Ibn al-Haytham's Theory of Conics, Geometrical Constructions and Practical Geometry

A History of Arabic Sciences and Mathematics Volume 3

Roshdi Rashed

• Publisher: Routledge
• Year: 2013
• Print ISBN: 9780415582155
• eBook ISBN: 9781135072537
• Edition: 1
• Topic: History
• Subtopic: European Medieval History
• Index: History

Theory of Conics, Geometrical Constructions and Practical Geometry: A History of Arabic Sciences and Mathematics Volume 3, provides a unique primary source on the history and philosophy of mathematics and science from the mediaeval Arab world. The present text is complemented by two preceding volumes of A History of Arabic Sciences and Mathematics, which focused on founding figures and commentators in the ninth and tenth centuries, and the historical and epistemological development of 'infinitesimal mathematics' as it became clearly articulated in the oeuvre of Ibn al-Haytham.

This volume examines the increasing tendency, after the ninth century, to explain mathematical problems inherited from Greek times using the theory of conics. Roshdi Rashed argues that Ibn al-Haytham completes the transformation of this 'area of activity, ' into a part of geometry concerned with geometrical constructions, dealing not only with the metrical properties of conic sections but with ways of drawing them and properties of their position and shape.

Including extensive commentary from one of world's foremost authorities on the subject, this book contributes a more informed and balanced understanding of the internal currents of the history of mathematics and the exact sciences in Islam, and of its adaptive interpretation and assimilation in the European context. This fundamental text will appeal to historians of ideas, epistemologists and mathematicians at the most advanced levels of research.

CHAPTER I

THEORY OF CONICS AND GEOMETRICAL CONSTRUCTIONS: ‘COMPLETION OF THE CONICS’

1.1. INTRODUCTION

1.1.1. Ibn al-Haytham and Apollonius’ Conics

The theory of conics — and this point is worth emphasizing — had never held such an important position before the ninth and tenth centuries. This new importance is not only on account of the properties of the curves, but also because they can they can be applied in areas not foreseen by the original mathematicians, specifically not by Archimedes and Apollonius. That is, the theory of conics has ceased to be simply a powerful instrument in the hands of geometers; it now offers algebraists a means of solving cubic equations.¹ Ibn al-Haytham is no exception to the rule. As a geometer he investigates the geometrical properties of conics — as we may see, for example, in his treatise The Measurement of the Paraboloid. As a physicist he is concerned with reflection properties of some of the curves — we may think, for example, of his treatise Parabolic Burning Mirrors. He also uses conics to achieve success in problems of geometrical construction. In short, a glance at his works will show that conics and their properties run through them from beginning to end. So it is easy to understand his taking an interest in Apollonius’ treatise, the more so since — the works of his predecessors having been lost — Apollonius was the only authority. No other Greek mathematical work, apart from Euclid’s Elements, was so much consulted, studied and cited by Arabic mathematicians, among them Ibn al-Haytham. Not only did he know the Conics down to the smallest details, but he even went so far as to transcribe the work, as we know from the copy in his handwriting that has survived down the centuries.²

Ibn al-Haytham, as a mathematician and, this once, a copyist, knew the history of the text of the Arabic version of Apollonius’ treatise. This history is recorded by the people who oversaw the search for Greek manuscripts and then their translation, the Banū Mūsā. In a text Ibn al-Haytham knew well (since he made several corrections to it)³ the Banū Mūsā say that they had a first version made up of the first seven books, which was hard to understand, and that later on, while in Damascus, Ahmad ibn Mūsā found Eutocius’ version of the first four books, which is key for understanding the complete work.⁴ As we shall see later, the Banū Mūsā report that of...

Table of contents

• Cover
• Half Title
• CULTURE AND CIVILIZATION IN THE MIDDLE EAST
• Title Page
• Copyright Page
• Table of Contents
• Foreword
• Preface
• INTRODUCTION: CONIC SECTIONS AND GEOMETRICAL CONSTRUCTIONS
• CHAPTER I: THEORY OF CONICS AND GEOMETRICAL CONSTRUCTIONS: 'COMPLETION OF THE CONICS'
• CHAPTER II: CORRECTING THE BANŪ MŪSĀ'S LEMMA FOR APOLLONIUS' CONICS
• CHAPTER III: PROBLEMS OF GEOMETRICAL CONSTRUCTION
• CHAPTER IV: PRACTICAL GEOMETRY: MEASUREMENT
• APPENDIX I: A RESEARCH TRADITION: THE REGULAR HEPTAGON
• APPENDIX II: SINĀN IBN AL-FATH AND AL-QABĪSĪ: OPTICAL MENSURATION
• SUPPLEMENTARY NOTES
• ADDENDA (vol. 2) Al-Hasan ibn al-Haytham and Muhammad ibn al-Haytham: the Mathematician and the Philosopher
• BIBLIOGRAPHY
• INDEXES