Edmond Halley's Reconstruction of the Lost Book of Apollonius's Conics: Translation and Commentary (Sources and Studies in the History of Mathematics and Physical Sciences)
Michael N. Fried
Language: English
Pages: 134
ISBN: 1461401453
Format: PDF / Kindle (mobi) / ePub
Apollonius’s Conics was one of the greatest works of advanced mathematics in antiquity. The work comprised eight books, of which four have come down to us in their original Greek and three in Arabic. By the time the Arabic translations were produced, the eighth book had already been lost. In 1710, Edmond Halley, then Savilian Professor of Geometry at Oxford, produced an edition of the Greek text of the Conics of Books I-IV, a translation into Latin from the Arabic versions of Books V-VII, and a reconstruction of Book VIII.
The present work provides the first complete English translation of Halley’s reconstruction of Book VIII with supplementary notes on the text. It also contains 1) an introduction discussing aspects of Apollonius’s Conics 2) an investigation of Edmond Halley's understanding of the nature of his venture into ancient mathematics, and 3) an appendices giving a brief account of Apollonius’s approach to conic sections and his mathematical techniques.
This book will be of interest to students and researchers interested in the history of ancient Greek mathematics and mathematics in the early modern period.
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will be to N Ξ as Γ δ to ΔΓ , or as the difference between the axis and its latus rectum to the sum of the same. Therefore, the rectangle contained by the axis and the sum of the axis and its latus rectum is equal to the rectangle contained by N Ξ and the same difference [i.e., the difference between the axis and the latus rectum]. But the rectangle contained by the axis and the axis and latus rectum together is equal to the square on the axis and its figure together; in fact, this sum is equal
together with twice that [line] equal in square to the rectangle NM Ξ ; and so too the half-sum of the [conjugate] diameters to Θ Ξ together with the [line] equal in square to the rectangle NMΞΘ . Therefore, Θ Ξ together with that [line] equal in square to the rectangle NM Ξ is given. Let [the sum] be Θ K, from which the given [line] Θ Ξ may be removed: therefore, the given remainder is equal in square to the rectangle NM Ξ , that is (by Elements II.6),134 the difference of the squares on ΞΘ , Θ
“half ψ ” is in Halley’s text. 268 Apollonius’s Conics: Book VIII Restored 99 equal to a fourth the given sum of squares; and let EX be set up perpendicularly [to the axis]: having made Θ Π along the minor axis equal to Θ Ξ or Θ N, let N Π be joined and let EX be taken as equal to the same. Next, with center X [and] radius Ξ E let a portion of a circle be described, which cuts the axis at point M, at which MΛ is set up perpendicularly to the axis. The rest is as before. By the reverse of the
axis and its latus rectum is to MΞ , so too will the proposed sum of Indeed, EX = N Π , while sq.N Π =sq.Θ Π +sq.Θ N=2sq.Θ N=1/2sq.N Ξ . Furthermore, the radius of the circle Ξ E = ΞΘ + Θ E=1/2(N Ξ +2φ )=1/2(N Ξ +1/2ψ ). Hence, by the construction given in the 2nd lemma of the Scholium, 1/2sq.N Ξ =rect.M Ξ ,(2Ξ E − M Ξ )=rect.M Ξ ,(N Ξ +1/2ψ − M Ξ ), that is, 1/2sq.N Ξ has been applied to the line N Ξ +1/2ψ , deficient by a square. 274 Conics VII.44 states: “If there is a hyperbola, and the
ordinates, figure, and latus rectum in Apollonius’s Conics Let LAM be a conic section. A line AB which bisects all lines ML drawn parallel to a given line is called a diameter of the section and the lines ML are called ordinates (in fact, Apollonius does not use a noun, but an adverbial phrase: tetagmen¯os epi t¯en diametron kat¯exthai, “drawn in an ordinate manner to the diameter”). If the ordinates happen to be perpendicular to the diameter, then the diameter is called the axis of the section.
