How do you find the caliber of an American Gun Co double barrel shotgun serial 40052 with two triggers two hammers and damascus barrels?

Shotgun ammunition is measured by gauge (or bore in England), not caliber. In addition to the gauge, you will need to know the correct shell length and an old damascus barreled gun probably uses 2 1/2 inch shells which are no longer available. To determine the correct ammunition, you will have to have the chambers measured by a gunsmith. He probably won't even charge you for this simple service. Most likely, an old American Arms shotgun will be a 12 gauge. If a dime fits the muzzle, it's a 12 gauge modified choke, if it fits loosely, it's 12 gauge open choke, and if it will almost go in, it's 12 gauge full choke. You'll probably find that one barrel is full and the other modified. Even if you determine the correct ammunition for this gun, don't shoot it. Although damascus barrels were excellent for the proper ammunition when new they are subject to deterioration over time (and your gun is probably nearly if not over 100 years old) if they have not had proper care for their entire lifetime.