Invented by: John Napier Year: 1617 Picture of collection: Private collection of G.W.J. Beckers

The working principal of the bones is explained in "Rabdologiae". The bones are an aid for multiplication and division. Even squares roots and powers could be done.

Here is explained how you can reduce multiplication to simple additions with Napier's rods or bones. Each bone contains a multiplication table of a number from 0 to 9.

A multiplication 6 x 5 can be made with the bone with the table of five by looking at the sixth row. The number of a row can be indicated with an index rod. Multiplications of a numeral (i.e. 6) with large numbers (i.e. 739) can easely be made with the rods of 7, 3 and 9 placed next to each other and with the index rod in front. The row with the index numeral 6 gives the result of (6 x 739) if the units, tens, hundreds and thousands are added.

6 x 7 = 4 2 (6 times 700) 6 x 3 = 1 8 (6 times 30) 6 x 9 = 5 4 (6 times 9) ----------------- 6 x 739 = 4 4 3 4 | | | | | | | +- units | | +--- tens | +----- hundreds +------- thousandsWith the "/" on the rods you can see which numbers should be added.

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An Abacus consists of a rectangular frame carrying a number of rods or wires. A transverse bar (centre bar) divides each of these rods into two unequal portions. On the upper smaller portion of each rod are two beads and on the lower portion five beads. The whole acts as a numerical register, each rod representing a decimal order as in our "arabic" notation. There is rather more space on the rods than is occupied by the beads, which are accordingly free to slide a short distance towards or away from the centre bar. The position of the beads on a particular rod represent a digit in that particular decimal position.

- Zeroise the registers by pushing all the beads away from the centre
bar.

- Set the number 85607. The seven in the most right column. The 0 in the
second column, the 6 on the third column, the 5 on the fourth and the 8 on the
fifth column from the right. The seven will be indicated by pushing one bead
of the upper portion and two beads of the lower portion to the centre bar. The
bead of the upper portion represents a 5 and the two beads of the lower
portion represents 1+1 = 2. Together this will make 7. The 6 in the third
column from the right can be made with 5 + 1 and the 8 with 5 + 1 + 1 + 1 (one
bead of the upper portion and three of the lower portion).

- We split the addition of 439 in an addition with 9, 30 and 400.

- Add a 9 to the most right column. We can split the 9 as 5 + 1+1+1+1. First
add the 5 with a bead of the upper part. By doing this we can read the result
5+5 + 1+1 = 12 in this column. This is more than 9 and so we reduce this
column with 10 (5+5) and add 1 on the second column from the right. After that
we can add 1+1+1+1 at the most right column by moving the beads of the lower
portion one by one to the centre bar. After three beads all the beads of the
lower part are moved to the centre bar and we can not add the last bead. We
can egalise the five beads of the lower portion with one bead of the upper
portion (move the beads of the lower portion away from the centre bar and move
one bead of the upper portion to the centre bar). Finally we add the last bead
by moving one bead of the lower portion back to the centre bar again. We can
read the result of 85607 + 9 = 85616.

- Add 30 by adding 3 at the second column from the right. In this case we
can move 3 beads of the lower portion to the centre bar.

- Add 400 by adding 4 at the third column from the right. Move 4 beads of
the lower portion to the centre bar. The result in this column is 10. Zeroise
this column and add 1 add the fourth column from the right.

- Read the result: 86046.

Remark: For multiplications with an abacus the user should know the multiplication tables of 1 to 10.

- Zeroise the registers by pushing all the beads away from the centre
bar.

- Set the number 58 in register 13 and 12.

- Set the number 43 in register 10 and 9.

- Start the multiplication with 8 * 43. This calculation can be split in 8 * 3 = 24 and 8 * 4(0) = 32(0) . Put the result of 8 * 3 = 24 in register 2 and 1. The result of 8 * 4 = 32 can be added at register 3 and 2. The multiplication with 8 is now finished and so we can zeroise register 12.
- Then we can multiply with 5(0): 5(0) * 43. Again we can split this
multiplication in 5(0) * 3 =15(0) and 5(0) * 4(0) = 20(00). The result of 5 *
3 = 15 can be added at register 3 and 2 and the result of 5 * 4 = 20 can be
added at register 4 and 3. Register 13 can be zeroised.

- The result (2494) can be red at the registers 4 till 1.

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The method is as follow:

Behind the numbers that are multiplied (e.g. 8*6) the complement till ten is placed.We put the diagonal sign and calculate the result as: the first number is the substract of the numbers on both sides of one diagonal; the second number product of the complements. So 8-4=4 (or 6-2=4) is the first number of the result and the second number of the result is 2*4=8. The proof: a*b=10*(a-(10-b))+(10-a)*(10-b).

Remark: There is a theory that the multiplication sign originates from the cross.

Some other examples:

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