Elements of analytical geometry and the differential and integral calculus . we musthave y^=:^{^A^x)x\ Dividing one of these equations by the other, omitting com-mon factors in the numerator and denominator of the secondmember of the new equation, we shall have2/2 _{^A—x)xY^ {^A~x)x Hence y^ : y^=(2A—x)x : {—x)x. By simply inspecting the figure, we cannot fail to perceivethat (9,A—x)y and x, are the abscissas corresponding to the or-dinate y^ and (—x), and x, are the two corresponding to y\Therefore, the squares of the ordinates, &c. Q. E. D. Scholium. Suppose one of these ordinates, as
Elements of analytical geometry and the differential and integral calculus . we musthave y^=:^{^A^x)x\ Dividing one of these equations by the other, omitting com-mon factors in the numerator and denominator of the secondmember of the new equation, we shall have2/2 _{^A—x)xY^ {^A~x)x Hence y^ : y^=(2A—x)x : {—x)x. By simply inspecting the figure, we cannot fail to perceivethat (9,A—x)y and x, are the abscissas corresponding to the or-dinate y^ and (—x), and x, are the two corresponding to y\Therefore, the squares of the ordinates, &c. Q. E. D. Scholium. Suppose one of these ordinates, as y, to representhalf the minor axis, that is, y=B. Then the corresponding valueof « will be A, and (2A—x) will be A, also. Whence the lastproportion will become y^ : B^=(2A—x)x : AK In respect to the third term we perceive that if AH is repre-sented by X, -4^will be {2A—x), and if 6^ is a point in thecircle, whose diameter is AA, and GHi\iQ ordinate, then and the proportion becomes r : B^^QH.^ : A^.Or y : GE=zB : A. Or .4 : B^QH I y^BR THE ELLIPSE. 47. PROPOSITION IV. The area of an ellipse is the mean proportional between theareas of two circles, ike diameter of one being the major axis, andthe diameter of the other, the minor axis. Conceive GIT to be a practical as wellas a mathematical line; or rather, conceiveit be a ver^ narrow parallelogrom. Conceive also other lines Q-H\ QH,<fec, drawn so as to fill the whole spaceoccupied by the semicircle and semi-ellipse.* Then by scholium to Prop. Ill, we haveA : B=:Gff : DH, z=GH : GH : DH\ &G. &G. But as the sums of proportionals have the same ratio as thel?like parts, (see proportion in algebra,) therefore A : B :: (Gff+GJI+&c.) : {DH-\-DH+&,q,) But the sum of all the narrow parallelograms represented by(^^-|-6^^-j-&c.) is the area of the semicircle on ^^ : and thesum of all the parallelograms represented by {DH-\DII-\-&,g,^is the area of the semi-ellipse. But wholes are in the same proportion as their hal
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