Dang , all the good answers are taken . Well that’s what I get I guess for working late .
As I understand it the octagon does 3 things
1) if provides for a more rigid barrel while at the same time allowing the barrel material to be somewhat soft .
Normally if you measure from a flat to the bore and then from the peak to the bore , you will find about .025 or more in diameter
2) ease of welding . As steve mention , barrels were hammered by hand , around a mandrel somewhat smaller then bore size . As such the barrels are somewhat round as the anvil “ hope that’s the right name “ that is use has half round notches in it . If you get muzzleloader , you will see , if I recall Hershel House forge welding up a barrel using the half round anvels .. The barrel is then hammered , Not ground to a to a given wall thickness .
Rotated , hammered again , so on and so forth . then the flats are draw filed or ground to true them up .
later as feltwad stated , this changed to just making the barrels thicker and grinding the flats , not drawing thiem
3) as steve said again . It provides for a easy way to know the barrel is true .
Now that should not be confused with meaning that the bore is true . We are only speaking of the outside of the barrel
Now with the bore , special reamers are made to bring the bore to a given constant , clean bore size . Then the barrel is placed in a rifling machine.
If you go to the the link for Toad hall , given above , you will see examples of a rifling machine
This is where what feltwad said comes into play ..
Later when steam and water hammers and grinders came into play , the processes changed a lot . But it was not really a whole lot easier to produce a octagon barrel , then a round one. but a minsat had been instilled and people wanted what they had come to know as a proper made barrel
Now I was also once told , long ago by a learned foe, that it was much simpler then that . In his view the octagon was architectural IE #1
Basically people looked for strenght in nature .
Basically a true round was weak , and it was thought that if a tree would bend and sway in the wind , a round barrel would do the same .
Yet a bee’s honey comb is ridged , withstands many times its own weight upon its walls , without deforming as such , this was felt to be the best design for something like a barrel .
Now true or not . I cant say . There sure seems to be a lot of examples of round barrels . However I can see the logic in that explanation as well