How can you tell if you are exactly at the North Pole and not a few miles away from it?

You would need to take many 'sightings' of stars above your horizon at accurately times moments and compute your latitudes from their know coordinates. Because a shift of one mile causes an angular shift of about 8 arcseconds, you need to make your measurements with an accurate theodolite, not a hand-held sextant.

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All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the
NASA IMAGE/POETRY Education and Public Outreach program.