Ibn al-Haytham's Geometrical Methods and the Philosophy of Mathematics

A History of Arabic Sciences and Mathematics Volume 5

Roshdi Rashed

• Publisher: Routledge
• Year: 2017
• eBook ISBN: 9781351686006
• Topic: History
• Subtopic: Middle Eastern History
• Index: History

This fifth volume of A History of Arabic Sciences and Mathematics is complemented by four preceding volumes which focused on the main chapters of classical mathematics: infinitesimal geometry, theory of conics and its applications, spherical geometry, mathematical astronomy, etc.

This book includes seven main works of Ibn al-Haytham (Alhazen) and of two of his predecessors, Th?bit ibn Qurra and al-Sijz?:

• The circle, its transformations and its properties;
• Analysis and synthesis: the founding of analytical art;
• A new mathematical discipline: the Knowns;
• The geometrisation of place;
• Analysis and synthesis: examples of the geometry of triangles;
• Axiomatic method and invention: Th?bit ibn Qurra;
• The idea of an Ars Inveniendi: al-Sijz?.

Including extensive commentary from one of the world's foremost authorities on the subject, this fundamental text is essential reading for historians and mathematicians at the most advanced levels of research.

CHAPTER I

THE PROPERTIES OF THE CIRCLE

INTRODUCTION

The long list of mathematical works written by Ibn al-Haytham includes several that are still missing. Among these are three that, in the mathematics of infinitesimals, speak for themselves: The Greatest Line that can be Drawn in a Segment of a Circle, a Treatise on Centres of Gravity and a Treatise on the Qarasṭūn. These treatises are all concerned with the geometry of measure. Their absence not only deprives historians of mathematics of facts that would have helped them to appreciate more clearly the range of Ibn al-Haytham’s oeuvre, but also, more seriously, it makes it absolutely impossible for them to understand the structures of this oeuvre and the network of meanings that they carry. If the first of the treatises cited above had been at our disposal, we should have a better understanding of the distance the author of a treatise on problems of figures with equal perimeters, on figures with equal areas and on the solid angle, travelled along the road of what was later to be called the calculus of variations.

This state of affairs is not peculiar to the geometry of measure; it is found also in the other type of geometry developed by Ibn al-Haytham and his predecessors: the geometry of position and forms. Among the books that until very recently were still missing we have one with the title On the Properties of Circles. A book with such a title is of course intriguing and surprising.¹ We ask ourselves what Ibn al-Haytham might deal with in a book whose title appears so strikingly modern. His predecessors, his contemporaries and Ibn al-Haytham himself wrote books and papers on one or another aspect of a geometrical figure, for example triangles, but rarely on all its properties taken together as a whole. Furthermore, Ibn al-Haytham had written more than once on the circle, on finding its perimeter and finding its area. We may ask what reasons he might have had to return to the subject of the circle.

These were the kinds of questions that could have been asked, until I was able to produce a copy of the treatise, and establish a text of it, though a rather damaged one. Ibn al-Haytham’s short introduction could not fail to sharpen the reader’s curiosity and raise questions. The author indeed proposes to investigate the properties of the circle, or at least a certain number of them, since ‘the properties of circles are numerous, and their number is almost infinite’ (p. 87). He promises not to include in this treatise properties that have already been discovered. There is even a request to the reader that if, in the course of his reading, he happens to come upon a result that is already obtained elsewhere, he will see this as no more than a coincidence produced without the author’s knowledge. Thus, Ibn al-Haytham explicitly lays claim to novelty and originality.

Thus, the question becomes: where does Ibn al-Haytham see this novelty? A mathematician of his standing, of his universally inventive genius, could not describe as new a result that was secondary or partial: only an idea he considered fundamental could be called ‘new’. This last statement is not a petitio principi on our part, but the conclusion of a sufficiently long analysis of similar situations in the mathematical and optical works of Ibn al-Haytham. If it does sometimes happen that Ibn al-Haytham makes a mistake when proving a result, he always has a sharp eye in relation to the value of his programme of research.

And we shall in fact show that in this book Ibn al-Haytham did not confine himself to dealing with metrical properties of the circle, but also considered affine properties. It is as if he ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Foreword
• Preface
• Introduction Motion and Transformations in Geometry
• Chapter I: the Properties of the Circle
• Chapter II: The Analytical Art in the Tenth to Eleventh Eenturies
• Chapter III: Ibn Al-Haytham and the Geometrisation of Place
• Appendix: the Ars Inveniendi: Thābit Ibn Qurra and Al-Sijzī
• Supplementary Notes
• Bibliography
• Indexes