Mechanical Computing
FTL writes "Tim Robinson has built a computer capable of solving polynomial equations -- using Meccano. His difference engine (mirror) uses a similar approach to Babbage's design. He's also created a differential analyzer (mirror) complete with a GUI. Both could be scaled up indefinitely to handle larger problems. 'Computing by steam' is possible."
Mechanical computers were built and used over 60 years ago to solve differential equations and other analytical type problems. I know MIT and UCLA had pretty good mechanical computers in the WWII era. Check out MIT's famous mechanical differential analyzer for and idea of what was and is and awesome piece of hardware.
Digital computers are more suited being built electronically (small and fast), but in the early days of computers, many were analog. The transition to digital happened around the same time as the transition from mechanical to electronic. Nowardays, analog computing is virtually unheard of, but I think that sometime during the late 80's/early 90's, they had found an application for analog computing in Neural Networks. I can't remember what it was, but the revival in analog computing has failed to materialise, so it must not have been very important.