Heinlein described himself as a mechanical engineer with a specialty in linkages. The reason for that specialty was the fire control computers on the big guns of the ships he served on, especially while a midshipman at Annapolis.

is a good description of the technology.

Fascinating discussion of the mechanical Analogue computers that Heinlein used. In 1956 I took an Air Force correspondence course in guided missile technology and all the computer guidance systems discussed were mechanical analogue computers as described in the article. After getting out of the AF, I worked at McDonnell Aircraft and learned to program the electrical analogue computers used in the Mercury simulator to provide realistic capsule displays from launch to orbit. I don't remember anything about the simulator programming except that I did it.

Wow. I wonder when the last one was retired from service anywhere in the world.