Why gauge increase with thickness decrease?

The concept of the gauge originated with pistols, and later shotguns. The gauge described the size of the barrel by the fraction of a pound of a round iron ball that just fits the barrel. For example, a 12 guage shotgun has a barrel that will just fit a round iron ball that is one twelfth of a pound. This measurement stemmed from cannon measurements, which measured the cannon by the size of the cannon ball it fired (such as a 6 pound cannon, or 20 pound cannon).

There are different gauges that describe thickness, including the American Wire Gauge that measures the thickness (diameter) of electrical wire, US Steel Wire Gauge for steel wire, Sheet Metal Gauge, etc.