FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEA S EDITED BY YVONNE DOLD-SAMPLONIUS JOSEPH W. DAUBEN MENSO FOLKERTS BENNO VAN DALEN FRANZ STEINER VERLAG STUTTGART FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEAS BOETHIUS TEXTE UND ABHANDLUNGEN ZUR GESCHICHTE DER MATHEMATIK UND DER NATURWISSENSCHAFTEN BEGRIJNDET VON JOSEPH EHRENFRIED HOFMANN FRIEDRICH KLEMM UND BERNHARD STICKER HERAUSGEGEBEN VON MENSO FOLKERTS BAND 46 FRANZ STEINER VERLAG STUTTGART 2002 FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEAS EDITED BY YVONNE DOLD-SAMPLONIUS JOSEPH W. DAUBEN MENSO FOLKERTS BENNO VAN DALEN FRANZ STEINER VERLAG STUTTGART 2002 Bibliographische Information der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet these Publikation in der Deutschen National- bibliographic; detaillierte bibliographische Daten sind im Internet Uber <http:// dnb.ddb.de> abrufbar. ISBN 3-515-08223-9 ISO 9706 Jede Verwertung des Werkes auBerhalb der Grenzen des Urheberrechtsgesetzes ist unzulhssig and strafbar. Dies gilt insbesondere fair Ubersetzung, Nachdruck. Mikrover- filmung oder vergleichbare Verfahren sowie fi rdie Speicherung in Datenverarbeitungs- anlagen. ® 2002 by Franz Steiner Verlag Wiesbaden GmbH. Sitz Stuttgart. Gedruckt auf s3urefreiem. alterungsbesthndigem Papier. Druck: Druckerei Proff. Eurasburg. Printed in Germany Table of Contents VII Kurt Vogel: A Surveying Problem Travels from China to Paris .................... 1 Jens Hoyrup: Seleucid Innovations in the Babylonian "Algebraic" Tradition and their Kin Abmad ........................................................... 9 J. Lennart Berggren: Some Ancient and Medieval Approximations to Irrational Numbers and Their Transmission ................................... 31 Jacques Sesiano: A Reconstruction of Greek Multiplication Tables for Integers .......................................................................................... 45 An-lrea Rr ard Problems of Pursuit- Recreational Mathematics or Astronomy? ..................................................................................... 57 Karine Chemla & Agathe Keller: The Sanskrit karanis and the Chinese mian ..... 87 Sreeramula Rajeswara Sarma: Rule of Three and its Variations in India ........... ..................................133 Liu Dun: A Homecoming Stranger: Transmission of the Method of Double False Position and the Story of Hiero's Crown ................. 157 Kim Plofker: Use and Transmission of Iterative Approximations in India and the Islamic World -- 167 Jan P. Hogendijk: Anthyphairetic Ratio Theory in Medieval Islamic Mathematics ................................................187 Ulrich Rebstock: An Early Link of the Arabic Tradition of Prac- tical Arithmetic: The Kitab al-Tadhkira bi-usul al-hisab wa'I-fara'id wa-'awlihd wa-tashihiha ................................................203 Ahmed Djebbar: La circulation des mathematiques entre l'Orient et ('Occident musulmans: Interrogations anciennes et elements nouveaux ........213 Charles Burnett: Indian Numerals in the Mediterranean Basin in the Twelfth Century, with Special Reference to the "Eastern Forms" ........237 Raffaella Franci: Jealous Husbands Crossing the River: A Problem from Alcuin to Tartaglia .....................................................................289 Tony Levy: De I'arabe a l'hebreu: la constitution de la litterature mathematique hebrafque ()GIe-XVIe siecle) .....................................307 vi Table of Contents Benno van Dalen: Islamic and Chinese Astronomy under the Mongols: a Little-Known Case of Transmission ................................327 Mohammad Bagheri: A New Treatise by al-Kasha on the Depression of the Visible Horizon ......................................................357 Alexei Volkov: On the Origins of the Toan phap dai thanh (Great Compendium of Mathematical Methods) ................................369 Menso Folkerts: Regiomontanus' Role in the Transmission of Mathematical Problems..................................................................411 David Pingree: Philippe de la Hire's Planetary Theories in Sanskrit----- ------ 429 Index of Proper Names ....................................................................................455 Contributors ....................................................................................................469 Introduction The mathematical Daoshu is brief in expression but extensive in use, showing a wise perception of generality. That from one kind all others are arrived at is called knowledge of the Dao. Classification is indis- pensable for the sage to learn and master knowledge. - Chenzi to Rong Fang. Zhou Bi Suanjing Communication and the exchange of goods, information, and ideas between cul- tures has been a part of the world's history since the beginning of time despite barriers of geography, language, and local custom. While it is often possible to trace the fortunes of artifacts as they moved between different parts of the world, the exchange of ideas is less tangible and therefore more difficult to document. How is it possible, for example, to account for the fact that in various mathe- matical texts - from ancient China to medieval Europe - more or less the same problems arise, often using the same examples, parameters, and methods? If virtually identical problems and procedures were not rediscovered from culture to culture independently, then how were they transmitted, and to what effect? It was in hopes of beginning to provide answers to such questions that in July of 1997, a conference was held at the Mathematisches Forschungsinstitut in Ober- wolfach, Germany. There a group of eleven scholars began the task of examining together primary sources that might shed some light on exactly how and in what forms mathematical problems, concepts, and techniques may have been transmit- ted between various civilizations, from antiquity down to the European Renais- sance following more or less the legendary silk routes between China and Western Europe. As the Oberwolfach meeting did not include either a Sanskrit expert or an historian of Indian mathematics, it made the rationale for a second, larger meeting all the stronger. With this in mind we approached the International Commission on the History of Mathematics, the International Mathematical Union, the US Na- tional Science Foundation, and the Rockefeller Foundation, in hopes of attracting sufficient support to hold a subsequent meeting at the Rockefeller Foundation's Study and Conference Center in Bellagio, Italy. The purpose of the Bellagio meeting was to bring together a larger group of scholars than had been possible at Oberwolfach, and to focus attention on early mathematical works, especially those in China, India, Mesopotamia, the Arabic/Islamic world, and the late Middle Ages / Renaissance in Europe. The focus of both the Oberwolfach and Bellagio meetings was upon a core of rather simple practical or textbook problems that turn up in different cultures and times, often but not always in different guises. For example, the problem familiar in every schoolroom from ancient Babylonia to modern Beijing: how high will a VIII Introduction ladder of given length reach up against a wall if its base is a certain number of feet away from the wall? Other procedures that appear in many similar contexts are derived from problems of surveying, practical arithmetic problems involving the conversion of money between different currencies or amounts of grain of vari- ous grades or value, or geometric problems concerning right triangles, circles, or spheres. By examining such analogous methods, procedures, and problems, it is sometimes possible to determine or at least to suggest with some degree of cer- tainty where and how the problems and methods in question originated, and how they may have migrated from one location or text to others in different places and later periods. Did such information flow predominantly in one direction at certain times, or were ideas being freely exchanged between cultures on a more-or-less continuous basis? Only recently could such meetings as those held in Oberwolfach and Bellagio have been possible, for it is only in the last decade or two that numerous primary sources of Chinese, Indian, Mesopotamian, and Arabic or Persian origins, among others, and a variety of ancient and medieval Western works, have been suffi- ciently identified, studied, edited, and published to make such cross-cultural stud- ies possible. While no single individual might have a fluent command of Chinese, Sanskrit, Arabic, Persian, Greek, Latin, Hebrew, Italian, French, German, and Spanish - to name just the most obvious languages that would be necessary to study the complexities of transmission of ideas between East and West in any comprehensive and satisfactory way - it was possible to assemble an interna- tional group of scholars with shared interests who did have collectively a com- mand of all these languages, and who could in turn examine together paradigmatic cases which served to help us, as a group, to piece together the history of the transmission of mathematics in the early days when trade routes and patterns of intellectual migration were the primary loci for exchange of both material and in- tellectual goods. A paradigmatic example of the kind of study we wanted to pursue through the conference at Bellagio was written nearly twenty years ago by Kurt Vogel. In his article, "Ein Vermessungsproblem reist von China nach Paris," professor Vogel studied