Based on the latest historical research, Worlds Out of Nothing
is the first book to provide a course on the history of geometry in
the 19th century. 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 (Plucker's equations) and their
role in resolving a paradox in the theory of duality; to Riemann's
work on differential geometry; and to Beltrami's role in
successfully establishing non-Euclidean geometry as a rigorous
mathematical subject. The final part of the book considers how
projective geometry rose to prominence, and looks at Poincare's
ideas about non-Euclidean geometry and their physical and
philosophical significance.
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."
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