Mechanical calculator - AbsoluteAstronomy.com

A mechanical calculator is a device used to perform the basic operations of arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

. Mechanical calculators are comparable in size to small desktop computer

Computer

A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

s and have been rendered obsolete by the advent of the electronic calculator.

Description of the illustrated calculator:

This is a pinwheel rotary type, operated by the hand crank at the right. Tips of the setting levers extend through the slots in the brass cover; pinwheels are inside. The result dials are inside the carriage at the front, which can be shifted sidewise. The right array of dials shows sums and differences, as well as the dividend for division. The other set of dials shows the number of cycles at each carriage position. This displayed number serves as a check on multiplication, and shows the quotient from division. "Butterfly" knobs clear the dials to zeros when rotated.

The mechanical calculator was invented in 1642 and the first commercially successful device

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...

was manufactured from 1851. Machines with columns of keys were introduced in 1887 while 10 key calculators and electric motors appeared in 1902. The use of electric motors allowed for the design of very powerful machines during the first half of the 20th century. In 1961, A full-keyboard machine like the comptometer

Comptometer

The comptometer was the first commercially successful key-driven mechanical calculator, patented in the USA by Dorr E. Felt in 1887.A key-driven calculator is extremely fast because each key adds or subtracts its value to the accumulator as soon as it is pressed and a skilled operator can enter all...

, called the Anita, became the first mechanical calculator to receive an all electronic calculator engine, creating the link in between these two industries and marking the beginning of its decline. The Anita was the only full-keyboard electronic calculator of any commercial significance. The last mechanical calculators were built in the middle of the 1970s.

The mechanical calculator was preceded by and competed with clerical aids such as abaci

Abacus

The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...

, Napier's bones

Napier's bones

Napier's bones is an abacus created by John Napier for calculation of products and quotients of numbers that was based on Arab mathematics and lattice multiplication used by Matrakci Nasuh in the Umdet-ul Hisab and Fibonacci writing in the Liber Abaci. Also called Rabdology...

and slide rule

Slide rule

The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...

s, and various books of mathematical table

Mathematical table

Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation...

s. The true precursors to the mechanical calculator were machines made of toothed gears linked by carry mechanisms like odometer

Odometer

An odometer or odograph is an instrument that indicates distance traveled by a vehicle, such as a bicycle or automobile. The device may be electronic, mechanical, or a combination of the two. The word derives from the Greek words hodós and métron...

s, astrolabe

Astrolabe

An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...

s, clock

Clock

A clock is an instrument used to indicate, keep, and co-ordinate time. The word clock is derived ultimately from the Celtic words clagan and clocca meaning "bell". A silent instrument missing such a mechanism has traditionally been known as a timepiece...

s and pedometer

Pedometer

A pedometer is a device, usually portable and electronic or electromechanical, that counts each step a person takes by detecting the motion of the person's hips...

s. Stylus-operated adders with circular slots for the stylus, and side-by -side wheels, as made by Sterling Plastics (USA), had an ingenious anti-overshoot mechanism to ensure accurate carries.

Ancient history

Devices have been used to aid computation for thousands of years, using one-to-one correspondence with our fingers. The earliest counting device was probably a form of tally stick

Tally stick

A tally was an ancient memory aid device to record and document numbers, quantities, or even messages. Tally sticks first appear as notches carved on animal bones, in the Upper Paleolithic. A notable example is the Ishango Bone...

. Later record keeping aids throughout the Fertile Crescent

Fertile Crescent

The Fertile Crescent, nicknamed "The Cradle of Civilization" for the fact the first civilizations started there, is a crescent-shaped region containing the comparatively moist and fertile land of otherwise arid and semi-arid Western Asia. The term was first used by University of Chicago...

included clay shapes, which represented counts of items, probably livestock or grains, sealed in containers.

The first clerical aids were abathia, and were often constructed as a wooden frame with beads sliding on wires. Abathias were in use centuries before the adoption of the written Arabic numerals

Arabic numerals

Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...

system and are still used by some merchants, fishermen and clerks in Africa, Asia, and elsewhere. As well, "counting boards", with parallel grooves to hold beads, were used like the abacus.

The counter abacus was devised by Egyptian mathematicians in Egypt in 2000 BC. It was used for arithmetic tasks. The Roman abacus

Roman abacus

The Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors...

was used in Babylonia

Babylonia

Babylonia was an ancient cultural region in central-southern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in central-southern Mesopotamia (present-day Iraq), with Babylon as its capital. Babylonia emerged as...

as early as 2400 BC. Since then, many other forms of reckoning boards or tables have been invented. In a medieval counting house

Counting house

A counting house, or compting house, literally is the building, room, office or suite in which a business firm carries on operations, particularly accounting. By a synecdoche, it has come to mean the accounting operations of a firm, however housed...

, a checkered cloth would be placed on a table, and markers moved around on it according to certain rules, as an aid to calculating sums of money (this is the origin of "Exchequer" as a term for a nation's treasury).

Other precursors to the mechanical calculator

A number of analog computer

Analog computer

An analog computer is a form of computer that uses the continuously-changeable aspects of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved...

s were constructed in ancient and medieval times to perform astronomical calculations. These include the Antikythera mechanism

Antikythera mechanism

The Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100...

and other astrolabe

Astrolabe

s from ancient Greece

Ancient Greece

Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

(c. 150-100 BC), which are generally regarded as the first mechanical analog computers. Other early versions of mechanical devices used to perform some type of calculations include the planisphere

Planisphere

A planisphere is a star chart analog computing instrument in the form of two adjustable disks that rotate on a common pivot. It can be adjusted to display the visible stars for any time and date. It is an instrument to assist in learning how to recognize stars and constellations...

and other mechanical computing devices invented by Abū Rayhān al-Bīrūnī (c. AD 1000); the equatorium and universal latitude-independent astrolabe by Abū Ishāq Ibrāhīm al-Zarqālī (c. AD 1015); the astronomical analog computers of other medieval Muslim astronomers

Islamic astronomy

Islamic astronomy or Arabic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age , and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and...

and engineers; and the astronomical clock

Astronomical clock

An astronomical clock is a clock with special mechanisms and dials to display astronomical information, such as the relative positions of the sun, moon, zodiacal constellations, and sometimes major planets.-Definition:...

tower

Clock tower

A clock tower is a tower specifically built with one or more clock faces. Clock towers can be either freestanding or part of a church or municipal building such as a town hall. Some clock towers are not true clock towers having had their clock faces added to an already existing building...

of Su Song

Su Song

Su Song was a renowned Chinese polymath who specialized himself as a statesman, astronomer, cartographer, horologist, pharmacologist, mineralogist, zoologist, botanist, mechanical and architectural engineer, poet, antiquarian, and ambassador of the Song Dynasty .Su Song was the engineer of a...

(c. AD 1090) during the Song Dynasty

Song Dynasty

The Song Dynasty was a ruling dynasty in China between 960 and 1279; it succeeded the Five Dynasties and Ten Kingdoms Period, and was followed by the Yuan Dynasty. It was the first government in world history to issue banknotes or paper money, and the first Chinese government to establish a...

. The "castle clock", an astronomical clock

Astronomical clock

invented by Al-Jazari

Al-Jazari

Abū al-'Iz Ibn Ismā'īl ibn al-Razāz al-Jazarī was a Muslim polymath: a scholar, inventor, mechanical engineer, craftsman, artist, mathematician and astronomer from Al-Jazira, Mesopotamia, who lived during the Islamic Golden Age...

in 1206, is considered to be the earliest programmable

Computer programming

Computer programming is the process of designing, writing, testing, debugging, and maintaining the source code of computer programs. This source code is written in one or more programming languages. The purpose of programming is to create a program that performs specific operations or exhibits a...

analog computer.

Overview

The 17th century was a turning point in the history of mechanical calculators. On one hand, it saw the invention of logarithm

Logarithm

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

s, logarithmic tables and the slide rule

Slide rule

which, for their ease of use by scientists in multiplying and dividing, ruled over and impeded the use and development of mechanical calculators until the production release of the arithmometer

Arithmometer

in the mid 19th century, and, on the other hand, it saw the invention of the adding machine by Wilhelm Schickard

Wilhelm Schickard

Wilhelm Schickard was a German polymath who designed a calculating machine in 1623, twenty years before the Pascaline of Blaise Pascal. Unfortunately a fire destroyed the machine as it was being built in 1624 and Schickard decided to abandon his project...

, followed by Blaise Pascal

Blaise Pascal

Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...

and by the first four-operations mechanical calculator by Gottfried Leibniz

Gottfried Leibniz

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

. He invented the Leibniz wheel

Leibniz wheel

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 calculators...

and he was also the first to describe 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...

. Neither of them was successful in commercializing their machines.

Logarithms and slide rules

Scottish mathematician and physicist John Napier

John Napier

John Napier of Merchiston – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun. He was the son of Sir Archibald Napier of Merchiston. John Napier is most renowned as the discoverer...

noted multiplication and division of numbers could be performed by addition and subtraction, respectively, of logarithms of those numbers. While producing the first logarithmic tables Napier needed to perform many multiplications, and it was at this point that he designed Napier's bones

Napier's bones

, a device consisting of square rods positioned side by side, marked with digits, used for multiplication and division.

In 1622 William Oughtred

William Oughtred

William Oughtred was an English mathematician.After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is...

invented the slide rule

Slide rule

, which was revealed by his student Richard Delamain in 1630. Since real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s can be represented as distances or intervals on a line, the slide rule allows multiplication and division operations to be carried out significantly faster than was previously possible. The devices were used by generations of engineers and other mathematically inclined professional workers, until the invention of the pocket calculator. The engineers in the Apollo program that sent a man to the moon made many of their calculations on slide rules, which were accurate to three or four significant figures.

Invention of the mechanical calculator

In 1642, Blaise Pascal

Blaise Pascal

Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...

invented the mechanical calculator while looking for a way to help his father who had been assigned the task of reorganizing the tax revenues of the French province of Haute-Normandie

Haute-Normandie

Upper Normandy is one of the 27 regions of France. It was created in 1984 from two départements: Seine-Maritime and Eure, when Normandy was divided into Lower Normandy and Upper Normandy. This division continues to provoke controversy, and some continue to call for reuniting the two regions...

. After three years of effort and 50 prototypes he introduced his machine to the public. He built twenty of these machines (called the Pascaline) in the following ten years. This machine could add and subtract two numbers directly and multiply and divide by repetition.

Gottfried Leibniz

Gottfried Leibniz

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

invented the first calculator that could perform all four arithmetic operations automatically while adding direct multiplication and division to the Pascaline ; it was 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'...

. He built it around 1672, but careful examination at the end of the 19th century showed a problem with the carry mechanism. It used his Leibniz wheel

Leibniz wheel

s which, a century and a half later, will be at the heart of the arithmometer

Arithmometer

, the first calculator to be commercialized. He also was the first to describe 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 1685. Leibniz once said "It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used."

Unknown prototype

Wilhelm Schickard

, a German polymath

Polymath

A polymath is a person whose expertise spans a significant number of different subject areas. In less formal terms, a polymath may simply be someone who is very knowledgeable...

, designed a calculating clock in 1623; a fire destroyed it during its construction in 1624 and Schickard abandoned his project. Two sketches of it were discovered in 1957; too late to have any impact on the development of mechanical calculators.

The 18th century

The 18th century saw the first fully functional, four operations, mechanical calculators. Both pinwheel calculators and Leibniz wheel calculators were built with a few unsuccessful attempts at their commercialization.

Machines produced

in 1820 Thomas de Colmar patented the 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...

. It was a true multiplication machine with a one digit multiplier (the millionaire calculator
The Millionaire Calculator
The Millionaire calculator was the first commercially successful mechanical calculator that could perform a direct multiplication. It was in production from 1893 to 1935 with a total of about five thousand machines manufactured.- History :...

released 70 years later had a similar user interface). He spent the next 30 years and 300,000 Francs developing his machine. Notable was the calculator's internal mechanical product lookup table.

in 1851, Thomas de Colmar simplified the arithmometer by removing the multiplier. This made it a simple adding machine, but thanks to its moving carriage used as an indexed accumulator, it still allowed for easy multiplication and division under operator control. The arithmometer was now adapted to the manufacturing capabilities of the time; Thomas can therefore manufacture consistently a sturdy and reliable machine. Manuals are printed and each machine is given a serial number. Its commercialization launches the mechanical calculator industry. Banks, insurance companies, government offices started to use the arithmometer in their day-to-day operations.

Dorr E. Felt, in the U.S., patented the Comptometer
Comptometer
The comptometer was the first commercially successful key-driven mechanical calculator, patented in the USA by Dorr E. Felt in 1887.A key-driven calculator is extremely fast because each key adds or subtracts its value to the accumulator as soon as it is pressed and a skilled operator can enter all...

in 1886. It was the first successful key-driven adding and calculating machine. ["Key-driven" refers to the fact that just pressing the keys causes the result to be calculated, no separate lever or crank has to be operated. Other machines are sometimes called "key-set".] In 1887, he joined with Robert Tarrant to form the Felt & Tarrant Manufacturing Company. The comptometer-type calculator was the first machine to receive an all-electronic calculator engine in 1961 (the ANITA mark VII
Sumlock ANITA calculator
The ANITA Mark VII and ANITA Mark VIII calculators were launched simultaneously in late 1961 as the world's first all-electronic desktop calculators. Designed and built by the Bell Punch Co...

released by Sumlock comptometer of the UK).

in 1878 W.T. Odhner patented the Odhner Arithmometer
Odhner Arithmometer
The Odhner Arithmometer was a very successful pinwheel calculator invented in Russia in 1873 by W. T. Odhner, a Swedish immigrant. Its industrial production officially started in 1890 in Odhner's Saint Petersburg workshop...

which was a redesigned version of the Arithmometer with a pinwheel engine
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...

but with the same user interface. Odhner started manufacturing his machine in his Saint Petersburg
Saint Petersburg
Saint Petersburg is a city and a federal subject of Russia located on the Neva River at the head of the Gulf of Finland on the Baltic Sea...

workshop in 1890. Many companies, all over the world, manufactured clones of this machine and millions were sold well into the 1970s.

In 1892 William S. Burroughs began commercial manufacture of his printing adding calculator Burroughs Corporation became one of the leading companies in the accounting machine and computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

businesses.

The "Millionaire"
The Millionaire Calculator
The Millionaire calculator was the first commercially successful mechanical calculator that could perform a direct multiplication. It was in production from 1893 to 1935 with a total of about five thousand machines manufactured.- History :...

calculator was introduced in 1893. It allowed direct multiplication by any digit - "one turn of the crank for each figure in the multiplier". It contained a mechanical product lookup table, providing units and tens digits by differing lengths of posts. http://home.vicnet.net.au/~wolff/calculators/Tech/Millionaire/Intro.htm Another direct multiplier was part of the Moon-Hopkins billing machine; that company was acquired by Burroughs in the early 20th century.

Prototypes and limited runs

In 1822 Charles Babbage
Charles Babbage
Charles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer...

designed a mechanical calculator, called a difference engine
Difference engine
A difference engine is an automatic, mechanical calculator designed to tabulate polynomial functions. Both logarithmic and trigonometric functions can be approximated by polynomials, so a difference engine can compute many useful sets of numbers.-History:...

, which was capable of holding and manipulating seven numbers of 31 decimal digits each. Babbage produced two designs for the difference engine and a further design for a more advanced mechanical programmable computer called an analytical engine
Analytical engine
The Analytical Engine was a proposed mechanical general-purpose computer designed by English mathematician Charles Babbage. It was first described in 1837 as the successor to Babbage's difference engine, a design for a mechanical calculator...

. None of these designs were completely built by Babbage. In 1991 the London Science Museum
Science Museum (London)
The Science Museum is one of the three major museums on Exhibition Road, South Kensington, London in the Royal Borough of Kensington and Chelsea. It is part of the National Museum of Science and Industry. The museum is a major London tourist attraction....

followed Babbage's plans to build a working difference engine using the technology and materials available in the 19th century.

In 1842, Timoleon Maurel invented the 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...

, based on the Arithmometer, which could multiply two numbers by simply entering their values into the machine.

In 1845 Izrael Abraham Staffel
Izrael Abraham Staffel
Izrael Abraham Staffel , Polish inventor, watchmaker, mechanic, designer of calculating machines.Staffel was born in 1814 in Warsaw to an impoverished Jewish family. He received an elementary education in a Jewish school and was then sent to a watchmaker to continue his education. He taught himself...

first exhibited a machine that was able to add, subtract, divide, multiply and obtain a square root.

In 1853 Per Georg Scheutz
Per Georg Scheutz
Pehr Georg Scheutz was a 19th-century Swedish lawyer, translator, and inventor, who is best known for his pioneering work in computer technology.Scheutz studied law at Lund University, graduating in 1805...

completed a working difference engine based on Babbage's design. The machine was the size of a piano, and was demonstrated at the Exposition Universelle
Exposition Universelle (1855)
The Exposition Universelle of 1855 was an International Exhibition held on the Champs-Elysées in Paris from May 15 to November 15, 1855. Its full official title was the Exposition Universelle des produits de l'Agriculture, de l'Industrie et des Beaux-Arts de Paris 1855.The exposition was a major...

in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

in 1855. It was used to create tables of logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

s.

In 1872, Frank S. Baldwin
Frank Stephen Baldwin
Frank Stephen Baldwin was an American inventor who is best known for having invented a pinwheel calculator in 1874. He started the design of a new machine in 1905 and was able to finalize its design with the help of Jay R...

in the U.S. invented 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 1875 Martin Wiberg
Martin Wiberg
Martin Wiberg was born in Viby, Scania, Sweden, enrolled at Lund University in 1845 and became a Doctor of Philosophy in 1850....

re-designed the Babbage/Scheutz difference engine and built a version that was the size of a sewing machine.

Operating the calculator at the beginning of this article

Although this is an old machine, nevertheless it represents how one operates any basic rotary calculator. Facits have a pinwheel cylinder that shifts internally, instead of a moving carriage, but the principles still hold.

First, clear the result dials, and then move all setting levers to zero. Position the carriage appropriately. (Use the levers at the front.) The handcrank must be at home position, engaged with its positioning stop.

To add, enter the number into the setting levers. Pull the crank handle to the right, and then toward you, so that it's going away from you when the handle is at the top. One turn will add the number into the accumulator dials, and the counter register to the left will show [1].

Multiplication: If you continue turning, you'll multiply by the number of turns -- you're adding repetitively. If you need to multiply by several digits, it's simplest to start with the rightmost multiplier digit, then shift the carriage to the right one position for the next digit.

To subtract, pull the crank handle to the right, and push it away from you, so the handle is moving toward you when it's at its highest point. If you subtract more than the number in the accumulator dials, you'll get a complement, which you'll need to convert.

Short-cut multiplication

It's quite unnecessary to crank six or more times for a multiplier digit. Instead, you can shift the carriage one position to the right, add once, then back up the carriage and subtract, until the counter shows the correct digit. For instance, to multiply by 8, shift, add once, shift back, and subtract twice. (10-2 = 8) Thinking ahead, instead, you can subtract twice before adding; the calculator will keep track for you.

Division

Clear the machine and enter the dividend into the setting slides, starting at the left. Move the carriage to the right so the leftmost dividend digit aligns with the leftmost setting lever. Add once. Clear the counter. If you're lucky, the machine should have a counter-reverse control that will make the counter increment for subtraction and decrement for addition. (Some later, better machines do this automatically for the first turn after entering the dividend and clearing the counter.)

So, now you have the dividend in the accumulator, left-justified.

Change the setting levers to enter the divisor, again to the left.

Be sure the counter is clear, and start subtracting. If the machine has a bell, you can crank mindlessly until the bell rings, then add once. Otherwise, you'll need to watch the accumulator contents to note (or anticipate) an "overdraft" (subtraction too many times); you have to correct it if it happens, by adding.

Shift the carriage one position to the left, and resume subtracting.

Repeat for each quotient digit, until you either reach the machine's limits or have enough digits. The counter cantains your quotient; the accumulator contains the remainder (if any).

It's of some interest that essentially-automatic division generally appeared before automatic multiplication; as each quotient digit developed, overdraft was allowed to happen, and it triggered a single add cycle followed by a shift.

Square root is possible, by the "fives method", but the description is rather more complicated. This type of machine, in particular, is quite good for this kind of calculation.

Short-cut division

This takes some thought, but can save time. By watching the accumulator, you can anticipate a large quotient digit, and in a fashion similar to short-cut multiplication, you can add and shift, instead of simply subtracting, to save cycles. (The Marchant calculator contains a multi-digit analog magnitude comparator that prevents overdrafts! Changing from subtraction to addition and back is messy and slow in that machine.)

Mechanical calculators reach their zenith

Two different classes of mechanisms had become established by this time, reciprocating and rotary. The former type of mechanism was operated typically by a limited-travel hand crank; some internal detailed operations took place on the pull, and others on the release part of a complete cycle. The illustrated 1914 machine is this type; the crank is vertical, on its right side. Later on, some of these mechanisms were operated by electric motors and reduction gearing that operated a crank

Crank

-Mechanism:* Crank , in mechanical engineering, a bent portion of an axle, or shaft, or an arm keyed at right angles to the end of a shaft, by which motion is imparted to or received from it...

and connecting rod

Connecting rod

In a reciprocating piston engine, the connecting rod or conrod connects the piston to the crank or crankshaft. Together with the crank, they form a simple mechanism that converts linear motion into rotating motion....

to convert rotary motion to reciprocating.

The latter, type, rotary, had at least one main shaft that made one [or more] continuous revolution[s], one addition or subtraction per turn. Numerous designs, notably European calculators, had handcranks, and locks to ensure that the cranks were returned to exact positions once a turn was complete.

The first half of the 20th century saw the gradual development of the mechanical calculator mechanism.

The Dalton adding-listing machine introduced in 1902 was the first of its type to use only ten keys, and became the first of many different models of "10-key add-listers" manufactured by many companies.

In 1948 the miniature Curta calculator, which was held in one hand for operation, was introduced after being developed by 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....

in 1938. This was an extreme development of the stepped-gear calculating mechanism. It subtracted by adding complements; between the teeth for addition were teeth for subtraction.

From the early 1900s through the 1960s, mechanical calculators dominated the desktop computing market (see History of computing hardware). Major suppliers in the USA included Friden

Friden, Inc.

Friden Calculating Machine Company was an American manufacturer of typewriters and electronic calculators. It was founded by Carl Friden in San Leandro, California in 1934. Friden electromechanical calculators were robust and popular....

, Monroe

Monroe Calculator Company

The Monroe Calculator Company was a leading maker of adding machines and calculators founded in 1912 by Jay R. Monroe and now known as Monroe Systems for Business...

, and SCM/Marchant

Marchant Calculator

The Marchant Calculating Machine Co. was founded in 1911 by Rodney and Alfred Marchant in Oakland, California.The company built mechanical, and then electromechanical calculators which had a reputation for reliability. First models were similar to the Odhner arithmometer. In 1918, employee Carl...

. (Some comments about European calculators follow below.) These devices were motor-driven, and had movable carriages where results of calculations were displayed by dials. Nearly all keyboards were full — each digit that could be entered had its own column of nine keys, 1..9, plus a column-clear key, permitting entry of several digits at once. (See the illustration below of a Marchant Figurematic.)One could call this parallel entry, by way of contrast with ten-key serial entry that was commonplace in mechanical adding machines, and is now universal in electronic calculators. (Nearly all Friden calculators, as well as some rotary (German) Diehls had a ten-key auxiliary keyboard for entering the multiplier when doing multiplication.) Full keyboards generally had ten columns, although some lower-cost machines had eight. Most machines made by the three companies mentioned did not print their results, although other companies, such as Olivetti

Olivetti

Olivetti S.p.A. is an Italian manufacturer of computers, printers and other business machines.- Founding :The company was founded as a typewriter manufacturer in 1908 in Ivrea, near Turin, by Camillo Olivetti. The firm was mainly developed by his son Adriano Olivetti...

, did make printing calculators.

In these machines, addition

Addition

Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

and subtraction

Subtraction

In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...

were performed in a single operation, as on a conventional adding machine, but multiplication

Multiplication

Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....

and division

Division (mathematics)

right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

were accomplished by repeated mechanical additions and subtractions. Friden

Friden, Inc.

made a calculator that also provided square root

Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

s, basically by doing division, but with added mechanism that automatically incremented the number in the keyboard in a systematic fashion. The last of the mechanical calculators were likely to have short-cut multiplication, and some ten-key, serial-entry types had decimal-point keys. However, decimal-point keys required significant internal added complexity, and were offered only in the last designs to be made. The Marchant (Model SKA) also offered square root calculation. Handheld mechanical calculators such as the 1948 Curta continued to be used until they were displaced by electronic calculators in the 1970s.

Typical European four-operations machines use the Odhner mechanism, or variations of it. This kind of machine included the Original Odhner, Brunsviga and several following imitators, starting from Triumphator, Thales, Walther, Facit up to Toshiba. Although most of these were operated by handcranks, there were motor-driven versions. Hamann calculators externally resembled pinwheel machines, but the setting lever positioned a cam that disengaged a drive pawl when the dial had moved far enough.

Although Dalton introduced in 1902 first ten-key printing adding (two operations, the other being subtraction) machine, these feature were not present in computing (four operations) machines for many decades. Facit-T (1932) was the first 10-key computing machine sold in large numbers. Olivetti

Olivetti

Divisumma-14 (1948) was the first computing machine with both printer and a 10-key keyboard.

Full-keyboard machines, including motor-driven ones, were also built until the 1960s. Among the major manufacturers were Mercedes-Euklid, Archimedes, and MADAS in Europe; in the USA, Friden, Marchant, and Monroe were the principal makers of rotary calculators with carriages. Reciprocating calculators (most of which were adding machines, many with integral printers) were made by Remington Rand and Burroughs, among others. All of these were key-set. Felt & Tarrant made Comptometers, as well as Victor, which were key-driven.

The basic mechanism of the Friden and Monroe, described above, was a modified Leibniz wheel (better known, perhaps informally, in the USA as a "stepped drum" or "stepped reckoner"). The Friden had an elementary reversing drive between the body of the machine and the accumulator dials, so its main shaft always rotated in the same direction. The Swiss MADAS was similar. The Monroe, however, reversed direction of its main shaft to subtract.

The earliest Marchants were pinwheel machines, but most of them were remarkably-sophisticated rotary types. They ran at 1,300 addition cycles per minute if you held down the [+] bar. Others were limited to 600 cycles per minute, because their accumulator dials started and stopped for every cycle; Marchant dials moved at a steady and proportional speed for continuing cycles. Most Marchants had a row of nine keys on the extreme right, as shown in the photo of the Figurematic. These simply made the machine add for the number of cycles corresponding to the number on the key, and then shifted the carriage one place. Even nine add cycles took only a short time.

In a Marchant, near the beginning of a cycle, the accumulator dials moved downward "into the dip", away from the openings in the cover. They engaged drive gears in the body of the machine, which rotated them at speeds proportional to the digit being fed to them, with added movement (reduced 10:1) from carries created by dials to their right. At the completion of the cycle, the dials would be misaligned like the pointers in a traditional watt-hour meter. However, as they came up out of the dip, a constant-lead disc cam realigned them by way of a (limited-travel) spur-gear differential. As well, carries for lower orders were added in by another, planetary differential. (The machine shown has 39 differentials in its (20-digit) accumulator!).

In any mechanical calculator, in effect, a gear, sector, or some similar device moves the accumulator by the number of gear teeth that corresponds to the digit being added or subtracted — three teeth changes the position by a count of three. The great majority of basic calculator mechanisms move the accumulator by starting, then moving at a constant speed, and stopping. In particular, stopping is critical, because to obtain fast operation, the accumulator needs to move quickly. Variants of Geneva drives typically block overshoot (which, of course, would create wrong results).

However, two different basic mechanisms, the Mercedes-Euklid and the Marchant, move the dials at speeds corresponding to the digit being added or subtracted; a [1] moves the accumulator the slowest, and a [9], the fastest. In the Mercedes-Euklid, a long slotted lever, pivoted at one end, moves nine racks ("straight gears") endwise by distances proportional to their distance from the lever's pivot. Each rack has a drive pin that's moved by the slot. The rack for [1] is closest to the pivot, of course.
For each keyboard digit, a sliding selector gear, much like that in the Leibniz wheel, engages the rack that corresponds to the digit entered. Of course, the accumulator changes either on the forward or reverse stroke, but not both. This mechanism is notably simple and relatively easy to manufacture.

The Marchant, however, has, for every one of the its ten columns of keys, a nine-ratio "preselector transmission" with its output spur gear at the top of the machine's body; that gear engages the accumulator gearing. When one tries to work out the numbers of teeth in such a transmission, a straightforward approach leads one to consider a mechanism like that in mechanical gasoline pump registers, used to indicate the total price. However, this mechanism is seriously bulky, and utterly impractical for a calculator; 90-tooth gears are likely to be found is the gas. pump. Practical gears in the computing parts of a calculator can't have 90 teeth. They would be either too big, or too delicate.

Given that nine ratios per column implies significant complexity, a Marchant contains a few hundred individual gears in all, many in its accumulator. Basically, the accumulator dial has to rotate 36 degrees (1/10 of a turn) for a [1], and 324 degrees (9/10 of a turn) for a [9], not allowing for incoming carries. At some point in the gearing, one tooth needs to pass for a [1], and nine teeth for a [9]. There's no way to develop the needed movement from a driveshaft that rotates one revolution per cycle with few gears having practical (relatively small) numbers of teeth.

The Marchant, therefore, has three driveshafts to feed the little transmissions. For one cycle, they rotate 1/2, 1/4, and 1/12 of a revolution. http://home.vicnet.net.au/~wolff/calculators/Tech/MarchantDRX/Actuator.htm. The 1/2-turn shaft carries (for each column) gears with 12, 14, 16, and 18 teeth, corresponding to digits 6, 7, 8, and 9. The 1/4-turn shaft carries (also, each column) gears with 12, 16, and 20 teeth, for 3, 4, and 5. Digits [1] and [2] are handled by 12 and 24-tooth gears on the 1/12-revolution shaft. Practical design places the 12th-rev. shaft more distant, so the 1/4-turn shaft carries freely-rotating 24 and 12-tooth idler gears. For subtraction, the driveshafts reversed direction.

In the early part of the cycle, one of five pendants moves off-center to engage the appropriate drive gear for the selected digit.
If possible, see John Wolff's Web site http://home.vicnet.net.au/~wolff/calculators/Tech/MarchantDRX/Intro.htm for a superb collection of photos with some accompanying explanations. He has similar sets of photos for several other notable calculators.

Some machines had as many as 20 columns in their full keyboards. The monster in this field was the Duodecillion made by Burroughs for exhibit purposes.

For sterling currency, £/s/d (and even farthings), there were variations of the basic mechanisms, in particular with different numbers of gear teeth and accumulator dial positions. To accommodate shillings and pence, extra columns were added for the tens digit[s], 10 and 20 for shillings, and 10 for pence. Of course, these functioned as radix-20 and radix-12 mechanisms.

A variant of the Marchant, called the Binary-Octal Marchant, was a radix-8 (octal) machine. It was sold to check very early vacuum-tube (valve) binary computers for accuracy. (Back then, the mechanical calculator was much more reliable than a tube/valve computer.)

As well, there was a twin Marchant, comprising two pinwheel Marchants with a common drive crank and reversing gearbox.http://www.vintagecalculators.com/html/the_twin_marchant.html
The article at the link describes them and shows a twin Brunsviga (side-by-side machines). Twin machines were relatively rare, and apparently were used for surveying calculations (The CORDIC algorithm was invented later, but these machine might be able to execute it.) At least one triple machine (Brunsviga(?)) was made. It's likely that a given accumulator could be engaged with either half of the twin.

The Facit calculator, and one similar to it, are basically pinwheel machines, but the array of pinwheels moves sidewise, instead of the carriage. The pinwheels are biquinary; digits 1 through 4 cause the corresponding number of sliding pins ot extend from the surface; digits 5 through 9 also extend a five-tooth sector as well as the same pins for 6 through 9.

The keys operate cams that operate a swinging lever to first unlock the pin-positioning cam that's partof the pinwheel mechanism; further movement of the lever (by an amount determined by the key's cam) rotates the pin-positioning cam to extend the necessary number of pins. http://home.vicnet.net.au/~wolff/calculators/Tech/FacitC1-13/C113.htm#Rotor

The end of an era

Mechanical calculators continued to be sold, though in rapidly decreasing numbers, into the early 1970s, with many of the manufacturers closing down or being taken over. Comptometer

Comptometer

type calculators were often retained for much longer to be used for adding and listing duties, especially in accounting, since a trained and skilled operator could enter all the digits of a number in one movement of the hands on a Comptometer

Comptometer

quicker than was possible serially with a 10-key electronic calculator. In fact, it was quicker to enter larger digits in two strokes using only the lower-numbered keys; for instance, a 9 would be entered as 4 followed by 5. Some key-driven calculators had keys for every column, but only 1 through 5; they were correspondingly compact. The spread of the computer rather than the simple electronic calculator put an end to the Comptometer

Comptometer

. Also, by the end of the 1970s, the slide rule

Slide rule

had become obsolete.

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