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Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century

Jeremy Gray
Springer Verlag
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Bollman
, on

The 19th century was certainly an exciting time in geometry. To chronicle all of that excitement in one place is a monumental task; to have done so with real clarity and attention to detail, as Jeremy Gray has done, is an impressive achievement.

With that said, it must be noted that this book is not an easy read. Mathematical intricacy has been sacrificed for a high level of historical detail. This is a fine choice, as it brings a lot more history in than many geometry teachers know. The story of the discovery of non-Euclidean geometry, with full attention to the controversy over credit, is here with both primary sources and careful commentary. Also thoroughly presented are the development of differential and projective geometry.

The detail in the treatment of the historical development of geometric ideas is truly impressive. This, of course, stems from the fact that the author isn’t simultaneously trying to teach geometry. With the right group of students, this could be an excellent auxiliary textbook for a traditional geometry course.

An unusual feature of this book is three chapters on assessment activities appropriate for a course taught from it. I am borrowing ideas from here already for my spring course in geometry.

Gray has succeeded on several levels: as a historical chronicler, as a mathematical scholar, and as an advisor to teachers. Worlds Out Of Nothing is a first-rate addition to the geometry enthusiast’s bookshelf.

Mark Bollman ( is an associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.

 Mathematics in the French Revolution.- Poncelet (and Pole and Polar).- Theorems in Projective Geometry.- Poncelet’s Traité.- Duality and the Duality Controversy.- Poncelet and Chasles.- Lambert and Legendre.- Gauss.- Janos Bolyai.- Lobachevskii.- To 1855.- Writing.- Möbius.- The Duality Paradox.- The Plücker Formulae.- Higher Plane Curves.- Complex Curves.- Riemann.- Differential Geometry of Surfaces.- Non-Euclidean Geometry Accepted.- Writing.- Fundamental Geometry.- Hilbert.- Italian Foundations.- The Disc Model.- The Geometry of Space.- Summary: Geometry to 1900.- The Formal Side.- The Physical Side.- Is Geometry True?- Writing.- Appendix: Von Staudt and his Influence.- Bibliography.- Index.