Napier's Bones
and the rods and boards that came after them.
In 1617, the year he died, John Napier of Merchiston published a small book on reckoning with rods, the Rabdologiae. Each rod carries one column of the times table. Lay the rods for a number side by side and any row of its multiplication table can be read across them, so only the sums along the diagonals are left to do. Nearly three centuries later, Genaille and Lucas cut rods that do those sums as well. In the same book Napier turned a chessboard into a calculator that works in binary.
Plate I. The bones, after Napier's Rabdologiae (1617): a board, the index rod, and a box of rods.
Click a rod in the box to lay it; click a rod on the board to put it back, or drag it (or use the arrow keys) to change its place.
Plate II. The rulers of Henri Genaille and Édouard Lucas (1891). Follow the triangles from right to left, and the product reads off with no carrying.
Click a band on the index rod to choose the multiplier.
Plate III. Napier's chessboard (Rabdologiae, 1617). Each square is worth twice the square to its right and twice the square below it.
Click a square to put a counter on it, and click a counter to take it off. Drag a counter, or focus it and use the arrow keys, to move it.
¶Commentary
The lattice. The bones are a lattice multiplication cut into pieces. In the lattice, or gelosia, method you rule a grid with one column for each digit of the multiplicand and one row for each digit of the multiplier. You write each digit product in its cell, split by a diagonal into tens and units, and add along the diagonals. Italian arithmetic books of the fifteenth century, such as the Treviso Arithmetic of 1478, call it multiplying per gelosia, after the latticed screens set in Venetian windows. The method is older than that name, and appears earlier in Arabic arithmetic.
Napier's step. Each column of a lattice depends on only one digit, so it can be made once and kept. Napier carved each column of the times table on a rod. To multiply, you lay out rods for the multiplicand, and the multiplier digit picks the row. His own rods were square in section, with a column on each of the four faces, and the digits on opposite faces add to nine. The diagonal sums are never more than 9 + 8 + 1 = 18, so a carry is never more than 1.
The book. Rabdologiae, from the Greek for "rod" and "reckoning", appeared in Edinburgh in 1617, three years after Napier's tables of logarithms. Besides the rods for multiplication and division, it describes special rods that help extract square and cube roots. It also describes the promptuary, a box of strips that does whole multiplications without writing partial products, and, in an appendix, the location arithmetic of Plate III.
How the bones spread. Sets in wood, bone, and ivory were made all over Europe within a few years. Wilhelm Schickard's calculating machine of 1623 carried the bones on turning cylinders in its upper half and an adding mechanism below. Later makers mounted them on cylinders in boxes, as Gaspar Schott did in his Organum Mathematicum (1668). The bones also suggested a step to the slide rule: they show that a multiplication table can be built into a mechanism you read instead of memorize.
Genaille and Lucas. The bones leave the carrying to the user. Édouard Lucas asked whether rods could do it themselves, and Henri Genaille, a French engineer, answered with the rulers of Plate II, published in 1891. The key fact is that when you multiply by n, the carry into any place is at most n − 1, since 9n + (n − 1) is less than 10n. So each rod needs only n cells for each multiplier, one for each possible carry, and a triangle in each cell can show where its own carry sends you. Genaille also made rulers for division on the same plan.
Location arithmetic. On Napier's chessboard each square is worth a power of two, and a number is a set of counters, so this is arithmetic in base 2. Addition is putting counters together, and simplifying turns a heap into binary. Doubling is a shift one square left, and multiplying is the same placing and sliding as the lattice, done with powers of two instead of digits. Napier also did subtraction, division, and square roots on the board. It is binary arithmetic nearly ninety years before Leibniz wrote about the binary numerals in 1703, though Napier treated it as a device for reckoning, not as a way of writing numbers.
Further. John Napier, Rabdologiae, seu Numerationis per Virgulas Libri Duo (Edinburgh, 1617), and its English translation by William Frank Richardson, Rabdology (MIT Press and Tomash Publishers, 1990). Édouard Lucas, Récréations mathématiques (four volumes, 1882–1894). Frank J. Swetz, Capitalism and Arithmetic: The New Math of the 15th Century (1987), which translates the Treviso Arithmetic.