As we pen these words, it’s becoming increasingly difficult to conceive of a world in which the most complicated calculations were performed using only pencil and paper. But it’s not all that long ago that we were counting on our fingers (and toes in some cases) and using pebbles, sticks, and bones to store and record the results of our calculations …

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Fingers, Toes, and Pebbles
The first tools used as aids to calculation were almost certainly man’s own fingers, and it is not simply a coincidence that the word “digit” is used to refer to a finger (or toe) as well as a numerical quantity. As the need to represent larger numbers grew, early man employed readily available materials for the purpose. Small stones or pebbles could be used to represent larger numbers than fingers and toes and had the added advantage of being able to easily store intermediate results for later use. Thus, it is also no coincidence that the word “calculate” is derived from the Latin word for pebble.



 
Bones with Notches
Bones with Notches The oldest objects known to represent numbers are bones with notches carved into them. These bones, which were discovered in Western Europe, date from the Aurignacian period 20,000 to 30,000 years ago and correspond to the first appearance of Cro-Magnon man. (The term Cro-Magnon comes from caves of the same name in Southern France, in which the first skeletons of this race were discovered in 1868.) Of special interest is a wolf’s jawbone more than 20,000 years old with fifty-five notches in groups of five, which was discovered in Czechoslovakia in 1937. This is the first evidence of the tally system, which is still used occasionally to the present day and could therefore qualify as one of the most enduring of human inventions.

Also of interest is a piece of bone dating from around 8,500 BC, which was discovered in Africa and which appears to have notches representing the prime numbers 11, 13, 17, and 19. Prime numbers are those that are only wholly divisible by the number one and themselves, so it is not surprising that early man would have attributed a special significance to them. What is surprising is that someone of that era had the mathematical sophistication to recognize this quite advanced concept and took the trouble to write it down – not to mention that prime numbers would appear to have had little relevance to the everyday problems of the time such as gathering food and staying alive.




 
Tally Sticks - The Hidden Dangers
The practice of making marks on, or cutting notches into, things to represent numbers has survived to the present day, especially among school children making tally marks on their desks to signify the days of their captivity. In the not-so-distant past, storekeepers (who often could not read or write) used a similar technique to keep track of their customer’s debts. For example, a baker might make cuts across a stick of wood equal to the number of loaves in the shopper’s basket. This stick was then split lengthwise, with the baker and the customer keeping half each, so that both could remember how many loaves were owed for and neither of them could cheat.

Similarly, the British government used wooden tally sticks until the early 1780s. These sticks had notches cut into them to record financial transactions and to act as receipts. Over the course of time, these tally sticks were replaced by paper records, which left the cellars of the Houses of Parliament full to the brim with pieces of old wood. Rising to the challenge with the inertia common to governments around the world, Parliament dithered around until 1834 before finally getting around to ordering the destruction of the tally sticks. There was some discussion about donating the sticks to the poor as firewood; but wiser heads prevailed, pointing out that the sticks actually represented “top secret” government transactions. The fact that the majority of the poor couldn’t read or write and often couldn’t count was obviously of no great significance, and it was finally decreed that the sticks should be burned in the courtyard of the Houses of Parliament. However, fate is usually more than willing to enter the stage with a pointed jape – gusting winds caused the fire to break out of control and burn the House of Commons to the ground (although they did manage to save the foundations)!




 
The Abacus
The first actual calculating mechanism known to us is the abacus, which is thought to have been invented by the Babylonians sometime between 1,000 BC and 500 BC (although some pundits are of the opinion that it was actually invented by the Chinese).

The word abacus comes to us by way of Latin as a mutation of the Greek word abax. In turn, the Greeks may have adopted the Phoenician word abak, meaning “sand,” although some authorities lean toward the Hebrew word abhaq, meaning “dust.” Irrespective of the source, the original concept referred to a flat stone covered with sand (or dust) into which numeric symbols were drawn. The first abacus was almost certainly based on such a stone, with pebbles being placed on lines drawn in the sand. Over time, the stone was replaced by a wooden frame supporting thin sticks, braided hair, or leather thongs, onto which clay beads or pebbles with holes were threaded. A variety of different types of abacus were developed, but the most popular became those based on the bi-quinary system, which utilizes a combination of two bases (base-2 and base-5) to represent decimal numbers. Although the abacus does not qualify as a mechanical calculator, it certainly stands proud as one of first mechanical aids to calculation.




 
John Napier and Napier's Bones
In the early 1600s, a Scottish mathematician called John Napier (1550-1617) invented a tool called Napier’s Bones, which were multiplication tables inscribed on strips of wood or bone.

Napier and Napier's Bones

Napier, who was the Laird of Merchiston, also invented logarithms, which greatly assisted in a variety of arithmetic calculations. In 1621, an English mathematician and clergyman called William Oughtred used Napier’s logarithms as the basis for the slide rule (Oughtred invented both the standard rectilinear slide rule and the less commonly used circular slide rule). Once again, however, although the slide rule was an exceptionally effective tool that remained in common use for over three hundred years, it also does not qualify as a mechanical calculator.



Note: The material presented here was abstracted and condensed from The History of Calculators, Computers, and Other Stuff document provided on the CD-ROM accompanying our book How Computers Do Math (ISBN: 0471732788).