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Napier's Bones

Napiers Bones, Box, and Booklet
John Napier
John Napier (1550-1617)

John Napier is perhaps best remembered for his contributions to mathematics: natural logarithms and numbered rods for multiplication, later known as "Napier's Bones".

Napier's Bones

Napier's Bones are an arrangement of multiplication tables written on rectangular rods to facilitate multiplication, division, and the extraction of square and cube roots. Ten rods are divided into nine rows of digits corresponding to multiples of the digit at the top. An additional rod has columns of square and cube values to help calculate square roots and cube roots.

In his book Rabdologia John Napier began his description of the construction of the rods as follows:

“Construct from silver, ivory, boxwood, or other similar solid material a number of square rods. Make 10 rods for numbers less than 11,111; 20 rods for numbers lessthan 111,111,111; 30 rods for numbers less than 1,111,111,111,111; etc.”

While sets of silver might be destined for nobility, ivory or boxwood were more common choices. This set has been created by rotary engraving numbers into aircraft grade aluminum in recognition of the magnificence of a set made from silver. Each rod is carefully hand finished to give a special feel of history. The three dimensional engraving of the numbers and careful finish guarantee each set will endure for centuries.

Napier specified a specific layout for the number columns:

"... the third face of each rod is always opposite to the first and the fourth to the second, and that the simples on each face are opposed not merely in the sense that one is on the upper and the other on the lower face or that one is on the right-hand and the other on the left-hand face but also in that they are at opposite ends, one at the top and the other at the bottom of the rod, and that these two opposed simples always add up to nine."

In contemporary times, Napier’s original layout is frequently modified in favor a larger single digit at the top of the rod. There’s no arguing that it’s easier to read, however we chose to make this set according to Napier’s exact instructions. Napier notes that a multiplication may be validated by turning the rods over and manipulating the complement digits of the result. It’s interesting to realize that Napier’s layout using the arrangement of opposing numbers represents one of the first user interface optimizations for automated calculation in recorded history.

Napier's Bones in presentation box Detail photo of engraved numbers on Napier's Bones Detail photo of engraved numbers on Napier's Bones Detail photo of square root rod of Napier's Bones

Click a picture to see a larger size.

This set is of the highest presentation quality - supplied with an exquisitely finished walnut box with a clear top to showcase the rods, a tablet to hold the bones during calculations, and a booklet that provides an introduction to John Napier, the rods, and instructions for multiplication, division, square roots, and cube roots.

It's a great gift to recognize a colleague, an heirloom present for a young person, and a compelling conversation piece for collectors of all ages.

Using Napier's Bones

The booklet supplied with Napier's Bones describes multiplication, division, square roots, and cube roots. In this example we'll illustrate the multiplication of 2489 by 7. Arrange the rods to show the columns for 2, 4, 8, and 9 on the tablet, then look down to the 7th row.

Napiers Bones Multiplication Example

Starting from the right and proceeding left, add up the numbers in the diagonals. When the sum exceeds 9, write the least significant digit as the answer for that diagonal and add a carry digit to the left adjacent diagonal. In this illustration this happens twice. If the multiplier were to be 70, the procedure is the same, but with the extra step of adding a 0 to the end (think of this as shifting the decimal point).


John Napier (1550-1617) was born at Merchiston Castle near Edinburgh, Scotland into a wealthy family. He studied briefly at St. Andrews, travelled on the Continent, and returned to Scotland. In 1572 he married Elizabeth Stirling, and they had two children. Elizabeth died in 1579, and John later married Agnes Chiholm, with whom he had 10 children.

Napier had diverse interests in mathematics, theology, and agriculture. Like his father, who had been one of the earliest to promote the Protestant cause in Scotland, he was also a fervent Protestant and was appointed a commissioner to the Gneral Assembly on behalf of the Presbytery of Edinburgh in 1588. He later published the Plaine Discovery of the Whole Revelation of St. John, which he considered to be his most important work.

In 1614 Naper published his famous Mirifici Logarithmorum Canonis Descriptio, describing his work on logarithms. The book contained 90 pages of tables and descriptions. The invention of logarithms was hugely significant in that it made multiplication, division, and root extraction far less time consuming, particularly for very large numbers. His collaboration with Henry Briggs was crucial to Brigg's construction of log base 10 tables and their widespread adoption.

Cover page of Rabdology

In 1617, just before he died, Napier published Rabdologia, seu Numerationis per Virgulas Libri Duo: Cum Appendice de expecitissimo Multiplicationis Promptuario. Quibus accessit et Arithmeticae Loaclis Liber Unus, commonly referred to as "The Rabdologia", or Rabdology.

This book describes three devices to facilitate calculations:

Numbered rods The technique of making and using numbered rods to facilitate multiplication.
Promptuary A large set of numbered paper strips for multiplication and division.
Location arithmetic A technique using a grid and counters to perform binary arithmetic.

Only the first was widely adopted. Rabdology was translated into several languages and variations of the numbered rods were used for several centuries.

Rabdology is credited as containing one of the earliest references to the use of a decimal point, in this case a improvement of Simon Stevin's decimal notation.

Rabdology has been translated by Wiliam Frank Richardson as volume 15 of the Charles Babbage Institute reprint series for the history of computing. This volume is difficult to find, but well worth the effort.

For Sale

Armstrong Metalcrafts is building a limited number of these for sale. The retail price is $295. To inquire about a purchase, please use our contact form or send an email to "sales" at

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