Certainly, without delving into the math of elastic deformation of metals, there must be a lot of trial and error design to come up with the shape of the anneriod diaphram within the ASI. The pitot does its bit by deriving and transmitting total pressure to the inside of the diaphram with static pressure fed to the outside internals resulting in a direct reading of dynamic pressure. Gears and cams can easily derive a mechanical answer to a complex equation.
Basic...diaphram expands with increase in total pressure against the static pressure within the instrument. This movement is linked to a shaft that also has a lever, which acts upon a sector...this is where the math is done. The shaft turns according to movement of diaphram, the lever on the shaft acts on the sector...as the sector moves it presents more movement as the lever slides along the side of the sector. The sector acts through a gear to turn the needle on the dial. There is also adjustment available through a spring that can be adjusted to give more or less resistance to the expanding diaphram.