. 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]. G : : AD-2IK :IB2—AD-2IK,by alt. CK : IK :: AD-AG : ID2—AD-2IK, and, by comp. CK : CI :: AD-AG : ID2—AD-2IK—AG- But—AD-, —\G=— AD-IG+ID—AG=. ad-id--: a ;a;;—ag=— adid+ia, AndID = IA+AD, and ID2 = IA2--AD2 +2IA-AD j obnsequendy,—AD-iA-riL/=—AD-2iA-f AD=r—21AAD—ADS CONIC SECTIONS. 391 and ID2— AD-2:&—A^-IA2+AD2 + 2lA-AD — 2lA-AD —Ao2 = aA2, therefore, CK : CI : : AD-AG : AI . But when the line IH, by revolving about the poi
. 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]. G : : AD-2IK :IB2—AD-2IK,by alt. CK : IK :: AD-AG : ID2—AD-2IK, and, by comp. CK : CI :: AD-AG : ID2—AD-2IK—AG- But—AD-, —\G=— AD-IG+ID—AG=. ad-id--: a ;a;;—ag=— adid+ia, AndID = IA+AD, and ID2 = IA2--AD2 +2IA-AD j obnsequendy,—AD-iA-riL/=—AD-2iA-f AD=r—21AAD—ADS CONIC SECTIONS. 391 and ID2— AD-2:&—A^-IA2+AD2 + 2lA-AD — 2lA-AD —Ao2 = aA2, therefore, CK : CI : : AD-AG : AI . But when the line IH, by revolving about the point I,comes into the position of the tangent 1L, then the points Eand H meet in the point L, and die points D, K, G, coin-cide with the point M ; and then the last proportion becomesCM : CI : : AM2 : Ai2. Q. E. D. PROPOSITION VIII. If a tangent and ordinate be drawn from any point in thecurve, meeting the transverse axis ; the semitransverse willbe a mean proportional between the distances of the saidtwo intersections from the centre. That is, CA is a mean proportional between CD andCT ;or CD, CA, CT, are continued 292 MATHEMATICS. For, by Prop. VII. CD : CT :: AD* : A™, 3 2 that is, CD : CT :: CA—tU : CT — uA,or CD : CT :: CD*; CA2 : CA8+CT2?and CD : DT :: CD2-|-CA2 : CT2—CD2,or CD : DT :: CD*-uCA2 : CT~ CD2 : CD-DT :: CD2 -CA2 : CD-DT+CT *DT,hence CD2 : CA2 :: CDDT : CT-DT,and CD8 : CA2 :: CD : CT ; therefore CD : CA :: CA : CT. Q. E. D. Cor. 1. Since CT is always a third proportional to CD,CA ; if the points D, A, remain constant, then will the pointT be constant also, and therefore all the tangents will meetin this poin<: T, which are drawn from E, of every ellipsedescribed on the same axis AF5, where they are cut by thecommon ordinate DEE, drawn from the point D. Cor. 2. Hence a tangent is easily drawn to the curvefrom any point, either in the curve or without it. First, if the given point E be in the c
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