By Gerald J. Toomer
With the e-book of this publication I discharge a debt which our period has lengthy owed to the reminiscence of an outstanding 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 available merely 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's faraway from gratifying the necessities of contemporary scholarship. particularly, it doesn't include the Arabic textual content. i'm hoping that the current variation won't simply treatment these deficiencies, yet also will function a origin for the research of the effect of the Conics within the medieval Islamic international. I recognize with gratitude assistance from a couple of associations and other people. the loo Simon Guggenheim Memorial origin, via the award of 1 of its Fellowships for 1985-86, enabled me to dedicate 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 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
See Fig. 37, in which M is a minimum and rz, rH the asymptotes. Then LA~r < LrZH. Proven using II 3 (which states that if Be equals half the latus rectum, ZB2 = rB· Bel, the basic theorem on the minimum, and similar triangles. xlviii Summary of V 38, V 39, V 40 &. V 41-43 The theorem is a lemma necessary for the proof (in V 42) of whether a minimum in a hyperbola will meet the curve again. It is used there, but nowhere else. V 38 Two minima from the axis to a conic will meet on the other side of the axis.
It is used there, but nowhere else. V 38 Two minima from the axis to a conic will meet on the other side of the axis. This is an immediate consequence of Props. 35 &. 36. It is cited explicitly only in Prop. 40, but, as noted above, is implicit in the core theorems of the book, V 44 ff. Apollonius does not discuss the general problem of whether the meeting-point of two minima is inside or outside of the conic; however, he does treat this question for the particular case of the ellipse in V 39 &.
Then it is easily shown, by V 40, that no third minimum can be drawn to the same quadrant from the point of intersection of the original two minima. As it stands, however, the enunciation and proof are incomplete: see n. 563. This is used in Props. 47, 54, 57, 76 &. 77. V 47 considers minima drawn to an ellipse, not just to one quadrant (as in V 45), but to the whole semi-ellipse,. The theorem states that no four minima can meet in a single point. Apollonius uses reductio ad absurdum, considering three possibilities: (1) One of the lines coincides with the minor axis.