Very quickly two different and competing approaches became visible, as pursued by André-Marie Ampère on the one hand and Jean Baptiste Biot on the other. Both of the two researchers worked experimentally and theoretically at the same time, and both dealt with the more or less the same realm of effects. But significant differences were visible from the beginning, concerning the experimental approach, the mathematical methods, and the scope of the resulting claims. A detailed study of the episode well provides insight into the early history of one of the most important research fields of 19th century physics, but also into characteristic forms, conditions and implications of processes of mathematization.