Elements of the conic sections

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PI ATE VI.

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PLATE VI.

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I from a potnt of an ellipsis astraight line e ordinately applied to a diameter, and romthe vertex of the diameter astraight linei drawn at right an-glecto it, and equa tociis alus reclum the quare of the ordinate is qua to the rectangle P plied to the latus reclum hau ingse iis readth the abscissa be- tween the ordinate and the Vertex of the diameter, ut deficient by a figure similar, and similarly 3itualed, to the figure contain-

ed by the diameter and the latus

reclum.

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Fig. Is Let te a poliat in the ellipsis; rom

dra FG, an ordinate to the diameter AB; and rom the verte of AB dra a perpendicu Iar BH, equa to the latus rectum of An thenthe square of FG is quai to the rectangi applied to ΒΗ havin the abscissa BG se iis

parallel o BH, and meetin AH in Κ, Complete the parallelogram KLΗΜ, ABHΝ:then, ecatis the rectangi AG is to thesquare os FG, as 22 2. AB to ΒΗ, that is,as AG to GK, that is, a the l. 6. Elem. rectangi AGB to the rectangle ΚGB; there re ex aequali the rectangle AGB is to the quareos FG, a the fame AG to the rectangle ΚGν and thus the quare of FG is equat tolli rectangle ΚGB, applied to the latus rectum BH hicli rectangi has for iis breadth the abscissa GB, but is es than the rectangle BM,

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by the figure LHM, similar, and similarly silualed, o BΝ 21. 6. Elem.)Fro the square of the ordinate heing thusequat to the deficient rectangle o that under the abscissa andini a partis the latus rectum, Apollonius called his curve line the ellipsis. COR. I seo the vertexi of the diameter AB any traight in Boebe drawn qua tolli latus rectum, though nolint right an gles to

PROP. XXIV., PROB.

I Wo nequat straight lines hichbisect ach other at right angies, hei nugiven in position and magnitude, to describe an ellipsis of

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the XC S. rig. I9. Let two nequa strat glit lines AB, DE, whicli hi sect ach other at right an gles in thepoin C, beatve in position and magnitudesit is require to describe an ellipsis,hich mayliave AB and DE se the XeS. Fro the extremit reos DE, the lassis thestWo, place DF equat o CB, the half of AB ille greater; and Do the centrem, illi the distance DF describe a circle hici, ill meet AB in t w potnis G, H in hicli scili end sona string of the sanae tength with the traight line AB, an describe an ellipsis, as as directe in the rs definitionis this book thestraight lines AB, DKwillae the axes of this ellipsiS. For since the potnis G, H are the foci of theellipsis, and that Cris 3 3. Elem. equat toCH, the poin C is it centre 3 des 2; and since C o CB is quai to the tength of the

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tion and magnitude, an a pDint without it ein gi ven to deScribe an ellipsis of hicli that stra ight line hal be ne of the axes, and whichihallias throughtha givei potnt; ut the givenpoint mustae so situaled that a perpendicular drawn fronicit may fati e tween the extremities of the give Straight line.

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ina be to the square os DE, a the rectangle ALB to the square of ΚL and place DE perpendicula io AB so that the mutuali hisecteach ther: and by the last prop. illi the Same AB, DE, a the Xes, de Scribe an ellipsisu his ellipsis illias through 2 Cor 15.

2. the poliat . PROP. XXVI. PROB.

Torand a diameter, the centro, thoaXes, and the focii an ellipsis given in poSition.