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

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

Published $\text {1986}$, Springer

ISBN 978-0-387-96318-1.

### Contents

$1$. Introduction
$\S 1$. The Beginnings of Islam
$\S 2$. Islam’s Reception of Foreign Science
$\S 3$. Four Muslim Scientists
Al-Khw?rizm?
Al-B?r?n?
‘Umar al-Khayy?m?
Al-K?sh?
$\S 4$. 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
$\S 2$. K?shy?r’s Arithmetic
Survey of The Arithmetic
Subtraction
Multiplication
Division
$\S 3$. The Discovery of Decimal Fractions
$\S 4$. Muslim Sexagesimal Arithmetic
History of Sexagesimals
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-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
$\S 7$. 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
$\S 1$. Euclidean Constructions
$\S 2$. Greek Sources for Islamic Geometry
$\S 3$. Apollonios’ Theory of the Conics
Symptom of the Parabola
Symptom of the Hyperbola
$\S 4$. 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
$\S 5$. The Construction of the Regular Nonagon
Verging Constructions
Fixed Versus Moving Geometry
Ab? Sahl’s Trisection of the Angle
$\S 6$. 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
$\S 7$. 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-Khw?rizm?’s Algebra
The Name “Algebra”
Basic Ideas in Al-Khw?rizm?’s Algebra
Al-Khw?rizm?’s Discussion of x2 + 21 = 10x
$\S 4$. Thabit’s Demonstration for Quadratic Equations
Preliminaries
Th?bit’s Demonstration
$\S 5$. Ab? K?mil on Algebra
Similarities with al-Khw?rizm?
A Problem from Ab? K?mil
$\S 6$. 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
$\S 7$. ‘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
$\S 8$. 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 l-Waf?’s Proof of the Addition Theorem for Sines
$\S 4$. Nas?r al-D?n’s Proof of the Sine Law
$\S 5$. Al-B?r?n?’s Measurement of the Earth
$\S 6$. Trigonometric Tables: Calculation and Interpolation
$\S 7$. Auxiliary Functions
$\S 8$. Interpolation Procedures
Linear Interpolation
Ibn Y?nus’ Second-Order Interpolation Scheme
$\S 9$. Al-K?sh?’s Approximation to Sin(1°)
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