Elements of the conic sections

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havin the Same ratio to achother a the parallel AC, BD, and lying on the fame fide of AB, or On the contrar Sides fit, acCording a the parallel AC, BD arein ille sanie side of ABO O the Contrary Si de oscit; and i a traight line e rawneither through the extremittes of the two irst, o tW last fines paralleis, the potiato herei meet AB, is in the samestraight line illi the extremities of the thera o parallel S.

Let a traight lineae drawn for Xample, through C, D the extremities of the paralleis AC, BD, and et i meet the traight line AB in the poliat the extremittes E, F of theothe two parallel AE, F, and the oin G, are in the fame traight line. For since the

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mittes E, F of the parallel AE, F the potnt

where it meet AB, and the Xtremittes C, of the the two paralleis, re in the Samestraight line.

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extremities of the the two paralleis, are like is parallel.

wis parallel.

Elem. parallelao FK.

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Fig λῖ I srom any potnt of an ellipsis Ε, whicli is no the vertex o etther axis, a straight line EF be drawn parallel to cither axis CD, Cet-ing a circle describe lipon theother axis in the olim , and fanother traight line e rawntouching the circle in the potnt where the parallel meet it: illis ther traight line meetsilie common diameter of the ellipsis and the circle an astraight line drawn froni thepoliat here it meet that diameter, to the oin E in the ellipsis, ouches the ellipsis also

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Let AB be the other axis of the ellipsis, and C the centre of the ellipsis and circle, and oin CG Iet G touch the circle in G, and meet the common diameter AB in H; the stra ight line thalcioin the poliati and the oin Etaken in the ellipsis, ouches the ellipSis. Becaus CG is a right angse, and CBIes than the right angi DCB GN, CB,illnecessarii meel: et theni meet in H, and oin HE HE,ill ouch the ellipsis in the oint E forci not, et it i possibie, meet the ellipsis again in Κ, and throughm drama straight line parallel o EF, o meet Amini, and the Circle in then, bccaus EF suo KL as Fio ML 2. cor. 6. 2. and that the potnis Η, Κ, are in ne straight in the oinis H, M, Gire likewise in one lemma 2. stra ight line: theresere theatraight line, H meet the circle

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the famem touches the circle: hicli is absurdci there fore HE meet the ellipsis no herebui in the poliat E and consequently ouchesi in his pOint. Fig. l. ext through D, the vertex o et ther axis CD, et a traight lineae drawn parallel to theother axis AB; his stra ight line ouches theellipsis for is not, it ili meet the ellipsis in more potnis than orae, et it, is possibie meet the ellipsis again in O; and through O dra astra ight line parallel to CD, meet in AB in P, and the circle in Q; also let CD meet the circle in R. Then, ecause C is a parallelogram CD is equa to Oci ut a s CD Ο ΡΟ, s is C to PQ 2 cor. 6. 2 d CR, there re, an Pinare quat: ut the are also parallel; the refore, is QR beboined, it illi parallelao CP and the angi QR is, there re, a rightangle. An thus RQ, dra n perpendicularto a diameter of the circle rom the extremi tyo that diameter, salis illi in the circle; hichis absurd l6 3. Elem: there remo ouches the ellipsis.