Elements of the conic sections

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COR. I. I EH ouch the ellipsis, and GHbe o ined, it a b Shewn in the Same man- ne that GH ouches the circle. Co R. 2. From a potnt in an ellipsis, only onestra ight line an e rawn which ill ouchtha ellipsis: ere it possibi se two stra ight lines toriouch the ellipsis in ne and the Samep aint two could ais to uch the circle in onepotnt; hicli is against the cor to 16 3. Elem. COR. 3. A stra ight line cannot meet an ellipsi, in m re than twolo in is for could a stra ight line meet the ellipsis in more than wo polritS,a stra ight line Could also me et the circle in more that two p antM Whicli, ouldie an absurdit the ellipsis, there fore i Conve Onthe fide here trai ght lines ouch it, ut con-

COR. 4. The angi containe by an diameter, hici is no et ther axis of the ellipsis, and that par of the tangent a iis vertex hichmeet the gredier aXis, S greater than a right

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gent, and Anthe greater aXis, the other things remaining as in the proposition the an gleCEH is reater than GH, that is 2 l. 1. Elem. greater than a right an gle. COR. 5. The proposition potnis ut a method by hicli, is the greater axis beatven inposition and magnitude, a traight line an bedrawn that hali ouch an ellipsis described Iapon the aXis, in a givens Dint. Coll. 6. No stra ight lines ouch in an ellipsis in the vertices of a diameter are parallel. Fig. 12. Let EH ST to uch the ellipsis in the vertices of the diameter CS and et them me et theaxis AB in H, T dra EF SV perpendicular to the fame axis, meet in the circle des

by Cor. I. ill ouch the circle in G, X and because GF, XV are divided in the fame ratio 2 Cor. 6. 2. in the potnis E, S, and that ECSis a traight line theres ore the oinis G, C, a re also in a strat glit line lem. Q and be- cause GH, T, hicli ouch the circle in the

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vertices of the diameter X are paralleis EH, SI are also paralleis lem. 3.

Case 1. When the traight line bisecting the Fig. Langle is paralle to the greater axis Let AB be the reater Xis, C the Centre, and D, E

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Tahe the halves of the antecedenis, and CK

also a right angle and consequently the stra ight Iine ΜΚ ouches the circle, I 6 3. Elem.)

See prop. 8. Elem. Plane Trigon. annexed totur author' editionis Euclid' Elements.

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re together reater than the same DG thatis, than DF, F together, that is, tha thegreater axis AB the poliat inis, Consequently, without the ellipsis 3 Cor. I. 2; and ConSe

PROP. XII. PROB.

The greater axis of an ellipsis e inggive in position and magnitude,

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Every traight line parallel o astraight in that ouches the ellipsis, and termina ted both ways by the ellipsis, is bisected by the diameter that passes through thopoint of OnlaCt.1 the diameterie either of the a Xes a tangent drawn through iis verte is parallel to theother Sic IO. and 2 cor. 10. Q, and thus a