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The method of fluxions and infinite series : with its application to the geometry of curve-lines online

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and other Analytical Argumentations of that kind. Again, that is
often call'd a given quantity, which always remains conftant and in-
variable, 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,
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 my Examples from our Author's admirable Treatife of
Univerfal Arithmetick. The chief artifice of this manner of Solu-
tion will confift in this, that as faft as the Mafter propofes the Con-
ditions of his Queftion, the Scholar applies thole Conditions to
ufe, argues from them Analytically, makes all the aeceffary deduc-
tions, and derives fuch confequences from them, in the fame order
they are propofed, as he apprehends will be mcft fubfervient to the
Solution. And he that can do this, in all cafes, after the fureft, fim-
pleft, and readieft manner, will be the beft ex-tempore Mathemati-
cian. But this method will be beft explain'd from the following
Examples.

I. M. A Gentleman being 'willing to diftribiite Abns S. Let

the Sum he intended to diftribute be reprefented by x. M. Among
fbme poor people. S. Let the number of poor be _}>, then - would

have been the fhare of each. M. He wanted 3 fiillings S. Make

3 = rf, for the lake of univerfality, and let the pecuniary Unit be
one Shilling ; then the Sum to be distributed would have been x-{-a,

and"



and INFINITE SERIES. 32"

and the fhnre of each would have been ^^- . M. So that each

y

might receive $ fallings. S. Make 5=^, then ^ = b, whence

x = by a. M. "Therefore he gave every ot.e 4 fallings. S. Make

4=f, then the Money diftributed will be cy. M. And he has 10

fallings remaining. S. Make io = d, then cy -f- d was the Money

he intended at firft to diftribute; or cy -+- d = (x =) by- a, or

y =5 ^jt- f . M. J^rf* w<2J the number of poor people ? S. The

number was y = 7 = 3 "*"' = 1-2. M. ^W /60w much Alms

'' ? 4
tf/tf Of tff ^ry? intend to diftribute ? S. He had at firft x = by a

= 5x13 3 = 62 fhillings. M. How do you prove your Solution?
S. His Money was at firft 62 fhillings, and the number of poor
people was 1 3. But if his Money had been 62-4-3 === ^5 ^ r 3 x 5
fhillings, then each poor perfon might have received 5 millings. But
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
c= 10 fhillings.

II. M. A young Merchant, at his firjl entrance npon bufmefs, began
the World with a certain Sum of Money. S. Let that Sum be x, the
pecuniary Unit being one Pound. M. Out of which, to maintain
himfclf the Jirjl year, he expended 100 pounds. S. Make the given
number ioo = tf; then he had to trade with x a. M. He
traded with the reft, and at the end of the year had improved it by a
third part. .S. For univerfality-fake I will aflume the general num-
ber n, and will make ^ = n i, (or n = ;) then the Improve-
ment was n i xx a = nx na x -f- a, and the Trading-
fiock and Improvement together, at the end of the firft year, was
MX na. M. He did the fame thing the fecond year. S. That is,
his whole Stock being now nx na, deducting a, his Expences for
this year, he would have nx na a for a Trading-ftock, and
n ix nx na a, or n'-x n*a nx -f- a for this year's Im-
provement, which together make n'-x n*a na for his Eftate at
die end of the fecond year. M. As aljo the third year. S. His
whole Stock being now ;< a x n x a na, taking out his Expences
for the third year, his Trading-ftock will be n*x n'-a na a,
^~ and the Improvement this year will be n i X*A- n'-a na a,
or J x n=a n'-x -f- a, and the Stock and Improvement together,
or his whole Eftate at the end of the third year will be n*x n*a

_ n 1 a na, or in a better form n"'x -+ "-^na. In like manner

T 2 if



324 Th* Method of FLUXIONS,

if he proceeded thus the fourth year, his Eftate being now n*x
n i a _ rf a na, taking out this year's Expence, his Trading-flock
will be n>x n>a if a na a, and this year's Improvement is
n Fx n=x is a if a na a, or n*x n^a n*x -f- a,
which added to his Trading-ftock will be n*x n*a tfa 72-^2

na, or 11* x -f- * na, for his Eftate at the end of the fourth

year. And fo, by Induction, his Eftate will be found n s x -4- - na
at the end of the fifth year. And univerfally, if I aflume the ge-

m _

neral Number m. his Eftate will be nx -f- ~ 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. S. Make 2 = , then n m x -t-

m m

- l _^na = bx, or x = - ==-#. M. At the end of 3 years.

n i x n" b

S. Then #2=3, a-=. 100, b = 2, n = %, and therefore x=s..




64

400 = 1480. M. j%!2<z/ <was his Eftate atjirji? S. It was 1480
pounds.

III. M. Two Bvdies A and B are at a given diftance from each
other. S. As their diftance is faid to be given, though it is not fo.
actually, I may therefore aflume it. Let the initial diftance of the
Bodies be 59 = e y and let the Linear Unit be one Mile. M. And
move equably towards one another. S. Let x reprefent the whole
fpace defcribed by A before they meet ; then will e x be the
whole fpace defcribed by B. M. With given Velocities. S. I will
aflume the Velocity of A to be fuch, that it will move 7 = c Miles
in 2 =f Hours, the Unit of Time being one Hour. Then be-

caufe it is c : f : : x : - , A will move his whole fpace x in the

time Y Alfo I will aflume the Velocity of B to be fuch, that it
will move 8 = d Miles in 3 ==:g Hours. Then becaufe it is d :
g :: e x : j-g> B will move his whole fpace e x in the time
'-7%. M. But A moves a given time - - - S. Let that time be
i = b Hour. M. Before B begins to move. S. Then A"s time is
equal to U's time added to the time h y or ^ = '^g -f- h.

M,'



and INFINITE SERIES, 325

M. Where will they meet., or what will be the fpace that each
have defcribed ? S. From this Equation we fhall have x =

* s*i X7== 1^ X7== r x 7 = 35 Miles, which will

' ' J '



x2



oxz-r7 x 3' / 37

be the whole fpace defcribed by A. Then e x= 59 35 =

24 Miles will be the whole fpace defcribed by B.
. IV. M. Jf 12 Oxen can be maintained by the Pafture 0/37 Acres
of Meadow-ground for 4 weeks, S. Make 12 = #, 3! = ^, 4=cj
then aiTuming the general Numbers e, f, h, to be determin'd after-
wards as occafion Ihall require, we mall have by analogy

Oxen



If





a '


Then


(J j


f ae


Alfo '


t_

Q .


ae

T


And "


^"


ace

T


Alfo


.1


ace


S-L, i or


\ "
. . r <-> ace

Alf J^J^J



> require <



'afture Time


" b '




c ~


be




c


e




c




. during <


i


e




/


e




b



M, ^4^ tf, becaufe of the continual growth of the Grafs after the
four weeks, it be found that 2 1 Oxen can be maintain d by the
pafture of \ o fuc h Acres for 9 weeks, S. Make 2i=J, e= io 3
f= = 9 ; then becaufe on this fuppofition, the Oxen d require the

pafture e during the time f; and in the former cafe the Oxen ~
required the fame pafture during the fame time : Therefore the
growth of the Grafs of the quantity of pafture e, (commencing
after 4 or c weeks, and continuing to the end of the Time f, or
during the whole time f c,) is fuch, as alone was fufficient to

maintain the difference of the Oxen, or the number d ^ , for
the whole time f. Then reciprocally that growth would be fuffi-
cient to maintain the number of Oxen df ' ' for the time i,



or the number of Oxen - ~

h bb



ace
J

for the time h. And becaufe
this growth will be proportional to the time, and will maintain a
greater number of Oxen in proportion as the time is greater j we
ihall have

Time



3 26 7%e Method ^FLUXIONS,

- ~* 1

Time Oxen Time Oxen

,, df ace , h c . df ace

t - c r - Ti *' " - c 7 - m tO T ' TZ

* bh f c h bh '

which will be the number of Oxen that may be maintain'd by the
growth only of the pafture e, during the whole time h. But it
was found before, that without this growth of the Grafs, the Oxen

^ might be maintain'd by the pafture e for the time h. There-



fore thefe two together, or ^ _f_ -^-^ x - ~ "" , will be the

ber of Oxen that may be maintain'd by the pafture e, and its growth
together, during the time b. M. How many Oxen may be main-
tain d by 24 Acres of fitch pafture for 1 8 week s ? S. Suppofe x to
be that number of Oxen, and make 24=^, and h= 18. Then
by analogy

Oxen Pafture

If

Then ex requ ij. e J <g during the time b.




. , r i ex ace

And conlequently = j-r

T J g bh

dft a^j ac /> r

x f ~ jf = T +




21*9 1 2 x 4 i 14 fi

I O J-j

V. M. If I have an Annuity S. Let x be the prefent value

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 re-
ceived 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

+ . r

- =x-{- x 1 -f- x'' H-A' 4 , &c. continued to fo many Terms

I X

as there are Units in in, (as may appear by Divifion 3) therefore
*~ y will reprefent the Amount of an Annuity of i pound,

to be continued for ;;/ years. M. Of Pounds. S. Make

= a*



and INFINITE SERIES. 327

= a, then the Amount of this Annuity for m years will be

^ a. M. To be continued for 5 years fuccejji'vely. S. Then
m = 5. M. Which I Jell for pounds in ready Money. S. Make



= c, then



-a = c, or x"" 1 -



I x



In any particular cafe the value of x may be found by the Refolu-
tion of this affected Equation. M. What Interefl am I alfav'd per
centum per annum? S. Make 100 = ^; then becaufe x is the
prefent value of i pound to be received i year hence, or (which
is the fame thing) becaufe the prefent Money x, if put out to ufe,
in i year will produce i pound; the Intereft alone of i pound
for i year will be i x, and therefore the Intereft of 100 (or K)
pounds for i year will be b bx y which will be known when x
is known.

And this might be fufficknt to (hew the conveniency of this Me-
thod ; but 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 A and B.
S. Then their diftance AB = m is given alfo. M. As likenaife the
two points C and D out of the Line AB, S. Then conlequently the
figure ACBD is



in mag-




given

nitude and fpe-
cie ; and pro-
ducing CA and
CB towards d
and <T, I can
takeA</=AD,
and Bf=KD.
M. Aljb I give
you the indefi-
nite right Line
EF in po/iticn,
pajjing thro' the

green point D. S. Then the Angles ADE and BDF are given, to which
(producing AB both ways, if need be, to e and f y ) I can make the
Angles h2e and B<f/~ equal refpedlively, and that will determine the
points e and f, or the Lines Ae = a, and Ef=c. And becaufe
de and <T/"are thereby known, I can continue de to G, fo that^/G.
= Sj\ and make the given line eG c= b. Likewife I can draw CH

and



228 The Method of FLUXIONS,

and CK parallel to ed and f$ refpeftively, .meeting AB in H and
K ; and becaufe the Triangle CHK will be given in magnitude snd
fpecie, I will make CK = d, CH=e, and HK=/ M. Now
let the given Angles CAD and C BD be conceived to revolve about the
green points or Poles A .and B. S. Then the Lines AD and CA^
will move into another fituation AL and cAt, fo as that the Angles
DAL, </A/, and CAc will be equal. Alfo the Lines BD and CB^ will
obtain a new fituation BL and cBA, fo as that the Angles DBL, <fBAand
CBc will be equal. M. And let D, the Inter feflion of the Lines AD and
BD, always move in the right Line EF. S. Then the new point of In-
terfedtion L is in EF; then the Triangles DAL and </A/, as alfo DBL
andJ'BA, are equal and iimilar ; then^//= DL= cTA, and therefore
G/==/A. M. What will be the nature of the Curve defer ibed by the
other point of Inter feSt ion C ? S. From the new point of Interfection c
to AB, I will draw the Lines ch and ck, parallel to CH and CK refpec-
tively. Then will the Triangle chk be given in fpecie, though not
in magnitude, for it will be Iimilar to ^CHK. Alfo the Triangle
Bck will be fimilar to Btf. And the indefinite Line Bk=x may-
be aflumed for an Abfcifs, and ck = y may be the correfponding
Ordinate to the Curve Cc. Then becaufe it is Bk (x) : ck (y )

:: Bf(c) :/A = ; = G/. Subtraft this from Ge-=&, and there
will remain le=.b - . Then becaufe of the fimilar Triangles chk



and CHK, it will be CK (d) : CH (e) : : ck (y) : ch= 'j . And
CK (/O : HK (/) :: ck ( y] : hk = -\ . Therefore A/J = AB .
Bk hk = m X .. But it is A/6 (m x 'f ) : cb 2) ::



(a) : le (b c - v ) . Therefore m x f x b J = ^,



or



dc ae bft. xv demy bdx* + bdmx = o. In which
Equation, becaufe the indeterminate quantities x and y arife only
to two Dimenfions, it mews that the Curve defcribed by the point
C is a 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 par-
ticular. S. That may eafily be done in the following manner :
e + l 'f ctl __ 2 p t an( j t h en the foregoing Equation will be-



come fcf - zpcxj> demy bdx'- 4- bdmx = o, and by ex-

trading



and INFINITE SERIES. 329

trading the Square - root it will be y = -.x -f. -f-

V I'P ft I 1 *" 1 ''d " </*;* XT . . . , .

!Z + - x x* + -. . _ XA . + __. Now here it is plain,
that if the Term j 4- ~ x X L were abfent, or if jj -4- ^ = o, or
r = ; that is, if the quantity - (changing its fign) fhould

be equal to ^ , then the Curve would be a Parabola. But if the
fame Term were prefent, and equal to fome affirmative quantity,
that is, if ? -f- - be affirmative, (which will always be when

is affirmative, or if it be negative and lefs than -. >} the Curve
will be an Hyperbola. Laftly, if the fame Term were prefent and
negative, (which can only be when - is negative, and greater than



y> the Curve will be an Ellipfls or a Circle.

I mould make an apology to the Reader, for this Digreffion
from the Method of Fluxions, if I did not hope it might contribute
to his entertainment at leaft, if not to his improvement. And I am
fully convinced by experience, that whoever fhall go through the
reft of our Author's curious Problems, in the fame manner, (where-
in, according to his ufual brevity, he has left many things to be
fupply'd by the fagacity of his Reader,) or fuch other Queflions
and Mathematical Diiquifitions, whether Arithmetical, Algebraical,
Geometrical, &c. as may eafily be collected from Books treating
on theie Subjects ; I fay, whoever fhall do this after the foregoing
manner, will find it 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
great end to be aim'd at by thefe Studies.



U u SECT,




330 7%e Method of FLUXIONS,

SECT. VII. The Conclufion ; containing a Jhort recapitu-
lation or review of the whole.

E are now arrived at a period, which may properly enough
be call'd the conclujion of tie Method of Fluxions ami Infinite
Series ; for the defign of this Method is to teach the nature of Series
in general, and of Fluxions and Fluents, what they are, how they
are derived, and what Operations they may undergo ; which defign
(I think) may now be faid to be accompliili'd. As to the applica-
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 ap-
ply'd and purfued in the fequel. I have continued my Annotations
in a like manner upon that part of the Work, and intended to have
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 work was already grown
to a competent fize; I refolved to lay it before the Mathematical
Reader unfinifh'd as it is, referving the completion of it to a future
opportunity, if I mall find my prefent attempts to prove acceptable.
Therefore all that remains to be done here is this, to make a kind
of review of what has been hitherto deliver'd, and to give a fum-
mary account of it, in order to acquit myfelf of a Promiie I made
in the Preface. And having there done this already, as to the Au-
thor's part of the work, I (hall now only make a fhort recapitula-
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-
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 com-
puting after a definite and particular manner, the latter after a ge-
neral and indefinite manner : So that both together compofe but
one uniform Science of Computation. For as in common Arith-
metick 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. 331

difpos'd as is prefcribed, we reckon by any other Root and its
Powers, or we may take any general Number for the Root of our
Arithmetical Scale, by which to exprefs and compute any Numbers
required. And as in common Arithmetick we approximate continually
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 ; fo in our Author's improved Algebra, or in
the Method of infinite converging Series, we may continually ap-
proximate to the Number or Quantity required, by an orderly fuc-
cefiion of Fractions, which are compofed of the reciprocal Powers
of any Root in general. And the known 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.

Hence I proceed to make fome Inquiries into the nature and
formation of infinite Series in general, and particularly into their
two principal circumftances of Convergency and Divergency; where-
in 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 ai-
certained and admitted, will render the Series finite, perfect, and
accurate. That in diverging Series this Supplement muft indifpen-
fablv be admitted and exhibited, or otherwise the Conclufion will be
imperfect and erroneous. But in converging Series this Supplement
may be neglected, becaufe it continually diminifhes with the Terms
of the Series, and finally becomes lefs than any affignable quantity.
And hence arifes the benefit and conveniency of infinite converging
Scries ; that whereas that Supplement is commonly fo implicated and
entangled with the Terms of the Series, as often to be impoiliblc to
be extricated and exhibited ; in converging Series it may fafely be neg-
lected, and yet we mall continually approximate to the quantity re-
quired, And of this I produce a variety of Inftances, in numerical
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
undergo. As firft when fimple fpccious Equations, or purs Powers,
are to be refolved into fuch Series, whether by Divifion, or by Ex-
traction of Roots ; where I take notice of the ufe of the afore-men-
tion'd Supplement, by which Scries may be render'd finite, that is,
may be compared with other quantities, which are confider'd as
given. I then deduce feveral ufeful Theorems, or other Artifices,

Una for



332 tte Method of FLUXION s,



for the more expeditious Multiplication, Divifion, Involution, and
Evolution of infinite Series, by which they may be eafily and rea-
dily managed in all cafes. Then I fhew the ufe of thefe in pure
Equations, or Extractions; from whence I take occasion to intro-
duce a new praxis of Refolution, which I believe will be found
to be very eafy, natural, and general, and which is afterwards ap-
ply'd to all fpecies 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 Theo-
rem is form'd, and diftributed into feveral fubordinate Cafes, by
which the Root of any Numerical Equation, whether pure or af-
fected, may be computed with great exactnefs and facility.

From Numeral we pafs on to the Refolution of Literal or Speci-
ous 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 Approxima-
tions. Thefe circumftances will depend upon fuch Powers of the
Relate and Correlate Quantities, with their Coefficients, as may hap-
pen to be found promifcuoufly in the given Equation. Therefore
the Terms of this Equation are to be difpofed in longum & in latum,
or at lenft the Indices of thofe Powers, according to a combined
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 fe-
veral Series that may be derived for the Root, as alfo their initial
Coefficients, Terms, or Approximations.

We then farther profecute the Refolution of Specious Equations,
by diverfe Methods of Analyfis -, or we give a great variety of Pro-
cefTes, by which the Series for the Roots are eafily produced to any
number of Terms required. Thefe ProcefTes are generally very lim-
ple, and depend chiefly upon the Theorems before deliver'd, for
finding the Terms of any Power or Root of an infinite Series. And
the whole is illustrated and exemplify 'd by a great variety of In-
ftances, which are chiefly thofe of our Author.

The



and INFINITE SERIES. 333

The Method of infinite Series being thus fufficiently dilcufs'd,
we make a Tranfition to the Method of Fluxions, wherein the na-
ture and foundation of that Method is explain'd at large. And fome
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 Nota-
tions are deliver'd and explain'd, which equally concern both thefe Me-
thods.

I then proceed with my Annotations upon the Author's firft Pro-
blem, or the Relation of the flowing Quantities being given, to de-
termine the Relation of their Fluxions. I treat here concerning
Fluxions of the firft order, and the method of deducing their Equa-
tions in all cafes. I 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
of Solutions. This is alfo apply'd to Equations involving feveral
flowing Quantities, by which means it likewife comprehends thofe
cafes, in which either compound, irrational, or mechanical Quan-
tities may 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-
tisfactory as I was able, and to remove the difficulties and objections



Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 29 of 30)