Editing Napier's bones (section)

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[[File:Napier's Bones.JPG|thumb|A set of Napier's bones]]
[[File:An 18th century set of Napier's Bones.JPG|thumb|An unusual 18th-century set of Napier's bones in which the numbers are on rotating cylinders rather than rods of square cross-section]]
{{Rabdology}}
'''Napier's bones''' is a manually operated calculating device created by [[John Napier]] of [[Merchiston]], [[Scotland]] for the [[calculation]] of products and [[quotient]]s of numbers. The method was based on [[lattice multiplication]], and also called ''[[rabdology]]'', a word invented by Napier. Napier published his version in [[1617]].<ref name="rabdologiae">{{cite book | title=Rabdologiæ | author=John Napier | date=1617 | location=Edinburgh, Scotland | language=Latin }}</ref> It was printed in [[Edinburgh]] and dedicated to his patron [[Alexander Seton, 1st Earl of Dunfermline|Alexander Seton]].

Using the [[Multiplication table|multiplication tables]] embedded in the rods, [[multiplication]] can be reduced to [[addition]] operations and [[division (mathematics)|division]] to [[Subtraction|subtractions]]. Advanced use of the rods can extract [[square root]]s. Napier's bones are not the same as [[logarithm]]s, with which Napier's name is also associated, but are based on dissected multiplication tables.

The complete device usually includes a base board with a rim; the user places Napier's rods and the rim to conduct multiplication or division. The board's left edge is divided into nine squares, holding the numbers 1 to 9. In Napier's original design, the rods are made of metal, wood or [[ivory]] and have a square cross-section. Each rod is [[engraved]] with a multiplication table on each of the four faces. In some later designs, the rods are flat and have two tables or only one engraved on them, and made of plastic or heavy [[cardboard]]. A set of such bones might be enclosed in a carrying case.

A rod's face is marked with nine squares. Each [[square]] except the top is divided into two halves by a [[diagonal]] line from the bottom left corner to the top right. The squares contain a simple [[multiplication table]]. The first holds a single [[Numerical digit|digit]], which Napier called the 'single'. The others hold the [[Multiple (mathematics)|multiples]] of the single, namely twice the single, three times the single and so on up to the ninth square containing nine times the number in the top square. Single-digit numbers are written in the bottom right triangle leaving the other triangle blank, while double-digit numbers are written with a digit on either side of the diagonal.

If the tables are held on single-sided rods, 40 rods are needed in order to multiply 4-digit numbers – since numbers may have repeated digits, four copies of the multiplication table for each of the digits 0 to 9 are needed. If square rods are used, the 40 multiplication tables can be inscribed on 10 rods. Napier gave details of a scheme for arranging the tables so that no rod has two copies of the same table, enabling every possible four-digit number to be represented by 4 of the 10 rods. A set of 20 rods, consisting of two identical copies of Napier's 10 rods, allows calculation with numbers of up to eight digits, and a set of 30 rods can be used for 12-digit numbers.