  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.
