Download e-book for kindle: Textual Studies in Ancient and Medieval Geometry by Wilbur Richard Knorr (auth.)

By Wilbur Richard Knorr (auth.)

For textual reviews on the subject of the traditional mathematical corpus the efforts by means of the Danish philologist, 1. L. Heiberg (1854-1928), are particularly major. starting along with his doctoral dissertation, Quaestiones Archimedeae (Copen­ hagen, 1879), Heiberg produced an striking sequence of variations and important stories that stay the basis of scholarship on Greek mathematical four technological know-how. For comprehensiveness and accuracy, his versions are exemplary. In his textual experiences, as additionally within the prolegomena to his versions, he rigorously defined the extant facts, equipped the manuscripts into stemmata, and drew out the consequences for the kingdom of the textual content. five with reference to his Archimedean paintings, Heiberg occasionally betrayed indicators of the philologist's occupational disorder - the tendency to rewrite a textual content deemed on subjective grounds to be unworthy. 6 yet he did so much less usually than his admired 7 contemporaries, and never as to detract extensively from the price of his versions. In analyzing textual questions relating the Archimedean corpus, he tried to take advantage of up to attainable facts from the traditional commentators, and in a few situations from the medieval translations. it's right here that possibilities abound for brand spanking new paintings, extending, and in a few circumstances superseding, Heiberg's findings. For at his time the provision of the medieval fabrics used to be restricted. lately Marshall Clagett has accomplished a substantial serious version of the medieval Latin culture of Archimedes,8 whereas the bibliographical tools for the Arabic culture are in strong order due to the paintings of Fuat Sezgin.

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Thus, the (rectangle) under AE, EG with the (square) on ZE is equal to the (rectangle) under DE, EB with the (square) onZE. Let the (square) on ZEbe subtracted in common. Thus, the remainder, the rectangle contained by AE, EG is equal to the rectanglecontained by DE, EB . 22 2. 4b [q] III 36: If there be taken a point outside a circle, and from it [sc. the point] two lines fall on the circle, and one of them cuts the circle, the other touches it, the (rectangle) under the whole secant (line) and the outside (part of it) intercepted between the point and the convex arc (of the circle) will be equal to the square of the tangent.

In particular, the project of classifying geometric problems and assigning priority to those solutions of simpler type, as is implicit in Apollonius' studies of planar constructions, would obscure the point of Nicomedes' application of the concho ids to the cube duplication. Apollonius, like Menaechmus and Diocles before him, perceived how to solve this problem by "solid" means, that is, via the intersection of conics. Nicomedes' method, by contrast, is of the higher "linear" type, since it makes use of the motion-generated conchoid curve.

It is at the same time clear how form (b), the version in Aj, is an equivalent alternative. If we now consider how to complete the synthesis, one route might be to introduce the circle cirumscribing ABGI, to which lines EBA, DBG are secants, whence it can be shown that AE'EB = GD'DB, by reversing the sequence in the analysis. That will entail establishing the identities AE'EB + BK2 = EK2 and GD·DB + BK2 = DK2, which the analysis derives from the secants to the 38 I. Ancient Texts on Geometric Problems circle.

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