.1l 0  0iL5U                             (  66
'Ajence to find the Fluxion of any kind of Power, proceed
.,ultiply the Power given by its Index or Expoiient, and
~xcn that Produt by the Flulxin of the Root of the. Power
yen; and afier t~hat, fubtra& One, or Unity, from' the In-
abof the Power.
V. To find the Fluxions of Surd kQuantities.
Suppofe it requir'd to find the Fluxion of V2 r X -X  ,
b2r 2 X-r X -   x   Suppofe z 7-x      ,    z, 2 ; then
b  r  x -  X x  z; andconfequently r x - x xz Z;
and by Drvifion, *       = ..  =   ( by Subilitution)
7r X    X X
:; 2  x-  x   to the~luxio   of Vz2rx-xx
If it be required to find the Fluxion of a y-x x
for ay- x|x put z5 then, ay - xx        z3, and a y
-2   x    I- 2  t Z: And multiplying by 3, 3 a y-6
.  -  .                  ~~~   ~   ~~2 s  2
X    X =   z :; and confequently, 3 a z  y- 6 z X X
z equal (fubflituting a y-x xlz    z.) 3a3 y2ay-.6
n xi y yj4- 3a xy-6a'y' xx          i2xzayx3x-.6x5
x    to the Fluxion of a y - x x.
VI. l7o find the Fluxion of 5Quantities compounded of Ra-
tional and Surd 5Quantities.
Let it be requir'd to find the Fluxion of b x2 _I c a x
.i-ea2x   xxx  4-aa-Z. Putbx'_1-cax -, ea'=p
and 4/ x x -I- a a = q. Then the given Quantity is p q
.-z, and the Fluxioi thereof is p q 4- q p-z: But q
xx
is 4 x x -1- a a, and p is _   71 b  x - c a x; therefore in
the Equation p q +-q p-z, if in the place p, q, P, q,
we rei ore the Quantities they reprefent, we Iall ave
x  71- c a x' -4- e a2 x x x; 4. z b x x if x x 4- a a X x
4/ x xl p 4 -


x ± a a x x = z. Which beihg reduc'd to
nation, gives  b xl -1-2acx-- e a2 x.l-
V x x 4- a a


x al x x = z- to the Fluxion of the given


rethod of FILAXIONS, or Calculus Integralis, con-
ding finite Magnitudces from the infinitely frnall
ds, as already obferv'd, from infinitely fmall
:o finite; and recompounds and fums up what
as refolved; whence it is alfo call'd the Sunzma-
S.
that has decompounded, this does not always
fo that the Inverfe Method is limited, and im-
leaft, hitherto. If it were once cdtnpleat, Geo-
ld be arrived at its lafi Perfekion.
in Idea of its Nature and Office, take the In-
dy fropofed in the diredt Method: In that the
ial Quantities of the Ordinates and Abfcifs,
i, give the Subtangent required. In this, on the
e Subtangent of an unknown Curve being had,
finitely fnall Quantities of the Abfcifs and Or-
:h produc'd it, and of confequence the Abfcifs
e themfelves; which are finite Magnitudes, in
tion the whole Effence of the Curve i4 founded.
fiinguifhing Province of this Methodis in meafur-
eof a Parallelogram tnultiplied by the infinitely
ent of its Weight, gives an infinitely fmall Pa-
; which is the Element of the finite Parallelo-
s repeated an Infinity of times therein, i, e . as
as there are Points in the Height of the Pa-
the finite Parallelogram, therefore, by mieans
ent,- the Element muil be multiplied by the
hch is the inverfe Method of RtucionS, re-
rom the infinitely fMall Quantity, to the finite.
Circuit of Infinitefimals, 'tis true, were imperti-
mple a Cafe; but when we have to do with
erninated by Curves, the Method then becomes
or at lea{k fuprior to any other.
e. &r. in a Parbola, the Space incluW  by-


iL


I)


;ween two infinitely near Ordinates, an infinitely fmtaIl 0
tion of the Axis, and an infinitely little Arch of the Curvidt
'Tis certain, this infinitely frnall Surface is no Parallelograms
fince the two parallel Ordinates which terminate it on one
Side, are not equal; and the Arch of the Curve, opt"
pofite to the little Portion of the'Axis, is frequently neither
equal nor parallel thereto. And yet this Surface, which is
no Parallel ram, may be confider'd, in the firidfeft Geo-
metry, as ifit really were one; by reafon it is itifinitel9
fmall, and the Error, of confequence infinitely little, i. e6
none.
So that, to meafure it, there needs nothing but to mul '
tiply an Ordinate of the Parabola by the infinitely fmall
Portion of the Axis corresponding thereto. Thus we have
the Element of the whole Parabola; which Element being
rais'd by the inverfe Method to a finlie Magnitude, is thd
whole Surface of the Parabola.
This Advantage is peculiar to the Geometry of Infinitesf
of being able without any Error to treat little Arches of
Curves, as if they were Right-lines; curvilinear Spaces, as
if relilinear ones, &c. enables it not only to go with more
Eafe and Readinefs 'than the antient Geometry, to the
fame Truths; but alfo a great Number of Truihs inaccefZ
fible to the other.
Its Operations, in eff1ef, are more eafy, and its Dioveried
more extenfive: And Simnplicity and Univerfality are its di-'
ilinguifhing Charadersa
to find the flo/w>  tcintty belonging to any Flit.
xion given.
To have the Dodrine of the Inverfe Method correfpond
and keep pace with that of the Dire&, we will apply it in
the fame Cafes.
10 Then, in the general: l1o exprefs. the variable Xuan-
tity of a Fluxion, there needs nothing but to write the  et-
ters without the Dots.
Thus the flowing Quantities of x y z, are x y z.-
110 Lo find the flowing SQuantities belonging to the Flu-
xion of the Produa of two puantities;
Divide each Member of the Fluxion by the fluxionary
Quantity, or Letter; or change the fluxionary Letter into
the proper flowing Quantity of which it is the Fluxion.
The Quota's conneded by their proper Signs will be the
flowing Quantities fought.
Only, if the Letters be all exaaly the fame, the flows
ing Quantity will be a fimple one, whofe Parts are not to
be conneaed together by the Signs-l-and-
111  To find the flowingg Quantity belonging to the Flu-
xion of any Power, either perfetC, or imperferl.
Take the fluxionary Letter or Letters out of the Equa-
tion: Then augment the Index of the Fluxion by I, or
Unity: Lafily, divide the Fluxion by the Index of its Power
fo increased by Unity.
Thus fuppofe 3 x x x propofed; by taking away x it wil
be 3 x x: and by increafing its Index by Unity, it will be
2 x X X: Then dividing it by 3, its now (augmented) In-
dex, the Quotient will be x x x, the flowing Quattity re6
quired.
n -I
Again, fuppofe   x   a Fluxion propos'd: By taking
away the fluxionary x, it will be-x  : By augmenting
m
the Index by Unity (ii. e. taking away-i ) it will be
And il0ly, by dividing the remaining Part of the
Fluxioii by I, prefix'd to, or multiplied into x; the Quo-
n
Eient will be x' ; which is the flowing Quantity fought;
71'Ze Ufes of the. dired Method of Fluxions, fee fpecij6ed,
under the Articles MAXIMIs and Mi~mnisTANGENTS, EC.
Th~5ofe of the Inverfe Method, fee under QUADRATURF.
of Carvees; RECTIFICATION   of Curves; CUSATURE of
Solids, &c.
FLYBOAT, a large Vefel, with a broad Bow, us'd by
Merchants in the coafting Trade.
Some of theta will carry goo Ton of Goods.
To FPLY Grofs, in Falconry, is faid of a Hawk when 1hd
flies at the great Birds, as Cranes, Geefe, Herons, &G.
FLY on HRead4 is when the Hawk miffing her 0;;iarrys
bekakes her felf to the next Check, as Crows, Ejc..
FLYERSf in Architedure, fich Stairs, as go ftrait; ana
do not wind round, nor its Steps made tapering; ibut the
fore and back part of each Stair and the Ends reftivl
parallel to one another.
So that if one Flight do not carry you to yOur aefign'
* Heitiibere is a broad half Space; an   tihen
Height, there is a                       yqg


4


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