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

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

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