Bachmann studied mathematics at the University of Berlin and received his doctorate in 1862 for his thesis on group theory. He then went to Breslau to study for his habilitation, which he received in 1864 for his thesis on complex units.
Bachmann was a professor at Breslau and later at Münster.
At the outbreak of Lutheranism, Bachmann sprang into prominence as one of its most energetic opponents.
He was one of that distinguished group of scholars composed of Cochlaeus, Emser, Peter Forst, and Augustin von Alveldt, who, under the direction of John of Schleinitz, Bishop of Meissen, fought the movement in Saxony.
Bachmann gave special attention to the reformation of monastic life and to a defense of the veneration of the Saints.
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