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Napier's Bones - a 17th Century Aid to Calculation

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A set of Napier's Bones

Napier's Bones or Napier's Rods are a calculating device for doing multiplication and division, invented by Scottish laird1, mathematician and Apocalypse-predictor John Napier (1550–1617). Napier is most famous for his invention of logarithms, but the Bones are not based on logarithms - they are a sort of dissected multiplication table.

At the time the device was invented, arithmetic was not common knowledge - most people could add and subtract but multiplication and division were considered difficult and were prone to mistakes. For example, famous diarist Samuel Pepys (1633–1703) did not learn the multiplication table until he was 29. The Bones simplified part of the processes of multiplication by eliminating the need to remember the multiplication table, although the full process was still fairly manual.

The device has been constructed in a number of ways, but typically consists of a wooden frame and a set of 10 or 20 rods. The rods may be made of wood, metal or plastic. In the past they would often be made of ivory2, which looks like bone, and it is this that gave them the nickname Napier's Bones. Each rod has a square cross-section and has a set of numbers written on each face. Rods are selected and placed in the frame, and the answer can then be read off on a particular row of the device, with a small amount of addition required to calculate the final answer.

Napier published his invention in 1617 in a book in Latin called 'Rabdologiæ, seu Numerationis per Virgulas'. This is usually known by the single-word title 'Rabdologia'. Napier wrote the book in Latin because it was the language of scientific discourse in the 17th Century and could be understood by scientists and mathematicians across Europe. The title translates as 'Of Rabdology, or Calculating with Rods'. The term Rabdology was one that Napier invented himself, from the Greek words 'rhabdos' (a rod) and the ending '-ology' (the knowledge or study of something).

The book included two other inventions by Napier. One was a more complicated device which he called the Promptuary. This also did multiplication and did not require intermediate results to be written down, but it couldn't do division and contained a lot of separate parts (typically more than a hundred) making it difficult to construct, so it never became popular. The other invention was a very simple device, basically just a large chess board with counters. It relied on binary or base 2 notation, writing a number in terms of just ones and zeroes. Napier showed how addition, subtraction, multiplication and division were all much easier to do in binary, but the disadvantage was that numbers had to be converted into binary before the operations and the results had to be converted back to decimal.

Incidentally, the book also includes the first published use of a decimal point, although the 'point' is actually a comma. Napier is thought to have invented this notation although he didn't invent the principle of decimals.

Description of Napier's Bones

Napier's Bones use a dissected multiplication table so first we will show the multiplication table for digits up to 9 in a slightly unusual way:

The multiplication table with diagonal lines across each cell and one digit in each triangle

Each of these vertical strips is the multiplication table for a single digit, showing the digit in the top square and the multiples of it in the squares below. (Napier used the term 'simple' for the number at the top of the strip.)

  • Each square containing a multiple is divided into two by a diagonal.

  • If the multiple is a single digit, it is written to the right of the diagonal. The space to the left remains blank.

  • If it is a two-digit number, the first digit is written to the left and the second digit to the right of the diagonal.

We can use these strips for multiplication and division3. Napier's invention uses rods of square cross-section with the faces engraved with these strips. An example is shown in the picture at the top of this Entry. The diagonal lines and the top and bottoms of the squares should be marked on the rods in ink, but the left and right sides of the squares should not be marked on the rods. Ten rods engraved with 40 of these strips are enough to handle multiplication of a four-digit number, or indeed any number less than 11,111, as long as each rod contains four different strips. Alternatively, you can put the strips on two faces of a flat strip which can be made of wood, metal, plastic or cardboard. For best results it should be possible to put the strips beside each other so that they touch without overlapping, so thick card is better than paper.

Napier himself suggested that the rods could be made from silver, wood or ivory. Boxwood is a suitable light-coloured hard wood, with the numbers being marked in black, while ebony (a dark hard wood) can be used if the numbers are white.

Simple Multiplication using Napier's Bones

A set of 10 rods can easily be used to multiply any four-digit number by a single digit. For example, we want to multiply 4383 by 7. We select rods with the faces for the digits 4, 3, 8 and 3 and place then side by side in the frame, up against the left side of the frame. The 'singles' across the top will show the number 4383 to be multiplied. The frame will look as follows:

The multiplication table for the digits 4383

We now look at the row 7 as given by the number 7 on the left of the frame, ignoring all the other rows. We also ignore the lines formed where the rods touch each other. We see numbers across the row as follows:

Row 7 - 2 / 8 2 / 1 5 / 6 2 / 1

We read from right to left:

  • There is a 1 in the rightmost position. This becomes the rightmost digit of the result.

  • The next digit is in the parallelogram formed in the space between the diagonals: 6 2. We add them to get the digit 8.

  • The next digit from the right is in the next space between diagonals: 1 5 which gives us 6.

  • Next is 8 2 which gives us 10, so the digit is 0 with a carry of 1.

  • This carry is added into the next space which in this case contains just the digit 2 so the final, left-most digit is 3.

  • The result of the multiplication is therefore 30681.

Multiplying Two Multi-digit Numbers Together

The device is designed to multiply a long number by a single digit. If you want to multiply a long number by another long number, you will have to use the device in combination with pen and paper.

  • Select the rods for the bigger number and place them in the frame.

  • Use the device to multiply the bigger number by the rightmost digit of the smaller number. Write the result on a piece of paper, against the right margin.

  • Use the device to multiply the bigger number by the next digit of the smaller number, writing the result below the first number and shifted one digit to the left. Write a 0 at the end of it to line it up with the first number.

  • Use the device to multiply the bigger number by the next digit of the smaller number, writing the result below the others and again shifting one digit to the left. Again add zeroes at the end to get it to line up with the results already written.

  • Keep going until you have used all the digits of the smaller number.

  • Now add up all the numbers you have written down. The result is the result of the multiplication.

Note that this is almost exactly the same as the traditional pen and paper method of multiplication, but the Bones remove the need to remember the multiplication tables.

Example: To Multiply 137 by 4383

We set up the bigger number 4383 on the Bones exactly as in the previous example and multiply it by each of the digits of 137 starting with the last one. Select the rods for 4, 3, 8 and 3 from the set and place them at the left of the frame.

To multiply 4383 by 7, we read off row 7 of the Bones, adding as appropriate: 2, 8+2, 1+5, 6+2,1 giving 30681. We write this on our sheet:


Next we multiply 4383 by 3, reading off row 3 of the Bones: 1, 2, 9+2, 4, 1 giving 13149. We write this below the first result, shifted one digit to the left and add a zero to line it up:

30681 131490

Finally we multiply 4383 by 1. We obviously don't have to refer to the Bones for this. We write it under the other results, shifted by another digit and with two zeroes after it:

30681 131490 438300

We now add the three numbers to get the result of the multiplication:

30681 131490 438300 ------ 600471

So 137 × 4383 = 600471.

Popularity of the Bones

Napier explained the Bones to his friends and they became popular. In 1617 when he was nearing the end of his life, he decided to publish a description of them in book form, mainly to prevent anyone else from claiming that they had invented them. This was 'Rabdologia' which we've already described earlier.

In the following 50 years or so, many other authors published descriptions of the Bones, or suggested improvements to them. Some of these credited the invention to Napier, while others described the Bones without any mention of their inventor.

As an example of an improvement, one author suggested writing the tables on cylinders and mounting them in a box in such a way that they can be rotated to show the table for any digit. This allows a number to be set up on the device very quickly, although the reading off of the result is not quite as easy. Another author suggests flat strips with two tables on each strip rather than the square-cross-section rods. The number of rods needed is doubled, but the manufacturing is simpler.

By the 18th Century, however, most accounts of the Bones seem to describe them as a device which is no longer in popular use, even though no other calculating device is yet available. Accounts generally say that the rods are troublesome to use and that it is less trouble to memorise the multiplication table than to always carry around a set of rods and fiddle around with finding the right one for the occasion.

In the 19th Century, mechanical calculators started to appear that could perform multiplication and division without any need for the operator to add up results or record intermediate values, making the Bones redundant. Nevertheless there were two new developments:

  • someone came up with the idea of sloping the rods at an angle of about 60 degrees. This meant that instead of reading numbers from a parallelogram, you could read them from a rectangle, making it easier to see which digits had to be added together. There seem to be no examples of rods made in this manner, so it may have never caught on.

  • a French mathematician called Henri Genaille invented a set of rods for multiplying a long number by a single digit in which the result can be read off without any need to add any digits together at all. These 'Genaille-Lucas Rulers' appear to have been purely an intellectual exercise rather than as a genuine aid to multiplication.

Appendix - Napier's Design

Napier's original description differs in a few details from the most popular versions made during the 17th and 18th Centuries:

  • The rods were very small - only 6mm wide and 60mm long. Napier specified them as roughly 3 'fingers' long, where a finger is 3/4 of an inch.

  • Although there are only 9 squares on the rod, Napier specified that the rods are 10 times as long as they are wide. He left a half square gap at the top and bottom of each rod.

  • He didn't use a frame or an index rod like the one shown in the picture at the top of this Entry. He assumed people would count down the rod to the appropriate row.

  • He didn't have a big digit in the top square. It was divided by a diagonal like all the other squares and the 'simple' was the same size as any other digit on the rod.

  • He arranged the tables on the faces of the rod so that simples adding up to 9 were always on opposite faces - 0 being opposite 9, 1 being opposite 8 and so on. A really unusual feature is that the tables for the digits 5, 6, 7, 8 and 9 were written in the opposite direction along the rod, from bottom to top. This was so that the entire block of rods could be turned over top to bottom and the other side would display what Napier called the 'opposite' of the original number. Nowadays we call this the 'nines complement' of the number, the number got by replacing each digit by 9 minus the digit. It can be used to do an additional check on the calculation to make sure no mistakes were made.

  • Napier's design had a couple of digits engraved on the end of the rod, to help in finding the correct rod.

A set of 10 rods will contain each digit four times as a 'single', and Napier arranged them on the rods as follows:

0198, 0297, 0396, 0495, 1287, 1386, 1485, 2376, 2475, 3465

This arrangement ensures that a set of 10 rods can multiply any number of up to 4 digits in length and also many bigger numbers, as long as they don't have too many repeated digits. A set of 20 rods (2 identical sets of 10) can multiply any number of up to 8 digits in length and 30 rods (3 identical sets of 10) can handle numbers up to 12 digits.

1In Scotland, a laird is the owner of a large estate of land.2It is no longer legal to make things from ivory, as the ivory industry is driving elephants to extinction.3They assist with the multiplication involved in long division.

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