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Episodes in the Mathematics of Medieval Islam by J.L. Berggren

Episodes in the Mathematics of Medieval Islam by J.L. Berggren

Episodes in the Math

Episodes in the Mathematics of Medieval Islam by J.L. Berggren

Title:

Episodes in the Mathematics of Medieval Islam

Subtitle:

With 97 Figures and 20 Plates

Author:

J.L. Berggren

ISBN 10:

0387963189

ISBN 13:

978-0387963181

Language:

English

Publisher:

Springer-Verlag

Genre/Subject:

History / Mathematics

Place of publication:

Berlin

Year Published:

December 10, 1986

Edition:

First edition

Binding:

Hardcover

Number of Pages:

197 pages

Dimensions (mm):

18 x 160 x 24 mm (0.8 x 6.5 x 9.8 inches)

Shipping Weight (g):

500 gram (1.1 pounds)

Price (SEK):

250 SEK

Description:

This book presents episodes from the mathematics of medieval Islam, work which has had a great impact on the development of mathematics. The author describes the subject in its proper historical context, referring to specific Arabic texts. Among the topics discussed are decimal arithmetic, plane and spherical trigonometry, algebra, interpolation and approximation of roots of equations. This book should be of great interest to historians of mathematics, as well as to students of mathematics. The presentation is readily accessible to anyone with a background in high school mathematics.

Episodes in the Mathematics of Medieval Islam

 

"[The] first book of its kind . . . Very interesting. It is definitely the product of a skillful mathematician who has collected over the years a reasonably large number of interesting problems from medieval Arabic mathematics. None of them is pursued to exhaustion, but all of them arranged in such a way, together with accompanying exercises, so that they would engage an active mind and introduce a subject."

 

―ZENTRALBLATT MATH

 

"This is a most scholarly book. The presentation is in the style of a textbook; each of the six chapters being followed by a set of exercises and a bibliography. … There is a good table of contents and a comprehensive index. … This is an excellent book full of information and thought-provoking ideas. It is worthy of careful study which will lead to a greater understanding of what the Islamic world has contributed to mathematics." (D.Stander, The Mathematical Gazette, Vol. 89 (515), 2005)

 

"Written in 1986 and inspired by Asger Aaboe’s classic Episodes in the Early History of Mathematics, this book contains a wealth of classroom-ready examples of much of the mathematics one finds in high school and early college … . Springer has taken the right step by issuing a paperback edition to get the book into the hands of a more general readership. … The re-issue of this gem is significant and welcomed. It will enrich your classes and deepen your perspective on mathematics and culture." (Glen van Brummelen, The MAA Mathematical Sciences Digital Library, January, 2004)

 

...as a textbook, this work is highly commendable...It is definitely the product of a skillful mathematician who has collected over the years a reasonably large number of interesting problems from medieval Arabic mathematics. None of them is pursued to exhaustion, but all of them arranged in such a way, together with accompanying exercises, so that they would engage an active mind and introduce a subject, which I am sure the author agrees with me is, at this stage, very difficult to introduce.

- G.Saliba, Zentralblatt

 

The book is, in spite of the author’s more modest claims, an introductory survey of main developments in those disciplines which were particularly important in Medieval Islamic mathematics:

Arithmetic (especially “Hindu reckoning”, including the handling of fractions and the extraction of roots), geometrical constructions (with emphasis on conic sections, verging constructions and constructions with a fixed compass opening), algebra, trigonometry (including the computational techniques in use) and spherics. All chapters include a discussion of “the Islamic dimension”, i.e., the application of mathematics to some problem of Islamic jurisprudence or custom, a set of exercises, and select, partially annotated bibliography. The presentation is mainly made through report of key works and paraphrase or quotation of central passages; the fullness thus achieved in restricted space can be illustrated by the subjects covered in only 28 pages on algebra: Presentation of the problem of unknown quantities and of the Greek and Indian background (including essential

details from Elements II); al-Khwarizmı’s basic ideas and his algorithm and proof for the case x2 +21 = 10x; Thabit’s Euclidean demonstrations; Abu Kamil’s advances, e.g., in the treatment of combined expressions, and a sketch of a refined combination of the principle of “false position” with normal algebra; al-Karajı’s beginnings of an arithmetization of algebra and al-Samaw-al’s full treatment of polynomials; and ,Umar al-Khayyamı’s classification of all equations of degree 1, 2 and 3, with a paraphrase of his solution and discussion of the case x3 +mx = n; an example from al-Khwarizmı’s algebra of inheritance; and a set of exercises.

In cases where modern symbolism is used in paraphrases the style of the original argument is always explained, and anachronisms are nowhere to be found. Broad historical descriptions are avoided, but enough background in biographical information is always given to put things into context. No knowledge of mathematics (or of the history of mathematics) beyond normal highschool level is presupposed, and everything required beyond that (be it Apollonian theory of conics or the definitions of celestial circles) is explained carefully and clearly. Scattered throughout the work are a number of lucid remarks on the character of Islamic mathematics or of mathematical work in general. The book will hence not only be an excellent textbook for the teaching of the history of mathematics but also for the liberal art aspect of mathematics teaching in general.

Reviewed by Jens Høyrup

Contents:

Chapter 1. Introduction

§1. The Beginnings of Islam

§2. Islam's Reception of Foreign Science

§3. Four Muslim Scientists

     .- Introduction

     .- Al-Khwarizmi

     .- Al-Biruni

     .- 'Umar al-Khayyami

     .- Al-Kashi

§4. The Sources

§5. The Arabic Language and Arabic Names

     .- The Language

     .- Transliterating Arabic

     .- Arabic Names

Exercises

Bibliography

1

1

2

5

5

6

9

12

15

21

24

24

25

25

26

27

Chapter 2. Islamic Arithmetic

§1. The Decimal System

§2. Kushyar's Arithmetic

.- Survey of The Arithmetic

.- Addition

.- Subtraction

.- Multiplication

.- Division

§3. The Discovery of Decimal Fractions

§4. Muslim Sexagesimal Arithmetic

.- History of Sexagesimals

.- Sexagesimal Addition and Subtraction

.- Sexagesimal Multiplication

.- Multiplication by Levelling

.- Multiplication Tables

.- Methods of Sexagesimal Multiplication

.- Sexagesimal Division

§5. Square Roots

Intruduction

.- Obtaining Approximate Square Roots

.- Justifying the Approximation

.- Justifying the Fractional Part

.- Justifying the Integral Part

§6. Al-Kashi's Extraction of a Fifth Root

.- Laying Out the Work

.- The Procedure for the First Two Digits

.- Justification for the Procedure

.- The Remaining Procedure

.- The Fractional Part of the Root

§7. The Islamic Dimension: Problems of Inheritance

.- The First Problem of Inheritance

.- The Second Problem of Inheritance

.- On the Calculation of Zakat

.- Exercises

Bibliography

29

29

31

31

32

33

24

35

36

39

39

42

43

43

44

45

47

48

48

49

50

50

52

53

53

54

54

57

60

61

63

63

65

67

68

Chapter 3. Geometrical Constructions in the Islamic World

§1. Euclidean Constructions

§2. Greek Sources for Islamic Geometry

§3. Apollonios' Theory of the Conics

.- Symptom of the Parabola

.- Symptom of the Hyperbola

§4. Abu Sahl on the Regular Heptagon

.- Archimedes' Construction of the Regular Heptagon

.- Abu Sahl's Analysis

.- First Reduction: From Heptagon to Triangle

.- Second Reduction: From Triangle to Division of Line Segment

.- Third Reduction: From the Divided Line Segment to Conic Sections

§5. The Construction of the Regular Nonagon

.- Verging Constructions

.- Fixed Versus Moving Geometry

.- Abu Sahl's Trisection of the Angle

§6. Construction of the Conic Sections

.- Life of Ibrahim b. Sinan

.- Ibrahim b. Sinan on the Parabola

.- Ibrahim b. Sinan on the Hyperbola

§7. The Islamic Dimension: Geometry with a Rusty Compass

.- Problem 1

.- Problem 2

.- Problem 3

.- Problem 4

.- Problem 5

.- Exercises

Bibliography

70

70

72

74

75

76

77

77

78

79

80

81

82

82

84

84

85

86

88

89

92

92

93

93

94

96

96

98

chapter 4. Algebra in Islam

§1. Problems About Unknown Quantities

§2. Sources of Islamic Algebra

§3. Al-Khwarizmi's Algebra

.- The Name "Algebra"

.- Basic Ideas in Al-Khwarizmi's Algebra

.- Al-Khwarizmi's Discussion of x2 + 21 = 10x

§4. Thabit's Demonstration for Quadratic Equations

.- Preliminaries

.- Thabit's Demonstration

§5. Abu Kamil on Algebra

.- Similarities with al-Khwarizmi

.- Advances Beyond al-Khwarizmi

.- A Problem from Abu Kamil

§6. Al-Karaji's Arithmetization of Algebra

.- Al-Samaw'al on the Law of Exponents

.- Al-Samaw'al on the Division of Polynomials

.- The First Example

.- The Second Example

§7. 'Umar al-Khayyami and the Cubic Equation

.- The Background to 'Umar's Work

.- 'Umar's Classification of Cubic Equations

.- 'Umar's Treatment of x3 + mx = n

.- Preliminaries

.- The Main Discussion

.- 'Umar's Discussion of the Number of Roots

§8. The Islamic Dimension: The Algebra of Legacies

.- Exercises

Bibliography

99

99

100

102

102

102

103

104

104

106

108

108

108

110

111

111

113

115

115

117

118

118

119

120

121

122

124

125

126

Chapter 5. Trigonometry in the Islamic World

§1. Ancient Background: The Table of Chords and the Sine

§2. The Introduction of the Six Trigonometric Functions

§3. Abu l-Wafa's Proof of the Addition Theorem for Sines

§4. Nasir al-Din's Proof of the Sine Law

§5. Al-Biruni's Measurement of the Earth

§6. Trigonometric Tables: Calculation and Interpolation

§7. Auxiliary Functions

§8. Interpolation Procedures

.- Linear Interpolation

.- Ibn Yunus' Second-Order Interpolation Scheme

§9. Al-Kashi's Approximation to Sin(1°)

.- Exercises

Bibliography

127

127

132

135

138

141

144

144

145

146

148

151

154

156

Chapter 6. Spherics in the Islamic World

§1. The Ancient Background

§2. Important Circles on the Celestial Sphere

§3. The Rising Times of the Zodiacal Signs

§4. Stereographic Projection and the Astrolabe

§5. Telling Time by Sun and Stars

§6. Spherical Trigonometry in Islam

§7. Tables for Spherical Astronomy

§8. The Islamic Dimension: The Direction of Prayer

Exercises

Bibliography

157

157

161

164

166

172

174

177

182

186

188

Index

186

About the Author(s):

John Lennart Berggren is professor in the Department of Mathematics on Simon Fraser University, Burnaby, Canada. His activities centre on the history of mathematics, so included here is information related to my teaching and research in this area. His research interests in the history of mathematics include ancient Greece and medieval Islam, as well as the history of such scientific instruments as the sundial and the astrolabe.

http://people.math.sfu.ca/~berggren/main.html

Pertinent URL:

http://akira.ruc.dk/~jensh/publications/1987%7BR%7D05_Berggren_Episodes.pdf

 

If you have any questions or need additional information, please feel free to contact Mezerah:

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