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Textual Studies in Ancient and Medieval Geometry

For textual studies relating to the ancient mathematical corpus the efforts by the Danish philologist, 1. L. Heiberg (1854-1928), are especially significant. Beginning with his doctoral dissertation, Quaestiones Archimedeae (Copen- hagen, 1879), Heiberg produced an astonishing series of editions and critical studies that remain the foundation of scholarship on Greek mathematical 4 science. For comprehensiveness and accuracy, his editions are exemplary. In his textual studies, as also in the prolegomena to his editions, he carefully described the extant evidence, organized the manuscripts into stemmata, and drew out the implications for the state of the text. 5 With regard to his Archimedean work, Heiberg sometimes betrayed signs of the philologist's occupational disease - the tendency to rewrite a text deemed on subjective grounds to be unworthy. 6 But he did so less often than his prominent 7 contemporaries, and not as to detract appreciably from the value of his editions. In examining textual questions bearing on the Archimedean corpus, he attempted to exploit as much as possible evidence from the ancient commentators, and in some instances from the medieval translations. It is here that opportunities abound for new work, extending, and in some instances superseding, Heiberg's findings. For at his time the availability of the medieval materials was limited. In recent years Marshall Clagett has completed a mammoth critical edition of the medieval Latin tradition of Archimedes,8 while the bibliographical instruments for the Arabic tradition are in good order thanks to the work of Fuat Sezgin.
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Table of Contents

Introduction: Philologist, Heal Thy Text.- I Ancient Texts on Geometric Problems.- 1 The Hero-Apollonius Method of Cube Duplication.- 2 The Hero-Apollonius Lemma in Nicomedes and Euclid.- 3 The Philonian Method of Cube Duplication.- Appendix: Philoponus' Account of Cube Duplication.- 4 Pappus' Texts on Cube Duplication.- 5 Eutocius' Anthology of Cube Duplications.- The Platonic Construction.- The Methods of Hero, Philo, and Apollonius.- The Method of Diodes.- The Methods of Pappus and Sporus.- The Method of Menaechmus.- The Method of Archytas.- The Methods of Eratosthenes and Nicomedes.- Summary.- 6 Eutocius' Text of Eratosthenes: A Thesis of U. von Wilamowitz.- Appendix: Two Accounts of Eratosthenes' Method.- 7 On Eutocius: A Thesis of J. Mogenet.- Appendix: Four Texts on Compound Ratio.- 8 Angle Trisections in Pappus and Arabic Parallels.- 9 The Ancient Commentators and Their Methods: Pappus and Eutocius.- II Arabic Geometric Texts and Their Ancient Sources.- A The Cube Duplication by Abu Bakr al-Harawi.- B The Angle Trisection by Ahmad ibn Musa.- C The Angle Trisection by Thabit ibn Qurra.- D The Angle Trisection by al-Sijzi.- E The Cube Duplication and Angle Trisection by Abu Sahl al-Quhi.- F The Cube Duplication by Abu Jacfar in the Manner of Nicomedes.- Appendix: Texts in Transcription.- Facsimile.- III The Textual Tradition of Archimedes': Dimension of the Circle.- 1 Versions in the Ancient Commentators.- Appendix I. Archimedes' Circle Theorem in Pappus and Theon: Translations and Facsimiles.- II. Pappus' Text of the Sector Theorem: Translation and Facsimile.- 2 Origin of the Extant Text of the Dimension of the Circle.- Appendix I. The Extant Greek Text of Dimension of the Circle: Translation and Facsimile.- II. Theon's Lemma to the Circle Theorem: Translation and Facsimile.- 3 The Medieval Tradition of Dimension of the Circle, Prop. 1.- Appendix I. Translation from the Arabic Text.- II. Variant Readings.- III. On the Arabic and Hebrew Translators.- Facsimiles of the Arabic, Hebrew and Latin Texts.- 4 Versions of Dimension of the Circle, Props. 2 and 3.- Appendix I. The Ancient and Medieval Texts of Props. 2 and 3.- II. Variants in the Medieval Versions.- 5 Lost Propositions of the Archimedean Prototype.- 6 Eutocius' Text of Dimension of the Circle.- Appendix: Extracts from Eutocius and Theon.- 7 Arabic Elaborations of the Dimension of the Circle.- Banu Musa.- Abu '1-Rashid.- Al-Tusi.- I. The Version of Abu 'l-Rashid: Translation and Facsimile.- II. The Version of al-Tusi: Translation and Facsimile.- 8 The Latin Tradition: De curvis superficiebus.- 9 The Latin Tradition: De quadratura circuli.- The Florence Version.- The Cambridge Version.- The Naples Version.- The Gordanus Version.- The Munich Version.- The Vatican, or ps.-Bradwardine, Version.- The Abbreviated Version.- The Version by Albert of Saxony.- The Corpus Christi Version.- The Glasgow Version.- Synthesis.- 10 The Anonymous Tract On Isoperimetric Figures.- Appendix: Propositions of the Anonymous Tract.- 11 On Hypatia of Alexandria.- Appendix I. Four Accounts of Division from Theon's Commentary on Ptolemy.- II. Pappus' Method of Long Division.- Facsimiles from Theon.- 12 The History of a Text: Tradition, Time and Opportunity.

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