A Survey of the Development of Geometry up to 1870

摘要

This is an expository treatise on the development of the classicalgeometries, starting from the origins of Euclidean geometry a few centuries BCup to around 1870. At this time classical differential geometry came to an end,and the Riemannian geometric approach started to be developed. Moreover, thediscovery of non-Euclidean geometry, about 40 years earlier, had just beendemonstrated to be a "true" geometry on the same footing as Euclidean geometry.These were radically new ideas, but henceforth the importance of the topicbecame gradually realized. As a consequence, the conventional attitude to thebasic geometric questions, including the possible geometric structure of thephysical space, was challenged, and foundational problems became an importantissue during the following decades. Such a basic understanding of the status ofgeometry around 1870 enables one to study the geometric works of Sophus Lie andFelix Klein at the beginning of their career in the appropriate historicalperspective.