Unifying Theories in Mathematics - Geometrical Theories

Geometrical Theories

A well-known example was the development of analytic geometry, which in the hands of mathematicians such as Descartes and Fermat showed that many theorems about curves and surfaces of special types could be stated in algebraic language (then new), each of which could then be proved using the same techniques. That is, the theorems were very similar algebraically, even if the geometrical interpretations were distinct.

At the end of the 19th century, Felix Klein noted that the many branches of geometry which had been developed during that century (affine geometry, projective geometry, hyperbolic geometry, etc.) could all be treated in a uniform way. He did this by considering the groups under which the objects were invariant. This unification of geometry goes by the name of the Erlangen programme.