QUADRATURE. 



= — : whence the area itfelf may 



fluxion of the area ARC will be juftly reprerented by the 

 uniformly generated triangle Q C S ; which, cxpreffing C P 



QSxCP_ji 

 by X, will be = -^ — — ■ 



be determined. But fincc, in many cafes, the value of z can- 

 not be computed (from the property of the curve) without 

 trouble, the two following expreilions, for the fluxion of the 



sy y J J"' •*' 



area, will be found more commodious, iitz. - — and — ; 



2 t 2 a 



where / = R P, and .v = the arc B N of a circle, defcribed 

 about the centre C, at any diftance a= C B. Thcfe ex- 

 preflions are derived from that above in the following man- 

 ner ; w«. i :} -.-.y (C R) : / (R P) ; therefore 2 = y ; con- 



fequently -~ — • Moreover, becaufc the celerity of R 



in the direftion of the tangent is denoted by i, that in a di- 

 reftion perpendicular to C Q (whereby the point R revolves 



C P si.- 



about the centre C) will, therefore, be = —-=5- x 2 = — ' 



UK y 



which, being to x the celerity of the point N about the 

 fame centre as the diftance or radius C R (j/) to the radius 

 C N (a), we (hall, by multiplying extremes and means, have 



. — - = y X, and, confequcntly, — = - — . In the exam- 



y 2 2 H 



pies fubjoincd, the letters x, y, z, and u will be ufed to de- 

 note the abfcifle, ordinate, curve-line, and area refpeftively. 

 Quadrature of a Right-angled Triangle. Let the bafe 

 A H [fg. 12.) = a, the perpendicular H M — b, and let 

 A B (x) be any portion of the bafe, confidered as a flowing 

 quantity, and B R \y) be the corrtfponding ordinate. Then, 

 the triangles A H M and A B R being fimilar, we fliall 



bx 

 have a : b :: X : y =^ . — . Whence ji x (the fluxion of the 



b X X 



area A B R) = ; and its fluent (fee Inverfe Method 



a 



c/* Fluxions) or the area itfelf = — ^, which, when x = a, 

 ■' ' 2a 



,„„ . ., .,TT»;r n, «* AHxHM 



and B R coincides with H M, will become ■ — = 



2 2 



= the area of the whole triangle A H M. See Menfura- 



tien o/" Triangles. 



Quadrature of a Circular SeHor. Let A O R (Jig. 13.) 



be the feftor ; A O or O R, its radius, = a, the arc A R, 



confidered as variable by the motion of R, =; c, and R r 



= i ; the fluxion of the area will be — ^ die triangle 

 O R >• : whence the area itfelf is = — =:AOxiAR: 



= 3' «r» r 



refolved into an infinite feriee, we fhall have u 



/ «_ _ x^ _ *' 5k» » \ _ 



^ V 2a ~ 8a' i6a' ~ izST^' ) ~ 



, . X^ X X^ X x^x\ 



x^ X — — - I, &c. whence, the 



2 a Ha' lOa'/ 



fluent of every term being taken, there will arife u = at 



(2 .-CT xT 



28 a' 



72a' 



5"' 



704 a''' 



&c 



) = 



^ ax X 



Sx- 



28a' 



&c.^ 



72 a' 704 a' 



= the area A B R. When x = -i a, the ordinate B R 

 will coincide with the radius O E ; in which cafe the area 



It to -m — -i^-tr — 



a 



becomes = 

 T-rircT> S:c.) =: 



.^ i aa 



X (0.6666 — O.I 



0.0089 



that is, the area of any circle is exprefled by a reftangle 

 under half the circumference and half the diameter. See 

 Circle. 



Quadrature of a Semldrek. Let the femicirck be of y. Thus x being 

 AREH (fg. 14.) ; its diameter AH = a, A B = ar, 

 and B R = y, &c. and we have y"- (B R^) = ax — x'' 

 (AB X BH), and, confequcntly, u {y x) = it ^/ (ax 



— 0.0017 ~ 0.0004, &c.) = 0.1964a"; which, mul- 

 tiplied by 2, gives 0.3928 a' for the area of the femi- 

 circlc, nearly. In order to obtain a more converging feries, 

 let the arc A R be ^ 4 A E = 30°, and the fine B R 

 (being half the chord of double tiie arc or half the fide 

 of a hexagon, i. e. half the radius), will be = i A O • 

 and AB(.%-) = AO - OB = AO - ,/ (O R' -'B R'); 

 which, radius being I, will be ~ 0.1339746 nearly: fub- 

 ftitute this quantity, with the value of a in the above feries, 



X J-' 



^a.r' X (f — — f, — i — &c.) and we fiiall have 



c a 2o a 



0.0693505 X (0.6666666 — 0.0133975 — 0.0001603 — 

 0.0000042 — &c.>— 0.0693505 X 0.6531046= 0.0452931 

 1= the area A B R : which, added to the area O B R 

 (=OB X :^BR= \/|-x 1= 0.2165063) gives 

 0.2617994 for the area of the feftor A O R ; the treble of 

 which, or 0.7853982 (A R being = ^ AE), will be the 

 content of the whole quadrant A O E ; which number, 

 found by taking only four terms of the feries, is true to the 

 lail decimal place. 



We might have found a feries of more rapid convergency, 

 but fliall referve that part of our inveitigation for the 

 article Rectification. 



Quadrature of the Lunes. See Lunes. 



Quadrature of the Ellipfe. The ellipfe, alfo, is a 

 curve whofe precife quadrature in definite terms is not yet 

 effefted. Indeed the area of an elhpfe, being an exaft 

 mean proportional between the areas of the circles defcribed 

 upon Its two axes, is obvioufly dependent upon the latter ; 

 and it is, therefore, ufelefs to repeat the operation. 



Quadrature of the Parabola. Let the curve A R M H 

 {Jig. 15.), be the common parabola, in which ^' (BR') 

 = a.v (AB x a, the parameter). See Conic Seaioni. 

 Whence we have^ = in x'^, and u (yx) := a^ x'^ x ; and, 



therefore, u = .= x a'^ x'''' = \ a'^ x^ y. x := -I y x = ^ 

 X A B X BR. Hence a parabola is -5- of a reftangle of 

 the fame bafe and altitude. 



The value of tlie area may alfo eafily be found in terms 



hav 



2yi- ' 



= , and u 



a 



(y-) 



2y\v 



: whence a = 



2y 

 = -^ X 



»') = a"* x^ .i X 



( I — - 1 } which expreflion being 



a 

 =ixABxBR 

 In the cubic parabola, whofe equation is f 



3^ 3 

 See Parabola. 



2y 

 = -^ X 



3 



; we 

 have 



