AbstractIn this paper, I propose the idea that the French mathematician Michel Chasles developed a foundational programme for geometry in the period 1827–1837. The basic concept behind the programme was to show that projective geometry is the foundation of the whole of geometry. In particular, the metric properties can be reduced to specific graphic properties. In the attempt to prove the validity of his conception, Chasles made fundamental contributions to the theory of polarity and also understood that a satisfactory development of projective geometry has to overcome the specificity of that theory itself. In this perspective, he developed his ideas on duality and homography, showing their dependence on the concept that he was going to pose at the basis of geometry: the anharmonic ratio, today called cross-ratio. The conceptual itinerary that starts from Chasles’ studies on polarity and ends with his results based on the cross-ratio represents the itinerary of his foundational programme. I will follow this complex and interesting line of thought, place it in the context of what was then called descriptive geometry, and outline a connection with the later great results of Von Staudt, Cayley, and Klein.

中文翻译：

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Michel Chasles 的几何基础课程，直到他的 Aperçu historique 出版

摘要在本文中，我提出了法国数学家米歇尔·查斯勒 (Michel Chasles) 在 1827 年至 1837 年期间开发的几何基础程序的想法。该程序背后的基本概念是表明射影几何是整个几何的基础。特别是，度量属性可以简化为特定的图形属性。为了证明他的概念的有效性，Chasles 对极性理论做出了根本性的贡献，并且还理解射影几何的令人满意的发展必须克服该理论本身的特殊性。从这个角度来看，他发展了对偶性和单应性的想法，表明它们依赖于他将在几何基础上提出的概念：非谐比，今天称为交叉比。从 Chasles 对极性的研究开始并以他基于交叉比的结果结束的概念路线代表了他的基础计划的路线。我将遵循这一复杂而有趣的思路，将其置于当时所谓的描述几何的背景下，并概述与 Von Staudt、Cayley 和 Klein 后来的伟大成果的联系。