Computing by machine started (as near as we now know) in the mid-east with the use of counting stones in channels. This was the precursor of a counting instrument invented by the Babylonians and normally associated with the Chinese: The abacus.

All about the abacus

The abacus reigned supreme for a great while because to use it you really didn't have to know anything about the theory of numbers. The uneducated could be trained to use it easily.

Math with Arabic numbers entered Europe in the 8^th and 9^th centuries. It did not catch on because the user had to understand theory. To help, various mechanical devices were invented.

In the early 1600s Napier (the inventor of logarithms) developed a series of rods that could be used for multiplication. Partial products appeared on the rods and all the user had to do was add them to get the final product.

Napier's rods

This led to ever more complicated mechanical devices based on gears and rods, with Blaise Pascal's mechanical Pascaline being the most famous.

Pascaline

These all lead us to 1791...

Babbage