A Mathematical History of the Golden Number

by Roger Herz-Fischler

Overview

The first complete, in-depth study of the origins of division in extreme and mean ratio (DEMR)-"the Golden Number"-this text charts every aspect of this important mathematical concept's historic development, from its first unequivocal appearance in Euclid's Elements through the 18th century.
Readers will find a detailed analysis of the role of DEMR in the Elements and of its historical implications. This is followed by a discussion of other mathematical topics and of proposals by modern commentators concerning the relationship of these concepts to DEMR. Following chapters discuss the Pythagoreans, examples of the pentagram before 400 H.C., and the writings of pre-Euclidean mathematicians. The author then presents his own controversial views on the genesis, early development and chronology of DEMR. The second half of the book traces DEMR's post-Euclidean development through the later Greek period, the Arabic world, India, and into Europe. The coherent but rigorous presentation places mathematicians' work within the context of their time and dearly explains the historical transmission of their results. Numerous figures help clarify the discussions, a helpful guide explains abbreviations and symbols, and a detailed appendix defines terminology for DEMR through the ages.
This work will be of interest not only to mathematicians but also to classicists, archaeologists, historians of science and anyone interested in the transmission of mathematical ideas. Preface to the Dover Edition. Foreword. A Guide for Readers. Introduction. Appendixes. Corrections and Additions. Bibliography.

Product Details

ISBN-13:	9780486400075
Publisher:	Courier Corporation
Publication date:	01/29/1998
Series:	Dover Books on Mathematics Series
Edition description:	Unabridged
Pages:	224
Product dimensions:	8.28(w) x 11.20(h) x 0.45(d)

Table of Contents

 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

Editorial Reviews

An in-depth study of the origins of division in extreme and mean ratio (DEMR). Provides a detailed analysis of the role of DEMR in Euclid's , historical implications, and its relationship to other mathematical topics and modern proposals. Also discusses the Pythagoreans, examples of the pentagram before 400 B.C., pre-Euclidean wrings, and post-Euclidean development through the later Greek period, the Arabic world, India, and into Europe. Annotation c. by Book News, Inc., Portland, Or.

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