We've all heard about the 1822 Babbage Difference Engine and other mechanical computer ideas of the 19th century. Sadly, Babbage's computer and printer ideas were never implemented in our world†, and computers did not exist until the 1940s.
Now imagine a world where this was accomplished, as was in short order his more elaborate Analytical Engine, and the industrial and mechano-computing revolutions went hand in hand. According to the Lovelace law of computation (1843), the amount of computation columns per troy pound of steel machinery doubles every 5 years.
Now it is important to note that the Analytical Engine (in Babbage's theoretical machine design as well as in our alternate world's everyday practice) was what we in our world have come to call Turing complete, i.e. is a universal computer. Digital, fully programmable.
Leaving aside the world-changing implications of such an advent, I'm curious to think about the limits of the 'Lovelace Law' -- just how miniaturized and just how powerful could a mechanical computer conceivably be, before some Kuhnian revolution would be required, such as moving to electromechanical devices?
We're not starting from a high bar. For a starting reference, the initial Analytical Engine design had the equivalent of a 16.7 kB memory, and the central processing mill could handle a multiplication of two 20-digit numbers in about 3 minutes. Thomas de Colmar's first arithmometer was not a general purpose computer, but could multiply two eight-digit numbers in 18 seconds.
Would we be able to reach, say, 1950 era computer levels?
† Outside a partial reconstruction in a museum, 170 years later