Blacksmithing is no longer a secret art and the techniques used are well-known. The square to octagon to etc. is a great way to turn a piece of bar stock into a round profile, but isn't the way a forge-welded barrel is made.
Forge-welding a barrel:
The purpose of the mandrel is to serve as a backer to the hammer blows and keeps the tube from collapsing. If a piece of flat stock is hammered into a swage block to make it U-shaped, the edges overlapped and forge-welded with the mandrel inserted, the end product is roughly cylindrical. After straightening as much as possible, the bore is then reamed so it is the same diameter from end-to-end. The octagonal shape must be intentionally created afterwards, serving to center the bore on the barrel profile which makes mounting sights and sighting-in easier (an off-center bore can cause problems). The barrel is likely straightened a few more times during the process.
...that wasn't the only way it was done they also forge welded by wrapping the iron around the mandrel and welding the edge to edge using like scrap iron like old horse shoes. to make a barrel of flat iron it would need to a little over 3 wide to make a one inch single seam Barrel....