. 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]. CK : KL, or, by mult. IE : EK :: CK* >. CK-KL. Bat, by Prop. II. Cor. EK is as the rectangle CK • KL,and therefore, IE is as CK3, or as CI*. Q. E. D. Cor. 1. As this property is common to every positionof the tangent, if the lines IE, TA, ON, &c. be append-ed to the points I, T, O, &c. and moveable about them,and of such lengths, that their extremities E, A, N, & in the curve of a parabola, in some one position oi
. 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]. CK : KL, or, by mult. IE : EK :: CK* >. CK-KL. Bat, by Prop. II. Cor. EK is as the rectangle CK • KL,and therefore, IE is as CK3, or as CI*. Q. E. D. Cor. 1. As this property is common to every positionof the tangent, if the lines IE, TA, ON, &c. be append-ed to the points I, T, O, &c. and moveable about them,and of such lengths, that their extremities E, A, N, & in the curve of a parabola, in some one position oi thetangent ; then making the tangent revolve about the pointC, the extremities E, A, N, &c. will always form thecurve of some parabola, in every position of the tangent. Cor. 2. The parameter of the axis is also a thirdproportional to IE and CK. For, by this Prop. EK : KL :: IE : CK ; and, by Prop. II. EK : KL :: CK : P; therefore, by equality, IE : CK :: CK : P. proposition x. The abscisses of any diameter are as the squares oftheir ordinates. That is, CQ, CK, CS, as QE», HA8, SN», , CQ : CK :: QE» : KA2, &c. CONIC SECTIONS. 333. For, draw the tangent CT, and the externals EI, AT,NO, &c. parallel to the axis, or to the diameter CS. Then, because the ordinates QE, RA, SN, &c. areparallel to the tangent CT, by the definition of them,therefore all the figures IQ, TR, OS, &c. are parallelo-grams, whose opposite sides are equal,namely, IE, TA, ON, & equal to CQ, CR, CS, &, by Prop. IX. CO, CR, CS, & as CI8,CT*, CO3, & as their equals, QES, RA2, SN2. &c. Q. E. D. Cor. Here, as in Prop. II. the difference of the ab-scisses is as the difference of the squares of their ordi-nates, or as the rectangles under the sum and difference ofthe ordinates, the rectangles under the sum and differenceof the ordinates being equal to the rectangle under the difrference of the abscisses and the parameter of that diameter,or a third proportional to any absci
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