Elements of the conic sections

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Let ACB, DCE be two conjugate diameterS, Fig. 27. and Iet a stra ight line hich ouches the hyperbola in F meet them in the potnis G, H, and Iet CK be the semidiameter conjugate to CF; the rectangle GFH is equat to the square of CK.

hecause of the parallels, Μ is to C, as GPto FH consequently the rectan gles G MC, GFH are similar an hecause the trian gles GΜF, CL are quiangular, M is to GF, as CL to Κ theres ore the rectangle GM C, GFH, and the quares of CL, CK, hichare seu similar an similari situaled rectilinealfigures, describe upon the seu proportionaIstra ight Iines M, GF, CL, C are likewise proportionals but the rectangle M C is equallo 35. 3. the rectangle AMB, that is to the 43. 3. square of CL; the refore the rectangle GFH is equat to the quare of Κ.

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1 frona a potnt of an hyperbola astraight linei drawn ordinatelyapplied to a transverse diameter, the rectangi contained by the segments of the diameter intercepte between is vertices and the ordinate, is to the quare fili Segment of thes Ordinate intercepte between the hyperbola and the diameter, a the diameter is to iis latus reclum but fa strato line e rawn ordinately applied to the secondit ameter of the transverse, the Sumof the quares of half the seconddiameter, and of iis segmentie- tween the ordinate and the cen-

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tre, is to the quare of the ordinato, a the econ diameter isto iis alus reCliam.

Let therei a transverse diameter AB, and Fig. M. D the secon diameter Conjugate torii, and let AH e the latus rectum o AB, and Dom

sum of the quare o CD, C is to the quareo 1Κ, as DKto iis latus rectum L. Caseo. Since AB, DE AH re propor tionals des 12. AB suo AH, a tho quare of

AB ci tho quare os DE, that is 28 3. as therectangle AGB to the quare of FG. Case 2. And since DE AB L are proportionals DE suo L, a the quare of DE suo the quare of AB, that is 29. 3. a the sumof the squares of CD, C is to the square of ΚF.

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rse dis Is rom a poliati in an hyperbola straight line Gae ordinatelyapplied to a transverse diameter AB, and rom the vertex of that diameter a traight line AH dilrawn perpendicular o AB, and equa toriis alus rectum; the Square of the ordinate is equa tolli rectangi applied to the lalus rectum havinisor iis breadth the

abscissa belween the ordinate and the verteX, ut Xceedin by a figure similar, and Similarly sit- ualed, to that hicli is contain-ed by the diameter and the latus

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BOOK III. RE HYPERBOLA. 26 I

and meeting AH in, and complete the rect

The hecause the rectangi AG is to thesquare os FG, as AB io AH, that is, asGBto GΜ, that is, a the rectangle AGB to the rectangi AG Μ oheresere AGB is to the quareos F, a the fame AG to the rectangle AGM consequently the square of GF is equalto the rectangi AGM, havin the abscissa AG for iis breadth, and applied to the latus rectum AH, and Xceedin the rectangle HAGO by the rectangle ΜΝHO, simila to BAHΡ. Fro the quare of the ordinate being thusequat to the eae redis rectangle o that Under the abscissa an a line reuter than the latus

rectum, Apolloniu calle this curve line the hyperbola. COR It is evident, that the quare of Fwouldie equat to the rectangle AGM, though AH were nolint right angies to AB.

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PROP. XLIX PROB.

Pist. dis A Strato lineo belliginive inpositio an magnitude, an apoin Fieing given to describean hyperbola, of hich AB may

See note prop. 25. B. II.