By Gerald J. Toomer
With the book of this booklet I discharge a debt which our period has lengthy owed to the reminiscence of an exceptional 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 merely in Halley's Latin translation of 1710 (and translations into different languages totally depending on that). whereas I yield to none in my admiration for Halley's variation of the Conics, it truly is faraway from pleasing the necessities of recent scholarship. specifically, it doesn't comprise the Arabic textual content. i'm hoping that the current version won't purely therapy these deficiencies, yet also will function a origin for the research of the effect of the Conics within the medieval Islamic global. I recognize with gratitude the aid of a couple of associations and other people. the toilet Simon Guggenheim Memorial beginning, via the award of 1 of its Fellowships for 1985-86, enabled me to commit an unbroken 12 months 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 vacationing Fellowship in Trinity time period, 1988, which allowed me to make sturdy use of the wealthy assets of either the collage Library, Cambridge, and the Bodleian Library.
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Extra info for Apollonius: Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banū Mūsā
3, 5, 6, 9, 10, 16 &. 20, and V 3 in Props. 16, 17 &. 18. V 4-10 deal with the minimum from points on the axis of a conic. The general case is enunciated in Props. 8-10, but the special cases for a point whose distance from the vertex of the section is equal to or less than half the latus rectum are treated in Props. 4-6 and Prop. 7 respectively. V 4 In a parabola, if (Fig. 4) point Z is marked at a distance from the vertex r equal to half the latus rectum, then zr is the minimum to the parabola from Z.
E. l6~. 5 If that point is B, the condition for touching is, by II 12, Be· eH = EZ· ZH. I Summary of V 44 &. V 45 can be drawn passing through E, and, as ZE increases beyond that, there will be no intersection of parabola and hyperbola, and no minimum can be drawn passing through E. The situation for hyperbola and ellipse is considerably more complicated, but here too the locus is a rectangular hyperbola which intersects, touches, or fails to intersect the original curve. Apollonius draws the determining hyperbola only in Props.
62. V 9 The basic theorem on minima in the hyperbola. See Fig. 9. For a point E on the axis [where Er > 1/2 R), the minimum is found by marking a point, Z, towards the vertex such that, if H is the center, the ratio (HZ:ZE) equals the ratio of transverse diameter to latus rectum, and erecting the perpendicular from Z, ze, to cut the section in e. g. e. (D ~ R). (3a) As in V 5, Apollonius first proves (3), using the lemma V I, then [3a), whence (2) and [1) follow immediately. Used in Props. 14,43,45,50,59,60,61 and [implicitly) 63.