Leibniz wheel
Encyclopedia
A Leibniz wheel or stepped drum was a cylinder with a set of teeth of incremental length which, when coupled to a counting wheel, was used in the calculating engine of a class of mechanical calculator
Mechanical calculator
A mechanical calculator is a device used to perform the basic operations of arithmetic. Mechanical calculators are comparable in size to small desktop computers and have been rendered obsolete by the advent of the electronic calculator....

s. Invented by Leibniz in 1673, it was used for three centuries until the advent of the electronic calculator in the mid 1970s.

Leibniz built a machine called the Stepped Reckoner
Stepped Reckoner
The Step Reckoner was a digital mechanical calculator invented by German mathematician Gottfried Wilhelm Leibniz around 1672 and completed in 1694. The name comes from the translation of the German term for its operating mechanism; staffelwalze meaning 'stepped drum'...

 based on that design in 1694. It was made famous by Thomas de Colmar when he used it, a century and a half later, in his Arithmometer
Arithmometer
An Arithmometer or Arithmomètre was a mechanical calculator that could add and subtract directly and could perform long multiplications and divisions effectively by using a movable accumulator for the result. Patented in France by Thomas de Colmar in 1820 and manufactured from 1851 to 1915, it...

, the first mass produced calculating machine. It was also used in the Curta calculator
Curta calculator
The Curta is a small, hand-cranked mechanical calculator introduced in 1948. It has an extremely compact design: a small cylinder that fits in the palm of the hand. It can be used to perform addition, subtraction, multiplication, division, and — with more difficulty — square roots and other...

, a very popular portable calculator introduced in the second part of the 20th century.

Concept

By coupling a Leibniz wheel with a counting wheel free to move up and down its length, the counting wheel can mesh with any number of teeth.

The animation on the side shows a nine-tooth Leibniz wheel coupled to a red counting wheel. It is set to mesh with three teeth at each rotation and therefore would add or subtract 3 from the counter at each rotation.

The computing engine of an Arithmometer has a set of linked Leibniz wheels coupled to a crank handle. Each turn of the crank handle rotates all the Leibniz wheels by one full turn. The input sliders move counting wheels up and down the Leibniz wheels which are themselves linked by a carry mechanism.

Eventually these wheels were replaced by pinwheels
Pinwheel calculator
A Pinwheel calculator was a class of mechanical calculator popular in the 19th and 20th century using, for its calculating engine, a set of wheels that had an adjustable number of teeth...

 which are similar in function but with a more compact design.

Machines built using this principle

Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 built his first stepped reckoner in 1694 and another one in 1706.
– Philipp-Matthaüs Hahn, a German pastor, built two circular machines in 1770.
– J.C. Schuster, Hahn's brother in law, built a few machines of Hahn's design into the early 19th century.
Lord Stanhope
Charles Stanhope, 3rd Earl Stanhope
Charles Stanhope, 3rd Earl Stanhope aka Charles Mahon, 3rd Earl Stanhope FRS was a British statesman and scientist. He was the father of the great traveller and Arabist Lady Hester Stanhope and brother-in-law of William Pitt the Younger. He is sometimes confused with an exact contemporary of his,...

 designed a machine using Leibniz wheels in 1777. He also designed a pinwheel calculator
Pinwheel calculator
A Pinwheel calculator was a class of mechanical calculator popular in the 19th and 20th century using, for its calculating engine, a set of wheels that had an adjustable number of teeth...

 in 1775.
– Johann-Helfrich Müller built a machine very similar to Hahn's machine in 1783.
– Thomas de Colmar invented his Arithmometer in 1820 but it took him 30 years of development before it was commercialized in 1851. It was manufactured until 1915. Louis Payen, and Darras were successive owners and distributors of the Arithmometer.
– Timoleon Maurel invented his Arithmaurel
Arithmaurel
First patented in France by Timoleon Maurel, in 1842, the Arithmaurel was a mechanical calculator that had a very intuitive user interface, especially for multiplying and dividing numbers because the result was displayed as soon as the operands were entered. It received a gold medal at the French...

 in 1842. The complexity of its design limited its capacity and doomed its production, but it could multiply two numbers by the simple fact of setting them on its dials.
– About twenty clones of the Arithmometer were manufactured in Europe starting with Burkhardt in 1878 then came Layton, Saxonia, Gräber, Peerless, Mercedes-Euklid, XxX, Archimedes, TIM, Bunzel, Austria, Tate, Madas etc... These clones, often more sophisticated than the original arithmometer, were built until the beginning of WWII.
– Joseph Edmondson invented and manufactured a circular calculator in 1885.
– Friden and Monroe calculators used a biquinary variant of this mechanism. Both were made in large numbers; Monroe started early in the 20th century; Friden in the 1930s. (The Marchant used a radically different and unique mechanism.) The variant mechanism worked with numbers 1..4 were as shown in the animation; numbers 5..9 engaged a five-tooth gear as well as 0..4 teeth of the Leibniz wheel. This made it unnecessary for the sliding gear to travel longer distances for the higher-number digits. Otherwise, pressing a 5..9 key would require either a longer stroke (as in a Comptometer) or excessive force combined with a gently sloping cam surface. Neither was desirable.
Curt Herzstark
Curt Herzstark
Curt Herzstark was born on July 26, 1902 in Vienna, and died October 27, 1988 in Nendeln, Liechtenstein. During World War II, Curt Herzstark's plans for a mechanical pocket calculator literally saved his life....

 introduced his Curta
Curta calculator
The Curta is a small, hand-cranked mechanical calculator introduced in 1948. It has an extremely compact design: a small cylinder that fits in the palm of the hand. It can be used to perform addition, subtraction, multiplication, division, and — with more difficulty — square roots and other...

portable calculator in 1948, it was very popular until the introduction of electronic calculators in the 1970s.

External links

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