You can read patents 2778907 and 2522337. The first one has a nice mainly schematic diagram and explanation as prelude to replacing the indicator with a resistor divider for electronic reading; the second is the patent for the idea using strictly analog methods and is the basis for mechanical meters.
You can find them at
https://ppubs.uspto.gov/pubwebapp/st...s/landing.html Look at Basic Search and type in the numbers or enter machmeter (which finds the first one) and maybe "mach meter" which is in the second one, which I found as a reference in the first one.
See
https://en.wikipedia.org/wiki/Machmeter for the some of the math.