罗巴切夫斯基建立非欧几何的动机

On the Motivation of Lobachevsky’s Establishing Non-euclidean Geometry

通过剖析对比罗巴切夫斯基的几何学思想与拉普拉斯的力学理论,探析罗巴切夫斯基建立非欧几何的动机。拉普拉斯在力与速度存在任意函数关系的假设下,提出更一般的力学定律,罗巴切夫斯基在承认长度与角度存在函数关系的基础上,建立了更一般的几何。罗巴切夫斯基的非欧几何为拉普拉斯的力学理论提供了相应的空间背景;反之,拉普拉斯的力学理论为罗巴切夫斯基的非欧几何提供了实在的物理应用。非欧几何的建立不仅来源于数学问题的解决,而且具有现实问题的需要。

This paper analyzes and compares Lobachevsky’s geometrical thoughts and Laplace’s mechanics theory,so that the motivation of Lobachevsky’s establishing of non-euclidean geometry would be clarified.Laplace admitted that there existed all possible mathematical relations between the force and velocity,and then investigated the general form of the basic mechanical principles.Lobachevsky established a new geometry more general than Euclidean geometry,based on the mutual dependence of line and angle.Lobachevsky’s non-euclidean geometry served as the space background for Laplace’s mechanics,and the latter in turn was taken as a real application case of the former.It can be seen that the establishment of non-euclidean geometry not only resulted from mathematical problem solving,but also from the needs of reality.