Elements of the conic sections

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in ille direction of the perpendi- Cular, e parallel to ne of thesides of the triangle through the axis the line hicli is the common sectior of the plane uti ingtheia se, and of the conica sur-faCe, is a parabola, havin for a diameter the traight line hichis the common section O the tri

angi through the a Xis, and of the

Same plane Lutting the a se,

Let thereae a cone, the verteX of Whicli is Fig. 15. theloin A, and the bas the circleio; et tbe ut through the axisi a plane, and let thasectioni the triangle ABC; et ille also cui by another plane cuiliniit base in the dire tio of the stra ight linem perpendicular tolli straight lineio; et the line DF be thesectio made in iis ursacen and let FG, the Common sectio of the triangle through the axis, and that other plane, e parallel o A

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one of the fides of that triangle the line DFEis a parabola, and FG one of iis diameters. In the sectio DF take any oin H, and through H dra ΗΚ parallel o D to meet FG in and through, dra LΜ parallello C: there re the plane passing throughΗΚ, LM is l5. II. Elem. parallel to the plane through DE, BC, that is to the base of

the Cone : and Consequently the plane through

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anda so the vertex o GF, and in hich DG may be ordinatet applied to the fame GF and hecause the poin D, b construction, is in theparabola described the potnim is likewiserinthis fame parabola 2 cor 13. I. And thesam thing may be demonstrate with regar . to allati potnts of the section DFE.

The secon Lemma fias ius, ascitis extant in the frs book of AsoLlonius Conio SectionS. Let ABC be a line, and te AC e FiDiria traight line given in position; and et ali the traight lines Mawnfrom the line ABC, o as to beat right angies to AC, e Such, that ach of them may have iis

tained by the segmenis, into hich

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fere Ce of a circle, and AC a dia

ameter of that Circle.

Frona the oinis D, B, E dra perpendicular DF, BG, H then the quare o DF sequa to the rectangle AFC, the quare of BG to the rectangi AG C, and the quare of EHto the rectangi ALIC. Bisec AC in , oining D, B, E then, Since the rectangle AFC, together With the quare of FΚ is qual 5. 2. Ele in . to the quare of ΑΚ, and that thesquare si is, b hypothesis, qua to therectangle FC; the quare o DF, togetherwith the quare of FK, is qua to the quare of Κ: heres ore the quare of D. 4T. I. Elem. his equat to the Same square of AK ΑΚ, theres ore, is equa to D. In like manner, ea ch of the traight lines Κ, Κ may be provexto e qua to AK or C; theres ore ABC is the circumferen ce of a circle, hichias the potnim for iis centre, an is describeda bout AC as a diameter.

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ELEMENTS

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II. The potnis D, E are named the foci. III. The sint C whicli bisecis the straight Iine etween the foci, is named the centre of the ellipsis. IV. A straight line passing through the centre, an terminated both way by the elIipsis,

is named a diameter , and the potnis here a diameter meet the ellipsis, re named the ertices of that diameter. V. The diameter hicli passes through therici, is named the greaterietis. VI. The diameter perpendicular to the greater axis, is named the leue azis. VII. Two diameters, ach of hich bisecis ali straight lines in the ellipsis that are parallelto the ther, are named conjugale diametera.

VIII. A traight in no passing throughthe Centre, ut terminate both ways by the ellipsis, an bisected by a diameter, is said to