Felix Klein

The proof that π is a transcental number will forerver mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number π by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone.

The theory of binary forms and the projective geometry of systems of points on a conic are one and the same, i.e., to every proposition concerning binary forms corresponds a proposition concerning such systems of points, and vice versa. ... Elementary plane geometry and the projective investigation of a quadric surface with reference to one of its pointa are one and the same.

In ordinary geometry a surface is conceived as a locus of points; in Lie's geometry it appears as the totality of all the spheres having contact with the surface.

Next to the elementary transcental functions the elliptic functions are usually regarded as the most important. There is, however, another class for which at least equal importance must be claimed on account of their numerous applications in astronomy and mathematical physics; these are the hypergeometric functions, so called owing to their connecton with Gauss's hypergeometric series.

As regards quartic surfaces, Rohn has investigated an enormous number of special cases; but a complete enumeration he has not reached. Among the special surfaces of the fourth order the Kummer surface with 16 conical points is one of the most important. The models constructed by Plücker in connection with his theory of complexes of lines all represent special cases of the Kummer surface.

It has been the final aim of Lie from the beginning to make progress in the theory of differential equations ...

Es ist eine Mannigfaltigkeit und in derselben eine Transformationsgruppe gegeben; man soll die Mannigfaltigkeit angehören Gebilde hinsichtlich solcher Eigenschaften untersuchen; die durch die Transformationen der Gruppe nicht geändert werden. (Given a manifold with its associated transformation group, one should investigate those structures of the manifold that have properties which are invariant under the transformation group.)