In 1829 Göpel entered the University of Berlin where, in addition to his main area of study which was mathematics, he also took classes in physics, chemistry, and even philosophy, philology, history and aesthetics. He continued to undertake research in mathematics after taking his first degree and he was awarded a doctorate in 1835 for his thesis De aequationibus secundi gradus indeternatis Ⓣ. He had been advised at the University of Berlin by Enno Heeren Dirksen (1788-1850) who had, a few years before, been thesis advisor to Carl Jacobi. After completing his studies at the University, Göpel taught at the Werder Gymnasium and at the Royal Realschule. He then worked as an official in the Royal Library of the Humboldt University of Berlin and had little contact with his mathematical colleagues although, for a while, he was friendly with August Crelle. There seem to be little further information about Göpel's life, the information given above coming from . The authors of  were Carl Jacobi and August Crelle; Jacobi certainly was very familiar with Göpel's mathematics but only Crelle knew him personally. An encyclopaedia entry for Göpel in the Allgemeine deutsche Biographie (1879), written by Moritz Cantor, closely follows the biographical information given in .
Göpel's doctoral dissertation studied periodic continued fractions of the roots of integers and derived a representation of the numbers by quadratic forms. He wrote on Steiner's synthetic geometry and an important work, published after his death, continued the work of Jacobi on elliptic functions. This work was published in Crelle's Journal in 1847.
W Burau in  writes:-
Göpel owes his fame to 'Theoriae transcendentium Abelianarum primi ordinis adumbratio levis', published after his death in the 'Journal für die reine und angewandte Mathematik'. The investigations contained in this paper can be viewed as a continuation of the ideas of C G J Jacobi. The latter had taught that elliptic functions of one variable should be considered as inverse functions of elliptic integrals, but later he also explained them in his lectures as quotients of theta functions of one variable. Moreover, Jacobi had formulated the inverse problem, named for him, for Abelian integrals of arbitrary genus p. From this arose the next task: to solve the problem for p = 2. This was done by Göpel and Johann Rosenhain in works published almost simultaneously. In 'Theoriae transcendentium' .., Göpel started from 16 theta functions in two variables ... and showed that their quotients are quadruply periodic. Of the squares of these 16 functions, four proved to be linearly independent. Göpel linked four more of these quadratics through a homogeneous fourth degree relation, later named the 'Göpel relation' which coincides with the equation of the Kummer surface. Göpel ... finally, after ingenious calculations, obtained the result that the quotients of two theta functions are solutions of the Jacobian problem for p = 2.
Article by: J J O'Connor and E F Robertson