Elements of the conic sections

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stra ight line parallel to the tangent is also parallel to this ther axis an consequently is bisected by the diameter that passes throughth potnt of contaci T. 2.)Νoru et Ele an other diameter, and et Fig. 5. EF ouch the ellipsis in iis verteX; et GH n. l. 2. parallel o EF, e termina ted in G, H, twopoints of the ellipsis, and meet the diameter

pendicular to AB, and let them meet the cir

louches the ellipsis, FH I. cor. IO. . t Ouches the circleri an diecause the parallel Q Ν, Οare Cut in the fame ratio in the poliat G, H,

and that HGL is a traight line, the poliat L, Q, R are in a stra ight line lemma 2. Again, since the parallels ΡΜ, Q are cui in the Sameratio in the potnis E G, through,hicli the pa-

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to ΚH. COR. 1. On the Contrary an Straight line GH, terminate both ways by the ellipsis, and bisected by the diameter CE or, in ther WOrd S, an Straight line ordinatet applied tolli diameter CE is parallelo EF, the tangent a iis vertex. For, is not, dra a tangent

that shallie 12 2. parallel o GH and GHwilli bisected by the diameter hicli passes through the sint of contact: ut, b hypothesis, the sam GH is bisected by another diameter Cri; hicli is absurd.

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Coll. 2. Ali straight lines ordinatet appliedio the fame diameter are parallel O ne nother.

COR. 3. I severa paralleis e terminated both way by an ellipsis, the diameter hichbi Secis ne of them, bisecis also the res of them for that one hicli is bisected by a diameter, is parallel to the tangent in the vertex of that diameter an consequently the re Stare parallela the fame tangent; and there re re bisected by the fame diameter. COR. 4. 1 the contrar a traight linewhich bisecis severat paralleis terminated both Waysi an ellipsis, is a diameter For fit benot dra in diameter bisecting one of the pasralleis, and this diameter ill also biseci theothers: ut, b hypothesis, each of them is bisected by another traight line 'hicli is a surd. An is fro the o in o Contact, astraight line be drawn bisecting another straight line parallel to the tangent, an terminated both way by the ellipsis that straight line is a diameter For, i ii emot drama diameter

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through the oint of contact this diameter, by the prop. also biseci the parallel to the tangent hicli is absurd. Co . . A traight line hicli is drawnthrough the verteX of a diameter, an is parallel o an ordinate to that diameter, ouches theellipSiS.Coll. 6. wo traight lines in an ellipsis, whicli passio through the centre, do notit- feci eachither: for is the bisected achither, the would e ach of them parallel to the tangent in the verte of the diameter drawnthrough the poliati bisection, and of consequenc they ould e parallel o achitheruwhicli is absurd. PROP. XIV. THEOR.

Two diameter of an ellipsi S either of hicli is parallel to a traight line ouching the ellipsis in ei-

ther verte of the ther, are

conjugate diameterS.

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quently C an a are paralleis theresere

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straight Iines parallelao CE ill likewisei parallel o Va, hicli ouches the ellipsis in V; and of consequenc the willae bisected 13. 2 the diameter V and becaus CV, byhypothesis, is parallel o EF, ali strat glit lines parallelao CV, ill likewiseie parallelao EF, and wil consequently be bisected by the diameter CE the refore E, V are conjugato diameters T def. 2. Coll. I. An since Wis the oni diameter that caniisectatraight lines parallelao CE, and terminated both ways by the ellipsis, Waloneis the conjugate to E. Co R. 2. Is there re, E, V be conjugato

diameters, ach of them is parallel to the tangent drawn through the vertex of the other. COR. 3. On the theriand aratraight linedrawn through the vertex o a diameter, parallel to the conjugate diameter, ouches theellipsis in that verteX.

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Co R. q. atraight line parallelao a diameter, and terminated both way by the ellipsis, is bisected by the conjugate diameter se it is parallel to the stra ight line hicli ouches the ellipsis in the vertex of this diameter and consequently it is bisected by this sanie conjugato diameter On the otheriand a traight linebisected by a diameter, is parallel to the conjugate diameter. PROP. XV THEOR.

I frona a potnt in an ellipsis to ei ther os Wo conjugate diameterS, a traight line e rasn parallelto the other, the square of the diameter it me eis, S to the quare of the other diameter, a the re Ctangi contained by the segmentS

into hicli the traight in divides the sirst, is to the quareo that traight line.