John Napier revolutionized mathematics in the 17th century with his discovery of logarithms and the decimal point. He was able to devise a method of computation using logarithms that could easily be solved with tables. He simplified this further by creating "Rabdologia", which were popularly called Napier's Bones. These were a series of ten sticks inscribed with numbers, one for each digit from 0 to 9. These were used for centuries by anyone who had to do complicated mathematics, although they are somewhat obsolete in the computer age. But for the historically curious who are willing to pardon my poor ASCII, a set of Napier's Bones would be laid out something like this.

+--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0 | | 1| | 2| | 3| | 4| | 5| | 6| | 7| | 8| | 9| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0/| |0/| |0/| |0/| |0/| |1/| |1/| |1/| |1/| |1/| |/0| |/2| |/4| |/6| |/8| |/0| |/2| |/4| |/6| |/8| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0/| |0/| |0/| |0/| |1/| |1/| |1/| |2/| |2/| |2/| |/0| |/3| |/6| |/9| |/2| |/5| |/8| |/1| |/4| |/7| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0/| |0/| |0/| |1/| |1/| |2/| |2/| |2/| |3/| |3/| |/0| |/4| |/8| |/2| |/6| |/0| |/4| |/8| |/2| |/6| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0/| |0/| |1/| |1/| |2/| |2/| |3/| |3/| |4/| |4/| |/0| |/5| |/0| |/5| |/0| |/5| |/0| |/5| |/0| |/5| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0/| |0/| |1/| |1/| |2/| |3/| |3/| |4/| |4/| |5/| |/0| |/6| |/2| |/8| |/4| |/0| |/6| |/2| |/8| |/4| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0/| |0/| |1/| |2/| |2/| |3/| |4/| |4/| |5/| |6/| |/0| |/7| |/4| |/1| |/8| |/5| |/2| |/9| |/6| |/3| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0/| |0/| |1/| |2/| |3/| |4/| |4/| |5/| |6/| |7/| |/0| |/8| |/6| |/4| |/2| |/0| |/8| |/6| |/4| |/2| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ |0/| |0/| |1/| |2/| |3/| |4/| |5/| |6/| |7/| |8/| |/0| |/9| |/8| |/7| |/6| |/5| |/4| |/3| |/2| |/1| +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+

Merchants would carry a set of these, usually made of wood or ivory (hence the term Bones in the name), and with these could quickly perform multiplication with simple addition. After Napier's death in 1617, improvements were made which allowed division, square roots, and cube roots, with a set of logarithm tables. A quick study of the rods shows that they are simply multiplication tables for each of the digits. Take, for example, 37826 * 7. We line up the sticks for each digit and get 21, 49, 56, 14, and 42 on the seventh row down.

3 * 7 = 21 7 * 7 = 49 8 * 7 = 56 2 * 7 = 14 6 * 7 = 42 ---------------- 264782

With practice this could be done very easily, visually, with a set of rods. All you have to do is add the numbers in the same diagnal row for each digit.

2/ 4/ 5/ 1/ 4/ /1 + /9 + /6 + /4 + /2 2/1+4/9+5/6+1/4+4/2

Then, from right to left, carrying the extra digits, it very easily comes out to 264782.

To multiply by a value greater than 9, simply get the result for each digit and multiply to get into the correct place. For example,

547 * 56 = 547 * 50 = 2/5 + 2/0 + 3/5 = 2735 * 10 = 27350 547 * 6 = 3/0 + 2/4 + 4/2 = 3282 27350 +3282 ----- 30632

These were very popular in Napier's day, and were what he was primarily known for at the time. Elaborate sets with leather cases and logarithm tables inside the case were not uncommon among the wealthier merchants. Slide rules are often (incorrectly) called Napier's Bones, which in reality are the ancestors of the slide rule. There were several refinements made in between, including a set which required no addition at all, simply following lines marked on the rods.

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The Joy of Mathematics - Theoni Pappas
http://www.cee.hw.ac.uk/~greg/calculators/napier/index.html - Undusting Napier's Bones
http://www.tased.edu.au/schools/rokebyh/curric/infotech/stage1/assign2/napier.htm - Make your own Napier's Bones
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