Download Apollonius: Conics Books V to VII: The Arabic Translation of by Gerald J. Toomer PDF

By Gerald J. Toomer

With the ebook of this ebook I discharge a debt which our period has lengthy owed to the reminiscence of a superb mathematician of antiquity: to pub­ lish the /llost books" of the Conics of Apollonius within the shape that's the nearest we need to the unique, the Arabic model of the Banu Musil. Un­ til now this has been obtainable in basic terms in Halley's Latin translation of 1710 (and translations into different languages completely depending on that). whereas I yield to none in my admiration for Halley's version of the Conics, it truly is faraway from pleasant the necessities of recent scholarship. specifically, it doesn't comprise the Arabic textual content. i'm hoping that the current variation won't in simple terms therapy these deficiencies, yet also will function a beginning for the examine of the effect of the Conics within the medieval Islamic global. I recognize with gratitude the aid of a few associations and other people. the toilet Simon Guggenheim Memorial beginning, through the award of 1 of its Fellowships for 1985-86, enabled me to commit an unbroken yr to this venture, and to refer to crucial fabric within the Bodleian Li­ brary, Oxford, and the Bibliotheque Nationale, Paris. Corpus Christi Col­ lege, Cambridge, appointed me to a traveling Fellowship in Trinity time period, 1988, which allowed me to make reliable use of the wealthy assets of either the college Library, Cambridge, and the Bodleian Library.

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Additional resources for Apollonius: Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banū Mūsā

Example text

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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