Episodes in the Mathematics of Medieval Islam

1. Introduction.- 1. The Beginnings of Islam.- 2. Islam's Reception of Foreign Science.- 3. Four Muslim Scientists.- Al-Khw?rizm?.- Al-B?r?n?.- 'Umar al-Khayy?m?.- Al-K?sh?.- 4. The Sources.- 5. The Arabic Language and Arabic Names.- The Language.- Transliterating Arabic.- Arabic Names.- Exercises.- 2. Islamic Arithmetic.- 1. The Decimal System.- 2. K?shy?r'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.- Obtaining Approximate Square Roots.- Justifying the Approximation.- Justifying the Fractional Part.- Justifying the Integral Part.- 6. Al-K?sh?'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 Zak?t.- Exercises.- 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. Ab? Sahl on the Regular Heptagon.- Archimedes' Construction of the Regular Heptagon.- Ab? 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.- Ab? Sahl's Trisection of the Angle.- 6. Construction of the Conic Sections.- Life of Ibr?h?m b. Sin?n.- Ibr?h?m b. Sin?n on the Parabola.- Ibr?h?m b. Sin?n on the Hyperbola.- 7. The Islamic Dimension: Geometry with a Rusty Compass.- Problem 1.- Problem 2.- Problem 3.- Problem 4.- Problem 5.- Exercises.- 4. Algebra in Islam.- 1. Problems About Unknown Quantities.- 2. Sources of Islamic Algebra.- 3. Al-Khw?rizm?'s Algebra.- The Name "Algebra".- Basic Ideas in Al-Khw?rizm?'s Algebra.- Al-Khw?rizm?'s Discussion of x2 + 21 = 10x.- 4. Thabit's Demonstration for Quadratic Equations.- Preliminaries.- Th?bit's Demonstration.- 5. Ab? K?mil on Algebra.- Similarities with al-Khw?rizm?.- Advances Beyond al-Khw?rizm?.- A Problem from Ab? K?mil.- 6. Al-Karaj?'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-Khayy?m? 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.- 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-Waf?'s Proof of the Addition Theorem for Sines.- 4. Nas?r al-D?n's Proof of the Sine Law.- 5. Al-B?r?n?'s Measurement of the Earth.- 6. Trigonometric Tables: Calculation and Interpolation.- 7. Auxiliary Functions.- 8. Interpolation Procedures.- Linear Interpolation.- Ibn Y?nus' Second-Order Interpolation Scheme.- 9. Al-K?sh?'s Approximation to Sin(1 ).- Exercises.- 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.