A Mathematical History of the Golden Number (eBook)

PREFACE TO THE DOVER EDITION FOREWORD A GUIDE FOR READERS A. Internal Organization B. Bibliographical Details C. Abbreviations D. Symbols E. Dates F. Quotations from Primary Sources INTRODUCTION CHAPTER I. THE EUCLIDEAN TEXT Section 1. The Text Section 2. An Examination of the Euclidean Text A. Preliminary Observations B. A Proposal Concerning the Origin of DEMR C. "Theorem XIII,8" D. "Theorems XIII,1-5" E. Stages in the Development of DEMR in Book XIII CHAPTER II. MATHEMATICAL TOPICS Section 3. Complements and the Gnomon Section 4. Transformation of Areas "Section 5. Geometrical Algebra, Application of Areas, and Solutions of Equations" A. Geometrical Algebra-Level 1 B. Geometrical Algebra-Level 2 C. Application of Areas-Level 3 D. Historical References E. Setting Out the Debate F. Other Interpretations in Terms of Equations G. Problems in Interpretation H. Division of Figures I. "Theorems VI,28,29 vs II,5,6" J. Euclid's Data K. "Theorem II,11" L. "II,11-Application of Areas, Various Views" i. Szabó ii. Junge iii. Valabrega-Gibellato Section 6. Side and Diagonal Numbers Section 7. Incommensurability "Section 8. The Euclidean Algorithm, Anthyphairesis, and Continued Fractions" "CHAPTER III. EXAMPLES OF THE PENTAGON, PENTAGRAM, AND DODECAHEDRON BEFORE -400" Section 9. Examples before Pythagoras (before c. -550) A. Prehistoric Egypt B. Prehistoric Mesopotamia C. Sumerian and Akkadian Cuneiform Ideograms i. Fuÿe's Theory D. A Babylonian Approximation for the Area of the Pentgon i. Stapleton's Theory E. Palestine Section 10. From Pythagoras until -400 A. "Vases from Greece and its Italian Colonies, Etruria (Italy)" B. Shield Devices on Vases C. Coins D. Dodecahedra E. Additional Material Conclusions CHAPTER IV. THE PYTHAGOREANS i. Pythagoras ii. Hippasus iii. Hippocrates of Chios iv. Theodorus of Cyrene v. Archytas Section 11. Ancient References to the Pythagoreans A. The Pentagram as a Symbol of the Pythagoreans B. The Pythagoreans and the Construction of the Dodecahedron C. Other References to the Pythagoreans Section 12. Theories Linking DEMR with the Pythagoreans i. The Pentagram ii. Scholia assigning Book IV to the Pythagoreans iii. Equations and Application of Areas iv. The Dodecahedron v. A Marked Straight-Edge Construction of the Pentagon vi. A Gnomon Theory vii. Allman's Theory: The Discovery of Incommensurability viii. Fritz-Junge Theory: The Discovery of Incommensurability ix. Heller's Theory: The Discovery of DEMR x. Neuenschwander's Analysis xi. Stapleton CHAPTER V. MISCELLANEOUS THEORIES Section 13. Miscellaneous Theories i. Michel ii. Fowler: Anthyphairesis Development of DEMR iii. Knorr: Anthyphairesis and DEMR iv. "Itard: Theorem IX,15" "Section 14. Theorems XIII,1-5" i. Bretschneider ii. Allman iii. Michel iv. Dijksterhuis and Van der Waerden v. Lasserre vi. Fritz vii. Knorr viii. Heiberg ix. Herz-Fischler CHAPTER VI. THE CLASSICAL PERIOD: FROM THEODORUS TO EUCLID Section 15. Theordorus i. Knorr ii. Mugler Section 16. Plato A. Plato as a Mathematician B. Mathematical Influence of Plato C. Plato and DEMR D. Passages from Plato i. The Dodecahedron in Phaedo 110B and Timaeus ii. "The "Divided Line" in the Republic 509D" iii. Timaeus 31B iv. Hippias Major 303B Section 17. Leodamas of Thasos Section 18. Theaetetus A. The Life of Theaetetus B. The Contributions of Theaetetus i. Tannery ii. Allman iii. Sachs iv. Van der Waerden v. Bulmer-Thomas vi. Waterhouse vii. Neuenschwander Section 19. Speusippus Section 20 Eudoxus A. "Interpreting "Section" i. Bretschneider ii. Tannery iii. Tropfke iv. Michel v. Gaiser vi. Burkert vii. Fowler B. Contributions of Eudoxus to the Development of DEMR i. Bretschneider ii. Allman iii. Sachs iv. Van der Waerden v. Lasserre vi. Knorr C. Commentary Section 21. Euclid Section 22. Some Views on the Historical Development of DEMR A. A Summary of Various Theories i. Equations and Appliction of Areas ii. Incommensurability iii. "Similar Triangles Development Based on XIII,8" iv. Anthyphairesis B. Summary of My Conclusions C. A Chronological Proposal D. A Proposal Concerning a Name CHAPTER VII. THE POST-EUCLIDEAN GREEK PERIOD (c -300 to 350) Section 23. Archimedes A. Approximations to the Circumference of a Circle B. Broken Chord Theorem C. Trigonometry Section 24. The Supplement to the Elements A. The Text B. Questions of Authorship C. Chronology Section 25. Hero A. Approximations for the Area of the Pentagon and Decagon i.. The Area of the Pentagram ii. The Area of the Decagon iii. The Diamenter of the Circumscribed Circle of a Pentagon iv. Commentaries B. "A Variation on II,11" C. The Volumes of the Icosahedron and Dodecahedron i. The Text ii. Commentary Section 26. Ptolemy A. The Chords of 36° and 72° in Almagest B. Chord (108°)/Diameter in Geography C. Trigonometry before Ptolemy Section 27. Pappus A. Construction of the Icosahedron and Dodecahedron B. Comparison of Volumes "CHAPTER VIII. THE ARABIC WORLD, INDIA, AND CHINA" Section 28. The Arabic Period i. Authors Consulted ii. Equations A. Al-Khwarizmi i. Algebra ii. Predecessors of al-Khwarizmi B. Abu Kamil i. On the Pentagon and Decagon ii. Algebra C. Abu'l-Wafa' D. Ibn Yunus E. Al-Biruni i. The Book on the Determination of Chords in a Circle ii. Canon Masuidius Section 29. Indix Section 30. China CHAPTER IX. EUROPE: FROM THE MIDDLE AGES THROUGH THE EIGHTEENTH CENTURY Section 31. Europe Through the 16th Century A. Authors Consulted i. The Middle Ages ii. Versions of the Elements and Scholia iii. Italy from Fibonacci through the Renaissance iv. 16th Century Non-Italian Authors v. Pre-1600 Numerical Approximations to DEMR vi. Fixed Compass and Straight-Edge Constructions vii. Approximate Constructions of the Pentagon B. Fibonacci i. Plannar Calculations ii. Volume Computations of the Dodecahedron and Icosahedron iii. Fibonacci and Abu Kamil iv. Equations from Abu Kamil's Algebra v. "The Rabbit Problem, Fibonacci Numbers" vi. Summary C. Francesca D. Paccioli E. Cardano F. Bombelli G. Candalla H. Ramus I. Stevin J. Pre-1600 Numerical Approximations to DEMR i. Unknown Annotator to Paccioli's Euclid ii. Holtzmann iii. Mästlin K. Approximate Constructions of the Pentagon Section 32. The 17th and 18th Centuries A. Kepler i. Magirus-The Right Triangle with Proportional Sides ii. Fibonacci Approximations to DEMR B. The Fibonacci Sequence C. Fixed Compass and Compass Only Constructions i. Mohr ii. Mascheroni By Way of a Conclusion "APPENDIX I. "A PROPORTION BY ANY OTHER NAME": TERMINOLOGY FOR DIVISION IN EXTREME AND MEAN RATIO THROUGHOUT THE AGES" A. "Extreme and Mean Ratio" B. "Middle and Two Ends" C. Names for DEMR "APPENDIX II."MIRABLIS...EST POTENTIA...": THE GROWTH OF AN IDEA" CORRECTIONS AND ADDITIONS BIBLIOGRAPHY