Mechanical Computer – Basic Mechanisms in Fire Control Computers (1953) [video]
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Multiplication is typically modeled using repeated addition, or surface areas. But when negative numbers are involved, it may be unclear. You can stick to areas if you have a Cartesian plane where the upper left and lower right quadrants are understood to contain negative area. But that may seem arbitrary, requiring explanations that seem like hand waving if you don't already have an intuition for it. The mechanical multiplier replaces hand-waving explanations of sign with a waving arm that clearly reverses slope under the right geometric conditions, to produce a positive or negative number.
That vid brought a few internal chats about computing. Since it's clear those things are computing advanced function in real time (with noise though), streaming physical changes downto the linkage graph, no side effects if you haskell-squint hard.. digital computers are a different take on computing, they were timeless, memoryfull at first.. and along the decades it seems we were all seeking to bring back automatic coupling (reactive recomputation) on the foreground.
https://en.wikipedia.org/wiki/Systolic_array
An early example of this (an electronic computer in the 1940s!) seems to have been:
Hollerith and discrete calculating machines were more about unordered data analysis. Static batch of input, chunk it the way you want, forward, backward.. doesn't matter. Very freeing in a way. I can understand why digital computation wiped the market, no noise and no constraint.. how nice. Until it starts to produce spaghetti bowls and crawling insects.
In that alternate universe, maybe Swiss watchmakers are the ones driving innovation in computing.