A mathematical history of division in extreme and mean ratio / Roger Herz-Fischler.

Format:

Book

Author/Creator:

Herz-Fischler, Roger, 1940-

Language:

English

Subjects (All):

Ratio and proportion--History.

Ratio and proportion.

Proportion.

Physical Description:

1 online resource (208 p.)

Edition:

1st ed.

Place of Publication:

Waterloo, Ont., Canada : Wilfrid Laurier University Press, c1987.

Language Note:

English

Contents:

Intro

TABLE OF CONTENTS

PREFACE

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D. Historical References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

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

D. A Babylonian Approximation for the Area of the Pentagon

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: An Anthyphairesis Development of DEMR

iii. Knorr: Anthyphairesis and DEMR

iv. Hard: 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. Theodorus

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

Section 17. Leodamas of Thasos

Section 18. Theaetetus

A. The Life of Theaetetus

B. The Contributions of Theaetetus

Section 19. Speusippus

Section 20. Eudoxus

A. Interpreting "Section

B. Contributions of Eudoxus to the Development of DEMR

C. Commentary

Section 21. Euclid

Section 22. Some Views on the Historical Development of DEMR

A. A Summary of Various Theories

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

B. A Variation on II,II

C. The Volumes of the Icosahedron and Dodecahedron

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

A. Al-Khwarizmi

B. Abu Kamil

C. Abu'l-Wafa'

D. Ibn Yunus

E. Al-Biruni

Section 29. India

Section 30. China

CHAPTER IX. EUROPE: FROM THE MIDDLE AGES THROUGH THE EIGHTEENTH CENTURY

Section 31. Europe Through the 16th Century

A. Authors Consulted

B. Fibonacci

C. Francesca

D. Paccioli

E. Cardano

F. Bombelli

G. Candalla

H. Ramus

I. Stevin

J. Pre-1600 Numerical Approximations to DEMR

K. Approximate Constructions of the Pentagon

Section 32. The 17th and 18th Centuries

A. Kepler

B. The Fibonacci Sequence

C. Fixed Compass and Compass Only Constructions

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

BIBLIOGRAPHY.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Bibliography: p. 176-191.

ISBN:

0-88920-777-1

OCLC:

243586220