Rabdology

In 1617 a treatise in Latin entitled "Rabdologiæ" and writtenby John Napier was published in Edinburgh. Printed three yearsafter his treatise on the discovery of logarithms and in the same yearas his death, it describes three devices to aid arithmetic calculations.

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

The first device, which by then was already popularly used and knownas Napier's bones, was a set of rods inscribed with themultiplication table. Napier coined the word rabdology (fromGreek ραβδoς [rabdos] , rod andλoγoς [logos] calculation or reckoning) to describethis technique. The rods were used to multiply, divide and even findthe square roots and cube roots of numbers.

The second device was a promptuary (Latin "promptuarium" meaningstorehouse) and consisted of a large set of strips that couldmultiply multidigit numbers more easily than the bones. In combinationwith a table of reciprocals, it could also divide numbers.

The third device used a checkerboard like grid and counters moving onthe board to perform binary arithmetic. Napier termed this techniquelocation arithmetic from the way in which the locations of thecounters on the board represented and computed numbers. Once a numberis converted into a binary form, simple movements of counters on thegrid could multiply, divide and even find square roots of numbers.

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

The promptuary was never widely used, perhaps because it was morecomplex to manufacture, and it took nearly as much time to lay out thestrips to find the product of numbers as to find the answer withpen and paper.

Location arithmetic was an elegant insight into the simplicity ofbinary arithmetic, but remained a curiosity probably becauseit was never clear that the effort to convert numbers in and out ofbinary form was worth the trouble.

An interesting tidbit is this treatise contains the earliest writtenreference to the decimal point (though its usage would not come intogeneral use for another century.)

The computing devices in Rabdology were overshadowed by his seminalwork on logarithms as they proved more useful and more widelyapplicable. Nevertheless these devices (as indeed are logarithms) areexamples of Napier's ingenious attempts to discover easier waysmultiply, divide and find roots of numbers. Location arithmetic inparticular foreshadowed the ease of and power of mechanizing binaryarithmetic, but was never fully appreciated.

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