Isaac Newton.

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

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that BD is to be taken the contrary way from BE.

7. Again if you affume c'-a 1 -+- c^x* = z 1 , you will have zc*x

= 2zz ; and z being eliminated, there will arife



8. Or if you affume



aa-^-xx



aa -J-.VA-



'Z.



\/aa -+- xx = z, make



- - , <z -}- ATA-

= v, and it will be ^ =s,and then (by Prob.i.) ^p ^ Alfo

the Equation aa -f- xx = 011; gives 2X = zvv, by the help of which
if you exterminate <u, it will become 3 -j^- = z = j- \/ aa-^-xx.

9. Laftly, if you affume 8 3^2 -f- ^&=. zz, you will obtain
32; 3x2; -f- $z = 2Z&. Wherefore by the affumed Equation
firflieek the Area z, and then the Ordinate z by the reiulting Equa-
tion.

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

P R O B. VIII.

To fad as many Curves as you pleafe, -wbofe Areas fiall
have a relation to the Area of any given Curve, a/fign-
able by finite Equations.

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





A C





.11



G,



/V



their Abfciffes or Bafes AB and AC. Then the Increments or Fluxions
of the Areas which they defcribe, will be as thofe Ordinates drawn

M into



82 fhe Method of FLUXIONS,

into their Velocities of moving, that is, into the Fluxions of their
Abfcifles. Therefore make AB = x, BD = v, AC = z, and
CE =y, the Area AFDB = j, and the Area AGEC = /, and let
the Fluxions of the Areas be s and t : And it will be xv : zy : : s : t.
Therefore if we fuppofe x = i, and v=s, as before; it will be

zy = t, and thence - =y.

2. Therefore let 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. i.) let the

Fluxions t and z be found, and then make - =>'.

3. Ex. i. Let the given Curve FDH be a Circle, exprefs'd by the
Equation ax xx = w, and let other Curves be fought, whofe
Areas may be equal to that of the Circle. Therefore by the Hy-

pothefis s=:f, and thence s = f, and y = - =^-. It remains

to determine z, by afluming fome relation between the Abfciffes
x and z.

4. As if you fuppofe ax=zz; then (by Prob. i.) a =. 2zz: So
that fubflituting : [ for z, then y = " = . But it is v =

(\/ax xx =) - \/ aa s.s, therefore \/ ' aa zz = y is the

a aa *

Equation to the Curve, whofe Area is equal to that of the Circle.

5. After the fame manner if you fuppofe xx =. z, there will

ariie 2x =s, and thence _)'= (==] ~ ; whence -j and x being

; I
exterminated, it will be y=- 7"-'

2Z 2 -

6. Or if you fuppofe cc = xz, there arifes o = z + xz, and

T-V (5 /

thence - = y = - , v az cc.

2 2^3

7. Again, fuppofing ax +- '- = z, (by Prob. i.) \t'isa + s=:z,
and thence -^- y ?> which denotes a mechanical Curve.

8. Ex. 2. Let the Circle ax xx = w be given again, and let
Curves be fought, whofe Areas may have any other aflumed relation
to the Area of the Circle. As if you afliime cx + s = t, and fup-



pofe alfo ax = ZZ. (By Prob. i.) 'tis c + s = t, and a =

Therefore



and INFINITE SERIES. 83

Therefore y = ~ = 2 ~~; and fubftituting ^ ax xx for j,
and 5 f for x, 'tis;'= 4- ^ v'^

9. But if you affume j =/, and x = z, you will have

s ^! =/, and i = z. Therefore y- - =j 2!2L O i

K fl '

= i; . Now for exterminating v, the Equation ax xx
= iJ'u, (by Prob. i.) gives ^ 2x= 2vv, and therefore 'tis y=.
. Where if you expunge v and A; by fubftituting their values



\/ ax xx and 2;, there will arile _)=-" \/tf;s

10. But if you affume ss = f, and x = zz, there will arife
2w=r^, and i = 2zz; and therefore _y = V = 4^- Anc

5 and x fubftituting \/ ax xx and &z, it will become y =
\Sa-zz;, which is an Equation to a mechanical Curve.

1 1. Ex. 3. After the fame manner Figures may be found, which
have an aflumed relation to any other given Figure. Let the Hyper-
bola cc -{- xx = wu be given ; then if you affume s = /, and
xx=cz, you will have s = f and 2X = cz; and thence _)' =

.r= -. Then fubftituting v/cc -+- xx for j, and C-z^ for x, it

s;

will be y =: - i/cz -{- zz.

* 2Z

12. And thus if you affume xv s=zf, and xx = cz, you
will have v-^-vx s=t, and 2X = cz. But v=.s, and thence

vx = i. Therefore y= - = ~. But now (by Prob. i.) cc-\-xx

% ***

= 1?^ gives x=^-ui;, and 'tis y = ^ Then fubftituting \/i<,-t-xx
for -u, and c*z* for x, it becomes y =. , ^ c ~ ^

13. Ex. 4. Moreover if the Ciffoid ^-^^_- - =1; were given, to
which other related Figures are to be found, and for that purpofe
you affume - ^/ax xx +- - s = t ; fuppofe - */ ax xx = h,

and its Fluxion /' -, therefore h +- - s = /. But the Equation *** -

M 2 =M



84 7%e Method of FLUXIONS,

=/j/j gives 3 * A ^ .V. ==2.^, where if you exterminate /&, it will be




And bcfides fuice it is - s -

3 3



xx ,



= t. Now to determine z and z, afTume
\/ aa ax = z ; then (by Prob. i .) a = 2zz, or z = -

V.. Jaxxx a A; '

v/tftf ~. And as this Equation belongs to the Circle, we mall
have the relation of the Areas of the Circle and of the Ciflbid.

14. And thus if you had aflumed V ">/ ax xx -h ~ s = f y

and x = z, there would have been derived y-=.\/as> .22-, an
Equation again to the Circle.

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

1 6. Ex. 5. Thus the Trochoid ADF being propofed, I refer it
to the Abfcifs

ABj and the
Parallelogram
ABDG being
compleated, I
leek for the
complemen-
tal Superficies
ADG,byfup-
pofing it to be
defcribed by
the Motion of
the right Line

GD, and therefore its Fluxion to be equivalent to the Line GD
drawn into the Velocity of the Motion ; that is, x*v. Now where-
as AL is parallel to the Tangent DT, therefore AB will be to BL
as the Fluxion of the fame AB to the Fluxion of the Ordinate BD,

that




and INFINITE SERIES. 85

that is, as r to -j. So that <u = and therefore xv == BL.

A h

Therefore the Area ADG is described by the Fluxion BL ; fince
therefore the circular Area ALB is defcribed by the fame Fluxion,
they will be equal.

17. In like manner if you conceive ADF to be a Figure of
Arches, or of verfed Sines, that is, whole Ordinate BD is equal to
the Arch AL ; lince the Fluxion of the Arch AL is to the Fluxion
of the Abfcifs AB, as PL to BL, that is, v : i :: a : \/ ax .v.v,
then -y = - / . Then vx, the Fluxion of the Area ADG,

2 v ax xx

will be 7=^=. Wherefore if a right Line equal to - .'

2V <,* xx ly . .x 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.

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

19. As to Spirals, the matter will be very fliort For from the
Center of Rotation A, the Arch DG being defcribed, with any
Radius AG, cutting the right Line AF in G, and the Spiral in D ;
fince that Arch, as a Line moving upon the

Abfcifs AG, delcribes the Area of the Spiral
AHDG, fo that the Fluxion of that Area is
to the Fluxion of the Rectangle i x AG, as
the Arch GD to i ; if you raife the perpen-
dicular right Line GL equal to that Arch,
by moving in like manner upon the fame
Line AC, it will defcribe the Area A/LG
equal to the Area of the Spiral AHDG :
The Curve A/L being a geometrical Curve.
And fartlirr, if the Subtenfe AL be drawn, then A ALG = |
xGL = |AGx GD = Sector AGDj therefore the complernental
Segments AL/ and ADH will alfo be equal. 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 may
be converted into equal geometrical Curves with the fame eale.

20.




86 tte Method of 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 mag-
nitude, other Curves may be found out that may be compared with
thefe, and that their defining 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 Su-
perficies of thofe Solids, and to the finding of any other flowing
quantity produced by a Fluxion analogous to it.

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 AFDB

is defcribed, move upright upon an Abfcifs AB
given in pofition, conceive (as before) the Paral-
lelogram ABEC to be defcribed in the mean time
on the other fide AB, by a line equal to unity.
And BE being fuppos'd the Fluxion of the Pa-
rallelogram, BD will be the Fluxion of the Area
required.

2. Therefore make AB = x, and then alfo ABEC=i \x=x,
and BE = x. Call alfo the Area AFDB = z, and it will be

BD=z, as alfo =~, becaufe x=i. Therefore by the Equa-

X

tion expreffing BD, at the fame time the ratio of the Fluions -




IS



and INFINITE SERIES. 87

is exprefs'd, and thence (by Prob. 2. Cafe i.) may be found the
relation of the flowing quantities x and z.

3. Ex. i. When BD, or z, is equal to fome fimple quantity.

4. Let there be given ~ = z, r , (the Equation to the Pa-

rabola,) and (Prob. 2.) there will arife - a = z. Therefore ^>
or -L AB x BD, = Area of the Parabola AFDB.

c. Let there be eiven = z, fan Equation to a Parabola of

J ^ aa *

the fecond kind,) and there will arife -^ = z, that is, ~ AB x BD
= Area AFDB.

6. Let there be given z

XX ~ '

or a^x 1 = x-:, (an Equation to
an Hyperbola of the fecond kind,)
and there will arife a 3 x 1



z




or 7 = z. That is, AB x BD

= Area HDBH, of an infinite length, lying on the other fide of

the Ordinate BD, as its negative value insinuates.

j. And thus if there were given ^ = z, there would arife



2XX



Z.



8. Moreover, let ax = zz, or a*x* = z, (an Equation again
to the Parabola,) and there will arife ~a^x^ = z,, that is, i-AB
x BD = Area AFDB.

9 . Let ~=zz- t then za*x = s, or 2 AB x BD = AFDH.

10. Let =zz', then ^ f = s, or 2 AB x BD = HDBH.

1 1. Let ax* = z~> ; then fV = z, or i AB xBD = AFDH.
And fo in others.

12. Ex. 2. Where z is equ.il to an Aggregate of fuch Quantities.

13. LetAT-H^ij then^-h *- = z>

, J

14. Let <z -{- ^ = ~ . then ax \ = &>

15. Let 3*i ^ z ; then 2x^ +- x 4** = 2r -

1 6. Ex. 3. Where a previous reduction by Divifion is required.

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



l%e Method of FLUXION s



,



x __ _ ^ 4. ?f ^l 5 , &c. And thence, (by Prob. 2.) as

11 1. a * x ^^

in the fecond Set of Examples, you will obtain z= -y - ^



/. "x a U~A ^

5^/3 A/4 *

1 8. Let there be given ^ ==*, and by divifion it will be

^ 1 ~J XX

~=i x*-{-x* x 6 , &c. or elfe s= -^ l - -f- -., &c. And

" X 1 X4 A. '

thence (by Prob. 2.) 2 = x ^ 3 -f-^ r 1# 7 , &c. =AFDBi
or 2 = -i H- ^- - 5 , &c. =HDBH.

X 3X* SA.''

L A

19. Let there be given , ~!Li X =z, and by divifion it will
be z = 2x^ 2X + 7^ I3AT 1 -f- 34*% 6cc. And thence (by

Prob. 2.) z = $x* x' -f- yx* ' T 3x3 + V 8 A< ^ &c -

20. Ex. 4. Where a previous reduction is required by Extraction
of Roots.

21. Let there be given z = \/ aa -\- xx, (an Equation to the
Hyperbola,) and the Root being extracted to an infinite multitude

of terms, it will be z=i a 4- * ~\ 7- - r, &c. whence

* ft a Q f ,9.*if. f ,l I I -7 fit*



. r . X X x ^x

as in the foregoing ss = ax+ 6 ~ , -h 77^ TT^ &c -



22. In the fame manner if the Equation z = \/aa xx were
given, (which is to the Circle,) there would be produced z=ax



b ^Oi.J ii2a s

23. And fo if there were given z-=\/x xx, (an Equation
alfo to the Circle,) by extracting the Root there would arife
z = x' f x* 4-** -r'-g-x^, 6cc. And therefore z = .ix*



1

V z

TT .



___ _

24. Thus s === v//z<* -^- AV xx, (an Equation again to the Cir-



bx



x .\

cle,) by extraction of the Root it gives z=a-\- - - gj s j occ.

, ^* Jf3 /'*v3 -

whence 2; = <7Ar -f- - -, - - , &c.

4 6< 24^ I

25. And thus v^~ZT7~ = ^, by a due reduction gives



z=i-+- T^-V* -h 43^4, &c. then 2 = AT -f- ^ 3 -f- T V^ S &c.
H-irf -f^ _l_^ + T V^

T ' - Vo^

26.



and INFINITE SERIES. 89



26. Thus finally z=l/a* -t-A' 5 , by the extraction of the Cubic
Root, gives z=a -+- ~ -+- ~ w &c. and then (by Prob. 2.)

*=** + -~, gfr + T> &c. = AFDB. Or elfe *=



C '' A 1 1 **

And thence * = 7



&c. = HDBH.



567*

27. Ex. 5. Where a previous reduction is required, by the refo-
lution of an affected Equation.

28. If a Curve be defined by this Equation z> + a*z



2a"' x 3 = o, extradl the Root, and there will arife z = a x .
j_ : : _j_ !4^-. &c. whence will be obtain'd as before z-=ax

64.2 5 i zaa







29. But if z~' cz* 2x*z c *z -f- 2x ? -+- c* = o were the
Equation to the Curve, the refolution will afford a three-fold Root;
either z = c + .v f? + Jl, &c. or S = c .v-f- !i' -,

V 32' 1 <-



or s = c A - f - (_ - &c. And hence will arife the

2" 2rc At T

values of tb,e three correfponding Areas, z = ex + x*
-f- T^t, &c. 2; = r^ i.v 1 + ^ ^0, &e. and x = ex

A 5 X 4 .,S

~ - - ! - flrr
6c 8. 1 24^' CCC>

30. I add nothing here concerning mechanical Carves, becaufe
their reduction to the form of geometrical Curves will be taught af-
terwards.

31. But whereas the values of z thus found belong to Areas
which are fituate, fometimes to a finite part AB of the Abfcifs,
fometimes to a part BH produced infinitely towards H, and fome-
times 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.

32. For Inflance ; to the Curve exnrefs'd bv the Equation

i-^-'xx

' m^^ JTC-




fhe Method of FLUXIONS,

~, it is found that z=x ^x }
_l_ 4_,vS &c. Now that I may de-
termine the quantity of the Area
MDll, adjacent to the part of the
Abfcifs /'B; from the value of z,
which arifes by putting AB = x,
I take the value of z, which arifes by putting Ab=x, and there
remains x -Lx* + ^-x', &c. x + x> -J-x', &c. the value of
that Area WDB. Whence if A*, or x, be put equal to nothing,
jqere will be had the whole Area AFDB = x x' -+- -^x', &c.

33. To the fame Curve there is alfo found z, ==. - -+

L, &c. Whence again, according to what is before, the Area

5**

I 1 1 ^ I I &/~f* *"T * ppr^TOt'f 1

1 ]V\T\ __ t _. I ,_ - oCC '- " T - 1 ' ' ^"" 1 -) OCC. J. ijCl CIUI C

if AB, or x, be fuppofed infinite, the adjoining Area bdH toward
H, which is alfo infinitely long, will be equivalent to - ^
-f- . &c. For the latter Series - -f- ~, &c. will

CA ^-35

vanifh, becaufe of its infinite denominators.

34. To the Curve reprefented by the Equation a-\- = Z,^ it

:s found, that z=.ax -. Whence it is that x - ax

i X X

-4- - = Area &/DB. But this becomes infinite, whether x be fup-
pofed nothing, or x infinite ; and therefore each Area AFDB and
&/H is infinitely great, and the intermediate parts alone, fuch as
&/DB, can be exhibited. And this always happens when the Ab-
fcifs x is found as well in the numerators of fome of the terms, as
in the denominators of others, of the value of z. But when x is
only found in the numerators, as in the firft Example, the value of
z, belongs to the Area fituate at AB, on this fide the Ordinate. And
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
points b and B, fuppofe in E, inftead
of the Area will be had the difference
&/E* BDE of the Areas at the diffe-
rent parts of the Abfcifs ; to which if
t here be added the Rectangle

he Area dEDG will be obtain'd.
t




G



and INFINITE SERIES.

36. But it is chiefly to be regarded, that when in the value of &
any term is divided by x of only one dimension ; the Area corre-
fponding to that term belongs to the Conical Hyperbola ; and there-
fore is to be exhibited by it felf, in an infinite Series : As is done in
what follows.

77. Let fl3 ~ glA '= z, be an Equation to a Curve ; and by divifion

J ax -f- xx J

it becomes z = - 2a +- 2X _ h^ &c. and thence



aa y



2X> l X*

Z = l^ j 2ax -f- x 1 ^T ' To* &c. And the Area &/DB

,&*.



aa



zax



xx



2*5

,



I denote the little Areas belonging



Where by the Marks and

1-1 1

- aa aa

to the Terms and

38. Now that |^ and |j| may be found, I make Kb, or x y to
be definite, and bE indefinite, or a flowing Line, which therefore I
call ;' ; fo that it will be -^; = to that Hyperbolical Area adjoin-
But by Divifion it will be - - = -
J x ~ \ y x

therefore,



ing to B, that is, j -






A4



or -

x



-* ' '



. and therefore the whole Area required



WDB =

X
2A3



21 3 .

xx H -, &c.

39. After the fame manner, AB, or x, might have been ufed for

a definite Line, and then it would have been



40. Moreover, if Z>B be bifefted in C, and AC be affumed to be
of a definite length, and Cb and CB indefinite ; then making AC

= i>, and C or CB =_)', 'twill be bd= -^ s=-\- ' -)- ~

-{- _i- ^-^-' &c. and therefore the Hyperbolical Area adjacent

N 2 to



'A Mt&od of FLUXIONS,



to the Part of the Abfcifs &C will be



a V



r I



.&c. Twill be alfo DB = -~ = ? - ~ + ~ - ^ +
&c. And therefore the Area adjacent to the other part of the Abfcifs CB

1 11 7* st^ 4 Si"*- 1

= " , + - -f- ' ' , &c. And the Sum of thefe

f 2f l Jf 1 4' * 5' 1

Areas 7- -\~ ~r ~r, &c. will be equivalent to -|



41. Thus in the Equation a 3 -f- z,* ~$- z x~= =o, denoting the
nature of a Curve, its Root will be z = ,v y

&c. Whence there arifes z, =-. Lxx -x



6cc. And the Area



' Y
T A

7



ox



8 J A X 8 I A i

_ _1 _ _!_

/^r^
Six' KC '



T, &c. that is,=:|.v



.'X ^'
T A Six



&c. _ - ^



&c. - i - -



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

in the former Example, where ? v + * v v = z was the Equation to
the Curve, if I fhould make b to be the beginning of the Ablcifs*
and fuppofmg Al> to be of any determinate length 4/7, for the re-
mainder of the Abfcifs B, I fliall now write x : Thst is, if I dimi-
nifti the Abfcifs by a, by writing x -f- a inftead of x, it will

-become ^~^,. = ~> and



_!! & Ci whence arifes s = \ax ' 4 ^ z 4 - ^' -' &c. =
273 j bia



Area .

43. And thus by affuming another and another point for the be-
ginning of the Abfcifs, the Area of any Curve may be exr-ivib'd an
infinite variety of ways.

44. Alfo the Equation rj-p = z might have been refolved
into the two infinite Series z, - - " -+- "-^ &c. a -f .v

.V 2 X 1 X }

** - }-; &c. where there is found no Term divided b} the fir ft

2 Power



and INFINITE SERIES. 93

Power of x. But fuch kind of Series, where the Powers of A* afcend
infinitely in the numerators of the one, and in the denominators
of the other, are not fo proper to derive the value of z from, by
Arithmetical computation, when the Species are to be changed in-
to Numbers.

45. Hardly any thing difficult can occur to any one, who is to un-
dertake fuch a computation in Numbers, after the value of the Area
is obtain'd in Species. Yet for the more compleat illufhation of the
foregoing Doctrine, I mall add an Example or two.

46. Let the Hyperbola AD be propofed,
whofe Equation is \/x-+-xx=z; its Vertex be-
ing at A, and each of its Axes is equal to Unity.
From what goes before, its Area ADB=-i.v>

-+- j'^ A*' 1 " -+- T'T* ? T^P*'"' &c ' that
is x* into Lx -+- x* T ' T .v * + y ' T .v 4 T 4 T - V s >
&c. which Series may be infinitely produced by
multiplying thelaft term continually by the fucceeding terms of this

Proereffion i- J #. 5 .v ^^r ~~'" q x ^^'v &c. That is

2 S 47 6-9 A - 8.-n Xi 10.15*.




the firft term ^..v 1 x I_ 3 x makes the fecond term -L.v* : Which

2 -5 '
multiply 'd by " l -~ x makes the third term T V- vl : Which mul-

tiply'd by ^ x makes T ' T .v ? the fourth term; and fo ad hifini-
tuin. Now let AB be affumed of any length, fuppofe ^, and writing
this Number for .v, and its Root 4 for x*, and the firft term ^x^
or y x T> being reduced to a decimal Fraction, it becomes

-^3333333> & c - This into '- ^ makeso.oo625 the fecond term.
This into ~ ' v makes 0.0002790178, &c. the third term. And

4-7 4

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



-i-o.



94 "The Method of FLUXIONS,

+ 0.0833333333333333 00002790178571429

62500000000000 34679066051

271267361111 834^65027

5135169396 26285354

144628917 961296

4954581 3 86 7 6

190948 1663

7963 75
35 2 _

1 1 0.0002825719389575

-f- 0.0896109885646518

4- 0.0896109885640518 "0^3284166257043

Then from the fum of the Affirmatives I take the fum of the ne-
gatives, and there remains 0.0893284166257043 for the quantity
of the Hyperbolic Area ADB ; which was to be found.

47. Now let the Circle AdF be propofed,
which isexpreffed by the equation \/x xx = z >
that is, whofe Diameter is unity, and from what
goes before its Area AdB will be -!#* ..#*
T ' T xi -fT^i & c> I n which Series, fince
the terms do not differ from the terms of the Se-
ries, which above exprefs'd the Hyperbolical Area, unlefs in the
Signs -4- and ; nothing elfe remains to be done, than to
conned: the fame numeral terms with other fignsj that is, by
fubtracting the connected fums- of both the afore -mention'd tables,
0.08989 3 560 503 6 1 93 from the firft term doubled 0.1666666666666,
&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
diameter. And hence we may obferve, that tho' the Areas of the
Circle and Hyperbola are not compared in a Geometrical confidera-
tion, 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
whole Area may be derived. For the Radius dC being drawn,
multiply Ed, or -^v/S? Ulto -^C, or i, and half of the product

s-Vs/3' or -54 12 ^5 8 773^5 2 75 w '" ^ e ^ e va ^ ue f the Triangle
cWB; which added to the Area AdB, there will be had the
Sector ACd = 0.1308996938995747, the fextuple of which

whole Area.

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



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