Why is it "on *the* one hand"?

Solution 1:

The definite article is used to signal that one is talking about specific items, not items in general.

For example, this exchange at the reception desk of the hotel.

Guest: I'd like a room.

Clerk: Will you be staying the weekend, sir?

Guest: No, just the one night.

The guest could have said "No, just one night," but adding the definite article makes it more emphatic and specific at the same time. This applies to "on the one hand" constructions as well.

Solution 2:

If you are talking to the person face to face, you'd emphasize the statement by waving one hand as you said "on the one hand", and by waving the other as you said "on the other hand". Then "the one hand" and "the other hand" are quite clear. And I'd be surprised if that was not the origin of the expression.

Solution 3:

On the one hand, it might be there for emphasis, just as in you slept with her on the one day I asked you to behave. Then the question would be: why is emphasis often felt to be needed with "hand", but normally not in similar phrases, such as on one side of the paper ... on the other side ... and tie one end of the rope to ... tie the other end to ...? On the other hand, I think it might be there not for local emphasis, but to make sure the reader gets that the two sentences introduced by each "hand" are to be taken as linked but opposite perspectives. Using "the" serves both to give them a common marker and to increase the visibility of the first "hand" as a signpost. I don't really feel that on one hand and on the one hand are used with less or more emphasis (but I could be wrong [duh]). Consider that you normally wouldn't say on the one side of the paper: the article is usually only added to "one" when there are three or more choices.