If you want to know about Tide-Predicting Machine No. 0 - read the following from Charles Pezold:
In the late 1860s and early 1870s, British scientist William Thomson—later known as Lord Kelvin—developed a technique based on Fourier analysis to mathematically predict the pattern of tides for particular seaports. Never just a theoretical physicist, Thomson then designed a machine to carry out these calculations. One of these tide-predicting machines is on view in the Science Museum in London. Today we categorize this machine as a special-purpose analog computer. While it’s certainly interesting from a technical perspective, it also has a surprising role in the history of 19th century science: William Thomson began his research into the tides in the midst of the Darwin Wars following the 1859 publication of The Origin of Species, and part of the impetus behind this astonishing machine was nothing less than to prove Darwin wrong. From the clash of thermodynamics, geology, and evolution, the era of analog computing was born.
I love mechanical computers like these. The engineering is amazing, and bugs are not considered a fact of life in these systems. One thing I often wonder about is how many mechanical computers/state machines are still in operation and why.
You might like this video about the US Navy fire control computer. Mechanical yet also programmable. Eg with cam-based calculation modules like 1/x or differential gearings to perform addition :
https://m.youtube.com/watch?v=gwf5mAlI7Ug
One probably still relevant usecase for mechanical state machines are elevators. Original design of push button controled elevators is surprisingly simple and involves mechanically implemented state machine where the carriage itself is part of the controller mechanism (the overall idea is that on each floor there is switch that gets flipped by the passing carriage that signals whether the carriage is above or below that floor), such systems are certainly still used (second generation of automatic elevator controllers implement the same logic "electronicaly" by means of johnson counters built from mechanical relays instead of the distributed electromechanical logic).
It's not in use today but if you haven't already seen it, I bet you would enjoy Bill Hammack's series explaining how a 100 year old mechanical harmonic analyzer works.
They're becoming less common, but mechanical automobile (and bicycle) odometers are little mechanical computers of a sort. Baseball and softball umpires (especially for rec leagues, not sure about professional games) often use a "clicker" (https://www.amazon.com/Easton-Umpire-Indicator/dp/B07BYX8XQL for example) to keep count of balls and strikes.
In the late 1860s and early 1870s, British scientist William Thomson—later known as Lord Kelvin—developed a technique based on Fourier analysis to mathematically predict the pattern of tides for particular seaports. Never just a theoretical physicist, Thomson then designed a machine to carry out these calculations. One of these tide-predicting machines is on view in the Science Museum in London. Today we categorize this machine as a special-purpose analog computer. While it’s certainly interesting from a technical perspective, it also has a surprising role in the history of 19th century science: William Thomson began his research into the tides in the midst of the Darwin Wars following the 1859 publication of The Origin of Species, and part of the impetus behind this astonishing machine was nothing less than to prove Darwin wrong. From the clash of thermodynamics, geology, and evolution, the era of analog computing was born.
You can read more here: http://www.charlespetzold.com/blog/2016/12/Computer-of-the-T...