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Charles Babbage got the idea for the Difference Engine in 1820s when he was verifying mathematical tables and decided that was the job for a steam engine. He built a small prototype. Do a search for the two working models of his final design. It is quite facinating. The Difference engine is a special purpose computer for polynomial interpolation.
He got the idea and made several designs for a general purpose numeric computer called the Analytic Engine that was never built.
The ideas that led to modern computers were triggered by the mathematical work of Alan Turing and his invention of the universal machine that could simulate any other machine of that type, in particular itself. The idea was taken up by John Von Neumann who got the idea of programs and data sharing the computer’s memory. This was the basis of all modern computers. At the time several teams were building computers, used for balistic calculations for the long-range and anti-aircraft guns being developed in WW II. The famous Colosus was not quite a general purpose computer, being specifically designed to break the German Enigma code in near real-time.
The answers to your question:
Stock control (the Leo built by the Lyons Tea Company)
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