Studies in the History of Mathematics and Physical Sciences

Transcription

1 Studies in the History of Mathematics and Physical Sciences 12 Editor G. J. Toomer Advisory Board R. P. Boas P. J. Davis T. Hawkins M. J. Klein A. E. Shapiro D. Whiteside

2 Sources in the History of Mathematics and Physical Sciences Vol. I: OJ. Toomer (Ed.) Diodes on Burning Mlrron: The Arabic Translation of the Lost Greek Original, Edited. with English Translation and Commentary by G.J. Toomer Vol. 2: A. Hermann. K.V. Meyenn. V.F. Weisskopf (Eds.) Wolfgang Pauli: Scientific Correspondence I: Vol. 3: J. Sesiano Books IV to VII of Diophantus' Arithmetica: In the Arabic Translation of Qusti Ibn Liiqi Vol. 4: Pol. Federisco Descartes on Polyhedra: A Study of the De Solidorum Elementis Vol. 5: o. Neugebauer Astronomical Cuneiform Texts Vol. 6: K. von Meyenn. A. Hermann. V.F. Weisskopf (Eds.) Wolfgang Pauli: ScientlOc Correspondence II: Vol. 7: J.P. Hogendijk Ibn AI-Haytham's Completioll of the Conics Vol. 8: A. Jones Pappus of Alexandria Book 7 of the Collectioll Studies in the History of Mathematics and Physical Sciences Vol. I: O. Neugebauer A History of Ancient Mathematical Astronomy Vol. 2: H. Goldstine A History or Numerical Analysis from the 16th through the 19th Century Vol. 3: C.C. Heyde/E. Seneta I.J. Bienayme: Statistical Theory Anticipated Vol. 4: C. Truesdell The Tragicomical History of Thermodynamics, Vol. 5: H.H. Goldstein A History or the Calculus of Variations from the 17th through the 19th Century Vol. 6: J. Cannon/So Dostrovsky The Evolution of Dynamics: Vibration Theory from 1687 to 1742 Vol. 7: J. LUtzen The Prehistory of the Theory of DIstributions Vol. 8: G.H. Moore Zermeio's Axlum of Choice Vol. 9: B. Chandler/W. Magnus The History of Combinatorial Group Theory Vol. 10: N.M. Swerdlow/O. Neugebauer Mathematical Astronomy in Copernicus's De Revolutionlbus Vol. II: B.R. Goldstein The Astronomy of Levi ben Gerson ( ) Vol. 12: B.A. Rosenfield The History of Non-Euclidean Geometry Vol. 13: B. Stephenson Kepler's Physical Astronomy

3 B.A. Rosenfeld A History of Non-Euclidean Geometry Evolution of the Concept of a Geometric Space Translated by Abe Shenitzer With the Editorial Assistance of Hardy Grant With 114 Illustrations Springer Science+Business Media, LLC

4 B. A. Rosenfeld Institute for the History of Science and Technology Moscow D.S.S.R. Abe Shenitzer (Translator) Department of Mathematics York Dniversity North York, Ontario M3J lp3 Canada Mathematics Subject Classifications (1980): 01A20, 01A30, 01A35, OIA40, 01A45, 01A50, 01A55, OIA60, 51-03, 51-MXX Library of Congress Cataloging-in-Publication Data Rozenfel'd, (Rosenfeld] B. A. (Boris Abramovich) A history of non-euclidean geometry. (Studies in the history of mathematics and physical seiences; 12) Translation of: Istoriîa neevklidovoi geometrii. Bibliography: p. Inc1udes index. 1. Geometry, Non-Euc1idean-History. 1. Title. II. Title: History of non-euc1idean geometry. III. Series. QA685.R '.9' Original Russian edition: lstoriya Neevklidovoi Geometrii: Razvitie ponyatiya o geometriceskom prostranslve. Nauka, Moscow, by Springer Science+Business Media New York Originally published by Springer-Verlag New York in 1988 Softcover reprint ofthe hardcover Ist edition 1988 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or seholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. Typeset by Aseo Trade Typesetting, Ltd., Hong Kong ISBN ISBN (ebook) DOI /

5 To the memory of my friend Isaac Yaglom ( ) one of the creators of modern non-euclidean geometry

6 Foreword The Russian edition of this book appeared in 1976 on the hundred-and-fiftieth anniversary of the historic day of February 23, 1826, when LobaeevskiI delivered his famous lecture on his discovery of non-euclidean geometry. The importance of the discovery of non-euclidean geometry goes far beyond the limits of geometry itself. It is safe to say that it was a turning point in the history of all mathematics. The scientific revolution of the seventeenth century marked the transition from "mathematics of constant magnitudes" to "mathematics of variable magnitudes." During the seventies of the last century there occurred another scientific revolution. By that time mathematicians had become familiar with the ideas of non-euclidean geometry and the algebraic ideas of group and field (all of which appeared at about the same time), and the (later) ideas of set theory. This gave rise to many geometries in addition to the Euclidean geometry previously regarded as the only conceivable possibility, to the arithmetics and algebras of many groups and fields in addition to the arithmetic and algebra of real and complex numbers, and, finally, to new mathematical systems, i.e., sets furnished with various structures having no classical analogues. Thus in the 1870's there began a new mathematical era usually called, until the middle of the twentieth century, the era of modern mathematics. Now, however, in the wake of the modem scientific and technological revolution, and the related appearance of computers and the significant development of finite mathematics, it is necessary to rename that earlier age. In view of the tremendous importance of the discovery of non-euclidean geometry we might call it the era of non-euclidean mathematics. In the present volume we investigate the mathematical and philosophical factors underlying the discovery of non-euclidean geometry and the extension of the concept of space, the history of these discoveries, and their subsequent development. Of the ten chapters in the book the sixth, "Lobaeevskian Geometry," plays a central role. It deals with the history of the discovery of hyperbolic geom-

7 viii Foreword etry, the struggle for its acceptance, and the history of its more important interpretations. Chapters 1-5 are devoted to the prehistory of hyperbolic geometry and of the ideas associated with the evolution of the concept of space. Specifically, in chapter 1 ("Spherical Geometry") we study the first geometry different from Euclidean geometry and the one which later served as the basis for elliptic geometry. In chapter 2 ("The Theory of Parallels") we give a detailed account of the attempts to prove Euclid's fifth postulate, the so-called parallel postulate; these attempts led directly to Lobaeevskirs discovery. Chapter 3 ("Geometric Tran.sformations") deals with the history of geometric transformations, an evolution that subsequently led to the theory of groups of transformations-the latter providing the basis for the contemporary non-euclidean geometries and others related to them. In chapter 4 ("Geometric Algebra and the Prehistory of Multidimensional Geometry") we study the history of geometric calculi, which later served as the basis for multidimensional geometry, and the theory of algebras, also closely related to non-euclidean geometry. In chapter 5 ("Philosophy of Space") we consider the history of the philosophical views on space that were important to Lobaeevskii's discovery as well as to later generalizations of non-euclidean geometry. Chapters 7 and 8 ("Multidimensional Spaces" and "The Curvature of Space") are devoted to the subsequent development of the concept of space-the appearance of the concepts of multidimensional flat and curved. spaces and their later generalizations. In chapter 9 ("Groups of Transformations") we study the history of groups of transformations whose importance for geometry was disclosed by Klein's famous "Erlangen Program," and the history of Lie groups-in particular, the theory of simple Lie groups-that has given rise to the most natural generalizations of the classical non-euclidean geometries. Chapter 10 ("Application of Algebras") deals with the history of associative and nonassociative algebras and their different geometric applications-in particular, the history of the emergence of spaces over various algebras. The English edition is a corrected and supplemented edition. In chapter 1, I added a section on spherical trigonometry in medieval India and in the treatises of al-khwarizml; this material provides the necessary links between ancient and later Arabic mathematics. In chapter 2, I added information on new publications devoted to the history of the theory of parallel lines. In chapter 3, I added a section on Apollonius' geometric transformations. In chapter 4, I added a section on three-dimensional geometric algebra. In chapter 5 of the Russian edition, I considered only pre-nineteenth century philosophy. In the English version of chapter 5, I brought the discussion up to our own time and added relevant material on medieval India. In chapters 9 and 10, I added sections on parabolic spaces (flag manifolds) and their connections with the theories of groups and algebras, on finite geometries, and on the applications of geometries to physics. The present bibliography inchides references to books and papers that have appeared in the last ten years.

8 Foreword ix The author is very thankful to the translator, Abe Shenitzer, for performing a very difficult task, to J. P. Hogendijk for his transcription of Arabic names, titles, and terms, and to Adolf Pavlovic Yushkevic, Piama Pavlovna Gaidenko, and the late Walter Kaufmann-Biihler for valuable advice.

9 Contents Foreword vii 1. Spherical Geometry The Theory of Parallels Geometric Transformations Geometric Algebra and the Prehistory of Multidimensional Geometry Philosophy of Space Lobaeevskian Geometry Multidimensional Spaces The Curvature of Space Groups of Transformations Application of Algebras References Index