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There are some old training videos that show how this worked: https://www.youtube.com/watch?v=gwf5mAlI7Ug

Also the Battleship New Jersey YouTube channel has some nice content on this: https://www.youtube.com/watch?v=szxNJydEqOs

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See [1] for the basic mechanical components. It's a better scan of the same film the Periscope Film archive sells, which is the first one linked above. No sprocket clatter.

[1] https://www.youtube.com/watch?v=s1i-dnAH9Y4

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Haven't been there in years but the Nike facility in Marin is well worth a visit if you're there when it's open. The control stations were originally on a higher ridge but they have one of the (basically) containers next to the missile sites now. The idea at the time is that they would explode ordinance (originally conventional, later nuclear) above incoming bombers causing a pressure wave that would make them crash.

Was also a Nike base on Angel Island but there's nothing left there but some old concrete pads.

We actually had one of the Nike bases defending Philadelphia literally next to where I grew up. Don't remember personally--was very young--but there were apparently troop manoeuvres on our property from time to time.

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One of my favorite internet links is an archive of manuals from this era. Especially the Torpedo Data Computer, another fire solution calculator.

Excellent illustrations!

https://maritime.org/doc/tdc/index.php

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If you're looking for more, the book "Between Human and Machine: Feedback, Control, and Computing before Cybernetics" is a detailed history of the development of electromechanical fire control computers and feedback systems.
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Off topic, but this is where I see AI going. A tool that condenses work down from requiring a team and a room to a box. We're decades away from that
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