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 (Plücker'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 Poincaré'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.
