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tained by the segments of the fit stmentioned Xi is to the quare of the perpendicular.
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s . . proportionais, heir quare are also
proportionals but the quare o CB is 5. , Elem. made up of the quare o C and therectangi AKB and the quare of of the 5. 2. Elem. square of CL and the rectangle DLE; theresere the quare of Bis of this
and 19 5 Elem. to the quare of CE, a theremaining rectangle ΑΚ to the remaining rectangi DLE; and by Conversion, the squareo CB is 5. 2. Elem. to the rectangi AEB, asthe rectangi AKB to iis Sces above the rectangi DLE, that is, a the rectangi AKB to precedin prop. the quare of ΚΗ: ut 2.2. the rectangle AEB is equa to the quare of CF there re, a the quare o C to the Square of CF so is the rectangi AKB to thosquare of ΚH; the quare, there re of AB isto that os FG, a the rectangi AKB to thesquare o HK. Fig. i. In the ther case, et Uribe perpendicularn, , , to the lesser axis the quare of FG is to the Square O AB, a the rectangle P to thesquare of VP The square of CB, astath been
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proved is to the quare of CF, a the rectangle AKB to the quare o HK o PC therelaretheri quare of CF prop. B. and I9. 5. Elem.)is to the quare o CB, a the rectangle FPGto the quare of Κ, that is, a the rectangle FP to the quare of HP. COR. I. Henc the quare os strat gli linos drawn Domioints of an ellipsis perpendicular
angles contained by the segments of that axis.
For let HM, PQ be perpendicular to the Xi Fig. 2. AB in M and u and by the proposition therectangle ΑΜΒ is to the quare of HM a thesquare of AB to the quare os FG, that is, asthe rectangle A A to the quare of PQ and, alternately the rectangi AMB is to the rectangi AQB as the quare of ΗΜ to the square
Con. 2. I a circleae described pontither Fig. 2. axis as a diameter, and H, P perpendicular to that axis, meet the circumferen ce in thepoinis N .R; these perpendicularchet cen theaXis, and the circumference of the circle, are to
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and the ellipsis for the rectan gles AMB, AOMare equat 35. 3. Elem. to the quare of ΜΝ, QR each tomach oheresere the square of MNis to the quare of QR a the quare of Hlo the quare of QP consequently the straight Iines N, R, H QP themselves are also 22 6. Elem. proportionalS. PROP. VII. THEOR.
Every traight line terminatet both Way in an ellipsis, and parallello ei ther Xis, is bisected by the
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Every traight in termina ted both Ways in an ellipsis, and bisected by one Xis, is parallel to the
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equat segments C QC of themther axis. PROP. IX THEOR.
O ali diameter the greater Xi is the realest, and the esse axis the least and a diameter hich
greater than ne more remotC. Fig. 2. Let Bae half the greater axis, and Chalf the essen; et Hie an other semi-di ameter, and dra minat right angies to Cn and hecause the quare of CB is to the quareo Cras the rectangle AMB to the quare fHM, and that 4 cor. I. 2. Cnis greater than
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Letio Crbe more remote sto the greater Fig. 2.
axis than CH and then CH illi greater than T dra TV parallelao HM, and Ietit meet AB in V, and the ellipsis again in X, and let HZie parallelao V, and mahe Q, equat o M. Because the rectangle A, B, hicli consi sis of the rectan gles AMB and QVM is to the
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square of VT, hicii 5. 2. Elem. is adempo the quare Unan rectangi TZX, as I. Cor. 6. . the rectangi AMB to the quare ofMHir Z the whol the rectangi AVB, isto the whol the quare of VT, as I9. 5. Elem. the re maining rectangle QVM to there maining rectangi TZX: ut the rectangle AV is 6 of this an prop. . . Elem. greater than the quate of VT; there fore therectangle QVM is also greater than TZX. To these nequa Is ad the quare of CV, and the quare of CM 5. 2. Elem. willi greater than the rectangi TZ together illi thesqua re of CV Superadd to the Same unequais,
aWay the Common square of V, and the remaining rect
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the quare of H o ZV, and the quare ofCM, Id together, ill e greater than thesquares of VT, C together that is, the squareo CH is greater than ille quare o CT and stere re H is greater than T.
paralleis have the fame ratio tocach other, as the segments of
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Fig. 3. In the ne case, Complete the parallelogram AB, AD , these re similar, and haveth angi at A common consequently therare bout the sanae 26. 6. Elem. diameter,
that is, the sints A, B, D are in the samestraight line. Fig. 4. In the ther case, oin A, DA and be- cause BC is to A as Eoo A, and theangle BCA equata DEA the triangles ABC, AD are quiangular 6. 6. Elem; Consequently the angi EAD is qua to the angleCAB and there re AB, A mahe I 4. 1. Elem. one traight line.