By Gerald J. Toomer
With the booklet of this publication 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 is the nearest we need to the unique, the Arabic model of the Banu Musil. Un til now this has been obtainable purely in Halley's Latin translation of 1710 (and translations into different languages solely depending on that). whereas I yield to none in my admiration for Halley's version of the Conics, it really is faraway from gratifying the necessities of recent scholarship. specifically, it doesn't include the Arabic textual content. i am hoping that the current version won't merely therapy these deficiencies, yet also will function a starting place for the examine of the impact of the Conics within the medieval Islamic global. I recognize with gratitude assistance from a few associations and folks. the loo Simon Guggenheim Memorial starting place, through the award of 1 of its Fellowships for 1985-86, enabled me to dedicate 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 stable use of the wealthy assets of either the collage Library, Cambridge, and the Bodleian Library.
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Extra resources for Apollonius: Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banū Mūsā
Proven from V 9 by reductio ad absurdum. Furthermore, LAiB is always acute. This is used in Props. 25, 28, 36 & 37. It should also have been invoked in Prop. 45 (p. 122), where the Banu Musa refer directly to V 9. V 15 The converse of V 11 & V 10. In Fig. 15, if I is the center of the ellipse ABi, then IB, the minimum from point I, is perpendicular to axis Ai; and if the minimum is drawn from any other point on the axis, as HZ, the perpendicular ZK from Z to the axis divides it in such a way that (IK:KH) equals the ratio of transverse diameter to latus rectum.
1 M lies between E and r, because l1r < lfzR. xlvi Summary of V 24-26, V 27, V 28 &. V 29 In Fig. 23 KE is a maximum. Hence AH: HK But =o:ii. O:R = R:D, by 115 (see (4) on p. xxxii), and, by similar triangles, AH:HK = 8Z:8A. Therefore 8Z:8A = R:D, whence, by V 10, ZE is a minimum. This is used in Props. 30, 40 &. 48. V 24-26 Only one minimum or maximum line can be drawn from a point on a conic to its axis. Apollonius proves separately, by reductio ad absurdum, the cases for minimum in the parabola (V 24), minimum in the hyperbola or ellipse (V 25) and maximum in the ellipse (V 26).
183 (for both sections), and VII 24 (for ellipse). I 27 If a straight line cuts the diameter of a parabola, it will, if produced, intersect the section on both sides of the diameter. This proposition, for which Apollonius gives a long and subtle proof, is not trivial, but is nevertheless omitted by Heath. It is used in V 41. I 30 In an ellipse, or the two branches of a hyperbola, any chord through the center4 is bisected at the center. This is used for the ellipse in V 71. 132 If a straight line is drawn through the vertex of a conic section parallel to an ordinate, it will be tangent to the section (d.
Apollonius: Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banū Mūsā by Gerald J. Toomer