Napier’s Bones

It has always been the endeavour of mathematicians to eliminate the laborious aspect of their work. John Napier was one such eminent mathematician whose logarithmic tables reduced multiplication to addition, division to subtraction and exponentiation to multiplication. In addition to this extremely useful mathematical tool, he is also credited with the invention of an apparatus, termed as Napier’s bones; which simplifies multiplication and some other mathematical operations, to a considerable extent (Mathematics and Computing. In The Hutchinson Unabridged Encyclopedia with Atlas and Weather guide.

Abington: Helicon ). John Napier was the scion of an affluent Scottish family, who pursued the study of mathematics as a hobby. He acquired knowledge in a number of subjects like agriculture, politics and religion. This ingenious invention for simplifying calculations, consisted of rods made of ivory, wood or metal; on which digits had been inscribed. This contrivance achieved considerable popularity amongst the denizens of the British Isles and Europe, on account of the ease with which it could be used and the general absence of mathematical skills among the populace of these countries (Napier’s Bones).

He was succeeded in this commendable effort by Blaise Pascal the inventor of the mechanical calculator, in the year 1642. Subsequently, Charles Babbage invented a calculating machine, which was the precursor to modern day computers. Some of the persons, who had been involved in the invention of computers, were Alan Turing and John Von Neumann, who were mathematicians of considerable repute (Mathematics and Computing. In The Hutchinson Unabridged Encyclopedia with Atlas and Weather guide. Abington: Helicon ).

An abacus was designed by Napier that employed numbered rods to perform various mathematical operations like multiplication, division and extraction of square roots. This device has also been termed as Napier’s bones (Weisstein). For instance, multiplication is performed by segregating the rods that represent the multiplier on the left. Thereupon, the rods that represent the multiplicand are placed adjacent and to the right hand side of the initial set of rods. The product is given by the sum of the pairs of numbers available in the row established by the multiplier.

This process is of historical significance, and finds mention, for the first time in Napier’s book Rabdologia, which was published in the year 1617. Consequently, this process is also known as rabdology (Weisstein). In this unique contraption, there are ten rods or bones that correspond to the digits 0 to 9. In addition, a eleventh bone or rod is present, which stands for the multiplier. As such, this multiplier is nothing more than the digits from 1to 9, placed vertically downwards.

The rest of the rods or bones, are inscribed with a digit in the uppermost square, and the beneath it is appended the multiplication table for that particular digit. In order to perform multiplication with repeating digits, several sets of these rods have to be utilized (Weisstein). Multiplication using Napier’s bones As an example, the multiplication of two numbers, namely, 7×4896 will be described below: Illustration of obtaining the product of 7×4896, with Napier’s bones (Weisstein) To obtain the product of 7×4896, the bones or rods are to be arranged in the manner shown in the diagram.

The calculation proceeds from right to left. The first bone to be considered is the rod on the extreme right of the multiplier row in the diagram. The number 2, is to be written down, as it comprises the last digit in the 7’s row of the 6 rod (Weisstein). After this, the sum of the digits in the parallelogram formed by the two contiguous numbers in the same row to the left are to be added; this gives 3+4 = 7, which constitutes the next digit. Hence, we obtain 72. The next addition yields, 6+6 = 12, therefore, the first digit 2 is to be written down to yield 272 and the remaining digit 1 is to be carried over (Weisstein).

In the next digit, we have 8+5+1(the carried over figure), which yields 14. Accordingly, the 4 is to be written down and once again the 1 is to be carried over. This yields 4272. Now, the last remaining digit on the left is 2 and to this the carried over numeral, namely 1 is to be added, which gives 2+1 = 3. Thus, we finally obtain 34272 as the product of 7 and 4896 (Weisstein). This apparatus was improved upon by the so called Genaille rods, which preclude the need to carry over numbers and to add numbers in the parallelograms.

With these Genaille rods, products can be directly read off (Weisstein). Division using Napier’s bones Division using Napier’s bones (Hansen) The Napier’s bones will now be employed to divide 46785399 by 96431. The rods for the divisor 96431 are to be placed alongside each other as shown in the diagram appended above. After this the displayed are to be noted, which provides the products of the divisor from 1 to 9. It is to be observed that the dividend comprises of eight digits; but that the partial products, with the exception of one of them, have just six.

This necessitates temporarily disregarding the last two digits of the number to be divided, namely 46785399, i. e. , 99. This results in 467853. After this it is required to find a partial product that is less than the truncated dividend obtained above, and which is also the greatest amongst such partial products. This number is obtained as 385724 (Hansen). On account of the fact that the number 385724 is to be found in the 4 row, 4 must be designated as the digit that occupies the left most position in the quotient.

After this the left aligned partial product is to be placed beneath the initial dividend and a subtraction is to be performed. This subtraction operation yields the number 8212999 (Hansen). Now the following process has to be continued iteratively. First the number is to be shortened to six digits and the partial product that is just lesser than this shortened number is to be selected. The corresponding row number is to be taken as the quotient’s next digit and the partial product is to be subtracted from the difference obtained in the first repetition.

This process is to be halted, once the divisor is seen to be greater than the result of the subtraction. The remainder of the division process is given by the number that is left at the conclusion of this process (Hansen). Napier’s invention was of immense help to bookkeepers and shopkeepers. It provided them with much greater time to expand their business and attend to other activities. This invention presaged the slide rule and greatly facilitated basic mathematical operations. Works Cited Hansen, Jim.

John Napier’s Bones. 22 January 2009 <http://jimsmathandscience. com/Napiers%20Bones/NapiersBones. html>. Mathematics and Computing. In The Hutchinson Unabridged Encyclopedia with Atlas and Weather guide. Abington: Helicon . 2008. 22 January 2009 <http://www. credoreference. com/entry/8002850>. Napier’s Bones. 22 January 2009 <http://www. bookrags. com/research/napiers-bones-csci-01/>. Weisstein, Eric W. MathWorld – A Wolfram Web Resource. 20 January 2009. 22 January 2009 <http://mathworld. wolfram. com/NapiersBones. html>.

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