Romans numbering system was the only system used for ages, in the western world. Calculations, in this Roman numbering system, lacked the compactness in writing, were time consuming and more importantly: "it was not possible to automate calculations". (Here an interesting link to the several calculation systems) It turns out that the Chinese were the first (2000BC  300BC) who had a decimal systems, they used bamboo rot symbols and a horizontal rot as zero symbol.
The current decimal numbering system was coming into existence in India in the 5th  8th century.
In the 15th and 16th century the Arabs (Moors) took it to Spain, on their turn they spread it in Europe and in the rest of the world. That was the perfect condition for a world wide use of the decimal system. It became the foundation of our current method of calculating and obviously our calculating machines.
The essentials of the decimal numbering system are:
the use of only 10 symbols (the nine figures and a symbol for zero.
the value every digit represents, in the total number.
From right to left units, tens, hundreds etc.
that calculations of the same figures, have the equal results on each digit.
(Fi.: 2 + 3 has the same result on every digit).
We hardly think about it nowadays, but it is very smart and the breakthrough of this calculating method, to have a symbol for zero.
State administrations, tax officials, bank directors, astronomy and science professors had a strong need to automate calculations. The accuracy of the slide ruler, maximum 3 digits, was often not enough. Books with extensive tables were used to assist the user with multiplications, logarithms and sin and cos etc. Some universities in France and Germany, had one or two mechanical calculators, often these concepts were build under the direct responsibility of their professors, the pioneers.
The mechanical precision of the metal craft tools and the obvious correspondingly skills of labor, was suffering them at realizing their ideas.
These university pioneers surged for mechanical tools and their experts and find them in aria's with a great clocks building tradition, like the "Ertzgebirge" and the "Black forest" in Germany.
The Pioneers
Mechanical Calculators History
The Antikythera ship wreckage
More evidence is coming up that the ancient Greeks had already an calculation tool. A piece of wreckage is found near the Greek island Antikythera. Several universities analyzed the wreckage and it turned out to be an astronomical / planet positions tool.
The still interesting / remaining issues are:

Did they already used calculating tools?
 Which tools did they use to build these?
 Are there more mechanisms from that period?
 Are there
any documents?
These
will be sure subject for further studies. Please see later on on this page for very interesting explaining video's. *1)
Leonardo da Vinci Italy (1452 – 1519)
The first known attempts, to describe a mechanical calculator, based on gears,
are previously only attributed to Leonardo da Vinci.
Twenty years later, in 1641, Blaise Pascal (Paris), yes the man of various physical propositions, developed and build the Pascaline / Pedometer. For his father, who was a tax man and obviously needed a calculating machine. (More details). He build more machines and got the credit to be the man who designed and build first commercial available calculator.
The calculation failures of the machine where a drawback and suffered him to continuous modifications.
The biggest difficulty was the carry (from 9 to 10) and even more the multiple carry (fi from 999 to 1000 etc.).
The Pascal design is in 1675 strongly improved by Gottfried Wilhelm Von Leibniz (born in Leipzig1646).
There are probably two copies build of this machine, which were based on the special wheel, which has become known as the Staffelwalze in Germany, as Stepped drum, Stepped Reckoner or Leibniz Wheel, in the English speaking countries.
It is a turn able cylinder with nine teeth of different lengths on it. The gearwheel above can horizontally moved to the left and the right. Moving this small gearwheel, leads to a turning
of this gearwheel with a equivalent number of teeth. This is shown on the numbered wheel or the result register. Please play the animation on the left.
In early machines the gearwheels are moved by sliders The keyboard was introduced, as the method was developed to convert a key pressed, to the needed distance to move the small gearwheel.
The Gauss and Curta are using one central stepped drum for feeding all the circular surrounding small gear wheels / digits.
The Swede Mr. Odhner modified the "Pinwheel" construction, which was invented in 1709 in Padua, Italia
by Giovanni Poleni. The first reliable Pinwheel calculator came on the market in 1875. This calculator had no keyboard, but number of shifters instead. The Pinwheel consists of separate disks. Each disk has a slider with which a number from 0 to 9 can be entered. Each digit has a slider which controls a mechanism of pins internally in the disk. On the adjacent side of the slider a number of pins, according the slider value, will stick out of the disk. These pins will turn the numbergear when turning the pin wheel. The result and the number of times the wheel is turned are displayed in two separate Windows. A second time the wheel is turning the same amount is adding to the number gears. A adding took place. You subtract the number by turning the wheel in the opposite direction.
The pinwheel and stepped drum machines (see later on) have implemented a genius idea to make the number gears / result unit, horizontally movable. This gave the machines
the ability to easily multiply and divide by successive additions and subtractions.
Pin Wheel
In this context it is nice to mention that Von Leibniz also invented the binary calculation rules (1 and 0), the calculating rule for all our nowadays computers. (More details on Von Leibniz) I like to recommend reading his Biography with his embittered and neglected last years.
Around 1825, the human and mechanical skills, the tools and machinery, was coming to a level that production of reliable calculators was becoming possible. Which was also evidenced by the "Stepped drum" technology. The first commercially successful Stepped drum calculator, see the picture left, was the Arithmometer or the Thomas machine.
Mr. Charles Xavier Thomas de Colmar (1785–1870 Paris) holds the credits for the design, patents and manufacturing.
An other calculator from that time is the Mathieus Hahn machine.
The carriage made multiplication possible though successive additions and shift digits of the multiplicand.
However it was until 1875 that mechanical calculators are more considered as curiosities, then as practical usable machines. This changed, from around 1875 on, the time frame when volume production was established.
The four basic calculation technologies which made it:
Direct add
Stepped drum
Pinwheel
Rocking segment
( I temporary skip history information on the Direct add technology)
Around 1890, manufacturer Burroughs (USA) started to build calculators with this rocking segment technology. The first had only displays, but it turned out that this technology is efficient to add a printing function. On the other, above mentioned technologies, you will not / seldom find printing capabilities.
The machines had many keys, 1..9 for each digit. So the vertical key location represents the value (19) of the figure of that digit, the horizontal location represents the digits value, the right most are the units, then the tens, the hundreds and the thousands etc.
This machine has a weight of 33kg.
You noticed the glass windows, to see the mechanics true it. Nice but this was not the main purpose. They did it primarily for noise isolation. Glass does this much better than a thin steel or aluminum plate.
Later, in 1922, full keyboards are more and more replaced by the 10key ones. Their pin block, the shift mechanism which stores the individual digit value, gave them a much smaller size and weight.
Stepped drum
Source: WikiPedia
The first mechanical calculators for practical usage
The German Mr. Burghard was rebuilding and improved the the Thomas machine (left picture). He gathered the best people for designs together and due to his outstanding business skills and the stride along production capabilities, their machines became more and more a commercial success. But more than that: It turnout to be the native company for the German calculating industry (Stepped drum technology) in Dresden, Glashütte (Ertzgebirge), Sömmerda (Erfurt) and Berlin, with brands like: Archimedes, Ludwig Spitz & Co (TIM), Peerles, Rheinmetall, Saxonia, Seidel & Naumann (XxX).
The significance and complexity of the peace of messing, found in the wreck of a berried Greek ship in 19001901, near the Greek island Antikythera, was long not understood. When it became known that the ship was of the first century BC, the remaining's are very likely the oldest known scientific calculator, an ancient mechanical computer, designed to calculate astronomical planet positions.
Investigations still going on more evidence, documents or others instruments.
Links to the subject. Please visit the website of and email acquaintance of mine who build a replica. Here you will find more details of the used gears dimensions.
The video below tells us a the story of a
comprehensive study / analysis. Starting with the process of getting the numbers of the gears known. It turns out to be an planet eclipse computer. Very useful for warfare to predicting darker nights . It tells us that probably Mr Archimedes was able to give it so much detail. For instance the ellipse shaped track of the moon is taken into account. That the knowledge was lost due to wars and the Arabs got the knowledge and the Moors brought it back to Europe centuries later. It is a long (~2h) but very much recommended to follow story.
A complete different functional concept, than what we've seen sofar in mechanical calculators, was designed in 1833 in London, England by Mr. Charles Babbage.
He had an clear application for his "Difference engine", he was developing. His machine should be able to calculate and produce "error free table books", sinus, cosines and logarithms etc.
He used the "difference method", from the numerical mathematics, which is very useful for this type of sequential calculations. Here the polygon is not calculated by its self, but by the differences between the two sequential ones. This eliminates the need for multiplication and division and the table can be build by only add and subtract type of calculations. Details on the difference method you can find here. The parameters used here can be set for various polygons.
His next project was the "Analytical Engine". He start working on this in 1834. It is considered as the first programming calculator where he was greatly influenced and assisted by misses Ada Lovelance, the first programmer.
The difference engine, the first "Programmable" calculator
Mr. C. Zuze his programmable calculator
Changing from science, to engineering, to mass production
After many professors had build concepts of calculators in an "university" environment, the "industry" and many engineers got many licences on their names. But "clever business men" at the end, making money in the calculator market.
The Gauss, deigned by Mr. C. Hamann. These pictures are made of the machine of the UT Delft, Prof. P. Breedveld. The cylindrical construction build in 1902. This was a predecessor *2) and an inspiration for Curt Herzstark to develop his famous Curta. But prior tho those two, Mr. Hahn's his calculator of 1774, was the first calculator with this cylindrical configuration, around one stepped drum.
Not so well known, as the other pioneers, is Mr. Christian Ludwich Gersten (born in Giessen Germany and studied also in London UK). He developed and build in 1709 and present it in London in 1722, the first "Rocking Segment" machine. Although it was not a segment of a gearwheel, but a toothed rack or Oscillating Rack. This rack moves back and forth by the shifters, engaging the counter register in one direction only, for adding. The shifters, are inputting the numbers by shifting the racks an equal number of teeth. The display reflecting the counter register, showing the results of the adding.
It is not known why this machine is not optimized for production in those years. Rebuilding it in later years showed no technical difficulties, considering the production skill in those years.
Designers of ASTRA / ASCOTA used that toothed or oscillating rack implementation of the rocking segment technology, very successful in their machines, much later (1920....).
Rocking Segment
*2) Predecessor of the Curta
Was it Christian Hamann, with his Gauss , the finding of Axel Jacob Petersson or the one of Paul Haacks, who is the predecessor or gave inspiration to Curt Herzstarks to develop his Curta, the first and very famous pocket calculator?
This PDF describes the patent search and many other original which might give you the answer.
In Germany Mr. Zuse worked on an "digital logic" approach, based on AND, OR and NOT mechanical building blocks he realized the first out of a range, the Z1 in 1937. It was not reliable at all, resulting in many fail calculation. In 1941 he build the next version with a number input of an 8 track cinema film and relay's. It had 64 registers of 22 bits.
In the UK in a covert operation (1943) a programmable calculator was build with radio valves. Mr Newmans, was in charge of the operation. It was 30 years after this secret operation, that the documents of that operation were disclosed.
Programming can be done with simple punched hole cards. This principle was already used at the textile industry.
This machine had:

1000 registers of 50 digits
 a decimal calculator (add) with carry
 control mechanisms for using results in next calculations
 printing unit
 a punched hole cart unit for in and output.
Although it was ample funded, his Difference engine No1 was not completed during his lifetime. With the support of his son and misses Ada Lovelance, Analytical engine No 2 became more mature, but he self never saw the complete machine working.
Between
19902002 two machines are rebuild. One is at the Science museum in London and one at the Computer History Museum in Mountain View, California (USA). (please see my report of my visit.)
In the US and Sweden approaches were made successfully.
In the US
the machine is used for the people counting. They used for the first time punch cards as input from the Hollerith company, much later merged with others manufacturers to finally the IBM company.
Preface: No machine, no matter how extraordinary, is as interesting as its maker.
Three exceptions
With the above
given three technologies the mainstream of mechanical calculators is set. But there are a few others who played an important role in the history of mechanical computing:
The Direct or instant multiplication machines: The millionaire (CH) and the Leon Bollee (Fr)
The concept of the later computers (programmable and the structure): The difference engine of Babbage (UK) and father Edvard and son George Scheutz (Sweden)
The computer with mechanical digital logic blocks: The Zuze machines (DL)
The Direct Multiplication Machines
Introduction A direct multiply machine does a multiply in one or two machine cycles in stead of performing multiple repetitive additions. These machines use a multiplier body. Edmund Barber patented one in 1875,
Ramón Vera in 1878, but both machines caused no break trough. The first mass production was produced by Leon Bollee (Le Mans, Fr.) The biggest success however had the Millionaire of Otto Steiger, build by H. Egli (Zurich, CH)
(please see the chapter "How do they calculate for more details on the working)
The Leon Bollee machine
Mr. Leon Bollee patented his machine in 1889. The unit, to the right of the machine, is the multiplier body. It is a block with an array of pins with specific length. This length determines the multiplying factor.
His passion was cars, so no follow up or updates were seen.
The Millionaire
This machine, called "the Millionaire", was designed in 1892 by Otto Steiger and produced by Hans Egli in Zurich Switzerland. This direct or instant multiplication machine is successfully used world wide.
In 1935, 4655 millionaires were build.
This machine was very popular and is popular by collectors nowadays, due to its rapid multiplications and the very nice internal mechanisms.
Despite the Millionaires popularity, the Egli company stopped the production and continued with the MADAS machines, which were more suitable for automatic calculations, driven by electric motors.
The Moon Hopkins billing machine
The Moon Hopkins machine is a type writer / billing machine which has a calculator unit with a direct multiplier enclosed. This machine, as others, developed from the billing machine area.
Burroughs has marketed this machine later.
An other encore of an mechanical calculator master piece
The scientist and Nobel price winner (in 1852), Mr. Albert Michelson, is known and has been awarded for his work on the speed of light measurements. It is not so well known that he had developed a mechanical harmonic analyzer. A machine who can perform Fourier Analysis and Synthesis. The video below shows that for instance a square wave can be seen as built up of several sine waves.
One machine is in "Das Deutches Museum" in Munich, the others showed at the Arithmeum in Bonn and the HNM in Paderborn (DL).
Wilhelm Schickard (1592–1635).
In Germany the first mechanical calculator was build in 1620 by Wilhelm Schickard. He was professor at the Tübingen university, somewhat north of the Black Forest. The documents were lost until the 19th century. It took until 1960 before replica's were build based on these drawings.