By Gerald J. Toomer
With the book of this publication 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 purely 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's faraway from gratifying the necessities of contemporary scholarship. specifically, it doesn't comprise the Arabic textual content. i'm hoping that the current version won't in basic terms therapy these deficiencies, yet also will function a beginning for the learn of the impact of the Conics within the medieval Islamic global. I recognize with gratitude assistance from a couple of associations and other people. the toilet Simon Guggenheim Memorial starting place, by means of the award of 1 of its Fellowships for 1985-86, enabled me to dedicate an unbroken 12 months to this undertaking, 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 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
G. Props. 25 & 28). If ~Br is the axis of a hyperbola, and ~ the center, it follows immediately from V 34 that the transverse diameters in the hyperbola increase in length as their distance from the transverse axis increases. Apollonius uses this, both implicitly and explicitly, in Book VII (see n. 107 on p. 603), but the Banli Mlisa never cite this proposition. V 35-40 deal with the intersection of minima. These are a necessary preliminary to V 44 ff. (which deal with drawing minima from the same point on the opposite side of the axis), although they are taken for granted rather than cited explicitly there.
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. For Figs. 45, the hyperbola in question passes through points Band r, and is determined by the asymptotes OT, TY; it also passes through the center N and the given point e.
I:d = d:r. (2) (3) in Book VII: (1) in Props. 24, summary following Prop. 31 implicitly in V 11. Furthern. 84 on p. 558) (4) This is used in V 23, VII 37 and VII 47. I 16 The fundamental proposition on conjugate diameters in the hyperbola. See Fig. 16 *. HA, aB are opposite branches of a hyperbola with common transverse diameter AB. If through the center r a line, :::rl\, is drawn parallel to the ordinates to diameter AB, that line will be a diameter conjugate with AB. This is proven by showing that it bisects any line He drawn parallel to the original diameter.
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