Guilty Mathematical War Pleasure.

Demonstrates beautifully the work of mechanical ballistic computers (Part 1) from the US Navy--a must see:

And modern (superb) explanation of mechanical integrators.  

As you can see, Jack Monaco in his presentation uses the same US Navy's training documentary. In the same time you can take a look an Space Navigation Indicator (INK) from 1970s Soyuz vehicle--also fascinating, here is Part 1. 

I highly recommend since the mathematics is explained beautifully and it also is the same math which is in the modern computing, despite its digital processing being highly advanced. But this is a great primer. In reality, of course, what is demonstrated here is not just mechanical but electro-mechanical computing, because it since 1920s involved the use of a huge number of electric machines.