A working model of an ancient computer was recently recreated in London:
The so-called "Antikythera mechanism" has puzzled historians since it was salvaged from an ancient shipwreck near the Greek island of Antikythera in 1901. It dates back to about 100 BC, and consists of more than 30 bronze gear wheels and pointers, enclosed in a wooden case.
The device is by far the most advanced scientific instrument to survive from antiquity - nothing else close to its complexity shows up in archaeological records for more than 1200 years, when mechanical clocks appeared in medieval Europe.
The Antikythera mechanism is thought to be a mechanical computer, which used sophisticated algorithms to calculate the motions of celestial bodies.
http://www.newscientist.com/data/images/archive/2686/26861602.jpg
http://www.newscientist.com/article/mg20026861.600-decoding-the-2000yearold-computer.html
http://www.decodingtheheavens.com/
Who built the first programmable robot? It's almost impossible to tell, and most people would put good money on Leonardo da Vinci. But now Noel Sharkey, a computer scientist at the University of Sheffield, UK, has traced the technology way back to ancient Alexandria.
In about 60 AD, a Greek engineer called Hero constructed a three-wheeled cart that could carry a group of automata to the front of a stage where they would perform for an audience. Power came from a falling weight that pulled on string wrapped round the cart's drive axle, and Sharkey reckons this string-based control mechanism is exactly equivalent to a modern programming language.
http://www.newscientist.com/blog/technology/2007/07/programmable-robot-from-60ad.html
Ben Crystall explains how he recreated a programmable robot dating from 60AD.
http://en.wikipedia.org/wiki/Hero_of_Alexandria
http://www.history.rochester.edu/steam/hero/
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