How does the mechanical linkage of the airspeed indicator convert dynamic pressure to KIAS?
(YouTube)

As the diaphragm inflates, it pushes a levered arm up, which drives a gear via a hollowed curved track (not via the cogs). The bigger the inflation, the smaller the rotation of the gear (arm increasing radius) compared to the piston displacement. It's the only variable in the mechanism that is not a fixed ratio.

Full disclosure: this is my analysis of the video, I haven't designed one before.