Napier's Bones

Napier's bones are an Abacus invented by John Napier for Calculation of products and quotients of numbers. Also called '''Rabdology''' (from Greek ραβδoς [rabdos], rod and λóγoς [logos], word). Napier published his invention of the 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. More advanced use of the rods can even extract Square Root s. Note that Napier's bones are not the same as Logarithm s, 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 . The Napier's rods consist of strips of wood, metal or heavy cardboard. '''Napier's bones''' are three dimensional, square in cross section, with four different '''rods''' engraved on each one. 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 9 rods corresponding to digits 1 to 9. The figure additionally shows the rod 0; although for obvious reasons it is not necessary for calculations.

MULTIPLICATION

Given the described set of rods, suppose that we wish to calculate the product of 46785399 and '''7'''. Place inside the board the rods corresponding to 46785399, as shown in the diagram, and read the result in the horizontal strip in row 7, as marked on the side of the board. 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. Note that in the hundred thousands place, where 5+9=14, we note '4' and carry '1' to the next addition (similarly with 4+8=12 in the ten millions place).

In cases where a digit of the multiplicand is 0, we leave a space between the rods corresponding to where a 0 rod would be. Let us suppose that we want to multiply the previous number by 96431; operating analogously to the previous case, we will calculate partial products of the number by multiplying 46785399 by 9, 6, 4, 3 and 1. 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 multipied 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 horizonatlly and vertically until they 'meet' at a diagonal line, the point the travells out of the grid in the same method and again 'falls' into the answer.

DIVISION

''Note: This section and those below it need revision.''

Division can be performed in a similar fashion. 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. Using the abacus, find all the products of the divisor and 1 to 9 by reading the displayed numbers. 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. Next, look for the greatest partial product that is less than the truncated dividend. In this case, it's 385724. You must mark down two things, as seen in the diagram: since 385724 is in the '4' row of the abacus, mark down a '4' as the left-most digit of the quotient; also write the partial product, left-aligned, under the original dividend, and subtract the two terms. You get the difference as 8212999. Repeat the same steps as above: truncate the number to six digits, chose the partial product immediately less than the truncated number, write the row number as the next digit of the quotient, and subtract the partial product from the difference found in the first repetition. Following the diagram should clarify this. Repeat this cycle until the result of subtraction is less than the divisor. The
number left is the remainder.

So in this example, we get a quotient of 485 with a remainder of 16364. We can just stop here and
use the fractional form of the answer

485rac{16364}{96431}

.

If you prefer, we can also find as many decimal points as we need by continuing the cycle as in
standard Long Division . Mark a decimal point after the last digit of the quotient and append a zero
to the remainder so we now have 163640. Continue the cycle, but each time appending a zero to the
result after the subtraction.

Let's work through a couple of digits. The first digit after the decimal point is
1, because the biggest partial product less than 163640 is
96431, from row 1. Subtracting 96431 from 163640, we're left with 67209.
Appending a zero, we have 672090 to consider for the next cycle (with the partial result
485.1)
The second digit after the decimal point is 6, as the biggest partial product less
than 672090 is 578586 from row 6. The partial result is now 485.16, and so on.

EXTRACTING SQUARE ROOTS

Extracting the square root uses an additional bone which looks a bit
different from the others as it has three columns on it. The first
column has the first nine squares 1, 4, 9, ... 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:

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