Elements of the conic sections

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vertices, e proportionalS; or is, third ly the four Segment be tween the ordinate and cach of

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gate diameters Wo traight lines

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be ordinately applied to nother

diameter, the quare Of the Seg

ment of that other diameter in

tercepted et e en 1 ther ordinate and the centre, is quai tothe rectangi contained by the Segment bet een the ther ordinate and the vertice of that sanae diameter.

Let A, B e the two conjugate diame Fig. riters of hicli the potnis A, B are vertices; fro A, B te AF, G be ordinatet appliedio another diameter DE; the quare of CG intercepte between the ordinate B and thecentre, is equat to the tectangle EFD contain-ed by the segment intercepte between theother ordinate AF and the vertices of DE: and likewis the quare of Cp is equa to the rect

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rectangle EF is equa to the sanae l. Cor. IT.

2. CFH; ence the quare of CG is qua tolli rectangle EFD; ake thes equat frona thesquare o CD, and there ill re main the rectangle EG equat to the quare of CF 4 and

5. 2. Elem. COR. I. Henc the semidiameter CD, towhicli the ordinates are drawn is to iis semidiameter conjugat CL, a the distance be- twee either ordinate and the centre is to theother ordinate the quare o CD is to thesquare of CL a the rectangle EF to the

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square of AF, that is by the proposition asthe quare of CG to the quare of AF there re Cinis to Liras CG to AF. In like man-ner, it ma be hewn that reis to CL, as Flo BG. COR. 2. The quares of the segments of the diameter to hicli the ordinates are drawn be- tween the ordinates and the centre, are together equat to the quare of the semidiameter. For since the quare of CG is equat to the rectangle EF there re the quare of CF toge- the with the quare of CG, is quai to the Square of CF togethe with the rectangle EFD, that is to the 5. 2. Elem. square of D. COR. 3. Hence the sum of the squares os any tW Conjugate diameter is equat to the sum os the square of the ages. et CD, CL be the semiaxes, and CA, CB conjugate semidiameters; Iet AF, G e perpondicular to CD, and ΑΜ, B perpendicular to CL then, because the quare o CD, as a provexin the precedin corollary, is qua to the square of

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by the fame corollary the square os CL is equalto the quare of M, togethe with that fCN, that is to the quare of AF, togethe with that of there re the quare o CD, CLare together equa to the quares of CF, CG, AF, BGci ut the quare of AC, BC are to-gether 4T. I. Elem. equat to the Same quares of CF, G AF, BG and there re the sumo the quare of AC, BC is equat to the sumo the quare of CD, L. PROP. XX. THEOR.

Is through the vertice of tW COnjugate diameter seu struight lines h drawn ouching the ellipsis the parallelogram containe bythes Straight Iines, is qua tolliat containe by the tangenis

drawn through the vertices of any othe two conjugate diameters.

I iet the traight lines V MY, YX, Utouch the ellipsis in the vertices A, B, an in

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the opposite vertices of two Conjugate diameters AC, BC in like manner, et ΡT, PR, RS, ST touch the ellipsis in the vertices of the conjugate diameter DC, C; the figures ΟVXY, RST are 6 cor. IO. 2. Parallel grams, and equatuo ach other. To the diameter Cindra AF, BG parallelio I and to the diameter CL dra AM, BN parallelo CD and et Ao, O meet thesam CD in the poliat H, Κ an having oin-ed BH, complete the parallelogram HCΝΩ Then, ecause AH ouches the ellipsis, and that AF is drawn ordinately applied to the diameter D there re H is to CD, a Clinto CD 17. 2. And by the first corollar of the regoin proposition Clinis to CL, as CF to B; there fore, et aequo, CH is o L, as DBG, or H and the an gles DCL, CH inare

equat for the are alternaten theres ore the parallelogram DL is equa to the parallelogram I 4. 6. Elena. NH: ut H is the doubie of the triangle CBH, upon theriam base CH, and bet Neen the fame parallely and like is the

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parallelogram ACB is the doubie of the samo triangle CBH, po the fame base CB, and be- tween the fame parallels CB, AH there re ACBO is equat o H and the parallelogram DL, achath been shewn is quai to the fame FH 'heresere the parallelogram DL AB areequat and thus the parallelogram PRST, OVXY, whicli are the quadruples of DL, AB, are likewis equat PROP. XXI. THEOR.

lipsis, meet two Conjugate diameter the rectangle Containe db iis segmenis, et Cen thepoliati contac and the diameters, is equa to the quare of the Semidiameter Conjugate to that

Whicli passes through the pollit

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Let the traight linem touch the ellipsis Fig. I . in the o in A, and let i meet the conjugate diameters CD, CL in H, Z , et the semidiameter Bae conjugate to A the the rectan-gle HAZ is equat to the quare of CB. For dra AF, G parallel to the diameter CL and sincem is to C, a H to AZ, the rectan gles def. 1. 6. Elem. HFC HAZare similar and H is to HA, a CG is to CB there re since the rectan gles ΗFC HAZare similar of which HF, H are homologous fides, and that the squares of G, CB are simi-Ia figures the rectangi HFC 22 6. Elem.)is to the rectangi HAZ, a the square of CG to the quare o CB: ut the rectangi HFCis equa to the rectangle I. Cor. II. 2. EFD, that is, to the l9.2. Square of CG and there re 14. 5. Elem. the rectangle HAZ i. also equat to the quare of CB. COR Hence it follows, is a straigdi line HAZtouch an ellipsis, and mee two diameter CH,

CZ i the rectangi HAZae qua to thesquare of CB the semiconjugate to CA, hic i

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passe through the oin os contaco then CH, CZ are two conjugate diameterS.

PROP. XXII. THEOR.

plied to a diameter the CCtangi contained by the Segments of the diameter is to the quare of the ordinate, a the diameteris to iis latu reClum.

Fig. is Letite a poliat in an ellipsis fronti dra FG ordinatet to the diameter AB. The rectangi AGB is to the quare of FG as the diameter Amto iis latus rectam. For let H e qua to the latus reclum; and since the diameter AB, it conjugate DE, an lalus rectum Bri 9 def. . are proportionals AB is to BH 2 cor. O. 6. Elem. asthe quare of AB to the quare os DE, that is,