Lord Kelvin and his analog computer
spectrum.ieee.org
spectrum.ieee.org
https://www.charlespetzold.com/blog/2016/12/Computer-of-the-...
https://www.charlespetzold.com/blog/2019/06/Computer-of-the-...
And the meccano integrator, from Manchester. And also Marc Brunel's block making machines, and possibly Henry Maudsleys screw cutting lathe (or maybe Whitworths) and his true-plates (always made in sets of 3 to avoid co-dependent bumps) as well as transatlantic cable, wheatstone bridge, you-name-it.
And of course Babbage-engine bits-and-bobs. It amazed me when the IETF had an open-night there, and were told to stick to the space display (ground floor) when it was geeks.. and upstairs were the ur-devices of .. computation and networking.
Here's how a mechanical harmonic synthesizer and analyzer work.[1] This is a very clear presentation of Michelson's design. It does forward and reverse Fourier transforms. Reverse (synthesis) is obvious enough, but the trick for analysis is clever.
The hardest part of the problem was the addition. Sine and cosine were easy. Multiplication by a constant was easy. But adding - that was hard. The Michaelson machine has an elegant solution. The positions to be added apply tension to springs, which pull on a common pull bar opposed by a single spring. So the addition happens in force space.
The USGS tide predicting machine #2 [2] added up the terms by running a long chain over many pulleys that can slide. The position of the end of the chain sums the pulley positions. This is a true positional sum, and is more accurate. But the mechanism is much bulkier and heavier.
[1] https://www.youtube.com/watch?v=GyYflzRVu6M
[2] https://en.wikipedia.org/wiki/Tide-Predicting_Machine_No._2
[1]: https://www.flickr.com/photos/ideonexus/5970013589/ [2]: https://pacificsciencecenter.org/exhibit/puget-sound-model/
Someone could argue that cutting/labelling all the pieces of spaghetti takes considerably more time. I'm sure we are all capable of seeing past such a boring technicality.
The book described how big a pile of spaghetti would need to be to sort large numbers, and I'm afraid I can't remember any of the details, but there's a certain point where you need to bang the pile of spaghetti against something the size of the moon.
Here is a list of his inventions: https://collection.sciencemuseumgroup.org.uk/people/cp37484