罗巴切夫斯基有他的“想象几何”模型吗？
Did Lobachevsky Have A Model Of His "imaginary Geometry"?
摘要：
非欧几里德几何的发明通常通过19世纪后期发展起来的希尔伯特形式公理化方法的光学来观察。然而，这样一个时代错误的做法未能提供一个健全的阅读罗巴切夫斯基的几何作品。虽然现代概念模型的一个给定的理论有一个对应的罗巴切夫斯基的著作中的作用，罗巴切夫斯基的几何理论变成非常不寻常的。罗巴切夫斯基并没有像现代读者所期望的那样考虑双曲几何的各种模型，而是使用了欧几里得平面的非标准模型（作为双曲三维空间中的一个特定曲面）。在本文中，我认为这罗巴切夫斯基的建设，并显示它如何可以更好地分析在一个替代的非希尔伯特的基础框架，这涉及到19世纪的几何历史的一些最近的事态发展领域。
英文摘要：
The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky's geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's geometrical theory turns to be very unusual. Lobachevsky doesn't consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.
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