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the quare of CP is to the quare of SQ, a thesquare o CV to the quare o GK but thesquare of CPis to the square of CS as the quareo C to the quare of and by Conver- Sion, and Prop. 7. I. Elem. the quare o CPis to the square of SQ a the quare o C to the rectangle EΚT the quare, there re, fCE is to the rectangle EΚT, a the quare of CV to that o GK alternat ely the quare fCE is to the quare o CV as the rectangle EΚrto the quare o GK theresere I 5.5. Elem. the quare of Tis to that of VX, as
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Universalty the quare o an diameter iato the quare oscit Conjugate, a the reClanglocontaine by the segments intercepte he tween iis vertices an a traight line ordinateIyapplied orit, is to the square of the segment of the fame straight line et ween the ellipsis and that diameter for an ordinate to a diameteris parallel to the tangent rawn through thevertex of that diameter and there re is a rallel to the conjugate diameter. COR. I. The quare si strat gli lines ordinatet applied to the fame diameter, re to ne another a the rectangles contained by the segments of that diameter, as a demonstratedit Cor. 6. 2. with regard to the Xes.
Coll. 2. I ET, X e conjugate diame ters of an ellipsis AT, and is frona a polia Gatiraight line GK b drawn parallelao X oneo the diameters, and meet in themther ET in and is the square of ET he to the quare of VX, a the rectangle KT to the quare ofGΚ the poliat G is in the ellipsis. For is thepoliat G is no in the ellipsis, the GK, ill
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me et it in omemther po in o that si de of the diameter Ea on hich G is; et it, is possi hie meet the ellipsis in de then, by the proposition the rectangle Κ is to the quare of iΚ, a the quare of ET to the square Os X, that is, b hypothesis, a the rectangle EΚTto the quare o GΚ: ence the quare of Kis equa to the square of Κ and thus thestraighi line, is equat to the traight line GK hicli is impos Sible. COR. 3. I frona two potnis G, e, ne of hich, e, is in the ellipsis, there e drawn to the diameter ET straight Iine GK, ef parallelio traight lines ordinatet applied to the fame ET; f the rectangle ENT, E . contained by the segments of the diameter ET Whicli are
interceptedi et Neen iis vertices, and the parallel dram to it have the fame ratio to achother a the quare o GK, ef then the therpoliat G is likewi se in the ellipsis This is de
monstrated rom Cor. I. in the Same manne ASthe se condiorollar frona the Proposition.
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Fig. 16. Co R. 4. I a circleae describedipo AB, adiameter of the ellipsis, an is straight lines DE, FG e drawn ordinatet applied to the diameter B and is frona the oinis , straight lines DH, Κ be drawn perpendicular to the fame AB, and meeting the circle in the potnis Η, Κ then the perpendiculars H, F shal have the fame ratio to acti Otherwhicli the ordinates DE, FG have. For thesquare os DE, FG have the fame ratio tomach
applied to a diameter, cui ostri belween the Centre and the potnis here the meet that diameter, qua Segments of it, the are equat andi equat, the cut ostri e tween the centre and theseloinis, equa Segments.
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to that diameter, meetin a Circle describedipon the sam diameter in H; and is the traight line ouch in the circle in II, meet that diameter in , anxifELae drawn oining the potnis L andi, the line EL ill ouchthe ollipsis in E and ConverSely.
Porci EL do not ouch the ellipsis it illcut it te it, is possibie meet it in anotherpolia Μ and through Μ to the diameter AB dra M parallelao ED and through, drawNO at right angies to the diameter AB, meet
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ing the circle in O o the sanae fide of AB with
H since the parallel DE, M are drawn frona the potnis D, Ν, as also the parallels DII, NO, hich have the fame ratio to ach ther
ista diameter of an ellipsis beatven in positionand magnitude, and the angle hicli that diameter malaes illi an straight line ordinatelyapplied o it Malven a traight line cantedra n that ill ouch the ellipsis in a iven
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lliat diameter thoe semidiameter B is a mean proportionalbetween CL and CD, the segments of thes diameter intercepted the one belWeen the Centre and the tangent, and the otherbe tween the centre and the ordinate; and the segments of the sanae diameter interceptes be- tween iis vertices and the tangent, have the Same ratio to ea chither
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as ille segments et e en it Vertices and the ordinate.
Havinidescribed a circle upon the diameter AB, and drawn rom helointi a traight linei H at right an gles io AB, meeting the circle in , oin HL then, hecause Hlolaches the 16 2. circle, and that Hreis perpendicular to the diameter, CD, B, CL are proportionals 8. 6. Elem. Whicli is the srst
Cop. I. Heiace the rectangi contained by the segments of the diameter intercepted e t en the ordinate and the centre, an die- tWeen the ordinate and the tangent, is equat tolliat contained by the segments et een theordinate and the vertices of the diameter Foras CD, B, CL are proportionals the quare
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o CB is equa to the rectangi DCL ut thesquare o CB is quanto the rectangi ADB, togethe with the quare of CD 5. 2. Elem. and the rectangle DCL is equa to the rectan-gle DL, togethe with the fame quare o CD 3. 2. Elem. Tahe way the common quareo CD, and there re main the rectangle ADB
COR. 2. And the rectangi contained by the segments of the diameter intercepte dieiweenthe tangen and the Centre, and et ween thetangen and the ordinate, is qua to that Contained by the segments et en the tangentand the vertices of the sam diameter for tharectangle ALB and quare os mare together equat 6. 2. Elem. to the quare of CL; and the rectan gles I CD, CLD are together equat 2. 2. Elem. to the sanae square of CL there re, ecause the quare of CB has been provexequat to the rectangi DCL, there re- main the rectangle ALB equa to the rectan-
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Fig si Frona a poliati os an ellipsis et astraight line I be ordinatelyapplied to a diameter AB, and froin the fame potnt et a traight line EL e rawn meetin that diameter in the segment of the diameter intercepte lae-
tWeen the Centre C and the oint L, the semidiameter B, and the segment DC etween thecentre and the ordinate e proportionals the traight line Liuill ouch the ellipsis in D: or