There is one major point we have not addressed. We know that we need to measure the zenith distance to objects, but how does a person actually do this? This is where the sextant is used. A sextant is actually a large-angle measuring instrument. That is, it is used to measure the angle between two objects. Generally the two "objects" we use are the horizon and a star.
We use the horizon instead of the zenith because there is no way to obviously tell precisely where the zenith is. But we can see the horizon (well, when it is clear and not too dark, anyway), and we know that the horizon is about 90 degrees away from the zenith, so the horizon makes an excellent proxy for the zenith.
We'll talk more about the sextant in a few moments.
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