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  • Print publication year: 2009
  • Online publication date: October 2011

Afterword

from Geometry, Topology and Foundations

Summary

Halsted's review is of Volume VII of Gauss's Werke, but in Volume VIII, the editors published Gauss's manuscript material on non-Euclidean geometry. These show that Gauss anticipated much of the work of Bolyai and Lobachevsky, as well as of Schweikart. The existence of these materials does not, however, challenge Halsted's conclusion that Gauss had no influence on the work of any of these men. Roberto Bonola's Non-Euclidean Geometry [2] is still a good source for details on Gauss's work; the English edition also contains Halsted's translations of the fundamental articles on non-Euclidean geometry by Bolyai and Lobachevsky.

More recent works on the history of non-Euclidean geometry include Jeremy Gray's Ideas of Space: Euclidean, non-Euclidean and Relativistic [10] and János Bolyai, Non-euclidean Geometry and the Nature of Space [11] and B. A. Rosenfeld's A History of Non-Euclidean Geometry [17]. The latter work, in particular, deals in great detail with the work of “certain Arabs and Persians” that Lewis skips over. In fact, it is now clear that some of the Islamic work on the parallel postulate was read in Europe in the seventeenth century and affected the European developments on the subject. Jeremy Gray has also written a historically-based textbook in geometry, Worlds out of Nothing: A Course in the History of Geometry in the 19th Century [12], that covers in detail not only non-Euclidean geometry but also projective geometry and many other topics. Julian Lowell Coolidge himself expanded on his article on projective geometry in A History of Geometrical Methods [4], first published in 1940.