b, zb, 3^, &c. underneath. ceffively",' Now to infert its proper. Term thefe Ave ma^f call primary Terms. in any other afitgiVd. Parallelogram, add the two primary 'Terms that' ftand over-againft if each- way, and write the Sum together..,

And-* thus all the Parallelograms bethe given Parallelogram. as oecafion there is as-far every way, the whole. Space ing fill's, in

will


and INFINITE SERIES. -will

become

a Table,

which may be

195

a combined Arithmetical

called

of the two general Numbers a and t\ ProgreJJion in piano, compofed will be the chief properties. of which thefe following of Terms, parallel to the primary Series o, a, za, ^a, Any

Row

&c. will be an Arithmetical Progreflion, whofe common Difference is a ; and it may be any fuch Progreflion at pleafure. Any Row or &c. will be an Column parallel to the primary Series o, zb, 3^, Arithmetical Progreflion, whofe common difference is ^j and it may be any fuch Progreflion. If a ftr-ait Ruler be laid on the Table, the Edge of which mall pafs thro' the Centers of any two Parallelograms whatever ; all the Terms of the Parallelograms, whofe Centers mail at the fame time touch the Edge of the Ruler, will conftitute an Arithmetical Progreflion, whofe common difference will coniiit of two parts, the firfl of which will be fome Multiple of a, and the other If this Progreflion be fuppos'd to proceed injeriora. a Multiple of b. or from the upper Term or Parallelogram towards the lower ; verjus, each part of the common difference may be feparately found, by fubthe primary Term belonging to the lower, from the primary ,

tracling

If this common diffebelonging to the upper Parallelogram. made to be when found, nothing, and thereby the Rerence, equal determined the b be and a lation of Progreflion degenerates into a Hank of Equals, or (if you pleafe) it becomes an Arithmetical ProgrefIn which cafe, if fion, whofe common difference is infinitely little. the Ruler be moved by a parallel motion, all the Terms of the Parallelomall at the fame time be found to touch the Edge grams, whofe Centers And if the motion of of the Ruler, fhall be equal to each other. the Ruler be continued, fuch Terms as at equal diftances from the found to touch the Ruler, fliall form firfl: fituation are fuccerTively

Term

;

an Arithmetical Progreflion. Laftly, to come nearer to the cafe in hand, if any number of thefe Parallelograms be mark'd out and dithe reft, or aflign'd promifcuoufly and at pleafure, flinguifh'd from. whofe Centers, as before, the Edge of the Ruler ihall fucthrough from any two (or more) in its parallel motion, beginning ceflively pafs initial or external Parallelograms, :whofe Terms are made equal ; an Arithmetical Progreflion may be found, which ihall comprehend and take in all thofe promifcuous Terms, without any regard had to the Terms that are to be omitted. Thefe are fome of the properties of this Table, or of a combined Arithmetical Progreflion in piano by which we may eafily underfland our Author's expedient, of Tabulating the given Equation, and may derive the neceflary Confequen~es from it. >,

.

C

c 2

For


The Method of

196

FLUXIONS,

For when the Root y is to be extracted out of a given Equation, confifting of the Powers of y and x any how combined togetherother known quantities, of which x is to be promifcuoufly, with the Root of the Scale, (or Series,) as explain'd before fuch a value ;

of y

is

to be found, as

when

fubftituted in the

Equation inftead of

and become equal to nothing. And y, the whole (hall be deftroy'd, firft the initial Term of the Series^or the firfl Approximation, is to be found, wtyich in all cafes may be Analytically reprefented by Ax* ; or we may always put y Ax m &c. So that we mail have y 1 A 4 * 4 *, &c. And fo of other A*x* m 6cc. 4 A*x, &c. Powers or Roots. Thefe when fubftituted in the Equation, and by that means compounded with the feveral Powers of x (or z} already found there, will form fuch a combined Arithmetical Progreffion in flano as is above defcribed, or which may be reduced to fuch, by and I. Thefe Terms therefore, according to making 1

3

_>'

=

=

,

=

=

,

_}>

=

a=m

the nature of the Equation, will be promifcuoufly difperfed in the Table j but the vacancies may always be conceived to be fupply'd, and then it will have the properties before mention'd. That is, the Ruler being apply'd to two (or perhaps more) initial or external Terms, (for if they were not external, they could not be at the beginning of an Arithmetical Progreffion, as is neceflarily required,) and thofe Terms being made equal, the general Index m will thereby be determined, and the general Coefficient A will alfo be known. If the external Terms made choice of are the loweft in the Table, which is the cafe our Author purfues, the Powers of x will proceed by increafing. But the higheft may be chofen, and then a Series will be found,

in

which the Powers of x

will proceed

by

decreafing.

be other cafes of external Terms, each of which will eommonly afford a Series. The initial Index being thus found, the other compound Indices belonging to the Equation will be known be found', in which alfo, and an Arithmetical Progreffion may they form of the Series wifll the and are all comprehended, confequently

And

there

may

be known.

of the Equation, as above, reduce if the Terms themfelves we fame in be the will effedt, thing to the form of a combined Arithmetical Progreffion, as was fhewn before. But then due care mufl be taken, that the Terms may be otherwife the Ruler cannot be acrightly placed at equal diftances j of the Indices, as may tually apply'd, to difcover the Progreffions be done in the Parallelogram.

Or

inftead of Tabulating the Indices

it

For


and INFINITE SERIES,

197

For the fake of greater perfpicuity, we will reduce our general Table or combined Arithmetical Progreffion in piano, to the partiwhich will th.n appear cular cafe, in which a-=.m and b=. i -,

thus

:

- 2M+0

m+6 _,,

+5 +

5

+ -

AO-f- 2

m-\-

5

M-6 "'"+5

"-+5

w+4 + 4 zm-if 3

JOT+6

2W+4

4'"+ 4

CT+2

2OT-4-2

4W+2

OT+I

2OT+

7

m +4

3

2

1

1

5^2 2m

3

m

3

3

<

3

>

3

3

3

601

3

Now

the chief properties of this Table, fubfervient to the prefent If any Parallelogram be feledted, and anwill be thefe. purpofe, other any how below it towards the right hand, and if their included Numbers be made equal, by determining the general Number m, which in this cafe will always be affirmative ; alfo if the Edge of the

Ruler be apply 'd to the Centers of thefe two Parallelograms ; all the Numbers of the other Parallelograms, whofe Centers at the fame time touch the Ruler, will likewife be equal to each other. Thus if the be feleded, as alfo the ParalleloParallelogram denoted by m -+- 4 and if we make m -t- 4 ^m -H 2, we mall have -f- 2

gram

377*

m=i.

jm

2,

;

=

m -f- 4, 3^7 -|- 2, $m, ;;; -h 6, Alfo the Parallelograms &c. will at the fame time be found to touch the Edge of

m=

i. the Ruler, every one of which will make 5, when And the fame things will obtain if any Parallelogram be felecled, and another any how below it towards the left-hand, if their included Numbers be made equal, by determining the general Number m, which in this cafe will be always negative. Thus if the Parallelogram denoted by 5/w-i-4be felecled, as alfo the Parallelogram 402 -f- 2; Alib 2. and if we make ^m-\-^.-=^.m -t-2, we fliall

have>=

the Parallelograms

6w+6, 5^4-4, 4^ + 2,

3?;;,

zm

2,

6cc.

will


7%e Method of

198 found

will be

which

will

fame time to touch the Ruler, every one of

at the

make

FLUXIONS,

when

6,

m

=

2.

The fame

things remaining as before, if from the firft fituation of move towards the right-hand by a parallel motion, it will continually arrive at greater and greater Numbers, which at equal diftances will form an afcending Arithmetical Progeffion. Thus if the two firft felected 1 whence Parallelograms be zm

the Ruler

(hall

=

m=.,

be

it

the

Numbers

Then

-j.

if

5;;;

3,

in all the

correfponding Parallelograms will the Ruler moves towards the right-hand, into the

I, &c. thefe Numbers will each be %m-\- i, 6m moves to the fame diftance, it will arrive at forwards 3. which will &c. each be If it moves forward +- i, 4/7; -{-3, 7/ 5^. to it will arrive the fame diftance, at yn -fagain 5, %m -f- 3, &c. which will each be 8f. And fo on. But the Numbers f, 3, 52., 8y, &c. are in an Arithmetical Progreffion whofe common diffe-

parallel fituation

If

rence

it

is

And

2-i.

the like, mutatis mutandis,

in

other circum-

fiances.

And hence of the Ruler,

contra, that if from the firft fituation moves towards the left-hand by a parallel motion, it will continually arrive at lefler and leifer Numbers, which at equal diftances will form a decreafing Arithmetical Progreffion. But in the other fituation of the Ruler, in which it inclines downit

will follow

<?

it

wards towards the left-hand, if it be moved towards the right-hand by a parallel motion, it will continually arrive at greater and greater Numbers, which at equal diftances will form an increafing Arith-

Thus

metical Progreffion. rallelograms be

Numbers

8m

+

i

if the

= $m

two i,

firft

~~ =Numbers

feleded

whence

m

}

in all the correfponding Parallelograms will be moves into the parallel fituation 5^-4-2,

the Ruler upwards thefe Numbers will each be

2m

+

i

f.

If it

move on

at the

or Pa-

and the 4!..

If

2;;;,

8fc.

fame diftance,

m-+- i, 6cc. which will each be i-i. If it move forward again to the fame diftance, it will arrive at m -f- 4, And fo &c. which will each be on. But the -+Num2, 4/;z 4^. or bers &c. .Ll, i, |, -L, &c. are in an in4,1, i|, i-i, 4.1, it

will arrive at

3,

creafing Arithmetical Progreffion, whofe common difference is , or 3. And hence it will follow alfo, if in this laft fituation of the Ruler

moves the contrary way, or towards the left-hand, it will continually arrive at lefler and lefler Numbers, which at equal diftances will form a decreafing Arithmetical Progreflion. Now if out of this Table we fhould take promifcuoufly any number of Parallelograms, in their proper places, with their refpeclive it

Num-


and INFINITE SERIES, Numbers

included, neglecYmg

tain Figure, fuch as this,

all

the reft

;

199

we mould form fome

cer-

of which thefe would be the properties.

"M-3

2OT-J-I

The Ruler

5;;;+ 1

two (or perhaps more) of the Ambit or Perimeter of the Figure, Parallelograms, and their Numbers

being apply'd to any

which are in the two of the external being made equal, by determining the Parallelograms

that

is,

to

general Number m ; if the over all the reft of the Parallelograms by a paffes parallel motion, thofe Numbers which at the fame time come to the Edge of the Ruler will be equal, and thofe that come to it fuccefllvely will form an Arithmetical Progreffion, if the Terms mould lie at equal diftan-

Ruler

be reduced to fuch, by fupplyingany Terms that may happen to be wanting. Thus if the Ruler fhould be apply'd to the two uppermoft and external Parallelograms, which include the Numbers 3/w-f-^ and o, fo that ^m ~}_ 5, and if they be made equal, we mall have m The next Numbers that the Ruler each of thefe Numbers will be 5. will arrive at will be m -f- 3, 4;;; +3, 6/ -f- 3, of which each will So that be 3. The la ft are zm -f- i, 5>-f- i, of which each is i. ces

;

or atleaft-they

may

=

=

and the Numbers

which form a decreafing Arithmetical Progreffion, the common difference of which And if there had been more Parallelograms, any how difpofed, is 2. would have been comprehended by this Arithmetical Numbers their or at leaft it might have been interpolated with other Progreffion, Terms, fo as to comprehend them all, however promifcuoufly and have been taken. irregularly they might if the Ruler be apply'd to the two external PaThus fecondly, and 6m-}- 3, and if thefe Numbers be made rallelograms 5/72+ 5 we mail have m 2, and the Numbers themfelves will be equal, each ic. The three next Numbers which the Ruler .will arrive at

here

#2

0,

arifing are

5, 3, i,

=

will


The Method of

20O will be each 11,

bers 15,

n>

and the two

5. will be

laft

FLUXIONS,

will be

comprehended

Progreffion 15, 13, 1 1, 9, 7, 5, whofe Thirdly, if the Ruler be apply'd to the

6m

^ach

But the

5.

Num-

in the decreafing Arithmetical common difference is 2.

two

external Parallelograms

and 5*0-4-1, and if thefe Numbers be made equal, we fhall The two next 2, and the Numbers will be each 9.

=

-f- 3 have tn

Numbers

that the Ruler will arrive at will be each

5, the next the next and the laft i. All which will be i, -+3, in the afcending Arithmetical comprehended Progreffion 9, 7,

will be

whofe common

-+- i,

i,

3,

5,

difference

is

2.

Fourthly, if the Ruler be apply'd to the two loweft and external Parallelograms 2m-\-i and 5/77 -+- i, and if they be made equal, we fhall have again m o, fo that each of thefe Numbers will be i The next three Numbers that the Ruler will approach to, will each be 3, and the laft 5. But the Numbers i, 3, 5, will be

=

.

compre-

hended

in

difference

is

an afcending Arithmetical Progreffion, whofe 2.

Ruler be apply'd to the two external Parallelograms

Fifthly, if the in -f- 3

have

m

common

2m =Numbers and

2,

+- i, and if thefe Numbers be made equal, and the Numbers themfelves will be each

we

fhall

The

5.

that the Ruler will approach to will each be 1 1, three next the next will be each 15. two and But the Numbers 5, 1 1, 15, will be comprehended in the afcending Arithmetical Progreffion 5, 7, 9, II, 13, 15, of which the common difference is 2.

Laftly, if the Ruler be apply'd to the two external Parallelograms pn -f- 5 and m-\- 3, and if thefe Numbers be made equal, we fhali

m=.

have The I, and the Numbers themfelves will each be 2. next Number to which the Ruler approaches will be o, the two next are each All which Numbers i, the next 4. 3, the laft will be found in the defcending Arithmetical Progreffion 2, I, p,

common

i, 2, 4, whofe 3, fix are all the poffible cafes of external

difference

is

And

i.

thefe

Terms.

Now

to find the Arithmetical Progreffion, in which all thefe refulting Terms fhall be comprehended ; find their differences, and the greateft common Divifor of thofe differences fhall be the common

difference of the Progreffion.

Numbers were

5,

common Divifor

is

1

the refulting differences are 6, 4, and their greateft Therefore 2 will be the common difference of

1,15, 2.

Thus in the fifth cafe before,

whofe

the Arithmetical Progreffion,

which

will include

all

Numbers 5, n, 15, without any fuperfluous Terms. plication of all this will be beft apprehended from the are to follow.

the

refulting

But the Examples

.ap-

that

30


and INFINITE SERIES. 30.

We

have before given the form of 1

6<? 3 .Y 5

ja*x )* +-

_j_ I!y4

4-^Ar*

=

o,

this

201

Equation, y<

when

the

Terms

$xy* are dif-

to a double or combined Arithmetical Scale, in orpofed according Or obferving the fame difpofition of the Terms, der to its Solution.

they

may

requires. dices of

A;

be inferted in their refpedive Parallelograms, as the Table Or rather, it may be fufficient to tabulate the feveral Inonly,

when

they are derived as follows. Let Ax" repreSeries to be form'd for y, as before, or let

Term of the Then by &c. y=;Ax'", A 6m tion, we fhall have fent the

firft

6

.\-

=

fubftituting this for y in the given Equam l a tIB s x$ .v

$A

+

-+-

+

-f. 7*A -^xv+s of AT, when felected from

o. Thefe Indices will ftand their with the general Table, refpective Parallelograms, 6fl 3 x J -f-^.x'4,

thus:

4

&c.


Tfo Method of

2 02

will give the

This here

A

=

Equation

FLUXIONS,

A6

7rt*A*

A

=

5 -j- 6<z

o,

which,

A=v/

has thefe fix Roots, 3*7,. ,/tf, ^/2a, the are Of two laft and to be the of which impoffible, rejected. be for taken others any one may A, according as we would profecute this or that Root of the Equation.

Now

that this Ax m

is

a legitimate

Method

for rinding the firft that when

Ap-

Confequently they

may

may appear from confidering,

the proximation Terms of the Equation are thus ranged, according to a double Arithmetical Scale, the initial or external Terms, (each Cafe in its turn,) become the moil confiderable of the Series, and the reft continually decreafe, or become of lefs and lefs value, according as they recede ,

more and more from

Terms.

thofe initial

rejected, as leaft confiderable, which will make thofe initial or external Terms to be (nearly) equal to nothing ; which Suppofi-

be

all

tion gives the Value of A, or of

And

Ax n

,

for the

fir ft

Approximation,

afterwards regularly purfued in the fubfeSuppofition and proper Supplements are found, by means of quent Operations, which the remaining Terms of the Root are extracted. this

We

try here

may

for the

^m

is

_j_ y

likewife, if

we

can obtain a defcending Series

Root y, by applying the Ruler to the two external Terms and 6m ; which being made equal to each other, will give

The Ruler in an d hence each of the Numbers will be 9. Then at zm -f- 2, or its motion will next arrive at $m-\- i, or 8f. Then at 4. And laftly at 3. But thefe Numbers 9, 8f, 5, 4, 5. be comprehended in an Arithmetical Progreffion, of which 3, will So that the form of the Series here the common difference is i.

m =T>

y =A.v* -f- Ex -+- Cx^ -f- D^, &c. external Terms equal to nothing, in order

But

will be

proximation,

we

mail have

A

6

+

A4

=o,

we

if

put the two

to obtain the or

A

I

1

-f-

-

=

firft

o,

Apwhich

So that we can have no will afford none but impoffible Roots. this fuppofition, and confequently tial Approximation from

ini-

no

Series.

But

laftly,

we may

to try the third and laft cafe of external Parallelograms, the Ruler to 4 and 4^2-4-3, which being made equal,

=

apply

m

The next -, and each of the Numbers will be 4. will be 3 ; the next 2m -\- 2, or 2| ; the next 50* -{- i, or But the Numbers 4, 3, af, 27, the laft will be 6m, or if.

will give

Number 27;

if, will

all

common 6cc. may

be found in a decreafing Arithmetical Progreffion, whofe

+ Bx

So that Ax* reprefent the form of this Series,

difference will be

f

.

H- Cx~*

if the

-+-

Dx~s

circumftances of the


and INFINITE SERIES.

203

the Coefficients will allow of an Approximation from hence. if

we make

the

initial

Terms

equal to nothing,

we

But

mall have a,

which will give none but impoflible Roots. So that we can have no initial Approximation from hence, and confequemly no Series for the Root in this form. 1 # =o, when the Terms by -+- qbx* 3 i. The Equation y

-\- b*

o,

s

;

are difpofed according to a double Arithmetical Scale, will have the form as was (hewn before ; from whence it may be known, what cafes of external Terms there are to be try'd, and what will be the

circumftances of the feveral Series for the Root y, which may be Or otherwiie more explicitely thus. Putting derived from hence. " of the Series y, this Equation will become firft Term the for Ax 1

if

we

M* 1

A'A.-?"

by Subftitution

thefe Indices

take

of

=

1"

o. So that x*, 6cc. gbx* out of the general Table, they will -f-

x

ftand as in the following Diagram. in order to have an afcending we may apply the Ruler to Series for

Now

y,

the two external Parallelograms

2W, which give m will be 2.

i,

The Ruler then

will firft

progreis bers 2, 3,

are 5,

will here be

or

;'

Coefficient

A

a

=

come

to 3, and then to yn, or But the Num5. contain'd in an afcending Arithmetical Progrefiion, difference is i Therefore the form of the Series AA; -f- B* 1 -f-(*', &c. And to determine the

=we

A,

that

9,

in its parallel

all

whofe common firft

2 and

made equal, will and each of the Numbers

therefore being

is

A

.

fhall

have the Equation

=+

So that

3.

either

bfcx 1

-f-

4-3*,

or

qbx*

--

o,

3^ may

Approximation, according as we intend to extract the affirmative or the negative Root. We mall have another cafe of external Terms, and perhaps another afcending Series for_y, by applying the Ruler to the Parallelograms 2; and 5;^, which Numbers being made equal, will g;ive be the

initial

2;=

m =zo.

(For by the way, when we put 5/77, we are not at that to becaufe this would Diviiion, argue by liberty bring And the laws of Argumentation require, that no us to an absurdity. Abfurdities muft be admitted, but when they are inevitable, and the are of ufe to of fome when

2=5,

mew

they

fliould therefore here argue

then

5//f

tion

I

2:>i

thought

=

falfity

Supposition. by Subtraction, thus: Becanfe cm

=

m

=

o, and therefore o, or pn the more necellary, becaufe I have

D

d 2

o.

We ^t>i t

This Cau-

obferved f >mc,

who


Method of FLUXIONS,

"The

204

lay the blame of their own Abfurdities upon the AnalyBut thefe Abfurdities are not to be imputed to the Art, Art. tical the unikilfulnef of the Artift, who thus abfurdly apbut rather to

who would

=

the Principles of his Art.) o, we {hall Having therefore have the Numbers 2/77. in its parallel The o. Ruler 577*' motion will next arrive at 2 ; and then at 3. But the Numbers o,

= =

plies alfo

777.

2, 3, will be comprehended in the Arithmetical Progreffion o, i, 2, 3, whofe common difference is i. Therefore y A -+- Ex -+- CAT*, &c. will be the form of this Series. Now from the exterior Terms A* 3 bA* o, or A by or A fi, we {hall have the firft Term

=

=

=

=

of the Series. There is another cafe of external Terms afford a defcending Series for y.

bly may the Parallelograms 3 and

7/7=4, an d ea ch of will come to 2 and

5777,

and making thefe equal, we

Numbers

thefe

to be try'd, which poffiFor applying the Ruler to

or

will be

3.

have

(hall

Then

the Ruler

But the Numbers

--

3, 2, if, common whofe in a comprehended defcending Progreffion, Ax^ difference is f. Therefore the form of the Series will be y laftly 2777,

;

will be

_f.

BA"T

=

-|-

CA^

A=

-f-

And

D, &c.

the external

Terms

A

r

.v

3

=

A: 3

o

Now

as the two former' will give i for the firft Coefficient. cafes will each give a converging Series for y in this Equation, when is lefs than .v Unity ; fo this cafe will afford us a Series when x is

than Unity ; which will converge fo much the fafter, the is greater fuppofed to be. have 32. already feen the form of this Equation y> -\-axy -f-

greater

x

We

=o,

A? 3 2# 3 aay double Arithmetical

Ax*

Scale.

to reprefent the have by fubftitution

=

when

the

And

Terms if

we

firft

are difpofed according to a

take the fictitious quantity to the Root ;', we {hall

Approximation 2^ 3 Sec, A' 3 A'X= m -f- aAx m + -+- a'-Ax" Thefe Terms, or at leaft thefe Indices of x, being felecled

o.

l

,

out of the general Table, will appear thus. to obtain an afcending Series for the Root y, we may apply the Ruler to the three

Now

external equal,

Terms

o,

777,

m

will

give are each o.

=

3777,

o.

which being made Therefore

thefe

In the next place the Numbers Ruler will come to 777.4- i, or i ; and laftly But the Numbers o, i, 3, are contain'd in the Arithmetical to 3. Progreffion o, i, 2, 3, the form of the Root if the

Equation

A

3

+

whofe

difference

y=. A -+- Ex -{-Cx -+A <2a' =o, (which is

is

a1

common

1

is

i.

Dx>,

Therefore 6cc.

derived

Now

from the initial


initial

the

is

Term?,)

A

Quotient

and INFINITE SERIES. divided by the factor A -f- ah. ~t-

=

a

1

o, or

A=.a

205 2a*,

for the initial

will give

it

Term

of the

Root^y.

we would

If

may being

alfo derive a

defcending Series for this Equation,

apply the Ruler to the external Parallelograms 3, yn, i alio thefe made equal to each other, will give m

=

;

we

which

Num-

Then the Ruler will approach to m-\- i, or 2 ; bers will each be 3. But the Numbers 3, 2, i, o, are a dethen to //;, or i ; laftly to o. difference creafing Arithmetical Progreflion, of which the common So that the form of the Series will here be -+- B -+is i.

y=Ax

And

the Equation form'd by the external Terms , 3 3 or i. x3 .v will be o, 1 c'x 3 -f- c 7 of the Equation x*)' s o, y+X} 33. The form as exprefs'd by a combined Arithmetical Scale, we have already feen, us all the varieties of external Terms, with which will eafily ra But for farther illuftration, putting A,v for Circumftances. other their '

CA,

&c. -f- Dx~-

=

A

A=

=

-

mew

the

Term

firft

+

of the Root

y,

we

I i" c'x* -+- c\ &c. 36--A ,v will ftand thus. tabulated, to have an afcending I

I

Now

we mufl

Series,

m have by fubftitution A t x^ + l Thefe Indices of x being

("hall

=o. 2

the

apply

Ruler to the two external Terms o and yn -\- 2, which being

made

m

.*-,

Number

equal, will give

and the two Numbers anting

that the Ruler arrives at

is

zm

+

will be each o.

or

r,

.J.

;

The

and the

next

la ft is 2.

But the Numbers o, i, 2, will be found in an afcending ArithmetiTherefore y =. Ax~ cal Progreffion, whofe common difference is the Root. To deter-f- C -f- D.x^, &c. will be the form of _l_ B.v from the exterior have Terms mine the firft Coefficient A, we fhall -i-.

'>

A'-f-6the

7

firft

=

o,

Term

which

will give

A

=

or Approximation to the

y^c

Root

=

7

c'\

will be

Therefore

y ==.

,

J/-^

&c.

We

may

try

if

we

can obtain a defcending Series, by applying

two external Parallelograms, whofe Numbers o, and and 5;-f-2, which being made equal, will give ;;;

=

the Ruler to the

Numbers i

;

and

will each be 2. laftly

at

o.

whofe

ProgreiTion, Series will here be

But

thefe

The Ruler

will next arrive at 2///-J- i, or the Numbers 2, i, o, form a de Icon cling

common difference is i. CvA -f. B,v y

=

are 2

+

So that die form of the

J ,

&c,

And

putting the initial


The Method of

206 initial

to nothing, as they ftand in the Equation, we or for the firft Approximation o, <r, this Series will be accommodated to the cafe of Con-

Terms equal

have A'* And to the Root. 1

ihall

FLUXIONS,

=

c*x*

A

=

Series is accommovergency, when x is greater than c as the other other when x is lefs than c. to the dated cafe, 6 If the o, 27^ 34. propofed Equation be 8z, f> -\- a^y* thus refolved without When reduced it be to any preparation. may -,

=

our form,

8z 6y/ -}-az 6J\*

will ftand thus,

it

3

*

*

,, 1 f=o; and by* 3 27^9

* B

putting_y=A

',&:c.it willbecome

8A*z*"

!

+ +aA 6

1

z*'

m +' s

&c.7

*

*

*

27^3

*

The

Terms

cafe of external

firft

=

m=s

will give

$A*z* m

+

s

27.^'

=

o,

Thefe Indices or Numbers But 0,2, therefore will be each o ; and the other 2/-f- 6 will be 2. will be in an afcending Arithmetical Progreffion, of which the common difference is 2. So that the form of the Series will be y=. Az~~1 And bccaufe 8A' -|- B -h Cs. -+- Dz*, &c. 27^9, or 2A=3^3,

whence 3/^-1-6

it

Root

the

3

J-0

Therefore the

.

Term

firft

or

Approximation

to

3

-^2 2. *

be

will

2.

=

=

A

will be

or

0,

But another cafe of external Terms will give aA*-z~- mJc 6 Thefe Indices or Numo, whence 2w-f-6 o, or /;; 3. bers therefore will be each o j and the other yn -+- 6 will be 3. found in a defcending Arithmetical P/ogrefiion, But o, 3, will be So that the form of the Series will whofe common difference is 3

=

=

= Az~* A=+

be y J tis

=

=

.

-f-

Ez~

3v/3 x^

6

4 >

Cs-'

-f-

f

r

^

,

And

ccc.

^A 1

becaufe

27^',

^''^ Coefficient.

external Terms, which may Laftly, there is another cafe of pom"a 6 m 6 afford a us -f- aA*z"- ^~ Series, by making SA*z3 defcending bly And the Numbers will be each equal to 6 ; o. o ; whence

=:

m

+

=

is o. But 6, o, will be in a of which the common difference defcending Arithmetical Progreffion, 6 is 6. Therefore the form of the Series will be _y= -f- Ez~ -f1 Oc- 11 &c. Alib becaufe 8A -+- a A o, it is A {a for the

the other

or Index

Number,

of z,

,

firft

=A

=

Coefficient.

produce one Example more, in order to fhew what variety may be derived from the Root in fome Equations; as alib all the cafes, and all the varieties that can be derived, in the to fhew Let us therefore affume this Equation, prefent ftate of the Equation. I fhall

of

Series

y*

--_ + 1,vl

rather y 3

z

a~ y x 1

~

--_ + - _ _ 3^ a

x

i

i

-+- a=

l

-\- x>

=

o.

ClI

/.

a>y~- x

Which

3-

if

-+-

6

_ I

_j_

a\)

__ fl\*

}'

=

=

^

a \y~ z A.

3

we make

.+.

A.\

1

o,

or

a 6 x~ s &c. and

-}-

m ,

difpofe


and INFINITE SERIES. difpofe the Terms according to a fion, will appear thus

207

combined Arithmetical Progref-

:

*** *

.*x"

*

m

+**

*

Now here it is plain by the difpofition of the Terms, that the Ruler can be apply'd eight times, and no oftner, or that there are eight cafes of external Terms to be try'd, each of which may give a Series for the Root, if the Coefficients will allow it, of which four And firft for the four cafes afcending, and four defcending. of afcending Series, in which the Root will converge by the afcending Powers of x ; and afterwards for the other four cafes, when the Series converges by the defcending Powers of x. I. Apply the Ruler, or, (which is the fame thing,) afTume the s 1 " -* 1 a"' A- *Equation a A~=x~^ o, which will give 3/77 will be

=

1

=

2in

from

or

2,

thefe Indices

7/7= 2;

next come to the Index to o ; then to zm 2, or 2

then to 3

But the Numbers

or 6.

in its parallel

refulting

motion

will or 2 then zm-{- 2, ; and laftly to 3/7; and 2/774- 2, 2, o, 2, 3, 6, are in an af-

then to

.

3 ;

The Number

A=^.

But the Pailer

6.

is

alfo

6,

;

3,

cending Arithmetical Progrellion, of which the common difference is i ; and therefore the form of the Series will be Ax --Bx*

y

-f-

C.v, &c. and II.

its firft

Term

will be

Affume the Equation a x~

=

give 3 ber refulting

6

zm hence

2, is

or

m

l

=f

-

1-

.

a

a''A }

=

alfo

x--

t

A

=

i

==z o, which will

The Num-

a*.

the next will be

or

iJL ; the 37/7, the next ; 2, ; 2/>-f- 2, or j or i ; the two laft zm 4- 2 and are each 3. But 3/7;, 3, the Numbers will be in i, found an i, 3, j, o, i|, 3, Arithmetical of which the common difference afcending Progreffion,

next 2/72 the next

is

f

;

or

3

i

;

the next o

and therefore the form of the Dx% &c. and its firft Term

+

;

Series will be will be

y

+ ^/ax.

= Ax^

+-

Bx

+-

III.


208

7?je

III.

Method of FLUXIONS,

=

or;;;=

2?/7

=

a* A. 1 .* 11""-

Aflame the Equation a 6 x~*

A

alfo

=+

o,

which

will

The Num2m 2,

a*.

2, f; 3 give ber refulting is 3 ; the next 3;?;, or if ; the next i ; the next o ; the next 2m -+- 2, or i ; the next or 3z, or the two laft and 2m which are each But the if; -f- 2, 3 3. Numbers be all if, i, o, i, if, 3, will 3, comprehended in an afcending Arithmetical Progreiiion, of which the common dif-

ference

~h B

f and therefore the form of the Series will be y - A.y~ Cx* -f- Dx, &c. and the firft Term will be a*x~'f or

is

-f-

;

,

"v/;-

A

IV. Affume the Equation give

=

3

=

3

2, or ;/z the next will be

2;

= = The

^'A 1 *-'*- 2

A: 3

A

alfo

2;

o,

which

a*.

will

Number

2m -{-2, or 2 ; the next 3 ; the two laft and 2#?4-2, each of which is 6. But the Numbers 6, 3/tf 2, o, 2, 3, 6, belong to an afcending Arithmetical Progref3, Therefore the form of fion, of which the common difference is i. the Series will be y Ax~- +- Bx~' -+- C -f- Dx, &c. and its firft 6 ; the next o; the next is

refulting

2m

3

2,

the next

;

or 2

;

=

Term will be ^ The four defending

Series are thus derived.

Afllune the Equation

I.

give

3;;z

fulting

O; the 3/72

=

is

6

2/w -4- 2, or #2 ;

=

2;;z-f-2, or

2m

and

2,

2,

greflion, of the Series will X Term will be .

be/

<

;

A

alfo

=

-

.

the next 2m 25 the next

each of which

is

which

will

The Number

re-

o,

2,

6.

or 2

;

the next

3; the two laft But the Numbers

belong to a defcending Arithmetical Prodifference is i. Therefore the form 1 C Ex D.*Ax* -i&c. and the firft ~f-f, 6,

3,

of which the

a-'A 1 x"- m + l

3

2;

the next will be 3

next

6, 3, 2, o,

Au

common

=

S,

a

Affume the Equation 2m -+- 2 3, or ;;:= f II.

=

;

x* alfo

a~

A

l

A

i

x im Jri

=+

~

a*.

=

which

will give

The Number

refulting

o,

be 3;^, or if; the next 2;/z-f-2, or i ; the ,/ or i ; the next 2, if; the yn, or two laft and 2m 2 are each But the Numbers 3 3. 3, if, i, o, i, if, 3, belong to a defcending Arithmetical Therefore the ProgreiTon, of which the common difference is i. form of the Series will be_)' Ax^-i-Ex+Cx~^-{- DAT*', &c. and the firft Term will be ^/ax. the next wi: 3 next o ; the next is

;

+

=

III.


and INFINITE SERIES.

=

Aflume the Equation x <7A-** *+ + 2 w H- 2, or TW f alfo A gve 3 ber refulting from hence is 3 the next will be III.

=

=

5

=

;

whichare

i

the next o

;

if

;

two Numbers the

;

But the

the next laft

3;;?,

2m and 2m

3 if, i, o,

which

o,

a*'.

;

next 2m -+-2, or the next 3777, or

209 will'

The Numor if 2, or

;

the i

;

each of

2,

3, are a Arithmetical comprehended defcending Progreflion, of which the common difference is f Therefore the form of the Series will 3.

3,

if,

i,

in

.

bcy=Ax~*-t-Bx~'-i-Cx~~ l -l-Dx- %

+ a*x~*

be

or

+a

-

&c

an d the

-

Term

will

.

IV. Laftly, aflume the Equation a 6 A-ix~i m

which

firft

=

2m -f-

m

=

rfA.- I

A

==='#. alfo ; next zm 2, refulting is 6 ; the next will be 3 ; the or 2 ; the next o ; the next 2m -f- 2, or 2 ; the next the ; 3 two next 3#; and 2m But the Numbers 6, 3, 6. 2, are each 2, o, 2, 6, belong to a defcending Arithmetical Progref3,' of which the -common difference is r. Therefore the form of iion, will give

3;;;

2,

or

2

The Number

the Series will

1

be/=A x <

H-BA

3

4-Cx 4-t-Dx-5 &c. and ,

the

firft

'

rn

Term is And this may .

fuffice

in all Equations of this kind, for finding

the farms of the feveral Series, and their

we muft finding

all

firft

Approximations.

proceed to their farther Refolution, or to the the reft of the

Terms

fucceffively,

Now

Method of

no

.SECT. V. The Refolution of Affe&ed Specious Equations, firofecuted by various Methods of Analyfis.

TTT ITHERTO

has been fhewn, when an Equation is ~J_ propofed, in order to find its Root, how the Terms of the Equation are to be difpoied in a two-fold regular fucceffion/fo as thereby to find the initial Approximations, and the feveral forms of the Scries in all their various circumftances. the Author proceeds in like manner to difcover the fubfequent Terms of the Series, which may be done with much eafe and certainty, when the form 35.

it

Now

of the Series is known. For this end he finds Refidual or Supplemental Equations, in a regular fuccefTion alfo, the Roots of which are a continued Series of Supplements to the Root In required. one of which the every Supplemental Equations Approximation is

E

e

found,


The Method of

2io

FLUXIONS,

found, by rejecting the more remote or lefs confiderable Terms, andfo reducing it to a fimple Equation, which will give a near Value of the Root. And thus the whole affair is reduced to a kind of Comparifon of the Roots of Equations, as has been hinted already. The Root of an Equation is nearly found, and its Supplement, which, ihculd make it compleat, is the Root of an inferior Equation > the Supplement of which is again the Root of an inferior Equation ; and fo on for ever.

retaining that Supplement, we may flop where we pleafe. Author's Diagram, or his Procefs of Refolution, is very

Or

36. The eafy to be underflood ; yet however it may be thus farther explain'd. Having inferted the Terms of the given Equation in the left-hand therefore are equal to nothing, as are alfo all the fubfequent Columns,) and having already found the firft Approximation to the Root to be a ; inflead of the Root y he fubflitutes its

Column, (which

equivalent a-\-p in the feveral Terms of the Equation, and writes the Refult over-againfl them refpedtively, in the rightrhand Margin. Thefe he collects and abbreviates, writing the Refult below, in the .

of which rejecting all the Terms of too high a ; i i he retains only the two loweft Terms ^.a p-\-a x=.o^ compofition, x for the fecond Term of the Root. Then which give p -%x-}-q, he fubflitutes this in the defcending Terms afluming/> to the left-hand, and. writes the Refult in the Column to the righthand. Thefe he collects and abbreviates, writing the Refult below Of which rejecting again all the higher in the left- hand- Column. i o i which T T-cx Terms, he retains only, the two loweft ^a*q left-hand

Column

==

I

for the third

give a

Or

Term

And

of the Root.

=

fo on.

imitation of a former Procefs, (which may be feen-, ,pag; the Refolution of this, and all fuch like Equations, may be in

165.) thus perform'd.

i)3_|_tf^y=

-+.axy-h

2fl'= (if y=-a-irp} A?

a*

+

!L

'$a*'p-\-T> ap -}-p*

+ +ay a*

3

Or collecting > andexpung}

J ing,

I r

collecting and expunging,

By which

Procefs the

Root

will be

found

_y

=

X <z

7* 4-

^,

&c.

Or


and INFINITE SERIES. Or in imitation of may thus refolve the

Method

before taught, (pag. 178, &c.) we Supplemental Equation of this Example

the firft

3

>a

=

;

a'-x -+ x*

w>. W-p -f- axp -f- 3^/ -4-/ muft be difpos'd in the following manner. it

briefly denotes, that a for the Value of y.

the

Term

the fecond

-,

=

Term

of the Series, to be derived f#, &c. infinuates, that fx

firft

Alfo^=*

is

where the Terms

But to avoid a great deal a, &c. may be here obferved, that y

of unneceffary prolixity, is

211

of the fame Series

y.

Alfo y

=

*

* -f-

644

&c.

that -4- r

infinuates,

is

1

the third

643

Term And fo

out any regard to the other Terms. Terms ; and the like is to be underftood of

of the Series

y, **

with-

all the fucceeding other Series what-

for all

ever.

==

4*'/l

a*x

+*

*

^

&c.

,

40963 13 1x4

c &c

7T7T

To

it a*x is may be obferved, that here the Series, into which ^a l p is to be rea*x, &c. and therefore p x, &c. which

this Procefs,

explain the firft

made folved is fet

-

or 4 .a*p

;

t

down a

_f-3^)

Term of

=

= Then

below,

-t-Tz-ax

is

axp ^ax &c. each of which are

l ,

per Places.

which with

a contrary Sign

;

or 4d a/>

=

*

+

&c. Then axp=.*-^-^ &c. and (by cubing) />

,

5

1 ,

&c. and (by fquaring)

down in their procollecled, will make -V^S muft be fet down for the fecond Term

Thefe Terms being

of ^a*p

=

=

-f-

l

-r'^ax

}

let

&c. and therefore

p

= =

* -f-

-?-

>

il. > &c. and (by fquaring) 3<?/> a * W^" ' ^ c T^efe being collected

=

3

-

be wrote down with a contrary Sign; and with A: 3 , one of the Terms of the given Equation, this, together * * -f- --'x* } &c. and therefore />= * * -f- -^- ? will make a*p '*" a

will

make

^,

to

=

l

~~ i

&c.

Then

axp= *

* -J-

= s

1*4 ,

&c. and (by fquaring)

E

e

2

3^/1*

* *


Ih* Method of

212 1&22

&c

FLUXIONSJ

=

and (by * -f- -1^1 &c. all which cubing) * N ] ft! 1024* 1 ***-fbeing collected with a contrary Sign, will make 4tf /> 59i_* &c. and therefore And by the -f,' &c. _,

'

.

, '

=

4096.1

^

/=***

,

40961*

we may

fame Method

163841

'

we

continue the Extraction as far as

pleafe.

The

Rationale of this Procefs has been already deliver'd, but as it will be of frequent ufe, I fhaM here mention it again, in femewhat a .different manner. The Terms of the Equation being duly order'd, fo as that the Terms involving the Root, (which are to be refolved into their refpecttve Series,) being

and

fide,

known Terms on

t,he

all

Column on one

in a

the other fide

any adventitious

;

Terms may be introduced, fuch as will be neceffary for forming the feveral Series, provided they are made mutually to deftroy one another, that the integrity of the Equation may be thereby preferved. Thefe adventitious Terms will be fupply'd by a kind of Circulation, which ceffary

pofe the Equation, foregoing Theorems.

one by one, by any of the

muft be derived -

-

,

are willing to avoid too many, and to0 high Powers in thefe Extraction's, we may proceed' in the following manner.

Or

we

if

The Example mall be the fame Supplemental Equation as before, which may be reduced to this form, 4a* -f- ax -+- ^ap -4- pp -x.p =s of which the Refolution may be thus a*x * 4-# 3

j

"

:

'

,

.' '

-'!

,

make the work eafy and pleafant enough ; and the neTerms of the fimple Powers or Roots, of fuch Series as comw^ill

'

4rf

a

__, 3

H- ax .

a

h TV**

-

X*

I

64

The Terms

.

$

5

-4-/

-

.-

.

i

2a

77^

&c:

,

I*S

3

5i2i

16384^3^

I call the aggregate Factor, of or which I place the known part parts 4<2* -{- ax .above, and the unknown, parts ^ap -f- pp in a Column to the left-hand, fa as that their refpeclive Series, as they come to be known, may be placed

4^*

-f-

ax-\-

them. regularly over-againft

^p-^-fp

Under 2,

thefe a

Line

is

drawn, to receive the

-

,


and INFINITE SERIES.

2*3

the aggregate Series beneath it, which is -form'd by the Terms of the as become Under this aggregate Seknown. Factor, they aggregate ries comes the fimple Factor />, or the of the Root to be fymbol

Terms become known alfo. Laftly, under all are the known Terms of the Equation in their proper Now as places. thefe laft Terms (becaufe of the Equation) are equivalent to the Product of the two Species above them from this confideration the Terms of the Series p are gradually derived, as follows. extracted, as

its

;

Term 4^

(of the aggregate Series) is brought no other Term to be collected with having Term, multiply 'd by the firft Term of />, firft Term of the is to the fuppofe q, Product, that is, ^.a'-q equal a*x, it will be q ~x, cr p -L.v, &c. to be put down in its we have T Thence (hall place. ap-=. %ax, &c. which towith will make the fecond Term for -}-ax above, gether -^^'ax Firft,

the

down into it. Then

initial

place, as becaufe this its

=

=

=

>

;

of the aggregate Series. Now if we fuppofe r to reprefent the le-of and to be cond Term wrote in its place accordingly ; by crofsp, -'y-ax^ =^ o, becaufe the fecond multiplication we lhall have ^.a^r _v^

Term of the Product is abfent, or=ro. Therefore r-=. which 64*' * -f- -^x may now be fet down in its place. And hence yap l &c. and p* ^x*, &c. which being collected will make ^~x Now if we fuppofe for the third Term of the aggregate Factor. of then s to third Term the p, by crofs-multiplication, (or reprefent

=

^=

s

=

&c. 5

I

3

(for

,

x

3

to be fet

512^*

and

i

3

^2l

Multiplication

l

2a

_j_

for

256

fore

3 -,

,

by our Theorem * ; 256

,

=

1

=

is

the

down

third Term

,

&c.

1

infinite

Series,)

q.a ;

of the Product.)

in its place.

Azea

*

of

Then

-lap

which together

=

4-

There* * -4-

will

make

Term of the aggregate Series. Then putting fourth Term of p, by multiplication we fliall have

for the fourth

/.to reprefent the

^=

= -^L

to be o, whence / ' 4096* 2048^ would proceed any farther in the ExIf fet down in its place. of the Setraction, we mufl find in like manner the fourth

we

Term

ries

3/, and the third

Term And

of p*-,

in order to find the fifth

Term

thus we may eafily and furely carry of the aggregate Series. on the Root to what degree of accuracy we pleafe, without any danger of computing any fuperfiuous Terms ; which will be no mean advantage of thefe Methods.

Or


Method of FLUXIONS, Or we may

we The

proceed in the following manner, by which

avoid the trouble of

fliati

Supany fubfidiary Powers of the fame -f+axp ^ap* -fExample, ^cfp plemental Equation all of others in imitation a*x-{-x*, (and this,) may be p= at

railing

=

reduced to

this

form,

which may be thus

/\.a*-

-+-

ax-+- ^a

-f-

The Terms being difpofed as in firft Term of the aggregate

or^=

this

Series.

;

~#x

the Refult

Then

firft

Series,

we

fhall

may

it

ax

the Term above, Term of the aggregate Term of />, and we fhal'l collect

1

-r-s-^x

Term

have 3^ x ~~a

=

o,

of^>.

-I- -rV-v

or r

Then 1

of the aggregate Series. Again, fuppofing of p, we ftiall have by Multiplication, third s

+

,

',

256

256

'

=x

i

.

=

that

^

s

1

be

by crofs-

^or tne third

to reprefent the

(fee the

is,

to

,

as above,

= ^V^

Term

for that purpofe,) A.a

4^*

call'd,

Term

Term

1

be

ftill

will be the fecond

have by Multiplication q.a'-r wrote every where for the fecond multiplication

as

r reprefent the fecond

let

x>,

down

Paradigm, bring

of p. with of 3#-f-/>x/>, which product

ax

+

a*x

of the Series p. Then will ~x, which is to be wrote every where x for the firft Term Multiply +- 3*2 by

q to reprefent the

a*x, 4^*5' for the firft Term or -+-

=

x/> x/>

ax

for the

=

p

refolved.

4#*

and fuppofe

-{-

all.

Theorem,

s=^^ 5iz

a

. '

to

be wrote every where for the third Term of p. And by the lame of the Term of the fourth way Multiplication aggregate Series will be

found to be

\-2L. }

And

Among

all

muft not omit

which

will

make

the fourth

Term

-of

p

to be

fo on. this

variety

to ftipply

of Methods

for

thefe Extractions,

the Learner with one more, which

is

we

com-

mon


and INFINITE SERIES/ and obvious enough, but which fuppofes the form of the Series required to be already known, and only the Coefficients to be unknown. This we may the better do here, becaufe we have alhow to determine the form and number of fuch Seready fhewn cafe This Method confifts in the aflumption in propofed. any ries,

mon

of a general Series for the Root, fuch as may conveniently reprefent it, by the fubftitution of which in the given Equation, the geThus in the prefent Equaneral Coefficients may be determined. A' 5 2a 3 tion y= 4- axy 4- aay o, having already found (pag. A 4- Bx -+- Cx*, &c. 204.) the form of the Root or Series to be y for of of the Methods any Cubing an infinite Series, by the help we may eafily fubrtitute this Series inftead of y in this Equation, which will then become

=

A

4~

3

3

A

l

4~ ^n.D l x 1 4~ 4^ 3A*C 4-

ijX

=

4~

-D'AJ" 3

^"*> *^c*

3

6ABC4- 36^ }

4-

6ABD

-f-

aDx*, &c.

o.

aBx

aA.x 4-

Now

becaufe

known.

Thus,

4- aCx*

1-

an indeterminate quantity, and muft continue' fo to be, every Term of this Equation may be feparately put equal tonothing, by which the general Coefficients A,B, C, D, &c. will be determined to congruous Values ; and by this means the Root^ will be as

x

is

A

( i.)

3A B a

before.

(2.)

3

1 4- a A

-+-

aA

2a~>

-+- a*-B

= =

o,

which

o, or

will give

A=/r,

B==

=

i B -4-6ABC-j-3A*D4-rfC-H a D o, orD'=^_> aD 4- ^E A*E 6ABD AO 44o, or 3 3B-C 3 (5.) _^2_ And. fo on, to determine F, G, H, &c. Then by fubfti163^4^^ Values of A, B, C, D, &c. in the aflumed Root, we tuting thefe 3

(4.)

+

+

=

=

.

(hall

have the former Series y

we may

=ax + ^4- '+ ;%^>

&c.

conveniently enough refolve this Equation, or the fame kind, by applying it to the general Theorem, any other of ra CT 1 90. for extracting the Roots of any affedted Equations in NumFor this Equation being reduced to this form ;i3 * 4- a 1 4-^x bers.

Or

laftly,

.

2.

x/


The Method of

2l b

=

c

2rt 3 -+-A:'

x

x.y

we

o,

fliall

FLUXIONS,

have there

#2

=

And

3.

inftead

of the firft, fecond, third, fourth, fifth,- &c. Coefficients of the Powers x , o, 2# 5 of y in the Theorem, if we write 1,0, aa -f- ax, ?

.&c. refpectively

and

;

A= a and B =

or

mation

B

we make

if i

4^

Approximation

4"

A , a

4

if

.t5 -(-

48a*.v4

-f-

i

-

>

for a nearer

A=

-f-

=

-

*

*

,

we make Again, by Subftitution we fliall

ax,

-f-

firft

we. fliall have

;

the Root.

to

the

Zfi4 ,* -f-

Approxi4^' -f- x*, and

have zqSjt

.*

the Fraction *

+1*9

,.

And taking this Numerator we fliall approach nearer ftill. near, that if we only take the firft

nearer Approximation to the Root. for A, and the Denominator for B,

But

this laft

Approximation

is

fo

Terms of the Numerator, and Terms of the Denominator, (which, troublefome Operation,) we fliall Series, fo often found already.

them by the

divide

five

firft five

no managed, of the have the fame five Terms will be

if rightly

Theorem will converge fo faft on this, and fuch like ocA a, (ma-cafions, that if we here take the firft Approximation ** have we fliall a &c. i ~x, &cc. y ,) king B -^ ^ a .And if again we make this the fecond Approximation, or A we fliall have y i,) t*, (making B

And

=

=

if

a

ax ~T

-i

=

*

we make

again

=

,

=

4

=

the

4

this the third

&c. (making

5

Approximation, or

'B==

we

1

z 4

A=:a

fliall

have the Value of the

to eight Terms at this Operation. the number of Terms., that double will ation

For every new Operwere found true by the

_

D ti&

^

*

*-**

true

Root

laft

Operation. proceed

To

J,)

with the fame Equation we have found before, we might likewife have a defcending Series in this AA'H-B -j-Cx- &c. for the Root y, which we fliall ftill

;

that pag. 205,

form,

v

=

1

,

or three ways, for the more abundant exemplification of It has been already found, that this Doctrine. i, or that x Make therefore y =. x is the firft Approximation to the Root. x> and fubftitute this in the given Equation jy 3 -f- axy -f-

extract

*a= -4-

-^

=

two

A=

o,

which

ax 1

-f-

a^x

_t_

ax

will then

2

3

=

-+- a* -f- 3.v/>

be refolved as follows.

any

^p

become -f- axp -f- a?p -\- ^xf This may be reduced to this form o.

-{-/* x/>

=

ax"-

a*x

+

2a*,

and

may OAT*


and INFINITE SERIES. 1

4_

/,.. '

_

*

3-v* "

_i_^

rt*

-f-

ax _ .

p*

y

aX

-f-

3A'

3

217

a*

_+_

+^

_+_

-f-

t rfl

~~ -U_

*""*

+~ _i-

IE1

_;

^

t

Sec.

s

&c>

&c.

,

64 " 4

c --

81^^

3-v

The Terms of the aggregate Factor, as alib the known Terms of the Equation, being down ^x l difpofed as in the Paradigm, bring for lire firft Term of the aggregate Series ; and fuppofing q to reprelent the firft Term of the Series p, it will be ax*, or

=

3^^

q=-

y,

for the

firft

Term of p. put down in

ax

Therefore

will be the

Term of 3^ to be its This will make the place. fecond Term of the aggregate Series to be nothing fo that if rethe fecond Term fliall have we of prefent by multiplication 3vV p, of p, to be put down a 1 *;, or for the fecond Term "_ in its a 1 be the fecond Term of $xp, as alfo Then will place. 1 to be fet down each in their places. ^d"~ will be the firft Term of/ The Refult of this Column will be -^z 1 which is to be made the third Term of the Then putting s for the third aggregate Series. Term of/, we mall have by Multiplication ^x^s -V rt3 == 2(l or s= $52-1 the next Operation we fhall have / And thus by J firft

;

;

=

=

/

,

,

'

>

=

.

and "Or

if

fo on.

we would

Equation by one of the fore-

refolve this reildual

the railing of Powers was avoided, and going; Methods, by which wherein the whole was performed by Multiplication alone ; we may

reduce

d^X

it

to

_j_

this

2n* ,

1

+

1

fix -f- a -f- 3* -j-/ x/> x/ form, 3* the Refolution of which will be thus

=

^.v

1

:

F

f

j.v-


tte Method of

2I 8

3**

-f-

--+

FLUXIONS,

ax -h 3*

a*

T*

~

fa*

+

--8ix*

3*-

*

a

&c.

,

.

243*3'

The Terms being difpos'd as in the Example, bring down 3*'* for the firft Term of the aggregate Series, and fuppofing q to reprefent the firft Term of the Series p, it will be ax*, or q yx^q Put down -+- 3* in its proper place, and under it (as alfo after La. it) put down the firft Term of/, or La, which being multiply'd, and collected with -j- ax above, will make o for the fecond Term of the aggregate Series. If the fecond Term of p is now reprefented to be put a'-x, or r by r, we fhall have ix^r * down in its feveral places. Then by multiplying and collecting we mail have -f- a* for the third Term of the aggregate Series. And putting s for the third Term of p, we fhall have by Multiplication or j= T T rf3 =2d 3Ar*j |^ And fo on as far as we pleafe. Laftly, inftead of the Supplemental Equation, we may refolve the

=

=

=

=

,

3^*

'

3

.

,

given Equation itfelf in the following

ax

-----f-

1

%a*x

ax 1

y= x

manner

La*x

-f-

:

-

* 28*4

\a* La* +-

,

Sec.

,

&c.

'a 243^5

=

Here becaufe it is y~> =x*, &c. it will be y x, &c. and therefore -I A which &c. muft in be fet down its =-fThen _t_ xy place. wrote with a it muft be again contrary fign, that it may be y= == * * rfx*, &c. and therefore (extracting the cube-root,) a * &c. Then -+- a*y 4- a*x, &c. and + axy j-^^, &c. which ,

=

= /= ,


and INFINITE SERIES. being collected with a contrary fign,

which

=

will

219

=

* *

makers &c. Hence

l &c. and (by Extraction) y * * -f- a y 3 &c. and -f- ^v>'= * * # frt a*, &c. which being cola and united with -f- 20 J above, will lected with contrary fign,

JLa*x,

=

,

,

make y"'

=

Then

&c.

* * *

&c. which being * * * *

177

And

&c.

fo

'

5 ,

=

7

a

-+-

f^

= =

&c. whence (by Extraction) y &c. and

* *

j*,

-f-

axy

* * *

* * *

+ ^7'

=

with a contrary fign, will make y* and l ^ en (by Extraction)^' ==* * * *

collected

&c

'

>

4-^

+

on.

need not trouble the Learner, or myfelf, with of the Author's giving any particular Explication (or Application) Rules, for continuing the Quote only to fuch a certain period as {hall be before determined, and for preventing the computation of fuperfluous Terms ; becaufe mod of the Methods of Analyfis here deliver'd require no Rules at all, nor is there the leaft danger of making 37, 38.

I

think

I

any unneceflary Computations. 39.

When we

are

1

this,

y

t)'

-+-

fj

3

to find the

t

v4

Root y of fuch an Equation

+ t>"> &c

-

=

*>

tllis is

ufually

as

call'd

z

For as here the Aggregate is the Reverfion of a Series. exprefs'd by the Powers of y; fo when the Series is reverted, the Aggregate

y

This Equation, as now it will be exprefs'd by the Powers of z. the of z the Series) to be unknown, (lands, fuppofes (or Aggregate it are to to we that and indefinitely, by means of the approximate

known Number y and its Powers. Or otherwife the unknown Number z is equivalent to an infinite Series of decreafing Terms, an Arithmetical Scale, of which the known Number y exprefs'd by ;

is

the Root.

This Root therefore muft be fuppofed

to be lefs

than

And thence it will folUnity, that the Series may duly converge. be much lefs than will This is ufually calthat z, alfo low, Unity. led a

Logarithmick

Series,

becaufe in certain circumftanp.es

it

ex-

between the Logarithms and their Numbers, as preffes the Relation If we look upon z, as known, and therefore will appear hereafter. mull be reverted; or the Value of y muft the Series as unknown, y be exprefs'd by a Series of Terms compos'd of the known NumThe Author's Method for reverting this Seber z and its Powers. ries will be very obvious from the confideration of his Diagram and we mall meet with another Method hereafter, in another part of ;

Works. It will be fuffiqient therefore in this place, to perform after the manner of fome of the foregoing Extractions. his

F

f 2

it

y


Method of FLUXIONS,

= a + |~

1

y

a

-+- f:i

+ TV-

3

4

4- T5o-3 }

&c

>

-

*

fV-

h f*

>

5

-4-

M

A./

AS',' &c. fa', &c.

'f^ 4 -H 4 _. fS

AX* J

L.

-s

&C.

*

Sec.

In this Paradigm the unknown parts of the Equation are fet down defcending order to the left-hand, and the known Number z is

in a fet

down

the

to

=

y

over-againft and therefore fj*

Then

right-hand.

is

y

=

z, Sec.

Sec. which is to be fet down in its fa with a place, contrary fign, fo that _}'= * -f- f % & c 1 And therefore (fquaring) * f 2', Sec. and (cubing) f^ 3 3 -h fy fa Sec. which Terms collected with a contrary fign,

and

y=

=

=:

4

&c

,

which Terms

Sec.

1

y

And

Sec.

an d (cubing)

-

= ***-{= f^% make f.?.

-

,

* * -f- -.^s,

rV24

* *

,

alfo

=4-

make

1

-j -?-

54 '

^c

-4-

therefore (fquaring) 3

f_)'

=

4

a

y_y

=

f_y*

,

=

and

f/

fign,

make

Sec.

with a contrary

collected

Therefore

-

fa

* -|-

* # *

f.s

f ,

4

&c.

&c. which Terms collected with a contrary fign, ***-{- -4-a Sec. And fo of the reft. if we were to revert the Series y -f- f/ -f- ^V>' + TT-T^ Thus 40. -f- T .fy T y' -h TTTS->''S ^ c ^, (where the Aggregate of the Seunknown Number ries, or the a, will reprefent the Arch of a Circle, whole Radius is i, if its right Sine is reprefented by the known Number y,) or if we were to find the value of r, confider'd as unknown, to be exprefsd by the Powers of a,, now confider'd as known

H- f^

5

-{-

y=

s

#-

,

-

5

3

=

;

we may proceed

thus

:

Lo*3

,^1^

f

]_

o*

5

____'__

*?"

1

^9

-.{

+ Sec.

being difpofed as you fee here,

= fa we

and therefore (cubing) fj 5

fs Sec.

3 ,

and

3 5 i>>9 3 T"T"3""a"

ATr* vVv

j

The Terms Sec.

o^C

Sec.

alfo

fo

-^y

that (cubing) 1

=-5

5-a

?

!r

,

Sec.

3

lhall

,

Sec.

have

we

mall have

jy==a, which makes y *

+

3

f_>'

=

*

=

-rV^'j

and collecting with a contrary

fign,

y


and INFINITE SERIES, '

r==* *

*

-+-TT.T-'.

TV~

7 ,

&c

-

&c- and TTT.v

=

= TTT~=

Hence 7

\<->

>

WT^"'

7

&

c. and and collecting with a Anct lo on '

* *

7

221

T TV

>

^y

&<--.

&c<

* * * contrary tign, v It" we fhould defire to perform this Extraction by another of the the Equation to be reduced foregoing Methods, that is, by fuppoiing; 6 4 &<' x to this- form i -+- j-_v* 4- -rV +- TTT.' ;==;, it TTTT^'^ may be fufHcient to let down the Praxis, as here follows.

+

I

'


The Method of

222

FLUXIONS.

but perhaps moft readily by fubftituting the Value of y now found in the given Equation, and thence determining the general Coefficients as before. By which the Root will be found to be _)'

or Z.O

J

-

Z^ gt**

I-

3

'_ *;7

T ~ *

9 9

fy7 6

6

I

i

i

i

~3

=

f, _

TT 3 TTTT' T^ T JTT^rT 42. To refolve this affected Quadratick Equation, in which one of the Coefficients is an infinite Series ; if we fuppofe y =. Ax m , &c. we (hall have (by Subftitution) the Equation as it ftands here below.

Then by whence in

its

m

I

i

parallel

4,

I

i

applying the Ruler,

=

*_2-

and

A

=~

motion will

we .

{hall

The

arrive at,

have

aAx m

4

=o,

next Index, that the Ruler

is

m

-+- I,

or 5;

m-\-2, or 6; &c. fo that the common difference fion is i, and the Root may be reprefented by y Cx 6 , &c. which may be extracted as here follows.

the next

is

of the ProgrelAx* -{-Ex* -f-

=

x"

-+-

> **-*-.


and INFINITE SERIES. Methods taught

=

+

a

x

-f-

_,_

&c.

g,

Now

*

1

found y

will be

it

before,

223

in the given Equation,

-4-

becaufe the infinite Series a

x

-\-

-+-

v4

vj

4-

ji

y

&c.

,

as

^r

a Geometrical

is

and therefore

ProgrcfTion,

is

a

-

equal to

the Equation will

the

tract a

become

-x

-

*

or

lax

ia~

y -f-

_>*

the

in

fquare-root a ...4+

-~^

be proved by Divifion

may

^

fubftitute this,

And

o> it

way,

ordinary

exa

-

^

we

if

;

will

R ootj And

if

we

ex-

y

r=

give

if this

Radical

be refolved, and then divided by this Denominator, the fame two Series will arife as before, for the two Roots of this Equation. And this fufficiently verifies the 43. In Series that are

whole Procefs.

very remarkable, and of general ufe, the not (if obvious) mould be always affign'd, when that can be conveniently done; which renders a Series ftill more ufeful and elegant. This may commonly be difcover'd in the Computation, by attending to the formation of the Coefficients, efpecially

Law

if

as

of Continuation

we put Letters to reprefent them, and thereby may be, defcending to particulars by degrees. for instance,

Series,

Confecution

z=y

{y

1

keep them as general In the Logarithmic y4, &c. the Law of

-f- .lys

Term, tho' ever fo remote, be to reprefent any at if we put For aflign'd may eafily pleafure. Term indefinitely, whcfe order in the Series is exprefs'd by the natural

is

very obvious, fo that any

Number

4- or

,

Term

is

m

is

L-y

l

= ,

Reverie of this Series, or y &c. the Law of Continuation indefinitely,

T= tinued for the

_

whofe order

=+

-j", where the Sign muft be an odd or an even Number. So that the

then will

according as

hundredth

T

T

in

the next

is

-j-_J_^

thus.

Let is

3

101 ,

&c. 4

In the

TTo-^S reprefent any Term m ; then is exprefs'd by

z,~\- f-s* +- fa;

the Series

m

is

T

-+- -y^a

-4-

^p- , which Series in the Denominator mud: be conto as many Terms as there are Units in m. Or if c ftands Coefficient of the Term immediately preceding, then is

T=

m <y

m

=

+

z fz"> -+- T TO-S' ToV^ 27 Again, in the Series y _r' the &c. (by which the Relation between Circular Arch _ rT TT a*nd its light Sine is exprefs'd,) the Law of Continuation will be thus. I

>


If

T

if c

Method of FLUXIONS,

*?%*

224 be any

Term

of the

Series,

whofe order

be the Coefficient immediately before

And

in the Reverie

of

or

is

then

;

=

and

exprefs'd by w,

=

T

_ f _im

I

". ,

zm

I

x zm

2

A

3

-f- -?)' -f- T T y~ be thus. If of will Confecution _f-_|^- T r reprefents any Term, the Index of whole place in the Series is ;//, and if "" ,,i-i_ And c be the preceding Coefficient j then 9

,

cc.

the

this Series,

z.

y -f-

^v

T

Law

T

=

.

.

2/11

I

X 2

2

the like of others.

44, 45, 46. If we would perform thefe Extractions after a more and general manner, we may proceed thus. Let the given Equation be v*_ -}- a\v -\- rf.vv z<jj r , j5 the Terms of which Ihould be difpofed as *, * Q * * * l> -}in the Margin. p, where Suppofe y _ ; y is to be conceived as a near /; Approximation to the Root y, and p as its fmall Supplement. When this is fubftiftand the will as it tuted, Equation a ~P I becaufe .v and f> ? ^'f" does here. f=~\

Indefinite

2^x-=o, _ +

+

fl

+

=

Now

+

-^

are both fmall quantities, the moil confiderable quantities are at the beginning of the Equation, from

_i_

+ _

'/ 3

/*

"

^

+

+

f

+

*

= */.

a

s

*

^ '

x

whence they proceed gradually dias oug;ht minifliing, both downwards and towards the right-hand when the to be Terms of an are fuppos'd, always Equation dilpos'd ;

And becaufe inftead of according to a double Arithmetical Scale. we have unknown here introduced two, If and one quantity _v, />, we may determine one of them b, as the neceffity of the Relblution

To

remove therefore

the moft confiderable require. Quantities out of the Equation, and to leave only a Supplemental Equation, whofe Root is/>; we may put 6* -+- a*b 2^ o, which Equation will determine b, and which therefore henceforward we are to And for brevity fake, if we put a 1 -+- 3^* look upon as known. mail have the we c, Equation in the Margin. Now here, becaufe the two initialTerms \. aox+axp -f- cp -+- abx are the moft con fiderable of ? * = * r the Equation, which might be removed, if

iliall

=

for the nrft Approximation

^

afiume prefs'd -flwll

,

and the

lower ; therefore

have

this

to^ we

fiiould

refulting Supplemental Equation

make p

=

_f- q,

would be de-

and by fubftitution

we

Equation following.

Or


and INFINITE SERIES. Or e

make

i

-+-

this

in this

=

if

form.

^+

.

3

*

^,

itwillaffume

'

3*?*

.e.

Equation,

^

=/;

+ J7

225

a

^

> V

1

+

*

**?

< 3

x*

Here becaufe the Terms put y

=

ftitution

-x

we

l

+- r,

fhall

to be next

removed

and by Sub-

mall have the Root y

=

the Root of this Equation

+ 3V _ 1^-^*

+?

have another

Supplemental Equation, which will be farther deprefs'd, and fo on as far as we pleafe. Therefore

we

are-f- cq -j-^-x^we

+<?

*

may

3

-f? l |

^

?

*

a

-x

<

--x*

c

b* -+.

c

a*b

Or by another Method of Solution, if A -\-Bx -}- Cx* +- Dx (as before) y

=

za*

=

=

a

-f-

3^*

affumc

&c. and fubflitute

this in

-= ,

c

o,

will be

1

we

in this 3

where b

Sec.

Equation

the Equation, to determine the general Coefficients, we fhall have e . a\ c a , . , -ja' c is the x"' t &c. wherein -x -tx*-\ y ,.-

= A --

--

Root of the Equation AS

-f. a*

A

art 5

o,

and

c

1

=

A

3 A*

+

a*.

47. All Equations cannot be thus immediately refolved, or their Roots cannot always be exhibited by an Arithmetical Scale, whofe Root is one of the Quantities in the given Equation. But to perform the Analyfis it is fometimes required, that a new Symbol or Quantity fhould be introduced into the Equation, by the Powers of which the Root to be extracted may be exprefs'd in a convergAnd the Relation between this new Symbol, and the ing Series. of the Equation, mu ft be exhibited by another Equation. Quantities were if it Thus propofed to extradl the Root y of this Equation, 1 4 &c. it would be in vain to expedt, x fi-\-y 4/ -Hy}' ^_}' that it might be exprefs'd by the fimple Powers of either x or a.

=

3

,

For the Series itfelf fuppofes, in order to its converging, that y is fome fmall Number lefs than Unity but x and a are under no fuch ;

limitations.

Powers of

And x, may

introduce a

therefore

a

compofed of the afcendiag

therefore neceflary to mall alfo be fmall, that a Series

Series.

new Symbol, which

G

4-

Series,

be a diverging

g

It is

form'd


Ibe Method of

226

FLUXIONS, to y. Now

Powers may converge it is plain, that x fo muft becaufe be each near other, rf, great, always is a therefore difference &.c. (mall Aflame their y y* } quantity. a the Equation x z, and z will be a fmall quantity as required; and being introduced inftead of x a, will give z-= y y* -f4 &c. whofe Root be extracted will z->t-^^ -ly being y ^y* -j-y.2 4-TV s4 ^cc> as before. x o, to find 48. Thus if we had the Equation _> -f-j* -h_y

form'd of

its

tho' ever

ami

=

,

3

>

i3

Root y Powers of

3

= =

we might have

a Series for y compofed of the afcending would which x, converge if x were a fmall quantity, lels but would than Unity, diverge in contrary Circumftances. Suppox to be a large Quantity in this cafe the was known that then fing

the

;

;

Author's Expedient

is

x=

pofing the Equation

Making &

this. l

,

the Reciprocal of x, or fup-

of x he introduces

inftead

z

into the

Equation, by which means he obtains a converging Series, confining of the Powers of z afcending in the Numerators, that is in reality, of the Powers of x afcending in the Denominators. This he does, to keep within the Cafe he propofed to himfelf ; but in the Method here purfued, there is no occafion to have recourfe to this Expedient, it being an indifferent matter, whether the Powers of the convergor the Denominators. ing quantity afcend in the Numerators

Thus king y

in the given

= Ax

Equation y> 4-j

m

A**'" 4-

6cc.)

,

A

1

5-

*""

4-

* ?

jy

4-

Ax m

* j

by applying the Ruler i, and o, or

= is

m=.

The

3.

or 2; the

next

laft

of which the Root will be y thus extract.

+y

&c.? .3

5 A' mall have the exterior Terms A 3 A* i. Alfo the refulting Number or Index to which the Ruler approaches will give 2/11,

A

i.

common

,

we

Term

m, or

v*

'

= Ax

=

But

make

3, 2, i,

difference

is

a defcending Progreffion, Therefore the form of the 4- DAT"* &c. which we may

i.

4- B 4- Cx"

1

,

'

=A

1 1 ,&c.it will be _y=x,&c.and therefore _y =A- , &c. which will make y* * * #*, &c. and (by Extraction)/ -^ with which ~x A* Then &c. * &c. below, } (by fquaring)^* ~x } &c. and therefore * * and changing the Sign, makes j 3

Becaufe

3

>)'

l}

=

=

=

=

y


and INFINITE SERIES.

=

=

==

227

Then ;* * * * * .1, &c. and y , }*~", &c. * * * -4&c. which together, changing the Sign, make y> * * &c. and TV*~S &c. Then y-f- 44*-', _ and * therefore &c. * ^ _^_ ^.^-^ ^ # Sec. and *""'> 75 ~ &c * * * * -f- -pT x 3 &c. and _v Now as this Series is accommodated to the cafe of convergency when x is a large Quantity, fo we may derive another Series from v

;'=***

_>'

= =

+

*

,

,.

-

>

hence, which will be accommodated to the cafe when .v is a fmall For the Ruler will direct us to the external Terms Ax* quantity. and and the refulting Numi ; x5 o, whence 3, or will 6 ; and the lair, is next Term The ber is 3. give 2m,

=

A=

m=

3*77,

9 will form an afcending Progremon, of which the common difference is 3. Therefore v =Ax'' -+- Ex 6 -t-Cy 9 &c. will be the form of the Series in this cafe, which may be thus

But

or 9.

3, 6,

,

derived.

y

=

-}

x>

x6 h x

#

-+-

s

11

2X> +-

=Then = = &c.

4*" 2x"

* 3AT

-f-

8

14*'

,

8

7A-

,

&c. &c.

6

&c. and therefore and &c. v , 2x, y^5=^9, &c. and therefore y Then y* * H- .V, &c. * * -j- 3^'*, 6cc. and 11 3 &c. and therefore7= * * * -f. o, &c. * S-^ The Expedient of the Ruler will indicate a third cafe of external m K Terms, which may be try'd alfo. For we may put A=x= -{- A*x*' m -f- Ax o, whence m o, and the Number refulting from the other Term is 3. Therefore 3 will be the common difference of the Progrelfion, and the form of the Root will be _y= A -{- Bx' -{A Cx 6 &c. But the Equation A 5 -f- A a o, will give A o, this to the former Series. And the other two which will reduce Roots of the Equation will be impofftble. If the Equation of this Example jy 3 -f- y* -{- r x"' o be the factor we mall have the Equation y* i, multiply'd by y y ... * ) y o, or r+ * # X~'y -f- x'

Here

= y =

*

becaufe_)'

x

6

Ar 3

,

it

will

bej>*=x

*

=

&c.

,

)

=

=

=

+

,

=

-

-

--

t

A'_)

AT

C=o,

=

which when

re-

J

only afford the fame Series for the Root y as before. l 1 This zy -+- i 49. Equation \* 2f' o, when x\y -h xy reduced to the form of a double Arithmetical Scale, will ftand as in

'folved, will

the Margin.

+

C

g

2

=

Now


The Method of

228

FLUXIONS,

Now

the finl Cafe of external Terms, fhewn by the Ruler, in order for an afcending Series, will i" 2 A*" make A'.**" _|- 2

= Number= o, or

i

-+-

;;/

zm -h

M^

where the

;

*

L

_

Axm>

fc

zAljc

#

tm jri

The 11-1

\4-s ^

alfo o.

is

refulting; i

o

i

*-

Or making y m

2>-f

+2."-

_

A^

1

*

_>

A 1*

the third the Arithmetical Progreffion will be o, Therefore or 2. zm-}- 2, i ; and is difference common confequently it will be i, 2, whofe 1 the But +- zA* Equation v == A -f- Bx -+ Cx -+- Dx*, &c. of the firft Coeffithe Value mould "_ zA -H i which o, give

fecond

is

i,

or

i ;

A

none but impoffible Roots ; fo that y, the Root of this Equation, cannot be exprefs'd by an Arithmetical Scale whofe Root is x, or by an afcending Series that converges by the Powers of x, when x is a fmall quantity.

cient,

As Ruler

for defcending Series, there are 1 l im AT will give us A**** i

fff

us with

will fupply

and

}

A=+

two

cafes to be try'd

= The Number +

A

i.

o, whence ^m

;

firft

the or

-f- 2,

4; the next will

is

arifing

= zm

be zm -f- i, or 3 ; the next 2w, or 2 ; the next m, or i ; the laft o. But the Arithmetical Progreffion 4, 3, 2, i, o, has^ i for its common Series will be y Ax -+ difference, and therefore the form of the ufual our Method, it B 4- CAT"', 8cc. But to extract this Series by this x* to the to reduce 4- x -+- z form, beft be will Equation thus to then __ 2 l and o, proceed

=

y~ 4- y~*

/

=

:

-_ x i

x

2 -f- ZX~*

| A:

97' h

Becaufe

_ x y _j_

JL

fo that

&c.

= x %x~ y= Then

jy

x*

l

,

&c.

2,

'tis

~1 >

-VAT-% &c.

-

A;~ I ,

&C.

77

c

therefore

==

(by Extradlion)

Then (by Divifion) zy~* and &c. 2*(by Extradion) y -f-

&c.

* * *

&c

1

= ,

l

zy~ which being united with a contrary '~ I & c> ant^ therefore by Extraction TA >

* * *

&c and y~* == *"*, 1 * * * fign, make^ ****- 4-i-s-v~3 y

* -f- -i-^"

&c.'

zx-*,

1

?

-

=*

=

.

,

A

In the other cafe of a defcending Series whence zm +- z A 1 *""^ -f- i hence arifing is o Number The i

=o,

.

we

= ;

mall have the Equation o,

or

m

=

the next will be

i,

zm

and

+

r, Cjf


and INFINITE SERIES.

229

But and the laft 4w, or 4. the Numbers o, I, 2, 4, will be found in a defcending Arithmetical Progrelfion, the common difference of which is i. Therefore the form of the Root is y A.x~' -+- Bx~'- -+- Cx~*, &c. and the Terms of the Equation mufl be thus difpofed for Refolution.

or

i

the next

;

2//v,

or

2

-,

=

- 2X -

---

= = =

1 I-f- A*""

x 1 &c. it will be by Extraction of the y~&c. and by finding the Reciprocal, y x~', Square-root y~ a contrary Sign, &c. Then becaufe &c. this with 2X, zy~ and collected with x above, will make_y 1 * -{- x, &c. which &c. makes and * -+ i, y~ (by Extraction) by taking the Recii becaufe Then * i, &c. ^^~ 6cc. procal, /== * zy~* this with a 2 above, will make contrary fign, and collected with and therefore * * 8ec. * * i, y~* (by Extraction) y~ 3 &c. Then becaufe and &cc. * * -f- ^x~ y 4*"" , (by Divifion) jy l # * * * * -}- -^"S ^ w iH be y %x~*y &c. and 2y~' " * * * -V*~J > & c Then 4* , &c and >'=*** j1 &c. and /* becaufe * * * -f- -fA;2y~' x~~-, &c. thefe

Here becaufe l

it

is

=x,

=

,

l

=

I

=

,

1

= = =

collected with &cc.

and y~'

l

=

1

= = a

1

-

=

= make y~ = & -V*~S 7 =

-

,

z

contrary fign will * * * *

=

c

-

an ^

V-v ~%

* * * *

** **

-f-

rlT*"4

*

&c.

Thefe are the two defcending Series, which may be derived for the Root of this Equation, and which will converge by the Powers of x, when it is a large quantity. But if x mould happen to be fmall, then in order to obtain a converging Series, As if it were known that the Root of the Scale.

we may

we much change x

differs

but

little

conveniently put z for that fmall difference, or 2 &. That is, irulead of x we rmy aflame the Equation .v and mall we have a new Equain this Equation fubftitute 2 2,

from

2,

+

tion y*

--

zy*

^zy*

2y

+

I

= =

o,

which

will appear

as

in

the Margin.

Here


The Method of FLUXIONS,

230

an afcending SezAz'" muft we put A+z*? ries, whence m o, and * _ T-l__ -KT The Number hence i. A arifing is o ; the next is 2/H-i, or i ; and the laft 2m -f- 2, or 2. But o, 1,2, are in an afcending

Here

to have

=

1=0, +=

Progreffion, the Series is

1

y

*

~

2;

4-

'

>

= l>

?*' .

Or

k

1

= Acommon

whole

-f-

*

.'*

B;s

A

4

difference

-f-

Cs a

,

A4.,

-+-

is

i.

Therefore the form of

D;s 3 , 6cc.

And

if the

Root y

be extracted by any of the foregoing Methods, it will be found y =. 1 Alfo we may hence find two defcending Se6cc. i -+--iz -^s ries, which would converge by the Root of the Scale z, if it were ,

a

large quantity.

field for the Solution 50, 51. Our Author has here opened a large indeterminate of thefe Equations, by Shewing, that the quantity, or what we call the Root of the Scale, or the converging quantity,

be changed a great variety of ways, and thence new Series will be derived for the Root of die Equation, which in different circum-

may

/tances will converge differently, fo that the the preSent occafion may always be chofe.

moft commodious for And when one Series

does not fufEciently converge, we may be able to change it for another that (hall converge falter. But that we may not be left to uncertain interpretations of the indeterminate quantity, or be obliged to make Suppositions at random j he gives us this Rule for finding initial Approximations, that may come at once pretty near the Root

Which required, and therefore the Series will converge apace to it. to what amounts this: are find when Rule to fubquantities,

We

ftituted for the indefinite Species in the propofed Equation, will it divifibk by the radical Species, increaSed or diminished by

make

The fmall diffeanother quantity, or by the radical Species alone. rence that will be found between any one of thofe quantities, and the indeterminate quantity of the Equation, may be introduced inftead of that indeterminate quantity, as a convenient Root of the by which the Series is to converge. the Equation propofed be y= -f- axy -f- cSy x* 2# o, and if for x we here Substitute #, we Shall have the Terms i 3 -f- 2a y a, the Quotient be3^*, which are divisible by y _y Therefore we ing y* -h ay -f- 3*2*. may fuppofe, by the foregoing x & is but a fmall quantity, or inftead of x we Rule, that a a in the Substitute z may propoied Equation, which will then o. become y* -f- 2a*y 2# 5 A azy -\- y-z 3"* -t- z= Scale,

Thus f

=

;

=

=

Series


and INF NITE SERIES, j

231 Powers of z9

from hence, compofcd of the afcending mull converge faft, crtfcris parifats, becaule the Root of the Scale z is a (mail quantity. Or in the fame Equation, if for x we fubftitute a, we fliall a have the Terms \* which are divifible by y a, the QuoSeries derived

3

,

tient being y* -4- ay -f- a*.

a and

rence between

Therefore

z

a

we may

to be but little,

and therefore

fmall quantity,

a

.v

in (lead

fuppofe the diffe-

or that

of

.v

x

-a

we may

=

z

fubftitute

is its-

in the This will then become given Equation. z a* where the Root y will con-f- 303* o, azy -f- T,a the Powers of the fmall quantity z. verge by Or if for x we fubftitute za, we lhall have the Terms which are divilible by y-\- za, the Quotient being _)* a*? -4- 6^

equal

=

l

r3

3

_>'

3

,

x Wherefore we may fuppofe there is but a fmall difzay-i- 3rt between is a fmall za and x, or that x ference za quantity ; and therefore infread of x we may introduce its equal z into the Equation, which will then become jv* za a'-y } 6a> o. -4-f- iza*z -f- 6az* -f-s azy .

=z

=

Laftly, if for

x we

fubftitute

we

z~*a,

fliall

have the Terms

z^a'-y -4-tf*y, which are divifible by y, the Radical Species alone. Wherefore we may fuppofe there is but a fmall difference between z^a x z^'a and x, or that z is a fmall quantity ; and 3

jy

=

-

therefore inflead of

x we may

fubfthute

its

equal

2?a

z,

which

Equation to y* -f- i azy -+- 3^4 x a^z ^z x a"y o, wherein the Series for the Root y may -\- 3^2 x az -f- Z' converge by the Powers of the fmall quantity z. But the reafon of this Operation ftill remains to be inquired into, which I mall endeavour to explain from the prefent Example. In x3 za* the indeterminate the Equation y~> -\- axy -f- a*y muft be fufceptible of all poffible quantity x, of its own nature, Values at leaft, if it had any limitations, they would be fhew'd by will reduce the

=

1

=o,

;

impoflible Roots. -rza,

+- za*y

z~*a, 6cc.

30*

=

Among

which

in o,

other values,

7

;

a1

will receive thefe, a,

the Equation would a 1y -4- 6a* o, y*

cafes

=

it

a,

become

=

o,

y*

3

_y

Now as thefe Equations admit 2^a*y -f- a'-y =. o, &cc. refpedtively. anof jull Roots, as appears by their being divifible by y -f- or other quantity, and the laft by y alone; fo that in the Refolution, the whole Equation (in thofe cafes), would be immediately exhaufted And in other cafes, when x does not much recede from one of thofe :

Values,


The Method of

232

FLUXIONS,

Therefore the Values, the Equation would be nearly exhaufted. is of the between x and any which fmall difference z, introducing and itfelf mull muft z the of thofe Values, pne deprefs Equation ; be a convenient quantity to be made the Root of the Scale, or the converging Quantity. I (hall give the Solution

which mall be

ples,

Here becaufe

5 _>'

this,

=

#;

,

of one of the Equations of thefe Exam3 a* o, or _y azy -f- y-z -4- ^az*

=

&c.

make Then

united with 3^*2 above, c ==. * -f-*' y

&

therefore

to

-

V

"

ssz # * 217Z 5

-f- .Is

= ==

&c.

a,

Then

azy and fign, 2a*z, &c. and 1 6cc. and -f- ^az

&c. and

3 ,

_)'*

=

y*

s>'

* *

= ***

*

J-.ss,

&c.

,

3a

*

,

Then

&c. and y

=

* * *

<>

OCC.

,

The Author Series

hints at

many

from the fame Equation

;

other ways of deriving a variety of as when we fuppofe the afore-men-

z

to be indefinitely great,

and from

find Series, in

which the Powers of z

(hall

tion'd difference

we

y

which muft be wrote again with a contrary

fl2r, 6cc.

*

will be

it

that Suppofition afcend in the Deno-

This Cafe we have all along purfued indifcriminately with minators. other the Cafe, in which the Powers of the converging quantity afcend in the Numerators, and therefore we need add here

nothing converging quantity fome other quantity of the Equation, which then may be confider'd

about

it.

Another Expedient

is,

to affume for the

So here, for inftance, we may change a into x, Or laftly, to affume any Relation at pleafure, (fup-

as indeterminate.

and x

into a.

=

x

= ~-

J^

az -f- bz\ x 3 , 5 pofe x &c.) between the indeterminate quantity of the Equation x, and the quantity z we would introduce into its room, by which new equivalent Equations may be form'd, and then their Roots may be extracted. And afterwards the value of z may be exprefs'd x} means of the afby

by

fumed Equation. 52.

The


and INFINITE SERIES.

233

The Author here, in a fummary way, gives us a Rationale of whole Method of Extractions, proving a priori, that the Series thus form'd, and continued in infinitum, will then be the juft Roots 52,

his

And if they are only continued to a of the propofed Equation. of number Terms, (the more the better,) yet then will competent near a be very Approximation to the juft and compleat Roots. they an when For, Equation is propofed to be refolved, as near an Approach is made to the Root, iuppofe y, as can be had in a lingle Term, compofed of the quantities given by the Equation and be*. ;

Relidual or Secondary Equation is Remainder, is whole the Root thence form'd, p Supplement to the Root of the Then as near an approach given Equation, whatever that may be. as can done a be is made to by lingle Term, and a new Relidual /, is form'd from the Remainder, wherein the Root q is the Equation the Relidual Equations Supplement to p. And by proceeding thus, are continually deprefs'd, and the Supplements grow perpetually lels and lefs, till the Terms at laft are lefs than any affignable quantities. may illuftrate this by a familiar Example, taken from the ufual caufe there

is

a

a

We

At every Operation we Divifion of Decimal Fractions. in the Quotient, as the Dividend and Divifor put as large a Figure Then this will permit, fo as to leave the leaft Remainder poflible. of we the a new which are to Remainder (applies Dividend, place as be done one we can exhauft as far by Figure, and therefore put the greateft number we can for the next Figure of the Quotient,

Method of

and thereby

Remainder we whole Dividend is exhaufted,

leave the leaft

can.

And

fo

we go

on,

can bz done, or a fufficient till we have obtain'd Approximation in decimal places or And the fame of way Argumentation, that proves our Aufigures. thor's Method of Extraction, may ealily be apply'd to the other ways of Analylis that are here found. either

till

the

if that

Here it is feafonably obferved, that tho' the indefinite fhould not be taken fo fmall, as to make the Series conQuantity faft, yet it would however converge to the true Root, verge very more And this would obtain in tho' by fteps and flower degrees. even if it were taken never fo large, provided we do proportion, not exceed the due Limits of the Roots, which may be difcover'd, 53, 54.

from the Root when exhibited deduced and illuftrated by fome Series, may by Geometrical Figure, to which the Equation is accommodated. So if the given Equation were yy =. ax xx, it is eafy to obbe that nor x can but infinite, ferve, neither^ they are both liable to h flv.rul

either a

from the given Equation, or

or

be farther

H


The Method of

FLUXIONS,

For if x be fuppos'd infinite, the Term ax Limitations. xx, which would give the Value ofjyy would vaniih in refpedt of Nor can x be negative; for then the on this Supposition. impoffible be would of Value negative, and therefore the Value of_y would yy

ieveral

x

=

again become impoffible. one Limitation of both quantities. If

ax and xx, when

that difference

is

=

then is^ o allb ; which is As yy is the difference between greateft, then will yy, and con-

o,

=

But this happens when x a, as fequently^, be greateft alfo. from the following Prob. 3. And in alfo y ftf, as may appear number of Terms, whether is exprefs'd by any general, when y to its come Limit when the difference then it will or finite infinite, and the affirmative is negative Terms j as may apgreateft between from the fame Problem. This laft will be a Limitation for y t

=

pear

but not for x. both x and y. gative

Term

Laftly,

For

if

=

when x a, we fuppofe x

then_y

=

o; which will limit

to be greater than a, the ne-

will prevail over the affirmative, will make the Value of

which

and give the Value of y impoffible. So that

yy negative, upon the whole, the Limitations of x in this Equation will be thefe, that it cannot be lefs than o, nor greater than a, but may be of any intermediate magnitude between thofe Limits. Now if we refolve this Equation, and find the Value of y in an infinite Series, we may ftill difcover the fame Limitations from

For from the Equation yy

thence.

=

ax

xx, by extracting the 3.

as

fquare-root, X

before,

1

-

c.

,

that

is,

i6a z

x cannot be

Nor

we

y

negative

== ;

'

fhall

have y

i

d*x* into

for then

can x be greater than a

;

i

X -

=. a^

_5.

L za*

**

*

3

O

,

~

&c.

Sa 1

TT

Here

x? would be an impoffible quantity. for then the converging quantity ~

by which the Series is exprefs'd, would be and confequently the Series would diverge,, and greater than Unity, The Limit between converging and not converge as it ought to do. and therefore y o diverging will be found, by putting x=a, i Series Numeral i identical the have fhall cafe we in which of fome which we _' nature with thofe, r &c. of the fame _l_ ^ -if. So that we may take x of any have elfewhere taken notice of. intermediate Value between o and a, in order to have a converging But the nearer it is taken to the Limit o, fo much fafter Series. nearer it is taken the Series will converge to the true Root and the But it will to the Limit a, it will converge fo much the flower. however or the

Root of the

Scale

==

;

,

;

4


and INFINITE SERIES.

235 And by

'however converge, if A: be taken never fo little lefs than a. Analogy, a like Judgment is to be made in all other cafes. The Limits and other affe&ions of y are likewife difcoverable from When x o. When x is a nafcent quanthis Series. o, then y to be or all but the Terms but the rirft juft beginning pofitive, tity, will and be a mean proportional between a and x. y may be negledted, Alfo y o, when the affirmative Term is equal to all the negative

=

=

=

i=

Terms.or when

=

-

-f-

z

-

-?

u-

, '

ib3

8a*

&c. that

when x

is,

=

a.

_ rj &c. as above. Laftly, y will be a Maximum when the difference between the affirmative Term and all the negative Terms is greateft, which by Prob. 3. will be found

For then

-f.

^a. the Figure or Curve that

Now

be adapted to this Equation, and which will have the fame Limitations that

to this Series,

=

is

they have,

AB

/cifs

_f_

4.

=

when x and

i

f

x,

Ordinate

BC=^

between

the Segments

AB

rrn x ==. ax yy nate BC

=

going

And

xx. _y

mean

a

=

BD

and

=

Diameter is BC =.}' Ordinate perpendicular

its is

AD

ACD, whofe

the Circle

and

may

a,

its

For

Ab-

as the

proportional

of the

a

Diameter

will be x, therefore the Ordiit

will be exprefs'd

Series.

But

it

is

by the forefrom the naplain

AB

ture of the Circle, that the Abfcifs cannot be extended backfo as to become neither can it be continued forwards, negative ; wards beyond the end of the Diameter D. And that at and D,

A

where the Diameter begins and ends, the Ordinate the greateft Ordinate

SECT. VI. 55.

'

|

"^HE

is

at the Center,

'Trqnfitton fo the

learned and

plifh'd

fagacious

one part of

or

when

is

Method of

=

nothing.

AB

And

^

Fluxions.

Author having thus accomwhich was, to teach the

his deiign,

Method of

converting all kinds of Algebraic Quantities into fimplc Terms, by reducing them to infinite Series He now goes on to fhew the ufe and application of this Reduction, or of thefe Series, in the Method of Fluxions, which is indeed the principal defign of :

this Treadle.

For

dependence upon defective without

this

Method

has fo near a connexion with, and that it would be very lame and

the foregoing, it.

He

lays

down

H

h 2

the fundamental Principles of this


The Method of

FLUXIONS,

very general and fcientiflck manner, deducing them from the received and known laws of local Motion. Nor is this inverting the natural order of Science, as Ibme have pretended, by introducing the Doctrine of Motion into pure Geometrical Spethis

Method

in

a

For Geometrical and. Analytical Quantities are belt conceived as generated by local Motion; and their properties may as well be derived from them while they are generating, as when their culations.

fuppos'd to be already accomplifh'd, in any other way. right line, or a curve line, is defcribed by the motion of a point, a fmface by the motion of a line, a folid by the motion of a furface, an angle by the rotation of a radius ; all which motions we

generation

is

A

conceive to be performed according to any ftated law, as occafion (hall require. Thefe generations of quantities we daily fee to obtain in rerum naturd, and is the manner the ancient Geometricians had often recourfe to, in confidering their production, and then de-

may

ducing their properties from fuch adhial defcriptions. And by analogy, all other quantities, as well as thefe continued geometrical be conceived as generated by a kind of motion or quantities, may of the Mind. progrefs The Method of Fluxions then fuppofes quantities to be generated by local Motion, or fomething analogous thereto, tho' fuch generations indeed may not be eflentially neceflary to the nature of the They might have an exiftence independent of thing fo generated. thefe motions, and may be conceived as produced many other ways, and yet will be endued with the fame properties. But this conception, of their being now generated by local Motion, is a very fertile notion, and an exceeding ufeful artifice for discovering their proand a great help to the Mind for a clear, diftincl:, and meperties, For local Motion fuppofes a notion thodical perception of them. of Ideas. a fucceffion time of time, and eafily diflinimplies what is, and what will be, in thefe geguifh it into what was, nerations of quantities ; and fo we commodioufly confider thofe be too much for our faculties, and exthings by parts, which would Mind take in the whole together, without for the to difficult tream fuch artificial partitions and distributions. Our Author therefore makes this eafy fuppofition, that a Line may be conceived as now defcribing by a Point, which moves either uniform motion, or elfe accorequably or inequably, either with an Velocity ding to any rate of continual Acceleration or Retardation. and like all fuch, it is fufceptible of is a Mathematical Quantity, infinite gradations, may be intended or remitted, may be increafed

We

or


and INFIN ITE SERIES.

237

or dlminifhfd in different parts of the fpace delcribed, according to it is infinite variety of fluted Laws. plain, that the fpace thus defcribed, and the law of acceleration or retardation, (that is,

Now

an

the velocity at every point of time,) mufl

have a mutual relation to each other, and muft mutually determine each other ; fo that one of them being affign'd, the other by neceflary inference may be derived from it. And therefore this is ftrictly a Geometrical Proand And all Geometrical blem, capable of a full Determination. Propoluions once demonftrated, or duly investigated, may be fafely made ufe of, to derive other Proportions from them. This will 1

the prefent Problem into two Cafes, according as either the or Velocity is affign'd, at any given time, in order to find the Space Arid this has given occasion to that diftin<5lion which has other. divide

each of irrcerje Method of Fluxions, confider apart. 56. In the direct Method the Problem is thus abftractedly proFrom the Space deferibed, being continually given, or affumed, pofed. or being known at any point of Time ajjigrid ; to find the Velocity of the in equable Motions it is well known, Motion at that Time. lince obtain'd,

which we

fhall

of the dirctt and

now

Now

always as the Velocity and the Time of is directly as the Spice dedefcription conjunclly ; or the Velocity And even in fcribed, and reciprocally as the Time of defcription. or as are accelerated or retarded, fuch continually inequable Motions, that the Space defcribed

is

according to fome ftated Law, if we take the Spaces and Times very fmall, they will make a near approach to the nature of equable MoBut if we tions ; and flill the nearer, the fmaller thole are taken. be and to the Times indefinitely fmall, or if Spaces may fuppofe then we fhall have the Veor evanefcent nafcent quantities, they are locity in

any

infinitely little Space,

as that

Space directly, and as the all inequable Mo-

This property therefore of

tempufculum inverlely. tions being thus deduced, will afford us a medium for folving the So that the Space as will be fhewn afterwards. prefent Problem, and the whole time of its defcribed being thus continually given, at the end of that time will be thence dedefcription, the Velocity terminable. abflract Mechanical Problem, which amounts to 57. The general the lame as what is call'd the inverfe Method of Fluxions, will be From the Velocity of the Motion being continually given, to dethis.

For the termine the Space defcribed, at any point of Time affign'd. afTiflance have the of we fhall this of which Mechanical Solution Theorem, that in inequable Motions, or when a Point defcribes a Line


2

Method of FLUXIONS,

*!}

<>8

any rate of acceleration or retardation, the indefidefcribed in Spare any indefinitely little Time, will be in nitely ratio of the Time and the Velocity ; or thejpafiolum will a compound be as the velocity and the tempiijculum conjunctly. This being the Line according

to

little

Law

of

all

equable Motions, it

quantities,

when

will obtain allb

in

the Space all

and Time

are

any

finite

Motions, when

inequable

the

For by this means all Space and Time are diminiih'd in infinitum. as it were, to equability. are Motions Hence the reduced, inequable Time and the Velocity being continually known, the Space delcribed may be known alfo as will more fully appear from what follows. ThisTroblem, in all its cafes, will be capable of a juft determina;

tion

;

tho' taking

be a very

difficult

good reafon

for

it

in

its

full extent,

we mult acknowledge

and operofe Problem.

calling

it

moleftijfimum

&

it

to

So that our Author had

omnium

difficilltmum pro-

blema. 58.

To

fix

the Ideas of his Reader, our

general Problems by

a particular

Example.

Author illuftrates his two Spaces x and y

If

fuch manner, that the Space x being uniformly increafed, in the nature of Time, and its equable velocity being reprefented by the Symbol x ; and if the Space y increafes in=. xx ihall equably, but after fuch a rate, as that the Equation y always determine the relation between thofe Spaces j (or x being continually given, y will be thence known ;) then the velocity of That is, if the the increafe of y fhall always be reprefented by 2xx. the of increafe of y, then the be to velocity reprefent put fymbol y will be (hewn hereafter. will the Equation y =. zxx always obtain, as Now from the given Equation y xx, or from the relation of the are defcribed

by two points

in

=

or its representative,) (that is, the Space and Time, Spaces y of the Velocities relation y=.2xx is being continually given, the found, or the relation of the Velocity y, by which the Space increafes, to the Velocity x, by which the reprefentative of the Time increales. And this is an inftance of the Solution of the firft general Problem,

and x,

or of a particular Queftion in the direct Method of Fluxions. But 2xx were given, or if the Ve-.vice versa, if the kit Equation y

= =

which the Space y is defcribed, were continually known locity y, by from the Time x being given, and its Velocity x , and if from thence. we ihould derive the Equation y xx, or the relation of the Space and Time This would be an inftance of the Solution of the fecond :

Problem, or of a particular Queftion of the inverfe Method And in analogy to this defcription of Spaces by movof Fluxions. ing points, our Author confiders all other quantities whatever as ge-

.general

nerated


and INFINITE SERIES.

239

nerated and produced by continual augmentation, or by the perpetual acceffion and accretion of new particles of the fame kind. the Laws of his Calculus of Fluxions, our Author 59. In fettling

very fkilfully and judicioufly difengages himfelf from all confideration of Time, as being a thing of too Phyfkal or Metaphyfical a nature to be admitted here, efpecially when there was no abfolute

For

Motions, and Velocities of Motion, compared or meafured, may feem neceflarily to include a notion of Time; yet Time, like all other quantities, may be reprefented by Lines and Symbols, as in the foregoing ex-

neceffity

when

for

it.

they come

tho'

all

to be

efpecially when we conceive them to increafe uniformly. thefe reprefentatives or proxies of Time, which in fomc meafiire may be made the objects of Senfe, will anfwer the prefent purSo that Time, in fome fenle, may pofe as well as the thing itfclf. be laid to be eliminated and excluded out of the inquiry. By this

ample,

And

is no longer Phyfical, but becomes much more and as Geometrical, being wholly confined to the defcription fimple and with their Lines of Spaces, comparative Velocities of increafe and decreafe. Now from the equable Flux of Time, which we conceive to be generated by the continual acceflion of new particles, or Moments, our Author has thought fit to call his Calculus the Method of Fluxions. 60, 6 1. Here the Author premifes fome Definitions, and other Thus Quantities, which in neceflary preliminaries to his Method. or Equation are fuppos'd to be fufceptible of continual Problem any increafe or decreafe, he calls Fluents, or flowing Quantities ; which are fometimes call'd variable or indeterminate quantities, becaufe they are capable of receiving an infinite number of particular values, in The Velocities of the increafe or dea regular order of fucceilion. fuch are call'd of creafe their Fluxions ; and quantities in quantities the fame Problem, not liable to increafe or decreafe, or whofe Fluxions are nothing, are call'd conftant, given, invariable, and determinate This diftindlion of quantities, when once made, is carequantities. obferved through the whole Problem, and infinuated by proper fully Symbols. For the firft Letters of the Alphabet are generally approfor denoting conftant quantities, and the laffc Letters compriated monly lignify variable quantities, and the fame Letters, being pointed, repreient the Fluxions of thofe variable quantities or Fluents refpecThis diftinction between thefe quantities is not altogether tivcly.

means the Problem

has fome foundation in the nature of the thing, at the Solution of the prefent Problem. For the flowing during or

arbitrary, but leafl


7#* Method

24-O

of

FLUXIONS.

or variable quantities may be conceived as now generating by Motion, and the conftant or invariable quantities as fome how o other alThus in any given Circle or Parabola, the Diame.ready generated. ter or Parameter are conftant lines, or already generated ; but the

Ordinate, Area, Curve-line, &c. are flowing and variable quantities, becaufe they are to be underftood as now defcribing by

Abfcifs, local

Motion, while

their

properties are

derived.

Another

diftinc-

may be this. A conftant or given Irne in any Problem is tinea qtitzdam^ but an indeterminate line is line a qua-vis Or laftiy, vel qutzcunque, becaufe it may admit of infinite values.

tion of thefe quantities

conftant quantities in a Problem are thofe, whole ratio to a common Unit, of their own kind, is fuppos'd to be known ; but in variable quantities that ratio cannot be known, becaufe it is varying perpe-

This diftinction of quantities however, into determinate and indeterminate, fubfifts no longer than the prefent Calculation requires;

tually.

a diftinftion form'd by the Imagination only, for its own conveniency, it has a power of abolifhing it, and of converting determinate quantities into indeterminate, and vice versa, as occaiion for as

it is

In require ; of which we fhall fee Inftances in what follows. a Problem, or Equation, theie may be any number of conftant quantities, but there muft be at leaft two that are flowing and indeterminate ; for one cannot increafe or diminifh, while all the reft con-

may

If there are more than two variable quantities in tinue the fame. a Problem, their relation ought to be exhibited by more than one

Equation.

ANNO-


(

241

)

ANNOTATIONS on Prob.i, O

R,

The relation of the flowing Quantities being given, to determine the relation of their Fluxions.

SECT.

I.

Concerning Fluxions of the

firft

orcler^

and

t(f

Jlnd their Equations.

HE

Author having thus propofed his fundamental Pro-' abftra<ft and general manner, and gradually them down to the form mod convenient for* brought blems in an s

he now proceeds to deliver the Precepts his Method of Solution, which he illuftrates by a fufficient variety of Examples,! referving the Demonftration to be given afterwards, when his Readers will be better prepared to apprehend the force of it, and when their notions will be better fettled and confirm'd. Theie Precepts of Solution, or the Rules for finding the Fluxions of any given' and appealEquation, are very fliort, elegant, and compreheniive to have but little affinity with the Rules ufually given for this purto their is But that of owing great pofe degree univerfality. are to form, as it were, fo many different Tables for the Equation, ;

;

We

:

flowing quantities in it, by difpofing the Terms accorthe Powers of each quantity, fo as that their Indices ding may' form an Arithmetical Progreflion. Then the Terms are to be multiply'd in each cafe, either by the Progreflion of the Indices, or as there are

to

'

the

by mould

Terms of any

.have the

fame

other Arithmetical Progreflion, (which yet common difference with the Progreffion of the Indices I

i

'

;)

as


Tfo Method

242

of

FLUXIONS.

by the Fluxion of that Fluent, and then to be divided by La ft of all, thefe Terms are to be collected, accorthe Fluent itfelf. ding to their proper Signs, and to be made equal to nothing; which will be a new Equation, exhibiting the relation of the Fluxions. This procefs indeed is not fo fhort as the Method for taking Fluxions, be given p relent ly v) which he el fe where delivers, and which is (to commonly follow' d ; but it makes fufficient amends by the univerlality of it, and by the great variety of Solutions which it will afford. For we may derive as many different Fluxional Equations from the lame given Equation, as we .(hall think fit to affume different Arithas

alfo

metical Progreffions.

.Yet

thefe Equations will agree in the main, the relation of

all

and

tho' differing in form, yet each will truly give the Fluxions, as will appear from the following

In the

2.

tion

firft

ax 1

x>

Example we

Examples.

are to take the Fluxions of the

=

Equao, where the Terms are always axy y"> one fide. Thefe Terms being difpofed according

-{-

brought over to to the powers of the Fluent x, or being conlider'd as a Number exax 1 -fprefs'd by the Scale whofe Root is x, will iland thus x> o; and affuming the Arithmetical Progrefiion 3, 2, y>x ayx* which is here that of the Indices of x, and ], o, multiplying each Term by each refpedlively, we fhall have the Terms jx 3 zax-

=

-

which again multiply'd by i or xx~ according to 1 2axx -f- ayx. Then in the fame Equathe Rule, will make ^xx tion making the other Fluent/ the Root of the Scale, it will ftand 5 o ; and affuming the Arith-f- oy*-i- axy thus, _y ax*y H- ayx

*

l

j

,

=

1

-

>

metical Progreffion 3, 2, I, o, which alfo Indices of y, and multiplying as before, ;

* -+-

3_)'

a

axy

axJ-

which multiply'd by

*,

Tlien

,

the Progreffion of the we fhall have the Terms is

--

,

or yy~*,

will

make

g the Terms, the Equation yxx o will ayx axj give the required relation of the tyy* if refolve For we this Fluxions. Equation into an Analogy, we fhall 2 ax i^x 1 have x y zax -h ay which, in all the values that 3>' x and y can affume, will give the ratio of their Fluxions, or the comparative velocity of their increafe or decreafe, when they flow according to the given Equation. Or to find this ratio of the Fluxions more immediately, or the 3i>'

zaxx

+ :

-+

-f-

colle(^i n

1

=

: :

-,

value of the Fraction

down

the Fraction

?

4'

by fewer

fteps,

we may

proceed thus.

with the note of equality

after

it,

and

Write in the

Numerator


and INFINITE SERIES.

243

Numerator of the equivalent Fraction write the Terms of the Equaeach betion, difpos'd according to x, with their refpective figns the Index of x in that Term, (increafed or diing multiply'd by ;

minifh'd, if you pleafe, by any common Number,) as alib divided by .v. In the Denominator do the fame by the Terms, when diiThus in the prepofed according to y, only changing the figns. fent Equation x"' ax 1 -f- axy we (lull have at once o, ;'

=

3

y ~ *

i,x*2ax-\-av ax

J>* *

+

*

Let us now apply the Solution another way. The Equation x ax* -f- axy o being order'd according to x as before, y* will be x ax* -(- ayx y*x =. o and fuppofing the Indices of x to be increas'd by an unit, or aifuming the Arithmetical Pro;

=

1

1

;

-~~

greffion

we

-j-

,

t

^

~

and multiplying the Terms

,

,

refpectively,

1 have thefe Terms ^.xx* Then y-xx^axx -}- zayx 1 the Terms ordering -f- oy according to /, they will become -\-axy -i- x*y =.0; and fuppofing the Indices ofy to be diminifli'd ax*

fhall

.

3

_)'

f

by an unit, or afluming the Arithmetical Progreffion ^ and multiplying the Terms refpecYively, we mall have So that collecting 2yy* * * x*yyax*yy~*.

,

.>

+

1

we

lhall

ax*yy~*

have

=

+- 2 ayx

-^axx for the Fluxional

4.v.v*

o,

c ^1 T>1 -11 y of the Fluxions will be i

x

may Or by

T

* *

3_J

Z)

-f-As,

J

x'>yy-' -+the ratio

zyy*

a -f-f- 2ay

-.

, 2

the Terms,

l

Or

Equation required. *

4X

Si iJ, ' ' y y y thefe Terms

_ *

v'v

:

ax l

.

.

l

.

w hich ,

ratio

\<

be found immediately by applying the foregoing Rule. contrary-wife, if we multiply the Equation in the fir ft form

~

the Progreffion

axx

*

-\-ytx\-

cond form by

H-

=

y>xx~

L

-

1

~ ^

v

? ,

}

,

And if we ^ l y- we .

,

,

5

,

zcixy -+-

x=j}~

flinll

have the Terms zxx 1

multiply the Equation in the

y

fiiall

have the Terms

Therefore collecting

!

+ rxx~>

we

,

cx-yy~'.

of the Fluxions

Or

>

will

be

|

4^*

'tis a.v.v

^v}*+ 2axy-i-x>j}- ~fix 1y}-'~o.

= ^ ^ ~^:^^.,-r

,

fc*,

1

^v.v the ratio

which might

been found at once by the foregoing Rule. And in general, if the Equation x"> - -ax % -- axy o, in y* a\- -f- <?.yv --> ,v the form xbe o, multiply'd by the Terms ";+ 3 "L+J. of this Arithmetical Progreffion v .v r ;n \> 11 JL O l.avc

;

=

)

produce the Terms

m -\-y.\-~ I

;

-v,

m-+-2n>:x-{i 2

m -\-

icxt

w

mj'xx-'-,

and


^

e

)e

244

Method of FLUXIONS,

the fame Equation, reduced to the form y*-\o, b; multiply 'd by the Terms of this Arithmetical Pro-

and

if

_f-

K\y= ax 1

"

Mjs

grerTion

we

m -\-

have

ilia 11

^

nax lyy~

H- iaxy-\-nx~>yy-*

H-

*

w *^ P ro( uce t ie Terms

^

7, ~7~7' "^'

vx 1 3.

Then

l .

arfxv

-i-

;

^

collecting the

Terms, i

H- m-\-.iaxy nax*yy*

my"'.\x~ w ~f- 3.X) * 4-^-t- irftfy -fo, for the nx*yy~~ Or the ratio of the. Fluxions will be Fluxional Equation required. 1

-

=m

-4-

*

l

- - --

3*" -

js -(-

z* -j- m -j-

^

-

*

I

*

m)$x

ay :

=

.

;

r

.

.

.

1

.

which might have been

:

n -j- D * -j- ax -f- nax^y nx'j found immediately from the given Equation, by the foregoing Rule. Here the general Numbers m and n may be determined pro lubitu, by which means we may obtain as many .Fluxional Equations as we ;?

which

pleafe,

commodated

-

x

all

And

belong to the given Equation.

;

-

before.

=.

fy-

i

and n

=

X

i,

Or

and

=.

-r- 7,'

3^4

^axi

a

n=

n have we ihall 11

i

,

,

ax

3j

-_ m=-

we make

if ax +_> A

zax

=

mall have 4

*'*-*"* + "> l

i,

before.

y

-

x

'-\~axij

we

n

l. V have -

11

ihall

thus

m

=

we make

as

found before. Or

=^ = 4*

Now

o, if

a

:

*-

we fhall we make m

if

i,

=

have ?,J J

= _

we

beft ac-

if

and n

Or

is

'

x*j

of Qthers y -(-

Thus

to the prefent exigence.

== o, we

;;

we make as

will

always find the fimpleft Expreffion, or that which

may and

I

l

th

j

s

var i ety of Solutions J

no ambiguity in the Conclusion, as poffibly might have been fufpected; for it is no other than what ought neceffarily to arife, from the different forms the given Equation may acquire, as will beget

If we confine ourfelves to the Progremon of will appear afterwards. the Indices, it will bring the Solution to the common Method of taking Fluxions, which our Author has taught elfewhere, and which, becaufe it is eafy and expeditious, and requires no certain order of the Terms, I mall here fubjoin. For every Term of the given Equation, fo many Terms mufr. be

form'd in the Fluxional Equation, as there are flowing Quantities in And this muft be done, (i.) by multiplying the Term that Term. by the Index of each flowing Quantity contain'd in it. (2.) By dividing

Fluxion.

=

o,

it

by the quantity

Thus

in

itfelf j and, (3.) by multiplying by the foregoing Equation x> ax* -f- ayx

the Fluxion belonging to

the

Term

.v 3

3

is

,

its

y

3

or ^x^x.

The


and INFINITE SERIES. The

ax 1

Fluxion belonging to

Fluxion belonging to ayx

-

is

or

,

avxv

.

245

ayxx

is

1-

,

The

zaxx.

And

or axy -f- ayx.

the

or Fluxion belonging to / 3 is y-y. So that the whole or Fluxion of the the whole Fluxional Equation, Equation, a 1 Ais zaxx -fThus the Equation -f3-v ayx ayx 3_>' _y=o. m m -* m x =}', will give mxx the and ; =.y Equation x z," y, will m t z" -f- nx mzz"~ give mxx y for its Fluxional Equation. And the like of other Examples. If we take the Author's funple Example, in pag. 19, or the Equation y or rather x* x ly ay xx, o, that is ayx o, in order to find its moft general Fluxional Equation ; it may be perform'd by the Rule before given, fuppofing the Index of x to be encreas'd by m, and the Index of y by ;;. For then we {hall have ,

=

l

= ?

diredtly x

=

=

"-'-g+-'*

nx zy

For the

_

n -|- \a

'

=

Term

firft

of the given

Equation being ayx, this multiply'd by the Index of x increas'd by l that is by ;;z, and divided by x, will give mayx~ for the firlt 7/7, Term of the Numerator. Alfo the fecond Term being x*y, this that is by w-f- 2, and of A- increas'd by the Index m, multiply'd by

m -h

for the fecond Term of the Nuof the given Equation may be now Index of y increas'd by n, that ,Y*J, which multiply'd by the l is by ;;, and divided by r, will give (changing the fign) nx y~ for Alib the fecond Term will the firft Term of the Denominator.

divided by ,v, will give merator. Again, the firft

2X

Term

l

then be

cyx, which multiply'd by

that

by n

the Index

of/

increas'd

by ;/, (changing the Sign) n -|- \a for die fecond Term of the Denominator, as found above. Now from this general relation of the Fluxions, we may deduce as many particular ones as we pleaie. Thus if we make ///= o, and is

and divided by

-f- i,

7/r=o, we

fhall

have

--

y,

= Or = -7^7 = =

--

r

will give

,

or

ay

Author's Solution in the place before cited. 1

aiid //v

;z=

=

o,

we make as before.

we

r,

and n

n

;/

=

1

lhall

= o.

II

have -

i,

we

and m-=.

2TA

(hall

2,

have

we

V

fhall

2xx, agreeable to our if

we

make;=

1

2tfl>

-77 -

X j

have v

.

Or

= =

^

we make

Or

.

'

All which, and innumerable other cafes,

proved by a fubftitution of equivalents.

-

2,

.,

if

if

A"

-^-^

. m

a

may

-1-,

v

,

be

Or we may prove

it

eafilv* c:

rally


e

tf>e Method of

246 thus.

rally

Becaufe by the given Equation

/-i value of the ratio /,

we

V ~-

have

The ^z*

V

4

= =

V

11

fhall

FLUXIONS,

=

~ mayx

--=

OT

x

m-\-zx c

gA& -.

+ 2X

*

na

n -+-

i

f

_7^ri7

a

=

r

it is

r

,

y=x a~ i

,,.

I

in

the

value,

and

,

.

7 mbmtute

its

,

,

2X

as above.

a

Equation of the fecond Example

3

is

2j -f- x*y 2cysi in Z' which are there three o, -4flowing quantities y, x, and z, and therefore there muft be three operations, or three Tables mufl be form'd. Firft difpofe the Terms according to y, thus ; z~>y= o, and multiply by the Terms of the Pro2j3 _j_ oja _{_ x*y 3.

- 2CZ 1

1

1

1

i xj//- , relpeclively, (where greffion 2 xjj"" , ixj/y" , oxj/y"" , the Coefficients are form'd by diminishing the Indices of y by the com-

and the refulting Terms will be qyy* * * -f- &yy*. Secondly difpofe theTerms according to x, thus-> yx*--}-ox-t-2y">x=o 3

mon Number

:,)

;

.

2cz and multiply by the Terms of the ProgreiTion 2xxx~\ i xxv~ r , oy.xx~ , (\vhere the Coefficients are the fame as the Indices of x,) and the only refulting Term here is -+- 2yxx * *. Laftly, difpofe l

the

Terms according

to z, thus

;

z=

2cyz--x*yz=o

-+-^y^

J

-4- 2}"

and multiply by the Progreffion

3xs~

I

2xzz~'

fx.zz~ ! oxzz*, the Indices of z,} and f

,

,

,

(where the Coefficients are alfo the fame as Then collecting the Terms will be ^zz* -h 6yzz-~-2cyz * 1 thefe Terms together, we fhall have the Fluxional Equation fyrj .

all

+

2cyz =. o. of our Author's dexterity, at Here we for abbreviating. For in every one of thefe Opefinding expedients rations fuch a Progreffion is chofe, as by multiplication will make the greateft deftrudtion of the Terms. By which means he arrives that the nature of the Problem will allow. at the fhorteft Expreffion, It we mould feck the Fluxions of this Equation by the ufaal mei

~3yy

_|_ av,v.v

yzz*

-+-

6yzz

have a notable inftance

thod, which

is

if we always a flu me the Prois, have 6yy* -+ 2xxy -\- xy 2cyz o ; which has two Terms 3'zz* And if the Progreffions of the Indices

taught above, that

oreffions of the Indices,

we

fhall

=

zcyz -+- ~}yz* ~r- dyzz more' than the other form. (-j increas'd, in each cafe, by any common general Numbers, we may form the moil: general Expreilion for the Fluxional Equation, that the Problem will admit of. t

3

4-

.


and INFINITE SERIES. On

247

occafion of the

laft Example, in which are three Fluents our Author makes an ufeful Obfervation, for and their Fluxions, the Reduction and compleat Determination of fuJi Equations, tho' the Rules of the vulgar Algebra it be derived from which matter thus. be confider'd of two flowing Every Equation, conlilling may is what correfponds to an indetcrmin'd Proor variable Quantities, Therefore one blem, admitting of an infinite number of Anfwcrs. of thofe quantities being afiumed at pleafure, or a particular value being affign'd to it, the other will alfb be compleatly determined. And in the Fluxional Equation derived from thence, thofe particular values being fubftituted, the Ratio of the Fluxions will be given in Numbers, in any particular cafe. And one of the Fluxions being taken for Unity, or of any determinate value, the value of the other may be exhibited by a Number, which will be a compleat Determi-

4.

;

nation.

But

if the

naged

in this

given Equation involve three flowing or indeterminate Quantities, two of them muft be a/Turned to determine the third ; or, which is the fame thing, fome other Equation muft be either given or aflumed, involving fome or all the Fluents, in order to a compleat Determination. For then, by means of the two Equations, one of the Fluents may be eliminated, which will bring this Alfo two Fluxional Equations may be derived, to the former cafe. three the Fluxions, by means of which one of them may be involving And fo if the given Equation mould involve four Fluents, eliminated. two other Equations fliould be either given or afTumed, in order to This will be fufficiently explain 'd by the a compleat Determination. two following Examples, which will alfo teach us how complicate Terms, fuch as compound Fractions and Surds, are to be ma-

5,

6.

Method.

Let the given Equation be y*

we are to take x we may introduce a of which

= = =

Let that be z

the Fluxions. third

x\/a*-

x~,

a*

To

the

=

xx*/ a* o, two Fluents y and

we aflume ;c, and we mall have if

another Equation. the two Equations

x* z* r= o. Then by the foreFluxional Equations (at leaft in one cafe) will zxx> o. Thefe two Fluenbe 2jy z zz o, and a*xx and their Fluxional Equations, may be reduced tial Equations, to one Fluential and one Fluxional Equation, by the ufual methods that is, we may eliminate z and z by fubftituting of Reduction y-

a-

&

going Solution

o, their

and

a'-x 1

=

:

their values yy

a a and zyy.

Then we fhall havej 1

a1

x\/ a 1

1

.v


fix Method of

248

" "

-

Q

!

"__

and 2yy

J

^4 -- za*

Or

7.

=

if

l

y* 4- 2a y

a"x z

'tis

= ==

=

FLUXIONS, Or by

o.

z rt

=

4

taking

the furds,

away

o,

and then a*xx

ay*

-f-

,

2xx=.

o.

the given Equation be x 5

-

x^^/ay

-\-x*-

corresponding Fluxional Equation ; to the two flowing quantities ,v and y we may introduce two others .z and i', and thereby remove the Fraction and the Radical, if we affume the 1

find

to

o,

its

-~

two Equations T.

+_>'

=

and x*~i/ay-t-xx=zv. ^

z,

Then we

(hall

z

i;=o, az-\-yz will give the three which by* V z Fluxional Equations ^xx* o, az +- yz -+- yz -+zayy and 2vv= o. 6xx o, Thefeby,ay'x* -+- ^.ayxx' -f"^byy* to on& of the be reduced common Algebra may known Methods Fluential and one Fluxional Equation, iavolving x and y and their

have the three Equations x= i<-~ r o, and ayx* -f- x 6

1

=

ay

-\-

=

o,

=

s

y

as

Fluxions,

is

required. the fame Method

we may

take the Fluxions of Binomial or other Radicals, of any kind, any how involved or compliAs for inflance, if we were to find the cated with one another. 8.

And by

xx, put it equal to y, or make ax-i~ xx s$. Then we fhall Alfo make </ aa

Fluxion oF-Vwf -\-*/aa

xx=yy.

==

1 y have the two Fluential Equations ax-\-z o, and a* AT*; 1 have the two Fluxional we mall whence z o, from Equations 2xx 2zz o.' o, or xx -f- zz o, and ax-}- z 2j/y ax~ This laft Equation, if for z and z we fubftitute their values^

=

and zyy o tute

;

its

=

=

will

ax,

whence value

=

y '

=

become xx

-f~" ~ - 2HX1 A

.

And

2\i

-J

-

an

axy*

here if for *y

-{-

we

A Jf

xx

-

,

,

y

----------

xx

7.1/fta

axz

4 .)'

=>

m

:

fubfti--

...,

.

And

other

Exam--

x

which

zz

there are three variable quantities x, y, the relation of the Fluxions will be 2zz -|-

z, and therefore ~ === o. _j_ ax 4j/j-3

re-

T^

1

many yax + y aa xx of a like- kind will be found in the fequel of this Work. pies 1 1, 12. In Examp. 5. the propofed Equation is 9, 10, quired

a^xx-

xx, we mall have the Fluxion

vax-+-\/aa ax

zaxyy

2yy*

A

'

=

But

-{-

and

axz

wants another Fluential EquaFluxional thence another and tion, Equation, to make a compleat Fluxional if another determination ; Equation were given or only afTurned, we mould have the required relation of the Fluxions x and y,.. as there

<

Suppofe


and INFINITE SERIES.

249

Fluxional Equation were i=.vv/^-v xx ; then by xx fubftitution we mould have the Equation zz -f- ax x x^/ax

Suppofe

axz

-f-

this

4)7

5

.vx -f-

v/rftf

=

rf;s,

2Z -4- ax x Analogy x :y :: 4_>' which can be reduced no farther, (or & cannot we have the Fluential Equation, from which the

o,

or the

3

:

be eliminated,) till xx is fuppos'd to be derived. Fluxional Equation And thus we may have the relation of the Fluxions, even in fuch cafes as \re have not, or perhaps cannot have, the relation of the

z=x\/ax

Fluents.

But

Reduction

not perhaps be conveniently perby Calculation, yet it may poffibly be perform'd Geometrically, as it were, and by the Quadrature of Curves ; as we may learn from our Author's preparatory Proportion, and from the following general Conliderations. Let the right Line AC, perpendicular to the right Line AB, be conceived to move always tho'

this

may

forni'd Analytically, or

parallel to itfelf,

C

fo as that

its

extremity

A may

defcribe the line

AB.

and fixt, or always at the fame diftance from A, how move from towards a let another with C, velocity any point does accelerated or retarded. The parallel motion of the line

Let the point

be

A

AC

not at all affect the progreffive motion of the point moving from A towards C, but from a combination of thefe two independent ; morions, it will defcribe the Curve while at the fame time the fixt point C will

ADH

defcribe

the right line

CE,

parallel

to

AB.

Let the line AC be conceived to move thus, Then till it comes into the place BE, or BD. the line AC is conftant, and remains the fame, while the indefinite or flowing line becomes BD. Alfo the Areas defcribed at the fame time, ACEB and ADB, are likewife flowing quantities, and their velocities of defcription, or their Fluxions, muft neceflarily be as their refpeclive defcribing Let AC or BE be Linear Unity, lines, or Ordinates, BE and BD. or a conftant

known

right line, to

be compared or refer'd are

refer'd

to

;

juft as

or

which

all

in

the other lines are to r.M other Numbers

Numbers, Numeral Unity,

to as being the fimi, all Numbers. of And let the Area ADB be pleft fuppos'd to be 'd to BE, or Linear which it will be reduced from Unity, by apply the order of Surfaces to that of Lines j ami let the refulting line be call'd z. That is, make the Area ADB z x BE ; and if AB be call'd x, then is the Area ACEB x x BE. Therefore the K k Fluxions tacitely

=

=


1

Ibe Method of

25 o"

FLUXIONS,

Fluxions of thefe Areas will be z x BE and x x BE, which are as z and x. But the Fluxions of the Areas were found before to be as x x BD. ED BE i, or z BD to BE. So that it is z x the Fluxion of the Area will be as the Consequently in any Curve, Ordinate of the Curve, drawn into the Fluxion of the Abfcifs. Now to apply this to the prefent cafe. In the Fluxional Equa:

:

:

:

=

=

z=x</ax

xx, if x reprefents the Abfcifs xx be the Ordinate ; then will this Curve of a Curve, and \/ ax So that will z and be a Circle, reprefent the corresponding Area. a of Circle can be exhibited whether the Area we fee from hence,

tion before

affumed

Terms, tho' in the Equation proppfed there which cannot be determined or exbe involved, fhould quantities Geometrical Method, luch as the Areas or Lengths prefs'd by any of Curve-lines ; yet the relation of their Fluxions may neverthelefs

or no, or, in general

be found.

We

now come

Demonftration of his Solutions the of Method of Fluxions, here laid proof of the Principles down, which certainly deferves to engage our mcft ferious attention. And more efpecially, becaufe thefe Principles have been lately drawn into debate, without being well confider'd or imderftoqd ; polfibly be T caufe this Treatife of our Author's, expreffly wrote on the fubjed, had not yet feen the light. As thefe Principles therefore have been treated as precarious at leaft, if not wholly inefficient to fupport the Doo trine derived from them ; I Shall endeavour to examine into every the moll: minute circumflance of this Demonstration, and that with

13. or to the

to the Author's

the utmoft circumipeclion and impartiality. have here in the firft place a Definition and a Theorem to-r Moments are defined to be the indefinitely jmall parts offoiv-

We

gether,

the acceflion of which, in indefinitely quantities, by The are continually increajed. of time, tboj'e quantities

itig

fmall portions

word Moment

a mevcoj by analogy feems to have been (momentum^ movimentum, borrow'd from Time. For as Time is conceived to be in continual and as a greater and a greater Time is generated flux, or motion, more and more Moments, which are conceived of by the acceffion So all other flowing Quantities as the fmalleit particles of Time :

may

be underitood, as perpetually, increafing, by the accellion of which therefore may not improperly be call'd

their fmallefr, particles, their

But what are here call'd their jmalleft particles, be underftood as if they were Atoms, or of any definite

Moments.

are not to

and determinate magnitude,

as in the

Method of

to be indefinitely fmall, or continually decreafing,

Indivisibles.} till

but

they are lefs

than


and INFINITE SERIES.

251

than any afiignable quantities, and yet may then retain all poffible That thefe Moments are varieties of proportion to one another. not chimerical, vifionary, or merely imaginary things, but have an existence Jut generis, at leaft Mathematically and in the Underftandfrom the infinite Divifibility of Quaning, is a neceflary confequence For all contity, which I think hardly any body now contefts *. tinued quantity whatever, tho' not indeed actually, yet mentally may be conceived to be divided in infinitutn, Perhaps this may be beft illuftrated by a comparative gradation or progrefs of Magnitudes. Every finite and limited Quantity may be conceived as divided into any finite number of fmaller parts. This Divifion may proceed, and thofc parts may be conceived to be farther divided in very litor particles, which yet are not Moments. tle, but flill finite parts, But when thefe particles are farther conceived to be divided, not as to become of a magnitude Ids than actually but mentally, fo far any afiignable, (and what can flop the progrefs of the Mind ?) then As are they properly the Moments which are to be understood here. includes no or condiminution of this gradation abfurdity certainly tradiction, the Mind has the privilege of forming a Conception of thefe Moments, a poffible Notion at leaft, though perhaps not an adequate one ; and then Mathematicians have a right of applying them to ufe, and of making fuch Inferences from them, as by any flrict way of reafoning may be derived. It is objected, that we cannot form an intelligible and adequate

Notion of thefe Moments, becaufe fo obfcure and incomprehenfible an Idea, as that of Infinity is, muft needs enter that Notion ; and therefore they ought to be excluded from all Geometrical Difquifitions.

It

may

indeed be allowed, that

Notion of them on that account, fuch

we

have not an adequate

as exhatifts the

whole nature

for a

of the thing, neither neceflary ; partial Notion, which is that of their Divifibility fine Jine, without any regard to is

it

at

all

There are many is fufficient in the preient cafe. other Speculations in the Mathematicks, in which a Notion of In-

their magnitude,

finity

is

a

neceflary ingredient,

which however

are admitted

Geometricians, as ufeful and dcmonftrable Truths.

The

by

all

Doctrine

of commenfurable and incommenfurable magnitudes includes a Notion of Infinity, and yet is received as a very demonftrablc Doctrine. We have a perfect Idea of a Square and its Diagonal, and yet we

K

k 2

know


The Method of FLUXIONS, of no finite common meafure, or that their they will admit probe exhibited in rational Numbers, tho' ever fo fmall, cannot portion but may by a feries of decimal or other parts continued ad

know

infini-

In common Arithmetick we know, that the vulgar Fraction the decimal Fraction 0,666666, &c. continued ad infinitum^ and 1., are one and the fame thing j and therefore if we have a fcientifick notion of the one, we have likewife of the other. When I deicribe a right line with my Pen, fuppofe of an Inch long, I defcribe firft one half of the line, then one half of the remainder, then one half of the next remainder, and fo on. That is, I actually run over all thofe infinite divifions and fubdivifions, before I have comI do not attend to them, or cannot diftinpleated the Line, tho' And by this I am indubitably certain, that this Series guifh them. of Fractions i -f- JL _j_ -.-}- _'r> &c. continued ad infinitum, is preto Unity. Euclid has demonflrated in his Elements, cifely equal that the Circular Angle of Contact is lefs than any aflignable rightlined Angle, or, which is the fame thing, is an infinitely little Angle in comparifon with any finite Angle And our Author fhews us fHll greater the infinite about fteries, gradations of Angles of ConIn Geometry we know, that Curves may continually approach tact. towards their Arymptotes, and yet will not a&ually meet with them; infinite diftance. till both are continued to an know likewife, or be that many of their included Areas, but of a finite Solids, will and determinable magnitude, even tho' their lengths mould be actually know that fome Spirals make infinite continued ad infinitum. about a Circumvolutions Pole, or Center, and yet the whole Line, thus infinitely involved, is but of a finite, determinable, and aflignable length. The Methods of computing Logarithms fuppofe, that between any two given Numbers, an infinite number of mean Proportionals maybe interpofedj and without fome Notion of Infinity tum.

:

My

We

We

and properties are hardly intelligible or difcoverable. many of the moft fublime and ufeful parts of muft be banifh'd out of the Mathematicks, if we are knowledge fo fcrupulous as to admit of no Speculations, in which a Notion

their nature

And

in general,

We

of Infinity will be neeeflarily included. may therefore as fafely and the Principles upon which the Method built, as any of the fore-mention'd Specula-

admit of Moments, of Fluxions is here

.

tions.

The nature and notion of Moments being thus eftablifli'd, we may pafs on to the afore -mcnticn'd Theorem, which is this.

,


and INFINITE SERIES.

253

(contemporary) Moments offairing quantities are as the Velocities of if this be flowing or increafing ; that is, as their Fluxions.

Now

proved of Lines, whatever, which

it

will equally obtain in

all

flowing quantities be always adequately rcprefented and exin Lines. But equable Motions, the Times being given, pounded by the Spaces defcribed will be as the Velocities of Defcription, as is known in Mechanicks. And if this be true of any finite Spaces whatever, or of all Spaces in general, it muft alfo obtain in infi-

may

nitely little Spaces, which we call Moments. tions continually accelerated or retarded, the

And

even

Motions

in

Mo-

in infinite-

So ly little Spaces, or Moments, muft degenerate into equability. that the Velocities of increafe or decreafe, or the Fluxions, will be Moments. Therefore the Ratio of always as the

contemporary

the Fluxions of Quantities, and the Ratio of their contemporary Moments, will always be the fame, and may be ufed promifcuoufly for each other. 14. The next thing to be fettled thefe Moments, which

a convenient Notation for be diftinguifh'd, reprefented, by they may It has been compared, and readily fuggefted to the Imagination. for ftand that when &c. variable or appointed already, x, y, z, v, flowing quantities, then their Velocities of increafe, or their Fluxions, fhall be reprefented by x, y, z, -j, &c. which therefore will be proBut as thefe are only portional to the contemporary Moments. or of another Species, they cannot be the MoVelocities, magnitudes ments themfelves, which we conceive as indefinitely little Spaces, or other analogous may therefore here aptly introquantities. duce the Symbol o, not to ftand for abfolute nothing, as in Arithrnetick, but a vanifhing Space or Qtiantity, which was juft now finite, but by continually decrealing, in order prefently to terminate is

We

is now become lefs than any affignable Qinintify. have certainly a right fo to do. For if the notion is intelligible, and implies no contradiction as was argued before, it may This is not furely be infinuated by a Character appropriate to it. the which be would to the hypothefis, contrary aligning quantity,

in

mere nothing,

And we

but the' ral

only appointing a mark to reprefent it.- Then multiplying Fluxions by the vanishing quantity we fhall have the fcveis

<?,

cc.

which

are vanifhing likewife, and pioportional to the Fluxions refpedlively. Thefe therefore may now reprefent the contemporary Moments- -of x, y, z, v, &c. And quantities

.\o,

yo,

zo,

r?,

in general, whatever other flowing .quantities, as well as Lines and I

Spaces,


2

"*flje

54

Method of FLUXIONS,

Spaces, arc reprefented by A-, y, z, -v, &c. as o may (land for a. of the fame kind, and as x, -vanishing quantity y, z, v, &c. may ftand for their Velocities of increafe or decreafe, (or, if

you

fpr

Numbers

zo,

i-o,

proportional to thofe Velocities,) then

&c. always denote

may

pleafe,

xo,

yo,

their

refpedive fynchronal Moments, .or momentary accefiions, and may be admitted into Computations And this we corne now to apply. .accordingly.

We

muft now have recourfe

and Equations that involve flowing that in the progrefs of is, Quantities. flowing, the Fluents will continually acquire new values, .by the accefilon of contemporary parts of thofe Fluents, and yet the Equation will be equally true in all thcfe, cafes. This is a neceffary refult from the Nature and Definition of variable Quantities. Confequently thefe Fluents be .how or increafed diminifh'd any .rnay by their contemporary or Increments Decrements ; which Fluents, fo increafed or dimiAs if niihed, may be fubflituted for the others in the Equation. an Equation mould involve the Fluents x and _y, together with any and Y X and are fuppofed to be of their conquantities, given any 15.

extenfive property,

belonging Which property

to a very notable, ufeful,

to. all

temporary Augments reflectively. Then in the given Equation we may fubflitute x -f- X for x, and y -+- Y for -y, and yet the Equation will be .good, or .the equality of the

X

and

Y

Terms

will be prefer ved. inflead of x and

were contemporary Decrements, x X and y Y reflectively. And as this inuft hold good of all contemporary Increments or Decrements whatever, whether finitely great or infinitely little, it will be true likewife of contemporary Moments. That is, in flea d of .r and y in xo and y we fubflitute .v-fany Equation, may -t-jo, and yet we

.So if

y we might

fubflitute

have a good Equation. The tendency of this will appear from what immediately follows. 16. The Author's fingle Example is a kind of Induction, and the a.\* proof of this may ferve for all cafes. Let the Equation x be given as before, including the variable a xy quantities x and r, inftead of which we may fubflitute thefe ihall

flill

s

+

5

_>'

=o

quancontemporary Moments, or x -- xo and y -i-yo respectively. Tlien we ihall have the Equation x -+- xo 3 a x x AO a -f- a x x -|- xo x y -Jo~ o. Thefe T+"}f Terms .being expanded, and reduced to three orders or columns, according as the vanifhing quantity o is of none, one, or of more tities

increas'd

+

/limenfions,

by

their

>

i

will ftand as in the

Margin.

=

|


and INFINITE SERIES; 18. Here the Terms of the fir ft or column, remove or deftroy one order, as another, being absolutely equal to noThey bething by the given Equation. ing therefore expunged, the remaining

17,

*3

255

+

_ i_ +a.rj> + _

* ?* w +3 A *"* r

2f,^

v

fl .

ox _ a ll'l

a.\iy

\

)>=o,

-\-axjs-

^

)3

_,;>,

_ "j.

1

|

j

be divided by the comThis being- done, all the Terms Multiplier <?, whatever it is. ftiil be affecled of the third order will by o, of one or more dimenfions, and may therefore be expunged, as infinitely lels than the remain thofe of the fecond order or others. Laftly, there will only

Terms may

all

mon

that

column,

is

i

zaxx

3.vA.'

axy 4- ayx

-+-

Tjy-

=

which

o,

Q^. E. D. in fomething a difbe thus derived, may be any fynchronal Augments of the Let and ferent manner. variable quantities A* and y, as befoie, the relation of which quanwill be the Fluxional

The fame

Equation required.

Conclufions

X

Y

Y

Then may tf-J-X and y 4exhibited by any Equation. be fubfKtuted for x and y in that Equation. Suppofe for inftance 3 o ; then by fubftitution we flwll that x> ax* 4- axy _y tities

is

-

=

-

3 a x .v 4- X a 4-#x.v4-Xx/4-Y have x 4- X y 4- V 1 z 1 ax or in termini* expanfis .v 5 -f- 3X X -f- 3xX -+- X o a 4 aX* ^XY Y 2rfxX -t- axy -\- <?.vY4- aXy -fj 3j 3;'Y 3 o will vaY o. But the Terms ,v ax* -+- axy _y niHi out of the Equation, and leave 3# 1 X 4- 3xX a 4-X 2axX 3

=

|

|

|

3

;

5

3

=

3

=

3

Y- == o, for XY 3/i y* Y of the let their the relation magnitude be contemporary Augments, what it will. Or refolving this Equation into an Analogy, the ratio Y /7X -Lv ,. 2 ..* ?r*-|- ^rX-L. X 1 A of thele Augments may be this, * X =. a* ..v _|-j* r 3.., + Now to find the ultimate rc.tio of thefe Augments, or their ratio when they become Moments, fuppofe X and V to diminil'h till they become vanishing quantities, and then they may be expunged out Or in thofe circumftances it will be of this value of the ratio. 7

aX* 4- axV 4- aXy 4-

,-

,

,.

,

*

-

P

=

y

-

this

-

~^ax

the

is .V

1

fame

,

which

ratio

is

that

as

2f>x--ai

now

.

or

of the .

a 3_)'

the ratio of the

axy

=

l

And

Moments. or

Fluxions,

it

will

be

-

$x-x

zaxx 4- ayx,

as

wss

found before. is no aflumption made, but what is of the ancient and modern both Methods iuflifiable by the received We only defend from a general Proportion, which Geometricians. is undeniable, to a particular cafe which is certainly included in ir. That

In this

way of

arguing there


The Method of FLUXIONS,

256

having the relation of the variable Quantities, we thence the relation or ratio of their contemporary Augdeduce da-eddy and ments having this, we directly deduce the relation or ratio of thofc contemporary Augments when they are nafcent or evanefcent, in a word, when they are Mojuft beginning or juft ceafing to be or To evade this realbning, it ought ments, vanilliing Quantities. be can be conceived lefs than afiignto proved, that no Quantities able Quantities; that the Mind has not the privilege of conceiving Quantity as perpetually diminiiLingy/w^w ; that the Conception of a .vanishing Quantity, a Moment, an Infinitefimal, &c. includes a In fhort, that Quantity is not (even mentally) divificontradiction ble ad infinitum ; for to that the Controverfy mufb be reduced at laft. But I believe it will be a very difficult matter to extort this been fo Principle from the Mathematicians of our days, who have long in quiet poiTefTion of it, who are indubitably convinced of the evidence and. certainty of it, who continually and fuccefslully apply it, arid who- are ready to acknowledge the extreme fertility and ufefulnefs of it, upon fo many important occalions. but to account for thefe two cir19. Nothing remains, I think, of Fluxions, which our AuMethod the to .cumilances, belonging

That

is,

;

;

:

Firft that the given Equation, whofe thor briefly mentions here. Fluxional Equation is to be found, may involve any number of This has been fufficiently proved already, and flowing quantities. feveral have feen we Examples of it. Secondly, that in taking Fluxions we need not always confine ourfelves to the progreffion of the Indices, but may affume infinite other Arithmetical Progreflions, as

conveniency

may

This

require.

will deferve a little farther illu-

than what muft neceiTarily refult from any given Equation may afTume, in an 3 ax 1 -4- axy Thus the Equation x 3 infinite variety. j o, m will become #"+*>' being multiply'd by the general quantity x y", m x my"~^^ -r- ax -$- 1y" -h ax m+ly"'t' o, which is virtually the fame as it was before, tho' it may aiTume infinite forms, accorEquation And if we take the ding as we pleafe to interpret m and n. ufual in the Fluxions of this Equation, way, we mall have m m i 1 nax m ^yy n~^ -fm -+- zaxx ^y" y* -j- nx^rty}*B n mxxm~ y a ''* -f- n + irf.Y" 'j/)ftration, tho' it is no other the different forms, which

=

=

1

+

.5= o.

4-

Now

nx*j>y~*

if

we

m -f-

n -f- 3j/y /xx~*y* derived before. was

Examples.

I

!

l

divide this again

2axx a

by

x"}",

l

nax*yy~~

?= o,

which

And

the like

-+-

we

m -+-

mail have

m

laxy 4- n-\- \axy

the fame general Equation as may be underftood of all other

is

SECT.


and INFINITE SERIES.

SECT.

II.

Concerning Fluxions the

257

of fuperior orders^

and

method of deriving their Equations.

our Author confiders only fir ft Fluxions, and has not thought fit to extend his Method to fuperior orders, as not diwithin his prefent purpofe. For tho' he here rectly foiling this Treatifc

IN

purfues Speculations which require the ufe of fecond Fluxions, or higher orders, yet he has very artfully contrived to reduce them to firft Fluxions, and to avoid the necefTity of introducing Fluxions of fuIn his other excellent Works of this kind, which perior orders.

have been publifh'd by himfelf, he makes exprefs mention of them, he difcovers their nature and properties, and gives Rules for deriving Therefore that this Work may be the more fertheir Equations. viceable to Learners, and may fulfil the defign of being an Inftitution, I mall here make fome inquiry into the nature of fuperior Fluxions, and give fome Rules for finding their Equations. And in its proper place, I mail endeavour to (hew fomething their application and ufe. as the Fluxions of quantities which have been hitherto con-

afterwards,

of

Now fider'd,

or their comparative Velocities

of increafe and decreafe, are

themfelves, and of their own nature, variable and flowing quantities alfo, and as fuch are themfelves capable of perpetual increafe and decrea&, or of perpetual acceleration and retardation ; they may be treated as other flowing quantities, and the relation of their Fluxions

be inquired and difcover'd. In order to which we will adopt our Author's Notation already publifh'd, in which we are to confo thefe ceive, that as x, y, z, &c. have their Fluxions #, z., &c.

may

likewife have their Fluxions x, /, of x, v, z, &c. And thefe again,

z,&c.which being

ftill

j, are the fecond Fluxions

variable quantities, have

j.

their Fluxions denoted

of x,

y, z, &c.

y, z, &c. which are the third Fluxions thefe again, being ftill flowing quantities,

by x,

And

have their Fluxions x, /, z, &c. which are the fourth Fluxions of And fo we may proceed to fuperior orders, as far as x, y, z, &c. there mall be occafion. Then, when any Equation is propofed, conof variable futing quantities, as the relation of its Fluxions may be found by what has been taught before ; fo by repeating only the fame and the Fluxions as operation, confidering flowing Quantities^ the

L

1

relation


The Method of

258

FLUXIONS,

of the fecond Fluxions may be found. And the like for all of Fluxions. higher orders have the Equation y* we ax if o, in which are the Thus we and fhall have the firft Fluxional x, two Fluents y Equation zyy And here, as we have the three Fluents j>, y, and x, o. ax if we take the Fluxions again, we fhall have the fecond Fluxional

relation

=

ax= o.

Equation zyy -+- zy* y,

and x,

y, y,

we

if

And

here, as there are four Fluents

take the Fluxions again,

third Fluxional Equation zyy

ax

bjy if

we

=

+

if

=

we

zyy

And

o.

zyy -f-

ax

^.yy

have the

fhall

=

or zyy

o,

here, as there are five Fluents y, y, y, y,

take the Fluxions again,

Equation zyy

ax

And

o.

+

we

.

..

-f-

-+-

6yy

we

fhall

l

ax

6y

we

and x,

have the fourth Fluxional

=

or

o,

zyy -+- Syy

here, as there are fix Fluents y, y, y, y, y,

take the Fluxions again,

4-

fhall

have zyy

+

zyy

-f-

=

=

-f-

6y*

and

xy

8yy -{-

ax ax o, for the o, or zyy +- i oyy -f- zoyy zyy And fo on to the fixth, feventh, 6cc. fifth Fluxional Equation. the Demonftration of this will proceed much after the manner as our Author's Demonftration of firft Fluxions, and is indeed ax o} For in the given Equation^* it. virtually included in if we fuppofe y and x to become at the fame time y -f- yo and x-)- xo, if we fuppofe yo and xo to denote the fynchronal Moments (that is, * of the Fluents y and x,) then by fubftitution we fhall have ~y +yo\ fyy

i

_j_

Now

=

= = Where

axx axo

-f-

xo

o,

or in termini* expanjis,

o.

ding the reft

by

o,

Now

Equation.

Moments of

expunging y be zyy

it

will

in

this

the Fluents

for thofe Fluents

Equation,

y,

we may

the kft Equation, and

it

1

y, and

= = we

ax ax

y

1

ax

-f- zyyo -+-y*o*

o, andj/

1

o for the

^

1

and

,

firft

divi-

fluxional

the fynchronal fuppofe and xo refpedively ; beyo } yo t and xo in -f-jj/o, y -+-yo,

if

x, to

fubftitute

y

x+

become zy-t-zyoxy-l-yo ax zyyo +- zyyo -+- zy'yoo

will

axx

+ xo

=

axo o. becaufe and o by the given Equation, Here becaufe zyy vanishes ; divide the reft by o, and we fhall have zy* + zyy zy'yoo o for the fecond fluxional Equation. ax Again in this Equar.

or expanding, zyy -f-

o,

ax=

=

we

tion, if

y, and

xt

fuppofe the Synchronal

to be yo, yo, yo,

Moments of

and xo refpedively

;

the Fluents y, for

y

t

thofe Fluents

we


and INFINITE SERIES. we may

+

y+yo, y

fubftitute

yo, y-t-yo, a^id

axx

_j_

xo

will

it

_}_ 2yy -+-

2yyo -t-

_l_ 2/_y

rfx

expunging have 6yy in like

-+-

Terms ax

2yy

Q^

2x7 -\-yo

ax

1

o by the

the

manner

Examples.

To

=

s;^

laft

in

+- zy -+- 2yo x

\

=

axo

Equation

which

o

+

y

-f-

yo

6yyo -t- 2y*o

s

But here becaufe

o.

l

2j'

dividing the reft by o, and ftill be found, we fliall

;

will

=

for all

+- xo in the lad

j

become

or expanding and collecting, 2j*

o,

all

x

a

..

Equation, and

259

o for the third fluxional Equation. And other orders of Fluxions, and for all other

E. D.

illuftrate the

method of

rinding fuperior Fluxions

by another

ax -{-axy let us take our Author's Equation # y> of the Fluxions relation he the has found which o, fimpleft o. Here we have the to be 3x^ a zaxx -h axy +- axy 3^/7* the Rules and fame the Fluxion of by flowing quantities x, y, x, y ; 3-

5

=

Example,

this

in

=

Equation,

when

contracled, will be

3#w

=

i

+ 6x*x

2axx

o. And in this Equazax* H- axy -+- 2axy -\- axy 3vy 6jf y tion we have the flowing quantities x, y, x,y, x, y, fo that taking the Fluxions again by the fame Rules, we fhall have the Equation, s

when

contracted,

^xx

l

-f-

iSxxx

!L

-{-

6x 3

i 3 yy* fyyy %axy -+- T,a.\y -f. axy are found the there flowing Equation

6axx -f- axy

2axx 6y

s

=

o.

And

quantities x, y, x,

as in

y

y

-f-

this

x, y,

x, y, we might proceed in like manner to find the relations of the fourth Fluxions belonging to this Equation, and all the following orders of Fluxions. And here it may not be amifs to obferve, that as the propofed

Equation expreffes the conflant -elation of the variable quantities x and as the firft fluxional Equation exprefles the conftant relation of the variable (but finite i.nd alTignable) quantities x and y, which denote the comparative Velocity of increafe or decreale of x and y in the propcfed Equation So the fecond fluxional Equation will exprefs the conftant relation of the variable (but finite and afligdenote x and which the nable) quantities y comparative Velocity of

and y

-,

:

y

And in and_y in the foregoing Equation. the conftant relation ot the variable the third fluxional Equation we have the increafe or decreafe ot

(but finite

and

.v

aflignable) quantities

L

1

.v

2

and

r,

which

will denote

the

com-


The Method of FLUXIONS,

260

comparative Velocity of the increafe or decreafe of "x and "y in the And fo on for ever. Here the Velocity of a foregoing Equation. Velocity, however uncouth it may found, will be no abfurd Idea when rightly conceived, but on the contrary will be a very rational and intelligible Notion. If there be fuch a thing as Motion any how continually accelerated, that continual Acceleration will be the Velocity of a Velocity ; and as that variation may be continually varied, that is, accelerated or retarded, there will 'be in nature, or at leafl in the Understanding, the Velocity of a Velocity of a Velocity. in Or other words, the Notion offecond, third, and higher Fluxions, muft be admitted as found and genuine. But to proceed : may much abbreviate the Equations now derived, the

We

by

known Laws of Analyticks.

From the given Equation x* ax 1 -+axy ^ere is found a new Equation, wherein, becaufe of y"' two new Symbols x and y introduced, we are at liberty to aflume another Equation, belides this now found, in order to a jufl Determination. For fimplicity-fake we may make x Unity, or any other conftant quantity that is, we x to flow equably, may fuppofe and therefore its Velocity is uniform. Make therefore x i and

=

=

;

the

firft

fluxional Equation will become 2ax -+- ay 3^* a So in the o. 2axx Equation 3x.v -f- 6x*x

3j)/)'

=

will

amount

1

}

+ axy

2ax* -+. o there are four new 3 vj* axy -i- zaxy -h axy 6y\y Symbols introduced, x, y, x, and r, and therefore we may afiume two other congruous Equations, which together with the two now found,

=

fimplicity .v

=o

;

and

Equation to thus in x,

}',

take

to a compleat Determination.

we make one

to be x

this,

x, y, x, i,

Thus

the other

if for the fake

of

i, neceflarily be thefe being fubftituted, will reduce the fecond fluxionaj

6x

2.0.

-f-

the next Equation,

x=

=

y

;

iay -f- axy ^yywherein there are

befides the three Equations

and thence

x=o,

x=

o,

will'

now

which

And

o.

6y*y fix

new Symbols

found, will

we may

reduce

it

i And the like of yy* $yyy 6f> == o. of orders. Equations fucceeding But all thefe Reductions and Abbreviations will be beft made as the Equations are derived. Thus the propofed Equation being x~> ax* o, taking the Fluxions, and at the fame time axy y=

to 6 -f- $ay -+axy

= + making x= (and confequently zax + ay + axy 3** zyy* = i,

x,

x, o.

&c.

And

=o,) we taking

(hall

have

the Fluxions again.


and INFINITE SERIES.

And

be 6x

will

it

again,

20. -f-

taking the Fluxions again, 6y*>

3^

axy far as there

And

o.

4

zay +- axy

3yy*

will be 6 -f- $d'y -+-

it

taking the Fluxions

=

1

2 4-yy'y

i%y y

3677*

=

6y*y

axy

again,

it

o.

%yy*

be

will

And fo

o.

i

on, as

occafion.

is

now

But

=

261

for the clearer apprehenfion of thefe feveral orders of endeavour to illuftrate them by a Geometrical

(hall

I

Fluxions, Let us allume Figure, adapted to a iimple and a particular cafe. the Equation y 1 r=ax, otyzs=ia*x*, which will therefore belong to

ABC, whole

the Parabola

LD =y

and Ordinate

;

Parameter

AP

where

is

= y=

AP a

is

Abfcifs

tf,

at the

Tangent

AD

=

x,

Vertex A.

And fupyaPsve~~*. taking the Fluxions, we fhall have motion of the the Parabola to be defcribed the equable pofing by Ordinate upon the Abfcifs, that equable Velocity may be expounded Then

by the given Line or Parameter

\t\v]\ibey=(a*x

a, that

*= ~k = zx

this Conftrudtion.

zx

=

2X

Make x (AD)

DG

and the Line

Jy,

=

"? :

we may

is,

y (BD)

y or BD. And done every where upon AE, (or

if

this be

if

the Ordinate

with a

DG

, '

=

Then

will give

us

DG

=

a (|AP)

::

a.

:

will therefore

reprefent the Fluxion of

AE

which

-?-

') 2X

put x

be fuppos'd to

parallel motion,) a

move upon Curve

GH

will be conftiucted or delcribed, whofe Ordinates will every where expound the Fluxions

of the correfponding Ordinates of the PaThis Curve will be one of rabola ABC. the Hyperbola's between the Afymptotes 3.

AE

and

Or yy

=

AP

;

for

its

=

from the Equation y the Fluxions again, and putting x -{-

,

.

Again,

zay

-11

Equation isjx=

2xy=aj,ory

=J

j

" ,

=a

or as

2 *y

=

before,

where the negative

ay,

we

by taking fhall

have

fign {hews only,

that_y is to be confider'd rather as a retardation than an acceleration, or an acceleration the this will give us the contrary way.

Now

following


202 following

DI

=

DG, And

?2* Method of FLUXIONS, Make x (AD) y (DG) Conftruaion. :

:

\a (iAP)

:

;

and the Line DI will therefore reprefent the Fluxion of y, or of j, and therefore the fecond Fluxion of BD, or of/.

be done every where upon AE, a Curve IK will be whofe Ordinates will always expound the fecond Fluxions comlructed, if this

of the correfponding Ordinates of the Parabola ABC. This Curve likewife will be one of the Hyperbola's, for its Equation is y

=

fl*

/Jy

G.

^

a*

1

from

Again,

the

*

6*5

we

^by taking the Fluxions

~y=~ y (DI)

::

which

,

y

Equation

=

mail have

v

^-

or

,

2ay

zxy

:

DL

and the Line

DL=y,

=:

=

ay' t

ay.,

or

Make x (AD)

will give us this Conftrudlion.

\a (|AP)

2xy

will

:

therefore

reprefent the Fluxion of DI, or of y, the fecond Fluxion of DG, And if this be or of y, and the third Fluxion of BD, or of^. be will a done every where Curve conflructed, whofe AE,

LM

upon

Ordinates will always expound the third Fluxions of the correfponding Ordinates of the Parabola ABC. This Curve will be an Hyper-

and

bola,

And

fo

its

y=. '=-1

Equation will be

we might

; ,

proceed to conftrucl Curves,

or

yy=

"

64*"*

the Ordinates of or reprefent the

(in the prefent Example) would expound fourth, fifth, and other orders of Fluxions. might likewife proceed in a retrograde order, to find

which

We

Curves whofe Ordinates mall reprefent the Fluents of any of

= =

the.

thefe

3.

Fluxions, the Curve

when

given.

GH

As

were given

taught in the next Problem,) ,

.v

which

(AD)

::

will give us

y (DG)

:

DB

we had y

if

La*xx~*} or

by taking the Fluents,

;

it

this

=

,

would be y

Conftruction. ,

will be

= (a^x*= =

2

-J

(as

and the Line

^-r

Make \a (|AP)

DB

if

)

:

will reprefent

or of y. And if this be done every where upon Curve AB will be con ftru died, whofe Ordinates AE, will always expound the Fluents of the correfponding Ordinates of the Curve GH. This Curve will be the common Parabola, whofe i Parameter

the Fluent of

the Line

DG, a


and INFINITE SERIES. Parameter or

yy=ax. So

if

AP

the Line

is

=

its

Equation

is

y

=

a*x'* t

the Parabola ABC, we might conceive its Ordinates Fluxions, of which the correfponding Ordinates

we had

to reprefent of fome other Curve, fiippofe

To

For

a.

263

which Curve, put y

find

QR, would

reprefent the Fluents.

for the Fluent of y, n I .: ..

y

for the Fluent

Iff

/

let, &c. _/, y, y, /, j/, y, y, &c. be a Series of both Terms proceeding ways indefinitely, of which every fucceedthe Fluxion of the preceding, and vice versa ; ing Term reprefents a Notation of our Author's, deliver'd elfewhere.) Then to according

of

y,

&c.

becaufe

(That

it '

will be y

is,

= (div*=<z^x^ =) ^r = [, W = =y is_y

'

\

%

.x

Z

2f!i!

J2. 3*

)

3

ftrudion.

Make $a (|AP)

:

ft

(AD)

taking the Fluents

,

which

;

will give us this

it

Con-

=y =

DQ^ -^ Fluent of DB, or of y. And of the Line AE, a Curve QR ::

y (BD)

:

and the Line DQ^will reprefent the if the fame be done at every point will be form'd, the Ordinates of which will always expound the Fluents of the correfponding Ordinates of the Parabola ABC. This a be of alfo will but a higher order, the Equation Curve Parabola, I I

3.

*

of which

is^=

,

3^

Again, becaufe "

ents

it

will be

=

= f ~ == zx

y

\ $a?

/

y=(

Make

Conftruaion.

or yy

.

3ilJL $a

v.

JfL.sff!|x"= |

(|AP)

a

= J\

W

2

^fl

.

taking the Flu-

which

will give us this

->.a.'-

7

i

"*

l

:

x (AD)

,

: :

y

(

DQ^J

:

=

= And

_y /

DS, and the Line DS will reprefent the Fluent of DQ^, or of_y. if the fame be done at every point of the Line AE, a Curve ST will thereby be form'd, the Ordinates of which will expound the Fluents of the correfponding Ordinates of the Curve QR. This //

Curve will be .

^-.

And

a Parabola,

fo

whofe Equation

we might go on

as far as

is

we

jy=

i

////

1^1 ,

or yy

=

pleafe,

Laftlv,


The Method of

264 we

Laftly, if

conceive

DB,

FLUXIONS, common

the

Ordinate of

all

thefe

Curves, to be any where thus conftrucled upon AD, that is, to be thus divided in the points S, B, G, I, L, 6cc. from whence to are drawn Ss, Qtf, B^, Gg, I/, L/, 6cc. parallel to and ; if this Ordinate be farther conceived to move either backwards or

Q^

AP

AE

with an equable Velocity, (reprefented by and as it defcribes thefe Curves, to x,) carry the aforeThen the points s, q, b,g, faid Parallels along with it in its motion fuch a manner, in the Line AP, as i, /, &c. will likewife move in that the Velocity of each point will be reprefented by the diflance of the next from the point A. Thus the Velocity of s will be reprefented by Aq, the Velocity of q by A, of b by Ag, of g by A/, of / by A/, &c. Or in other words, Aq will be the Fluxion of A.S ; Al> will be the Fluxion of Ag, or the fecond Fluxion of As ; Ag will be the Fluxion of Ab, or the fecond Fluxion of Aq, or the third. Fluxion of As ; Ai will be the Fluxion of Ag, or the fecond Fluxion of Ah, or the third Fluxion of Aq, or the fourth Fluxion of As ; and fo on. Now in this inftance the feveral orders of Fluxions, or Velocities, are not only expounded by their Proxies and Reprefentatives, but alfo are themfelves actually exhibited, as far as may be done by Geometrical Figures. And the like obtains wherever elfe forwards

AP

upon AE,

= = tf

:

a beginning ; which fufficiently mews the relative nature of all thefe orders of Fluxions and Fluents, and that they differ from each other by mere relation only, and in the manner of conceiving. And in general, what has been obferved from this Example, may be eafily accommodated to any other cafes whatfoever. Or thefe different orders of Fluents and Fluxions may be thus explain'd abftractedly and Analytically, without the afliftance of CurveLet any conflant and lines, by the following general Example. known quantity be denoted by a, and let a" be any given Power or Root of the lame. And let x n be the like Power or Root of the variable and indefinite quantity x. Make a m x m a y, or

we make

:

=^ = m

y

a

l

~ mx m

.

Here y

will be

alfo

a

which

become known

:

:

:

an indefinite quantity,

foon as the value of x is affign'd. Then taking the Fluxions, it will be y ma l ~ m xx m~ 1 ; and fuppofing x to flow or increafe uniformly, and making its constant ma* mx m -*. Here if a, it will be y Velocity or Fluxion x will

as

= =

=

for

ma

a :

1

:

n m

x

y

:

we y.

write

its

value y,

So that y will be

it

will be

alfo a

=

y known and

,

that

affignable

is,

x

:

Quantity,


and

whenever x (and therefore y)

tity,

we

Fluxions again, irtS"""^*""

;

y

N FINITE SERIES.

I

= ~~

ta v -

come

m

known

a

Then

;

x

x

is,

m

:

where

alfo

will be

y

Fluxions again, we

=

-

-

that

;

y

:

that

,

x

have y

=

m

30

:

:

be

will

it

y.

have y

fhall

is

x

i

::

m

:

za

And

m

2 x

m

x

::

:

x

i

y

y

:

taking the

m

So that y will

y /. And from

known, whenever x

=m

given.

m

rnx

x

is,

known, when x

fhall

is,

:

we

y=.-^~

or

value y,

its

writing

taking the Fluxions again, ?.a*- m \ m -*,

=

So that y will bex (and therefore y and y) is affign'd.

when

quantity,

taking the ia* "xx"- 1 ;;; x -

l

la

Then

affign'd.

have^=wxw m m~

mall

or for ma"-~

that

,

x

1

is

265

be

alfo

is this Inductipn we given. may conclude in general, that if the order of Fluxions be denoted by any integer number ?/, or if n be put for the number of points over the n

^_____

x

m

ll-l-i

na y y y ; or from the Fluxion of any order being given, the Fluxion of the next immediate order may be hence found. Letter

t

be

will always

it

:

:

:

:

ate 'tis

Fluent.

m

\a

:

As x

=

if ::

y

:

'tis

2,

If

y.

72

obferving the fame analogy, if

m

=

za

:

'tis

0,

n==

i,

A;

y

: :

ma 'tis

:

x

:

:

y

:

y

: :

:

y

ia

:

}

immedi-

If n -

y.

m--

n

"+t

_______ thus invert the na : x proportion m and then from the Fluxion given, we fhall find its next

Or we may

And

y.

:

i

x

y

::

:

y where y is put for the Fluent of; or for y with a negative point. And here becaufe y=.a - mx m it will be m 4- la x :: a -" *" 1

;

,

l

'

y, or caufe

y y

= =

I

_

~V+ m-\-\a

{a*-<"x*>

1

= =

.

--

tism-{-2a

it

x

::

:

^

__Zj__T

will be

y

which

y

=) il

=^ =

alfo

may

thus appear.

Be-

y}

or

,

taking the Fluents, (fee the

.

Again, if we

m-f,,

-

m I

II

;

:

m-\-\a

I :

1

1 1 v" -*-

/

next Problem,)

:

,

l

y

v ..

Mm

..

=

*

make

+...

-

= .

2,

For

a

becaufe


The Method of

266 becaufe

y

=

-x

.f

m

_

-.

m+

i

-

.

FLUXIONS,

== Again,

,

if

taking w the Fluents

we make

=

3,

'tis

it

m -|-

And -t-3

m

+

l

X

m

-\- 2

x

l

j+3a

will be

fo for

"~'~*

other fuperior orders of Fluents.

all

And

this

may

fuffice

and properties of thefe

mew the comparative nature of Fluxions and Fluents, and

in general, to feveral orders

to teach the operations by which they are produced, or to find their As to the ufes they may be apply 'd refpeftive fluxional Equations. when that will come more properly to be confider'd in found, to,

another place.

SECT.

III.

Tfte

Geometrical and Mechanical Elements of Fluxions,

foregoing- Principles of the Doftrine of Fluxions being chiefly abftradted and Analytical I mail here endeavour, after a general manner, to (hew fomething analogous to them in Geo-

THE

a.nd Mechanicks ; by which they may become, not only the of the Underftanding, and of the Imagination, objeft (which will only their but even of Senfe poffible exiftence,) too, by making prove them adually to exift in a vifible and fenfible form. For jt is now become neceffary to exhibit them all manner of ways, in order to give a fatisfaclpry proof, thai they have indeed any real exiftence at

metry

all.

And tion, fider

it

fir ft,

by way of prepara-

~

will be convenient to con-

Uniform and equable motions,

as alfo fuch as are alike inequable.

Let the right Line AB be defcribed by the equable motion of a point, which is now at E, and will preAlfo let the Line fently be at G. to the CD, parallel former, be dethe fcribed by and K, at equable motion of a point, which is in the farne times as the former is in E and G. Then will EG and be contemporaneous Lines, and therefore will be proportional to

H

HK

the


and INFINITE SERIES.

267

Draw the indefithe Velocity of each moving point refpedlively. of like Tri-^ becaufe in L then and GK, meeting nite Lines ; and E and HLK, the Velocities of the points H, which angles and HK, will be now as EL and HL. Let were before as the defcribing points G and K be conceived to move back, again, and C, and before they apwith the fame Velocities, towards

EH ELG

EG

A

H

let g and ^, at any fmall proach to E and diftance from E and H, and draw gk, which will pafs through L ; then ftill their Velocities will be in the ratio of Eg and H/, be thofe Let Lines ever fo little, that is, in the ratio of EL and HL. with the moving points g and k continue to move till they coincide

E

H

them be found in

H

will pafs decreeing Lines Eg and that are lefs and lefs, and will finally polYible magnitudes through become vanishing Lines. For they muft intirely vanifh at the fame moment, when the points g and k mall coincide with E and H. In all which ftates and circumftances they will ftill retain the ratio Let of EL to HL, with which at laft they will finally vaniih. have coincided with after ftill continue to thofe points move, they E and H, and let them be found again at the fame time in y and

and

;

in

which

cafe the

all

Still the Velocities, which are any diftance beyond E and H, and efteemed be and H*, Ey negative, will be as EL may of any finite magnithofe whether Lines and are Hx and HL, Ey if the Line yx.L, by its tude, or are only nafcent Lines ; that is, to be but and divaricate from juft motion, beginning emerge angular EHL. And thus it will be when both thefe motions are equable motions, as alfo when they are alike inequable ; in both which cafes the common interfedlion of all the Lines EHL, GKL, gkL, &c. But when either or both thefe motions -will be the fixt point L.

K, at

now

as

are fappos'd to be inequable motions, or to be any how continually accelerated or retarded, thefe Symptoms will be fomething different for then the point L, which will ftill be the common interfeclion ;

of thofe Lines

when they

firft

begin to coincide, or to divaricate,

no longer be a fixt but a moveable point, and an account muft be had of its motion. For this purpofe we may have recourfe to will

the following

AB

Lemma.

be an indefinite and

fixt right Line, along which anothe: moveable right Line DE may be conceived to move or roll in fuch a manner, as to have both a progreflive motion, as alfo aa That is, the common angular motion about a moveable Center C. interfection C of the two Lines AB and DE may be fuppofed to move with any progreffive motion from A towards B, while at the fame 2

Let

indefinite but

Mm


The Method of

26S

FLUXIONS,

DE

tame time the moveable Line revolves about the lame point C, with any angular motion. Then as the Angle continually decreafes, and at two Lines laft vanifhes when the and

ACD

ACB

DCE

yet even then the point of interfection it C, (as may be ftill call'd,) will not be loft and annihilated, but will appear again, as foon as the Lines begin to divaricate, or to feparate from each That is, if C be the point of interfeclion other. before the coincidence, and c the point of interfeccoincide

;

when the Line dee {hall of AB out there will be fome inter; again emerge mediate point L, in which C and c were united in the fame point, at the moment of coincidence. This tion after the coincidence,

point, for diftin&ion-fake, may be or the point of no divarication. to inequable Motions :

call'd

Now

the Node,

to apply this

AB

be defcribed by the continually accelerated motion of a point, which is now in E, and will be prefently found inr G. Alfo let the Line CD, parallel to the former, be defcribed

Let the Line

the equable

by

mo-

tion of a point, is found and K, at the fame times as

which

H

in

the is

in

other point

E

and G.

Then willEG and

HK be

contem-

poraneous Lines ; and producing

and GK till they meet in I, thofe contempo-

EH

raneous Lines will be as El and and be conceived to points each with the fame degrees of

HI

Let the defcribing and C, again towards in of their moVelocity, every point tion, as they had before acquired ; and let them arrive at the fame time at g and k, at fome fmall diftance from E and H, and draw Then Eg and Hk, being contemporary Lines gki meeting EH in /. and little alfo, very by fuppofuion, they will be nearly as the Ve-

G

K

refpedlively.

move back

A

locities


and INFINITE SERIES,

H

E

269

and which contemporary Let the points g and k continue their motion till they coincide with E and H, or let the Line GKI or gki continue its progremve and angular motion in this manner, till it coincides with EHL, and let L be the Node, or point of no

g and

at

locities

Lines will be

k, that

is,

at

as E/'

and

H/'.

now

divarication, as in the

foregoing

Lemma.

;

Then

will the laft ratio

of the vanifhing Lines Eg- and lik, which is the ratio of the Velo' cities at E and H, be as EL and refpe&ively. Hence we have this Corollary. If the point E (in the foregoing be fuppos'd to move from A towards B, with a Velocity figure,) moves from how accelerated, and at the fame time the point any if an with D C towards equable Velocity, (or inequable, you pleafe ) will be refpectively as the Lines EL and thofe Velocities in E and

HL

H

;

H

HL, which

EG

Lines

L

to be found, by fuppofmg the contemporary continually to dkninim, and finally to vanim. to move with a the moveable indefinite Line

point

and

is

HK

Or by fuppofmg

GKI

EG and and angular motion, fhall always be contemporary Lines, till at laft GKI mall coincide with the Line EHL, at which time it will determine the Node L, or the point of no divarication. So that if the Lines AE and in

progreffive

fuch manner, as that

HK

CH

fcription ratio of

And low cus

and HL.

hence

it

^

will fol-

alfo, that the Loof the moveable

point or is,

E

at

EL

Fluents, any how related, their Velocities of deand H, or their refpe&ive Fluxions, will be in the

two

reprefent

of

all

Node

L-,

that

IH

C

the points of

T>

no divarication, will be fome Curve-line L/, to which the Lines EHL and

GK/

always be Tangents in L and /. And the nature of this Curve L/ may be deterwill

mined by

the given relation of the Fluents or Lines

however the

relation of

be determined in

its

all cafes

;

AE

and

CH

and vice versa. Or intercepted Tangents EL and HL may that is, the ratio of the Fluxions of the ;

given Fluents.

For


tte Method of FLUXLONS,

270

Make the us apply this to an Example. x, and let the relation of thefe be always x". Make the contemporary Lines this Equation y exprefs'd by are contempoand Y and HKs=X.; and becaufe and by fuppofition, we fhall have the whole Lines For

let

illuftration-fake,

Fluents AE= y and CH ;=

=

EG

=

CH

AE

AG

rary

contemporary alfo, and thence the Equation y -f-Y= by our Author's Binomial Theorem will produce y -nx"~ X -+- n x"-^-x*~*X* , &c. which ( becaufe y

come

y

::

-+-

=

1

Y= i

I

nx n ~

:

lation of the

x

x"- X-Jl

+

1

^-^Ar'-^X

,

&c. or

x ^-^""^X, &c. which

in

CK

x -j-X

Y

x"

)

= |

This

.

x"

will

an Analogy,

+ be-

X

:

will be the general re-

EG

contemporary Lines or Increments

and HK.

Now

us fuppofe the indefinite Line GKI, which limits thefe contemand angular motion, porary Lines, to return back by a progrefiive and HK, and fo as always to intercept contemporary Lines

let

EG

and by that means to determine the we may fuppofe EG Y and X, to dibecome which and to in cafe minifli hi i-nfinitum, vanifhing Lines, i nx"~ . But then it will be like wife X we fhall have X Y EG :: HL EL x y, or i nx"~' x :y, ory=nxx'. Y And hence we may have an expedient for exhibiting Fluxions and Fluents Geometrically and Mechanically, in all circumftances, fo as to make them the objects of Senfe and ocular Demonftration. Thus in the laft figure, let the two parallel lines AB and CD be defcribed by the motion of two points E and H, of which E moves may be fuppos'd to move any how inequably, and (if you pleafe) and K correfpond to equably and uniformly ; and let the points E and G. Alfo let the relation of the Fluents AE =r y and

coincide finally to Node L; that is,

with

EHL,

=

HK =

l

:

: :

HK

:

: :

:

: :

:

:

:

:

:

:

H

H

CH

=x

be defined by any Equation whatever. Suppofe now the them to carry along with the indefinite defcribing points E and Line EHL, in all their motion, by which means the point or Node L will defcribe fome Curve L/, to which EL will always be a TanOr fuppofe to be the Edge of a Ruler, of an ingent in L. which moves with a progreffive and angular modefinite length,

H

EHL

combined together the moveable point or Node L in this will have the leaft angular motion, and which is which Line, always of no divarication, will defcribe the Curve, and the Line the point tion thus

;

be a Tangent to it in L. Then will the fegbe proportional to the Velocity of the points or will exhibit the ratio of the Fluxions refpeclively y to the Fluents and CF x. a-nd x, belonging

or

Edge

ments E and

itfelf will

EL

H

and

HL

;

AE=y i

=

Or


and INFINITE SERIES.

271

fuppofe the Curve L/to be given, or already conftmcled, we may conceive the indefinite Line EHIL to revolve or roll about it, and by continually applying itfelf to it, as a Tangent, to move be the and from the fituation EHIL to GK.ll. Then will

Or

if

we

AE

CH

H will HL will

of the defcribing points E and Fluents, be their Fluxions, and the intercepted Tangents EL and be the redlilinear meafures of thofe Fluxions or Velocities. the fenfible velocities

Or

it

may be reprefented thus If L/ be any rigid obftacle in form of a Curve, about which a flexible Line, or Thread, is conceived to be wound, part of which is ftretch'd out into a right Line LE, which will therefore touch the Curve in L ; if the Thread be conceived to be farther wound about the Curve, till it comes into the fituation this motion it will exhibit, even to the Eye, the fame L/KG ; :

by

of increafe, or their increafing Fluents as before, their Velocities of thofe Fluxions, as alfo the Tangents or rectilinear reprefentatives the done be fame Thread, And the Fluxions. by unwinding may we Thread of the inftead Or in the manner of an Evolute. may

by applying its Edge continually to the curved Obftacle L/, and making it any how revolve about the moveIn all which manners the Fluents, able point of Contadl L or /. Fluxions, and their rectilinear meafures, will be fenfibly and mechamuft be allowed to have a place nically exhibited, and therefore they And if they are in nature, even tho' they were but in rernm naturd. and conceiveable, much more if they are fenfible barely pofiible and vifible, it is the province of the Mathematicks, by fome method or other, to investigate and determine their properties and pro-

make

ufe

of a Ruler,

portions.

by one Thread EHL, perpetually winding about the curved obftacle L/, of a due figure, we mall fee the Fluents AE and CH at any rate aflign'd, by the mocontinually to increafe or decreafe, tion of the Thread EHL either backwards or forwards ; and as we (hall thereby fee the comparative Velocities of the points E and H, that is, the Fluxions of the Fluents AE and CH, and alfo the Lines EL and HL, whofe variable ratio is always the rectilinear meafure of So by the help of another Thread GK/L, windthofe Fluxions obftacle in its part /L, and then ftretching out into a the about ing or Tangent /KG, and made to move backwards or forright Line wards, as before ; if the firft Thread be at reft in any given fituation EHL, we may fee the fecond Thread defcribe the contempoIncrements EG and HK, by which the Fluents porary Lines or AE and CH are continually increafed ; and if GK/ is made to ap-

Or

as

:

proach


^

272

Method of FLUXIONS,

e

EHL, we may

contemporary Lines contiapproaching towards and continuing the motion, we may pretwo Lines actually to coincide, or to unite as one fently fee thofe then we may fee the contemporary Lines actually to vaand Line, the fame at ntfh time, and their ultimate ratio actually to become And if the motion be ftill continued, we mall that of EL to HL. fee the Line GK/ to emerge again out of EHL, and begin to defcribe other contemporary Lines, whofe nafcent proportion will be And fo we may go on till the Fluents are exthat of EL to HL. proach towards

imallv to diminim, and the ratio of EL to HL

of

;

All thefe particulars

haufted.

fee thofe

their ratio continually

may

be thus eafily

made the

objects

fight, or of Ocular Demonftration. This may ftill be added, that as we have here exhibited and re-

firft Fluxions geometrically and mechanically, we may do prefented the fame thing, mutatis mutandis, by any higher orders of Fluxions. Thus if we conceive a fecond figure, in which the Fluential Lines fhall

of the ratio of the intercepted Tangents (or the the firft Fluxions) of figure ; then its intercepted Tangents will expound the ratio of the fecond Fluxions of the Fluents in the firft increafe after the rate

Alfo if we conceive a third figure, in which the Fluential figure. Lines fhall increafe after the rate of the intercepted Tangents of the fecond figure ; then its intercepted Tangents will expound the And fo on as far third Fluxions of the Fluents in the firft figure. This is a neceflary confequence from the relative naas we pleafe. ture of thefe feveral orders of Fluxions, which has been fhewn before.

And

farther to

mew

the univerfality of this Speculation, and

how

accommodated to explain and reprefent all the circumftances of Fluxions and Fluents; we may here take notice, that it may be alfo adapted to thofe cafes, in which there are more than two Fluents, which have a mutual relation to each other, exprefs'd by one or more Equations. For we need but introduce a third parallel well

it is

any how moving, and that any two of thefe defcribing points carry an indefinite Line along with them, which by revolving as a Tangent, defcribes the Curve whofe Tangents every where determine the Fluxions. As alfo that any other two of thofe three points are connected by an-

Line, and fuppofc

it

to be defcribed

other indefinite Line, another fuch Curve.

by

a third point

which by revolving

And

in

like

manner

defcribes

may be four or more parallel All but one of thefe Curves may be affumed at pleafure, Lines. when they are not given by the ftate of the Queftion. Or Analyfo there

tically,


'

///. j //'//<'. i

,

i( //.

uvuutm

/v

/(v Yfa/?////

2-3.


and INFINITE SERIES. tically,

fo

many

may

Equations

as given by the Problem,)

the

is

273

except one, (if not the Fluents concern'd.

be aflumed,

number of

may not be difficult to give a pretty good notion of Fluents and Fluxions, even to fuch Perlbns as are not much verfed in Mathematical Speculations, if they are willing to be This iniorm'd, and have but a tolerable readinefs of apprehenfion. But

I

laftly,

believe

it

here attempt to perform, of a Fowler, who is aiming to I {hall

way, by the inftance two Birds at once, as is re-

in a familiar (lioot

Let us fuppofe the right Line AB prefented in the Frontifpiece. to or be the level with the Ground, in which to Horizon, parallel a Bird is now flying at G, which was lately at F, and a little be-

And let this Bird be conceived to fly, not with an equable or uniform fwiftnefs, but with a fwiftnefs that always increafes, (or with a Velocity that is continually accelerated,) according to fome known rate. Let there alfo be another right Line CD, parallel to fore at E.

the former, at the fame or any other convenient diftance from the Ground, in which another Bird is now flying at K, which was lately at I, and a little before at ; juft at the fame points of time as the

H

Bird was at G, F, E, refpectively. But to fix our Ideas, and to make our Conceptions the more fimple and eafy, let us imagine this fecond Bird to fly equably, or always to defcribe equal parts of the Line in equal times. Then may the equable Velocity of firft

CD

Bird be ufed as a known meafure, or ftandard, to which we may always compare the inequable Velocity of the firft Bird. Let us now fuppofe the right Line to be drawn, and continued to the point L, fo that the proportion (or ratio) of the two Lines EL and be the fame as that of the Velocities of the two may

this

EH

HL

E

H

and Birds, when they were at refpeclively. ther fuppole, that the Eye of a Fowler was at the

And

let

us far-

fame time at the and he that directed his or L, Gun, point Fowling-piece, according to the right Line LHE, in hopes to moot both the Birds at once. But not thinking himfelf then to be fufficiently near, he forbears to difcharge his Piece, but ftill pointing it at the two Birds, he advances towards them continually according to the direction of his Piece, till his Eye is prefently at M, and the Birds at the fame time in F and I, in the fame right Line FIM. And not being yet near enough, we may fuppofe him to advance farther in the fame manner, his Piece being always directed or level'd at the two Birds, while he himfelf walks forward according to the direction of his till his his

Eye Eye,

is

now

at

K

at

Piece, the Birds in the fame with Line right The Path of his Eye, delcribed by this

N, and

and G.

N

a

double


The Method of

274

FLUXIONS,

double motion, (or compounded of a progreffive and angular mobe ibme Curve-line LMN, in the fame Plain as the reft tion,) will of the figure, which will have this property, that the proportion of the diftances of his Eye from each Bird, will be the fame every where as that of their refpeftive Velocities. That is, when his Eye

L, and the Birds

was

at

EL

and

at

E

and H,

their Velocities

were then

as

HL, by the Conftruftion. And when his Eye was at M, and the Birds at F and I, their Velocities were in the fame proporand IM, by the nature of the Curve LMN". tion as the Lines

FM

And when cities

are

his

in

Eye

is

at

N, and

the proportion of

G

and K, their Veloto KN, by the nature of the of all other fituations. So that be the fenfible meafure of always

the Birds at

GN

fame Curve. And fo univerfally, the Ratio of thofe two Lines will Now if thefe Velocities, the ratio of thofe two fenfible Velocities. or the fwiftneffes of the flight of the two Birds in this inflance, are call'd Fluxions; then the Lines defcribed by the Birds in the fame time, may be call'd their contemporaneous Fluents; and all inftances whatever of Fluents and Fluxions, may be reduced to this Example, and may be illuflrated by it. And thus I would endeavour to give fome notion of Fluents and Fluxions, to Perfons not much converfant in the Mathematicks j but fuch as had acquired fome fkill in thefe Sciences, I would thus to inflrudl, and to apply what has been now deliver'd. proceed farther The contemporaneous Fluents being EF=_y, and Hl=.v, and their rate of flowing or increafing. being fuppos'd to be given or known their relation may always be exprefs'd by an Equation, which will be compos'd of the variable quantities x andjy, together with any known quantities. And that Equation will have this probecaufe of thofe variable quantities, that as FG and IK, EG perty, and HK, and infinite others, are alfo contemporaneous Fluents; it will indifferently exhibit the relation of thofe Lines alfo, as well as of EF and HI ; or they may be fubflituted in the Equation, inftead of x and y. And hence we may derive a Method for determining the Velocities themfelves, or for finding Lines proportional to them. X ; in the given Equation I may For making FG =Y,.and IK fubftitute y -}- Y inftead of ^y, and x -f- X inftead of x, by which I fhall obtain an Equation, which in all circumftances will exhibit ;

=

Now it may be plainly the relation of thofe Quantities or Increments. if the Line MIF is conceived continually to approach perceived, that nearer and nearer to the Line NKG, (as jufl now, in the inftance of the Fowler,)

till it

finally coincides

with

it;

the Lines

FG

=

Y, and


and INFINITE SERIES. and IK

275

=

X, will continually decreafe, and by decreafing will apand K, and nearer to the Ratio of the Velocities at nearer proach and will finally vanifh at the fame time, and in the proportion of

G

GN

to KN. thofc Velocities, that is, in the Ratio of Confequently in the Equation now form'd, if we fuppofe to decreafe and and at laft to vanifh, that we obtain their ultimate continually, may

X

Y

GN

mail thereby obtain the Ratio of to KN. But when and vanifh, or when the point F coincides with G, and I with .x'; fo that we fhall have H, then it will be y, and : x :: And hence we mall KN. obtain a Fluxional Equay tion, which will always exhibit the relation of the Fluxions, or Ve-

Ratio

we

;

X

Y

EG

GM for

Thus,

Lines y and

may

y*

o

= Y

;

and x

(fee before,

aX

= IK =

if EF=j', and HI x, and the indefinite are fuppofed to increafe at fuch a rate, as that their

1

-f-

for

we

pag, 255.)

axY

-\-

aXj

which may be thus exprefs'd 1 aX .+. ay +- 3tfX -h X Analogy, when

Y

X

and

Ratio, will become becaufe it is then

Y

Y

:

X

this

FG=Y,

X

-+-

1 Equation x

ax* + axy X, by fubftituting x, and reducing the Equation that will

always be exprefs'd by then making

for j,

zaxX

=

Example,

A:

relation

arife,

HK

belonging to the given Algebraical or Fluential Equation.

locities,

y -f-

=

:

have ^x"-X

fhall

-+-

rfXY

in

an Analogy,

^

:

and

-f-

3y*Y

Y

Z

X

-f-

Y X :: 3** 37 Y -+- Y*. 3jyY*

:

aX -+-

ax

3#X

!

3

-

=

o,

2 ax

This

are vanishing quantities, or their ultimate And ax. ^ax -f- ay 3** 3^*

X

:

:

:

GN

KN

x :: x, it will be y the of the Which gives ax. -+- ay 3X 3^* proportion And the like in all other cafes. Q^. E. I. Fluxions. might alfo lay a foundation for thefe Speculations in the fol1

zax

:

::

:

::

v

:

:

:

We

lowing manner.

Let

ABCDEF,

6cc. be the of a Polygon, Periphery or any part of it, and let

the Sides

AB, BC,

CD, DE, &c. be magnitude

of any

whatever.

In the fame Plane, and at

any

diftance,

draw

the two parallel Lines /6, and bf\ to which continue the right Lines

BCcy, DEes, &c. meeting the AB4/3,

parallels as in the figure,

N

n 2

Now

if

we

fuppofe


7%e Method of FLUXIONS,

276

or bodies, to be at $ and b, and to move pofe two moving points, in the fame time to y and c, with any equable Velocities ; thofe Velocities will be to each other as @y and be, that is, becaufe of the as /3B and bE. Let them fet out again from y and c, parallels,

and

arrive at the

fame time

at

^ and

thole Velocities will be as yfr

them depart again from e, with any equable

and and

de,

that

is,

as

and

how many Polygon may be.

d,

with any equable Velocities ; Let is, as yC and cC. and arrive in the fame time at g

foever,

d,

cd,

Velocities

;

And and how

J^D and dD.

where,

and

that

thofe Velocities will be as it

will be the

S-t

fame thing every

fmall foever, the Sides of the

Let their number be increafed, and their magnitude be diminim'd in infinitum, and then the Periphery of the Polygon will continually approach towards a Curve-line, to which the Lines AB^/3, ECcy, CDd, &c. will become Tangents as alfo the -,

Motions may be conceived

to degenerate into fuch as are accelerated Then in any two points, fuppofe and d,

or retarded continually. where the defcribing points are found at the fame time, their Velocities (or Fluxions) will be as the Segments of the refpeclive Tan-

dD

and the Lines /3^ and bd, intercepted by any two Tangents J>D and /SB, will be the contemporaneous Lines, or Fluents. Now from the nature of the Curve being given, or from gents cTD and

;

the property of its Tangents, the contemporaneous Lines may be And vice versa, from the found, or the relation of the Fluents. Rate of flowing being given, the correfponding Curve may be found.

ANNO-


and INFINITE SERIES.

277

ANNOTATIONS on Prob.iO

The

Relation of the Fluxions being to 7 & o given, find the Relation of the Fluents.

SECT. the

R,

I.

A particular Solution

general Solution,

with a preparation for

;

by 'which

it is

diftribitted

into-

three Cafes.

E

now come

of the Author's fecond fundamental Problem, borrow'd from the Science of Rational Mechanicks Which is, from the Veloare

to the Solution

:

of the Motion at

all times given, to find the defcribed or find to the of the Fluents from the ; Spaces quantities In difcuffing which important Problem, there will given Fluxions. And firft it may be occafion to expatiate fome thing more at large. not be amifs to take notice, that in the Science of Computation all the Operations are of two kinds, either Compolitive or Refolutative.

cities

The Compolitive rectly, in

way of

or Synthetic Operations proceed neceffarily and dicomputing their feveral qit(?fita> and not tentatively or by

tryal.

Such

are Addition, Multiplication, Railing of Powers, But the Refolutative or Analytical

and taking of Fluxions.

Opera-

tions, as Subtraction, Divifion, Extraction of Roots, and finding of Fluents, are forced to proceed indirectly and tentatively, by long or deductions, to arrive at their feveral qutefita ; and

fuppofe require the contrary Synthetic Operations, to prove and confirm every llep The Compofitive Operations, always when the of the Procefs. data are finite and terminated, and often when they are interminate i

or


The Method of FLUXIONS, or infinite, will produce finite conclufions ; whereas very often in the Refolutative Operations, tho' the data are in finite Terms, yet the quafita cannot be obtain'd without an infinite Series of Terms. Of this we mall fee frequent Inftances in the fubfequent Operation, of returning to the Fluents from the Fluxions given.

Author's particular Solution of this Problem extends to fuch <afes only, wherein the Fluxional Equation propofed either has been, or at leafl might have been, derived from fome finite Algebraical

The

the necefTary Terms Equation, which is now required. Here be are neceflary, it will not being prefent, and no more than what difficult, by a Procefs juft contrary to the former, to return back But it will moft commonly happen, again to the original Equation, either if we aflume a Fluxional Equation at pleafure, or if we arrive at one as the refult of fome Calculation, that fuch an Equation is to be refolved, as could not be derived from any previous finite Alredundant or defigebraical Equation, but will have Terms either all

and confequently the Algebraic Equation required, or its be had by Approximation only, or by an infinite Series. mufl Root, cafes we mult have recourfe to the general Solution of all which In

cient

this

;

Problem, which

we

fhall find afterwards.

The

(i.) All fuch Precepts for this particular Solution are thefe. Terms of the given Equation as are multiply 'd (fuppofe) by x, muft be difpofed according to the Powers of x, or muft be made a

Num-

ber belonging to the Arithmetical Scale

whofe Root

is

x.

(2.)

Then

they muft be divided by A-, and multiply'd by x ; or x muft be changed into A', by expunging the point. (3.) And laftly, the Terms muft be feverally divided by the Progreilion of the Indices of the Powers of x, or by fome other Arithmetical ProgrerTion, as need mail require. And the fame things muft be repeated for every one of the flowing quantities in the given Equation. zaxx -f- axy Thus in the Equation $xx^yy- -f- ajx -Q^ 1 zaxx -\-axy by expunging the points become the Terms -^xx zax*- -+- axy, which divided by the Progreffion of the Indi^x'' .

a 3.X)'

which

ax*

Alfo the Terms -+- axy. the become ayx by expunging points 3j * -f- ayx, divided by the Progreffion of the Indices 3, 2, i,

ces 3, 2,

I,

reflectively,

will give

A'

5

* -+-

will give

redundant required.

3

refpectively,

y> * -+- ayx.

The

Term ayx, is x* Where it muft be

more than once, mult

aggregate of thefe,

ax*

-\-

axy

_}"

=

neglecting the o, the Equation

noted, that every Term, which occurs be accounted a redundant Term.

So


and INFINITE SERIES. So ;//

propoied Equation were

if the

m

-f-

279 m-\- 2(jyxx

^yxx*

1

-f-

4 -v

ay* xx

n-\- $xyy> -\-n-\- lax^yy -+- nx+y nax>y m} values n whatever the and m Numbers o, general may acquire ; thofe Terms in which x is found are reduced to the Scale whofe

-+-

i

=. if

Root

is

they will ftand thus

x,

:

m

-+-

m

yyxx'

1

-+-

zayx*.

-+

my**-, or expunging the points they will become m -f- Ziivx* +- m -+- \ay-x 1 Thefe being dim -+- %yx+ m\*x. vided refpedtively by the Arithmetical Progreffion m -f- 3, m-\-2,.

m-\-\ay*xx

1 Alio m, will give the Terms yx+ y+x. ayx -f- ay'-x whofc Scale the Terms in which y is found reduced to the being 1

m-\-

i,

Root

isy, will ftand thus

;

n

:

-4-

^xyy* * +- n -+ iax*jy-{- nx*y; nax"=y

^

or expunging the points they will become n -\~^ x * -^~ n ~^~ iaxi)" Thefe being divided reipeclively by the Arithmetical Pro+- nx+y. grefTion ^-{-3, 1

ax*}' -)-

mull

all

x+y ax*y. be confider'd

So that yx* Equation x*

Thus

ayx*

?z-|-i, thefe

will

give the

Terms

xy* -\-

as

-f-

Terms, being the fame as the former, redundant, and therefore are to be rejected. i

ay

xi

ax -\-ayx

-+- ;z-f-

;;,

But

y^x=o

1

we had

if

nx*yy~* it

?i-{- 2,

y*

=

this Fluxional

\ay

=. o,

or dividing by yx, the

)

o will

arife as before.

Equation mayxx~

to find the Fluential

m -+- 2xx

l

Equation to which

the Terms mayxx~ * m -f- 2xx, by expunging the and dividing by the Terms of the Progreffion m, m-\- 1, w-t-2, I

belongs

;

points, will give the

Terms ay

x*.

Alfo the

Terms

nx^yf

1

-+-

n-\-iay,

by expunging the points, and dividing by n, n-\- i, will give the x 1 -f- ay. Now as thefe are the fame as the former, they Terms are to be efteem'd as redundant, and the Equation required will be o. And when the given Fluxional Equation is a genex1 ay and ral one, adapted to all the forms of the Fluential Equation, as cafe of the two laft Examples is the then all the Terms ariling from the fecond Operation will be always redundant, fo that it will be fufficient to make only one Operation. Thus if the given Equation were ^.yy 1 -f- z 3yy~ J -f- 2yxx 3:32* H- 6}'z.z o, in which there are found three flowing quan2cyz tities j the only Term in which x is found is 2yxx, in which exthe and then point, punging dividing by the Index 2, it will be1 Then the Terms in which y is found are 4^*4- z*yy~~ t come^* which expunging the points become # * 4-s 3 , and dividing

=

;

=

l

.

^

by,


72k Method of

280

FLUXIONS,

s i, give the Terms aj Laftly by the Progreffion 2, i, o, the Terms in which z is found are zcyz, which yzz* -J- 6yzz 32;"' -f- 6yz* 29-2, and dividing expunging the points become ;

5

by the Progreffion

Now we

if

we

3,

2,

collect thefe

mall have

yx

z

-+- 2y>

i,

give

the

.

&+

Terms

Terms, and omit the redundant z"'

-f-

yz

1

2cyz

=

T.yz

1

zcyz.

Term

z*,

o for the Equa-

tion required. 3, 4. But thefe deductions are not to be too

they are verify 'd by a proof; and

till

we

much rely'd upon, have here a fure method

of proof, whether we have proceeded rightly or not, in returning from the relation of the Fluxions to the relation of the Fluents. For every refolutative Operation mould be proved by its contrary comSo if the Fluxional Equation xx pofitive Operation. <xy xy-\o were given, to return to the Equation involving the Fluents ny'= by the foregoing Rule we fliall firft have the Terms xx xy, which will the become x* and .vy, by expunging points dividing by the 1 will the Terms the Terms, or Alfo ^x Progreffion 2, i, give xy. ;

xy -+- ay, by expunging the points will become xy. by Unity. So that leaving out the l redundant Term xy, we fhall have the Fluential Equation x xy Now if we take the Fluxions of this Equation, we -+- ay == o. iliall find by the which foregoing Problem xx xy xy -+- ay fame as the we the are to conclude our work is Equation given, being But if either of the Fluxional Equations xx true. xy -f- ay or xx o had been propofed, tho' by purfuing the xy -f- ay foregoing method we fhould arrive at the Equation x* xy-\-ay o, for the relation of the Fluents ; yet as this conclulion would not fland the teft of this proof, we muft reject it as erroneous, and have recourfe to the following general Method ; which will give the value of y in either of thofe Equations by an infinite Series, and therefore for ufe and practice will be the moil commodious Sorather

-+- ay,

Term, which

are only to be divided

=o,

=o,

=

=

lution. 5.

As

Velocities can

be compared only with Velocities,

and

all

other quantities with others of the fame Species only ; therefore in every Term of an Equation, the Fluxions muft always afcend to the

lame number of Dimenfions, that the homogeneity may not be deWhenever it happens otherwife, 'tis becaufe fome Fluxion; ftroy'd. taken for Unity, is there underftood, and therefore muft be fupply'd

when

making

z=i,

vice versa.

And

=

The

az'-x* o, by Equation xz -+- xyx ax*==o and like wile may become -x -f- xyx as this Equation virtually involves three variable

occafion requires.

>

quantities,


and INFINITE SERIES. it

quantities,

will

another

require

yx=yy;

this

or

as has been already ob-

=

manner

Fluential

either

Equation,

Fluxionai, for a compleat determination, So as the Equation yx ferved. xyy, by putting x in like

281

=

i becomes and fuppofes the Equation requires

other.

Here we

are taught fome ufeful Reductions, in As when the Equation Equation for Solution. their with Fluxions, the ratio flowing Quantities

6, 7, 8, 9, 10, II. order to prepare the

contains only two of the Fluxions may always be reduced to fimple Algebraic Terms. The Antecedent of the Ratio, or its Fluent, will be the quantity to

be extracted ; and the Confequent, for the greater fimplicity, may o is be made Unity. Thus the Equation zx +- 2xx yx y y'tis 2 2 -+ 2X reduced to this, y y, or making

=

_^_ 2 x

king _!_

2L

So the Equation ya

y.

x= _j_

i,

f!?

will _|_

^

ticular Solution to

__ ##

=

_{_ tfy

=

o,

become y

=

xa -f-

yx a (

~^+

= = x=i, = maxx = jdb = +

y

i

xy

-f-

o,

)

i

But we may apply the par1 this Example, by which we mail have {x xy and thence y =."- ^~Thus the Equation ,

&c. by Divilion.

.

=y

x=i,

becomes yy -+- xx, and ex-i ~ the Series 'tis \/ -j- xx y tracting the fquare-root, 8 6 1 &c. either i -jthat x*-{-2X is, 2.-4-X y 5X -f- I4x', 8 10 1 6 4 * 4 x -f- x zx 6 x -{-2x x jx -f- I4* , &c. or y 1 1 10 &c. 8 , Again, the Equation y> -+-axx*y-{-a x y _j_ rx I4-.V 3 ? 3 x 2x'tf i, becomes _y -\-axy -\-ay X 3x putting 3 affected an Cubic of form this has o. Now 2<7 Equation a been refolved before, (pag. 1 2.) by which we mail have y ^x + ? 4 xx ^9^ c, iji* " ' ~*~ uz" 1 16384^5 6^ of the fake 12. For perfpicuity, and to fix the Imagination, our Author here introduces a diftinction of Fluents and Fluxions into The Correlate is that flowing Quantity which Relate and Correlate. he fuppofes to flow equably, which is given, or may be arTumed, at any point of time, as the known meafure or ftandard, to which

yjr

xy-{-xxxx, making

=

=

=

=

i

x=

=o,

=

'

It the Relate Quantity may be always compared. may therefore and or denote Time its ; Fluxion, Velocity being an very properly uniform and conftant quantity, may be made the Fluxionai Unit, or the known meafure of the Fluxion (or of the rate of flowing) of

the Relate Quantity.

The

Relate Quantity, (or Quantities if ieve-

O

o

ral


The Method of

282

FLUXIONS,

concern'd,) is that which is fuppos'd to flow inequably, with; of acceleration or retardation ; and ts any degrees inequability may combe meafured, or reduced as it were to equability, by

ral are

conihntly This correfponding Correlate or equable Quantity. is the to be therefore found by the Proble'm, or whofe Quantity Root is to be extracted from the given Equation. And it may be conceived as a Space defcribed by the inequable Velocity of a Body or Point in motion, while the equable Quantity, or the Correlate, or meaiures the time of This may be illureprefents defcription. ftrated by our common Mathematical Tables, of Logarithms, Sines, In the Table of Logarithms, for Tangents, Secants, &c. inflance, the Numbers are the Correlate Quantity, as proceeding equably, or while their as a Relate differences, equal by Logarithms, Quantity, And this refemblance proceed inequably and by unequal differences. would more nearly obtain, if w e mould fuppofe infinite other Numbers and their Logarithms to be interpolated, (if that infinite Numparing

it

with

its

r

ber be every where the fame,) fo as that in a manner they may become continuous. So the Arches or Angles be confider'd as

may

the Correlate Quantity, becaule they proceed by equal differences, while the Sines, Tangents, Secants, &c. are as fo many Relate Quantities, whofe rate of increafe is exhibited by the Tables. 13, 14, 15, 16, 17. This Diflribution of Equations into Orders, or Gaffes, according to the number of the flowing Quantities and their Fluxions, tho' it be not of abfolute for the Solution, neceflity

make it more expedite and methodical, and yet ferve to convenient with us places to reft at. fupply

may

SECT.

II

.

may

Solution of the Jirft Cafe of Equations.

18, 19, 20, 21, 22, 23.

r

~|~^HE -i.

the

firft

Cafe of Equations

Quantity

?

or

,

is,

what

wherr

fupplies

te found in Terms compofed of the Powers place, can always of x, and known Quantities or Numbers.. Thefc Terms are to be multiply'd by x, and to be divided by the Index of .v in each Term,, which will then exhibit the Value of jr. Thus in the Lquationj/ a .xi/

its

l

-+-

x x*, 10

it

has been found that

~

=

=

i

-t-x*

-

x*

Therefore =^-4- x* x'-f- 2,v I4* , &cc. x jX* &c. and confequently y -fx -f- ^x" be as &c. may ealily proved by the direct Method.

=

-f-

2x

s

7

+

1

5^'

1

9

J-x

$x* -f-t-

14*'*,

-j-l^A"

3

,

But


and INFINITE SERIES. But

and the

this,

by a Method form'd after

like Equations, may be refolved more readily in imitation of fome of the foregoing Analyfes,

In the given Equation

manner.

this

=j/-l-.v*, which

will bej)*

H=

X*

thus refolved

is

-+-

J $

4

y y*-

283

X* .V 4

2X S 2X &

-f-

=

make x

i

;

then

it

:

9 &C. pC 9 5X , &C.

-f-

,

of y ; then will x 4 be the firft Term of which is to be put with a contrary Sign for the fecond j/ Term of y. Then by fquaring, -f- 2X 6 will be the fecond Term of 2x* will be the third Term of y. Therefore j/, and 8 will be the third Term of and -f- 5*" will be the

Make

the

AT*

firft

Term

1

,

5#

=

j/,

Term

and fo on. Therefore taking the Fluents, y ix 7 -f-4-x, &c. which will be one Root of the I..V -+-fx* x -+And if we fubtradt this from x, we (hall have y Equation. 7 &e. for the x other -i-A; Root. AX', ^.v* -f-

fourth

of y

;

= ^ = ax + ^,

5

3

So

v

if -

f

=

a

04^

c

&c

'

6?

>

&c

that

-

A

yx

-^yii:

&c

ya

==f c

then

_y=^ v

Laftly,

dex

o,

it

if

ing quantity,

#

I?IA.'4

.

&c.

If 4

*.

*

s .

=

-,

=

^--**. x

or

,

c

ex?

f.

=

will be

or value of y,

or

'-

j. v

then Y

or

if

^

jj-T i

is,

A'

v=^->^4-

then

li

*

2*

I

5

, "

-^

h

r

4-r -4-

~,

=

y mufl be

or

lefs

or

'

-

= = ^

^v

;

dividing

by the In-

a

-

,

or

y

infinite,

is

is

infinite.

very plain.

That For

than any affignable quantity,

muft be bigger than any

affignable

finite.

O

o 2

this

as

o

its

quantity,

Expreffion, is

a vanim-

Reciprocal that

is,

-

in-


The Method of

284

Now and

in

CE

fcribe the

and

its

let

4

=

-

a point

Line

CDE,

x

,

let

AB

infinite,

may

be thus proved.

reprefent the conftant quantity a,

move equably from C towards E, and of Avhich

let

any

indefinite part

equable Velocity in D, (and every where

A c

ought to be

that this quantity

In the Equation

FLUXIONS,

o,

elfe,)

E

is

CD

de-

be x,

reprefented


and INFINITE SERIES, x we may write b +- x, and we r

vx

r

-r-v

i

i

Multiplication and Divifion

~

we

X, and

-^

1-

fhall

-

-J- 3

4

,

jx

X

1

xj

=

may

,

= y= =

y^

T

"-

we may

~X

1

-J-

4.x -f2X"' -{-

4.V

-

2x*

4X

ax

"X 1

-r b

tl

~,

j

6cc.

which

-' ,

inftead of y xx, or X

=

4x*

xj

-f-

|x 4

3

-+- ^.x

that

,

-f.

x

-f-

4x

x1

zx

y

is

-f-

-f-

xy

:

A: J

3*

4X h 4^ 4

=

yx

11

*

/

zx

-4-2

y=.^x

J

Term

write

|

-

\

~

\ -f-

^-

z -^^ +-

4

H- xyj

=

ax :. V*-}-* f

be thus refolved

^"

,

(

or by Divifion

&c. and therefore

s

y the Equation ~

Or

=

-

x"',

&c.

,

x

=

and then by

--

xx, becaufe of the

'

then have

2X 1

2X zx

-fr

=

=

-4-

give an infinite value for ^,

would

^x

if ~

So

2.6.

is

}'=

&c. and therefore

,

-77

it

=

have ~

fhall

285

1

AT 3

4x* 4- x AT* 2x

_ ^X3

2X 4 , &C, 2x 4 6cc. 2x 4 &c. x * t &c

-j-

3

,

3

-f-

iX4

,

_|_

.

Term

of j, then 4x will be the firfl Term of 4* will be the fecond Term of j. Then xy, and confequently a be the fecond Term of .vy, and therefore -(- 4x* will 4x 3x , or x*, will be the third Term of_y ; and fo on.

Make 4

the

firft

fc

27. So

if

change x into the by * -+-

=

-.

x, then

i

foregoing;

x5 ,

6cc.

AT~^ -f-

x~

J

==

X

-.'

-+-

yi v Methods of Reduction

and v/i

x

=

'tis

X

I

I

s/

X

I

,

Therefore collecting thefe according to their Signs,

ix

2.v-|-

1

-t-

T^-x

3 ,

=

&c. that

is-^

=x4-2x*

x 4- x -f- ix 4- ^x and therefore y 28. So if the given Equation were X .

t

'^ A

'

;

a

s

change the beginning of x.

4 ,

-f-

-f-

|x

'tis

3

x

-+-

a"d

&c.

4-

+

x*

4-

i

^x 4 &c. ,

&c.

==

that

i

-rr x ^

r-^

But

x.

i

=

4

4-^

Term x~'

of the

becaufe

x'~,

i.

is.

*

~^

~ ii^^C -!- 3t"A*

inftead

of

~

~"

x

X%

"

write

|

x,


-

y

--

x, then

c

c*x-

1

,

=

f3

=

c*-X

=

Al

and therefore

SECT. 2 9>

Method of FLUXIONS,

7$

286

_>'

c">x~*

c

l

x-

^c=x-~ -\-c*x~

yx

1

or

,

^

=

c*x~*

.

l .

Solution of the fecond Cafe of Equations.

III.

belonging to this fecond cafe are thofe, wherein the two Fluents and their Fluxions, fuppofe x and y, x and j, or any Powers of them, are promifcuoufly inAs our Author's Analyfes are very intelligible, and fee'm to volved. want but little explication, I mall endeavour to refolve his Examples in fomething an eafier and fimpler manner, than is done here ; by applying to them his own artifice of the Parallelogram, when needful, or the properties of a combined Arithmetical Progreffion in

3- TT^Quations

M^

as explain'd before : As alfo the Solution of afTeclcd Equations.

The Equation yax ~ -+- ".becomes ~ 31.

=

,

Methods before made aax

piano, ufe of, in the

=

o by a due Reduction in which, becaufe of the Term there

xxy

-

occafion for a Tranfmutation, or to change the beginning of the x. therefore the conftant Quantity ArTurning quantity b, we may put 4whence Divifion will be had -f- -^ , by

is

Correlate

y

-j

==

=^ ax

a

v

-I-

-^

y

-+-

ax z

ax*

-ji

77

e &c which i

>

i

'

.

-.-,

,

Equation

is

then

prepared for the Author's Method of Solution. But without this previous Reduction to an infinite Series, and the Reiblution of an infinite Equation confequent thereon, we may The given Equaperform the Solution thus, in a general manner. tion

now 4

is

/y ~+\

aby

xy

yx

=

-f-

= a

1 ,

j-

-|-

,

or putting

x

=

which may be thus refolved

:

i,

it

is

aby

-f-

axj


and INFINITE SERIES. fee Difpoiing the Terms as you ~ will be the Term of aby, then

will be the

firft

ax

and

axv, /*

down

Term

fecond

abx

Thefe

-^~

three together

of

by.

Term

firft

Thefe two

of to-

with a contrary Sign, mud be put of aby. Therefore the fecond Term of x,

like

Term '-'-~ b"

and the

*,

will be the

/

-

and the

firft

of j, and thence -|x

x

b

Term

firfl

i

of ..\y will be

ly will be

Term

firft

So that

*y.

Term

for the fecond

will be '-~-x, t>-

'y

=

ax

or -,x

gether,

of

will be the

i

done here, make a 1 the

is

a

Term

287 -

make

Then

A*. ^b-

a

Term

~

2*

-

A*,

_

xy will be

of

which with

the

Term

and the fecond

A*,

firft

~

of -'y will be

a

of

yA*. contrary

Therefore the third Sign muft be made the third Term of aby. ~-^-'A* and the third Term of y will be Term of ry will be ZaL *

'

t

b

/

,

=.

*

_^_

a,

7

-

we

A'

3

.

And

fo on.

Here

mall have the fimple Series

we would have a defcending Equation, we may proceed as follows Or

[,y\

=<!*

a

---

~f-

a**a*bx~

b x

_f_

1

-+-

y =. x

* -+-

Series for the

if

xy~\

cafe if

a particular

in'

we

^

7

i

x a*x-*, &c. a-i-l) y.a -bx~-, &c,

zu\ +- zab

-+-

!

l

-~t '-,

3

a

A~

;

-

this

:

,

rt

&c.

,

Root y of

~'

JJ/=:

make.

&c. &c. 6cc.

^ -4-/>X 2rt*A-~ 3 , &C.

and make a* the firft Term of the Difpofe the Terms as you fee, - be the firft Term of y, and a*x~"- will Series xy, then will be the firft Term of of axy which together by, l make a-\- b x a x~ ; this therefore with a contrary Sign muft be the fecond Term of Then the fecond Term of y will xy.

be the

firft

Term

and a"'X~

I

of y.

Then

will be

the

will -f- a"-bx~

firft

l

Term

y

t

be a-\-by.a x~~ ) and the fecond 3i

Therefore the fecond

Term

of

Term

be ofj/ will

by will

be

a-\-

a

l>-x.2a

-f- b

1

x~ 3

,

x tf*Av~*,

and


2

TZe Method of

88

and the fecond Term of axy

Term

firft

of aby will

be

FLUXIONS, a-\-b* 2a*x~*, and tlie which three together make

be

will

a^bx~-

;

xa 1 *.*. ThiswithacontrarvSisnmuftbethethird ^za -I2a* -+- zab -+- b % x a 2 x~3 for the Term of xy, which will give Here if we make b=.a, thenj= third Term of y ; and fo on. zab

1

--

a1

-+- b*

ca4

za*

.

&C. +- *'r ^T x* 3 And thefe are all the Series, hibited in this Equation, as x

'

by which the value of y can be exmay be proved by the Parallelogram.

that Method may be extended to thefe Fluxional Equations, as well as to Algebraical or Fluential Equations. To reduce thefe within the we are Limits of that to confider, that Rule, Equations

For

Ax m may

reprefent the initial Term of the Root jr, in both thefe kinds of Equations, or becaufe it may be y Ax m , &c. fo in

as

=

~

Fluxional Equations (making #=1, we mall have a\foy=mAx m I ) 6cc. or writing y for Ax m , 6cc. 'tis myx~* , &c. So that in y the given Equation, in which y occurs, or the Fluxion every Term of

=

t

of the Relate Quantity, we may conceive it to take away one Dimenfion from the Correlate Quantity, fuppofe x, and to add it to the Relate Quantity, fuppofe y ; according to which Reduction we

may

inlert the

Terms

a like Reduction for

This

Quantity.

will

the Parallelogram. And we are to make the Powers of the Fluxion of the Relate bring all Fluxional Equations to the Cafe of in

all

Algebraic Equations, the Refolution of which has been fo amply treated of before. the Thus in the prefent Equation aby -+- axy by -f- yx +- aa, Terms mufl be inferted in the Parallelogram, as if yx~ ' were fubftituted inftead of y ; fo that the Indices will ftand as in the Margin,

=

and the Ruler will give only two Cafes of exterOr rather, if we would reduce this nal Terms. to the form of a double Arithmetical Equation Scale, as explain'ci before, we mould have it in this Here in the firft Column are contain'd thofe form.

Terms which have y of one Dimenfion, In the fecond Column is equivalent to it. Alfo in the or y of no Dimenfions. or fuch in Terms which x xy, .

fecond Line are the {iqiis

of

.v,

becaufe

in the third line

is

Terms -j-

aby t

axy

a

regarded

Term

a1

Line

+ axi i_ y

J

,

;

in

,

1

as

,

2

C~

is

of one Dimenfion.

<f

is

_

what

is

firft is

by~l

or the

or

.

In

the,

,

which have no Dimenif it

which x

were is

ay.

Laftly,

of one negative

Dimenlion


and INFINITE SERIES.

289

is confider'd as if it were -f- abx~~ )\ And thus dilpos'd, it is plain there can be but two Cafes J

Dimension, becaufe -\-aly

Terms being

thefe

of external Terms, which we have already difcufs'd. -TV ?2. If the oropofed Equation be 2 x -4O jy

=

making

x=

y

'tis

i,

we

Solution of which

xy~

2.v -f-

-f- 3_v

l

mall attempt without

y 1

xx

=

o the 2yx~ any preparation, or

without any new interpretation of the Quantities.

;

the

Firft,

Terms

are to be difpos'd according to a double Arithmetical Scale, the of which are y and .Y, and then they will Itand as in the Margin.

Method of doing in all cafes

I

obferve in 2.V

the Equation there are three powers of y,

which

y, and 7-'

1

are

y

,

which

are

.v

x,

,

*

there-

;

,

and x~

it

will be

I

A

at the right hand ; or Then I infert every

The

l

which

,

+p- Xj~

1

-

2JXfore I place thefe in order at the top of the Table. I obferve likewife that there are four 1

Roots

with certainty

this

as follows.

is

or

>

enough

Powers of

x,

I place in order in a Column to conceive this to be done.

Term

of the Equation in its proper place, acof y and x in that Term ; filling up the cording to its Dimenfions vacancies with Aflerifms, to denote the abfence of the Terms beThe Term _}'*""', y I infert as if it were longing to them. as

before.

is

explain'd Rukr to the exterior

duce

Series

;

for

Then we may perceive, that if we apply the Terms, we mail have three cafes that may prowhich

the fourth cafe,

is

that of direft afcent or

To

always to be omitted, as never affording any Series. defcent, with the defcending Series, which will arife from the is

begin

external

2x and

Terms

and the Analyfis

-f-

xy~

s .

The Terms

to be performed, as here follows

two

are to bsdifpos'd, :

-J44*-*, &c.

=

=

1 2X, 6cc. then y2, &c. and by Divifion and &c. Therefore 3>'=T, &c. confequently A-> " 1 and or &c. &c. , -I*" * -f# by Divifion y 4, y~1 l 1 x c an d &c. x\~ Therefore *^ y 2)' confequently -f.*- , c anc^ ^7 ^' v ^" * * r^ % * T*"~% T*" ) &c. So that y~

Make xy~

l

T=4,

=

=

y

=

&

= Then =

^

l

1

fion

'=

=

1

v

* * -f- r f x~'^c -

.

3v

P p

*

#

-j-

'

4r A ~%

^c

-

anc^

y


Method of FLUXIONS,

=

Thefe three&c. and A % 6cc. zyx~* i l * * * make 4- r^x- , and therefore xy~ 44*"% together J A on. c & "d fo * * that * fo -fV|T*~~ &c. y Another defcending Series will arife from the two external Terms 2X, which may be thus extracted -4- -y and

y

==

* -f- i*-"-,

=

-

:

zx

f

-f-

41*-' '

3

&c. &c. X -, &C. i-x-% &c.

i|*-

_ ^4x-*,

,

+^

Make

=

= =

* 2X, &c. then y 3/ ^x, &c. and (by Divifion) y 1 x~*, &c. and x>'~ =|,&c. and y=- T> c - ThereS &c. and _y fore 3_y * * T T) &c. and (by Divifion) * -g-*" , &c. and zyx~* _y= * o, &c. and xy~* and &c. * * * 4&c. Therefore 4, ~x~\ jy 3_y=* i-j..*J &c. 6cc. ^.i^ , The afcending Series in this Equation will arife from the two ex-

= = =

,

&

1

=

1

ternal

tion

Terms

by

y,

2yx~* and xy~ ; or multiplying the whole Equathe external Terms may be clear'd from (that one of l

we

mall have yy the Refolution is thus

y,)

=

3^*

4- zxy

- v\ # t^V

=

:

S

v* ^^*

Jl M^~

x 4- 2y i x~ 1

^S~" r

-1 ^vl/ ^

-

_I_ .

9

v3

"T"*\ *r

'

&r

LA/V *

o,

of which


and INFINITE SERIES.

=

=

Then y'y y. 1 * * 8cc. and y

of

=

Then yy &c. and therefore |.v

5 ,

l

2)'

x~ t

=

=

-f- o,

&c. and

2.vy

-^^S &c. and

* *

,

_>'

= =

-4--V-'

* *

*

&c.

&c.

33, 34.

291

a A* 1 * * **> &c. and confequently 2v*x~ 4-v*, &c. and by extracting the fquare-roor,

The

Author's Procefs of Refolution, in this and the fol-

very natural, fimple, and intelligible; it proterminatim, by p'afling from Series to Series, and by gathering Term after Term, in a kind of circulating manner, of which Method we have had frequent inftances before. By this means he collects into a Series what he calls the Sum, which Sum

lowing Examples,

ceeds Jeriatim

is

is

&

the value of

-

of the Fluxions of the Relate

or of the Ratio

,v

and Correlate in the given Equation and then by the former Problem he obtains the value of y. When I firft obferved this Method of Solution, in this Treadle of our Author's, I confefs I was not u little pleafed ; it being nearly the fame, and differing only in a few circumftances that are not material, from the Method I had hap;

pen'd to fall into feveral years before, for the Solution of Algebraical and Fluxional Equations. This Method I have generally purfued in the courfe of this work, and fliall continue to explain it farther by the following Examples.

The

=

Equation of this Example i 3^ -f- y + x -+- xj y o being reduced to the form of a double Arithmetical Scale, l

will (land as

here in

the

Margin

;

and the

Ruler will difcover two cafes to be try'd, of which one may give us an afcending, and the other a defcending Series for the firft for the afcending Series.

The Terms

Root

y.

~

v,

v<)

xI

And

being difpofed as you fee, makej/=i, &c. then Therefore x, &c. the Sign of which Term y it will x * * -{changed, 2.v, &c. bej/= being 3 AT, &c. Pp 2 and

y=x,

&c.

=

=


77->e

292

=

Method of FLUXIONS, x x &c. Then * y

=

* , -+- #% &c. and thefe each 'tis &c. * * -+**, other, *% .vy deftroying y therefore * &c. and Then &c. *-t-7.* _y=** ^x*, _y &c. and * * * .I* , &c. * -f- x', &c. it will be jxy &c. und therefore y * * * &c. y ^x*

and therefore y

=

=

= =

The

3

=

,

Analyfis in the fecond cafe will be thus

V

Make

= =

4

5

:

*

h x

I2X~ 3

-+

6*- 1

~t-

*

f~ 2

1

6*-" 1 +- 6.V~ Z *

4

AT

=

,

,

&C.

&c

.

2X~1, &C.

I2AT~*,

6CC.

= =

x l , &c. then ;' Therefore x, &c. xy _y x x, &c. and changing the Sign, 'tis % xy-=. 3*, &c. * Then * -h 4, &c. 4*, &c. and therefore y jy= * &c. 4, i, &c. and changing the Signs, 'tis andj xy * * H- 5 -f- i, 6cc. * # -l- 6, &c. and y * * 6x~*, &c. &cc.

= = =

=

=

35, 36. If the given

H

^

,

&c.

its

=

Equation were

Refolution

y

^ ==:i-f-^-f.^._f_^-

be thus perform'd

may

zx'i

&c

a

X 4

XV

:

>

&c

,

&c.

A* A*

-

-

1

1, &c.

-

a*

A4

A*

f

Make &c. and

y y

i,

*

*"

*

&.c.

+

then

;,

2^2

*^

y=

^^4

=

* *

,

"T~

x, &c.

&c. and therefore ~i

y

&c. and j

=

* *

&c.

,

i/)3

o>,4

Therefore _y

=

+

"I"

f

o

a

=*+-, &c. J7

,

2

therefore

j

&c.

Then

And

fo on.

Now


and INFINITE SERIES.

Now &c.

. '

4

in

+

is

this

becaufe the

Example,

equal to a

=?

x

'

it

will be

Series

y=

/

293 |

A;

^

-+-

h

I,

4-

^

or ay '

=

-+. .

xy J

o ; which that is, jx -f- rfx xx ay -+- j will the Solution before deliver'd, give the particular Equation, by o. Hence relation of the Fluents yx I** y ay -{-ax

=jy

tf

A

*",

=

=

a*

-_xx

an(j

x

a

y= x

,

Divifion J

* -f-

za

h

-,

za~

-f- 2a*r

, '

&c. as found

above. 37.

of

this

The Equation Example being

tabulated,

or

x-'

reduced

to a double Arithmetical Scale, will ftand as here in the Margin.

Where

it

may

be ob-

ferved, that becaufe of the Series proceeding both cafe

ways ad injinitum, there can be but one of exterior Terms, of which the Solution here follows:


Th* Method of

294

an affirmative Velocity.

to be

was fuppos'J

FLUXIONS,

as often as there place hereafter,

is

This

occafion for

Remark mull

take

it.

this Example the Author puts x to reprefent the Relate 38. In or the be Root to and to extracted, Quantity, y reprefent the Corto Bat the confufion of relate. Ideas, we mall here change prevent

into y, and / into A", fo that y (hall denote the Relate, and x the Let the given Equation there-fare be Correlate Quantity, as ufual.

.v

=

-

~x

4**

-\-

i

1

2xy'

-f^

-h 7#*

zx'

-+-

}

whofe Root y

is

to

Thefe Terms being difpoled in a Table, will ftand the Refolution will be as follows, taking y and -t- x for the two external Terms. be extracted.

And

thus:

* *

X1 X1

*

+ **

*

*

'*

I

5 J

_j_

i:

J X

=

1

I*'

*

-A'H-ZX

+ I*** -

1

4..1

x/

*

*

#

*

1

*

At

*

*

* .

1

-t

a

*x

2J34-4*

Z-

If

_|- 7A;

^i

*

* *

*

y

j/=|A;, &c. then^ &c.

* o,

&c.

it

will

it

it is

2Ay

5

be

zxy^

There are two other

into Surds.

a ,

&c.

* o, &c. x*, &c.

Now

And

becaufe

jx=s = is^ = y

whereas

and therefore

it is

it

|-x,

* # -f-

*

therefore us with

y= =

-l-x

x* then >r 3 , &c. &c. be* * Now _y= 3**, * -f- o, &c. it will be alfo y^ == * -\- o, &c. and * -f- o, &c. and confequently y E= ***-{- 7^^, &c. and

= y= = y = ***-{_ two

4**, &c. caufe

will be alfo

=

And fo on. 2**, &c. other cafes of external Terms,

Root y, but they This may be fufficient to (hew the Series for the

Method, and how we are 39. The Author mews

.34.

will

will run too

fupply

much

univerfality of the to proceed in like cafes. here, that the fame Fluxional

Equation

variety of Series for the Root, according as conftant quantity at pleafure. Thus the or # -j-f3* -\-y xy, may be re-

may often afford a great we fhall introduce any Equation of Art.

which

j/=i

folved after the following general

manner:


and INFINITE SERIES. r^3*+

** 1 a 4. x -fza*

Ji

+

ax 1

ax ax

y i<?*',

= +

<:.

*

295

*l

+ ax+ax

1

^

x*

axt, isV. ,

= == = = =

Here inftead of making i, 6cc. we may make _y=o, &c. becaufe then y and therefore y a, &c. y o, &c. then <z -+and 6cc. * &c. and -fi, a, confequently y therefore^ * ax =; * -+- ax -- x, 6cc. Then x, &c. and y xy * * -f- zax -f- x == _- ##, &c. and therefore y 3*, &c. * * -f- ax* x 1 &c. There* * -f- 2ax zx, &c. and then y 1 1 ax -f- x 6cc. and * * fore y xy= * ax* AT*, * and * tax* * &C. -f. -f- x*, 6cc. and y confequently y &c. &c. Here if we make a a, * * -f- .iflx -f- -i-* o, we fhall have the fame value of y as was extracted before. And by what-

=

ji/

=

=

=

,

ever

Number a

is

= =

,

,

3

5

1

interpreted,

fo

many

different Series

we

fhall

obtain for y.

40. The Author here enumerates three cafes, when an arbitrary Number mould be affumed, if it can be done, for the firft Term of the Root. Firft, when in the given Equation the Root is affected with a Fractional Dimenfion, or when fome Root of it is to be ex-

convenient to have Unity for the firft Term, or fome other Number whofe Root may be extracted without aSurd, As in the if fuch Number does not offer itfelf of its own accord. 1 &c. and tisA' therefore we fourth Example i}' , may eafily have

tracted

for then

;

it is

=

>

=

&

Secondly, it muft be done, when by reafon of the fquare-root of a negative Quantity, we fhould otherwife fall upon Laftly, we muft aflame fuch a Number, when impoflible Numbers. otherwife there would be no initial Quantity, from whence to begin that is, when the Relate Quantity, the computation of the Root or its Fluxion, affects all the Terms of the Equation. 41,42,43. The Author's Compendiums of Extraction- are very x^

-i->'>

c>

;

and fhevv the univerfality of his Method. As his feveral want no explanation, I lhall proceed to refolve his ExamMethod. As if the given Equation ples by the. foregoing general 1 x4 x or y the Refolu-tion might /-' werej=:-

curious, ProcciTes

,

be thus

=

,

:

y


The Method of

296

=

T

'y

f

1

O

*

f J

*

a.~ I -f-

'

<

X*

.

a~*x

l<3-7x 3

*

**

-

_1_ -t-

j

= y= =

_I_ -4-

'IL

1^

,77

ga,

&c.

,

a~ 7 x*, &c.

\a~$x*- -\-

4-^~*A" /7

Make

FLUXIONS,

;

AT see.

,

&c. then afluming any conftant quantity a, it may a~ &c. and 'Then by Divifion a, y~ l 1 a* therefore _y * -f* -4- a~ x, 6cc. and confequently _y , &c. Then by Divifion * -{- -3x, &c. and therefore y y~ 3 * * * * a~ix, &c. and confequently _y ^S & c Then 1 Divifion and therefore y &c. y-' 4^ s'x again by be

o,

_y

&c.

l

==

l

>

= =

,

l

1

* *

*H-|d~5 A fo

=

Or

i

-f-

'

J

x

IK*

==

-

=

a-*x*

A"

1

-f-

may

|-|-Ar

,

be thus refolved 8

-f- 2 AT" 3 -f-

AT" 2 -f- 2AT~ 7

y~'=

=

i,

^x',

we

ihall

&c.

4

.x*

I4AT-

:

-+- 2

l6x-

J

3,

'

8

= Make

,

-

* * * .vv&c.and confequently/ of the reft. Here if we make a

the fame Equation

- y~

4~

1

;

&c. And have y

=

= =

2i6x'~~ I 3,

+ l8x- + 28oxJZ

I

7 )

&C. &c. &C,

z

2Ar~ 3 , &c. Thenj/= Divifion y =z* -\-2x~ &cc. and confequently by 1 8 Then j/=* yi4x~ ,&c. and therefore 8 12 Then * *4-i4.v~ , &c. and by Divifion _)'= * *+i8^^ , &c. ^ 3 2 * l6x~ 21 6jf * * -f* * &c. and therefore y~ } y 17 fo on. &c. and by Divifion y 28o^~ , &c. And * * * Another afcending Series may be had from this Equation, viz.

A-S&c.

or_y=^~

l

2fc.and therefore -f-2 .v~ 7 , &c.

,

3

,

r=*

==

y=^/2x

J

\ X'

l

=

3

+

** -f-

^

-f-

then making i the firft 44. The Equation y

Term = 1 ,

l

&c. by multipying

it

by y, and

of yj.

x~ y- may J

3 -+- 2y

-4- ojc-

,

=

&c.

be thus refolved

:


IN FINITE SERIES.

Ma ke y

= =

=

297

= =

then y ^x, 6cc. Therefore zy 6x, &c. &c. and 6cc. * Therefore c)x, 3*, confequently_>' 1 Then * -f- 3* 6cc. and x~ y l === y == * IA*, &c. a_>' 4 * &c. and / * * -f- 6# * * -f- 2AT 3 , 9**, 6cc. Therefore^ &c. &c. Or the Refolution may be perform'd after thefe two following manners

and x~ y i I

= = =

3 ,Scc.

,

,

I

:

1=3

zy

*-' -f- IA

*,&c.

;'*-'\= _? v~~ l ^1 _j _/ T

Make

-*

=

* ' '4 "F^ v~"~ 2

v~~

^^^r.-v

4

3

2A--j-i

= j=

j

&"f* A'^*

&C.

A-~'

&c. or_y =r o, &c. and ?, 6cc. then &c. Therefore * , 2_y %x~ &c. or / ~ 1 z * -f- T*"", } &c. and l c anc & * %x by fquaring x~'j* y 1 -'''" 2 2 &c. * and therefore * I , -f- ^A*~ , 6cc. and 2_>'=r* 2 * * And fo on. I-*" } &c. y &c. Again, divide the whole Equation by y, and make x y "2, 1 &c. And &c. and becaufe &c. 2A;, thenj' j'^ni^ j/=2, "" "" 1 1 &c. and Aj ^ c therefore yx~* I-* 'tis^" 3>'~~

zy-=.

x -iy-- =-j- %x-

= ==

= *

==

*

+

H- T-^" o,

=

s 4-jrJ

=

,

-

>

1

1

,

^C

1 )

-

= y=

* *

an(l y I

1

*

=&

H-

T X ~*> ^cc>

x-*+x~ _!_*-

/;

1

,

c

&c

^c

,

-

-

}

2

* -+- -I-*"

=

1

T>

45, 46. If the propofed Equation be_y Solution may be thus :

77

l

J

&c. and

&c. and y

3,

1

-

-

Then 'tis jx~

=

becaufe j^y" 1 l

=

* *

y -\- x~'

T"v

X~-,

== ~4

>

its


The Method of

298

FLUXIONS,

we pleafe, of which the Fluents may be exprefs'd in finite Terms; but to return to thefe again may ibmetimes require particular Expe} Thus if we aflume the Equation y 2x x* -fdients. ,

as

taking the Fluxions, and putting 1

-f-

|x

as alfo

,

-for

=

~

i

^x we

the foregoing Equation, and

which here

the Solution of

x=

\

the Solution

fx

e

'

be thus

may

__

gx*

have

jr

Subtract this

have j/

=

ZX

=^ 2

from

laft

2#-f-l* t

i

,

follows.

47. Let the propos'd Equation be

which

{hall

~x*.

-+-

fhall

i,

= we

y=

i

-f-

,

2#-{- !#*, of

:

1

ex'

fx

= o.

tabulating the Terms of this Equation, as ufual, it may be l obferved, that one of the external Terms y -+- ^yx~ is a double Term, to which the other external Term i belongs in common.

By

Therefore to

= ==x 'tis

e,

-f-

ez=

v

=

that

* -+-

_y

=

l/x

== *

and/ = j

-f-A'%

gx

* * -f-

=#

*

1 ,

^s

|*-> -fSo &c. ,

/w

^r^ /= would be ubfurd * * * -f-

*

,

4 ,

&c.

-j-e,

2/x

5cc.

,

&c. and

* * +- zgx*,

So &c. then

or

i,

makejv'

* * -+-gx* -4-

*

/

*.

_y=

Butbecaufe here

And fo

i z

2x,

~\~fx

is,

=

i-

therefore y

-f-

&c. and therefore

3

.

-f- ex, or

&c. or 2g

if

then

&c. and

-f- e ,

&c. and therefore jj/

3

^-.v

-=*#*and

&c. that &c. So if

A-*,

&c.

zex,

y^j &c. therefore

* 1

s== * *

that

y

&c. and confequently i &c. That becaule 2ex ex, is, So if i, or _y= 2x, &c. i-

then

= y= =x ze=. we make _y=* y= 2/= |/ we = /= = = -jg+, ^ -_ we make =

afllime

thefe,

feparate

-i.

ix 1 ,

<Scc.

or

__. >

* 2/^x 4 ,

&c. then

* * * -f-

2^=^^,

/JA: 3

this

,

fo

&c.

Equa-

the fubfequent Terms except will vanifh in infinitum, and this will be the exact value of And y. the fame may be done from the other cafe of external Terms, as tion

o.

all

from the Paradigm. 48. Nothing can be added to illuflrate this Investigation, we would demonftrate it fynthctically. Becaufe^ =ex* as

will appear

}

unlefs is

here

found,


and INFINITE SERIES. found, therefore y

=

and we

fubftitute y,

e:

v+~', or

fhall

have_y

=

y

=~

Iff!

Here

.

x~49, 50. The given Equation y =yx~- 4thus reiblved after a general manner. be may

=

n

y

2 4A- 4- x1

2x 4- 3

'

/

-f-

-----

-~ x~ z )\

i

-f- 4-v

"~'

l

+a

-f- 4.v

= = = y

ficc.

j

1

y=

conlequently

^A,-"

y=.

=

1 ,

fix'

* * * 'tis

o,

^

i

x1

x"

&c.

=

-+- 3,

= * + 4* 1

=

*

* * * -+-

,

&c.

,

,

ax~ z

and therefore * * -+- a, &c.

1

4*-'

rtx~~ 3

; H-

*

fax" 3

fo

.

=

x~ zy

Therefore

6cc.

x~ zy

therefore 1

,

= = j And

be y

may

&c. and

* -f6cc.

* * -f- o,

it

=

1

Then

&c.

* -+- 4.x,

quently quantity a,

a

2* , &c. then;

2.V

.v-*

I

^A-"~' -j-

Make_y

4-

-f- 3

,

J

y= x

ex'(

4- i-*-* &c. 4- ax~* a *~* + *~ 3 T*~* &c. ax~* -\-ax~* #-' +- fx~ z .x~* ,.&c.

4-rf.v~

4A

*

of

in (lead

as given at firft.

,

x

299

i

,

and therefore and confe&c. 4-v~", afluming any conftant * 4, 6cc.

Then

#-*_y

=

* *

ax~* +- x~ z , &c. and on. Here if we make

^

^

3

&c.

,

this Example is y=. ^xy* -\-y, which our ufual we fliall refolve by Method, without any other preparation than dividing the whole by j*, that one of the Terms may be

51, 52.

The Equation of

from the Relate Quantity ; which will reduce thus 3r, of which the Refolution may be

it

clear'd

c=

TT *3 '

3x

y

Make or y^

&c.

&c.

=

it

5

^.x

^

yy~^

:

,

jJ;y"~^

fx

1 ,

-f-

=f

=

x

6

-f-

-V^ T'^

-+

-|-

3

^rx* TTTT^

7

-4-

+ -~-x

TTY*"> &c

==*_{- f^

&c. and y^

=

* -f-

Then becaufe therefore 37'

&c. and by cubing

= y"'

^x

TT4<T*'>

y$

y=

,

-

'

,

Qjl

2

'

|jc% 6co.

and therefore

l

&c.

&C

= = ^=

%y~'

&c. and by cubing * rV^'S & c 'tis ji/y""^ * # -+- T T * 4 &c. and ;-j And * * 4- T TT-v8 ) & c -

=

;=

,

&c.

,

-

=

1

s

4

3#, .&c. or taking the Fluents, 6 &c. or y And becaufe fAr , &c.

will be jj)T~7

&c.and

f X-X*

= =

*

*

4- TV

* * -f-

+T

* *

on

fx*, ^

4x7

>

-V

v? > '

Tr- v

4 >

-

53-


7%e Method of

-oo 53. Laftly,

xx*,

afTuming

in the c

Equation

y

=

FLUXIONS, zy^

-+-

i==

orjj/y

zx

-4-

whofe Fluxion therefore be 2V*= 2c -f- 2x -f- ~x'^, or

for a conftant quantity,

and taking the Fluents, it will y*=c -+- x -f- -i-x'. Then by fquaring, is

x ty*,

o,

+

3

_>

=

c 1 -+-

2cx

-f- A-* -f.

Here the Root

receive as many diffe_y may !** -+ f^ rent values, while x remains the fame, as c can be interpreted diffe-

-icx*

Make

rent ways.

The Author

-

=

o, then y pleas'd here to c

= make

x1

-+- -ix* -+- loc*.

an Excufe for his being fo minute and particular, in dilcuffing matters which, as he fays, will but feldom come into practice ; but I think any Apology of this kind is needlefs, and we cannot be too minute, when the perfecare rather much obliged to him tion of a Method is concern'd. for giving us his whole Method, for applying it to all the cafes that may happen, and for obviating every difficulty that may arife. The ufe of thefe Extractions is certainly very exteniive ; for there are no Problems in the inverfe Method of Fluxions, and efpecially fuch as are to be anfwer'd by infinite Series, but what may be reduced to is

We

fuch Fluxional Equations, and may therefore receive their Solutions from hence. But this will appear more fully hereafter.

SECT.

IV.

Solution of the third Cafe of Equations, with

fame TT* O R

neceffary Demonftrations.

more methodical Solution of what our Author a moft troublejbme and difficult Problem, (and furely the Inverfe Method of Fluxions, in its full extent, deferves to be call'd fuch a Problem,) he has before diftributed it into three Cafes. The firft Cafe, in which two Fluxions and only one flowing Quanin the given Equation, he has difpatch'd without much tity occur The difficulty, by the affiftance of his Method of infinite Series. in which two fecond Cafe, flowing Quantities and their Fluxions are any how involved in the given Equation, even with the fame affiftance is flill an operofe Problem, but yet is difculs'd in all its The third varieties, by a fufficient number of appofite Examples. Cafe, in which occur more than two Fluxions with their Fluents, is here very artfully managed, and all the difficulties of it are reduced to the other two Cafes. For if the Equation involves (for inftance) three Fluxions, with fome or all of their Fluents, another full DeEquation ought to be given by the Queftion, in order to a 54.

the

calls

terminationj


and INFINITE SERIES.

301

termination, as has been already argued in another place; or if not, the Queftion is left indetermined, and then another Equation may be affumed ad libitum, fuch as will afford a proper Solution to the And the reft of the work will only require the two Queftion. former Cafes, with fome common Algebraic Reductions, as we fhall fee in the Author's Example.

Now

to confider the Author's 55. Example, belonging to this third Cafe of finding Fluents from their Fluxions given, or when there are more than two variable Quantities, and their Fluxions, either exprefs'd or underftood in the This Example given

=

Equation.

z 4- yx zx o, in which becaufe there are three Fluxions A-, and therefore (and z, y, virtually three Fluents x, y, and z,) and but one Equation given ; I may affume (for inftance) x=y, whence x =JK, and by fubftitution zy z -\-yy o, and therefore zy Now as here are only two Equations x y== o & + T)'* l and zy z-\-^y =o, the Quantities x, y, and z are ftill variable Quantities, and fufceptible of infinite values, as they ought to be. Indeed a third Equation may be had, as zx z-\-x* o; but as this is only derived from the other two, it new limino brings tation with it, but leaves the ftill indetermiand quantities flowing is

=

=

=

nate quantities. Thus if I mould affume zy=a-\-z for the fccond Equation, then zy=z, and by fubftitution zx zjr-k-yx=;o, x -f- .Ixv -f- ^x'-x, &c. and therefore y or y x -+- ix 1 j^ H-TT# & c which two Equations are a compleat Determination. s and thence x=Z)y Again, if we affume with the Author 1 1 we mall have by fubftitution <\.yy z -^-yy o, and thence zy z -+o, which two Equations are a fufficient Determina-

=

= S

J

=

-

x=j

= ^ We

tion.

this

is

=

,

t

=

o ; but as z -+- ^x^ may indeed have a third, zx included in the other two, and introduces no new limitation,

the quantities will

ftill

of fecond Equations

remain

may

fluent.

And

be aflumed, tho

1

thus an infinite variety it is always convenient,

that the affumed Equation fliould be as fimple as may be. Yet fome caution muft be ufed in the choice, that it may not introduce fuch a limitation, as fhall be inconfiftent with the Solution. Thus if I fhould affume zx o for the fecond Equation, I mould have

=

z=

=

o to be fubftituted, which would make yx o, and therefore would afford no Solution of the Equation. 'Tis eafy to extend this reafoning to Equations, that involve four or more Fluxions, and their flowing Quantities but it would be And thus our Author has comneedlefs here to multiply Examples. this Cafe alfo, which at firft view might appear forpleatly folved midable

zx

z

,


7%e Method of FLUXIONS,

302

midable enough, by reducing

all

difficulties

its

to

the

two former

Cafes.

The

Author's way of demonstrating the Inverfe Method of Fluxions Short, but fatisfactory enough. have argued elfewhere, that from the Fluents given to find the Fluxions, is a direct and fynthetical Operation ; and on the contrary, from the Fluxions And in the given to find the Fluents, is indirect and analytical. order of nature Synthefis mould always precede Analyfis, or Commould before Refolution. But the Terms Synthefis and go pofidon often ufed are a in Analyfis vague fenfe, and taken only relatively, as in this For the direct Method of Fluxions place. being already demonftrated fynthetically, the Author declines (for the reafons he gives) to demonstrate the Inverfe Method alfo, that is, 56, 57.

We

is

fynthetically He contents primarily, and independently of the direct Method. himfelf to prove it analytically, that is, the direct Meby fuppofing

thod, as fufficiently demonstrated already, and Shewing the neceSTary connexion between this and the inverSe Method. And this will always be a full proof of the truth of the conclufions, as Multiplication is a good proof of Division. Thus in the firlt Example we if that the is x1 found, given Equation y -f- xy I,

we

x y=x the truth of which

have the Root

Shall

To

y=^x

1

-j-

fx

-*

3

-f-

6

^.x*

-^r*

,

conclufion, we may hence find, 1 direct i the 2x .i* 3 -f-f#* Method, -{-x _y by T T X', &c. and then fubStitute theSe two Series in the given Equation, as follows; cc.

prove

=

_

_{_

jf 4- ]_ X y -------f_ X Xy ---------1_ X - __ #3 r -f. 2 X A-* -{y 1

.

--

x

_

.

X< 3.4

, _J_ _>_ X

__

^ ^+ _j_

^f -rX'

__ _^ X 6 6 _{_ _?_ X 5

^X

6

1

Now by collecting thefe Series, we mall find the refult to produce the given Equation, and therefore the preceding Operation will be fufticiently proved. 58. In this and the fubfequent paragraphs, our Author comes to open and explain fome of the chief My Steries of Fluxions and Fluents, and to give us a Key for the clearer apprehenfion of their nature and properties. Therefore for the Learners better instruction, I Shall not think much to inquire fomething more into this circumstantially

order to which let us conceive any number of right ae &c. Lines, AE, t as, indefinitely extended both ways, along which a Body, or a defcribing Point, may be fuppofed to move in each matter.

In

Line,


and INFINITE SERIES.

303

Line, from the left-hand towards the right, according to any Law or Rate of Acceleration or Retardation whatever. Now the Motion thefe of of every one Points, at all times, is to be eftimated by its diftance from fome fixt point in the fame Line ; and any fuch Points may be chofen for this purpole, in each Line, fuppoie B, I), /3, in which all the Bodies have been, are, or will be, in the fame Mo-

ment of Time, from whence

to

compute

their

contemporaneous

Augments, Differences, or flowing Quantities. Thefe Fluents may be conceived as negative before the Body arrives at that point, as nothing when in In the rlrft Line

and as affirmative when they are got beyond it. AE, whole Fluent we denominate by x, we may the luppofe Body to move uniformly, or with any equable Velocity then may the Fluent x, or the Line which is continually defcribed, it,

;

B

A

C

J>

:E

a,

*

2,

1-19"

/?

!

reprefent

Time, or {land

II8

c/^

for the Correlate Quantity,

to

which the

feveral Relate Quantities are to be constantly refer'd and compared. For in the fecond Line ae, whofe Fluent we call y, if we fuppofe the Body to move with a Motion continually accelerated or retarded,

according to any conftant Rate or Law, (which Law is exprefs'd by any Equation compos'd of x and y and known quantities j) then will there always be contemporaneous parts or augments, defcribed in the two Lines, which parts will make the whole Fluents to be contemporaneous alfo, and accommodate themfelves to the Equation

So that whatever value is afiumed for the Circumftances. the Correlate x, correfponding or contemporaneous value of the ReOr from late y may be known from the Equation, and vice versa. the Time being given, here represented by x, the Space represented by y may always be known. The Origin (as we may call it) of the Fluent x is mark'd by the point B, and the Origin of the Fluent y If the Bodies at the fame time are found in and by the point b. BA and ba. If then will the contemporaneous Fluents be in

all its

A

,

fame time, as was fuppofed, they are found in their refpecthen will each Fluent be nothing. If at the tive Origins B and fame time they are found in ^ and c, then will their Fluents be And the like of all other points, in which the -1- BC and -\-bc. at the

,

i

moving


The Method of

304 moving Bodies

either

have been,

FLUXIONS, or

fliall

be found,

at

the fame

time.

of thefe Fluents, or the points from whence we tho' they muft be conceived to be variable begin to compute them, (for and indetermined in refpedt of one of their Limits, where the deat prefent, yet they are fixt and determined as to fcribing points are

As

to the Origins

w

their other Limit, which is their Origin,) tho' before appointed the Origin of each Fluent to be in B and b, yet it is not of abfolute mould begin together, or at the fame Moment of neceffity that they Time. All that is neceflary is this, that the Motions may continue as before, or that they may obferve the fame rate of flowing, and

have the fame contemporaneous Increments or Decrements, which will not be at all affected by changing the beginnings of the Fluents.

The Origins of the Fluents are intirely arbitrary things, and we may remove them to what other points we pleafe. If we remove them from B and b to A and c, for inftance, the contemporaneous

Lines will ftill be AB and ab, BC and be, &c. tho' they will change AB we fhall have o, inftead of B or o Inftead of their names. of -+- BC we fliall have -f- AC ; &c. inftead have we fliall -+- AB, ac -{-be, inftead of b or o we have ab we fliall So inftead of fliall have cd, &c. That be, inftead of-f- /Wwe fliall have -+- bc the general Law of determines which in the is, Equation flowing we may always increafe or diminifh x, or y y or both, or

+

increafing,

occafion may require, and yet the Equaquantity, as tion that arifes will ftill exprefs the rate of flowing ; which is all that Of the ufe and conveniency of which Reduction is neceffary here. we have feen feveral in fiances before. If there be a third Line a.e, defcribed in like manner, whofe

by any given

its parts correfponding with the others, as be another Equation, either given or muft & a/3, &y, y, of flowing, or the relation of z to the aflumed, to afcertain the rate Or it will be the fame thing, if in the two Equations Correlate x. For thefe the Fluents x, y, Z, are any how promifcuoufly involved. determine the Law of and limit two Equations will flowing in each And we may likewife remove the Origin of the Fluent z Line. And fo if there were more to what point we pleafe of the Line a. Fluents. more Lines, or what has been faid by an eafy inftance. Thus 59. To exemplify

Fluent

may

be z, having c - there

=

we may aflume y inftead of the Equation xy -+- xxy, or x is diminifli'd of x is the where by Unity ; for changed, Origin of which Reinftead of The lawfulnefs x, fubftituted j -J-- x is

y=xxy,

duction


and INFINITE SERIES.

305

may be thus proved from the Principles of Analyticks. Make which (hews, that xand2 flow or increale i -\-z, whence x alike. Subftitute thefe infteadof x and x in the Equation^':=xxy, and duftion

x=z,

=

become y zy -+- zzy. This differs in nothing elle from afTumed the Equation y xy -f- xxy, only that the Symbol x is the into which can make no real change in the z, Symbol changed So that we argumentation. may as well retain the dime Symbols it

will

=

were given at firft, and, becaufe z-=xfuppofe x to be diminiih'd by Unity.

as

-

we may

i,

as well

60, 6 1. The Equation expreffing the Relation of the Fluents will times give any of their contemporaneous parts for afluming of values different the Correlate Quantity, we ma'!, thence have the at all

;

correfponding different values of the Relate, and then by fubtradion we fhall obtain the contemporary differences of each. Thus if the given Equation were

y

= = x

-{-

-

where x

,

equably increafmg or decreafing &c. fucceifively, then y infinite, tity

;

2,

is

fuppos'd to be a quan-

make x

=

2|,

4^,

3.1,

And

o, i, 2, 3, 4, 5, 5-^-,

&c. refpec-

taking their differences, while x flows from o to i, tively. from i to 2, from 2 to 3, &c. y will flow from infinite to 2, from 2 to 2-i-, from 2| to 3.1, &cc. that is, their contemporaneous parts

be

will

have x 2-i-,

i,

i,

=

Likewife,

if

i,

i,

we

o,

go

&c. and

i,

infinite, backwards, or if

2,

i,

,

&c. refpeclively.

-{.I,

we make x negative, we &c. which will make _y= infinite,

mall 2,

&c. fo that the contemporaneous differences will be as be-

fore.

Perhaps tion,

GOH

may make

it

to reprefent this

and

KOL

a ftronger impreffion upon the Imaginaby a Figure. To the rectangular Afymptotes

let

be oppofite Hyperbola's

ABC ;

and

DEF

bifed the

An-

GOK

by the indefinite right Line gle yOR, perpendicular to which draw the Diameter BOE, meeting the Hyperbola's in B and E, from whence draw and EST, as alfo CLR and

BQP

DKU

rallel

to

GOH.

Now

if

OL

is

pa-

made

and equable the Equation y x -f- -'

to reprefent the indefinite

quantity x in

then

CR may

= CL = ^ = = OLR= CR =^ LR

reprefent y.

l

-

For

,

x

r

,

therefore

(fuppofing -4-

or


*The

306 or

x

y

=

x

-f-

^

= CR, = = OL,

Now

then

o,

Method of FLUXIONS, the Origin of

or y, will coincide with the

If

x=

then y the proceeding contrary way, to coincide with the Afymptote

and

2

If x

infinite.

= OK =

2,

O

j

if

and

OG, = = BP=2. = CR = OQ^ = And may = OS = OH, y = ET = = Dv =

therefore will be infinite.

If x

or x, being in

OL,

then

2 i.

if

x

i,

_y

Afymptote then

i

fo

o, then y

_y

of the

reft.

Alfo

be fuppofed

and therefore will be negative then 2. If x &c. And thus we 2~, may

purfue, at leaft by Imagination, the correfpondent values of the flowing quantities x and_y, as alfo their contemporary differences, through all their

poiTible varieties

;

according to their relation to each other,

=

x +- by the Equation y The Transition from hence to Fluxions is fo very eafy, that it may be worth while to proceed a little farther. As the Equation as exhibited

.

.

of the Fluents will give (as now obferved) of fo if thefe differences their contemporary parts or differences any are taken very fmall, they will be nearly as the Velocities of the expreffing the relation

;

moving Bodies, or points, by which they are defcribed. For Motions continually accelerated or retarded, when perform'd in very But if thofe diffefmall fpaces, become nearly equable Motions. rences are conceived to be dirninifhed in infiriitum, fo as from finite differences to become Moments, or vanifhing Quantities, the Mo-

equable, and therefore the Velocities of their Defcription, or the Fluxions of the Fluents, will be accurately as thofe Moments. Suppofe then x, y, z, &c. to reprefent or Equations, and their Fluxions, or VeFluents in tions in

them

will be perfectly

any Equation, of increafe or decreafe, to be reprefented by x, y, z, &c. and their refpedlive contemporary Moments to be op, oq, or, &c. where p, q, r, &c. will be the Exponents of the Proportions of the Moments, and o denotes a vanifhing quantity, as the nature of Moments requires. Then x, y, z, Sec. will be as op, oq, or, &c. So that ,v, y, z, &c. may be ufed inflead that is, as p, g, r, &c. That is, the fync ni th e designation of the Moments. r ^ of/>, ?> of be Moments &c. chronous x, y, z, may reprefented by ox, oy, in the Fluent x may be fuppofed Therefore &c. oz, any Equation to be increafed by its Moment ox, and the Fluent y by its Moment &cc. or x -+- ox, y -{- oy, &c. may be fubftitnted in the Equation oy, inflead of x, y, &c. and yet the Equation will flill be true, becaufe From which Opethe Moments are fuppofed to be fynchronous. locities

-

->

ration


and INFINITE SERIES,

307

Equation will be form'd, which, by due Redu&ion, muft the relation of the Fluxions. neceflarily exhibit x -+- z be given, by Thus, for example, if the Equation y Subftitution we fliall have y -f- oy =. x -f- ox -+- z -+- oz, which, bex -f- z, winch ox -f- oz, or y x -+- z, will become oy caufe y ration an

=

=

= zx = we =

Here

the relation of the Fluxions.

is

or

i,

or ~x-f-

the others, refpect of

Now

as the

as

increafes

A-

Equation y be

=

y

And

x

we

z

afllime

zx -f- A z

'tis

=

=

o,

+

or z ===

Fluxion of z conies out negative,

=

-+-

z

will decreafe,

x

-+- z,

-

-

>

=

z

if

-f

=

=

an indication that Therefore in the

'tis

and the contrary. - or if the relation of the Fluents ,

then the relation of the Fluxions will be

,

from the Equation y

as before,

contemporaneous

parts,

Fluxional Equation y rate

= =

=

A-

if

again,

by increafing the Fluents by their contemporary Mohave z -+- oz x A- -f- ox ozx -f- oxz i, or zx i. Here becaufe zx o, i, 'tis ozx -f- o.\z -+- oozx o. But becaufe ozx is a vanifliing Term in -+- 02X fliall

ments, -f- oozx

=

=: x -+-

we

-

or differences of the Fluents

=

x

now

^

found,

we may

of flowing, or the proportion of the Fluxions

=

x

derived the fo

;

y

from the

obferve the

at different values

of the Fluents.

For becaufe

#

=

o,

when y locity

or is

it

when

x

is

:

y

wherewith x

it is

:

i

\

: :

x1

:

x1

i

;

when

but beginning to flow, (confequently That is, the Vei. y :: o defcribed is infinitely little in comparifon of the

the Fluent

infinite,)

I

: :

is

will be

x

:

:

and moreover it is infinuated, (becaufe velocity wherewith^ is defcribed; of increafes x that while i,) by any finite quantity, tho' never fo

by an

fame time.

This of from the the When appear infpeclion foregoing Figure. i That is, o. i, (and confequently _)'= 2,) then x y x will then flow infinitely faflrer than y. The reafon of which is, that y is then at its Limit, or the leaft that it can poflibly be, and little,

y

will decreafe

infinite quantity at the

will

x=

:

:

:

:

ftationary for a moment, or its Fluxion of that of x. So in the foregoing Figure, nothing comparifon BP is the lea ft of all fuch Lines as are reprcfented by CR. When Or 27,) it will be A(and therefore y 3. y :: 4

therefore in that place is

x=2,

it

is

in

=

:

R

r

2

:

the


Method of FLUXIONS, there greater than that

of^, in the ratio of 4 :: x 8. And fo on. So that then When to 3. 9 y tend towards equality, which the Velocities or Fluxions conftantly

the Velocity of x

is

x=3,

:

:

(or CL) finally vanishing, x and y become they do not attain till And the like may be obferved of the negative values of equal.

x and

y.

SECT. V.

Refolution of Equations^ whether Algebraical or Fluxionalt by the ajfiftance of fuperior orders 77je

of Fluxions. the foregoing Extractions (according to a hint of our Auand thor's,) may be perform'd fomething more expeditioufly,

ALL

we

have recourfe to To (hew this firft by an eafy Inftance. orders of Fluxions. fuperior Let it be required to extradl the Cube-root of the Binomial a 3 -\- x"' ; or to find the Root y of this Equation y 1 a> -f- x 3

without the help of fubfidiary Operations,

,

= = = =y = 3

or rather, for fimplicity-fake, let be_) &c. or the initial Term of y will be a. it

if

=a* -+- z.

Then y=.a,

Taking the Fluxions of l But i, or -$yy-^y~ Equation, 1 it will be and fubftitution 6cc. &c. is as it a, jtf" by y y 1 &c. a Here * -f- \a~ z^ vacancy is taking the Fluents, 'tis For left for the firft Term of y, which we already know to be a.

this

we

=

z

have

fhall

.

,

y=

another Operation take the Fluxions of the Equation jj/=: j/~* ; 5 Then becaufe y whence y a, &c. %yy~* jy~

=

=

=

-

= y = = two

?&"*, &c. and taking the Fluents, and &c. taking the Fluents again, 'tis $a~ z,

'tis

y 3

jy-

*

'tis

* *

l

^a-^z

,

firfl Terms of Here two vacancies are to be left for the are already known. For the next Operation take the y, which 6 s =. Fluxions of the Equation y=. -f^y~ that is, y 9-yy~

6cc.

=

,

8

_l_^.ji-

.

Or becaufe _7=c,&c.

the Fluents,

y

=

* * *

'tis s

-v

y=&c.

* L^. a -*z,

T a~*z*,

Fluxions of the

=

Iry"

Then

taking

11 -

Again,

Equation y

Or the

becaufe Fluents,

for

=

a,

*

8 ,

*

Then

&c.

-^a-^z

1 ,

taking

6cc.

and

another Operation take the

=: ~~y^

= y=

y

_y=if(S&c. y *

'tis

;

&c.

whence 'tis

^a-

"y 1

=

y

=

^, &c.

-*f^~

9

IT^"", &c. jy

=

* *


and INFINITE SERIES.

=And

1

&c. y '^-?-a-"z+, &c.

H-a-^z

* * *

,

have therefore found

^-

1*

C*

-

I

XI bi

'

Or

^1

that

laft,

v

we would

if

univerfally,

Term

=

make y

of the Series

y=a

=

'tis

a

a

-f-

or y,

ra

x

t/m

m

m

\

,

&c. and

,

Again, becaufe

6cc.

be y x

i~%

Again, becaufe will be

y

=m

And y

x

~y-3 X

~g

-f-

B

8i

~

-+-

^

(

-

y i

'^^-

=m

x

m

m

=m

x

=

'tis

_y

/==

'tis

17

*

have ~yy

fhall

y

= =

a m, &c.

m

it

*

y

'tis

j

m

* *

m

'tis

am

y

^

~

t

x

y

will

it

=- d m ~ 2

x

be

will

ma m

=

= Sec.

=

,

*

I

x% &c.

2

taking the Fluxions

"',

m

2j/y

be-

i

taking the Fluents,

_= //;

Then

,

and becaufe y

;

j

x

;//

a m , &c.

x

is

into an equivalent have a m for the m a &c.

fhall

taking the Fluxions

,

and therefore y

taking the Fluents,

=**/;/ CT

-

m

ly

&c.

,

is

7 :=

and becaufe

m ~'i

6cc.

it

\

y we

it

is

it

x

ia

/

We

pleafe. ,

taking the Fluents,

lyv

'tis/ ?

* *

*

m

x

=

will be

it

But becaufe

.

1

*

9

=

x-

and we

+- x, taking the Fluxions

or

i,

+

fa

refolve a -f-

= ~ y = my now = y y = my =m *= m = wxw And x

-.1

-\

y.

j_

caufe

we

l\f\^

'

I

infinite Series, firfl

at

=

_)'

as far as

309

-

~

"

n

,3

&c. and

,

we may go on

fo

&c. or for z writing x*.

,

"z

_+._ _.rt

5

x

2a -ix*, &c.

And

fo

i

/

w

m

m

x

x

i

and

v

x

w/

=

we might

it

*

we

far as

pleafe, proceed as maif the Law of Continuation had not already been fufficiently ~ m x -+a* -f- ma So that we fhall have here a -+- x nifefl. * ~~" ^ m ^ 2 ni __ m m x -+- m x m x !LU fl x l 4- w x - - x - a

x

3,

6cc.

I

2

2

~^

r

ftt

=

l

-*

3

?^-V-4;c<, &c. Vhis is a famous Theorem of our Author's,

tho' difcover'd

by

.,him after a very different manner of Investigation, or rather by lldufr *>v >'$ vntrrrU*t> Indudion. It is commonly known by the name of his Binomial a x; may JujJf&rcL f Theorem, becaufe by its amftance any Binomial, as be exof it Root or at Power to

+

be

railed

tradted.

any

And

it

pleafure,

is

obvious, that

may

any

when m

-^fajfifa m- Cff by any 1

is

interpreted

&'

.

**'*


r^

t

r .3 i


and INFINITE SERIES. /lull

+x

have a

ra I

=

a"

=^*=fX;0i.^ Now Hop

of

this Series will

*

mx

-f-

^.

its

-f-

own

311

"'+'

w

x

-7-

*'"*=

^7

~

,

number

accord, at a finite

of Terms, when m is any integer and negative Number that is, when the Reciprocal of any Power of a Binomial is to be found. But in all other cafes we mall have an infinite converging Series for the Power or Root required, which will always converge when a ;

and x have the fame Sign converging quantity,

By comparing

is

thefe

;

^ two

becaufe the Root of the Scale, or the

which

,

common

:

_

.

-+equivalent Series

11 x

,

x a

m

-

quantity

_ a-

TO

x

-x

-f-

a-i-x

3

,

have the two

fliall

x

-j-x

2

+ x |*

than Unity.

or by collecting from

we

,

+ 11 x "L_ x ==^=2

lefs

i

^

-j-

always

Series together, ,

each the

is

,

&c.

=

&c. from whence

we

5

I

might derive an infinite number of Numeral Converging Series, not inelegant, which would be proper to explain and illuftrate the nature of Convergency in general, as has been attempted in the former part of this work. For if we aflume fuch a value of i as will

make

either

of the Series become

finite,

the other Series will

exhibit the quantity that arifes by an Approximation ad infinitum. And then a and A; may be afterwards determined at pleafure.

Method, we fliall fhew (according to promife) how to derive Mr. de Moivre's elegant Theorem ; for to any indeterminate Power, or for extractraifing an Infinitinomial fame. The way how it was derived from the of the Root ing any of and genefis of Powers, (which the nature abftract coniideration indeed is the only legitimate method of Inveftigation in the prefent cafe,) and the Law of Continuation, have been long ago communicated and demonftrated by the Author, in the Philofbphicai TranfYet for the dignity of the Problem, and the betactions, N 230.

As another Example of

ter to

deduce

illuftrate it

the prefent

this

Method of Extraction of Roots,

I

lhall

here as follows.

b~ -+- cz* -f- itz* -4- ez* &.c. * where the value of y is to be found by an infinite Series, of )', which the firfl: Term is already known to be a m or it is y a", a -f- l>z -f- cz, 1 -J- dz> -f- ez*, &c. and putting &c. Make v z i, and taking the Fluxions, we mail have -y r^zi b -+- 2cz -+-

Let us aflume the Equation a

=

=

=

-+-

t

,

|


Tie Method of

I2

,

7</jz

-f-

4s*. &c

we make

if

=

-y

Then

-

becaufe

= =

y

&c. and V

<*,

FLUXIONS, u",

t>,

ma'-'b, &c. and taking the Fluents,

For another Operation, becaufe x

_l_

;,;

y,

v,

ii>i;"'~

;

.

and -yfubftituting

we

tively,

jy

1

&c.

zmca" -+mz

m

1

1

=

_>

it is

,

2f -f- 6<fe

their values a,

= 7 =

have

fliall

= mvv"=

mviT-', where have _y

=

fliall

be

_y

And becaufe <u

z

we

will

it

=

^

it is

&c.

+

_y

maF*kst

*

=

12^2;*,

^,

&c. and 2c,

x

OT

i6*a

6cc. refpec-

m-z

,

&c. and

* 2fnca -\-m-x. m i*tf*-**, taking the Fluents m l z* -\-m~x. * * mca taking the Fluents again, y J

=

..

For another Operation, becaufe/

m v"j m ~ + yn

And

becaufe

w

\bca

Fluents

w

-u

&c.

6cc. b,

2c, m ~'i

&c.

m

=

And

We

if the

length,

it

i

*

l/

m

=

fliall

-+-

mxm

v,

v, v,

z1 )

2 tv

m

~1 t

*v">.

v, fubftituting a,

~

have )"=, 6mda m l -f- 6m x m 2l> a ~*, &c. And taking the \bca m -*z -+- m x z -+- 6m x m 3

l

* *

and/

=

* * *

x

.

therefore have

and

m- 2

iv 1 v m

ixw

fliall

6mda m ~

8cc.

m- 1

-vv-{-m\m

&c. for

we x

= mw

~z

m 1 2foa -iz , &c.

m finitum.

m

be y 2foa m ~*z,

^^ x w

x

&c.

6^/, x.

will

Km

i

iv m

/

= 6^+24^2;,

-\-

it

x

l

i

y

6cc.

^^^a

&c.

'tis

&c. for

mda m- l z*

And

fo

-+

on

z'

m

x

/-

tf

whole be multiply'd by s", and continued to a due form of Mr. de Moivres Theorem.

will have the

Algebraical or Fluential Equations may be exFor an Example let us take the Cubick tradled by this Method. x"' 2 3 fo often before refolved, Equation y* -\~axy -\-a*y the &c. Then in which y Fluxions, and making a, taking

The Roots

=

of

all

=o,

=

=

z Here o. we fliall have ^yy- -+- ay +- axy -f- a*-y 3x 1 if for y we fubftitute a, &c. we fliall have ^y -\- a 4- axy 3**, &C A " d &c &c. =o, or>= -j:+^" =.--the Fluxions v &c. * Then the Fluents, taking -i-x, taking

x

i,

=

= ^' j

-

'

again


and INFINITE SERIES. again of the 4- a ly

lafl

6x= o.

we

&c. *y

= ^

and 7

,

'

-f-

3^7

=

3 2a

&c. and therefore 7 * * *

^7x7

4- a*y

&c. and 7 :,

&c.

=

&c.

5

=

-

|^

=

* * * *

Therefore the Root

y

is

"

&c. 7

6

4

5j; C-

4a

^

&c. 7

,

= = 7=

Make 7

* *

=

&c. 7

a,

^

=

,

= __ 6cc.

&c./

^,

=

&c. then 7

,

&c.

'-

,

^-.

r68a

'

^C

* *

^ nc

^

'

^ on as ^ar as

= ai + f- H-H^ + i68' x

12^-

643

The

a,

^ + ^~ ^-

^^

*

=

-{- iSyy'y ^

Again, 377* -1-24777-1- 18^*74- 3677*

o.

and7=#

&c. and y

,

^

-

=

=

Again, ?yy*

-

Make 7

0.

then 7

6cc.

,

=

6

2 ay

'""

'

-\- a*-j

axy

'

1

(3 -I.'?

"\2.ii

=

a,

4-

4- axy &c. and 7 ^., &c. and therefore 6y y

-f-

_!_ 32a

CSV.

* *

,

4- 4^7 4-

__ y= ZZif+j''-^: 43% &c. and y= &c -h z

&c. and '7 ,

=

we make 7

if

1

have 37}*

fliall

AT

4

*

_j_ 675 -f-

^|j,

Where

have

fliall

we

Equation,

313

-|2|fl

'

J

we &c.

when found by this Method, muft always have its Powers afcending but if we defire likewife to find a Series with defcending Powers, it may be done by this eafy artiAs in the prefent Equation y* -\- axy -f- a*y x"' fice. 2a* o, we may conceive x to be a conftant quantity, and a to be a flowing or rather, to prevent a confufion of Ideas, we may change quantity Series for the

Root,

;

=

;

a

and x into a, and then the Equation will be _y 5 -f- axy -+ ax 3 o. In this we mall have y a, &c. and tax*y 6x* o, king the Fluxions, 'tis 3j/y* -4- ay -f- axy -f- 2xy +- x*y z "~" #& ay 2X],"-r- 6-V n r o But becauie y=a, &c. tis ^ or ficc. 7-1 r into x,

=

rt 3

y=

== 'tis

-

=

H

.

.

y +<**+*

i, &c. and therefore/

=

i

>

-f- 6)/

3_)^

3

Again taking the Fluxions I2x o, or / 4- zay +- axy -\-2y-\- ^.xy -+- x*y *

/

-yj>-^-'j-4y+"* yJ+flX-f-*

-i-.v,&c.

=

king

_y

-6^-2^-2,

1

Again

e>

,

r

1

4

=s

= y=

=

y

,

it

is

rt

a,

&c. and

&c. and

_y

= =

_y

y* |, &c.

3vy* -t- i8;7/ -f- 6y

^ 5

,

-f-

S f

'tis

y

==

&c. and

Qr

~^ + f~

2a

3^

;'=**

3^} 4- axy

-+.

67

^

,

&c.

,

&c.

H- 6x/ H-

x*_y

12


The Method of FLUXIONS,

_ /v

=

12

= -rv- -= &c = -, &c.)~ y ?

Or

O,

= = -^ = ^T a t &c.

y

&c. y

,

* *

fore

we

y

2"<Z

Then

&c. ,

and

-

4-'

&c. and y

mall have y

=

y

taking the Fluents,

=

* *

2

And

2

,

3**

=

7

c.

,

+ v -T + + 12 *

&c.

,

There-

fo on.

Gift

x

.

<z

|^

4

/

3>

+ 7^7

7^

Or now we

&c.

,

=

x into #, and a into x ; then it will be y x again change -. &c. for the Root of the given Equation, as itf _i_ ,

may

_ _

ii^j

3*

was found before, pag.

&c.

2 16,

Alfo in the Solution of Fluxional Equations, we may proceed in As if the given Equation were ay a*x -{- xy the fame manner. Radius of a if the Circle be o,. (in which, reprefented by a, and the of the Arch fame, if y be corresponding Tangent will be

any

x 3) let it be required to extradl the Root y out of reprefented by this Equation, or to exprefs it by a Series compofed of the Powers a 1 -Ji, then the Equation will be of a and x. Make x

=

=

ay

= -yr=o. Then = we ay xy = y= we Then we may y='~^^ = a

Here becaufe_y' i, &c. taking the Fluents x * x, &c. it will be y taking the Fluxions of this EquaBut (hall have -f- 2xy -+o, or tion, -^T 1 a

are to have a conftant

becaufe

fuppofe

'tisj/=

a'y -f- 27

^

7 ==

,

fore

y

=

Q, &-C.

y =.

+ 4*7

* * o,

&c.

xy

-{-

&c. whence y

6xy

* o,

Again,

=

write their values

* * *

ay +6y +

&c.

+xy

&c.

=

_y=

ay

-+-

o,

* *

Then or

o,

.

I2y 4-

~*?~^

&c. j.'= * * * 8^,7 -f-

A'

4

j

= =

J

o.,

&c.

we

=

o,

o,

mall have

^

*

Fluxions

the

Here

"J'^.'T*

o,

&c y

,

01-7

of y,

taking the Fluxions

=

y

&c. and

Taking

-

Term

taking the Fluents

il

*

^c

>

,71

i,

.

for the firft

quandty

o,

i

&c. and

y and_y we

}'= and

* o,

'tis

again, if for

_,

o,

or y

again,

<5cc.

&c.

,

c. 'tis

There-

y =. **#

=~

2

l

^!.''

1

-


and INFINITE SERIES.

=+

_>>=:***-{-

"'

-

5

&c.

}

*****

&c.

,

o,

&c.

-f-

loxy

2 4 rf-

9

-, &c.

=

xy

-f-

i2*y

&c.

,

_}=*****

-^

^

o, or

_ !^2f

.

&c

,

= ***

&c. ;

,

y=. .

*

*6

o

j)/=******

c^c.

,

whence y=s

o,

-+- A,"^

4x301-3

* * * -

# * * * *

_y

&c. and

,

Then > =-

&c.

,

6*5

30*4

= **4- ^

;<

r56 7? =

&c.

,

4-

a*j+ 307 -f-

Again,

12x30**

&c.

45

* * * *

--^ y= = = ******* -H y= ^ +

==- * *

--

=

jy

Again, a*y-\- zoy

y~^=:- 30 x j

Jr^*^^,

Then

&c.

,

A' 7

and _/

c.

have here

,

fl + 5^ P This

*

'

ox- 1

#

*

.

,

&c. '

+

^;

>

&c

we

So that

fo on.

that

-

is,

_y

=

&c

*!

/'

OA 4

And

'

is only to {hew the universality of this Method, are to proceed in other like cafes ; for as to the Equamight have been refolved much more fimply and ex-

Example

how we

and tion

it

itfelf,

in

peditioufly,

Divifion

will be

it

Becaufe

the following manner.

= =

y

~

i

x king the Fluents, y In the fame Equation

+^

^ a*x

6

-f-

-+-

^

*-

; 1

=

6

y

=

*L

4-

,

-^-

&c

-

by

}

And

.

ta-

&c. ^ were >

4- x ^ o, if it requir'd to exprefs x by y, (the Tangent by the Arch,) or if x were made the Relate, and y the Correlate, we might proceed thus. Make x

=

y J

Then x ~-^

,

a?x +- x*

then a*

i,

&c.

=

&e.

or

-

si

5c c

i

G,

And = = 04-^, whence * y,

= 14 = = ^ =s = = jc

o,

taking the Fluxions,

&c.

A-

* o,

'tis

x

&c. and x

* * o,

So that the Terms of this Series will be alternately deficient, and therefore we need not compute them. Taking the Fluxions

&c.

.

again,

A-

=

,

tis

* *

x

j-,

,

=

*

Zxx -j-

&c. and

^==

=:

# * *

2

n ,

&c.

&c. ^ S f 2 ,

i-i->i

r

Therefore x

Again,

=

2V

* -^

x=z -^

,

&c.

-l-^r

>

and


Method of FLUXIONS, and again,

=

A-

c

<j4

8v'

c

_,

6

Again, x

Here

=

A-

"

,

&c.

,

&c.

J

fi

7.1 Cfl

^

= **

gy*

=

&c. x

, '

* # *

--

*

=

x

will be

.

&c..

'

=

7

J and again, x

,

2n*-

6

5

::

I

--

-4--;o.;x4-i2.v.v-(-2.vA-

l

&c.

i,

6

8

^U.

*,

it

#,

>-

21*

and x

'

* * * * *

&c. and

i,

==-***#,

J

C

&c.

,

*

occ.

,

J

= ..i4\

^=

4

c

-

**

,2

tf

# and

for

6cc.

i6y

*

c ,n_.

Subitituting

.

iQAr- 4" 2A ;xr

=

x

'tis

2XX XX

and x writing

for x, x,

v

ig

o,

>'

* * * *

20xx+

fpeaively,

^i

alfo

2 4

'

.

h

=- -

&c.

,

and

t. whence x =.

&c.

,

-^

-f-

&c. for # and x } 16

xx

>X

,

&c

^

&c. and

,

^

.

=

Then x

Sec.

,

&c. re-

,

J

x=***i^- &c. 2==..... &c. and 7^ &c. x= ****** JJpr, &c.

,

3^

,

That

&c.

=y+.^ + -l. +

X

is,

6

l us take the Equation ^*jj* .y j> if the Radius of a Circle be denoted which, /z*x* o, (in by a, then the the of Arch be fame,, and if y correfponding right Sine any which are to extract the Root from we x will be denoted by j) y.

For another Example,,

let

,

Make

x=i,

then

it

will be

M

&c. or

j/=

i,

&c.

J}

the Fluxions

we

Xy __ x y

0, or "y

again,

=

'tis

Therefore

&c.

fore y

a*y

}

Then x

fhall

*

a ;y

^=o,

:=

*

,

^r~7i

'

/=

}

&c. ^

4^

5^'

J

=

y=.

=

&c.

ta ^ing. the

=

__.

o,

or ,

=

o,

^

&c. and

_y

=

* *

^

>

&c.

.

and again

<7

* * *

4

1-

* * *

,

a

9_y

j/

= i^=,&c. = J,& ^= y '

a*y>

Fluxions

y

orj &c,

Taking

o ) or

^^ ^ nc^

* *

x*y

* x,

2X yy

zxy*

=

= -^^

a*, or ;}*

1

^a

3*

j/

=

and therefore

have za*yy

=

=

1

x^y

a*y*~

,

C

.

,

There-

&c. *

^5^ *

*


INFINITE SERIES, * * ,

,

j^-4

and

6cc.

.

= ***#*

r

Fluxions again, 1

z$y

^

6

i6y

g.y

x*y

=

y

*

,

2^Y

= ****** ~ Or = = = y =

and again,

a*jr

7

o,

i

i

xv

c

&c.

&c. Jv

* * *

*

_)'

the

;<

a*

&c. y

r

^-|x

&c.

t

=

Taking

= ~ +-^ = -/, = = ^V, ^.v,

or

Therefore J -2-3 x r

* * *

= j= =

&c.

,

S

o,

x*y

1*7

&c.

, '

n ly

'tis

-IL

::

6

317

r 6

gJT

^ c>

x4 >

and

&c.

,

* *

y ===

=

y

&c.

2a 6

* * * * *

*****#*

&c. 4- -^- 4- -^ II2* 40^.4 If we were required to extraft the Root x out of the fame Equa1 1 l the: rt * x 1)/ 1 o, (or to exprefs the Sine by tion, a y* -^-.6

&c.

.

J

TI2a

and therefore x

i,

Therefore x

x

'tis

again,

=

= =

2^1

=

and x

axx

'tis

* o,

1 t

-^ ,

===

occ.

x

* *

- v* i

?r/24

, *

=:

=

i

6cc.

6 ,

&c. x

_

* j.J

^

-+-

)

&c.

x Again, * O

,

t

I

*__

2Ofl*

-^

-

6

,

Therefore x

6cc.

=

=

..

And

,

6cc.

= =

&c. x

7

x

therefore

-

=

x a

-

,

and

,

=

x

g

,

,

&c,-

,

-

o,

x

&c. and

&c.

=

^>

&c. x

=

occ.

fi

and x

=

x fl

~ 1

y

*

&c. x J

f

=:

^,

,

-1

V*

the-

*

^ j4

?

v

*

*

x=

* * *

^j

i

Taking

Again, x

&c.

Therefore x

J ,

,

&c.

=

Taking the Fluxions

Thence

Oi

or x*

o,

* y,

&c.

-

* *

=

x=

or

&c.

,

&c.

,

=

o,

* * o,

* * *

-^-j

&c. x

-^-

=

^

6 -

and x

&c.

&c. x

a** 1

x1

a1

andx=*

= =

~

,

2a*xx

^

&c.

,

l

t><i

then

put

Fluxions

x

-i-

i,

Arch,) -*,

x

v *

=

* *

*

*

x= **

72Ofl

x= y> ^ <

-f-

&c. If *y

.

were required to extraft the Root y out of w1 o, (where x =s i,) .x*y* + m*y*

it

^

=

this

Equation, proceed

we might


The Method of

3i 8

FLUXIONS,

'"

Becaufe y~ ==.

ceed thus.

an d_y=

x 1/

3xy 'tis

^r=

p

y=

OTX

= '

~

or

OT

y

=

=

&c. and

cfy

{- in*

'tis

a*y -+-

9 x y

,--x 9 -^

;*, &c.

ay

or

o,

*y

&c>

;/

=

xy -+- my

xy

See.

;;;,

have 2a*yy

fhall

v

^^_ jf^l; and making y

-I

=

m

7xy

4

'tis

o, or

Therefore taking the Fluxions again,

-

za''

Fluxions again,

=

&c. and therefore Xy

^

i^LV,

^c

j

o,

-

-

^ ~^

Taking the Fluxions, we

w*> &c.

2x*yy -bJzr$*yy

2xy*

r=

-#

v

1

OT

* * *

4 x2

x

Therefore

'"

*

y=

"~

'

;

A

/2

&c. -^y

"'x,

~^\-', &c.

=

Taking the

ya

=

xy

$xy o,

x

m, &c.

'

zx.

x_y

i/

=

=

'tis

Q

or y ;"

*

x

'

~

g'

lx 4

o

"'' x

9

~

and again,

j

->'

~t~

CT

\y

" vv

___

&c

'

^v

equivalent to a Theorem of our Author's, which (in he For if A- be the another place) gives us for Angular Sections, Sine of any given Arch, to Radius a ; then will y be the Sine of another Arch, which is to the firft Arch in the given Ratio of m to Here if m be any odd Number, the Series will become finite j i. and in other cafes it will be a converging Series. And thefe Examples may be lufficient to explain this Method of Extraction of Roots ; which, tho' it carries its own Demonftration along with it, yet for greater evidence may be thus farther illustrated. In Equations whofe Roots (for example) may be reprefented by the A -+- Ex -f- Cx 4 Dx 3 6cc. (which by due Regeneral Series y duction may be all Equations whatever,) the firfc Term A of the Root will be a given quantity, or perhaps o, which is to be known from the circumftances of the Queilion, or from the given

This

Series

is

=

+

,

=

Equation,


and INFINITE SERIES.

319

have been abi ^antly explain'd already. Equation, by Methods that we flrall have have y B -f- 2C.v -+- 30**, Then making i, &c. where B likewife is a conftant quantity, or perhaps o, and

=

x=

=

firft Term of the Series This therefore is to be y. reprefents the firft Fluxional the derived from Equation, either given or elfe to be found ; and then, becaufe it is y B, &c. by taking the Fluents * Ex, ccc. whence the fecond Term of the Root it will be y

=

=

will be

Then becaufe it is_y= zC quantity zC will reprefent the

known.

-f-

6D.v,

&c. or becaufe

Term of y ; this is the conftant to be derived from the fecond Fluxional Equation, either given or And then, becaufe it is y to be found. zC, &c. by taking the Fluents

it

which the

will be

third

y

firft

=

=

zCx, &c. and again y

*

Term

=

of the Root will be known.

is_y=6D, &c. or becaufe the fent the firft Term of the Series y

it

conftant quantity ;

this

Cx 1 Then

* *

6D

cc.

by

becaufe

will repre-

to be derived

is

,

from the

= =

Fluxional Equation. And then, becaufe it is y 6D, &c. the it will be Fluents v * 6Dx, &c. y * * 3D*'-, by taking See. and _)'==.* * * D* 3 &c. by which the fourth Term of the Root will be known. And fo for all the fubfequent Terms. And hence it will not be difficult to obferve the compofition of the Co-

third

=

,

moft

efficients in

cafes,

and thereby difcover the Law of Continuaare notable and of general ufe.

tion, in fuch Series as If you ihould defire cal

to

Equations are derived

know how

from the

AB

=

the foregoing Trigonometrion Circle, it may be fhewn thus :

the Center A, with Radius a t let the Quadrantal Arch BC be Draw the Tangent BK, and defcribed, and draw the Radius AC. of the Circumthrough any point c ference D, draw the Secant ADK,

meeting the Tangent

in

At any

K.

other point d of the Circumference, but as near to as may be, draw the Secant A.tte, meeting BKin/ ; on

D

Center A, with Radius the

Arch

Then

K/,

fuppofing

AK,

meeting

A

the point

defcribe in

/,

d con-

tinually to approach towards D, till it finally c<- .ncides with it, theTri:

lineum K//6 will continually approach to a right-lined Triangle, and to funilitude w/ith the Triangle ABK So that when Dd is a :

Moment


The Method of

320 Moment

^

x

BD=y

= BK = Moments Kk =

of the Circumference,

=

Make AB

.

a,

the

and inflead of the

;

=AB

a*x

x*y

D

From

will be K-! Da

it

Tangent

~

AB

and the Arch

x,

"

-

will be

it

=

^4 x L)J

&.1

and Dd,

proportional Fluxions x and Jy, and

+

FLUXIONS,

+ *-

or a 2Jv

-

,

a*

y

the

fubftitute

o.

and de

to

let

the Perpendiculars

fall

DE

and

Dg-,

which Dg meets

Then the ultimate form de, parallel to DE, in g. of the Trilineum Ddg will be that of a right-lined Triangle fimilar to DAE. Whence "Dd : dg :: v/

AD

Make Dd, be -

dg,

y

AD

BD=_>', and

a,

:

:

a

:

a'-x

x^y*

1-

\/ a

=

1

AT*.

AE

:

DE=x;

Or^

1

x

:

1

by

right Sine

is

;

to

i?_

by

vice versa.

,.

right

x,

3-

D&q. Moments

and x

Sine

Therefore

Ja

if thefe

;

m, their Fluxions will be "x

and

a

:

the

it

will

or a^y 1

1 ,

is

x,

being

and likewife the Fluxion of an Arch, whofe

y, being exprefs'd i

for

o.

f^_^

each other as

F

/iJJ$

y and

a*

::

Hence the Fluxion of an Arch, whofe exprefs'd

=

and

Fluxions

fubftitute their proportional

x

:

=

in the

Arches are to

fame proportion, x

v

x

= -T37i

:

,

*/

}

a

x

: :

y

or putting

i

:

#=

;,

i,

or 'tis

da .

x

.

l

a*y

==

'

We

thefe,

might which

o ; the fame Equation as before refolved. derive other Fluxional Equations, of a like nature with would be accommodated to Trigonometrical ufes. As

if/ were the Circular Arch, and x its verfed Sine, we mould have a^x* o. Or if y were the Arch, the Equation zaxy* x'-y'l it would be # 4 x* and x the correfponding Secant, x^y* a*-x*y o. Or inftead of the natural, we might derive Equations for

=

=

But I fhall leave thefe Tangents, Secants, &c. that others fuch and might be propofed, to exmany Difquifitions, of the Learner. ercile the Induftry and Sagacity

the

artificial Sines,

SECT,


and INFINITE SERIES. 'SECT. VI.

An

Terms and

Analytical Appendix

',

321

explaining fome

Expreffions in the foregoing work.

mention has been frequently made of given Equations, I mall take ocad libitum, and the like BEcaufe calion from hence, by way of Appendix, to attempt fome kind of explanation of this Mathematical Language, or of the Terms giver/, and required Quantities or Equations, which may afligjfd, affiimed, fome to things that may otherwife feem obfcure, and give light may remove fome doubts and fcruples, which are apt to arife in and others aframed

;

Now the origin of fuch kind of ExpreiTions feems to be this. The whole affair of purfuing probability or of Mathematical Inquiries, refolving Problems, is fuppofed (tho' be tranfacled between two Perfons, or Parties, the Protacitely) to the Refolver of the Problem, or (if you pleafe) between the and pofer Hence this, and fuch like Mafter (or Inftruclor) and his Scholar. the in

Mind of a

Learner.

all

Phrafes, datam reffam, vel datum angtthim, in iniperata rations JeAs Examples inftrudl better than Precepts, or perhaps when care. both are join'd together they inftrucl beft, the Mafter is fuppos'd to

propofe a Queftion or Problem to his Scholar, and to chufe fuch Terms and Conditions as he thinks fit ; and the Scholar is obliged

Problem with thofe limitations and reftriclions, with Indeed it is required thofe Terms and Conditions, and no other. on the part of the Mafter, that the Conditions he propofes may be

to

folve the

confident with one another ; for if they involve any inconfiftency or contradiction, the Problem will be unfair, or will become abNow furd and impomble, as the Solution will afterwards difcover. thefe Conditions, thefe Points, Lines, Angles, Numbers, Equations, Gfr. that at firft enter the ftate of the Queftion, or are fuppofed to

be chofen or given by the Mafter, are the data of the Problem, and the Anfwers he expects to receive are the qii(?/ita. As it may fometimes the data may be more than are neceffary for determining that happen, the^Qiu ft'.on and lo perhaps may interfere with one another, and the Problem (as now propofed) may become impolTible fo they may be fewer than are neceffary, and the Problem thence will be indetermin'd, and may require other Conditions to be given, in order to a compleat De,

;

termination, or perfectly to fulfil the quafita. In this cafe the Scholar is to fupply what is wanting, and at his difcretioa miy a(Jit me fuch and fo

many otherTerms and Conditions, Equations and Limitations, as he finds

T

t

will


7&? Method of

322

FLUXIONS,

will be neceffary to his purpofe, and will befl conduce to the fimthe eafieft, and neateft Solution that may be had, and pleft, yet in

For it is convenient the Problem fhouM the moft general manner. as as be propofed may be, the better to fix the Imaginaparticular tion ; and .yet the Solution mould be made as general as poffible, that it may be the more inftrucHve, and extend to all cafes of a like nature.

Indeed the word datum

is

often ufed in

a fenfe

which

is

fome-

but which ultimately centers in it. As datum, when one quantity is not immediately given, but however is neceffirily infer'd from another,, which other perhaps is neceffarily infer'd from a third, and fo on in a continued Series,

thing different that is call'd a

from

this,

from a quantity, which is known or neceffarily infer'd given before This is the Notion of Euclid'?, fenfe the in explained. data, and other Analytical Argumentations of that kind. Again, that is often call'd a given quantity, which always remains conftant and invariable, while other quantities or circumftances vary ; becaufe fuch as thefe only can be the given quantities in, a Problem, when taken in the foregoing fenfe. To make all this the more fenfible and intelligible, I /hall have, till it is

recourfe to a few pradlical inftances, by way of Dialogue, (which, was the old didadlic method,) between Mafter and Scholar; and this only in the common Algebra or Analyticks, in which I fhall borrow Examples from our Author's admirable Treatife of The chief artifice of this manner of SoluUniverfal Arithmetick.

my

tion will confift in this, that as faft as the Mafter propofes the Conditions of his Queftion, the Scholar applies thole Conditions to

them Analytically, makes all the aeceffary deducfuch confequences from them, in the fame order derives and tions, they are propofed, as he apprehends will be mcft fubfervient to the And he that can do this, in all cafes, after the fureft, fimSolution. readieft manner, will be the beft and ex-tempore Mathematipleft, But this method will be beft explain'd from the cian.

ufe,

argues from

following

Examples.

M.

I.

the

Sum

A

Gentleman being 'willing to diftribiite Abns he intended to diftribute be reprefented by x.

S.

Let

M. Among

S. Let the number of poor be then - would fbme poor people. _}>, M. He wanted 3 fiillings have been the fhare of each. S. Make for lake of the and let the rf, univerfality, 3 pecuniary Unit be then the be to Sum distributed one Shilling ; would have been x-{-a,

=

and"


and INFINITE SERIES. and the fhnre of each would have been

^^-

.

y

might receive $ fallings.

x

=

by

4=f, fallings

=5

M.

So that each

^=

then

b,

whence

M. "Therefore he gave every ot.e 4 fallings. S. Make Money diftributed will be cy. M. And he has 10 S. Make io d, then cy -f- d was the Money remaining. a.

then the

he intended

y

Make 5=^,

S.

32"

^jt-f

at

=

firft

M.

.

number was y

= =

to diftribute; or cy -+-

J^rf* w<2J 7 ''

=

?

=

number of poor

the

3

d

=

"*"'

4

M.

1-2.

(x

=) by-

people ?

a, S.

or

The

^W /60w much Alms

He

= ^

had at firft x a by 62 M. How do Solution? 3 fhillings. you prove your S. His Money was at firft 62 fhillings, and the number of poor But if his Money had been 62-4-3 === ^5 r people was 1 3. 3 x 5 each poor perfon might have received 5 millings. But fhillings, then as he gives to each 4 fhillings, that will be 13x4=52 fhillings diftributed in all, which will leave him a Remainder of 62 52

= 5x13 tf/tf

Of

c=

10

II.

the

tff

^ry? intend

to

diftribute ?

S.

fhillings.

M.

A young Merchant,

World with a

certain

Sum

at his firjl entrance npon bufmefs, began S. Let that Sum be x, the of Money.

one Pound. M. Out of which, to maintain pecuniary Unit being S. Make the the Jirjl year, he expended 100 pounds. given himfclf he had to trade with then x ioo a. M. He number tf; traded with the reft, and at the end of the year had improved it by a .S. For univerfality-fake I will aflume the general numthird part. n then the Improveber n, and will make ^ i, (or n ;)

=

=

= =

a nx na x -f- a, and the Tradingi xx ment was n fiock and Improvement together, at the end of the firft year, was M. He did the fame thing the fecond year. S. That is, na. MX

whole Stock being now nx na, deducting a, his Expences for na a for a Trading-ftock, and he would have nx na ix nx n*a nx -f- a for this year's Imn a, or n'-x which make n'-x n*a na for his Eftate at together provement, M. As aljo the third year. S. His die end of the fecond year. nx a whole Stock being now ;< a x na, taking out his Expences his

this year,

n'-a na a, Trading-ftock will be n*x i X*Abe n n'-a na a, Improvement n=a n'-x -f- a, and the Stock and or J x Improvement together, n*a or his whole Eftate at the end of the third year will be n*x

for the third year,

^~

and the

_

n1 a

his

this year will

na, or in a better

form

n"'x

T

2

-+

"-^na.

In like manner if


Th* Method of

324

FLUXIONS,

he proceeded thus the fourth year, his Eftate being now n*x rf a na, taking out this year's Expence, his Trading-flock n a na n>a if a a, and this year's Improvement is will be n>x if a na or n*x is a n^a n*x -f- a, Fx n=x a, n his will be n*x n*a tfa 72-^2 which added to Trading-ftock if

_

i

na, or

x

11*

*

na, for his Eftate at the end

-f-

year.

And

at the

end of the

by Induction,

fo,

his Eftate will be

found n s x

And

if I

fifth year.

univerfally,

m

Number m.

neral

of the fourth

nx -f-

his Eftate will be

_ ~

-4-

-

na

aflume the ge-

l

-na at the end of i

any number of years denoted by m. M. But he made his Eftate double to -what it was at firft. then nmx -tS. Make 2 m m ==-#. M. At the end of 3 years. bx, or x l _^na

= Then #2=3, a-=. =

=

n

S.

i

=

n"

x

100, b

b

2,

n

=

%,

,

and therefore x=s..

64

400

=

M.

1480.

j%!2<z/ <was his

pounds.

M. Two

III.

As

Bvdies

A

B

and

S. It

Eftate atjirji?

are at a given diftance

was 1480

from each

be given, though it is not fo. other. Let the initial diftance of the actually, I may therefore aflume it. M. And e y and let the Linear Unit be one Mile. Bodies be 59 move equably towards one another. S. Let x reprefent the whole x be the before they meet ; then will e fpace defcribed by whole fpace defcribed by B. M. With given Velocities. S. I will to be fuch, that it will move 7 c Miles aflume the Velocity of of one in 2 Unit Time Hour. Then the beHours, being S.

their diftance

is

faid to

=

A

=f

caufe

it is

c

:

f

time

Y

Alfo

will

move

8

g

::

'-7%. i

=

equal

x

e

b

=

A

= :

: :

x

:

-

,

A will move

his

aflume the Velocity of B to be fuch, that it d Miles in 3 ==:g Hours. Then becaufe it is d : j-g>

B

will

move

his

whole fpace -

U's

in the

I will

M. But A moves a given time - Hour. M. Before B begins to move. to

whole fpace x

time added

to

the

time hy

e

x

in the time

S.

Let that time be

S.

Then A"s time

or

^ = '^g

is

-f- h.

M,'


and INFINITE SERIES, M. Where

will they meet., or

have defcribed ? * s*i

S.

From

this

what

325

will be the fpace that each

Equation we

fhall

=

have x

=

1^ X7== Jr x 7' X7== 35 Miles, which will ' / ' x 3' x2 37 oxz-r7 35 be the whole fpace defcribed by A. Then e 59 be the whole fpace defcribed by B. will Miles 24 IV. M. Jf 12 Oxen can be maintained by the Pafture 0/37 Acres S. Make 12 ^, 4=cj #, 3! of Meadow-groundfor 4 weeks,

=

.

then aiTuming the general

wards

Numbers

as occafion Ihall require,

Oxen If

we

e,

x=

=

=

f, h, to be determin'd after-

mall have by analogy


7%e Method

26

3

^FLUXIONS, -

t

-

-

df *r

c

ace

*'

Ti bh

-

,

"

Oxen

-m

h

c

1

c

f7

ace

df h

.

T

tO

c

'

TZ bh

'

number of Oxen

will be the

which

Time

Oxen

Time ,,

~*

that may be maintain'd by the of But it the e, only growth pafture during the whole time h. was found before, that without this growth of the Grafs, the Oxen

might be maintain'd by the pafture

^

fore thefe

two

^

together, or

ber of Oxen that

_f_ -^-^

e

x -

time

for the

~ "" ,

h.

There-

will be the

be maintain'd by the pafture e, and its growth the time b. M. How many Oxen may be maintogether, during S. Suppofe x to tain d by 24 Acres offitch pafture for 1 8 week s ?

may

make

be that number of Oxen, and

24=^,

and

h=

18.

Then

by analogy

Oxen

Pafture

If

Then

ex

r And conlequently T J .

,

i

requ ij. e

ex

g

=

f ~ jf = T + a^j

dft

x

21*9 I

1

O

2

ac

x 4

during the time

b.

ace j-r bh r

/>

14

i

J <g

fi

J-j

S. Let x be the prefent value have an Annuity of i pound to be received i year hence, then (by analogy) x* will be the prefent value of I pound to be received 2 years hence, &c. and in general, x" will be the prefent value of i pound to be received m years hence. Therefore, in the cafe of an Annuity, the Series x -f x* -+- x"' ~+- x*, &c. to be continued to fo many Terms as there are Units in m, will be the prefent value of the whole Annuity of i pound, to be continued for m years. But becaufe

M. If I

V.

-I

+

X

.

=x-{- x

as there are Units

*~ y

will

1

-f- x''

in in,

reprefent

to be continued for

;;/

H-A' 4

,

&c. continued to

r

fo

many Terms

may appear by Divifion 3) therefore Amount of an Annuity of i pound, S. Make M. Of Pounds.

(as

the

years.

=

a*


and INFINITE SERIES.

= m=

then

a,

the

M.

a.

^

Amount

of

this

=

c,

then

-a

=

will

years

be

S.

Then

pounds in ready Money.

S.

Make

or

c,

m

fuccejji'vely.

To be continued for 5 years

M. Which IJell for

5.

for

Annuity

327

x""

--

1

I

x

In any particular cafe the value of x tion of

this affected

may be found by the RefoluM. What Interefl am I alfav'd per

=

Equation.

centum per annum? S. Make 100 ^; then becaufe x is the value of i to be i received prefent pound year hence, or (which the fame thing) becaufe the prefent Money x, if put out to ufe, i will year produce i pound; the Intereft alone of i pound for i year will be i x, and therefore the Intereft of 100 (or K) for i will be b bx y which will be known when x pounds year is

in

is

known.

And

might be fufficknt to (hew the conveniency of this Mebut I mall farther illuftrate it by one Geometrical Problem, which mall be our Author's LVII. VI. M. In the right Line AB I give you the ftuo points and B.

thod

S.

this

;

Then

their diftance

two points figure

C

and

ACBD is

D

AB

=m

out of the

A

is

given

Line AB,

M. As likenaife Then conlequently

alfo.

S.

the the

given in magnitude and fpe-

and producing CA and CB towards d and <T, I can

cie

;

takeA</=AD, and

Bf=KD.

M.

Aljb

I give

you

the

indefi-

nite right Line in po/iticn,

EF

pajjing

thro' the

ADE

S. Then the Angles and BDF are given, to which green point D. (producing AB both ways, if need be, to e and y ) I can make the Angles h2e and B<f/~ equal refpedlively, and that will determine the and f, or the Lines Ae And becaufe a, and Ef=c. points e de and <T/"are thereby known, I can continue de to G, fo that^/G.

f

=

=

Sj\

and make the given

line

eG

c=

b.

Likewife

I

can draw

CH

and


The Method of FLUXIONS,

228

CK parallel to ed and f$ and becaufe the Triangle

and

K

;

fpecie, let

green points

move

DAL,

obtain a

AL

into another fituation

</A/,

and

new

CAc will be equal.

fituation

BL

H

in

Angles CAD and C BD be conceived to revolve or Poles A .and B. S. Then the Lines AD

the given

will

=

CK

make

will

I

AB

refpeftively, .meeting

and

CHK will be given in magnitude snd M. Now d, CH=e, and HK=/

and

and cAt,

about the

and

fo as that the

BD

CA^

Angles

CB^ will cBA, fo as that the Angles DBL, <fBAand Alfo the Lines

and

CBc will be equal. M. And let D, the Interfeflion of the Lines AD and BD, always move in the right Line EF. S. Then the new point of Interfedtion L is in EF; then the Triangles DAL and </A/, as alfo DBL andJ'BA, are equal and iimilar

;

then^//=

DL=

cTA,

and therefore

M. What

will be the nature of the Curve defer ibed by the G/==/A. S. From the new point of Interfection c other point of InterfeSt ion C ? and to AB, I will draw the Lines ch and ck, parallel to refpec-

CK

CH

Then

tively.

though not

will the Triangle chk be given in fpecie,

Alfo the Triangle in magnitude, for it will be Iimilar to ^CHK. to the And indefinite Line Bck will be fimilar mayBtf. be aflumed for an Abfcifs, and ck y may be the correfponding Then becaufe it is Bk (x) ck (y ) Ordinate to the Curve Cc.

Bk=x

=

:

::

= =

Bf(c) :/A

will remain

and

CHK,

CK

(/O

=m

ck

::

(/)

Then

.

CK

will be

it

Subtraft

G/. -

le=.b

HK

:

hk

Bk

;

(

y]

..

X

hk

:

le

:

-v

(b

But

Therefore

bft. xv

ae

dc

.

)

(e)

=

c

(a)

there

becaufe of the fimilar Triangles chk

CH

:

(d)

Ge-=&, and

from

this

demy

it is

m

:

ck (y)

:

:

ch=

= AB

Therefore A/J

.

-\

A/6

(m x

bdx*

x

fxb

:

'f )

+ bdmx =

J o.

And

.

'j

cb

2)

= ^,

.

::

or

In which

becaufe the indeterminate quantities x and y arife only Equation, to two Dimenfions, it mews that the Curve defcribed by the point

C

Conic Section. M. Ton have therefore folved the Problem in general, but you fionld now apply your Solution to the feveral fpecies of Conic Sections in parthe following manner S. That may eafily be done in ticular. is

a

:

e

+

come fcf

l'f

ctl

-

__ 2p

zpcxj>

t

h en the foregoing Equation will bebdx'- 4- bdmx o, and by exdemy an(j

t

=

trading


and INFINITE SERIES. the

trading

V

+ - x x* + ft

I'P

!Z

that if the

be equal to

fame that

^

,

1

is,

- be

affirmative,

or if

?

if

_ XA + __.

''d

"

</*;*

.

the quantity

prefent,

-f-

if

-..

be

will

it

y

=

XT

Now

abfent, or if --

-.x

-f.

.

.

.

here

it

is

-4jj

(changing

its

,

.

plain,

=

o, or

But

if

the

and equal to fome affirmative quantity, (which will

affirmative, it

^

-f-

fign) fhould

then the Curve would be a Parabola.

Term were is,

is

that

;

I

*"1

- root

~ x X L were j 4-

Term

=

r

Square

329

be negative and

lefs

always

than

-.

>}

be

when

the Curve

an Hyperbola. Laftly, if the fame Term were prefent and is negative, (which can only be when negative, and greater than

will be

the Curve will be an Ellipfls or a Circle.

y> I

mould make an apology

the Reader, for this Digreffion did not hope it might contribute if not to his improvement. And I am

from the Method of Fluxions, to his entertainment at leaft,

whoever fhall go through the of our Author's curious Problems, in the fame manner, (where-

fully convinced reft

to

if I

by experience,

that

according to his ufual brevity, he has left many things to be or fuch other Queflions fupply'd by the fagacity of his Reader,) whether and Mathematical Diiquifitions, Arithmetical, Algebraical, Geometrical, &c. as may eafily be collected from Books treating in,

fhall do this after the fay, whoever foregoing a very agreeable as well as profitable exercife : As being the proper means to acquire a habit of Investigation, or of arguing furely, methodically, and Analytically, even in other Sciences as well as fuch as are purely Mathematical ; which is the

on

theie Subjects ; manner, will find

great

I

it

end to be aim'd

at

by thefe

U

Studies.

u

SECT,


7%e Method of

330

SECT. VII. The Conclufion

be

Series

;

are

now

containing a Jhort recapitu-

;

review of the whole.

lation or

E

FLUXIONS,

which may properly enough

arrived at a period,

conclujion of tie Method of Fluxions ami Infinite for the defign of this Method is to teach the nature of Series call'd the

how

in general, and of Fluxions and Fluents, what they are, are derived, and what Operations they may which ;

they

undergo

defign As to the applica(I think) may now be faid to be accompliili'd. tion of this Method, and the ufes of thefe Operations, which is all that now remains, we mall find them infilled on at large by the Author in the curious Geometrical Problems that follow. For the

whole that can be done,

either

by

Series or

by Fluxions, may

eafily

be reduced to the Refolution of Equations, either Algebraical or Fluxional, as it has been already deliver'd, and will be farther apI have continued my Annotations ply'd and purfued in the fequel. in a like manner upon that of the Work, and intended to have part added them here ; but finding the matter to grow fo faft under my hands, and feeing how impoffible it was to do it juftice within fuch narrow limits, and alfo perceiving this to a

I refolved

competent fize; Reader unfinifh'd as it

opportunity,

to

lay

it

work was already grown before the Mathematical

referving the completion of it to a future if I mall find my prefent attempts to prove acceptable. that remains to be done here is this, to make a kind is,

Therefore all of review of what has been hitherto deliver'd, and to give a fummary account of it, in order to acquit myfelf of a Promiie I made

And having there done this already, as to the Auof the work, I (hall now only make a fhort recapitulapart tion of what is contain'd in my own Comment upon it. And firft in my Annotations upon what I call the Introduction, or the Refolution of Equations by infinite Series, I have amply pur-

in the Preface. thor's

fued a ufeful hint given us by the Author, that Arithmetick and Algebra are but one and the fame Science, and bear a ftridl analogy to each other, both in their Notation and Operations ; the firft computing after a definite and particular manner, the latter after a general and indefinite manner So that both together compofe but :

one uniform Science of Computation. For as in common Arithmetick we reckon by the Root Ten, and the feveral Powers of that

Root

;

fo in Algebra,

or Analyticks,

when

the

Terms

are orderly dilpos'd


and INFINITE SERIES. as

difpos'd

is

we reckon by any Number

prefcribed,

we may

Powers, or

take any general

331

other Root and its for the Root of our

Arithmetical Scale, by which to exprefs and compute any Numbers And as in common Arithmetick we approximate continually required. to the truth, by admitting Decimal Parts /;; infnititm, or by the ufe of Decimal Fractions, which are compofed of the reciprocal

Powers of the Root Ten

our Author's improved Algebra, or in the Method of infinite converging Series, we may continually apNumber or Quantity required, by an orderly fucproximate to the cefiion of Fractions, which are compofed of the reciprocal Powers ;

fo in

Operations in common Arithmetick, having a due regard to Analogy, will generally afford us proper patterns and fpecimens, for performing the like Operations in this Univerfal Arithmetick.

of any Root

And

in general.

the

known

proceed to make fome Inquiries formation of infinite Series in general, and particularly into their two principal circumftances of Convergency and Divergency; wherein I attempt to (hew, that in all fuch Series, whether converging or diverging, there is always a Supplement, which if not exprefs'd is however to be underftood which Supplement, when it can be aicertained and admitted, will render the Series finite, perfect, and

Hence

into the nature

I

and

;

diverging Series this Supplement muft indifpenand fablv be admitted exhibited, or otherwise the Conclufion will be But in converging Series this Supplement imperfect and erroneous.

accurate.

That

in

be neglected, becaufe it continually diminifhes with the Terms the of Series, and finally becomes lefs than any affignable quantity. And hence arifes the benefit and conveniency of infinite converging

may

Scries

;

that whereas that Supplement is commonly fo implicated and with the Terms of the Series, as often to be impoiliblc to

entangled

be extricated and exhibited ; in converging Series it may fafely be neglected, and yet we mall continually approximate to the quantity reAnd of this I produce a variety of Inftances, in numerical quired, and other Series. I then go on to mew the Operations, by which infinite Scries are either produced, or which, when produced, they may occasionally As firft when fimple fpccious Equations, or purs Powers, undergo. are to be refolved into fuch Series, whether by Divifion, or by Extraction of Roots ; where I take notice of the ufe of the afore-mention'd Supplement,

by which

Scries

may

be render'd

finite,

that

is,

may be compared with other quantities, which are confider'd as I then deduce feveral ufeful Theorems, or other Artifices, given.

Una

for


tte Method of

332

FLUXION s,

more expeditious Multiplication, Divifion, Involution, and infinite Series, by which they may be eafily and reain all cafes. Then I fhew the ufe of thefe in dily managed pure or from whence I take occasion to introExtractions; Equations, duce a new praxis of Refolution, which I believe will be found to be very eafy, natural, and general, and which is afterwards ap-

for the

Evolution of

of Equations. Then I go on with our Author to the Exegefis numerofa, or to the Solution of affefted Equations in Numbers ; where we mall find his Method to be the fame that has been publifh'd more than once in other of his pieces, to be very {hort, neat, and elegant, and was a great Improvement at the time of its firft publication. This Method is here farther explain'd, and upon the fame Principles a general Theorem is form'd, and diftributed into feveral fubordinate Cafes, by which the Root of any Numerical Equation, whether pure or affected, may be computed with great exactnefs and facility. From Numeral we pafs on to the Refolution of Literal or Specious affected Equations by infinite Series ; in which the firfl and chief difficulty to be overcome, confifts in determining the forms of the feveral Series that will arife, and in finding their initial ApproximaThefe circumftances will depend upon fuch Powers of the tions. Relate and Correlate Quantities, with their Coefficients, as may happen to be found promifcuoufly in the given Equation. Therefore in latum, the Terms of this Equation are to be difpofed in longum of the Indices thofe lenft to a or at combined Powers, according

ply'd to

all

fpecies

&

Arithmetical Progreffion in p/ano, as is there explain'd ; or according to our Author's ingenious Artifice of the Parallelogram and Ruler, the reafon and foundation of which are here fully laid open. This will determine all the cafes of exterior Terms, together with the Progreffions of the Indices ; and therefore all the -forms of the feveral Series that may be derived for the Root, as alfo their initial Coefficients,

We

Terms, or Approximations.

then farther profecute the Refolution of Specious Equations, by diverfe Methods of Analyfis or we give a great variety of ProcefTes, by which the Series for the Roots are eafily produced to any number of Terms required. Thefe ProcefTes are generally very lim-,

and depend chiefly upon the Theorems before deliver'd, for And finding the Terms of any Power or Root of an infinite Series. is illustrated a and 'd whole of Inthe by exemplify great variety ftances, which are chiefly thofe of our Author.

ple,

The


and INFINITE SERIES. The Method of infinite Series being we make a Tranfition to the Method of ture

and foundation of that Method

is

thus

333 dilcufs'd,

fufficiently

Fluxions, wherein the na-

And fome

explain'd at large.

general Observations are made, chiefly from the Science of Rational Mechanicks, by which the whole Method is divided and diftinguiih'd into its two grand Branches or Problems, which are the Diredt and Inverfe Methods of Fluxions. And fome preparatory Notations are deliver'd

and explain'd, which equally concern both

thefe

Me-

thods.

then proceed with my Annotations upon the Author's firft Problem, or the Relation of the flowing Quantities being given, to determine the Relation of their Fluxions. I treat here concerning Fluxions of the firft order, and the method of deducing their EquaI tions in all cafes. explain our Author's way of taking the Fluxions of any given Equation, which is much more general and fcientifick than that which is ufually follow'd, and extends to all the varieties I

of Solutions.

This

to

alfo apply'd

is

which means

flowing Quantities, by cafes, in which either

compound,

it

Equations involving feveral likewife

irrational,

comprehends thofe

or mechanical

Quan-

be included. But the Demonftration of Fluxions, and of the Method of taking them, is the chief thing to be confider'd here; which I have endeavour'd to make as clear, explicite, and fa-

tities

may

tisfactory as I

was

and to remove the difficulties and objections it But with what fuccefs I muft leave

able,

that have been raifed againft to the judgment of others.

:

then treat concerning Fluxions of fuperior orders, and give the For deriving their Equations, with its Demonftration. tho' our Author, in this Treatife, does not expreffly mention thefe orders of Fluxions, yet he has fometimes recourfe to them, tho' taI have here ("hewn, that they are a necelTary citely and indirectly. and that refult from the nature and notion of nrft Fluxions not abfolutely all thefe feveral orders differ from each other, and effentially, but only relatively and by way of companion. I

Method of

;

And and

this I

I

prove

actually

as

from Geometry as from Anaiyticks ; and make fenfible thefe feveral orders ot

well

exhibit

Fluxions.

But more efpecially in what I call the Geometrical and Mechanical Elements of Fluxions, I lay open a general Method, by the help of Curve-lines and their Tangents, to reprefent and exhibit Fluxions and Fluents in all cafes, with all their concomitant Symptoms and AffecYions,


^

f

334

3e

Method of FLUXIONS,

Aiic&ions, after a plain and familiar manner, and that even to ocular view and infpedlion. And thus I make them the Objects of Senfe, which not only their exiitence is proved beyond all poflible con-

by

tradiftion,

but alfo the

fully evinced,

verified,

Method of and

deriving

them

is

at the

fame time

illuftrated.

Then

follow my Annotations upon our Author's fecond Problem, or the Relation of the Fluxions being given, to determine the Relation of the flowing Quantities or Fluents ; which is the fame thing And firft I explain (what outas the Inverfe Method of Fluxions. Author calls) a particular Solution of this Problem, becaufe it cannot be generally apply 'd, but takes place only in fuch Fluxional Equa-

might have been, previoufly derived or Fluential Equations. Whereas the Algebraical that and whofe Fluents or Roots Fluxional Equations ufually occur, are required, are commonly fuch as, by reafon of Terms either re-

tions as have been, or at leaft

from fome

finite

dundant or deficient, cannot be refolved by this particular Solution ; but muft be refer'd to the following general Solution, which is here distributed into thefe three Cafes of Equations.

The firft Cafe of Equations is, when the Ratio of the Fluxions of the Relate and Correlate Quantities, (which Terms are here exbe exprefs'd by the Terms of the Correlate plain'd,) can Quantity in which Cafe the Root will be obtain'd by an eafy alone proIn finite Terms, when it may be done, or at leaft by an cefs And here a ufeful Rule is explain'd, by which infinite Series. infinite an Expreffion may be always avoided in the Conclufion, which otherwife would often occur, and render the Solution inexpli;

:

cable.

The

fecond Cafe of Equations comprehends fuch Fluxional Equations, wherein the Powers of the Relate and Correlate Quantities, with their Fluxions, are any how involved. Tho' this Cale is much more operofe than the former, yet it is folved by a variety of eafy

and fimple Analyfts, (more fimple and expeditious, I think, than thofe of our Author,) and is illuftrated by a numerous collection of Examples. The third and laft Cafe of Fluxional Equations is, when there are more than two Fluents and their Fluxions involved j which Cafe, without much trouble, is reduced to the two former. But here are alfo explain'd fome other matters, farther to illuftrate this Dodlrinej as the Author's Demonftration of the Inverfe Method of Fluxions, the Rationale of the Tranfmutation of the Origin of Fluents to other i

places


and INFINITE SERIES. at

places

pleafure,

the

way of

ments of Fluents, and fuch

finding the contemporaneous Incre-

like.

Then to conclude the Method of Fluxions, is propofed and explain'd, for general Method kinds of Equations,

335

or

a very convenient

Fluxional,

and

the Refolution of

all

by having recourfe

Algebraical This Method indeed is not conto fuperior orders of Fluxions. Author's tain'd in our prelent Work, but is contrived in purfuance of a notable hint he gives us, in another part of his Writings.

And

this

Method

is

exemplify 'd by feveral curious and ufeful Pro-

blems. Laftly, by way of Supplement or Appendix, fome Terms in the Mathematical Language arc farther explain'd, which frequently occur in the foregoing work, and which it is very neceflary to apprehend rightly. And a fort of Analytical Praxis is adjoin'd to this in which is Explanation, to make it the more plain and intelligible ; exhibited a more direct and methodical way of refolving fuch Algebraical or Geometrical Problems as are ufually propofed ; or an attempt is made, to teach us to argue more cloiely, dhtinctly, and Analytically. this

is And chiefly the fubftance of my Comment upon this part of our Author's work, in which my conduct has always been, to endeavour to digeft and explain every thing in the moft direct and natural order, and to derive it from the moft immediate and genuine I have always put myfelf in the place of a Learner, and Principles. have endeavour'd to make fuch Explanations, or to form this into fuch an Inftitution of Fluxions and infinite Series, as I imagined would have been ufeful and acceptable to myfelf, at the time when I fidl Matters of a trite and eafy nature enter'd upon thefe Speculations. But in things of more animadverfion over with a have I flight pafs'd or greater difficulty, I have always thought myfelf obliged novelty, to be more copious and explicite ; and am conlcious to myfelf, that :

Wherever have every where proceeded cumjincero ammo docendi. have fallen fhort of this defign, it fliould not be imputed to any want of care or good intentions, but rather to the want of fkill, or I (hall be to the abftrufe nature of the lubject. glad to fee my deand (hall be abler fects hands, by always willing and thank-

I

I

fupply'd

ful

to be better

What

instructed.

and

may furnifli perhaps will give the greateft difficulty, be the Explanations mod matter of objection, as I apprehend, will before given, of Moments, -vanifiing quantities, infinitely little quantitles,


The Method of

236 fjfies,

and

and the

FLUXIONS,

which our Author makes ufe of in this Treatife, deducing and demonftrating hisMethod of Fluxions. here add a word or two to my foregoing Explana-

like,

elfe where, for

I fhall

therefore

hopes farther to clear up this matter. And this feems to be the more necefTary, becaufe many difficulties have been already ftarted about the abftracl nature of theie quantities, and by what name they ought to be call'd. It has even been pretended, that they

tions, in

are utterly impoffible, inconceiveable, and unintelligible, and it may therefore be thought to follow, that the Conclu lions derived by their means muft be precarious at leaft, if not erroneous and impoflible. to remove this difficulty it mould be obferved, that the only

Now

by our Author to denote thefe quantities, is the But this o, by itfelf, or affected by fome Coefficient. a and finite ordinary quantity, which Symbol o at firft reprefents mu ft be understood to diminim continually, and as it were by local Motion ; till after fome certain time it is quite exhaufted, and termiThis is furely a very intelligible Notion. nates in mere nothing. But to go on. In its approach towards nothing, and juft before it becomes abfolute nothing, or is quite exhaufted, it muft neceflarily of all proportions. For it cannot pafs through vanifhing quantities to at once an affignable quantity that were ; nothing pafs from being to proceed per fa/turn, and not continually, which is contrary to the While it is an affignable quantity, tho' ever fo little, Suppofition. exact truth, in geometrical rigor, but only an Apthe not is it yet and to be accurately true, it muft be lefs than it ; proximation to

Symbol made

letter

ufe of

either

a vanifhing is, it muft be the Conception of a Moment, or Therefore vanishing quantity. muft be admitted as a rational Notion. quantity, But it has been pretended, that the Mind cannot conceive quanbe fo far diminifh'd, and fuch quantities as thefe are repretity to Now I cannot perceive, even if this impoflifented as impoffible. that the Argumentation would be at all affected bility were granted, would be the lefs certain. The imby it, or that the Concluiions of Conception may arife from the narrownefs and imperpoffibility fection of our Faculties, and not from any inconfiftency in the naSo that we need not be very folicitious about ture of the thing. the pofitive nature of thefe quantities, which are fo volatile, fubas to efcape our Imagination ; nor need we be tile, and fugitive, in much pain, by what name they are to be call'd j but we may confine ourfelves wholly to the ufe of them, and to difcover their

any

that affignable quantity whatfoever,

properties,


and INFINITE SERIES.

337

They are not introduced for their own fakes, but only properties. as fo many intermediate fteps, to bring us to the knowledge of other are real, intelligible, and required to be known. quantities, which It is fufficient that we arrive at them by a regular progrefs of diminution, and by a juft and neceflary way of reafoning and that they are afterwards duly eliminated, and leave us intelligible and For this will always be the confequence, indubitable Conclusions. when we argue let the media of ratiocination be what they will, And ftriet Rules of Art. it is a to the very common according ;

thing in Geometry, to make impoffib'e and nbfurd Suppofitions, which is the fame thing as to introduce iinpoffible quantities, and

by

their

We

tities,

means

to difcover truth.

have an inftance fimilar to

which, though

as

this,

in

another fpecies of Quan-

inconceiveable and

as

impofTible as thefe

can be, yet when they arife in Computations, they do not affect the Conclufion with their impoffibility, except when they ought fo to do; but when they are duly eliminated, by juft Methods of Reduction, the Conclufion always remains found and good. Thefe. Quantities are thofe Quadratick Surds, which are diftinguifh'd by the name of impoffible and imaginary Quantities ; fuch as ^/ i,

^/a, number

v/ is

3,

&c. For they import, that a quantity or which multiply'd by itfelf mall produce a which is manifeftly impoffible. And yet thefe

v/

4,

to be found,

negative quantity

;

have all varieties of proportion to one another, as thofe quantities aforegoing are proportional to the poffible and intelligible numbers 8cc. respectively -,. and when I, ^/2, v/3, 2, they arife in Compuare eliminated and excluded, they always leave tations, and regularly a juft and good Conclufion. lax" Thus, for Example, if we had the Cubick Equation x~> from whence we to were extract the Root x; 42 =o, -J-4IX we to mould have this fiird ExRule, by proceeding according the x for Root, 4 -f- y'3 4- v/ -"fr-f- ^J^ ,/ preffion -~ is involved -^, in which the impoffible quantity ^/ and this not to be as abfurd and ufelefs, yet Expreffion ought rejected becaufe, by a due Reduction, we may derive the true Roots of the Equation from it. For when the Cubick Root of the firft innis culum rightly extracted, it will be found to be the impoffible Number i -+as may appear by cubing ; and when the ^/ Cubick Root of the fecond vinculum is extracted, it will be found

=

;

,

to be

i

\/-

j-

Then by

collecting

X.x

thefe

Numbers, the im-

-


338 impoffibie

77je

Method of FLUXIONS,

Number </ will be

will be eliminated,

found x

the Equation Or the Cubick Root of the

=

4

i

i

=

and the Root of 2.

' be A -f- y/ T T) as may likewife appear by Involution ; and of the fecond vincu_' So that another of the Roots of lum it will be | T </ Or the Cuthe given will be x 4 -f- 1 -f- A 7.

firft

.

vinculum will

=

alfo

=

Equation bick Root of the fame firft vincuhtm will be \ i| J v/ i H- ^/ and of the fecond will be .11. So that the third And Root of the given Equation will be x T 4 - 4 3in like manner in all other Cubick Equations, when the furd vincula include an impoffible quantity, by extracting the Cubick Roots, and then by collecting, the impoffible parts will be excluded, and the three Roots of the Equation will be found, which will always be But when the aforefaid furd vincula do not poffible. include an impoffible quantity, then by Extraction one poffible Root only will be found, and an impoffibility will affect the other two Roots, or will remain (as it ought) in the Conclufion. And a like judgment may be made of higher degrees of Equa-

=

=

tions.

So that thefe impoffible quantities, in all thefe and many other inftances that might be produced, are fo far from infecting or dethat they are the neceflary ftroying the truth of thefe Conclufions, means and helps of difcovering it. And why may we not conclude the fame of that other fpecies of impoffible quantities, if they muft needs be thought and call'd fo ? Surely it may be allow'd, that if thefe Moments and infinitely little Quantities are to be elteem'd

a kind of impoffible Quantities, yet neverthelefs they may be made in findufeful, they may affift us, by a juft way of Argumentation, or or other poffible Quanof Velocities, Fluxions, ing the Relations And finally, being themfelves duly eliminated and tities required. excluded, they may leave us finite, poffible, and intelligible Equa-

of Quantities. Therefore the admitting and retaining thefe Quantities, however impoffible they may feem to be, the investigating their Prowith our utmoft induftry, and applying thofe Properties to perties ufe whenever occafion offers, is only keeping within the Rules of Reafon and Analogy; and is alfo following the Example of our aud illuftrious Author, who of all others has the greateffc

tions, or Relations

fagacious 'Tis enlarging the numto be our Precedent in thefe matters. right ber of general Principles and Methods, which will always greatly i

con-


[

'43

]

v

THE CONTENTS I

of the following Comment.

on the Introduction ; or the Refolution of /JNnotations *-* Equations by Infinite Series. pag. 143 Sedt. I. Of the nature and conjlruttion of or infinite converging Series. Sedt. II.

P-H3

The Refolution offimple Equations, or ofpure Powers,

by infinite Series.

The Refolution of Numeral Affected Equations,

Sedt. III. Sedt.

~

=

IV. The Refolution of Specious Equations by ; andfirjifor determining the forms of the

II.

V. The Refolution ofAJfe&ed

Annotations on

Prob.

i.

ing Quantities being given-t their Fluxions.

.

.

p.

the Relation to

I.

9

1

209

P-235

of the flow-

determine the Relation p. 241

of

Sedt.

ami 1

fpecious Equations proje-

Method of Fluxions.

or,

p.

86

P-

cuted by various Methods of Analyjis. Tranfition to the

59

1

Series,

their initial Approximations.

Sedt. VI.

1

infinite Se-

ries

Sedt.

p.

Concerning Fluxions of the firft Order , and

their Equations.

Fluxions of fuperior Orders, method of deriving their Equations.

Sedt. II. Concerning

to

find p.24i

and

the

>

Seft. III.

The

]

Geometrical and Mechanical Elements

Fluxions,

i

[T]

of

p.266 III.


CGNTENTa ,111.

Annotations on Prob.

or y the Relation of the Fluxions to determine the Relation of the Fluents.

being given,

A

Se. I.

2.

p.2 77 particular Solution

general Solution,

by which

;

a preparation

'with it

is

dijlributed

Cafes.

Solution of the firft Cafe of Equations.

Sedl. II. Sedt. III. Seft.

three

p.a// p. 282

-p.286 IV. Solution of the third Caje of Equations, with fome

-

Sedt V. The Refolution of Equations, Fluxional, .

r lux ions. Sedt. VI. An

by >

the

afliftance diJ

-

P-3OQ

.

whether Algebraical Orders of of j j[uperior r j

-

p-3o

Analytical Appendix, explaining Jome Terms in the P-3 2 I foregoing Work.

and ExpreJJiom ,Se<5t.

-

to the

Solution of the fecond Caje of Equations.

neceffary Demonftrations.

or

into

VII. The Conclufwn

or review of the whole.

,

-

-

containing a j}:ort recapitulation

P-33


Reader is defired to correfl the following Errors, which I hope will be thought but few, and fuch as in works of this kind are hardly to be avoided. 'Tis here neceflary to take notice of even literal Miftakes, which in fubjefts of this nature are often very That the Errors are fo few, is owing to the kind affillance of a flcilful Friend or material. the IVefs ; as alfo to the care of a diligent Printer. two,_ who fupplyfd my frequent abfence from,

THE

ERRATA. In tie Preface, pag. xiii< lirt, 3. read which P. 1 19. I. 1 2. read here fubjoin'd. Ibid. 1. 5. for matter read read Hyperbola. to the Fluxion of the' Area, manner. Pag. xxiii. 1. alt. far Preface, &e. nWConclufton of this Work. P- 7. \.T,i.for lDxIP P. 15. 1.9. ready P.I3J.1.8. readJf !>**+ -&*' ~{- read =.. is

=

=

&c.

-',4,

l,'2j.

P'.

read -

17.

1.

read

17.

P. 35,

.

P. 32.

.

\.

9* 1.

3:

for

CE x \Q

ACEG -

19, read. !

P. 135.

.

.

);

and

Ibid.

15. read

1.

P.

9"

lOtfjr*

read

P. 145. \.fenult. 13.8. I. 9. ^WAb&Jifs^AB. P. 149. 1. 2O. read whkh irt; read 7\~~ 3 P. 157. I.i3./-f^ ax. P.i68. l.j. retd^ax. P. 1 77.' P-I7I. \.\j.fir Reread $*. \.l$.rcait .

loxty.

~~r'~

P. 62. 1.4. read

firyreatl-y.

.

2

A>.

\,2-[. read.

*

read

y~-

....

.

6. read to 2m, P. 213^ [.-j. far5 P. 229, 1. 21* for x Species read Series. retu( t. read 30. x 4. /i/V. 1. 24. for x 4'readx *. P. 234. P. 87. 1. 22. read 1. 2. P. 236. 1. 26. ;vW genera^or yy ready. P.t P. 243. 1. 29. read. Ibid. 1.22,24. reaJAVDK. ting. ax*yi*. P. 284.

P. 64. \.q. for 2 read z. P. 82. 1. 17. read zzz.

+

y-

Ibld:\.ult.for

P. 63.!. 31.

P.-gz. 1.5.

Ibid.

\.

read-\-

.7

-,',

.

P. 204.

Ibid.

1.

uit.

1.

1

read

i

P. 289.

.

1.

17.

fur right read

j^

\.z\.for z read P. 109.

1.

x.

33. dele as

P. 104. 1.8. read 6;jt 1 P. I 10. 1. 29. read

left.

.

ofen.

read 1. 1

and v/ ^ 1

x l '=.

P.

nj,

1

1

7.

for Parabola

P. 295.

1.

2.-

i,

'

P. 297. \.ig.forjx

8.

y.

read

1\3O4.

m a ~t>

.

1.

read'

read y*

20, 21

P. 3

1

7.

.

dil:

',

'.

x 4 --^-J-ax*. P. 298.

-(- be.

1. tilt,

read

1.14.

P^og. 1 a'-j

.


ADVERTISEMENT. Lately publijtid by the Author,

BRITISH HEMISPHERE, or a Map of a new contrivance, proper for initiating young Minds in the firft Rudiments of

THE

It is in the form of a Halfof Inches but about Diameter, Globe, comprehends the whole 15 known Surface of our habitable Earth ; and mews the iituation of all the remarkable Places, as to their Longitude, Latitude, Bearing and Diftance from London, which is made the Center or Vertex of It is neatly fitted up, fo as to ferve as well for ornament the

Geography, and the ufe of the Globes.

Map.

as ufe

j

and

fufficient Inftructions are annex'd,

to

make

it

intelligible

to every Capacity.

Sold by

W. REDKNAP,

Church.

at the

Leg and Dial near the Sun Tavern

SEN EX, ; and by ]. Price, Haifa Guinea.

in Fleet-jlreet

at

the Globe near

St.

Dunftan's


and INFINITE SERIES.

339

contribute to the Advancement of true Science. In fhort, it will enable us to make a much greater progrefs and profkience, than we othervvife can do, in cultivating and improving what I have elfe-

where

call'd

The Philofopby of Quantity.

FINIS.


3T

.

..

I


Methodoffluxions00newt  
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