# 19th-Century Geometry

19th-Century Geometry Robin Wilson Stamp Corner is equivalent to the parallel postulate: given any line l and any point p not lying on l, there is a unique line through p 19th-Century that is parallel to l. For more than 2000 years, mathemati- cians tried to deduce this result from the ﬁrst four postulates, Geometry but they were unable to do so. This is because there are ‘‘non-Euclidean geometries’’ satisfying the ﬁrst four postulates but not the ﬁfth. Such geometries have inﬁnitely many lines through p that are parallel to l, and they were discovered independently around 1830 by Nikolai Lobachevskii (1792–1856) in Russia arl Friedrich Gauss (1777–1855) lived in Go ¨ ttingen and Ja´nos Bolyai (1802–1860) in Hungary. When Bolyai’s and worked in many areas, ranging from complex father outlined Ja ´nos’s work to his friend Gauss, the latter numbers to astronomy and electricity. He gave the dismissed it as something he had discovered previously but ﬁrst satisfactory proof that every polynomial equation has a never published; the Bolyais never forgave Gauss for this. complex root, and in number theory he initiated the study Neither Bolyai nor Lobachevskii gained widespread recog- of congruences and proved the law of quadratic reciprocity. nition for his efforts until after his death. He also determined which regular polygons could be con- Another interesting geometrical object is the Mo¨bius structed with ruler and compasses: his regular 17-gon is band, which has only one side and one boundary edge. First shown on the East German stamp. discovered by Johann Benedict Listing (1808–1882), it is Gauss was also interested in the foundations of geome- named after the German mathematician and astronomer try. Euclid’s Elements commences with ﬁve ‘‘postulates.’’ August Mo ¨ bius (1790–1868), who described it in 1858. Four are straightforward, but the ﬁfth is different in style. It Gauss’s 17-gon Gauss Gaussian plane Lobachevskii Mo¨bius band Bolyai’s geometry Column editor’s address: Robin Wilson, Mathematical Institute, Andrew Wiles Building, University of Oxford, UK e-mail: r.j.wilson@open.ac.uk 94 THE MATHEMATICAL INTELLIGENCER  2017 Springer Science+Business Media, LLC https://doi.org/10.1007/s00283-017-9756-4 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Mathematical Intelligencer Springer Journals

# 19th-Century Geometry

, Volume 40 (1) – Dec 22, 2017
1 page