QUA
($28)
QUA
whofe Sum may be found— ——We have two infinite Series, exprefling the Ratio of the Circumference to the Diameter, tho' indefinitely, as above' The firft difco-
ver'd by Sir Ifaac Newton , where the Diameter being put I, the Circumference is 4— 5 — ^~— -^-fg, &c. ' The
iecond, difcover'd by M. Leibnitz 5 where the Diameter being 1, the Circumference is 4 — |_|_| _j-£ — ± £j?g. The Investigation of each of which Series, by the Calculus integralis, is as follows.
Sir Ifaac Newton V Quadrature of the Circle', or the In- veltigation of his Scries, for fquaring the Circle.
If the Kadius of the Circle, A C = 1 (Tab. An AtYsis, fig.s4-) C?=5?,^ ==1 / (i — x *) and y' (1 — **J =1 — | #* — vx*- _^#* _ ^-ji* — Jfr-" — #*°> ^ c " t0 Infinity. Then will
ydte =dx — t*«^* — |**^a? — iV#*^# — » If — x* dx — ¥ J ff — x 19 dx—,i$c. to Infinity.
Jydx — x — \x* — 4 V* S — tt?# 7 — £&e*' — rf*
- , £jfo to Infinity.
When * becomes equal to the Radius C A, the Space D C P IV1 degenerates into a Quadrant. Substituting, there- fore, r for x, the Quadrant will be 1 — i — $5 — ^3 '—
- V s t — Wrs, £ife in infinitum* — Which fame Series will mea-
sure the entire Area ot the Circle, the Diameter being I* M* Leibnitz'* Quadrature of the Circle.
Let the Tangent KB (Tab. Analysis, fig. 25.) == x i BC = i j and the Secant A C, infinitely near another C K, and the little Arch K L be drawn with the Radius C K j then will AK=^*, KC = / (1 +**)• Now fince the Angles at B and L, are right Angles 5 and by reafon of the infinitely frrtall Angle KCL, the Angle BKC = K A C 3 we /hall have
KC;BC::KA :KL dx
Further CK:KL::CM;wM
V(i+**)i
Therefore the Se£lor CM m = zdx-.{ .--#*) = 1 (d X
— x*dse-\-x+d# — x* dx-- x* dx — x'*dx, &c. whence by the Summary Calculus, we find the Sector B C M, whofe Tangent KB—x'^x — %x* -f- T *- vS — t* x7 ~\ m ~h x<> -j'tx 11 , @C. in infinitum, find therefore it ii M be the Octant of the Circle, or an Arch of 45 s ", the Sector will bt£ — i-jrrg "tV. Sifc. in infinitum. The Double, therefore, of this faeries 1 — f.-4-j — y-j-y — it. & c - in infinitum, is the Quadrant of the Circle j or it the Diameter be = 1, the entire Area of the Circle.
Quadrature of the Lunes- Tho' a definite Qua- drature of the entire Circle, was never yet given ; yet
there have been various Portions of it fquar'd The firft
partial Quadrature was given by Hippocrates of Cbio ; who Tquared a Portion call'd, from its Figure, the Lune % or 1m~ nula. See Lu n e, where the Qtiadrature is fhewn.
This Quadrature has no depennance on that of the Circle j but then it only extends to the entire Lune, or its half: tf you would fquare any Portion thereof, at plea- fure, the Quadrature of the Circle comes in the way.
Yet fome of the modern Geometers have found the Quadrature of any Portion of the Lune at pleafure, inde- pendently of the Quadrature of theCircle ; tho' ftill fubject to a certain restriction which prevents the Quadrature from being perfect, and, as the Geometricians call it, abfolute and indefinite.
In 1701, the Marquis (& rHopitalyubliftid a new man- ner of fquaring the Parts of the Lune taken different ways, and under different Conditions tho' this, too, is imper- fect in the fame manner as the others.
QyAT>RhTvR?.oftfce Ellipfis The Ellipfis, too, is
a Circle whole precife Quadrature in definite Terms is not yet effected. We have here therefore, as before, recourfe to a Series.
^to find the Quadrature of the Ellipfis.
Let AC (Tab. Analysis, F/g. itf.) =5«GC = cPC
— x. Then will
y =: c y/ (a*— x x ) : a
- x*~ x* — <f#*
But •[« -<1**> s«.. ,'«* infinitum. Therefore, ydx=cdx 5 cx'dx icit'°dz
-7 at"
12b# 2;
ix Cx*dx
ia
iSc.in
'dx
&c. in infinitum.
If then for x be put a ; the Quadrant of the Ellipfis will be ac—\ ac—f~ ac — ria ac— rr's*- ac— ts tt * c, (5c. in infi- nitum. Which lame Series exhibits the c.itue Area of the Ellipfis, if a denote the entire Axis.
Hence, i. If */ ac=i ; the Area of the Ellipfis=i — J ■__ T |-i— rr 5 s— ?Tif, &?£• in infinitum: Whence it is
evidentthat an Ellipfis isequal to a Circle whofe Diameter is a mean Proportional between the conjugate Axes of the Ellipfis. 2. Hence, alfo, an Ellipfis is to a Circle whofe Diameter is equal to the greater Axis, as ac to »' ; that is, as c to £, or as thelefs Axis to the greater. Hence, kllly, having the Quadrature of the Circle, weflialllikcwifehave that of the Ellipfis, and on thecontrary.
Quadrature of the 'Parabola For the Parabola,
we have a Qnadratrix or tranfeendent Curve, which gives its Square. See Quadratrix.
But it may belikewifehad thus : gx=y Seel'ARABQLA.
Sidx
'.dx
fydx — ia' :zx': z = . </ax =f ■/ x ^* = | xy.
Hence, the Parabolic Space is to the Keflangleof the S»- miordinate into the Abfciflie as | xy to xy ; that is, as 2 to 3.
Quadrature of the Hyperbola For this, too, we
have a Quadratrix, invented by Mr. 'Perks. See Qua- dratrix.
The analytical Quadrature was firft given by IV. Mercator of Holjlein, the firlt Inventor of infinite Series. But Mer- cator finding his Series by Divifion ; Sir Jfaac Newton and M. Leibnitz improv'd upon his Method ; the one feeking 'em by the Extracf ion of Roots, the other by a Series pre- fuppofed. See Series.
Mercator's Quadrature of the Hyperbola between its Jfymptotes.
Since in an Hyperbola within the Afymptotes, a' = by + xy ; or, if a — b=i, (which may be fuppofed, fince the determination of b is arbitrary.) 1 = 1'+ xy Then will 7-fTw.i
That is (theDivifiun being actually performed) y=a _ x-f x'_y > +x*—x '+**, iSc.
ydx==dx—xdx+x'~dx—x , dx' + ~ i dx—xi—dx +x'J::, Sgc. fydx —x—i X'+}X ' — i **-j-i*i — i jt* + fP7'iSc. Ill
infinitum.
Quadrature of the Cycloid Since TP (Tab.
Analysis, Fig. 27.) =PM ; in .the Triangle I'M T, the Angles M and T will be equal; and confequemly, T P Q = 2 M. But the Meafure of the Angle A P Q_is the half Arch A P j which likewife meafures the Angle TPA. Therefore A PQ_=T MP =3 M j»S, by reafon M P and m q are parallel.
Wherefore, fince the Angles at S and Q_are right Angles, we have
AQjQ_P::MS:»«S.
Let, then, A Q==«, AB=i ; then will PQ=:/ (* — xx') and mS=dx y(x — xx) : x. But it is fhewn, that
V (X—XX) = X 1 : '— i X> : ' § X < . * dx— 4 *' : ' dx, &C
in infinitum. Therefore, dx y* (x — xx) : x =3 (the N u- merarors of the Exponents being diminifli'd by two Units in the Divifion by x)x — ' : 'dx — ' : x' ' *fy — | x> : * dx — ^ x s : z dx, £5?c. in infinitum. Whofe Sum 2 x * * — ^ x 1 - » — ts x ':' — rr* 7 :*, S$c. in infinitum, is the Semiordi- nate of the Cycloid Q M refer'd to the Axis A E. Hence Q_M^x, or the Element Q_MS# of the Cycloidi' cal Space AMQ = 2i'- '<(* — ±x"'dx — ^v ! :« dx — ^x 7 -'dx, (Sc. in infinitum. Whofe Sum =3 }**: *— A ,:! -k' 7: *-A«' : ', tSc. in infini- tum, exprelies the Segment of the Cycloid AMQ;
If then mS =g, G =dx / (x— xx) : x be multiply'd into GM = AQ_=x; we fhall find the Element GMHG of the Area AMG = *y / (x—xx). Which being thes fame with the Element of the Segment of the Circle APQ, the Space A MG will be equal to the Segment of the Circle APQ_; and confequemly, the Area A "DC equal to the Semicircle A P B.
Hence, Since C B is equal to the Semiperiphnry of the Circle ; if that =f and AB =a ; the Reflangle B CD A = ap; and the Semicircle AP B ; and, confequently, the external Cycloidical Space A D C = | ap. Therefore the Area of the Semicycloid A C B=|- ap, and A M C B P A =-\ap. Confequently, the Area of the Cycloid is triple of the generating Circle.
Quadrature of the Logiftic, or Logarithmic Curve— - Let the Snbtangent PT (Tab. Analysis, Fig. 28.) =<j P M==»P^=3 4x 5 then will ydx -,dy=:a ydx = ady Jydx = ay Wherefore the indeterminate Space H P M I, is equal to the Reflangle of P M into PT.
Hence, 1. LetQJ>=2i; then will the indeterminate Space ISQH = »2>; and, confequently, SMPQ_=«y — a z = a (y — z) -, that is, the Space intercepted between the two Logiftic Semiordinates is equal to the Reef angle of
the
