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I really can't figure out where the masu form comes from. I thought it might be from kobun volitional む + causative す but that's -(a)ます not -(i)ます.
Both conjugations of ません and ましょう are identical to how the Kyoto dialect conjugates す. That's as much as I could figure out, I'm really stumped here.
According to Frellesvig (A History of the Japanese Language, p338), the masu auxiliary is derived from the polite auxiliary marase, which apparently is a humble word for 'give'. The marase form emerged in Late Middle Japanese (1200-1600) and is itself a causative form derived from the earlier mawir ('come').
Beyond that, I don't know anything, but I suppose you can imagine the concept of 'giving' being applied to one's own actions to suggest a humbleness/politeness in a similar way that the current ～てあげる is used.
