一，译文

第五个公设的第四个结果，也是最显著的，就是非欧几里得几何的产生。已经提到的发起人：萨奇里(Saccheri)、朗伯(Lambert)、高斯(Gauss)。由于第五个公设不能被证明，我们没有义务接受它；因此，让我们有意拒绝它。第一个在相反的公设上建立新的几何学的是俄国人尼古拉-伊万诺维奇-洛巴切夫斯基 (Nikolai Ivanovich Lobachevski, 1793-1856），他假设通过一个给定的点可以画出一个以上的平行线到一个给定的直线，或者假设一个三角形的角之和小于两个直角。大约在同一时间，特兰西瓦尼亚人(Transylvanian) 博尔亚伊 (Janos Bolyai, 1802-1860）发现了一种非欧几里得几何。一段时间后，德国人伯恩哈德-黎曼（Bernhard Riemann）（1826-1866）概述了另一种几何学，他不了解洛巴切夫斯基和博尔亚伊的著作，并提出了根本性的新假设。在黎曼的几何学中，没有平行线，而且一个三角形的角之和大于两个直角。伟大的数学老师费利克斯-克莱因（Felix Klein, 1849-1925），展示了所有这些几何学的关系。欧几里德的几何学指的是零曲率的表面，介于洛巴切夫斯基的正曲率表面（如球体）的几何学和黎曼的负曲率表面几何学之间。简而言之，克莱因称欧几里得几何为抛物线，因为它一边是（黎曼的）椭圆几何，另一边是（洛巴切夫斯基的）双曲几何的极限。

对欧几里得要求过高是不明智的，他在《几何原本》的开头提出了数目相对较少的公设，就是非常了不起的，特别是当我们考虑到时代，例如公元前300年，但他不可能也没有深入了解公设思维的深度，就像他不可能深入了解非欧几里得几何的深度一样。然而，他是大卫-希尔伯特（ David Hilbert，1862-1943）的遥远先驱，甚至是洛巴切夫斯基的精神祖先。

二，原文

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.

参考文献：

【1】乔治·萨顿（George Sarton）与《希腊化时代的科学与文化》 ，http://blog.sciencenet.cn/blog-2322490-1292301.html

【2】张卜天译本，兰纪正、朱恩宽译本。