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The first calculators
Logarithms and slide rules are a step towards making arithmetic easier but they are not really mechanical calculators. The credit for the first such calculator usually goes to Pascal but recently an earlier machine was discovered. Wilhem Schickard built an adding machine based on cogs and gears in 1623 for the astronomer Kepler.
Wilhem Schickard (1592-1635)
Wilhem Schickard's calculator made use of a set of Napier's Bones to work out the partial products and then a simple counter mechanism to add them up. Unfortunately the one made for Kepler was destroyed in a fire and Schickard's original machine has gone missing. However despite not having the evidence of the machine itself there is evidence that one existed - i.e. Schickard actually built it rather than just drew it and it worked.
A replica Schickard machine
Even though he wasn't first, the French mathematician, physicist and philosopher, Blaise Pascal does deserve the credit of inventing, in 1642, an adding machine that was produced in small numbers. It is said that he invented it to help his father, an accountant, do the tedious manual work of adding up the books.
Blaise Pascal (1623-1662)
The Pascaline was a very simple "mileometer" type counter. You dialled in the numbers to be added and the result appeared in windows along the top of the machine. The only real innovation was the use of 9's complement arithmetic to avoid having to wind the wheels backwards to perform subtraction. The complement idea only works if you are using a numeric representation that wraps round. For example, if you have a two digit counter that rolls over at 99 then adding 70 to 30 gives the result 00 - 70 plus 29 gets the counter to 99 and one more makes it roll over to 00. If you think about this in a slightly different way, the way a mathematician like Pascal would think about it, 70 behaves like -29 because when you add it to 29 you get 0. You should also be able to see that the complement of any number can be found by subtracting it from 99 or however many 9s the counter goes up to - hence "9s complement".
The Pascaline was of great interest and Pascal and his father thought that it would make their fortunes. But it didn't -partly because clerks and accountants were frightened that it might threaten their jobs! Although the Pascaline could have speeded up many tasks and reduced tedium, clerks and accountants were relatively cheap and so the employers had no real incentive to pay good money to improve their working conditions.