Napier's Rods, Abacus and Slide Rule

Napier's rods or bones

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
           +------- thousands

With the "/" on the rods you can see which numbers should be added.

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Abacus

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.

Addition

85607 + 439 = 86046

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.

Of course the above description can be simplified by an user with expirience.

Multiplication

For this example of multiplications we will number the columns from the right. The most right column is register 1. The second column from the right is register 2 etc. The most left column is register 13.
Remark: For multiplications with an abacus the user should know the multiplication tables of 1 to 10.

The example: 58 * 43 = 2494

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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Multiplication with complement

In the sixteenth century on belonged to the intelligent people, if you could read write or calculate. Many people could learn the calculation tables or they forgot them quickly. That is why the calculation books of that time began with multiplication with complements. This is an aid to multiply digits from 5 to 10, by using only digits from 0 to 5. So, the only tables of multiplication that one should know were the tables of 1 to 5.

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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Slide rule

Slide rules are based on the properties of logarithms. For examples logarithms reduce a nultiplication to addition, therefore markinig distances proportional to the logarithms of numbers, instead of to the numbers themselves, enables the user to use distance addition to display a multiplication result. Smart!

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