QUADRATURE, in geometry, denotes the squaring, or reducing a figure to a square. Thus, the finding of a square, which shall contain just as much surface, or area, as a circle, an ellipsis, a triangle, &c. is the quadrature of a circle, ellipsis, &c. The quadrature of rectilinear figures, or method of finding their areas, has been already delivered. See MENSURATION. But the quadrature of curvilinear spaces, as the circle ellipsis, parabola, &c. is a matter of much deeper specula tion, making a part of the higher geo metry; wherein the doctrine of fluxions is of singular use. We shall give an ex ample or two.
Let A It C (Plate xni. Miscell. fig. 5) be a curve of any kind, whose ordinates R b, C B, are perpendicular to the axis A B. Imagine a right line, bR g, perpen. dicular to A B, to move parallel to itself from A towards B; and let the ve locity thereof, or the fluxion of the ab sciss, A b, in any proposed position of that line, be denoted by 6 d, then will b n, the rectangle under h d and the ordinate, b R, express the corresponding fluxion of the generating area, A b R ; which fluxion, if A b = x, and b R =y, will be From whence, by substituting for y or a, according to the equation of the curve, and taking the fluent, the area itself, A 6 R, will become known.
But in order to render this still more plain, we shall give some examples, where in x, y, z, and u are all along put. to de note the absciss, ordinate, curve-line, and the area, respectively, sinless where the contrary is expressly specified. Thus, if the area of a right angled triangle be re quired; put the base A H (fig 6) a, the perpendicular H M 6, and let A B x, be any portion of the base, con sidered as a flowing quantity ; and let B R = y be the ordinate, or perpendicular corresponding. Then because of the similar triangles, A H M and A B R, we shall have a: vx: ,• Whenc u a', the fluxion of the area A B It is, in this case, equal to b and conse quently the fluent thereof, or the area itself, = —which, therefore, when x 2 a a, and B It coincides with II M, will be.
come 11 the area of the whole triangle A H M : as is also demon. strable from the principles of common geometry.
Again, let the curve A It M H, (fig. 7.) whose area you would find, be the com mon parabola ; in which case, if A B = x, and B R = y, and the parameter = a; 1 1 we shall have ya x, andy = a Txf: 1 I and therefore, a (=y 3. 3 2 1 whence a =_-1X X x = x AB X BR. Henceapa rabola is two thirds of a rectangle of the same base and altitude.
The same conclusion might have been found more easily in terms of y ; for x = 2 , and S: = — y --; and consequently II a a 2 ylit 2 y3 ) = wnence,u3 a 2 y — y X e x=3 X AB XB a -)It, as before.
To determine the area of the hyper. bolic curve A M R 13, (fig. 8) whose • equation yn = ant+n ; whence we m+n m+n have y = a nan X 01; and m+n m—n • therefore u (= y a ) = x xn m+n an 1—n X whose fluent is u, — 1 — inn = m+n n—m nan n X which, when x = 0, will — alsit be = 0, if n be greater than m; therefore the fluent requires no correc tion in this case ; the area, A M R B, in cluded between the asymptote, A M, and the ordinate B R, being truly defined by Mtn ti—nt n an X x as above. But if n be less than m, then the fluent, when x = 0, QUA will be infinite, because the index a being negative, 0 becomes a divisor to n amfn ; whence the area, A M It B, will also be infinite.
But here, the area, B It H, compre hended between the ordinate, the curve, and the part, B H, of the asymptote, is finite, and will be truly expressed by m+n nn an + an , the same quantity with m — 72 its signs changed ; for the of the m+n m part A M It B, being an X an that of its supplement B R H must con M+11 —M sequently be — nn X xn .21% whereof m+n 1—m m+n c—m _ — an X Xn X xn the fluent is — 1,7 = the area, B R H, which wants no cor rection ; because when x is infinite and the area B R H = 0, the said fluent will also entirely vanish ; since the value of —n in+ nXn , which is a divisor to an , is then infinite.
For further examples see Simpson's Fluxions, vol i. sect. vii.