Gus, the form of the A with the little v-shaped crossbar is usually 18th century. A collector friend tells me the form of the base plug, tapered then flattened, was common in Rhode Island in the 1700s. I never trust that the pulls are original to an old horn. Often they come loose and get replaced during use or “before sale”. The circles are a mystery. Is it possible they were scribed with a compass? Decorative? The horn is pinned with small iron nails. It’s a smallish horn for the period if it is indeed late 1700s. Probably holds 3/4 of a pound of powder. Not ideal for militia use.
It's possible the circles were scribed by a compass for decoration, though it isn't much of a decoration, unless the guy had to stop before he finished it?
Personally, I think the base plug was made from a "recycled" wood "something" and those marks were made by a period center bit, as shown below.
The above type center bits were commonly used to bore holes in wood during the 18th and early 19th century. From buying original ones and using them, I know that when you sharpen the cutting edge on the left side of these bits, if you don't think about shortening the point on the right side to agree with it, you will get that kind of circle in the wood with the point in the center. Actually in many of the original bits I've seen that actually had been used, they didn't think to shorten the point for a number of sharpening's.
Gus