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Basically, a air gauge acts like a piston in an automobile engine. One end of the barrel is pressurized and a precision plug passed through it. The amount of air escaping around the plug is measured.
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Not exactly.
A carefully dimension cylinder is passed through the hole while air is pumped into passages inside the cylinder that exit on the circumference.
The back pressure created by passing the cylinder through the bore indicates clearance between the bore and the cylinder.
It is more of a uniformity check than an absolute measure.
Changes in the pressure indicate variations in bore diameter.