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Dra two traight Iines parallelmo achother, and terminate both ways by the ellipsis , the traight line hich bisecis them is 4 cor. I 3. 2. a diametera and the potnt bisecting that dianae terris 3 2. the centre. In orde to find the aXes, find the Centre C, Fig. 20. and in the ellipsis tali an potia A, and oin C, and stomuli centre C, and with the distance AC, describe a circle AF is his circle salis holly without the ellipsis, C is thegreates of the semidiameters and theresere 9. 2. the half of the greater aXis. eXt, talae any oin D, and let a circlei described stom
lipsis, the traight line CG is tho hal neither
of the greater nor of the les aXis the circle, Consequently mu Si meet the ellipsis agaim: Iet
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PROP. XXVII. PRos. I Wo conjugate diameter of an et
lipsis being given in position and magnitude, to find the Xes, and describe the ellipsis.
Rig. 21. et AB, CD, e the give diametersu et them meet each ther in the Centre E. Suppos the problem solved that is, let FG, HI be the axes tote found and through A draruthe stra ight lineis paralleloo CD AL,illtoucli 3 cor. 14. 2. the ellipsis in A, and willhealven 28 dat. in position Iet the same
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AL meet the axes in the potnisi, M ohere- fore the rectangi LAM is quai to the square. of CE, the semiconjugate to 2I. 2. AB: ut CE is iven, and consequently iis quam is givem theresere the rectangle LAAlcis givendiet the rectangle EA be equat o LAM; and hecause EA salven in position and magnitude, A is also ive in positio an magnitude; and hecause the rectangle LAM, AN areequat, the potnisi, E M, Mare C OnV. 35. 3.
Elem. in the circumference of a Circle there-sore i E is hiseciud in o the centre of the circle illi in the traight line OR hicli is a right an gles to EN cor. I. 3. Elem in ut LEM eing a right angle the Centre of the circle is lihewis in cor. 5. 4. Elem. thestraight line LM : Dis here re in the potnt
Ρ, where OΡ LM interfeci each other: there fore the centre mi given, and the oin E isgiven: ence the circle describe from theceniret, illi the distance ΡΕ, salven in 6.def. dat. position so likewiserare the potnis L, Μ, here iis circumferen Ce meet theatraight line AM given in position there rethe axes EL EM arealven in position dra
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Aia a right angies to the Xis FG and he- cause AL ouches the ellipsis, Eu, EF, Lare IT. 2. proportionals and Eu, L aregiven theresere EF is ive in magnitude. It ma in like manneri proved that ΕΚ sgive in magnitude therei ore the axes FG, ΗΚ arealven in positio an magnitude anda ellipsis described through theio intra, illithe axis FG 25. 2. williave AB, CD two fit conjugate diameters. The composition is a follows produce EAto Ν, so that the rectangle EA may be equalto the quare of CE: bisect E in , and dra OP at right angies t it, meetin thestraight line AL, whicli is parallel O CE in thepol ni P and rom the centrei, distanc PE,
describe a circle, an let AP meet iis circum
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this ellipsis AB, CD are conjugate diameters. For since in is perpendicular to the axis FG, an Eu, EF, EL proportionais, AL ouches the ellipsis l8. . in the poliat an die cause Cinis parallel to the tangent AL, it is in hesam position with the conjugate diameter o AB and the angi LEM Minitia a semicircle is a right angle consequently EM is theother axis : ence the rectangi LAM is qualto the quare of the semidiameter conjugate toAE 2I 2: but the fame rectangi LAM sequat 35. 3. Elem. to the rectangle EAN, thatis, by the construction, to the quare of CE; theres rex is the semiconjugate to AE theellipsis theresere passes through C; an be- cause Ereis equat o EC and EB equa to EA, it passe Iikewise through the sints D, B. Hence AB, CD are conjugate diameters in theellipsis described.
The positio an magnitude os a diameter an ellipsis te ing
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given, and the position os astra ight line, passing through agi veri potnt in the ellipsis, anciordinatet applied to that diam e ter e in also given to deScribe the ellipsis.
M. et L et Ambe the give diameter, to whicli RS, a traight line give in position is ordinatelyapplied froni a give pol ni R of the ellipsis tobe described. Bisect AB in the poliati, and draw through a traight line parallelao S, an in that parallel tali equa strat glit lines M, ED, so that the rectangi ASB may be to the square of RS, a the quare of AE to the square o EGor ED. I a mean proportionalie found I 3. 6. Elem.)be tween AS and B then 22 6. Elem. the mea proportionat is to S, as Eoo C, whicli is there re found I 2 6. Elem. then, by the precedin proposition deScribe an ellipsis of hich AB, CD may be conjugate di-
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ameters this ellipsis illias through the 2 Cor. I 5. . potnt R, and RS illi ordinatelyapplied 4 cor. Is 2. to the diameter AB. PROP. XXIX. THEOR.
I a cone ut by a plane paSSingthrough in axis e ut also byanother plane, meet in both the Sides, of the triangle through theaXiS, but Dei ther parallel to the
rily siluated i that other plane, and the plane in hicli the baseo the cone is silualed, meet in the direction os a traight line Perpendicular eithe to the baseos the triangle through the axis,
O to that a se produc ed theline hicli is the Common CCtion of this ther plane, and the
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conical ursace, is an ellipsis, whicli has se one of iis diameter the Common section of the triangle through the Xis, Withthis fame plane.
Fig. 22. Let thereae a cone the verie of hicli stheioin A, and the bas the circle BC et ithe cuti a plane through the axis and let thesectioni the triangle ABG let ille cui like-wi se by another plane, meetiniboth the fides AB, AC, of the triangle through the axis, butnei ther parallel to the base of the Cone nor subcontrarii situatos; et the line DEF bo the
tio of this plane, illi the base of the cone continued be perpendicular o BC: the thelitae DEF is an ellipsis and DF, the commonsectio of the triangle through the Xis, and this a me plane, sine of iis diameters.
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o these ratios are the fame to ne an Others
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ex aequali, the quare o E is to the quare of ΝΟ, a the rectangi DKF to the rectan gleDOF. Describe, there re, an ellipsis 28. 2.)of hicli DF may be a diameter, and in hichΕΚ may be ordinatet applied to DF and be-Caus the oin E by Construction, is in this ellipsis, the poliat likewis incit 3 cor. I 5. 2. An the fame thing may be demonstrate&with regar to ali the oints of the section DER