Napier,+John

=Napier, John=

toc Read here about John Napier and his work on logarithmic calculation and rules.

=Napier’s Bones= Napier's bones is an abacus created by John Napier for calculation of products and quotients of numbers that was based on Arab mathematics and lattice multiplication used by Matrakci Nasuh in the Umdet-ul Hisab and Fibonacci writing in the Liber Abaci. Also called Rabdology (from Greek ῥάβδoς [r(h)abdos], "rod" and -λογία [logia], "study"). Napier published his version of 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 roots. Note that Napier's bones are not the same as logarithms, 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, quintuple, 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 10 rods corresponding to digits 0 to 9. The rod 0, although it may look unnecessary, is obviously still needed for multipliers or multiplicands having 0 in them.

=Rabdology=

Rabdology (from Greek ῥάβδoς [r(h)abdos], "rod" and -λογία [logia], "study"). In 1617 a treatise in Latin entitled Rabdologiæ and written by John Napier was published in Edinburgh. Printed three years after his treatise on the discovery of logarithms and in the same year as his death, it describes three devices to aid arithmetic calculations.

The devices themselves don't use logarithms, rather they are tools to reduce multiplication and division of natural numbers to simple addition and subtraction operations.

The first device, which by then was already popularly used and known as Napier's bones, was a set of rods inscribed with the multiplication table. Napier coined the word rabdology (from Greek ραβδoς [rabdos], rod and λoγoς [logos] calculation or reckoning) to describe this technique. The rods were used to multiply, divide and even find the square roots and cube roots of numbers.

The second device was a promptuary (Latin promptuarium meaning storehouse) and consisted of a large set of strips that could multiply multidigit numbers more easily than the bones. In combination with a table of reciprocals, it could also divide numbers. The third device used a checkerboard like grid and counters moving on the board to perform binary arithmetic. Napier termed this technique location arithmetic from the way in which the locations of the counters on the board represented and computed numbers. Once a number is converted into a binary form, simple movements of counters on the grid could multiply, divide and even find square roots of numbers.

Of these devices, Napier's bones were the most popular and widely known. In fact, part of his motivation to publish the treatise was to establish credit for his invention of the technique. The bones were easy to manufacture and simple to use, and several variations on them were published and used for many years.

The promptuary was never widely used, perhaps because it was more complex to manufacture, and it took nearly as much time to lay out the strips to find the product of numbers as to find the answer with pen and paper. Location arithmetic was an elegant insight into the simplicity of binary arithmetic, but remained a curiosity probably because it was never clear that the effort to convert numbers in and out of binary form was worth the trouble. An interesting tidbit is this treatise contains the earliest written reference to the decimal point (though its usage would not come into general use for another century.)

The computing devices in Rabdology were overshadowed by his seminal work on logarithms as they proved more useful and more widely applicable. Nevertheless these devices (as indeed are logarithms) are examples of Napier's ingenious attempts to discover easier ways to multiply, divide and find roots of numbers. Location arithmetic in particular foreshadowed the ease of and power of mechanizing binary arithmetic, but was never fully appreciated.