Napier's bones

The method was based on lattice multiplication, and also called rabdology, a word invented by Napier.

In Napier's original design, the rods are made of metal, wood or ivory and have a square cross-section.

In some later designs, the rods are flat and have two tables or only one engraved on them, and made of plastic or heavy cardboard.

If the tables are held on single-sided rods, 40 rods are needed in order to multiply 4-digit numbers – since numbers may have repeated digits, four copies of the multiplication table for each of the digits 0 to 9 are needed.

The answer is read off the row corresponding to the single-digit number which is marked on the left of the frame, with a small amount of addition required, as explained in the examples below.

An intermediate result is produced by the device for multiplication by each of the digits of the smaller number.

To demonstrate how to use Napier's bones for multiplication, three examples of increasing difficulty are explained below.

The numbers lower in each column, or bone, are the digits found by ordinary multiplication tables for the corresponding integer, positioned above and below a diagonal line.

(A blank space or zero to the upper left of each digit, separated by a diagonal line, should be understood, since 1 × 1 = 01, 1 × 2 = 02, 1 x 3 = 03, etc.)

In this example, there are four digits, since there are four groups of bone values lying between diagonal lines.

In this diagram, the third product digit from the yellow and blue bones have their relevant values coloured green.

Each row is evaluated individually and each diagonal column is added as explained in the previous examples.

The sums are read from left to right, producing the numbers needed for the long hand addition calculations to follow.

The rows and place holders are summed to produce a final answer.

Using the abacus, all the products of the divisor from 1 to 9 are found by reading the displayed numbers.

A decimal point is marked after the last digit of the quotient and a zero is appended to the remainder which leaves 163640.

Then, the number in the second column from the sixth row on the square root bone, 12, is set on the board.

At this stage, the board and intermediate calculations should look like this: The numbers in each row are "read", ignoring the second and third columns from the square root bone; these are recorded.

Like before, 8 is appended to get the next digit of the square root and the value of the eighth row, 1024, is subtracted from the current remainder, 1078, to get 54.

The second column of the eighth row on the square root bone, 16, is read and the number is set on the board as follows.

When the board is rearranged, the second column of the square root bone is 6, a single digit.

Now, the largest value on the board smaller than the current remainder, 136499, is 123021 from the ninth row.

The row that has the answer may be guessed by looking at the number on the first few bones and comparing it with the first few digits of the remainder.

If all the digits have been used, and a remainder is left, then the integer part is solved, but a fractional bit still needs to be found.

The ninth row with 1231101 is the largest value smaller than the remainder, so the first digit of the fractional part of the square root is 9.

To find the square root of a number that isn't an integer, say 54782.917, everything is the same, except that the digits to the left and right of the decimal point are grouped into twos.

Thus, in the picture it is immediately clear that: In 1891, Henri Genaille invented a variant of Napier's bones which became known as Genaille–Lucas rulers.

By representing the carry graphically, the results of simple multiplication problems can be read directly, with no intermediate mental calculations.