By Gerald J. Toomer
With the book of this e-book 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 is the nearest we need to the unique, the Arabic model of the Banu Musil. Un til now this has been available in basic terms in Halley's Latin translation of 1710 (and translations into different languages fullyyt depending on that). whereas I yield to none in my admiration for Halley's version of the Conics, it truly is faraway from enjoyable the necessities of contemporary scholarship. specifically, it doesn't comprise the Arabic textual content. i am hoping that the current version won't simply treatment these deficiencies, yet also will function a beginning for the learn of the impression 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 origin, by way of the award of 1 of its Fellowships for 1985-86, enabled me to dedicate an unbroken yr 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 reliable use of the wealthy assets of either the college Library, Cambridge, and the Bodleian Library.
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Additional info for Apollonius: Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banū Mūsā
Hence NE:QE = :::N:NII, and (addenda or dividendo) NQ:QE = :::II:NII = ep:PM. Therefore BE is a minimum only if (ep:PM) equals the ratio of transverse diameter to latus rectum. If, then, 9P:PM = NO:OM = D:R, the lines BE and rz are minima, but no other line passing through 9 can be a minimum. For in the case of points like "'>I, which lie between rand B, the auxiliary hyperbola passes below them, hence "'>I~·~T > 9p·PT Id. p. 1:Jl'I' < D:R. And for points like K, which lie outside of rB, the auxiliary hyperbola passes above, hence Ky·yT < 9p·PT Id.
12), then the minimum from any other point, Il, on that line is the portion of the original minimum, IlA, and other lines from Il to the curve on either side of M increase as their distance from M increases. Proven by considering the angles in the pairs of triangles ilEA, lEA and IlZE, iZE etc. This proposition, which is presumably inserted for the sake of completeness, is the only one in the book dealing with minima from a point above the axis;! hence it is never used again. v 13 The converse of V 8.
The proof, which is not trivial, uses reductio ad absurdum. Used in VI 25. d. Book V: For a deep analysis of the central topic of Book V (which culminates in Props. 51 & 52), the reader should consult the admirable discussion of Zeuthen, Kegelschnitte pp. 284-309, where it is treated from a modern viewpoint. Here I give only a brief analysis of the content and interconnections of the individual propositions. 342). Book V; summary of V 1-3 xxxix the book may be stated as follows. For a given conic section and a given point P: (1) to construct all minimum and/or maximum straight lines from P to the conic; (2) to determine how the distance between P and a variable point X on the conic changes as PX moves away from the position of minimum or maximum.