Elements of analytical geometry and the differential and integral calculus . 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 halves, whenceA : jB=area circle : area the area of the circle on the major
Elements of analytical geometry and the differential and integral calculus . 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 halves, whenceA : jB=area circle : area the area of the circle on the major axis, is rtA^.Substituting this, and the proportion becomesA : B=irtA^ : area ellipse. Or area ellipse=rt^^, which is the mean proportional between {jtA^) and (ttB^,) theexpressions for the areas of the circles, one on the major axis,the other on the minor axis. Q. E. D. * These narrow parallelograms are called differentials, in the differentia]calculus—and the sum of them is called the integral, in the integral calcidus. 48 ANALYTICAL ScHDUUM. Hence the common rule in mensuration to findthe area of an ellipse. Rule—Multiply the semi-major and semi-minor axes togethert aridmultiply that product iy PROPOSITIOlf V. To find the product of the tangents of two supplementary chordswith the axis of X. Let X, y, be the co-ordinates ofany point, as P, and x, y, the co-ordinates of the point A. Then the equation of a linewhich passes through the twopoints A and P, ( Prop. Ill, ,) will be y—y—a{x^x). (1) Th^ equation of the line which passes through the points Aand Pf will be of the form y-y=a\x—x). (2) For the given point A\ we have y=0, and x=—A,Whence (1) becomes y=a{xJrA). (3) For the given point A we have y=0, and x=Ay whichvalues substituted in (2) give y=a{x—A), (4) As y and x are the co-ordinates of the same point P in bothlines, we may combine (3) and (4) in any manner we them, we have y^ =aa{x^—A^ ). (5) Because P is a
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