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 thoughtprovoking 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 classroomready 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 reissue 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); alKhwarizmı’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; alKarajı’s beginnings of an arithmetization of algebra and alSamawal’s full treatment of polynomials; and ,Umar alKhayyamı’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 alKhwarizmı’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

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. AlKashi'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

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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

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chapter 4. Algebra in Islam
§1. Problems About Unknown Quantities
§2. Sources of Islamic Algebra
§3. AlKhwarizmi's Algebra
. The Name "Algebra"
. Basic Ideas in AlKhwarizmi's Algebra
. AlKhwarizmi's Discussion of x2 + 21 = 10x
§4. Thabit's Demonstration for Quadratic Equations
. Preliminaries
. Thabit's Demonstration
§5. Abu Kamil on Algebra
. Similarities with alKhwarizmi
. Advances Beyond alKhwarizmi
. A Problem from Abu Kamil
§6. AlKaraji's Arithmetization of Algebra
. AlSamaw'al on the Law of Exponents
. AlSamaw'al on the Division of Polynomials
. The First Example
. The Second Example
§7. 'Umar alKhayyami 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

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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 lWafa's Proof of the Addition Theorem for Sines
§4. Nasir alDin's Proof of the Sine Law
§5. AlBiruni's Measurement of the Earth
§6. Trigonometric Tables: Calculation and Interpolation
§7. Auxiliary Functions
§8. Interpolation Procedures
. Linear Interpolation
. Ibn Yunus' SecondOrder Interpolation Scheme
§9. AlKashi's Approximation to Sin(1°)
. Exercises
Bibliography

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