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Mathematical Treasure: Klein's "Not Euclidean" Geometry

Frank J. Swetz (The Pennsylvania State University)

Image of Title of Klein's "Not Euclidean" Geometry

Felix Klein (1849-1925) was one of the most influential mathematicians and mathematics educators of his time. Nicht-Euklidische Geometrie (Not Euclidean Geometry; or, later Non-Euclidean Geometry), Lectures by Felix Klein, Winter Semester 1889-90, vol. 1 (1892), is a very unusual treatise. First, it is a facsimile of a handwritten set of notes by Frederick Schilling, a student of Klein. Secondly, the German word nicht was actually read, at this time, as “not” rather than “non.” The book would appear in formal type in 1893. In these lectures, Klein made a case for his “Erlangen Program,” a new synthesis of geometry which holds that the study of the invariant properties of Euclidean space are best understood as images of a set of transformations.

Image of pages 228 and 229 of Klein's "Not Euclidean" Geometry

On pages 228-229, Klein discussed the projection of a spherical region onto a plane.

Image of pages 234 and 235 of Klein's "Not Euclidean" Geometry

Pages 234-235 contain examples of stereographic projections and conformal mappings of solids onto planes.

The Special Collections staff at the Linderman Library of Lehigh University in Bethlehem, Pennsylvania, is pleased to cooperate with the Mathematical Association of America to exhibit this and other items from the Library’s holdings in “Mathematical Treasures.” In particular, Convergence would like to thank Lois Fischer Black, Curator, Special Collections, and Ilhan Citak, Archives and Special Collections Librarian, for their kind assistance in helping to make this display possible. You may use these images in your classroom; all other uses require permission from the Special Collections staff, Linderman Library, Lehigh University.

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Klein's "Not Euclidean" Geometry," Convergence (August 2014)