Episodes in the Mathematics of Medieval Islam

Softcover Nachdruck of the original 1. Auflage 1986
ISBN: 978-0-387-40605-3
Verlag: Springer

Buch, Englisch, 197 Seiten, Book, Format (B × H): 155 mm x 235 mm, Gewicht: 690 g

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.

Zielgruppe

Professional/practitioner

Autoren/Hrsg.

Berggren, J.L.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Geschichte der Mathematik
• Geisteswissenschaften Islam & Islamische Studien Islam: Philosophie & Wissenschaften
• Geisteswissenschaften Islam & Islamische Studien Geschichte des Islam Geschichte des Islam: 7. - 14. Jahrhundert
• Interdisziplinäres Wissenschaften Wissenschaften: Allgemeines Geschichte der Naturwissenschaften, Formalen Wissenschaften & Technik

Inhaltsverzeichnis

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.