Isaac Newton.

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

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the Diameter.

50. To thefe we mail add the calculation of the Area compre-
hended between the Hyperbola dfD and its Afymptote CA. Let
C be the Center of the Hyperbola, and putting



'twill be -^ =BD, and -^ -=.bd; whence



a+x






"



the Area AFDB = bx - - 4- 4 -*,
&c. and the Area







4- , &c. and the fum 0aL>&=. 2ox-\ ? c AB /'/.<

4- ~ 4- ^ ? , &c. Now let us fuppofe CA = AF=i, and Kb
or AB = T L., Cb being 0.9, and CB = i.i ; and fubftituting thefe
numbers 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
you fee in this Table.

O.2OOOOOOOOOOOOOOO

6666666666666

40000000000
285714286

2222222
l8l82



The fum 0.200670695462151 1= Area bdDB.
51. If the parts of this Area Ad and AD be defired feparately,
fubtract the lefler BA from the greater dA, and there will remain

3-+ -^4- - h - &c. Where if i be wrote for a and b,

and -jig. for x, the terms being reduced to decimals will iland
thus;

O.O IOOOOOOOOOOOOOO

500000000000

3333333333

25000000

2OOOOO
1667



The fum o.



= A^ AD,



5 2 -



96 The Method of FLUXIO N s,

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

53. By the fame tables thofe Areas AD and hd will be obtain'd
alfo, when AB and Ab are fuppos'd T ~, or CB=i.oi, and
d> = o.gg, if the numbers are but duly transferr'd to lower places,
as may be here feen.

O O2OOOOOOOOOOOOOC0 O.O30ICOOOOOOOO3OO

66666666666 50020000

4000000 3^

28



Sum o 020000(5667066(195 =



Sum 0.0001000050003333 AJ AD.



==AD.

54. And fo putting AB andA=-~o-> orCB=i.oor, and'
0^ = 0.999, there will be obtain'd Ad= 0.0010005003335835,
and AD = o. 0009995003330835.

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

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

56. From thefe Areas thus found it will be eafy to derive others,

I f \ 2.

by addition and fubtradtion alone. For as it is ' into -^ = 2,
the fum of the Areas 0.693 I 47 I ^5599453 belonging to the Ratio's
^|and ^- 2 , (that is, infifting upon the parts of the Abfcifs 1.2 o 8
and 1.2 o.9,)will be the Area AFcPjS, C/3 being = 2, as is known.
Again, fince ^ into 2 = 3, the fum 1.0986122886681097 of the

Area's belonging to ^-| and 2, will be the Area AFcT/3, C/3 being 3..
Again, as it is ~ = 5, and 2 x5= 10, by a due addition of
Areas will be obtain'd 1.6093379124341004 = AF^/3, when
c /3=5; and 2.3025850929940457 =AF < T/3, when C/3 = 10.
And thus, fince 10x10=100, and 10x100=1000, and ^5

x 10 xo.98 = 7, and lox i.i = n, and .' x ' ' I ^ ) and



- =499 ; it is plain, that the Area AF^/3 may be found by
the compofition of the Areas found before, when C/3 = i oo j i ooo i

7>



and IN FIN ITE SERIES, 97

7; or any other of the above-mention'd numbers, AB = BF being
llill unity. This I was willing to infinuate, that a method might
be derived from hence, very proper for the conftrudtion of a Canon
of Logarithms, 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
not very troublefome. But whereas that Canon feems to be deriva-
ble from this fountain more commodioufly than from any other,
what if I mould point out its contraction here, to compleat the
whole ?

57. Firfl therefore having affumed o for the Logarithm of the
number i, and i for the Logarithm of the number 10, as is gene-
rally done, the Logarithms of the Prime numbers 2, 3, 5, 7, 1 1,
13, 17, 37, are to be inveftigated, by dividing the Hyperbolical
Areas now found by 2.3025850929940457, which is the Area cor-
refponding to the number 10: Or which is the fame thing, by mul-
tiplying by its reciprocal 0.4342944819032518. Thus for Inftance,
if 0.69314718, &c. the Area correfponding to the number 2, were
multiply'd by 0.43429, &c. it makes 0.3010299956639812 the Lo-
garithm of the number 2.

58. Then the Logarithms of all the numbers in the Canon,
which are made by the multiplication of thefe, are to be found
by the addition of their Logarithms, as is ufual. And the void
places are to be interpolated afterwards, by the help of this
Theorem.

59. Let be a Number to which a Logarithm is to be adapted, A-
the difference between that and the two neareft numbers equally
diflant on each fide, whofe Logarithms are already found, and let d
be half the difference of the Logarithms. Then the required Loga-
rithm of the Number n will be obtain'd by adding d-\- +- gr^,

&c. to the Logarithm of the leffer number. For if the numbers
are expounded by C/>, C/3, and CP, the rectangle CBD or C,&T=i,
as before, and the Ordinates pq and PQ^being raifed ; if n be wrote

for C/3, and x for p or /3P, the Area pgQP or ^ -+- ~ } + ~,
&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;



g 8 Tie Method of FLUXIONS,

dx dx* dx*

- -+- -f- &C.

one: which therefore will be . , that is, when the

x A' 3 A"* a

- +- -4- &c.
divifion is perform'd, d-\- -4- - &c.

* 2n i Zfj s

60. The two firft terms of this Series d-\- I think to be accu-

2n

rate enough for the construction of a Canon of Logarithms, even
tho' they were to be produced to fourteen or fifteen figures; pro-
vided the number, whofe Logarithm is to be found, be not lefs
than 1000. And this can give little trouble in the calculation, be-
caufe x is generally an unit, or the number 2. Yet it is not necef-
fary to interpolate all the places by the help of this Rule. For the
Logarithms of numbers which are produced by the multiplication or
divifion of the number laft found, may be obtain'd by the numbers
whofe Logarithms were had before, by the addition or fubtraction
of their Logarithms. Moreover by the differences of the Loga-
rithms, and by their fecond and third differences, if there be occa-
lion, the void places may be more expeditioufly fupply'd ; the fore-
going Rule being to be apply'd only, when the continuation of fome
full places is 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
Arithmetical progreffion.

62. Alfo by purfuing the fteps of this method, rules might be
eafily difcover'd, for the conftruction of the tables of artificial Sines
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
are exprefs'd by Equations confirming of complicate terms ; and that
by means of their reduction to Equations, which confift of an infi-
nite number of fimple terms. But whereas fuch Curves may fome-
times be fquared by finite Equations alfo, or however may be com-
pared with other Curves, whofe Areas in a manner may be confi-
der'd as known ; of which kind are the Conic Sections : For this
reafon I thought fit to adjoin the two following catalogues or tables
of Theorems, according to my promife, conflructed by the help of
the jtb and Bth aforegoing Propofitions.

64.



and IN FINITE SERIES. 99

64. The firft of thefe exhibits the Areas of fuch Curves as can be
fquared ; and the fecond contains fuch Curves, whole Areas may be
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
Areas, as before. The letters and 6, annex'd to the quantity z,
denote the number of the dimenfions of the fame z, whether it be
integer or fractional, affirmative or negative. As if =3, then

JZ1ZZZ2 3 , z l "=z s , z-=z-~> or-' 3 , &+' = z*, and z*-' =z*.

65. Moreover in the values of the Areas, for the fake of brevity,
is written R inftead of this Radical \Se-{-f& t or </e-t-fzi-\-gz**>,
and/ inflead of </b-t-iz* t by which the value of the Ordinate^ is
affected.



10O



"fhe Method of FLUXIONS,






,
t



I
I

i



s



1



I



a



CO

rt



ii

'5



CO

U

3



Curve



u



+



n

en

e*



H N










- 1 N CO



*

1 v,

N 1 S 1



'
T *~

1 *J-

~

V ^






and INFINITE SERIES.



101



T- *

II II



a



CO



bo



t

II






U



a

o
u



f ^

^ \

O ol \O

iM

""* I I

1 I



cno *?> oa'j j?
N cr>



N



II II



T
1



v

t



*



i



H-



T*

II II



?r



-f'



c






f



M



s



OJ






M M



x



o

G



* **

X



01



X"

i



X X

s
CO s:



x"



IO2 e jff>e Method -o^ FLUXIONS,

67. Other things of the fame kind might have been added ; but I
fhall now pafs on to another fort .of Curves, which may be com-
pared with the Conic Sections. And in this Table or Catalogue
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 a^, and the Area

defcribed a^EC. But the beginning of this
Area, or the initial term, (which com-
monly either 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-
pendicular a^/.

68. After the fame manner you have the Conic Sedlion repre-
fented by the Line PDG, whofe Center is A, Vertex a, rectangular





Semidiameters 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
DT meeting AB in T, the Subtenfe aD, and the Re&angle infcribed
or adfcribed ABDO.

69. Therefore retaining the letters before defined, it will be
AC = z, CE=y, a.%EC = t, AB or aB = x, BD = i;, and
ABDP or aGDB=j. And befides, when two Conic Sections are
required, for the determination of any Area, the Area of the latter
mall be call'd <r, the Abfcifs |, and the Ordinate T. Put p for



and INFINITE



103



S

S



u



o

en

_2
3



rt



-y

CO



5
U






o

U.



oa






a
O
Q






V






V



Tl



BL,
O



Q
O

14

o

c






.

+ i?



.V3
>*,



Q
O

rt
2



ea
Q
O

rt
>s| =



O

eg

Q



Q
O



M

OH

Q

pa



CO

Q
O









**












I

4-






- V-



104



Method of FLUXIONS,



OJ



m

(LI

3



Q

O
a
c
c

h

O

Q

Pi
O
C



O

rt
O



^1^



a

o

Q
O

o
.5

o
Q
O

2



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Q

rt
O
B



dina



ed



C

3



-
3

U



I

o

fe



4



I



a
ji



^

H



-



4-



.<v
fa.



cj

o
o

n



X

15 ^



V

4-



fr

SJ



t

3|

s






"T



II II



+



+






I

K
+

%



v






-V"



u

13



O



U

S)



u

O



and INFINITE SERIES.



U

<J-.

o

1=
o



$

N

4-
H

Mj



P s-



^ ^



It tt



*-



<

Q



CO

Q
O



4



i



'









.






I.Hf

Tf-

1






+ 1

5



+ 1



V



1 +



<5

+
b<j



+



)



I?






4-



E -V

H-|

^i t.



X



io6



Method of FLUXIONS,



s



J*



U



J

to



_u

o









I

3
U



t/>

o

fa






+



X



v.



i



s



+



x



and INFINITE SERIES, 107

71. Before I go on to illuftrate by Examples the Theorems that
are deliver'd in thefe claffes of Curves, I think it proper to obferve,

72. I. That whereas in the Equations reprefenting Curves, I have
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,
they muft be changed in the fubfequent values of the Abfcifs and Or-
ninate of the Conic Section, and alfo of the Area required.

73. II. Alfo the figns of the numeral Symbols and 0, when they
are negative, muft be changed in the values of the Areas. More-
over their Signs being changed, the Theorems themfclvcs may ac-
quire a new form. Thus in the 4th Form of Table 2, the Sign ot

d '

being changed, the 3d Theorem becomes -;_ iv ,.-j- I ~~~' ,-^ -}> ~~^



x , &c. that is, 7=^= =}', *"==*,

' cz -f-/a

into 2.w 3^===^. And the fame is to be obferved in others.

74. III. The feries of each order, excepting the 2d of the ift Ta-
ble, may be continued each way ad infinitum. For in the Series of
the -;d and 4th Order cf Table i, the numeral co-efficients of the
initial terms, (2, 4, 16, 96, 768, Sec.) are fonn'd by multi-
plying the numbers 2, 4, 6, 8, ro, &c. continually
into each other ; and the co-efficients of the fubfequcnt terms are de-
rived from the initials in the 3d Order, by multiplying gradually by
1 > A, , , -Li, &rc. or in the 4th Order by multi-
plying by * i, 4-, f, T> -rV. &C. But the co-efficients
of' the denominators i, 3, 15, 105, &c. a rife by multiplying the
numbers i, 3, 5, 7, 9, &c. gradually into each other.

75. But in the ad Table, the Series of the i ft , 2 d , 3'', 4 h , c; 1 ", and
io th Orders are produced in infinitum by diviiion alone. Thus having

= v, in the ift Order, if you perform the diviiion to a con-
venient period, there will arifo j~ ~z ^ 'j 7 ^

==.)'. The firft three terms belong to the ift Order of



t/x



.4 - 1-'




Table i, and the fourth term belongs to the ift Species cf this Order.

d 3n Jc - 4 <: ~ n

Whence it appears, that the Area is 7^- - ~ 1^f z + r ?r ~

__ _il s . } putting s for the Area of the Conic Section, whofe Abfcifi

*' d

is x=r , and Ordinate v = g - : r- -.

P 2



io8 7&e Method of FLUXIONS,

76. But the Series of the ^th and 6th Orders may be infinitely
continued, by the help of the two Theorems in the 5th Order of
Table i. by a due addition or fubtraction : As alib the 7th and 8th
Scries, by means of the Theorems in the 6th Order of Table i. and
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 far-
ther continued, fuppofe 6 = 4>j, and the ift Theorem of the

jth Order of Table i. wll become 8fts~ 4l| ~~ 1 . 5/b~ 3>1 ~ 1 into
=. -^-=^f. But according to the 4th Theorem of




this Series to be produced, writing ^ for </, it is ~ f%>

<x=v, and 'Qfr'-'S/*' __ t

ize

So that fubtrafting the former values of / and /, there will remain

4J ' / J- 1 10/1/3 Ii;/?} R S a _,, - , .

qnez v/^-h/ 2 =/> I2e ft Thefe being mul-

ij j

tiplied by - ; and, (if you pleafe) for -~ writing xv*, there will arife
a 5th Theorem of the Series to be produced,'



, 1 ! s -

= v, and -r- = f.



77. IV. Some of thefe Orders may alfo be otherwife derived from
others. As in the 2d Table, the 5th, 6th, 7th, and nth, from the
8th; and the 9th from the loth : So that I might have omitted them,
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.

78. V. If the defining Equation of any Curve is compounded of
feveral Equations of different Orders, or of different Species of the
fame Order, its Area mufl be compounded of the correlponding A-
reas ; taking care however, that they may be rightly connected with
their proper Signs. For we mufl not always add or fubtra<fl at the
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
conftitute a correfponding Area. And this muft be done, when the
constituent Areas are pofited on the contrary fide of the Ordinate.
Huf that the cautious Geometrician may the more readily avoid this

in-



and INFINITE SERIES. 109

inconveniency, I have prefix' d their proper Signs to the feveral Va-
lues of the Areas, tho' ibmetimes negative, as is done in the jth
and yth Order of Table 2.

70. VI. It is farther to be obferved, about the Signs of the Areas,
that -f- * denotes, either that the Area of the Conic Section, adjoin-
ing to the Abfcifs, is to be added to the other quantities in the value
of t , ( fee the ifl Example following ;) or that the Area on the other
fide of the Ordinate is to be fubtracled. And on the contrary, s
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
is to be added, as it may feem convenient. Alfo the Value of f, if
it comes out affirmative, denotes the Area of the Curve propoled ad-
joining to its Abfcifs : And contrariwife, if it be negative, it repre-
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 un-
certainty: But its initial Limit, or the beginning from whence its de-
fcription commences, may obtain various pofitions. In the following
Examples it is either at the beginning of the Abfcifs, or at an infinite
diftance, or in the concourfe of the Curve with its Abfcifs. But it
may be placed elfewhere. And wherever it is, it may be found, by
ieeking that length of the Abfcifs, at which the value of f becomes
nothing, and there erecting an Ordinate. For the Ordinate fo raifed
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 Ab-
fcifs.

82. IX. Whenever the dimenfions of the terms in the values of
.v, i;, and /, (hall afcend too high, or defcend too low, they may be
.reduced to a juft degree, by dividing or multiplying fo often by any

given quantity, which may be fuppos'd to perform the office of Uni-
ty, as often as thole dimenfions mail be either too high or too low.

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

84-



no



The Method ^FLUXIONS,




84. EXAMPLE I. Let QER be a
Conchoidal of fuch a kind, that the Q
Semicircle QH A being defcribed, and
AC being creeled perpendicular to R
the Diameter A Q^_ if the Parallelo-
gram QACI be compleated, the Dia-
gonal AI be drawn, meeting the Se-
micircle in H, and from H the'per-

pendicular HE be let fall to 1C ; then the Point E will defcribe a
Curve, whole Area ACEQJs fought.

^.Therefore make AQ^==a, AC=z, CE=y, and becaufe of the

continual Proportionals AI, AQ^, AH, EC, 'twill be ECor_>'= - ^

86. Now that this may acquire the Form of the Equations in the
Tables, make =2, and for z~- in the denominator write z*, and
a*z~-* * for



or



;]-' in the numerator, and there will arife_y =
flf > an Equation of the ift Species of the ad Order of Table 2,

a -\-x,

and the Terms being compared, it will be^ = rf 3 , e = a*, and
f= I j .fo that 4/ .J'' i

/ v ii T-<



x,



3 tf 1 .*; 1 = -u, and xv 2s



t.



87. 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

which, as unity, a* may be multiplied once in the value of x, and
in the value of v, a> may be divided once, and ^x 1 twice. And by

this means you will obtain s/"^niTr =^,^/a l .v 1 =1', and xv
2s, t: of which the conllradion is thus.

88. Center A, and Radius AQ^_ defcribe the Qigadrahtal Arch
QDP ; in AC take AB = AH ; raiie the perpendicular BD meeting
that Arch in D, and draw AD. Then the double of the Scclof

ADP will be equal to the Area fought ACEQ^ For






' AB.?=) BD, or-y ; and .vj 2s= 2 A ADB 2
or = 2*A ADB'-f- aBDP, that is, either = aOAD, or=2DAP:
Of which values the affirmative aDAP belongs to the Area ACEQ,
on this fide EC, and the negative aC^AD belongs to the Area
RE R extended ad infi.ritum beyond EC.

89. The folutions 'of Problems thus found may fometimes be
made more elegant. Thus in the prefent cafe, drawing RH the le-

midiameter



and INFINITE SERIES.



in



midiameter of the Circle QH A, becaufe of equal Arches QH and DP,
the Sector QRH is half the Sector DAP, and therefore a fourth part
of the Surface ACEQ^

90. EXAMPLE II. Let AGE be a Curve, which is defcribed by the
Angular point E of the Norma AEF, whilft one of the Legs AE,
being interminate, paffes continually through the given point A,
and the other CE, of a given length,
flides upon the right Line AF gi-
ven in pofition. Let fall EH per-
pendicular to AF, and compleat
the Parallelogram AHEC ; and
calling AC = z, CE =_y, and
EF = rf, becaufe of HF, HE, HA
continual Proportionals, it will be

HAor y=



r,




91. Now that the Area AGEC may be known, fuppofe = *,

tr-i
or 2 = , and thence it will be j== =}' Here fl "ce z in the

' a ~z^ 1

numerator is of a fraded dimenfion, deprefs the value of/ by di-

~ V) ~ I
viding by z&, and it will be 7=7= = S> an Equation of the

y a ~ * i

ad Species of the ;th Order of Table 2. And the terms being com-
pared, it is </= i, e= i, and /= a*. So that z 1 =
/- ' __ N A .i jV /^i _ .v 1 -u, and 5 xv = /. Therefore fince

\*~ ) _ ;

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



92. Ex-



The Method of FLUXIONS,



112

92. EXAMPLE III. Let AGE be the Ciflbid belonging to the
Circle ADQj defcribed with
the diameter AQ.. Let DCE
be drawn perpendicular to the
diameter, and meeting the
Curves in D and E. And na-
ming AC = z t CE =.y, and
AQj== a ; becaufe of CD,
CA, CE continual Proportio-
nals, it will be CE or y =

:, and dividing by z, 'tis

X

y = / ~~ Therefore zr~ l

' az I

== ^, or i = ,and thence



y =



V aa i-i



an Equation or




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

% = x, </ax xx = v, and 3^ 2x1; = /. Wherefore

it is *AC = x, CD = v, and thence ACDH = s ; fo that
3ACDH 4AADC = 3* 2xv = t = Area of the Ciflbid
ACEGA. Or, which is the fame thing, 3 Segments ADHA = Area
ADEGA, or 4 Segments ADHA = Area AHDEGA.

93. EXAMPLE IV. Let PE
be the firft Conchoid of the
Ancients, defcribed from Center
G, with the Afymptote AL,.
and diftance LE. Draw its
Axis GAP, and let fall the Or-
dinate EC. Then calling AC
=: z, CE =.y, GA = a, and
Ap . c ; becaufe of the Pro-
portionals A C : CE AL : :
GC : CE, it will be CE or y

04. * Now that its Area PEC may be found from hence, the
paits'of the Ordinate CE are to be confider'd feparately. And if

the Ordinate CE is fo divided in D, that it is CD = v/^ ,

and




and INFINITE SERIES.



and DE = *\/V ^ ; CD will be the Ordinate of a Circle de-

fcribcd from Center A, and with the Radius AP. Therefore the
part of the Area PDC is known, and there will remain the other
part DPED to be found. Therefore fince DE, the part of the Or-
dinate by which it is defcribed, is equivalent to -\/e* z* ; fup-

pofe 2 = w, and it becomes -^/e* z* = DE, an Equation of
the ift Species of the 3d Order of Table 2. The terms therefore
being compared, itisd=t>, f = c t , and/= i; and therefore

1 . j = x, \/ i -+- c* x 1 = v, and zbc l s - - = t.

1



Z Z.

95. Thefe things being found, reduce them to a juft number of
dimenfions, by multiplying the terms that are too deprefs'd, and
dividing thofe that are too high, by fome given Quantity. If this
be done by c, there will arife ~ = x, </ c * -t- x % = v, and



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