The evolution of the Balmer Series has not been well understood because of peculiar historical circumstances. Upon review of earlier publications of spectroscopy, an analysis of Balmer's papers, and a study of biographical information on Balmer, a clearer picture results. Much of the background information used by the mathematician Balmer was conveyed by the physicist Hagenbach and can be traced to a contemporary textbook by the spectroscopist Kayser, published one year prior to Balmer's historic lecture. Since Balmer did not describe all of his mathematical steps, they were reconstructed, using Balmer's own statements as much as possible. Balmer's success in establishing the series is connected with his ability in handling graphic methods of analysis; he referred to such analysis in his earlier paper, later published a geometrical construction of the hydrogen-line series and a text on projective geometry.