Book:J.L. Berggren/Episodes in the Mathematics of Medieval Islam

Subject Matter

 * History of Mathematics

Contents

 * $1$. Introduction


 * $\S 1$. The Beginnings of Islam


 * $\S 2$. Islam’s Reception of Foreign Science


 * $\S 3$. Four Muslim Scientists
 * Al-Khwārizmī
 * Al-Bīrūnī
 * ‘Umar al-Khayyāmī
 * Al-Kāshī


 * $\S 4$. The Sources


 * $\S 5$. The Arabic Language and Arabic Names
 * The Language
 * Transliterating Arabic
 * Arabic Names


 * Exercises


 * $2$. Islamic Arithmetic


 * $\S 1$. The Decimal System


 * $\S 2$. Kūshyār’s Arithmetic
 * Survey of The Arithmetic
 * Addition
 * Subtraction
 * Multiplication
 * Division


 * $\S 3$. The Discovery of Decimal Fractions


 * $\S 4$. Muslim Sexagesimal Arithmetic
 * History of Sexagesimals
 * Sexagesimal Addition and Subtraction
 * Sexagesimal Multiplication
 * Multiplication by Levelling
 * Multiplication Tables
 * Methods of Sexagesimal Multiplication
 * Sexagesimal Division


 * $\S 5$. Square Roots
 * Obtaining Approximate Square Roots
 * Justifying the Approximation
 * Justifying the Fractional Part
 * Justifying the Integral Part


 * $\S 6$. Al-Kāshī’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


 * $\S 7$. The Islamic Dimension: Problems of Inheritance
 * The First Problem of Inheritance
 * The Second Problem of Inheritance
 * On the Calculation of Zakāt


 * Exercises


 * $3$. Geometrical Constructions in the Islamic World


 * $\S 1$. Euclidean Constructions


 * $\S 2$. Greek Sources for Islamic Geometry


 * $\S 3$. Apollonios’ Theory of the Conics
 * Symptom of the Parabola
 * Symptom of the Hyperbola


 * $\S 4$. Abū Sahl on the Regular Heptagon
 * Archimedes’ Construction of the Regular Heptagon
 * Abū 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


 * $\S 5$. The Construction of the Regular Nonagon
 * Verging Constructions
 * Fixed Versus Moving Geometry
 * Abū Sahl’s Trisection of the Angle


 * $\S 6$. Construction of the Conic Sections
 * Life of Ibrāhīm b. Sinān
 * Ibrāhīm b. Sinān on the Parabola
 * Ibrāhīm b. Sinān on the Hyperbola


 * $\S 7$. The Islamic Dimension: Geometry with a Rusty Compass
 * Problem 1
 * Problem 2
 * Problem 3
 * Problem 4
 * Problem 5


 * Exercises


 * $4$. Algebra in Islam


 * $\S 1$. Problems About Unknown Quantities


 * $\S 2$. Sources of Islamic Algebra


 * $\S 3$. Al-Khwārizmī’s Algebra
 * The Name “Algebra”
 * Basic Ideas in Al-Khwārizmī’s Algebra
 * Al-Khwārizmī’s Discussion of $x^2 + 21 = 10 x$


 * $\S 4$. Thābit’s Demonstration for Quadratic Equations
 * Preliminaries
 * Thābit’s Demonstration


 * $\S 5$. Abū Kāmil on Algebra
 * Similarities with al-Khwārizmī
 * Advances Beyond al-Khwārizmī
 * A Problem from Abū Kāmil


 * $\S 6$. Al-Karajī’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


 * $\S 7$. ‘Umar al-Khayyāmī and the Cubic Equation
 * The Background to ‘Umar’s Work
 * ‘Umar’s Classification of Cubic Equations
 * ‘Umar’s Treatment of $x^3 + m x = n$
 * Preliminaries
 * The Main Discussion
 * ‘Umar’s Discussion of the Number of Roots


 * $\S 8$. The Islamic Dimension: The Algebra of Legacies


 * Exercises


 * $5$. Trigonometry in the Islamic World


 * $\S 1$. Ancient Background: The Table of Chords and the Sine


 * $\S 2$. The Introduction of the Six Trigonometric Functions


 * $\S 3$. Abu al-Wafā’s Proof of the Addition Theorem for Sines


 * $\S 4$. Nasīr al-Dīn’s Proof of the Sine Law


 * $\S 5$. Al-Bīrūnī’s Measurement of the Earth


 * $\S 6$. Trigonometric Tables: Calculation and Interpolation


 * $\S 7$. Auxiliary Functions


 * $\S 8$. Interpolation Procedures
 * Linear Interpolation
 * Ibn Yūnus’ Second-Order Interpolation Scheme


 * $\S 9$. Al-Kāshī’s Approximation to $\map {\mathrm {Sin} } {1 \degrees}$


 * Exercises


 * $6$. Spherics in the Islamic World


 * $\S 1$. The Ancient Background


 * $\S 2$. Important Circles on the Celestial Sphere


 * $\S 3$. The Rising Times of the Zodiacal Signs


 * $\S 4$. Stereographic Projection and the Astrolabe


 * $\S 5$. Telling Time by Sun and Stars


 * $\S 6$. Spherical Trigonometry in Islam


 * $\S 7$. Tables for Spherical Astronomy


 * $\S 8$. The Islamic Dimension: The Direction of Prayer


 * Exercises