Genaille–Lucas rulers (also known as Genaille's rods) are an

arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...

tool invented by Henri Genaille, a French railway engineer, in 1891. The device is a variant of

Napier's bones Napier's bones is a manually operated calculating device created by John Napier of Merchiston, Scotland for the calculation of products and quotients of numbers. The method was based on lattice multiplication, and also called ''rabdology'', a w ...

. By representing the

carry Carry or carrying may refer to: People *Carry (name) Finance * Carried interest (or carry), the share of profits in an investment fund paid to the fund manager * Carry (investment), a financial term: the carry of an asset is the gain or cost of ...

graphically, the user can read off the results of simple

multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...

problems directly, with no intermediate

mental calculation Mental calculation (also known as mental computation) consists of arithmetical calculations made by the mind, within the brain, with no help from any supplies (such as pencil and paper) or devices such as a calculator. People may use menta ...

History

In 1885, French mathematician

Édouard Lucas __NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him. Biography Luc ...

posed an arithmetic problem during a session of the

Académie française An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of tertiary education. The name traces back to Plato's school of philosophy, founded approximately 386 BC at Akademia, a sanctuary of Athena, the go ...

. Genaille, already known for having invented a number of arithmetic tools, created his rulers in the course of solving the problem. He presented his invention to the

in 1891. The popularity of Genaille's rods was widespread but short-lived, as

mechanical calculator A mechanical calculator, or calculating machine, is a mechanical device used to perform the basic operations of arithmetic automatically, or a simulation like an analog computer or a slide rule. Most mechanical calculators were comparable in si ...

s soon began to displace manual arithmetic methods.

Design

A full set of Genaille–Lucas rulers consists of eleven strips. On each strip is printed a column of triangles and a column of numbers.

Multiplication

By arranging the rulers in the proper order, the user can find unit multiples of short natural numbers by sight. Genaille-Lucas rulers example 1.png, Five rulers, corresponding to the digits of 52749, are arranged side-by-side, next to the "index" ruler. Genaille-Lucas rulers example 2.png, The unit multiplicand is 4, identifying the row to use. Genaille-Lucas rulers example 3.png, Evaluating the multiplication starts from the units column. Genaille-Lucas rulers example 4.png, The tip of each triangle points to the next digit of the result. Genaille-Lucas rulers example 5.png, The digits visited, by following the triangles from base to tip are those of the desired product: 210996.

Division

Soon after their development by Genaille, the rulers were adapted to a set of rods that can perform division. The division rods are aligned similarly to the multiplication rods, with the index rod on the left denoting the

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

, and the following rods spelling out the digits of the dividend. After these, a special "remainder" rod is placed on the right. The

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

is read from left to right, following the lines from one rod to the next. The path of digits ends with a number on the remainder rod, which is the

remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In a ...

given by the division.

Resources

References

External links

History of computers and computing: Napier's bones
Describes the use of Genaille–Lucas rulers. {{DEFAULTSORT:Genaille-Lucas rulers Mechanical calculators Multiplication French inventions