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Tycho had complete command of spherical trigonometry. He provided a detailed explanation of how to determine the declination of a star if one knew something about that star's neighbors. Here I borrow a diagram from Dr. Rosa's translation (from the Latin!) of Tycho's writings.
The following is standard spherical trig, so I will just give the essentials:
C is a zenith star whose declination (Dz) is not known and needs to be determined
P is the north celestial pole
A is a primary star whose declination (Da) is already known
B is a primary star whose declination (Db) is already known
The separations of the three stars have already been measured: BA, BC, and AC
The distance of star A from the pole (PA) is (90-Da)
The distance of star B from the pole (PB) is (90-Db)
These five angles (PA, PB, AB, BC, and AC) can be used to find PC, and the declination of the zenith star (Dz) is then (90-PC)
the process is:
Triangle BPA has all three sides known, so all its internal angles can be found by formulas. We want angle /ABP.
Triangle BCA has all three sides known, so all its internal angles can be found by formulas. We want angle /ABC.
/CBP is the difference of these two angles. /CBP = /ABP - /ABC
Triangle BPC has two sides known (PA and BC), and the included angle /CBP, which we just found. So PC can be calculated.
Dz = 90 - PC
Q.E.D.
A more detailed exposition of the geometry of this zenith star work-around is on a later page.
So I knew I had to set up and execute a specialized set of measurements to handle the zenith stars. The main effort would be to get separation measures between the zenith stars and those nearby primary stars that were more than three degrees from my zenith.
With my latitude of 39 degrees N, my zenith stars were those within the band of declinations from 36 degrees to 42 degrees.
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