Would a compass work on Arda? Or rather, beginning in what age would a compass work on Arda?
... and the follow-up question: Why don't Frodo and Sam use a compass in the Emyn Muil?
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I will shamelessly steal from Andres F's answer here:
Middle-earth is supposed to be the same world as our Earth, at a "different stage of imagination". Something like a fictional history of the real world Earth. Here is the relevant bit from a BBC interview with Tolkien:
G: I thought that conceivably Midgard might be Middle-earth or have some connection?
T: Oh yes, they're the same word. Most people have made this mistake of thinking Middle-earth is a particular kind of Earth or is another planet of the science fiction sort but it's just an old fashioned word for this world we live in, as imagined surrounded by the Ocean.
G: It seemed to me that Middle-earth was in a sense as you say this world we live in but at a different era.
T: No ... at a different stage of imagination, yes.
If that's the case, Arda, like Earth, has magnetic poles and thus compasses will work.
It's also confirmed indirectly by the fact that on Arda, birds can navigate great distances (e.g. birds spying for Saruman). They do that via sensing planetary magnetism, biology wise.
Ok, a guess would be that after the fall of Numenor, when the Valar detached Aman from the rest of Arda, and made the world geoid-shaped (not round! The world's not f'ing round), then magentic poles start to work, upto the issue of pole reversal.
That still does not solve the question of why Frodo and Samwise don't have a compass though.