From earliest antiquity, numerical calculations were devised in mnemonic or manual form, with a high risk of error. In Europe, the use of the abacus for calculation began in the twelfth century. The abacus was a frame fitted with parallel rods that carried sliding beads called calculi, which represented whole numbers. At the same time, "abacus books" began to circulate. These were manuals giving instructions for using the simple instrument, chiefly for money-changing and trade transactions.

In 1617, John Napier, the inventor of logarithms, devised a very simple manual calculation system using numbered rods. Blaise Pascal, in 1642, was the first to design a compact and entirely mechanical calculating machine. The pascaline, as it was called, could add and subtract—and, in a more complex mode, multiply and even divide. The lack of market demand, the cost, and the difficulty of producing the components for the machine explain the scant success of the pascaline, of which only about twenty copies were built. Practical applications also proved limited for the ingenious calculating machines later invented by Tito Livio Burattini and, most notably, by the famous mathematician and philosopher Leibniz.

In the first half of the nineteenth century, Charles Babbage, the pioneer of modern calculus, designed—but never managed to finish—a "differential machine" to automatically compute log tables. He later designed an "analytical machine," which, in some ways, was the forerunner of the modern computer. It was capable of performing arithmetical operations in any sequence. The first calculating machine to enjoy a wide diffusion was the arithmometer of François Charles-Xavier Thomas de Colmar, invented in around 1820 and later improved. It was used extensively by banks and insurance companies from the second half of the nineteenth century onward.