Worlds Out of Nothing is the first book to provide a course on the
history of geometry in the 19th century. Based on the latest
historical research, the book is aimed primarily at undergraduate
and graduate students in mathematics but will also appeal to the
reader with a general interest in the history of mathematics.
Emphasis is placed on understanding the historical significance of
the new mathematics: Why was it done? How - if at all - was it
appreciated? What new questions did it generate?
Topics covered in the first part of the book are projective
geometry, especially the concept of duality, and non-Euclidean
geometry. The book then moves on to the study of the singular
points of algebraic curves (PlA1/4ckera (TM)s equations) and their
role in resolving a paradox in the theory of duality; to Riemanna
(TM)s work on differential geometry; and to Beltramia (TM)s role in
successfully establishing non-Euclidean geometry as a rigorous
mathematical subject. The final part of the book considers how
projective geometry, as exemplified by Kleina (TM)s Erlangen
Program, rose to prominence, and looks at PoincarA(c)a (TM)s ideas
about non-Euclidean geometry and their physical and philosophical
significance. It then concludes with discussions on geometry and
formalism, examining the Italian contribution and Hilberta (TM)s
Foundations of Geometry; geometry and physics, with a look at some
of Einsteina (TM)s ideas; and geometry and truth.
Three chapters are devoted to writing and assessing work in the
history of mathematics, with examples of sample questions in the
subject, advice on how to write essays, and comments on what
instructors should be looking for.
Jeremy Gray isProfessor of the History of Mathematics and
Director of the Centre for the History of the Mathematical Sciences
at the Open University in England, and is an Honorary Professor in
the Mathematics Department at the University of Warwick. He is the
author, co-author, or editor of 14 books on the history of
mathematics in the 19th and 20th Centuries.
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