Isaac Newton.

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

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and IN FINITE SERIES. 127






it will be Ae : A/ :t A?P ' All* " Q"P : CA] 1 :: TT '
And A/: P/> :: CA : C P. Then ? a>quo t Ae:Pp :: ^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 per-
pendicular Ag-, by the decreafe of which it becomes fy. There-
fore Ae and Pp are as the fluxions of the lines Ag (z) and AP (x),

that is, as z and x. Wherefore 2, : x :: ~- : CP. And fmce it



is Cgl * = CAI a AgT = i &&, and CA = i ; it will be

CP_= * ~* z . Moreover fmce we may aflume any one of the



three x,y, and z for an uniform fluxion, to which the reft are to be
referr'd, if x be that fluxion, and its value is unity, then CP =



13. Befides it is CA (i) : Ag (z} :: CP : PL; alfo CA (i) : Cg
zz ) : : CP : CL ; therefore it is PL = 2Z , and CL =



~ z



j Z z. Laftly, drawing /^parallel to the 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) and PD (;'), that is, as i and y. Therefore becaufe of fimilar
triangles Ppq and CAg, fmce CA and Ag, or i and z, are in the
fame ratio, it will be y = . Whence we have this folution of
the Problem.

14. From the propofed Equation, which exprefles the relation
between x and^x, find the relation of the fluxions x and y, (by Prob. i.)
and putting x = i, there will be had the value of _)-, to which z
is equal. Then fubftituting z for/, by the help of the lafl Equa-
tion find the relation of the Fluxions x,y, and z, (by Prob. i.) and
again fubftituting i for x, there will be had the value of z. Thefe

being found make ^21= CP, z x CP = PL, and CP x v/ 1 yy

Z

= CL; and C will be a Point in the Curve, any part of which
KG is equal to the right Line CG, which is the difference of the
tangents, drawn perpendicularly to the Curve \)d from the points
C and K,



I2 8 7%e Method of FLUXIONS,

15. Ex. Let ax=yy be the Equation which exprefles the rela-
tion between AP and PD ; and (by
Trob. i.) it will be firft ax= 2yy, or
a = 2yz. Then zyz -f- zyz = o, or

= z. Thence it is CP =

y

I yy l_J



4-vv



aa.




c



And from CP and PL taking away y
and x. there remains CD = ,

aa

and AL = ?a ~ . Now I take

away y and x, becaufe when CP and
PL have affirmative values, they fall on the fide of the point P to-
wards D and A, and they ought to be diminiihed, by taking away
the affirmative quantities PD and AP. But when they have negative
values, they will fall on the contrary fide of the point P, and then
they muft be encreafed, which is alfo done by taking away the affir-
mative quantities PD and AP.

1 6. Now to know the Length of the Curve, in which the point
C is found, between any two of its points K and C ; we rauft ieek
the length of the Tangent at the point K, and fubtradt it from CD.
As if K were the point, at which the Tangent is terminated, when
CA and Ag, or i and z, are made equal, which therefore is fituate
in the Abicifs itfelf AP ; write i for z in the Equation a= 2yz,
whence a=2y. Therefore for y write ^a in the value of CD,

that is in , and it comes out a. And this is the length
of the Tangent at the point K, or of DG ; the difference between
which and the foregoing indefinite value of CD, is - -i#> that

is GC, to which the part of the Curve KC is equal.

17. Now that it may appear what Curve this is, from AL (hav-
ing firft changed its fign, that it may become affirmative,) take AK,

which will be ^a, and there will remain KL = %a, which
call /, and in the value of the line CL, which call v, write for



aa -> anc ^ l ^ ere






l a "fe \/^at = v ; or = vv, which
is an Equation to a Parabola of the fecond kind, as was found before.

i a,



and INFINITE SERIES. 129

1 8. When 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 Equa-
tion ^x-^-^y _}' 3 =o were affumed; from hence (by Prob. i.)
firft there arifes a 1 4-^*2 y*z = o, then aaz zyyz y*z=o,



and therefore it is z = , and z = . Whence are

yy aa ' aa yy

given PC = ""'- , and PL = 2rxPC, by which the point C is

determined, which is in the Curve. And the length of the Curve,
between two fuch points, will be known by the difference of the
two correfponding Tangents, DC or PC y.

19. For Example, if we make a= i, and in order to determine
fome point C of the Curve, we take y = 2 ; then AP or x becomes

.y 3"'.v_ _ . -_. - * PC 2 and PI -

Zaa T' z T> z T> 1V "- 2 > ana rLl ?

Then to determine another point, if we take ^' = 3, it will be
AP=6, =i, z = >ir, PC= 84, andPL= ioi.
Which being had, if y be taken from PC, there will remain 4
in the firil cafe, and 87 in the fecond, for the lengths DC j the
difference of which 83 is the length of the Curve, between the two
points found C and c.

20. Thefe are to be thus underftood, when the Curve is conti-
nued between the two points C and c, or between K and C, with-
out that Term or Limit, which we call'd its Cufpid. For when
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
points C or K, muft be feparately found, and then added together.

PROB. XI.

To find as many Curves as you pie of e, 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.

i. It is performed by involving the Length, or the Area of the
propofed Curve, in the Equation which is affumed in the foregoing
Problem, to determine the relation between AP and PD (Figure

Art. 12. pjg. 126.) Eut that z, and z may be thence derived, (by

S Prob.



130 7%4 Method of 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 Ab-
fcifs and of the Ordinate. For let RN be the perpendicular Ordi-
nate, moving upon the Abfcifs MN, and
let QR be the propofed Curve, at which
RN is terminated. Then calling MN
= s, NR=/, and QR='i>, and their
Fluxions s, /, and <u refpeclively ; con-
ceive the Line NR to move into the place
nr infinitely near the former, and letting _ ^
fall RJ perpendicular to nr, then RJ, sr, M" v N"

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




lines, and becaufe of the right Angle Rsr, it will be >/R/ -f-Tr*
= Rr, or \/V -f- f- = <v.

3. But to determine the fluxions s and t there are two Equations-
required; one of which is to define the relation between MN and NR,.
or s and /, from whence the relation between the fluxions s and t-
is to be derived ; and another which may define the relation be-
tween MN or NR in the given Figure, and of AP or x in that re-
quired, from whence the relation of the fluxion s or t to the fluxion
x or i may be difcover'd.

4. Then <u 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

defined. Then we are to take PC = '-^, PL =y x PC, and

DC = PC y, as in the foregoing Problem.

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

be as 2ss = 2tt, or a ~ 2 's=i. And thence - =v s*-i-t* ==: v.



zt zt



By the fecond it is 2X = as, and therefore - t =. v. And by the

third u=y, that is, ^ = z } and hence ^ ^'=2;. Which

being



and INFINITE SERIES. 131

being found, you muft take PC = 1 -^. , PL=/x PC, and DC

== PC y, or PC QR- Where it appears, that the length of
the given Curve QR cannot be found, but at the fame time 'the
length of the right Line DC muft be known, and from thence the
length of the Curve, in which the point C is found ; and fo on the
contrary.

6. Ex.2. The Equation as ss = ff remaining, make # = j,

an d irv ^ax-=.^ay. And by the firft there will be found ^ = -y,
as above. But by the fecond i = s, and therefore ^ = v. And
by the third 2iw 4^ = 407, or (eliminating -y) ^ i = z.

Then from hence " 3L == z ,

j. Ex. 3. Let there be fuppos'd three Equations, aa = st, a +
*s = x, and A: -f- v =}' Then by the firft, which denotes an

Hyperbola, it is o=rf+/i, or 7 = ', and therefore '-V" 4- "

V/M -f- tf = v. By the fecond it is 3* = i, and therefore
- v/w -+- = v. And by the third it is i + -u == y t or i +

3'

</ss-4-tt=:z; then it is from hence w =s, that is, putting w
3'

for the Fluxion of the radical -^ </" -t- ^, which if it be made

equal to iv, or | -f- ~ = 7C'i;, there will arife from thence ^
^ = 2W7i;. And firft fubftituting ~ for /', then 1. for s, and

dividing by aw, there will arife P^ 3 = iv = z. Now _>' and z

being found, the reft is perform'd as in the fivft Example.

8. Now if from any point Q_of a Curve, a perpendicular QV is
let fall on MN, and a Curve is to be found whofe length may be
known from the length which arifes by applying the Area QRNV

to any given Line ; let that given Line be call'd E, the length

which is produced by fuch application be call'd <y, and its fluxion v.
And fince the fluxion of the Area QRNV is to the Fluxion of the
Area of a reiTtangular parallelogram made upon VN, with the height
E, as the Ordinate or moving line NR = t, by which this is dc-
fcribed, to the moving Line E, by which the other is deicribcd in

S 2 the



132 tte Method of FLUXION s,

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

given Line E, are in the fame ratio ; it will be v= s ~ . Therefore

by this Rule the value of v is to be inquired, and the reft to be
perform'd as in the Examples aforegoing.

9. Ex. 4. Let QR be an Hyperbola which is defined by this

Equation, aa -+ = // ; and thence arifes (by Prob. I.) =tf,
or = t. Then if for the other two Equations are aflumed x=s
and y = v ; the firft will give i = j, whence v = ^ = } and
the latter will give y = v, or z = -g, then from hence z= ^ ,
and fubftituting or for t, it becomes z = ~ . Now y and z

ct ft hit

being found, make -r~ === CP, and_y x CP =n PL, as 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 fuffi-
ciently fliewn before.

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-
cal Curves ; of which thing there is a remarkable Example in fpiral
lines.

1 1. Let AB be a right Line given in pofition, BD an Arch mov-<
ing upon AB as an Abfcifs, and yet re-
taining A as its Center, AD^ a Spiral, at

which that arch is continually terminated,
bd an arch indefinitely near it, or the place
into which the arch BD by its motion next
arrives, DC a perpendicular to the arch bd t
dG the difference of the arches, AH an-
other Curve equal to the Spiral AD, BH a
right Line moving perpendicularly upon

AB, and terminated at the Curve AH, bh the ^ ~B~<T

next place into which that right lane moves, andHK perpendicular to

bb.




and INFINITE SERIES. 133

bb. And in the infinitely little triangles DG/ and HK, lince DC
and HK are equal to the fame third Line Bb, and therefore equal
to each other, and Dd and Hh (by hypothecs) are correfpondent
parts of equal Curves, and therefore equal, as alfo the angles at G
and K are right angles ; the third fides dC and hK will be equal
alfb. Moreover fince it is AB : BD :: Ab : bC :: hb AB (Qb) :
bC BD (CG) j therefore - A * B - = CG. If this be taken away
from dG, there will remain dG * & = dC = /6K. Call
therefore AB=*, BD=-y, andBH=>', and their fluxions
z, v, and y refpedtively, fince B, dG, and /jK are the contempora-
neous moments of the fame, by the acceflion pf which they become
A, bd t and bb, and therefore are to each other as the fluxions.
Therefore for the moments in the lafl Equation let the fluxions be
fubftituted, as alfo the letters for the Lines, and there will arife-y .

^ == -.y. Now of thefe fluxions, if z be fuppos'd equable, or the

~ " *'

unit to which the reft are refer'd, the Equation will be i; ^=)'-

12. Wherefore the relation between AB and BD, (or between z
and v,) being given by any Equation, by which the Spiral is defined,
the fluxion v will be given, (by Prob. i.) and thence alfo the fluxion
;', by putting it equal to v. ^ . And (by Prob. 2.) this will give
the line y, or BH, of which it is the fluxion.

i?. Ex. i. If the Equation jzrzr-u were given, which is to the

Spiral of Archimedes, thence (by Prob. i.) - 2 -^ = v. From hence
take - , or - , and there will remain - =y, and thence (by Prob. 2.)
2?_-r. Which fhews the Curve AH, to which the Spiral AD i

2U

equal, to be the Parabola of Apollonius, whofe Latus reclum is 2??;
or whole Ordinate BH is always equal to half the Arch BD.

14. Ex. 2. If the Spiral be propofed which is defined by the

5 }_

Equation a 3 =a'v 1 , or v =;^ , there arifes (by Prob. i.) =-r,

T 2^ T

_l_ I

from which if you take ^, or ~- , there will remain , = v, ano

fl x 2i T

i
thence (by Prob. 2.) will be produced ^l = v. That i. ; ; -BD nrr

3^

EU, AH being a Parabola of the fecond kind, t >



is



134 tte Method of FLUXIONS,

15. Ex. 3. If the Equation to the Spiral be z</"^ =-y, thence
(by Prob. i.) - a , . ?.~- = v ; from whence if you take away ""- or

' 2 V ac -\- cz K

^/- ?, there will remain , ~.. - = y. Now 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 Scho-
lium to Prob. 9. I reduce it to the form of the Equations in the firft
column of the Tables, by fubftituting z* for z, ; then it becomes

=.y, which Equation belongs to the 26. Species of the 4th

Orderof Table i. And by comparing the terms, it is d=,e=:ac,
andf=c, fo that -~ 2 - ^ ac -f- cz == f=y. Which Equation

belongs to a Geometrical Curve AH, which is equal in length to the
Spiral AD.

PROB. XII.

To determine the Lengths of Curves.

1. In the foregoing Problem we have fhewn, that the Fluxion of
a Curve-line is equal to the fquare-root of the fum of the fquares of
the Fluxions of the Abfcifs and of the perpendicular Ordinate.
Wherefore if we take the Fluxion of the Abfcifs for an uniform and
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
Curve, we find the Fluxion of the Ordinate, we mall have the
Fluxion of the Curve-line, from whence (by Problem 2.) its Length
may be deduced.

2. Ex. i. Let the Curve FDH be propofed, which is defined by

the Equation - -f- - '- =_y ; making the Abfcifs AB = s, and the

moving Ordinate DB =y. Then Jr

from the Equation will be had,

(by Prob. i.) 3 = y, the ^ J v -

\ s ' aa 12Z.S. - / '

fluxion of z being i, and y being

the fluxion of y. Then adding the X~

fquares of the fluxions, the fum

v/ill be -h |-f- -^ == it, and extracting the root,



and INFINITE SERIES. 135

= t, and thence (by Prob. 2.) ^ ^ = : t . Here / ftands for the

fluxion of the Curve, and / for its Length.

3. Therefore if the length </D of any portion of this Curve were
required, from the points d and D let fall the perpendiculars db and
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
Length required. As if Ab === ?a, and AB = a, writing La for #,






it becomes t = ; then writing a for #, it becomes / =
from whence if the firfl value be taken away, there will remain
^ for the length </D. Or if only h.b be determin'd to be ^a, and

AB be look'd upon as indefinite, there will remain - _i_ -1

aa 1 2ft 24

for the value of



4. If you would know the portion of the Curve which is repre-
fented by /, fuppofe the value of / to be equal to nothing, and there

arifes z* = , or z= - .. Therefore if you take AB=-^- >
12 V*z y, 2

and eredT: the perpendicular bd t the length of the Arch ^D will be

t or And the fame is to be underflood of all Curves

11%



aa



in general.

5. After the fame manner by which we have determin'd the

length of this Curve, if the Equation ^ -f- -^L =y be propofed,
for defining the nature of another Curve ; there will be deduced

^ . _lL -=.t\ or if this Equation be propofed, La*y?~*.
3" 1 * "*

2_

there will arife ^ -f-i^ 5 '= t. Or in general, if it is cz* -{-



* _ .- =_>', where 6 is u fed for reprefenting any number, either

,-8"

Integer or Fraction, we (hall have cz* = /.

o 4&Qi od<r

6. Ex.2. Let the Curve be propofed which is defined by this
Equation "" + ^ \/ #a -t- =t^,V; then 1 (by Prob. i.) will be had
_y = ^^-r ^f*-* + 4* ^ or exterminating y t y= '-'</~aa-{- zz.
To the fquare of which add i. and the fum will be i -J- ~ 4- 4 - 4 .

aa a*

and



136 ttt Method of FLUXIONS,

and its Root i -f- * = t. Hence (by Prob. 2.) will be ob-

aa * * /



tain'd 2 + ^

A



7. Ex. 3. Let a Parabola of the fecond kind be propofed, whofe
Equation is z* = ay 1 , or ~ =_y, and thence .by Prob. i. is derived



r==y. Therefore < 1 -+- 2f: = ~ i -+- yy ss . Now fmce the

2a a 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 belong-

ing to it, I have / = ' v/ 1 -t- . And thus you may find

the lengths of thefe Parabolas Z 1 = ay*, 2? r= ay*, z> = ay*,
&c.

8. Ex. 4. Let the Parabola be propofed, whofe Equation is *

4 *

= rfy 3 , or ^=:^; and thence (by Prob. i.) will arife 1^ = _y.

"



Therefore v/ 1 -f- i^ = </yy -+- i = t. This being found, I

ga 7

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

sF = x, v/i -f- 1 ^ = v, and |j=?. Where x denotes the Ab-

9 7

fcifs, y the Ordinate, and s the Area of the Hyperbola, and / the
length which arifes by applying the Area %s to linear unity.

9. After the fame manner the lengths of the Parabolas z 6 =ay',
z* :z=y 7 , z' =ay', &c. may alfo be reduced to the Area of the
Hyperbola.

jo. Ex. 5. Let the CuToid of the Ancients be propofed, whole

Equation is ^T^jL" __ ; . an d thence (by Prob. i.)

V az. 2.Z. '



22,*



v / az zz=y, and therefore -^ ^/"^ = ^ yy -f- i = t ;

which by writing 2? for ^ or z~\ becomes ^ v/ ' az" -f- 3 = /,
an Equation of the ift Species of the 3d Order of Table 2 ; then
comparing the Terms, it is ^ = d, 3 =. e, and ^ =^5 fo that

i /: 20;' 4</c i 3

= 'u, and 6; ___.l_into s=f.

AT My 2iA?

And



and INFINITE SERIES.



37



= v, and

a ax



And taking a for Unity, by the Multiplication or Divifion O f
which, thefe Quantities may be reduced to a juft number of Di-

menfions, it becomes az = xx, <
f : Which are thus conftructed.

1 1. The Ciflbid being VD, AV the Diameter of the Circle to
which it is adapted, AF its Afymptotc, and DB perpendicular to
AV, cutting the Curve in D ; with the
Semiaxis AF = AV, and the Semipara-
meter AG = jAV, let the Hyperbola
YkK be defcribed ; and taking AC a mean
Proportional between AB and AV, at C
and V let CA and VK drawn perpendi-
cular to AV, <:ut the Hyperbola in ,
and K, and let right Lines kt and KT
touch it in thofe points, and cut AV in
/and T; and at AV let the Rectangle
AVNM be defcribed, equal to the Space
TK&. Then the length of the Ciflbid
VD will be fextuple of the Altitude VN.



d




12. Ex. 6. Suppofing Ad to be an Ellipfis, which the Equation
i/az 2zz =y reprefents ; let the mechani-
cal Curve AD be propofed of fuch a nature, that v ''
if B</, or_)', be produced till it meets this Curve
at D, let BD be equal to the Elliptical Arch &d.
Now that the length of this may be deter-
min'd, the Equation \/ az 2.zz=. y will give

=y, to the fquare of which if i be added, there ariies

, the fquare of the fluxion of the arch A.J. To which



zy az
aa 4



02 Szz



if i be added again, there will arife -^ ^ ^- , whofe fquare-root
=.* __ is the fluxion of the Curve-line AD. Where if z be ex-

2y / az 2ZZ

tracted out of the radical, and for z ~ be written c", there will be
" - , a Fluxion of the ift Species of the 4th Order of

'



=rt; fo that z= = x, \/ ux _.v.v = <



Table 2. Therefore the terms being collated, there will arife d=.^a,
e = 2,



and -1 + ,= into,






J3 8



Method of FLUXION s.



i-i. The Conftruaion of which is thus; that the right line </G
being drawn to the center of the Ellipfis, a parallelogram may be
made upon AC, equal to the fedlor AC/, and the double of its
height will be the length of the Curve AD.

14. Ex. 7. Making A/3= tp, (Fig, i.) and CL being an Hyper-
bola, whofe Equation is v/ a -+ % = $&, and its tangent <TT
being drawn ; let the Curve
WD be propofed, whofe

Abfcifs is , and its per-
pendicular Ordinate is the
length BD, which arifes by
applying the Area a^To. to
linear unity. Now that the
length of this Curve VD
may be determin'd, I feek
the fluxion of the Areaa<rTa,
when AB flows uniformly,

and I find it to be -^

v/ ' b ax, putting AB =,
and its fluxion unity. For

'tis AT = = </z, and its fluxion is -rV , whofe half drawn

t>p o za v z

into the altitude /3<^, or v/ a + - , is the fluxion of the Area




, defcribed by the Tangent <TT. Therefore that fluxion is
-p v/ ' b az, and this apply'd to unity becomes the fluxion of the
Ordinate BD. To the fquare of this ~^~ add i, the fquare of



the fluxion BD, and there arifes ^~ fl ^+ a l6 ^ ta , whofe root -^

</a*b a>z-\- ibfrz*- is the fluxion of the Curve VD. But this
is a fluxion of the ift Species of the 7th Order of Table 2 : and

the terms being collated, there will be - = </, aab=e, a*=f,



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

and */i6bb - a*% + a&t- i = Y ) (an Equation to another Conic

Section,



and INFINITE SERIES. 139

Section, (hppofe ML (Fig. 3.) whofe Area IKLM is <r, where IK

i T/"T *w* \ T /XT 2aftbb^f fl5^Y*tf4y Aaabb? T.2abbs

g and Kl_/= TiJ L,aitiy " 2 /\

15. Wherefore that the length of any portion DJ of the Curve
VD may be known, let fall db perpendicular to AB, and make Kb
= z ; and thence, by what is now found, feek the value of t.
Then make AB=,s, and thence alfo feek for /. And the diffe-
rence of thefe two values of / will be the length Dd required.

16. Ex. 8. Let the Hyperbola be propos'd, whofe Equation is

=)', and thence, (by Prob. i.) will be had^ = - ( O r
To the fquare of this add i, and the root of the fum
= /. Now as this fluxion is not to be found



aa -\- bz.z. + bits.



\/aa 4- tzz

will be ^/

in the Tables, I 'reduce it to an infinite Series ; and firft by divifion

" y i / 3 y 4 / 1

it becomes t ;= </ 1 -f-jaS 1 ^ 2:4 H-r 2 ' 5 7* z * > & c - a d extracting



the root, t ==



a-




c A ,
z & , &c. And

*



hence (by Prob. 2.) may be had the length of the Hyperbolical Arch



17. If the Ellipfis \/aa bz,z=.y were propofed, the Sign of
b ought to be every where changed, and there will be had z 4-

& _f- - ^ *-z' -^ 1 i ^t_s 7 , &c. for the length of its

Arch. And likewife putting Unity for b, it will be z -+- -^ -f-
3ii_4_ Jil , &c. for the length of the Circular Arch. Now the

104 I I 2V.'' > O



numeral coefficients of this feries may be found adinfinitum t by mul-
tiplying continually the terms of this Progreflion j , , ^- >

S x 9 ' 10 x i i '

18. Ex. 9. Laftly, let the Quadratrix VDE be propofed, whole
Vertex is V, A being the Center, and AV
the femidiameter of the interior Circle, to



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