mechanical computer is built from mechanical components such as levers and gears, rather than electronic components. The most common examples are adding machines and mechanical counters, which use the turning of gears to increment output displays. More complex examples could carry out multiplication and division--Friden used a moving head which paused at each column--and even differential analysis. One model sold in the 1960s calculated square roots.
Mechanical computers can be either
analog, using smooth mechanisms such as curved plates or slide rules for computations; or digital, which use gears.
computers reached their zenith during World War II, when they formed the basis of complex bombsights including the Norden, as well as the similar devices for ship computations such as the US Torpedo Data Computer or British Admiralty Fire Control Table. Noteworthy are mechanical flight instruments for early spacecraft, which provided their computed output not in the form of digits, but through the displacements of indicator surfaces. From Yuri Gagarin's first manned spaceflight until 2002, every manned Soviet and Russian spacecraft Vostok, Voskhod and Soyuz was equipped with a showing the apparent movement of the Earth under the spacecraft through the displacement of a miniature Globus instrument terrestrial globe, plus latitude and longitude indicators.
Mechanical computers continued to be used into the 1960s, but were quickly replaced by electronic calculators, which--with
cathode-ray tube output--emerged in the mid-1960s. The evolution culminated in the 1970s with the introduction of inexpensive handheld electronic calculators. The use of mechanical computers declined in the 1970s and was rare by the 1980s.
In 2016, NASA announced that its
Automaton Rover for Extreme Environments program would use a mechanical computer to operate in the harsh environmental conditions found on Venus.
Antikythera mechanism, c. 100 BC - A mechanical astronomical clock.
Cosmic Engine, 1092 - Su Song's hydro-mechanical astronomical clock tower invented during the Song dynasty, which featured the use of an early escapement mechanism applied to clockwork.    
Castle clock, 1206 - Al-Jazari's castle clock, a hydropowered mechanical astronomical clock, was the earliest programmable analog computer.   
Pascaline, 1642 - Blaise Pascal's arithmetic machine primarily intended as an adding machine which could add and subtract two numbers directly, as well as multiply and divide by repetition.
Stepped Reckoner, 1672 - Gottfried Wilhelm Leibniz's mechanical calculator that could add, subtract, multiply, and divide.
Difference Engine, 1822 - Charles Babbage's mechanical device to calculate polynomials.
Analytical Engine, 1837 - A later Charles Babbage device that could be said to encapsulate most of the elements of modern computers.
Odhner Arithmometer, 1873 - W. T. Odhner's calculator who had millions of clones manufactured until the 1970s.
Ball-and-disk integrator, 1886 - William Thomson used it in his Harmonic Analyser to measure tide heights by calculating coefficients of a Fourier series.
Percy Ludgate's 1909 Analytical Machine - The 2nd of only two mechanical Analytical Engines ever designed.
Marchant Calculator, 1918 - Most advanced of the mechanical calculators. The key design was by Carl Friden.
István Juhász Gamma-Juhász    (early 1930s) 
Kerrison Predictor ("late 1930s"?)
Z1, 1938 (ready in 1941) - Konrad Zuse's mechanical calculator (although part imprecisions hindered its function) 
Mark I Fire Control Computer, deployed by the United States Navy during World War II (1939 to 1945) and up to 1969 or later.
Curta calculator, 1948
Moniac, 1949 - An analog computer used to model or simulate the UK economy.
Voskhod Spacecraft "Globus" IMP navigation instrument, early 1960s
Digi-Comp I, 1963 - An educational 3-bit digital computer
Digi-Comp II, mid 1960s - A rolling ball digital computer
Automaton - Mechanical devices that, in some cases, can store data and perform calculations, and perform other complicated tasks. Turing Tumble, 2017- An educational Turing-complete computer partially inspired by the Digi-Comp II
Early electrically powered computers constructed from
switches and relay logic rather than vacuum tubes (thermionic valves) or transistors (from which later electronic computers were constructed) are classified as electro-mechanical computers.
These varied greatly in design and capabilities, with some later units capable of floating point arithmetic. Some relay-based computers remained in service after the development of vacuum-tube computers, where their slower speed was compensated for by good reliability. Some models were built as duplicate processors to detect errors, or could detect errors and retry the instruction. A few models were sold commercially with multiple units produced, but many designs were experimental one-off productions.
Automatic Relay Computer
The Booths, experimental
later known as WITCH
Harvard Mark I
"IBM Automatic Sequence Controlled Calculator"
Harvard Mark II
"Aiken Relay Calculator"
Imperial College Computing Engine (ICCE)
  
Office of Naval Research ONR Relay Computer
6-bit, drum storage, but electro-mechanical relay ALU based on
Atlas, formerly Navy cryptology computer ABEL
  
Commercial use at Zeiss Optical in Jena
Hobbyist logic demonstrator magazine article
Bell Labs Model I
George Stibitz, "Complex Number Calculator", 450 relays and crossbar switches, demonstrated remote access 1940, used until 1948
Bell Labs Model II
"Relay Interpolator", used for wartime work, shut down 1962
Bell Labs Model III
"Ballistic Computer", used until 1949
Bell Labs Model IV
Navy "Mark 22 Error Detector", used until 1961
Two units delivered, general-purpose, built in trigonometric functions, floating-point arithmetic
General purpose, simplified Model V with several enhancements
Unnamed cryptanalysis multiplier
Harry Porter's Relay Computer, demonstrator/hobby, but integrated circuit memory.
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