. 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]. For, the two CA, CD, being equal to the two CG, CB,the third proportionals CT, CS, will lie equal also ; thenthe two sides CE, CT, being equal to the two CH, CS, andthe included angle ECT equal to the included angle HCS,all the other corresponding parts are equal : and so the Z T= Z S, and TE is parallel to HS. Cor. 3. And Hence the four tangents, at the four ex- tremities of any two conjugate diameters, form a parallel-
. 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]. For, the two CA, CD, being equal to the two CG, CB,the third proportionals CT, CS, will lie equal also ; thenthe two sides CE, CT, being equal to the two CH, CS, andthe included angle ECT equal to the included angle HCS,all the other corresponding parts are equal : and so the Z T= Z S, and TE is parallel to HS. Cor. 3. And Hence the four tangents, at the four ex- tremities of any two conjugate diameters, form a parallel- im circumscribing the ellipse, and the pairs of opposite sides are each equal to the corresponding parallel conjugate diameters. For, if the diameter eh be drawn parallel to the tangentTE or HS, it will be the conjugate to EH, by the defini-tion j and the tangents to eh will be parallel to each other,and to the diameter EH for the same reason. Vol, IL O o 298 MATHEMATICS. PROPOSITION XIII. All the parallelograms circumscribed about an ellipse areequal to one another, and each equal to the rectangle of thetwo axes. That is, the parallelogram PQRS = the rectangle V*- Let EG, e§*, be two conjugate diameters parallel to thesides of the parallelogram, and dividing it into four less andequal parallelograms. Also, draw the ordinates DE, de,and CK perpendicular to PQ ; and let the axis CA produc-ed meet the sides of the parallelogram, produced if necessa-ry, in T and t. Then, by Prop. VIII. CT : CA :: CA : CD,and Ct : CA :: CA : Cd;therefore, by equality, CT : Ct :: Cd : CD ;but, by sim. tri. CT : Ct :: TD : Cr/, therefore, by equality, TD : Cd :: Cd : CD,and .•. the rectangle TD*DC = the square Cd2. Again, by Prop. VIII. CD : CA :: CA : CT,or, by division, CD : CA :: DA : AT,and, by composition, CD : DI3 :: AD : DT ;Consequently, the rectangle CDDT = Cd% = AD-Dli. CONIC SECTIONS. 290 But, by Prop. II. CA2 : Ca2 :: (AD-DB or) C^2 : DE2 ; therefore, CA : Ca :: Cd : DE ;in like
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