非欧几里得几何 - 译自《希腊化时代的科学与文化》（4）

Non-Euclidean Geometry

The fourth consequence, and the most remarkable, was the creation of non-Euclidean geometries. The initiators have already been named : Saccheri, Lambert, Gauss. Inasmuch as the fifth postulate cannot be proved, we are not obliged to accept it ; hence, let us deliberately reject it. The first to build a new geometry on a contrary postulate was the Russian, Nikolai Ivanovich Lobachevski (1793-1856), who assumed that through a given point more than one parallel can be drawn to a given straight line or that the sum of the angles of a triangle is less than two right angles. The discovery of a non-Euclidean geometry was made at about the same time by the Transylvanian, Janos Bolyai (1802-1860). Some time later, another geometry was outlined by the German, Bernhard Riemann (1826-1866), who was not acquainted with the writing of Lobachevski and Bolyai and made radically new assumptions. In Riemann’s geometry, there are no parallel lines and the sum of the angles of a triangle is greater than two right angles. The great mathematical teacher, Felix Klein (1849-1925), showed the relation of all these geometries. Euclid’s geometry refers to a surface of zero curvature, in between Lobachevski’s geometry on a surface of positive curvature (like the sphere) and Riemann’s, applying to a surface of negative curvature. To put it more briefly, Klein called the Euclidean geometry parabolic, because it is the limit of elliptic (Riemann’s) geometry on one side and of the hyperbolic (Lobachevski’s) geometry on the other.

It would be unwise to claim too much for Euclid. The fact that he put at the beginning of the Elements a relatively small number of postulates is very remarkable, especially when one considers the early date, say 300 B.C., but he could not and did not fathom the depths of postulation thinking any more than he coud fathom those of non-Euclidean geometry. Yet he was the distant forerunner of David Hilbert (1862-1943), even as he was Lobachevski’s spitual ancestor.

1】乔治·萨顿（George Sarton）与《希腊化时代的科学与文化》

2】张卜天译本，兰纪正、朱恩宽译本。

https://wap.sciencenet.cn/blog-2322490-1310034.html

全部精选博文导读

GMT+8, 2022-1-20 16:02