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Napier’s bones

By David, 18 June 2013 Activity

How did people multiply before calculators were invented? You could try doing it in your head, but you could also use a mechanical calculator, such as this set of Napier’s bones.

You will need

four strips of paper with numbers on them. The top number on each strip is highlighted - 4, 6, 7, 0

Arrange strips to spell out your number on the first line.

What to do

  1. Thoroughly stick each copy of the Napier’s bones template to some cardboard. This isn’t necessary, but it makes the bones a lot easier to use.
    The third row of each of the trips. The last box is highlighted - it says 0/0

    The units digit of the answer is the last digit on the row.

  2. The template consists of 10 strips (or bones) separated by heavy black lines. Cut along all the heavy black lines to separate your bones.

Multiplying using Napier’s bones

  1. Think of a four digit number – something like 4670.
  2. Look at the top numbers on your bones. Grab a bone with the first digit of your four digit number, then one with the second, then one with the third and one with the fourth. Put them side by side in order, on the board if you wish.
  3. Think of a one digit number to multiply your four digit number by. Then count down the rows to get to that number – so if you picked 3, read the third row.
    the third and fourth columns are highlighted - 2/1 0/0

    Add together the next two digits on the bones to get the second-last digit of your answer.

  4. The row will look something like 1/2 1/8 2/1 0/0. Start from the right side. Write down the right-most digit on a separate piece of paper – in this case it is 0.
  5. Then read the next two digits to the left – they are located diagonally on different bones. Add them together and write the result down to the left of your last digit. In this case, the two digits add to get 1. This means our example will read ‘10’.
  6. Continue to read pairs of digits, adding them together and writing them to the left of the previous digits. If you get an answer with two digits, only write down the units on your paper. When you add the next two digits together, add a 1 to the answer before writing it down. As an example, for the third digit of 4670 x 3 we would add 8 + 2 = 10. Write down 0, and remember to carry a 1.
    These boxes add to 10. Write down the 0, and a note to carry the 1.

    These boxes add to 10. Write down the 0, and a note to carry the 1.

  7. When you reach the left side of the column, there will be a single digit by itself. Write this down, remembering to add 1 if you need to carry. You’ve finished the multiplication!
  8. If you have a calculator handy, you can check your answer. In our example, the answer is 14 010.

What’s happening?

Napier’s bones were invented about 400 years ago by John Napier. They were popular for many years, until much faster mechanical calculators started to appear around 1900.

Remember to carry 1 when the previous digits added up to 10 or more.

Remember to carry 1 when the previous digits added up to 10 or more.

They are remarkably simple – each bone is simply a times table for the digit at the top. Reading down the six rod, you see 6, 12, 18, 24, 30, etc. To simplify carrying, the diagonal lines separate the tens and ones digits.

Applications

Napier’s bones are great for multiplication, but you can also use them to do division. Napier’s bones can also be used to do larger multiplications, and even to find square roots, although you need a separate square root bone to make it possible.

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