Rāshid - Ibn al-Haytham’s geometrical methods and the philosophy of mathematics

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, Thbit 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: Thbit ibn Qurra;

The idea of an Ars Inveniendi: al-Sijz.

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

Roshdi Rashed is one of the most eminent authorities on Arabic mathematics and the exact sciences. A historian and philosopher of mathematics and science and a highly celebrated epistemologist, he is currently Emeritus Research Director (distinguished class) at the Centre National de la Recherche Scientifique (CNRS) in Paris, and is the former Director of the Centre for History of Medieval Science and Philosophy at the University of Paris (Denis Diderot, Paris VII). He also holds an Honorary Professorship at the University of Tokyo and an Emeritus Professorship at the University of Mansourah in Egypt.

J. V. Field is a historian of science, and is a Visiting Research Fellow in the Department of History of Art and Screen Media, Birkbeck, University of London, UK.

This series studies theMiddle East through the twin foci of its diverse cultures and civilisations. Comprising original monographs as well as scholarly surveys, it covers topics in the fields of Middle Eastern literature, archaeology, law, history, philosophy, science, folklore, art, architecture and language. While there is a plurality of views, the series presents serious scholarship in a lucid and stimulating fashion.

For a full list of books in the series, please go to https://www.routledge.com/middleeaststudies/series/SE0363

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A History of Arabic Sciences and Mathematics

Volume 5
Roshdi Rashed

Translated by
J. V. Field

First published 2017
by Routledge
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and by Routledge
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2017 Roshdi Rashed and CAUS

The right of Roshdi Rashed to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.

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Publishers Note

This book has been prepared from camera-ready copy provided by the author

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A catalogue record for this book is available from the British Library

Library of Congress Cataloging in Publication Data
The Library of Congress has cataloged volume 1 of this title under the
LCCN: 2011016464

ISBN: 978-0-415-58219-3 (hbk)
ISBN: 978-1-315-16845-6 (ebk)

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I. Fakhr al-Dn al-Rz:
Ibn al-Haythams critique of the notion of place as envelope

II. Al-asan ibn al-Haytham and Muammad ibn al-Haytham:
the mathematician and the philosopher On place

This book is a translation of Les Mathmatiques infinitsimales du IXe au XIe sicle, vol. IV: Mthodes gomtriques, transformations ponctuelles et philosophie des mathmatiques. The French version, published in London in 2002, also included critical editions of all the Arabic mathematical texts that were the subjects of analysis and commentary in the volume.

It also included: 1. Les emprunts dIbn Hd aux Connus et LAnalyse et la Synthse (Ibn Hds borrowings from The Knowns and On Analysis and Synthesis); 2. Al-Baghdd critique dIbn al-Haytham (Al-Baghdd as a critic of Ibn al-Haytham), with the critical edition of the Arabic text, French translation and mathematical commentaries.

The whole book, apart from these two appendices, has been translated, with great scholarly care, by Dr J. V. Field. The translation of the primary texts was not simply made from the French; I checked a draft English version against the Arabic. This procedure converged to give an agreed translation. The convergence was greatly helped by Dr Fields experience in the history of the mathematical sciences and in translating from primary sources. I should like to take this opportunity of expressing my deep gratitude to Dr Field for this work.

Very special thanks are due to Aline Auger (Centre National de la Recherche Scientifique), who helped me check the English translations against the original Arabic texts, prepared the camera ready copy and compiled the indexes.

Roshdi Rashed

In his magisterial Aperu historique, Michel Chasles, having given the titles of books and referred to the achievements of the great geometers of the Hellenistic period, writes:

[] then, for two or three centuries more, there came the writers of commentaries, who have passed on to us the works and the names of the geometers of Antiquity; then finally the centuries of ignorance, during which Geometry slumbered among the Arabs and the Persians until the Renaissance of learning in Europe.

In this peremptory judgement, Chasles, whose good faith cannot be doubted, was setting out what was known by historians in the mid nineteenth century rather than describing historical facts. At the time, investigations into the history of geometry in classical Islam were few and scattered, and it is not surprising that Chasles judgement became the received opinion. And indeed we find it repeated tirelessly in the historical introductions to manuals of geometry, such as that by Robert Deltheil and Daniel Caire, as well as in writings penned by historians of geometry, even sometimes up to the present day. Nevertheless, a little later a better, though still far from satisfactory, grasp of the historical facts struck the first blows against this general prejudice. Today, for the majority of historians of mathematics, the opinion for which Chasles was the spokesman has given way to a different one, less absolute without however being more accurate, which could be summarised as follows: Arab geometers, while they never reached the high level of the geometry of the classical Greeks, did at least have the merit of recognising the importance of their work and of having preserved both its spirit and its matter, even going so far as to add several notable details. For this, the names that are mentioned are those of Thbit ibn Qurra and of Nar al-Dn al-s. Although it is more nuanced, but also more eclectic, this way of looking at things in fact derives from the same logic: to stop at the threshold of questions, without indicating the criteria and setting out the reasoning that could have led to this modest contribution to geometry. It is not clear why, according to the proponents of this theory, the geometers of classical Islam should have thus confined themselves to the role of conscientious preservers of the Hellenistic geometrical heritage, while they were making major advances in all the other disciplines: algebra, number theory, trigonometry, and so on. It is inexplicable that the notable developments in these latter disciplines and in mathematised disciplines such as astronomy and optics should have had so little effect on work in geometry. It is not clear why the single exception made by the historians of mathematics should be for the development of the theory of parallel lines.