The Early History of Non-Euclidean Geometry

Abstract

IN a recent number of NATURE (June 30) there appeared a review of a book by G. Mannoury on the philosophy of mathematics, and the reviewer emphasised a statement of the author to the effect that the claim, for Gauss that he was the first to assert the possibility of a non-Euclidean geometry is threatened by F. K. Schweikart, who in December, 1818, sent a note, to Gauss asserting the existence of a geometry in which the sum of the angles of a triangle is less than two right angles. The facts about Schweikart were made known fifteen years ago by Stackel and Engel (“Theorie der Parallellinien”, p. 243), and the actual documents were published in Gauss's “Werke”, Bd. viii. (1900). It must be admitted that Schweikart arrived independently at this result, though it is not so obvious that he had forestalled the “giant mathematician”. Schweikart states his hypothesis very clearly, and explains that Euclidean geometry is a special case of a more general geometry. On the other hand, Gauss was interested in the theory of parallels from at least 1799; and some time between 1808 and 1816 he arrived at the belief that non-Euclidean geometry was possibly true, for in 1808 he asserted that the idea of an a priori linear constant (the “space-constant”) was absurd, while in 1816 he declared that, while seemingly paradoxical, this idea was in no way self-contradictory, and that Euclid's geometry might not be the true one. In his comments on Schweikart's note, he exhibits quite an extensive knowledge of non-Euclidean trigonometry.