Napier's Bones and the Genaille-Lucas Rulers


In 1617, John Napier published Rabdologia, a book that contained a description of the rods that he had developed to aid in multiplication, division, and the extraction of square roots. Napier's bones have since become popular as a way of demonstrating multiplication to school students.

One problem with Napier's rules is that the user is required to perform a number of additions and must keep track of the "carries" that occur during these additions. In the late 1800's, a frenchman named Henri Genaille invented an improved version of Napier's bones in response to a problem posed by Edouard Lucas. The Genaille-Lucas rulers allow for multiplication and division without the carries required by Napier's Bones.

This web page presents Napier's Bones and the Genaille Lucas ruler in ready to print form. To use these documents, print them out on paper or transparency stock, use a paper cutter to separate the strips, and then have fun using them!

Note that the files are available in both postscript and PDF format. Free software for viewing and printing PDF files is available from Adobe.

Note also that these figures were painstakingly drawn by hand. There is every possibility that somewhere in he long process of doing this, I may have made a mistake. Please contact the author if you find any errors.

Note added May 7, 2001. Jan Meyer identified two errors in the Genaille-Lucas rulers. I've corrected these errors.

Note added June 16, 2017. Finally corrected a bunch of small errors in the Genaille-Lucas rods that I've been told about over the years. Thanks to John Bayes, Jan Mayer, Tom Diffenbach, and Chris Staecker for comments.

Unfortunately, I haven't yet written up any instructions on how to use these. You'll want to refer to books lists at the bottom of this page for more informatin. If you can think of any traditional "dead trees" magazine or journal that might be intersted in this, I might be inspired to sit down and write something up.

Files to Download


Aspray, William ed., Computing Before Computers, Iowa State University Press, 1990. This book has been made available online in portable document format (PDF).

Behr, Achim, Extracting Square Roots by means of the Napier rods.

Gardner, Martin, Knotted Doughnuts and Other Mathematical Entertainments, W. H. Freeman, 1986.

Napier, John, Rabdology, translated by William Frank Richardson, MIT Press, 1990.

Williams, M. R., A History of Computing Technology, 2nd ed., IEEE Press, 1997.

This page and the postscript/pdf files that it links to are Copyright 2000, Brian Borchers. You may reproduce these pages for personal or classroom use, but you may not use these pages for any commercial purpose.

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