Rabdo
Also called Rabdology (from Greek ῥάβδoς , "rod" and -λογία , "study"). Napier published his version of rods in a work printed in Edinburgh, Scotland, at the end of 1617 entitled Rabdologiæ.
Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations and division to subtractions. Note that Napier's bones are not the same as logarithms, with which Napier's name is also associated.
The abacus consists of a board with a rim; the user places Napier's rods in the rim to conduct multiplication or division.
The board's left edge is divided into 9 squares, holding the numbers 1 to 9. A set of such bones might be enclosed in a convenient carrying case.
A rod's surface comprises 9 squares, and each square, except for the top one, comprises two halves divided by a diagonal line.
The first square of each rod holds a single-digit, and the other squares hold this number's double, triple, quadruple and so on until the last square contains nine times the number in the top square. The digits of each product are written one to each side of the diagonal; numbers less than 10 occupy the lower triangle, with a zero in the top half.
A set consists of 10 rods corresponding to digits 0 to 9.
To obtain the product, simply note, for each place from right to left, the numbers found by adding the digits within the diagonal sections of the strip (using carry-over where the sum is 10 or greater).
From right to left, we obtain the units place (3), the tens (6+3=9), the hundreds (6+1=7), etc. Then we place these products in the appropriate positions, and add them using the simple pencil-and-paper method.
This method can also be used for multiplying decimals.
For a decimal value multiplied by an integer (whole number) value ensure that the decimal number is written along the top of the grid. From this position the decimal point simply drops down the vertical line and 'falls' into the answer.
When multiplying two decimal numbers together, the decimal points travel horizontally and vertically until they 'meet' at a diagonal line, the point then travels out of the grid in the same method and again 'falls' into the answer.
The form of multiplication was also used in the 1202 Liber Abaci and 800 AD Islamic mathematics and known under the name of lattice multiplication.
Let's divide 46785399 by 96431, the two numbers we used in the earlier example. Put the bars for the divisor (96431) on the board, as shown in the graphic below.
Note that the dividend has eight digits, whereas the partial products (save for the first one) all have six. So you must temporarily ignore the final two digits of 46785399, namely the '99', leaving the number 467853.
Continue the cycle, but each time appending a zero to the result after the subtraction.
Let's work through a couple of digits. 64, 81, the second column has the even numbers 2 through 18, and the last column just has the numbers 1 through 9.
Let's find the square root of 46785399 with the bones.
First, group its digits in twos starting from the right so it looks like this:
46 78 53 99
Note: A number like 85399 would be grouped as 8 53 99
Start with the leftmost group 46.
Pick the largest square on the square root bone less than 46, which is 36 from the sixth row.
Because we picked the sixth row, the first digit of the solution is 6.
Now read the second column from the sixth row on the square root bone, 12, and set 12 on the board.
Then subtract the value in the first column of the sixth row, 36, from 46.
Append to this the next group of digits in the number 78, to get the remainder 1078.
At the end of this step, the board and intermediate calculations should look like this:
_____________
√46 78 53 99 = 6
36
--
10 78
Now, "read" the number in each row (ignore the second and third columns from the square root bone.) For example, read the sixth row as
0/6 1/2 3/6 → 756
Now find the largest number less than the current remainder, 1078. So you should set the board to
12 + 1 = 13 → append 6 → 136
Note: If the second column of the square root bone has only one digit, just append it to the current number on board.
The board and intermediate calculations now look like this.
_____________
√46 78 53 99 = 68
36
--
10 78
10 24
-----
54 53
Once again, find the row with the largest value less than the current partial remainder 5453.
This time, it is the third row with 4089.
_____________
√46 78 53 99 = 683
36
--
10 78
10 24
-----
54 53
40 89
-----
13 64
The next digit of the square root is 3. When you rearrange the board, notice that 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.
In practice, you often don't need to find the value of every row to get the answer.