A Survey of the Development of Geometry up to 1870

TL;DR

This paper surveys the historical development of classical and non-Euclidean geometries up to 1870, highlighting foundational shifts and the emergence of Riemannian geometry, providing context for early works of Lie and Klein.

Contribution

It offers a comprehensive historical overview of geometry's evolution up to 1870, emphasizing the transition from classical to modern geometric ideas.

Findings

Development of non-Euclidean geometry demonstrated as a valid geometry

Transition from classical differential geometry to Riemannian geometry

Foundational questions about the structure of physical space emerged

Abstract

This is an expository treatise on the development of the classical geometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a "true" geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective.