. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. PROPOSITION V. If a line be drawn from any point in the axis produced,to cut the curve in two points, then shall the external partof the axis be a mean proportional between the two ab-scisses of the ordinates to the two points of intersection. That is, AI a mean proportional between AD, AG ;or, AD : AI :: AI : ii For, by Prop. I. AD : AG :: DE2 : GH* ;and, by sim. tri. IDS : IG2 :: DE2 : GH» ;*therefore, by equality,
. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. PROPOSITION V. If a line be drawn from any point in the axis produced,to cut the curve in two points, then shall the external partof the axis be a mean proportional between the two ab-scisses of the ordinates to the two points of intersection. That is, AI a mean proportional between AD, AG ;or, AD : AI :: AI : ii For, by Prop. I. AD : AG :: DE2 : GH* ;and, by sim. tri. IDS : IG2 :: DE2 : GH» ;*therefore, by equality, AD : AG :: ID* : IG2 and, by div. AD : DG:: ID3 : IG2—ID* or DG-ID+IG, ©r AD : ID :: ID : ID + IG; and, by div. AD : AI :: ID : IG,* and again, by div. AD : AI :: AI : AG. Q. E. D. Cor. l. * Hence we have AD : AI :: DE : GH. CONIC SECTIONS. 327 Cor. 2. If the line be supposed to revolve about thepoint I, then as it recedes farther from the axis, the pointsE and H approach tow.;, ac • i r, the point E decend-ing, and the point H ascending, till at last they meet inthe point C, when the line becomes a tangent to the curveat C. And then the points D and G meet in the pointM, and the ordinates DE, GH, in the ordinate CM. Con-sequently AD, AG, becoming each equal to AM, theirmean proportional will be equal to the absciss is, the external part of the axis, cut off by a tan-gent, is equal to the absciss of the ordinate to the pointof contact.
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