Napier's Bones

Ten strips, each holding one column of the times table.

1. Cut ten card strips and divide each into nine cells.
2. Draw a diagonal across every cell, from lower left to upper right.
3. On rod n, write n×1 to n×9, putting the tens digit above the diagonal and the units below.

Lay out the number, read one row, add along the stripes.

1. Set out rods 4, 6, 8 side by side to represent 468.
2. Look along row 7.
3. Add the digits within each diagonal stripe, carrying left where a stripe exceeds 9.
4. Read off 3276.

Repeat per digit and shift, exactly as in long multiplication.

1. For 468 × 27, read row 7 and write the result.
2. Read row 2, write it shifted one place left.
3. Add the two partial products.

Multiply the same pair of numbers three ways and record the cost of each.

1. By the bones.
2. By doubling and halving, as in the Russian peasant method.
3. By ordinary long multiplication from memorised tables.

John Napier (1550-1617), laird of Merchiston, published Rabdologiae in 1617, describing the rods along with two other calculating devices. Sets were made in ivory, which is where the name bones comes from, and they sold widely because they needed no training in a new number system.

Two different answers to one problem, from one man. Napier's logarithms, published in 1614, turn multiplication into addition by mapping numbers onto a scale where distances add — the idea that becomes the slide rule. The bones turn multiplication into addition by carrying the times table as objects and letting a diagonal do the carrying. Same goal, different mechanisms, different costs: logarithms need a printed table and give approximate answers quickly for any numbers; the bones need a physical set and give exact answers for whole numbers. Both are in this catalogue, and comparing them is more informative than either alone.

The method is older than the rods. Diagonal-cell multiplication — lattice multiplication, or gelosia after the lattice windows it resembles — is described in Arabic and Indian sources and reaches Europe through Fibonacci. Napier's contribution was making the lattice reusable by putting each column on a movable rod.

Where it led: Wilhelm Schickard's calculating clock of 1623 incorporated a mechanised set of Napier's rods, making it arguably the first mechanical calculator. The bones sit exactly on the line between a written method and a machine — and lattice multiplication is still taught in some primary curricula today, because it separates the multiplying from the carrying.