A mechanical calculator, or calculating machine, was a mechanical device used to perform automatically the basic operations of arithmetic. Most mechanical calculators were comparable in size to small desktop computers and have been rendered obsolete by the advent of the electronic calculator.
Blaise Pascal invented the mechanical calculator in 1642; it was called Pascal's Calculator or Pascaline and was the only working mechanical calculator in the 17th century. Thomas' arithmometer, the first commercially successful machine, was manufactured two hundred years later in 1851; it was the first mechanical calculator strong enough and reliable enough to be used daily in an office environment. For forty years the arithmometer was the only type of mechanical calculator available for sale.
The comptometer, introduced in 1887, was the first machine to use a keyboard which consisted of columns of nine keys (from 1 to 9) for each digit. The Dalton adding machine, manufactured from 1902, was the first to have a 10 key keyboard. Electric motors were used on some mechanical calculators from 1901. In 1961, a comptometer type machine, the Anita mk7 from Sumlock comptometer Ltd., became the first desktop 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 production of mechanical calculators came to a stop in the middle of the 1970s closing an industry that had lasted for 120 years.
Charles Babbage designed two new kinds of mechanical calculators, which were so big that they required the power of a steam engine to operate, and that were too sophisticated to be built in his lifetime. The first one was an automatic mechanical calculator, his difference engine, which could automatically compute and print mathematical tables. In 1855, Georg Scheutz became the first of a handful of designers to succeed at building a smaller and simpler model of his difference engine. The second one was a programmable mechanical calculator, his analytical engine, which Babbage started to design in 1834 ; "in less than two years he had sketched out many of the salient features of the modern computer. A crucial step was the adoption of a punched card system derived from the Jacquard loom" making it infinitely programmable. In 1937, Howard Aiken convinced IBM to design and build the ASCC/Mark I, the first machine of its kind, based on the architecture of the analytical engine; when the machine was finished some hailed it as "Babbage's dream come true".
Ancient history 
The desire to economize time and mental effort in arithmetical computations, and to eliminate human liability to error, is probably as old as the science of arithmetic itself. This desire has led to the design and construction of a variety of aids to calculation, beginning with groups of small objects, such as pebbles, first used loosely, later as counters on ruled boards, and later still as beads mounted on wires fixed in a frame, as in the abacus. This instrument was probably invented by the Semitic races and later adopted in India, whence it spread westward throughout Europe and eastward to China and Japan.
After the development of the abacus, no further advances were made until John Napier devised his numbering rods, or Napier's Bones, in 1617. Various forms of the Bones appeared, some approaching the beginning of mechanical computation, but it was not until 1642 that Blaise Pascal gave us the first mechanical calculating machine in the sense that the term is used today.—Howard Aiken, Proposed automatic calculating machine, presented to IBM in 1937
The 17th century 
The 17th century marked the beginning of the history of mechanical calculators, as it saw the invention of its first machine, Pascal's calculator, in 1642. Blaise Pascal had invented a machine that could perform computations that where previously thought to be only humanly possible, but he wasn't successful in creating an industry.
In a sense, Pascal's invention was premature, in that the mechanical arts in his time were not sufficiently advanced to enable his machine to be made at an economic price, with the accuracy and strength needed for reasonably long use. This difficulty was not overcome until well on into the nineteenth century, by which time also a renewed stimulus to invention was given by the need for many kinds of calculation more intricate than those considered by Pascal.—S. Chapman, Pascal tercentenary celebration, London, (1942)
The 17th century also saw the invention of some very powerful tools to aid arithmetic calculations like Napier's bones, logarithmic tables and the 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 in the mid 19th century.
Invention of the mechanical calculator 
Blaise Pascal invented the mechanical calculator in 1642. After three years of effort and 50 prototypes he introduced his calculator to the public. He built twenty of these machines in the following ten years. This machine could add and subtract two numbers directly and multiply and divide by repetition. Pascal chose the most demanding method of re-zeroing for his machine since it propagates a carry right through the machine. In doing so, he proved, before each operation, that his calculator was fully functional. This is a testament to the quality of the Pascaline because none of the 17th and 18th century criticisms of the machine mentioned a problem with the carry mechanism and yet it was fully tested on all the machines, by their resets, all the time.
Pascal's invention of the calculating machine, just three hundred years ago, was made while he was a youth of nineteen. He was spurred to it by seeing the burden of arithmetical labor involved in his father's official work as supervisor of taxes at Rouen. He conceived the idea of doing the work mechanically, and developed a design appropriate for this purpose; showing herein the same combination of pure science and mechanical genius that characterized his whole life. But it was one thing to conceive and design the machine, and another to get it made and put into use. Here were needed those practical gifts that he displayed later in his inventions...—S. Chapman, Pascal tercentenary celebration, London, (1942)
In 1672, Gottfried Leibniz started working on adding direct multiplication to Pascal's calculator ; while first trying to simply interface with the pascaline, he eventually designed an entirely new machine called the Stepped Reckoner ; it used his Leibniz wheels, was the first two-motion calculator, the first to use cursors (creating a memory of the first operand) and the first to have a movable carriage. Leibniz built two Stepped Reckoners, one in 1694 and one in 1706. Only the machine built in 1694 is known to exist, it was rediscovered at the end of the 19th century having spent 250 years forgotten in an attic in the University of Gottingen.
In 1893, the German calculating machine inventor Arthur Burkhardt was asked to put Leibniz machine in operating condition if possible. His report was favorable except for the sequence in the carry.
Leibniz had invented his namesake wheel and the principle of a two motion calculator, but after forty years of development he wasn't able to produce a machine that was fully operational; this makes Pascal's calculator the only working mechanical calculator in the 17th century. Leibniz was also the first person to describe a pinwheel calculator. He 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."
Calculating clocks: unsuccessful mechanical calculators 
Both Pascal and Leibniz tried to design some kind of calculating clock before inventing their machines.
...I devised a third which works by springs and which has a very simple design. This is the one, as I have already stated, that I used many times, hidden in the plain sight of an infinity of persons and which is still in operating order. Nevertheless, while always improving on it, I found reasons to change its design...—Pascal, Advertisement Necessary to those who have curiosity to see the Arithmetic Machine, and to operate it, (1645)
When, several years ago, I saw for the first time an instrument which, when carried, automatically records the numbers of steps by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only counting but also addition and subtraction, multiplication and division could be accomplished by a suitably arranged machine easily, promptly, and with sure results—Leibniz, on his calculating machine, (1685)
The principle of a calculating clock (input wheels and display wheels added to a clock like mechanism) for a direct entry calculating machine couldn't be implemented with the technological levels of the 17th century because their heavier and more numerous gears could be damaged when a carry had to be moved several places along the accumulator. The only 17th century calculating clocks that have survived to this day do not have a machine wide carry mechanism and therefore cannot be called mechanical calculators. The first true calculating clock was built by the Italian Giovanni Poleni in the 18th century and was a two-motion calculating clock (the numbers are inscribed first and then they are processed).
- In 1623, Wilhelm Schickard, a German professor of Hebrew and Astronomy, designed a calculating clock which he drew on two letters that he wrote to Johannes Kepler. The first machine to be built by a professional was destroyed during its construction and Schickard abandoned his project in 1624. These drawings had appeared in various publications over the centuries, starting in 1718 with a book of Kepler's letters by Michael Hansch, but in 1957 it was presented for the first time as a long lost mechanical calculator by Dr. Franz Hammer. The building of the first replica in the 1960s showed that Schickard's machine had an unfinished design and therefore wheels and springs were added to make it work. The use of these replicas showed that the single tooth wheel, when used within a calculating clock, was an inadequate carry mechanism. (see Pascal versus Schickard).
- Around 1643, a French clockmaker from Rouen, after hearing of Pascal's work, built a calculating clock of his own design. Pascal fired all his employees and stopped developing his calculator as soon as he heard of the news. It is only after being assured that his invention would be protected by a royal privilege that he restarted his activity. A careful examination of this calculating clock showed that it didn't work properly and Pascal called it an avorton (aborted fetus).
- In 1659, the Italian Tito Livio Burattini built a machine with nine independent wheels, each one of these wheels was paired with a smaller carry wheel. At the end of an operation the user had to either manually add each carry to the next digit or mentally add these numbers to create the final result.
- In 1666, Samuel Morland invented a machine designed to add sums of money, but it was not a true adding machine since the carry was added to a small carry wheel situated above each digit and not directly to the next digit. It was very similar to Burattini's machine. Morland created also a multiplying machines with interchangeable disks based on Napier's bones.
- In 1673, the French clockmaker René Grillet described in Curiositez mathématiques de l'invention du Sr Grillet, horlogeur à Paris a calculating machine that would be more compact than Pascal's calculator and reversible for subtraction. The only two Grillet machines known have no carry mechanism, displaying three lines of nine independent dials they also have nine rotating napier's rod for multiplication and division. Contrary to Grillet's claim, it was not a mechanical calculator after all.
The 18th century 
The 18th century saw the first mechanical calculator that could perform a multiplication automatically ; designed and built by Giovanni Poleni in 1709 and made of wood, it was the first successful calculating clock. For all the machines built in this century, division still required the operator to decide when to stop a repeated subtraction at each index, and therefore these machines were only providing a help in dividing, like an abacus. Both pinwheel calculators and Leibniz wheel calculators were built with a few unsuccessful attempts at their commercialization.
Prototypes and limited runs 
- In 1709, the Italian Giovanni Poleni was the first to build a calculator that could multiply automatically. It used a pinwheel design, was the first operational calculating clock and was made of wood; he destroyed it after hearing that Antonius Braun had received 10,000 Guldens for dedicating a pinwheel machine of his own design to the emperor Charles VI of Vienna.
- In 1725, the French Academy of Sciences certified a calculating machine derived from Pascal's calculator designed by Lépine, a French craftsman. The machine was a bridge in between Pascal's calculator and a calculating clock. The carry transmissions were performed simultaneously, like in a calculating clock, and therefore "the machine must have jammed beyond a few simultaneous carry transmissions".
- In 1727, a German, Antonius Braun, presented the first fully functional four operations machine to Charles VI, Holy Roman Emperor in Vienna. It was cylindrical in shape and was made of steel, silver and brass; it was finely decorated and looked like a renaissance table clock. His dedication to the emperor engraved on the top of the machine also reads "..to make easy to ignorant people, addition, subtraction, multiplication and even division".
- In 1730, the French Academy of Sciences certified three machines designed by Hillerin de Boistissandeau. The first one used a single tooth carry mechanism which, according to Boistissandeau, wouldn't work properly if a carry had to be moved more than two places; the two other machines used springs that were gradually armed until they released their energy when a carry had to be moved forward. It was similar to Pascal's calculator but instead of using the energy of gravity Boistissandeau used the energy stored into the springs.
- In 1770, Philipp Matthäus Hahn, a German pastor, built two circular calculating machines based on Leibniz' cylinders. J.C. Schuster, Hahn's brother in law, built a few machines of Hahn's design into the early 19th century.
- In 1775, Lord Stanhope of the United Kingdom designed a pinwheel machine. It was set in a rectangular box with a handle on the side. He also designed a machine using Leibniz wheels in 1777. "In 1777 Stanhope produced the Logic Demonstrator, a machine designed to solve problems in formal logic. This device marked the beginning of a new approach to the solution of logical problems by mechanical methods."
The 19th century 
For 40 years, the arithmometer was the only mechanical calculator available for sale and was sold all over the world. By then, in 1890, about 2,500 arithmometers had been sold plus a few hundreds more from two licensed arithmometer clone makers (Burkhardt, Germany, 1878 and Layton, UK, 1883). Felt and Tarrant, the only other competitor in true commercial production, had sold 100 comptometers in three years.
The 19th century also saw the designs of Charles Babbage calculating machines, first with his difference engine, started in 1822, which was the first automatic calculator since it continuously used the results of the previous operation for the next one, and second with his analytical engine, which was the first programmable calculator, using Jacquard's cards to read program and data, that he started in 1834, and which gave the blueprint of the mainframe computers built in the middle of the 20th century.
Desktop calculators produced 
- In 1851, Thomas de Colmar simplified his arithmometer by removing the one digit multiplier/divider. 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 could therefore manufacture consistently a sturdy and reliable machine. Manuals were printed and each machine was given a serial number. Its commercialization launched the mechanical calculator industry. Banks, insurance companies, government offices started to use the arithmometer in their day-to-day operations, slowly bringing mechanical desktop calculators into the office.
- In 1878 Burkhardt, of Germany, was the first to manufacture a clone of Thomas' arithmometer. Until then Thomas de Colmar had been the only manufacturer of desktop mechanical calculators in the world and he had manufactured about 1,500 machines. Eventually twenty European companies will manufacture clones of thomas' arithmometer until WWII.
- Dorr E. Felt, in the U.S., patented the Comptometer 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 released by Sumlock comptometer of the UK).
- In 1890 W.T. Odhner got the rights to manufacture his calculator back from Königsberger & C, which had held them since it was first patented in 1878, but had not really produced anything. Odhner used his Saint Petersburg workshop to manufacture his calculator and he built and sold 500 machines in 1890. This manufacturing operation shut down definitively in 1918 with 23,000 machines produced. The Odhner Arithmometer was a redesigned version of the Arithmometer of Thomas de Colmar with a pinwheel engine, which made it cheaper to manufacture and gave it a smaller footprint while keeping the advantage of having the same user interface.
- In 1892 Odhner sold the Berlin branch of his factory, which he had opened a year earlier, to Grimme, Natalis & Co.. They moved the factory to Braunschweig and sold their machines under the brand name of Brunsviga (Brunsviga is the Latin name of the town of Braunschweig). This was the first of many companies which would sell and manufacture clones of Odhner's machine all over the world; eventually 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 businesses.
- The "Millionaire" 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.  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 1820, Thomas de Colmar patented the Arithmometer. It was a true four operation machine with a one digit multiplier/divider (the millionaire calculator released 70 years later had a similar user interface). He spent the next 30 years and 300,000 Francs developing his machine. This design was replaced in 1851 by the simplified arithmometer which was only an adding machine.
- In 1842, Timoleon Maurel invented the Arithmaurel, based on the Arithmometer, which could multiply two numbers by simply entering their values into the machine.
- In 1845, Izrael Abraham Staffel first exhibited a machine that was able to add, subtract, divide, multiply and obtain a square root.
- Around 1854, Andre-Michel Guerry invented the Ordonnateur Statistique, a cylindrical device designed to aid in summarizing the relations among data on moral variables (crime, suicide, etc.)
- In 1883, Edmondson of the UK patented a circular stepped drum machine
Automatic mechanical calculators 
- In 1822, Charles Babbage presented a small cogwheel assembly that demonstrated the operation of his difference engine, a mechanical calculator which would be capable of holding and manipulating seven numbers of 31 decimal digits each. It was the first time that a calculating machine could work automatically using as input results from its previous operations. It was the first calculating machine to use a printer. The development of this machine, later called "Difference Engine No. 1," stopped around 1834.
- In 1847, Babbage began work on an improved difference engine design—his "Difference Engine No. 2." None of these designs were completely built by Babbage. In 1991 the London Science Museum followed Babbage's plans to build a working Difference Engine No. 2 using the technology and materials available in the 19th century.
- In 1855, Per Georg Scheutz 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 in Paris in 1855. It was used to create tables of logarithms.
- In 1875, Martin Wiberg re-designed the Babbage/Scheutz difference engine and built a version that was the size of a sewing machine.
Programmable mechanical calculators 
- In 1834, Babbage started to design his analytical engine, which will become the undisputed ancestor of the modern mainframe computer with two separate input streams for data and program (a primitive Harvard architecture), printers for outputting results (three different kind), processing unit (mill), memory (store) and the first ever set of programming instructions. In the proposal that Howard Aiken gave IBM in 1937 while requesting funding for the Harvard Mark I which became IBM's entry machine in the computer industry, we can read: "Few calculating machines have been designed strictly for application to scientific investigations, the notable exceptions being those of Charles Babbage and others who followed him. In 1812 Babbage conceived the idea of a calculating machine of a higher type than those previously constructed to be used for calculating and printing tables of mathematical functions. ....After abandoning the difference engine, Babbage devoted his energy to the design and construction of an analytical engine of far higher powers than the difference engine..."
- In 1843, during the translation of a French article on the analytical engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers. This is considered the first computer program.
- From 1872 until 1910, Henry Babbage worked intermittently on creating the mill, the "central processing unit" of his father's machine. After a few setbacks, he gave in 1906 a successful demonstration of the mill which printed the first 44 multiples of pi with 29 places of figures.
Cash registers 
The cash register, invented by James Ritty in 1879, addressed the old problems of disorganization and dishonesty in business transactions. It was a pure adding machine coupled with a printer, a bell and a two sided display that showed the paying party and the store owner, if he wanted to, the amount of money exchanged for the current transaction.
The cash register was easy to use and, unlike genuine mechanical calculators, was needed and quickly adopted by a great number of businesses. "Eighty four companies sold cash registers between 1888 and 1895, only three survived for any length of time".
By 1900, NCR had built 200,000 cash registers and there were more companies manufacturing them, compared to the "Thomas/Payen" arithmometer company that had just sold around 3,300 and Burroughs had only sold 1,400 machines.
1900s to 1970s 
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 and connecting rod 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 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, Monroe, and SCM/Marchant. (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, did make printing calculators.
In these machines, addition and subtraction were performed in a single operation, as on a conventional adding machine, but multiplication and division were accomplished by repeated mechanical additions and subtractions. Friden made a calculator that also provided square roots, 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. 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 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  moves the accumulator the slowest, and a , 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  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 , and 324 degrees (9/10 of a turn) for a , not allowing for incoming carries. At some point in the gearing, one tooth needs to pass for a , and nine teeth for a . 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. . 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  and  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  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. 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. 
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.
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 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 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. Also, by the end of the 1970s, the slide rule had become obsolete.
Operating an Odhner calculator 
The Odhner arithmometer was the most produced mechanical calculator.
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 .
- 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.
- 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.)
See also 
- Jean Marguin (1994), p. 48
- See Pascal's calculator#Competing designs
- Beside two arithmometer clone makers from Germany and England, the only other company to offer calculators for sale was Felt & Tarrant from the USA which started selling their comptometer in 1887 but had only sold 100 machines by 1890.
- Ernst Martin p.133 (1925)
- Ernst Martin p.23 (1925)
- #MARG,Jean Marguin p. 171, (1994)
- Anthony Hyman, Charles Babbage, pioneer of the computer, 1982
- "The introduction of punched cards into the new engine was important not only as a more convenient form of control than the drums, or because programs could now be of unlimited extent, and could be stored and repeated without the danger of introducing errors in setting the machine by hand; it was important also because it served to crystallize Babbage's feeling that he had invented something really new, something much more than a sophisticated calculating machine." Bruce Collier, 1970
- I. Bernard Cohen, p. 66-67, (2000)
- Brian Randell, p.187, 1975
- Please see Pascaline#Pascal versus Schickard
- "The arithmetical machine produces effects which approach nearer to thought than all the actions of animals. But it does nothing which would enable us to attribute will to it, as to the animals.", Pascal, Pensées Bartleby.com, Great Books online, Blaise Pasdcal, Thoughts
- Magazine Nature, (1942)
- Scripta Mathematica, p.128 (1932)
- From the calculating machine of Pascal to the computer, p.43 (1990)
- (fr) La Machine d’arithmétique, Blaise Pascal, Wikisource
- Guy Mourlevat, p. 12 (1988)
- Courrier du CIBP, N°8, p.9, (1986)
- "...et si blocage il y avait, la machine était pratiquement inutilisable, ce qui ne fut jamais signalé dans les textes du XVIIIe siecle parmi ses défaults" Guy Mourlevat, p.30 (1988)
- Jean Marguin, p. 64-65 (1994)
- Scripta Mathematica, p.149 (1932)
- David Smith, p.173-181 (1929)
- As quoted in Smith 1929, pp. 180–181
- Translated from "j'en composai une troisième qui va par ressorts et qui est très simple en sa construction. C'est celle de laquelle, comme j'ai déjà dit, je me suis servi plusieurs fois, au vu et su d'une infinité de personnes, et qui est encore en état de servir autant que jamais. Toutefois, en la perfectionnant toujours, je trouvai des raisons de la changer" Avis nécessaire à ceux qui auront curiosité de voir la Machine d'Arithmétique et de s'en servir Wikisource: La Machine d’arithmétique, Blaise Pascal
- Quoted in David Smith, p.173, (1929)
- Michael Williams, p.124,128 (1997) for Schikard's machine and the fact that the machines built by Burattini, Morland and Grillet were calculating clocks without a complete carry mechanism.
- History of computer (retrieved on 01/02/2012)
- Michael Williams, p.122 (1997)
- Michael Williams, p.124,128 (1997)
- "The appearance of this small avorton disturbed me to the utmost and it dampened the enthusiasm with which I was developing my calculator so much that I immediately let go all of my employees..." translated from the French: "L'aspect de ce petit avorton me déplut au dernier point et refroidit tellement l'ardeur avec laquelle je faisais lors travailler à l'accomplissement de mon modèle qu'à l'instant même je donnai congé à tous les ouvriers..."
- "But, later on, Lord Chancellor of France [...] granted me a royal privilege which is not usual, and which will suffocate before their birth all these illegitimate avortons which, by the way, could only be born of the legitimate and necessary alliance of theory and art." translated from the French: "Mais, quelque temps après, Monseigneur le Chancelier [...] par la grâce qu'il me fit de m'accorder un privilège qui n'est pas ordinaire, et qui étouffe avant leur naissance tous ces avortons illégitimes qui pourraient être engendrés d'ailleurs que de la légitime et nécessaire alliance de la théorie avec l'art"
- "...a useless piece, perfectly clean, polished and well filed on the outside but so imperfect inside that it is of no use whatsoever." translated from the French: "...qu'une pièce inutile, propre véritablement, polie et très bien limée par le dehors, mais tellement imparfaite au dedans qu'elle n'est d'aucun usage"
- All the quotes in this paragraph are found in (fr) Wikisource: Avis nécessaire à ceux qui auront curiosité de voir la Machine d'Arithmétique et de s'en servir.
- Picture of Burattini's machine Florence, Istituto e Museo di Storia della Scienza, inv. 3179 (accessed on January, 09 2012)
- A calculator Chronicle, 300 years of counting and reckoning tools, p. 12, IBM
- Michael Williams, p.140 (1997)
- Picture of Morland multiplying machine Florence, Istituto e Museo di Storia della Scienza, inv. 679 (retrieved on January, 09 2012)
- They belong to the Musée des Arts et Métiers in Paris.
- "Grillet's machine doesn't even deserve the name of machine" translated from the French "La machine de Grillet ne mérite donc pas même le nom de machine", Jean Marguin, p.76 (1994)
- Copy of Poleni's machine (it) Museo Nazionale della Scienza e della Tecnologia Leonardo Da Vinci. Retrieved 2010-10-04
- Jean Marguin, p. 93-94 (1994)
- translated from the French: "De plus le report ne s'effectuant pas en cascade, la machine devait se bloquer au-delà de quelques reports simultanés", Jean Marguin, p.78 (1994)
- Jean Marguin, p.94-96 (1994)
- #MARG, Jean Marguin, pages 80-81 (1994)
- Marguin, p.83 (1994)
- Picture of Hahn's Calculator IBM Collection of mechanical calculators
- Jean Marguin, pages 84-86 (1994)
- Door E. Felt, p.15-16 (1916)
- Le calcul simplifié. Maurice d'Aucagne
- This is one third of the 120 years that this industry lasted
- Arithmometre.org (retrieved on 01/02/2012)
- Felt, Dorr E. (1916). Mechanical arithmetic, or The history of the counting machine. Chicago: Washington Institute. p. 4.
- "The calculating engines of English mathematician Charles Babbage (1791-1871) are among the most celebrated icons in the prehistory of computing. Babbage’s Difference Engine No.1 was the first successful automatic calculator and remains one of the finest examples of precision engineering of the time. Babbage is sometimes referred to as "father of computing." The International Charles Babbage Society (later the Charles Babbage Institute) took his name to honor his intellectual contributions and their relation to modern computers." Charles Babbage Institute (page retrieved on 01/02/2012).
- Ifrah G., The Universal History of Numbers, vol 3, page 127, The Harvill Press, 2000
- Chase G.C.: History of Mechanical Computing Machinery, Vol. 2, Number 3, July 1980, IEEE Annals of the History of Computing, p. 204
- Serial numbers and Years of manufacturing www.arithmometre.org, Valéry Monnier
- J.A.V. Turck, Origin of modern calculating machines, The Western Society of Engineers, 1921, p. 75
- G. Trogemann, pages: 39-45
- David J. Shaw: The Cathedral Libraries Catalogue, The British Library and the Bibliographical Society, 1998
- J.A.V. Turck, Origin of modern calculating machines, The Western Society of Engineers, 1921, p. 143
- A notable difference was that the Millionaire calculator used an internal mechanical product lookup table versus a repeated addition or subtraction until a counter was decreased down to zero and stopped the machine for the arithmometer
- L'ami des Sciences 1856, p.301 www.arithmometre.org (page retrieved on 09/22/2010)
- Larousse, P. (1886), Grand dictionaire universel du XIX siecle, Paris, entry for A-M Guerry
- Patent application in french from www.ami19.org scanned by Valéry Monnier (retrieved on January 12, 2012)
- James Essinger, p.76 (2004)
- "The better part of my live has now been spent on that machine, and no progress whatever having been made since 1834...", Charles Babbage, quoted in Irascible Genius, 1964, p.145
- "It is reasonable to inquire, therefore, whether it is possible to devise a machine which will do for mathematical computation what the automatic lathe has done for engineering. The first suggestion that such a machine could be made came more than a hundred years ago from the mathematician Charles Babbage. Babbage's ideas have only been properly appreciated in the last ten years, but we now realize that he understood clearly all the fundamental principles which are embodied in modern digital computers" B. V. Bowden, 1953, pp.6,7
- Howard Aiken, 1937, reprinted in The origins of Digital computers, Selected Papers, Edited by Brian Randell, 1973
- NCR Retrospective website accessed October, 02 2012
- History of the cash register retrieved October, 05 2012
- See the number of machines built in 1890 in this paragraph
- Dick and Joan's antique accessed October, 02 2012
- List of serial numbers by dates arithmometre.org retrieved October 10, 2012
- Before the computer, James W. Cortada, p.34 ISBN 0-691-04807-X
- De la machine à calculer de Pascal à l'ordinateur (in fr). Paris, France: Musée National des Techniques, CNAM. 1990. ISBN 2-908207-07-9.
- Trogemann, G.; Nitussov, A. (2001). Computing in Russia. Germany: GWV-Vieweg. ISBN 3-528-05757-2.
- Felt, Dorr E. (1916). Mechanical arithmetic, or The history of the counting machine. Chicago: Washington Institute.
- Marguin, Jean (1994). Histoire des instruments et machines à calculer, trois siècles de mécanique pensante 1642-1942 (in fr). Hermann. ISBN 978-2-7056-6166-3.
- Mourlevat, Guy (1988). Les machines arithmétiques de Blaise Pascal (in fr). Clermont-Ferrand: La Française d'Edition et d'Imprimerie.
- Taton, René (1969). Histoire du calcul. Que sais-je ? n° 198 (in fr). Presses universitaires de France.
- Turck, J.A.V. (1921). Origin of Modern Calculating Machines. The Western Society of Engineers. Reprinted by Arno Press, 1972 ISBN 0-405-04730-4.
- Ginsburg, Jekuthiel (2003). Scripta Mathematica (Septembre 1932-Juin 1933). Kessinger Publishing, LLC. ISBN 978-0-7661-3835-3.
- Martin, Ernst (1992). In The Charles Babbage Institute. The Calculating Machines translation from Die Rechenmaschinen (1925). Cambridge, Massachusetts: The MIT Press.
- Cohen, I. Bernard (200). In The MIT Press. Howard Aiken : Portrait of a Computer Pioneer. Cambridge, Massachusetts: The MIT Press. ISBN 9-780262-531795.
- Smith, David Eugene (1929). A Source Book in Mathematics. New York and London: McGraw-Hill Book Company, Inc.
- Moseley, Maboth (1964). Irascible Genius, Charles Babbage Inventor. London: Hutchinson & Co, Ltd.
- Bowden, B. V. (1953). Faster than thought. New York, Toronto, London: Pitman publishing corporation.
- Williams, Michael R. (1997). History of Computing Technology. Los Alamitos, California: IEEE Computer Society. ISBN 0-8186-7739-2.
- Randell, Brian (1973). The origins of Digital computers, Selected Papers. Springer-Verlag. Text "ISBN 3-540-06169-X" ignored (help)
- IBM. A calculator Chronicle, 300 years of counting and reckoning tools. New York.
- Collier, Bruce. The little engine that could've: The calculating machines of Charles Babbage. Garland Publishing Inc. ISBN 0-8240-0043-9.
- Essinger, James (2004). Jacquard's Web. Oxford University Press. ISBN 0-19-280577-0.
- Prof. S. Chapman (October 31, 1942). "Blaise Pascal (1623-1662) Tercentenary of the calculating machine". Nature (London) 150: 508–509.
- "Usage de la machine". Courrier du centre international Blaise Pascal (in fr) (Clermont-Ferrand) (8): 4–25. 1986.
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