While the operation of multiplication is presented by historiography as an unproblematic notion which did not see substantial evolution, a polemic by Ampère against Bézout's arithmetic textbook, highly popular in France over many decades, is used here to unravel decisive restrictions imposed on multiplication in various mathematical cultures since Old Babylonian times. The analysis illustrates not only the controversy about the existence of a Greek "geometric algebra," but makes accessible also reflections on non-commutativity of operations in the 18^th century already. The concept of multiplication thus underwent characteristic changes in the process of generalising mathematics.