QUADRATURE. 



have y ■= p'-r x' ; multiply this by x, and wo liaveji jf = 

 p^ x^ X, for the fluxion of the area. Therefore fluent y x 

 = f ^5 *j ^ A of xy, that is, area = ^^of circumfcribing 

 reiSlangle. 



And in the fame manner, it will be found that in the 

 general parabola, whofe equation is a" ^ ' x — ^" ; the area := 



X circumfcribing reftangle. 



QuADRAi'LiRE of the Hyperbola. The analytical quad- 

 rature of this curve was firli given by N. Mercator of Hol- 

 ftein, tlie firfl; inventor of infinite fericfes. But Mercator 

 finding hici feries by divifion, fir Ifaac Newton and M. Leib- 

 nitz improved upon his method ; the one feeking them by 

 the extradliou of roots, the other by a feries prefuppofed. 



See HYPEKBOi.A. 



Quadrature of the Afymptot'ic Spaces in an Hyperbola. 

 Let D E F {Jig. X^.) be an hyperbola, of which the afymp- 

 totes are C M and C N ; to find the area E G H F, com- 

 prehended between the orduiatcs G E and F H. 



Let C G = a, G E -= i, G H = A.-, F H ^ ji ; then by 

 the property of the hyperbola, CG x GE = CH x HF, 



OT ab = {a — x) y, or y =: ; and, therefore, ji a- 



ab ; 



a + X 

 the fluent of which is ab X hyp. log. (a + «), 



which fluent, however, requires a correftion, for when 

 X = o, the area = o ; but the above expreffion when x = o 

 h ab X hyp. log. of a, therefore the correftion is — ab 

 X hyp. log. of a, that is, the correct fluent which ex- 

 preffes the area ii ab x hyp. log. {a + x) — ab x hyp. 



log. a, or area ^G¥ H — ab x hyp. log. . 



If C G and G E each = i 



yx = 



the fluent of 



I + X 



which is hyp. log. (i -f x), which requires no correction. 



Quadrature of the Cycloid. Let CAL {Jig. 17.) be 

 a cycloid, AD the axis, ABD the generatmg circle, A F 

 a tangent at the vertex, C F parallel to A D. 



Take any point P in the arc, and draw P M perpendieular 

 to A M. Then the fluxion of the external area A M P =: 

 P M X the fluxion of A M. 



Let A E = *, A D = 2 a ; then B E =r ^ (2 a k — x') 

 {a — x) X 



and the fluxion of B E = 

 AlfoPB 



^/ {2 ax — x') 

 the arc B A ; therefore the fluxion of P B = 



and the fluxion of P B + BE, or of 



v' {zax — x^) ' 



. -, {2a — x\ X , r in. /- ■ 



AM = — S —T\ ; therefore the fluxion of the area 



^ (2ax — x^) 



APM,or PM, X by the fluxion AM = (^^^- =''')'' 



^{2ax — x^) 

 z= X ^/ {zax — a-'). 



But the fluent of this fluxion is the fame as that found 

 above for a circle, whofe radius is a, and verfed fine x ; that 

 is, the area ABE; and, therefore, when j: = 2 a, the whole 

 •xternal area C F A is equal to the area of the femicircle 

 ABD. But C D being equal to the femicircumference 

 A D, the whole reftangle C D A F =: four times the femi- 

 circle ABD, and confequently the internal area ACD = 

 three times the femicircle ABD; or the whole area of 



Vol. XXIX. 



the cycloid equal three times the area of its generating 

 circle. 



Quadratuiie of the Lo^yiic, or Logarithmic Curve. 

 Let the fubtangent PT (fig. 18.) = «, P M = a-, P/ 

 = dx ; then will 



^-^ = a 



y 



y i = ay 

 and fluent oi y x = ay. 



Wherefore the indeterminate fpace HPMI, is equal to 

 the rcdangle of P M into P T. 



Hence, i. Let QS = 2; ; then will the indeterminate 

 fpace I S Q H := az; and, confequently, S M P Q ^ ajr 

 — a% — a {y — %) ; that is, tlie fpace intercepted between 

 the two logiilic femiordinates is equal to the reClangle of 

 the fubtangent into the difference of the femiordinates. 



2. Therefore the fpace B A P M is to the Ipace P M S Q 

 as the difference of the femiordinates A B and P M is to 

 the difference of the femiordinates P M and S Q. See 

 Logarithmic Curve. 



QVADRATVRE of the Logarithmic Spira/. Let CBAC 

 (Jg. 19.), be the area propofed : let tlie right line AT 

 touch the curve at A, upon which, from the centre 

 C, let fall the perpendicular C T : then, fince by the nature 

 of the curve, the angle T A C is every where the fame, 

 the ratio of A T (t) to C T (t) will be conltant : and. 



therefore, the fluent of - x - 



t 2 



= - X — 



t 4. 



tlie 



quired area. 



Quadrature of the Spiral of Archimedes. Let C R R 

 {Jig. 20.) be the curve, whofe area C R^ C is required. Let 

 A C be a tangent at the centre C, about which centre, with 

 any radius AC (= a), fuppofe a circle A_^^ to be defcribed: 

 then the arc or abfcifs Ag correfponding to any propofed 

 ordinate C R, being to that ordinate in a conltant ratio 



{e. gr. as m to n) we have ^ (A^) = - ; therefore u = 



my'^y my' 



; confequently « = - "— 



zan 6an 



the 



2a 

 CRR^C. Set SiPlv. At of Archimedes. 



Quadrature of Defcartes^ Curve, whicli is defined by 

 the expreffion, b- : x' :: b — x : y. 



Since b'y=- b x'' — x^ 



y = [bx"- - x^) -i- b- 

 y X = {bx- i — x' x) -i- b^ 



flu. 



3b~x'^^b\ 



Quadrature of all Curves cs/nprehendtd under the general 



Equation, y -z^. \/ {x -f a). 



Since jf = (j: + a)" 

 y k ■= ic {x ■'r a)"' 



Make (^ -f a) "' = 1; 



Then X -\- a ■=■ v", or * = t>" — a 



Whence x — mv""' v 



y X = mv^v, the fluent of which 11 



-— — - = — — - {x + a) (,f -f- a)". = — — - (x + a)' 



Let jr = o ; the remainder will be 



m + I 



Whence, 



