Posted by: Wayne Robinson | July 27, 2010

Innovations of the Seventeenth Century – The Pocket Calculator

Scottish mathematician John Napier, when he wasn’t inventing logarithms, popularising the decimal point or dieing of gout invented one of the earliest pocket calculators.

Based on an arabic concept of lattice multiplication, Napier’s Bones reduced complex multiplications, divisions, squares, cubes and finding square and cube roots to a simple set of additions or subtractions.

Black and white image of original set of Napier's Bones from Computer History Museum's collection

An original set of Napier’s Bones, a palm-sized calculator designed by John Napier in 1617.
The Computer History Museum, Mountain View, California

The Rabdologiae provides a description of making of a set of rods. Napier suggested using “silver, ebony, boxwood or some strong material of a similar nature”. “…ten oblong rods are required for numbers up to five figures; twenty are needed for numbers up to nine figures; or thirty of numbers up to 13 figures”.

Each of the four faces of each rod is divided into nine squares. The top square is one of the digits 0 to 9 and the subsequent squares down are the multiples of that number as would appear on a times table. The multiple is written with the  tens on the left and units on the right of the diagonal line. As on modern dice, opposite faces of a rod have complementary numbers, For example, 0 and 9 or 1 and 8 so that the sum of the top number on opposite sides is always 9. The numbers at the bottom of each rod in the photo above separated by the vertical line show what the number is on adjacent faces. For example, the 9 rod on the right has 2 on the next face to the left, 7 on the face to the right and we know that as the upward face is 9, the downward one is 0. Glenda noticed the 9s in the photo above have different numbers on adjacent sides, two have 2 and 7, the other, 3 and 6.

Ann Unfold'd Rodde

An unfolded rod

There’s a good description of how to use the bones and an interactive example here. Napier’s Bones were widely used in Europe until the mid 1960s.

Napier, J, Rabdologiae, seu Numerationis per Virgulas Libri duo, 1617

Gibson, George A. Napier’s Life and Works. Proceedings of the Royal Philosophical Society of Glasgow, reprinted in: E.M. Horsburgh, Herbert Bell et al. Modern instruments and methods of calculation : a handbook of the Napier tercentenary exhibition. Bell, London, 1914

Hawkins, W.F. Napier’s mathematical works – translations by W.F. Hawkins, University of Auckland, 1978

Redin, J, A Brief History of Mechanical Calculators Part I – The Age of the Polymaths,



  1. Yet more proof, if proof be needed, that the Scots invented absolutely everything.

    Keith (the Scot)

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