1. By an isogonal (winkeltreu) representation of two areas on one another we mean a one-one, continuous, and continuously differentiable, representation of the areas, which is such that two curves of the first area which intersect at an angle a are transformed into two curves intersecting at the same angle a. If the sense of rotation of a tangent is preserved, an isogonal transformation is called conformal.
Disregarding as trivial the Euclidean magnification (Ahnlichkeitstransformation) of the plane, we may say that the oldest known transformation of this kind is the stereographic projection of the sphere, which was used by Ptolemy (flourished in the second quarter of the second century; died after A.D. 161) for the representation of the celestial sphere; it transforms the sphere conformally into a plane. A quite different conformal representation of the sphere on a plane area is given by Mercator's Projection; in this the spherical earth, cut along a meridian circle, is conformally represented on a plane strip. The first map constructed by this transformation was published by Mercator (1512-1594) in 1568, and the method has been universally adopted for the construction of sea-maps.
2. A comparison of two maps of the same country, one constructed by stereographic projection of the spherical earth and the other by Mercator's Projection, will show that conformal transformation does not imply similarity of corresponding figures. Other non-trivial conformal representations of a plane area on a second plane area are obtained by comparing the various stereographic projections of the spherical earth which correspond to different positions of the centre of projection on the earth's surface. It was considerations such as these which led Lagrange (1736-1813) in 1779 to obtain all conformal representations of a portion of the earth's surface on a plane area wherein all circles of latitude and of longitude are represented by circular arcs.
3. In 1822 Gauss (1777-1855) stated and completely solved the general problem of finding all conformal transformations which transform a sufficiently small neighbourhood of a point on an arbitrary analytic surface into a plane area. This work of Gauss appeared to give the whole inquiry its final solution; actually it left unanswered the much more difficult question whether and in what way a given finite portion of the surface can be represented on a portion of the plane. This was first pointed out by Riemann (1826-1866), whose Dissertation (1851) marks a turning-point in the history of the problem which has been decisive for its whole later development; Riemann not only introduced all the ideas which have been at the basis of all subsequent investigation of the problem of conformal representation, but also showed that the problem itself is of fundamental importance for the theory of functions.
4. Riemann enunciated, among other results, the theorem that every simply-connected plane area which does not comprise the whole plane can be represented conformally on the interior of a circle. In the proof of this theorem, which forms the foundation of the whole theory, he assumes as obvious that a certain problem in the calculus of variations possesses a solution, and this assumption, as Weierstrass (1815-1897) first pointed out, invalidates his proof Quite simple, analytic, and in every way regular problems in the calculus of variations axe now known which do not always possess solutions. Nevertheless, about fifty years after Riemann, Hilbert was able to prove rigorously that the particular problem which arose in Riemann's work does possess a solution; this theorem is known as Dirichlet's Principle.
Meanwhile, however, the truth of Riemann's conclusions had been established in a rigorous manner by Carl Neumann and, in particular, by H A Schwarz. The theory which Schwarz created for this purpose is particularly elegant, interesting and instructive; it is, however, somewhat intricate, and uses a number of theorems from the theory of the logarithmic potential, proofs of which must be included in any complete account of the method. During the present century the work of a number of mathematicians has created new methods which make possible a very simple treatment of our problem; it is the purpose of the following pages to give an account of these methods which, while as short as possible, shall yet be essentially complete.