The Jordan Curve Theorem and an Unpublished Manuscript by Max Dehn H. GUGGENHEIMER Communicated by C. TRUESDELL l* The JORDAN Curve Theorem is the basis of a correct development of the CAUCHY-RIEMANN approach to the theory of functions of a complex argument. CAUCHV himself discovered his integral formula [1] by formally adding real integrals; in his coherent development of function theory [2] he simply speaks of "points renferm6s dans une certaine aire qu'enveloppe un certain contour". RIEMANN'S thesis [3], contemporary with [1], contains a correct definition of simply and multiply connected domains but neither there nor in the theory of ABELIAN functions [4], published one year after the appearance of [2], is there any attempt to prove that certain curves (e.g., the circle) in fact form the boundary of simply connected domains. The geometric deficiencies seem to be one of the main reasons that moved WEIERSTRASSto develop function theory on an arithmetic basis; the canonical exposition of WEIERSTRASS'Stheory [5] manages to avoid the use of any integrals with the exception of integrals on real intervals and for the computation of the periods of elliptic functions on pp. 369-402. Of the textbooks of function theory written before JORDAN'S Cours d'analyse, only C. NEUMANN [6] and J. THOMAE[7] seem to have noted the topological problem. NEUMANN'S book is important for the development of geometric function theory since for the first time it introduces the representa- tion of complex numbers on the RIEMANNsphere. The author notes that this is "ein Gedanke, der mir aus Riemann's Vorlesungen durch mfindliche Ueberliefe- rung zu Ohren kam..." ([6], Vorwort zur ersten Auflage, p. v of the second edition). For the treatment of curves, the book is on the level of CAUCHY. As the author notes ([6], Vorwort zur zweiten Auflage, p. viii), "Absichtlich habe ich indessen in dieser Beziehung die Theorie in derjenigen Form, in welcher sie l)on CAUCHY und RIEMANNgegeben ist, zu conserviren gesucht .... Ueberhaupt dtirfte es ja bei der Darlegung einer mathematischen Theorie weniger auf eine durchweg strenge Darstellung, als vielmehr darauf ankommen, dab die angege- benen Methoden die zur strengen Darstellung erforderlichen Mittel gewiihren." THOMAE'S book [7] (his third text of function theory) gives RIEMANN'Sdefinition and (p. 5) declares it intuitively clear that the circle and its topological equivalents are boundaries of simply connected domains. (THOMAE is a co-discoverer of the DARBOUX integral and author of some early papers on set theory applied to analysis.)