Napier's bones

John Napier, having invented logarithms, turned to a simpler aid for multiplication: numbered rods that could be arranged to read off products directly. Published in 1617, Napier's bones used the ancient lattice multiplication method encoded on ivory or bone strips, allowing users to multiply large numbers by looking up partial products rather than calculating them.

Each rod displayed a single digit and its multiples from 1 to 9, arranged so that when rods were placed side by side, the user could read off partial products by adding diagonally. To multiply 425 by 6, place rods for 4, 2, and 5 adjacent, read the row for 6, and add the diagonal digits. The method reduced multiplication to addition—simpler, faster, and less error-prone.

The bones were far easier to use than logarithm tables for simple calculations. A merchant checking an invoice or a navigator computing a position could carry a set of rods and perform multiplications in seconds that would otherwise take minutes. The trade-off was precision: logarithms handled any number to any desired accuracy, while bones worked only with integers up to the rod set's capacity.

Variations proliferated. Genaille-Lucas rulers of 1891 eliminated even diagonal addition. Circular arrangements became the slide rule. The principle of reading off pre-calculated values from physical artifacts persisted until electronic calculators made such aids obsolete.

Napier's bones occupied the same niche as the abacus: a physical computing device that augmented human calculation without requiring mathematical sophistication. Anyone could learn to use them in an hour; the mathematical understanding encoded in the rods' construction remained invisible.

The bones remind us that computation has always sought shortcuts. From fingers to pebbles to rods to electronic chips, each technology made calculation faster by encoding mathematical relationships in manipulable form. Napier contributed two approaches—logarithms for the mathematician, bones for the merchant—both serving the same goal of making multiplication less laborious.