Elements of the conic sections

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of hal that ordinate is qua to the

rectangi contained by the abscissabe tween that sanie ordinate and the verte of that diameter, and thelalus rectum of the sam diameter.

Let AB be the directrix o a parabola, an Fig. . AD a diameter, to hicli EL, drawndromulie n. l. 2.poin E of the parabola, is ordinatet applied 3 and through the vertex D of the diameter AD, dra DK parallelao EL , DK, of Consequence, will ouch the parabola dra DM perpendicular to the aXis, and fro Q the verteX of the axis dra ordinately applied to the diameter DL and Consequently parallel o EL. Since QR is equat o DK, it square is equalto T4. I. Elem. the quare o DNI MK; but the quare of Dwis by the firs case of the foregoin proposition equa to seu timesthe rectangle uin and since Mucis equal IO. I. to QK, the quare of Κ is equat tofour times the quare of Q theresere the

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square o Q is qua to seu times the rectangle OB, together illi seu times thesquare of Q, that is, o four times 3. 2. Elem. the rectangle QMdi but Q or QK, is equat o DR and B to D there re thesquare o Qxis equa to seu times the rectan-gle RDA and since QR EL are ordinatelyapplied to the diameter AD, the quare of QRis to the quare o EL as 2 Cor of the re cedin proposition Rinto LD, that is, ac urtimes the rectangle RDA , ur times the rectangi LDA: ut the quare of OR achathheen proved is equat to four times the rectan-gle DA; there re the quare of EL is equalto four times the rectangi LDA, that is, toste rectangi contained by the abscissa Luanditie parameter of the diameter AD. It was seo the properi above demonstrat-ed that Appollonius nam e the curve line,

BOLA.

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parallel to straight lines ordinatet applied tolli diameter AD, and meeting the fame AD belo iis vertex D the square o EL is equalto the rectangi contained by the abscissa Dand the parameter of the diameter AD; thepoin E is in the parabola. For since the oin L is illi in s 4. l. the Fig. 4.

Parabola, the straight line EL necessarii meet D 2 Cor. 8. 1 the parabola: is, there re, I does no meet the parabola in the potnt Eoon the fame fide of the diameter with E, et tmee it, i possibie, in some ther potnt, earer, o more remote, fio the diameter, hannis: et this other potntie di then the quare of eL is quai to the rectangi containe byLD, and the parameter of the diameter, thatis, accordin to the hypothesis, to the quare of EL: hicli is absurd. Coll. 2. I from two potnis , Q, ne ofwhicli Q, is in the parabola, therea drawn

to the diameter Arestra ight lines EL QR parallel to strat glit lines ordinatet applied to AD; is the quares of the paralleliberio ne another

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as the abscissa be tween the parallel and the Verte of the fame AD; the ollier potnt Disalso in the parabola. Is the diameteri meet the directrix in A, seu times Amis it latus reclum then, incethe quare o Q is to that o EL a Rreis toLD, that is, si ur time the rectangle RDAt four times the rectangle LDA an since, by the proposition the quare of QR is equalto Dur times the rectangle DA the quare of EL is equat to four times the rectangle LDA; and theresere, by the precedin corollary thepsin E is in the parabola. PROP. XIV. THEOR.

dinatet appliedo a diameter, an is nother traight in be

dra n rom ille fame potnt o sto ouch the parabola, and me et

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BOOK I. THE . PARABOLA SD

that diameter the segment of the diameter intercepted e twixtili ordinate and the tangent sit sected in the verte of the diam

Atein a poliati a parabola, AC drawn Fig. 6.fro A, oras to e ordinatet applied to the diameter BC, and AD drawn frona the fame potnt, o a toriouch the parabola, and meet

BC in D; the segmen C is bisected in B, the vertex of BG. Froin the verte Bese Bribe rawn parallello AD; it,illae ordinately II. I.ὶ applied tolli diameter AE and the abscissa BC illae equat to the abscissa 8 cor I 2. l. AE thatis, o BD. COR. I. On the ther hand AC Ming ordinatet applied to the diameter BC, is rebedrawn aking BD equat o BC, D touches the parabola.

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Coll. 2. Is a traight line ouches a parabola, it segment e tween the ointi Contaci, and an diameter, is bisected by a straight Iine ouch in the parabola, in the vertex filia diameter. Let AD a traight Iine touChing the parabola, meet the diameter C in the oin D,

and the tangent B in σι et A be rarunparallel o BG AC,illi ordinatet appliedio the 11. 1. diameter B and since, by the proposition, Mis equa to BD, AG is like-wise equat o GD. PROP. XV. . PROB.

To findo diameter, the Xis, thelatus rectum of the aXis the focus, and the directrico a parabolagiven in position.