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Napier's bones


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Napier's bones were invented by John Napier (1550-1617), a Scottish mathematician and scientist. They help you to do multiplication. There are 9 different 'bones' or strips with numbers on - see below. You may need several copies of each strip.

Napier's bones

If you look at these strips, you will see that they are really the times tables. Each square gives 2x, 3x, and so on, but the tens and units are divided by a slanting line. You also need a frame to fit them in. The side with the numbers and the bottom are raised slightly, so you can slot the number strips next to them.


How to use Napier's bones

Napier's bones are good for multiplying a long number by a single digit number. Let's multiply 425928 by 7. First take the strips for 4, 2, 5, 9, 2 and 8, and fit them into the frame. They must fit snugly.

Since we are multiplying by 7, you need to look at the squares next to the 7 on the side. It is coloured yellow on this diagram, but of course you don't get this help with the real bones!

Now you can work out the answer very quickly. You read off the digits, but any numbers within slanting lines must be added. So the answer is 2 8+1 4+3 5+6 3+1 4+5 6 or 2 9 7 11 4 9 6. Most of these digits are fine, but the 11 needs to have 10 carried to the left. This makes 2 9 7+1 1 4 9 6 or 2981496, which is the right answer!

If you do the same sum on paper, you will see what is going on. When you start the calculation, you multiply the end digit, 8, by 7, which is 56. You put the 6 at the end, and carry the 7. Now you multiply the next digit, 2, by 7. This is 14, but you need to add the 5 carried from the previous calculation. So you add the 4 and 5, and carry the 1. You can see the same addition happening with the Napier's bones. They provide a mechanism for the carrying, and do the simple multiplication for you. Here are the rest of the calculations for you to compare. Remember that an extra 1 had to be carried at one point.

425928
x7
6
5
425928
x7
96
15
425928
x7
496
615
425928
x7
1496
3615
425928
x7
81496
14615
425928
x7
2981496
14615

Interactive Napier's bones

Try using the Napier's bones below. Click on 'Get multiplication'. Then click on the bones on the right to produce the bigger number in the frame. When you have them all, you will be asked for the answer, which you can read off the correct line of the frame (see above).

               


                                

Using Napier's bones for long multiplication and division

Napier's bones can be used for more serious long multiplication and long division as well. You write out the calculation as if you were doing the whole thing yourself, but each multiplication be done using Napier's bones, which saves quite a bit of work!