##### This picture shows what were known as "counting tables" [photo courtesy IBM]

##### A typical computer operation back when computers were people.

The

**was an early aid for mathematical computations. Its only value is that it aids the memory of the human performing the calculation. A skilled abacus operator can work on addition and subtraction problems at the speed of a person equipped with a hand calculator (multiplication and division are slower). The abacus is often wrongly attributed to China. In fact, the oldest surviving abacus was used in 300 B.C. by the Babylonians. The abacus is still in use today, principally in the far east. A modern abacus consists of rings that slide over rods, but the older one pictured below dates from the time when pebbles were used for counting (the word "calculus" comes from the Latin word for pebble).**

*abacus*##### A very old abacus

##### A more modern abacus. Note how the abacus is really just a representation of the human fingers: the 5 lower rings on each rod represent the 5 fingers and the 2 upper rings represent the 2 hands.

In 1617 an eccentric (some say mad) Scotsman named John Napier invented ** logarithms**, which are a technology that allows multiplication to be performed via addition. The magic ingredient is the logarithm of each operand, which was originally obtained from a printed table. But Napier also invented an alternative to tables, where the logarithm values were carved on ivory sticks which are now called

**.**

*Napier's Bones*

##### An original set of Napier's Bones [photo courtesy IBM]

##### A more modern set of Napier's Bones

Napier's invention led directly to the ** slide rule**, first built in England in 1632 and still in use in the 1960's by the NASA engineers of the Mercury, Gemini, and Apollo programs which landed men on the moon.

##### A slide rule

Leonardo da Vinci (1452-1519) made drawings of gear-driven calculating machines but apparently never built any.

##### A Leonardo da Vinci drawing showing gears arranged for computing

The first gear-driven calculating machine to actually be built was probably the ** calculating clock**, so named by its inventor, the German professor Wilhelm Schickard in 1623. This device got little publicity because Schickard died soon afterward in the bubonic plague.

##### Schickard's Calculating Clock

In 1642 Blaise Pascal, at age 19, invented the ** Pascaline** as an aid for his father who was a tax collector. Pascal built 50 of this gear-driven one-function calculator (it could only add) but couldn't sell many because of their exorbitant cost and because they really weren't that accurate (at that time it was not possible to fabricate gears with the required precision). Up until the present age when car dashboards went digital, the odometer portion of a car's speedometer used the very same mechanism as the Pascaline to increment the next wheel after each full revolution of the prior wheel. Pascal was a child prodigy. At the age of 12, he was discovered doing his version of Euclid's thirty-second proposition on the kitchen floor. Pascal went on to invent probability theory, the hydraulic press, and the syringe. Shown below is an 8 digit version of the Pascaline, and two views of a 6 digit version:

##### Pascal's Pascaline [photo © 2002 IEEE]

##### A 6 digit model for those who couldn't afford the 8 digit model

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