. 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]. foYy draw the ordinate ED to the transverse , by Prop. I. CA8 : C«8 :: CD8 —- CA8 : DE3. But CD2=^E8, and DE»=G/»,therefore, CA8 : Ca* :: d&* —-CA8 : G/8,or, by alt. CA8 : aE8—CA8 :: Ca8 : Cd*,and, by comp. CA8 : dE* :: Cg8 : Ca?+C^2,and, by alt. and inv. Ca2 : CA8 :: Ca2 -\-Cd2 : «/E like manner, CA8 : Caa :: CAa+CD8 : De*. Q. E. D, Cor. By the last Prop. CA8 : Ca* :: CD2—.CA3 : DE8,and by this Prop. CA8 : Ca
. 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]. foYy draw the ordinate ED to the transverse , by Prop. I. CA8 : C«8 :: CD8 —- CA8 : DE3. But CD2=^E8, and DE»=G/»,therefore, CA8 : Ca* :: d&* —-CA8 : G/8,or, by alt. CA8 : aE8—CA8 :: Ca8 : Cd*,and, by comp. CA8 : dE* :: Cg8 : Ca?+C^2,and, by alt. and inv. Ca2 : CA8 :: Ca2 -\-Cd2 : «/E like manner, CA8 : Caa :: CAa+CD8 : De*. Q. E. D, Cor. By the last Prop. CA8 : Ca* :: CD2—.CA3 : DE8,and by this Prop. CA8 : Ca8 :: CD8+CA8 : D<?8 ;therefore, DE * : De* :: CD8—CA8 : CD2+ like manner, de* : c/E 2 :: C^a—Ca8 : C</2+Ca2* PROPOSITION IV. The square of the distance of the focus from the centrej)B equal to the sum of the squares of the semiaxes. Vol II. P p 500 MATHEMATICS. Or, the square of the distance between the foci is equalt« the sum of the squares of the two axes. That is, CF* =CA«+Ca», or Ff*=AB*+ab*.. For, to the focus F draw the ordinate FE ; which, bythe definition, will be the semiparameter. Then by thenature of the curve CA.» : Ca3 :: CF2 — CA2 : FES jand, by the def. of the param. CA2 : Ca* :: Ca* : FE ;therefore, Ca* = CF2 — CA8 ; and, by add. CF* = CA2+C<z ; or, by doubling, F/» = AB» +ab*. Q. E. D- Cor. 1. The two semiaxes and the focal distance fromthe centre are the sides of a right-angled triangle CAa ;and the distance Aa = CF the f ocal distance. For, as above, CA2 + Ca = CF«, and, by right-angled triangles, Ca2 4- C A 2 = Aa2,therefore, CF = Aa, and Ff = Aa -f Ba. Cor. 2. The conjugate semiaxis Ca is a mean propor-tional between AF, FB, or between Af, f B, the distancesof either focus from the two vertices. ForCa9=CF»—CA»=Lr+CACF— CA=AFFB CONIC SECTIONS. 307 Cor. 3. The same rectangle AF • FB of the focal dis^tances from either vertex, is also equal to the rectangleAC*FE under the semitransverse and its semiparameter ;s
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