In the British Parliament members voted on major issues by those on one side standing and walking to a small room where they were counted by "tellers" as they went through the doorway. At a later period all members stood and walked through one of two doors, one for "yes" and one for "no". This process was called a "division", and the rooms are called "division lobbies". The house is said to "divide".
John Reed, the author of Ten Days, was an American (US) journalist, and so might not have picked up on this usage. But British political culture and vocabulary was much imitated in early twentieth century Russia. I strongly suspect this meaning of "division" is the source of the usage in Ten FaysDays.
I own a copy of Ten Days, but I haven't read it in perhaps 30 years, adn i don't recall the passage, so I am not sure.