Springer Undergraduate Mathematics Series
Advisory Board
M.A.J. Chaplain University of Dundee
K. Erdmann Oxford University
A. MacIntyre Queen Mary, University of London
L.C.G. Rogers University of Cambridge
E. Süli Oxford University
J.F. Toland University of Bath
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Fields, Flows and Waves: An Introduction to Continuum Models D.F. Parker
Further Linear Algebra T.S. Blyth and E.F. Robertson
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Metric spaces M. Ó Searcóid
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Special Relativity N.M.J. Woodhouse
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Vector Calculus P.C. Matthews
Jeremy Gray
Worlds Out of Nothing
A Course in the History of Geometry
in the 19th Century
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Mathematica in Education and Research Vol 4 Issue 3 1995 article by Richard J Gaylord and Kazume Nishidate ‘Traffic Engineering with Cellular Automata’
page 35 fig 2. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Michael Trott ‘The Implicitization of a Trefoil Knot’ page 14.
Mathematica in Education and Research Vol 5 Issue 2 1996 article by Lee de Cola ‘Coins, Trees, Bars and Bells: Simulation of the Binomial Process’ page 19
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Mathematics Subject Classification (2000): 01A05, 01A055, 01A60, 03-03, 14-03, 30-03, 51-03, 53-03
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2006934192
Springer Undergraduate Mathematics Series ISSN 1615-2085
ISBN-10: 1-84628-632-8
ISBN-13: 978-1-84628-632-2
eISBN-10: 1-84628-633-6
eISBN-13: 978-1-84628-633-9
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Springer Science+Business Media
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Jeremy Gray
Honorary Professor, The University of Warwick, United Kingdom
and
The Centre for the History of the Mathematical Sciences,
The Faculty of Mathematics and Computing,
The Open University, Walton Hall, Milton Keynes,
Buckinghamshire, MK7 6AA, United Kingdom
email: j.j.gray@open.ac.uk
Preface
In 1789 geometry was at a low ebb. Euler had written on it – he wrote on
everything – but his successor, Joseph-Louis Lagrange was much more of an
algebraist and an analyst. The great flowering of French mathematics that
may be said to have started with Lagrange’s arrival in Paris in 1787 at the
age of 51 and continued with Laplace, Legendre, and then the generation of
Cauchy, Fourier and Poisson, accomplished much and innovated widely, but less
in geometry than in other areas. Only Gaspard Monge stood out, both as an
original geometer and as an inspiring teacher. In 1914, at the other end of what
historians call the long 19th century, geometry was not only a major branch
of mathematics with several new branches springing from it, it could claim to
have been one of the most provocative and challenging in its implications for
the nature of mathematics. There were whole new geometries, some contesting
the centrality of Euclidean geometry and therefore the framework for all of
mechanics, others arguably yet more fundamental, and pervading more and
more of the domain of pure mathematics. This transformation of geometry is
the theme of this book.
This book discusses the ideas, the people, and the way they (the people and
the ideas) fit into larger pictures. It starts with the rediscovery of projective
geometry in post-revolutionary France by Poncelet, a student of Monge. It then
takes in the discovery of non-Euclidean geometry by Bolyai in Austria–Hungary
and Lobachevskii in Russia in the 1820s. The reception of Poncelet’s work was
poor, that of the work of Bolyai and Lobachevskii dismal, and we consider why
this was. With these clear examples in place showing the importance of the so-
cial in the reception of mathematical ideas, the book then considers how matters
turned around. For projective geometry, the algebraic methods of M¨obius and
Pl¨ucker are prominent because they not only led to the resolution of a major
paradox facing Poncelet’s work they also opened up the little-studied domain of
