Jacobi's work on Abelian and theta-functions was greatly extended by Adolph Göpel (1812–1847), professor in a gymnasium near Potsdam, and Johann Georg Rosenhain of Königsberg (1816–1887). Göpel in his Theoriœ transcendentium primi ordinis adumbratio levis (Crelle, 35, 1847) and Rosenhain in several memoirs established each independently, on the analogy of the single theta-functions, the functions of two variables, called double theta-functions, and worked out in connection with them the theory of the Abelian functions of two variables. The theta-relations established by Göpel and Rosenhain received for thirty years no further development, notwithstanding the fact that the double theta series came to be of increasing importance in analytical, geometrical, and mechanical problems, and that Hermite and Königsberger had considered the subject of transformation. Finally, the investigations of G. W. Borchardt of Berlin (1817–1880), treating of the representation of Kummer's surface by Göpel's biquadratic relation between four theta-functions of two variables, and researches of H. H. Weber of Marburg, F. Prym of Würzburg, Adolf Krazer, and Martin Krause of Dresden led to broader views. Researches on double theta-functions, made by Cayley, were extended to quadruple theta-functions by Thomas Craig of the Johns Hopkins University.
Starting with the integrals of the most general form and considering the inverse functions corresponding to these integrals (the Abelian functions of variables), Riemann defined the theta-functions of variables as the sum of a -tuply infinite series of exponentials, the general term depending on variables. Riemann shows that the Abelian functions are algebraically connected with theta-functions of the proper arguments, and presents the theory in the broadest form.[56] He rests the theory of the multiple theta-functions upon the general principles of the theory of functions of a complex variable.
