G'day GuestEd,I have seen a circa 1845 instruction book for the German Concertina - a German crossbreed from their accordion (aka melodion) and the first Wheatstone system concertinas. This new tuning was then picked up by English makers who subsequently made them in the standard English hexagonal shape.
I have seen one square "Anglo-German" by an English maker, but with improved German style reeds and keywork ... before Lachenal, Wheatstone's former toolmaker, revolutionised the design and brought in a combined cottage out-work / factory production in the 1850s. Some of Wheatstone's 1830s prototypes were square too, but this was when he was just feeling his way. The Germans, mostly went to hexagonal ... except for Arnold's "Bandoneon", which remains a square design to this day.
Regards,
Bob Bolton (all of whose concertinas are hexagonal or octagonal)