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See Fig. 10: if L1 is the center, the minimum from E, EH, is found by forming (L1Z:ZE) = ratio of transverse diameter to latus rectum, Summary of V 11, V 12, V 13, V 14 & VIS xliii and erecting the perpendicular at Z. Then, for any other point 8, the difference between the minimum and Ee is given by ES2 _ EH2 = zp2. (D ~ R ). (3a) This theorem is used in Props. 15,23,45,50,53,54,55,59 and (implicitly) 63. V 11 This is a special case of V 10, where the point on the axis is the center of the ellipse.

51. V 45 The same proposition for hyperbola and ellipse (for the latter the two minima must be drawn in the same quadrant, and to the major axis I. 1 For a reconstruction of the analysis see Zeuthen, Kegelschnitte pp. 288-293. Summary of V 45, V 46 &. V 47 Ii In Figs. B the two minima, BE, rz, meet at point El; the center is N. Then, by V 9 &. 10, NQ:QE = NH:HZ = ratio of transverse diameter to latus rectum. e. l Again, Apollonius' synthetic proof, which is long and cumbersome, uses essentially only the basic theorems on minima, but if we introduce the auxiliary hyperbola2 many of the steps in the proof are immediately obvious.

Hogendijk. Fig. ax) is tangent to the original curve at B. If we draw BcrS tangent to that curve, meeting OX in S, then by II 3 SX = Xo = KH. So BX:BK = SX:crK = KH:crK. 1K. e . 1cr. 1K, or M" is mean proportional between I1cr and 11K. 1H. As remarked above (p. xlix n. 3), in the limiting case the point from which the single minimum is drawn is the center of curvature of the section at the point B. Zeuthen (followed by Heath) notes that Apollonius' construction allows one to find the locus of the centers of curvature of the different points on the section: this locus is a curve of higher order which is known in modern times as the "evolute" of the conic.

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