FLU
(*5 )
F L Y
Hence to find the Fluxion of any kind of Power, proceed
Multiply the Power given by its Index or Exponent, and then that Product by the Fluxion of the Root of the Power given ; and after that, fubtract One, or Unity, from the In- dex of the Power.
V. To find the Fluxions of Surd Quantities.
Suppofe it requir'd to find the Fluxion of <J z r x — x x, or zrx — xx I i- Suppofe a r x _ x „ [i = z h then
is z r x ■ — **=:33; and confequently r x ■ — x xz=.zz;
and by Divifion.
T X — :
r= is =£ (by Subflitution )
z= to the Fluxion of ^ 2 r x — ■ x x
j a r x-
If it be required to find the Fluxion of a y — x x | * '■>
for a y — x x\ put z 3 then, ay — x x = z r , and a y
— a x x -y3 T z : And multiplying by 3, 3 a y — 6
x x — z s z -j and confequcntly, 3 ss * y — 6 z T x x
r-~ z equal ( fubftituting ay — • * x[* = 2; *- ) 3 a 1 y 1 y — 6
a 1 x x y y -\- 3 a x* y ~ 6 a' y* x x -J- 12 a y x i x~—5x*
x = to the Fluxion of a y ■ — • x x l .
VI. To find the Fluxion of Quantities compounded of Ra- tional and Surd Quantities.
Let it be requir'd to find the Fluxio n of b x* -L c a x _\- c a~ x x / x x -[- a a zn z. Put it x* - t - c a x -p e a 1 =^ and x / x x -\- a a = q. Then the given Quantity is $ q
szz z, and the Fluxion thereof is f q ~\- q f — z : But q
is V x x -\- a a, and p is rz: a £ £ ^ -L c a x 5 therefore in
the Equation p q \ q p =. z, if in the place ^5, <7, ^, ^, we reitore the Quantities they reprefent, we fhall have
b x l A- c a x' -\- e a~ x x x -\- z b x x ^ x x -\- a a x x V x x J- e a
X c a X V x x -\- a a x x — g. Which being redue'd to one Denomination, gives 3 b x i -4-aW£**4- e a~ x 4-
V x j: -|- a a r zz: s zz: to the Fluxion of the given
Quantity. /"
Jnvevfc Method 0/ Fluxions, or Calculus Intcgralis, con- filts in finding finite Magnitudes! from the infinitely imall Parts thereof.
It proceeds, as already obferv'd, from infinitely Imall Quantities to finite ; and recompounds and funis up what the other has refolved ; whence it is alfo call'd the Summa- tory Calculus. '
But what that has decompounded, this does not always re-eilablifh ; fo that the Invcrfe Method is limited, and im- perfca 5 at leaft, hitherto. If it were once compleat, Geo- metry would be arrived at its latt Perfeftion.
To give an Idea of its Nature and Office, take the In- flance already propofed in the direct Method: In that the infinitely fmall Quantities of the Ordinates and Abfcils, being known, give the Subtangent required. In this, on the contrary, the Subtangent of an unknown Curve being had, gives the infinitely imall Quantities of the Abfcifs and Or- dinate which produe'd it, and of confequence the Abfcils and Ordinate thcmfelves ; which are finite Magnitudes, in whofe Relation the whole Effence of the Curve is founded.
But the diflinguifhing Province of this Methodis in meafur- ing the Bafe of a Parallelogram multiplied by the infinitely fmall Element of its Weight, gives an infinitely fmall Pa- rallelogram ; which is the Element of the finite Parallelo- gram, and is repeated an Infinity of times therein; i.e. ^as many times as there are Points in the Height of the Pa- rallelogram.
To have the finite Parallelogram, therefore, by means of its Element, the Element muft be multiplied by the Height ; which is the invcrfe Method of Fluxions-, re- afcending from the infinitely fmall Quantity, to the finite. _
Such a Circuit ot" Infinitcfimals, 'tis true, were imperti- nent in fo fimple a Cafe ; but when we have to do with Surfaces, terminated by Curves ; the Method then becomes neceifary, or at leaft fuperior to any other.
Suppofe, e. £r. in a Parabola, the Space included be-
tween two infinitely near Ordinates, an infinitely fmall Poj-' tion of the Axis, and an infinitely little Arch of the Curves 'Tis certain, this infinitely fmall Surface is no Parallelogram; fince the tw : o parallel Ordinates which terminate it on on© Side, are not equal ; and .the Arch of the Curve, op- pofite to the little Portion of the Axis, is frequently neither equal nor parallel thereto. And yet this Surface, which is no Parallelogram, may be confider'd, in the ftricteft Geo- metry, as if it really were one ; by reafon it is infinitely fmall, and the Error, of confequence infinitely little, i. ct 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 correfponding thereto. Thus we have the Element of the whole Parabola ; which Element being rais'd by the inverse Method to a finite Magnitude, is tho whole Surface of the Parabola.
This Advantage is peculiar to the Geometry of Infinites,_ of being able without any Error to treat Tittle Arches of Curves, as if they were Right-lines ; curvilinear Spaces, as if rectilinear ones, tie. enables it not only to go with more Eafc and Readinefs than the antient Geometry, to the fame Truths ; but alfo a great Number of Truths inacctf- fible to the other.
Its Operations, in effect, ace more eafy, and its Difcoveries more extenfiye : And Simplicity and Univerfality are its di- flinguifhing Characters.
To find the jto-zving Quantity belonging to any Flu- xion given. To have the Doctrine of the Inverfe Method correfpond and keep pace with that of the Direct, we will apply it in the fame Cafes.
1° Then, in the general : To exprefs the variable Quan- tity of a Fluxion, there heeds nothing but to write the .Let- ters without the Dots.
Thus the flowing Quantities o£ x y z, are x y z.
11° To find the flowing Quantities belonging to the Flu- xion of the 'Produii of two Quantities ;
Divide each Member of the Fluxion by the fluxionary Quantity, or Letter 5 or change the fluxionary Letter into the proper flowing Quantity of which it is the Fluxion : The Quota's connected by their proper Signs will be the flowing Quantities fought.
Only, if the Letters be all exactly the fame, the flow- ing Quantity will be a fimple one, whofe Parts are not to be connected together by the Signs -i- and — ■
III To find the flowing Qtiantity belonging to the Flu- xion of any -Tower, either perfetl, or imferfeS.
Take the fluxionary Letter or Letters out of the Equa- tion : Then augment the Index of the Fluxion by 1, or Unity : Lafily, divide the Fluxion by the Index of its Power fo increafed by Unity.
Thus fupp'ofe 3 x x x propofed ; by taking away x it will be 3 x x : and by incrcafing its Index by Unity, it will be 3 xx x: Then dividing it by 3, its now (augmented) In- dex, the Quotient willbe x x *, the flowing Quantity re- quired.
Again, fuppofe-
a Fluxion propos'd : By taking
away the fluxionary *, it will be — x : By augmenting
m the Index by Unity ( /. c . taking away — 1 ) it will be
• — x : And lafily, by dividing the remaining Part of the Fluxion by — , prefix'd to, or multiplied into x, the Quo- tient will be x ™ ; which is the flowing Quantity fought. .
The Ufes of the direB Method of Fluxions, fee fpecified tinder the Articles Maximis<j»^ Minimis, Tangents, iSc.
Thofe of the Inverfe Method, fee under Quadrature of Curvet; Rectification of Curves ; Cubature of Solids, Sec.
FLYBOAT, a large Veffel, -with a broad Bow, us'd by Merchants in the coafting Trade.
Some of them will carry 800 Ton of Goods.
To Fly Grofs, in Falconry, is faid of a Hawk when flic flies at the great Birds, as Cranes, Geefe, Herons, Sjc
Fly on Head; is when the Hawk miffing her Quarry, betakes her feif to the next Check, as Crows, tic.
FLYERS, in Architecture, fuch Stairs, as go ftrait ; and do not wind round, nor its Steps made tapering ; but the fore and back part of each Stair and the Ends reipectiveiy parallel to one another.
So that if one Flight do not carry you to your defign d
Height, there is a broad half Space ; and then you fly
6 ' * R again
