Episodes in the Mathematics of Medieval Islam

by Berggren, Lennart; Berggren, J. L.

• ISBN13: 9780387406053
• ISBN10: 0387406050
• Format: Paperback
• Copyright: 2004-01-01
• Publisher: Springer Verlag

Summary

From the reviews: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...No knowledge of mathematics (or of the history of mathematics) beyond normal high-school 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.- Jens Høyrup, Mathematical Reviews...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

Table of Contents

Chapter 1. Introduction	1	(28)
§1. The Beginnings of Islam	1	(1)
§2. Islam's Reception of Foreign Science	2	(3)
§3. Four Muslim Scientists	5	(16)
Introduction	5	(1)
Al-Khwãrizmi	6	(3)
Al-Birüni	9	(3)
'Umar al-Khayyãmi	12	(3)
Al-Kãshi	15	(6)
§4. The Sources	21	(3)
§5. The Arabic Language and Arabic Names	24	(2)
The Language	24	(1)
Transliterating Arabic	25	(1)
Arabic Names	25	(1)
Exercises	26
Bibliography	21	(8)
Chapter 2. Islamic Arithmetic	29
§1. The Decimal System	29	(2)
§2. Küshyãr's Arithmetic	31	(5)
Survey of The Arithmetic	31	(1)
Addition	32	(1)
Subtraction	33	(1)
Multiplication	34	(1)
Division	35	(1)
§3. The Discovery of Decimal Fractions	36	(3)
§4. Muslim Sexagesimal Arithmetic	39	(9)
History of Sexagesimals	39	(3)
Sexagesimal Addition and Subtraction	42	(1)
Sexagesimal Multiplication	43	(1)
Multiplication by Levelling	43	(1)
Multiplication Tables	44	(1)
Methods of Sexagesimal Multiplication	45
Sexagesimal Division	41	(7)
§5. Square Roots	48	(5)
Introduction	48	(1)
Obtaining Approximate Square Roots	49	(1)
Justifying the Approximation	50	(1)
Justifying the Fractional Part	50	(2)
Justifying the Integral Part	52	(1)
§6. Al-Kãshi's Extraction of a Fifth Root	53	(10)
Introduction	53	(1)
Laying Out the Work	54	(1)
The Procedure for the First Two Digits	54
Justification for the Procedure	51	(9)
The Remaining Procedure	60	(1)
The Fractional Part of the Root	61	(2)
§7. The Islamic Dimension: Problems of Inheritance	63	(4)
The First Problem of Inheritance	63	(2)
The Second Problem of Inheritance	65	(1)
On the Calculation of Zakãt	65	(2)
Exercises	67	(1)
Bibliography	68
Chapter 3. Geometrical Constructions in the Islamic World	10	(89)
§1. Euclidean Constructions	70	(2)
§2. Greek Sources for Islamic Geometry	72	(2)
§3. Apollonios' Theory of the Conics	74	(3)
Symptom of the Parabola	75	(1)
Symptom of the Hyperbola	76	(1)
§4. Abú Sahl on the Regular Heptagon	77	(5)
Archimedes' Construction of the Regular Heptagon	77	(1)
Abú Sahl's Analysis	78	(1)
First Reduction: From Heptagon to Triangle	79	(1)
Second Reduction: From Triangle to Division of Line Segment	80	(1)
Third Reduction: From the Divided Line Segment to Conic Sections	81	(1)
§5. The Construction of the Regular Nonagon	82	(3)
Verging Constructions	82	(2)
Fixed Versus Moving Geometry	84	(1)
Abú Sahl's Trisection of the Angle	84	(1)
§6. Construction of the Conic Sections	85	(4)
Life of Ibrahim b. Sinãn	85	(1)
Ibrahim b. Sinãn on the Parabola	86	(2)
Ibrahim b. Sinãn on the Hyperbola	88	(1)
§7. The Islamic Dimension: Geometry with a Rusty Compass	89	(7)
Problem 1	92	(1)
Problem 2	92	(1)
Problem 3	93	(1)
Problem 4	93	(1)
Problem 5	94	(2)
Exercises	96	(2)
Bibliography	98	(1)
Chapter 4. Algebra in Islam	99	(28)
§1. Problems About Unknown Quantities	99	(1)
§2. Sources of Islamic Algebra	100	(2)
§3. Al-Khwãrizmi's Algebra	102	(2)
The Name "Algebra"	102	(1)
Basic Ideas in Al-Khwãrizmi's Algebra	102	(1)
Al-Khwãrizmi's Discussion of x2 + 21 = 10x	103	(1)
§4. Thãbit's Demonstration for Quadratic Equations	104	(4)
Preliminaries	104	(2)
Thãbit's Demonstration	106	(2)
§5. Abü Kãmil on Algebra	108	(3)
Similarities with al-Khwãrizmi	108	(1)
Advances Beyond al-Khwãrizmi	108	(2)
A Problem from Abü Kãmil	110	(1)
§6. Al-Karaji's Arithmetization of Algebra	111	(7)
Introduction	111	(2)
Al-Samaw'al on the Law of Exponents	113	(2)
Al-Samaw'al on the Division of Polynomials	115	(1)
The First Example	115	(2)
The Second Example	117	(1)
§7. 'Umar al-Khayyãmi and the Cubic Equation	118	(6)
The Background to 'Umar's Work	118	(1)
'Umar's Classification of Cubic Equations	119	(1)
'Umar's Treatment of x3 + mx = n	120	(1)
Preliminaries	120	(1)
The Main Discussion	121	(1)
'Umar's Discussion of the Number of Roots	122	(2)
§8. The Islamic Dimension: The Algebra of Legacies	124	(1)
Exercises	125	(1)
Bibliography	126	(1)
Chapter 5. Trigonometry in the Islamic World	127	(24)
§1. Ancient Background: The Table of Chords and the Sine	127	(5)
§2. The Introduction of the Six Trigonometric Functions	132	(3)
§3. Abul-Wafã's Proof of the Addition Theorem for Sines	135	(3)
§4. Nasir al-Din's Proof of the Sine Law	138	(3)
§5. Al-Birüni's Measurement of the Earth	141	(3)
§6. Trigonometric Tables: Calculation and Interpolation	144	(1)
§7. Auxiliary Functions	144	(1)
§8. Interpolation Procedures	145	(6)
Linear Interpolation	146	(2)
Ibn Yünus' Second-Order Interpolation Scheme	148	(3)
§9. Al-Kãshi's Approximation to Sin(1°)	151	(3)
Exercises	154	(2)
Bibliography	156
Chapter 6. Spherics in the Islamic World	151	(38)
§1. The Ancient Background	157	(4)
§2. Important Circles on the Celestial Sphere	161	(3)
§3. The Rising Times of the Zodiacal Signs	164	(2)
§4. Stereographic Projection and the Astrolabe	166	(6)
§5. Telling Time by Sun and Stars	172	(2)
§6. Spherical Trigonometry in Islam	174
§7. Tables for Spherical Astronomy	117	(65)
§8. The Islamic Dimension: The Direction of Prayer	182	(4)
Exercises	186	(2)
Bibliography	188	(1)
Index	189