25 Books to Read Before You Die
 
 

Recently Viewed clear list


The Powell's Playlist | August 6, 2014

Graham Joyce: IMG The Powell’s Playlist: Graham Joyce



The Ghost in the Electric Blue Suit is set on the English coast in the hot summer of 1976, so the music in this playlist is pretty much all from the... Continue »
  1. $17.47 Sale Hardcover add to wish list

spacer
Qualifying orders ship free.
$17.95
New Trade Paper
Ships in 1 to 3 days
Add to Wishlist
available for shipping or prepaid pickup only
Available for In-store Pickup
in 7 to 12 days
Qty Store Section
25 Remote Warehouse Science Reference- General

A Mathematical History of the Golden Number

by

A Mathematical History of the Golden Number Cover

 

Synopses & Reviews

Publisher Comments:

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.

Book News Annotation:

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 Elements, 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. Book News, Inc., Portland, OR (booknews.com)

Synopsis:

This comprehensive study traces the historic development of division in extreme and mean ratio ("the golden number") from its first appearance in Euclid's Elements through the 18th century. Features numerous illustrations.

Synopsis:

This comprehensive study traces the historic development of division in extreme and mean ratio ("the golden number") from its first appearance in Euclid's Elements through the 18th century. Features numerous illustrations.

Synopsis:

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 appearance in Euclid's ELEMENTS through the 18th century. Of interest not only to mathematicians but also classicists, archaeologists, historians of science, or anyone interested in mathematical ideas.

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

Product Details

ISBN:
9780486400075
Author:
Herz-Fischler, Roger
Publisher:
Dover Publications
Author:
Mathematics
Author:
Herz-Fischler, Roger
Location:
Mineola, N.Y. :
Subject:
General
Subject:
History
Subject:
Number Theory
Subject:
Ratio and proportion
Subject:
Ratio and proportion -- History.
Subject:
General Mathematics
Subject:
Science Reference-General
Edition Number:
Dover ed.
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
75
Publication Date:
19980131
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
216
Dimensions:
11 x 8.38 in 1.11 lb

Other books you might like

  1. Outwitting the Gestapo Used Book Club Hardcover $5.50
  2. Puzzles Old & New: How to Make &... Used Trade Paper $14.95
  3. Violence & Gender Reexamined (Law... New Hardcover $27.25
  4. Art and Geometry: A Study in Space... Used Trade Paper $3.95
  5. Spirituality by the numbers Used Trade Paper $7.95
  6. Life Before Us (75 Edition)
    Used Trade Paper $8.00

Related Subjects

Children's » General
Reference » Science Reference » General
Science and Mathematics » History of Science » General
Science and Mathematics » Mathematics » History
Science and Mathematics » Mathematics » Number Theory
Sports and Outdoors » Outdoors » General
Travel » Travel Writing » General

A Mathematical History of the Golden Number New Trade Paper
0 stars - 0 reviews
$17.95 In Stock
Product details 216 pages Dover Publications - English 9780486400075 Reviews:
"Synopsis" by ,
This comprehensive study traces the historic development of division in extreme and mean ratio ("the golden number") from its first appearance in Euclid's Elements through the 18th century. Features numerous illustrations.

"Synopsis" by ,
This comprehensive study traces the historic development of division in extreme and mean ratio ("the golden number") from its first appearance in Euclid's Elements through the 18th century. Features numerous illustrations.

"Synopsis" by , 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 appearance in Euclid's ELEMENTS through the 18th century. Of interest not only to mathematicians but also classicists, archaeologists, historians of science, or anyone interested in mathematical ideas.
spacer
spacer
  • back to top
Follow us on...




Powell's City of Books is an independent bookstore in Portland, Oregon, that fills a whole city block with more than a million new, used, and out of print books. Shop those shelves — plus literally millions more books, DVDs, and gifts — here at Powells.com.