Napier's bones Guide, Meaning , Facts, Information and Description

This article started off as a machine translation of an article from the Spanish Wikipedia. It needs lots of revision and editing before it is usable here. This is a work in progress.

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 find the product, simply add the numbers within the diagonal sections of the strip.

Division

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

         _____________
        √46 78 53 99    =    6
         36
         --
         10 78

         _____________
        √46 78 53 99    =    68
         36
         --
         10 78
         10 24
         -----
            54

         _____________
        √46 78 53 99    =    68
         36
         --
         10 78
         10 24
         -----
            54 53

         _____________
        √46 78 53 99    =    683
         36
         --
         10 78
         10 24
         -----
            54 53
            40 89
            -----
            13 64

         _____________
        √46 78 53 99    =    683
         36
         --
         10 78
         10 24
         -----
            54 53
            40 89
            -----
            13 64 99

         _____________
        √46 78 53 99    =    6839
         36
         --
         10 78
         10 24
         -----
            54 53
            40 89
            -----
            13 64 99
            12 30 21
            --------
             1 34 78

Similar to finding the fractional portion of the answer in long division, append two zeros to the remainder to get the new remainder 1347800. The second column of the ninth row of the square root bone is 18 and the current number on the board is 1366. So compute

1366 + 1 → 1367 → append 8 → 13678

The board and intermediate computations now look like this.

         _____________
        √46 78 53 99    =    6839.
         36
         --
         10 78
         10 24
         -----
            54 53
            40 89
            -----
            13 64 99
            12 30 21
            --------
             1 34 78 00

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.

         _____________
        √46 78 53 99    =    6839.9
         36
         --
         10 78
         10 24
         -----
            54 53
            40 89
            -----
            13 64 99
            12 30 21
            --------
             1 34 78 00
             1 23 11 01
             ----------
               11 66 99

Subtract the value of the ninth row from the remainder and append a couple more zeros to get the new remainder 11669900. The second column on the ninth row is 18 with 13678 on the board, so compute

13678 + 1 → 13679 → append 8 → 136798

         _____________
        √46 78 53 99    =    6839.9
         36
         --
         10 78
         10 24
         -----
            54 53
            40 89
            -----
            13 64 99
            12 30 21
            --------
             1 34 78 00
             1 23 11 01
             ----------
               11 66 99 00

You can continue these steps to find as many digits as you need and you stop when you have the precision you want, or if you find that the reminder becomes zero which means you have the exact square root.

There's only one more trick left to describe. If you want to find the square root of a number that isn't an integer, say 54782.917. Everything is the same, except you start out by grouping the digits to the left and right of the decimal point in groups of two.

That is, group 54782.917 as

5 47 82 . 91 7

Modifications

During the 19th century, Napier's bones underwent a transformation to facilitate the reading. The rods began to make with an inclination of the order of 65°, so that the triangles that had to be added were aligned vertically. In this case, in each square of the rod slogan the unit to the right and the ten (or the zero) to the left.

The rods were made of way so that the vertical and horizontal engraving was more visible than the meetings between the rods, facilitating themselves much the reading when being the pair of components of each digit of the result in a rectangle. Thus, in the figure it is appraised immediately that:

987654321 x 5 = 4938271605

The apparatus is a magnificent box of wood with inlays of bone. In the top section contains the "bones" abacus, and in the bottom section is the second, card abacus. This card abacus consists of 300 stored cards, in 30 drawers. One hundred of these cards are covered with numbers (referred to as the "number cards"). The remaining two hundred contain small triangular holes, which when they are laid on top of the number cards, allow the user to see only certain numbers. By the capable positioning of these cards, multiplications can be made up to the amazing limit of a number 100 figures in length, by another number 200 figures in length.

In the doors of the box are in addition the first powers to the numbers digits, the coefficients of the terms of the first powers of binomial and regular the numeric data of poliedros.

It is not known who was the author of this riquísima jewel, nor if it is of Spanish responsibility or it came from the foreigner, although it is probable that originally Academy of Mathematics belonged to the Spanish Academy of Mathematics created by Felipe II or that brought like gift Prince of Wales . The only thing that can make sure is that it was conserved in Palace, of where passed to National library and later to the National Archaeological Museum, where still it is conserved. In 1876, the Spanish government sent the apparatus to the exhibition of scientific instruments celebrated in Kensington , where it called the attention extraordinarily, until the point of which several societies consulted to the Spanish representation about the origin and use of the apparatus, which motivated that D. Felipe Picatoste wrote a monograph that later was sent to all the nations, surprising the fact that the abacus was only well-known in England, country of origin of its inventor.

Source: Hispano-American Encyclopedic Dictionary , Montaner i Simon (1887).

External links

• Java implementation of Napier bones in various number systems
• Large PDF scan of Rabdologiæ

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