Separation angle
I was surprised to find out that not only did Tycho have a detailed knowledge of spherical geometry (triangles drawn on the surface of a sphere), but that much earlier astronomers, such as Ptolemy and Hipparchus, were equally adept. All of these people had a better understanding of spherical trig than I ever will. But here is the basic idea:
The goal is to find the difference in right ascension (DRA ) between two measured stars (star1 and star2 in the diagram). CODEC is the distance (in degrees) from the North Pole to the star. CODEC = 90 - DEC. The specific equations for this calculation can be found here.
Here is an example in Tycho's own hand in late December 1598. He finds the position of Jupiter relative to nearby stars. If the stars have been cataloged (both RA and DEC found), then the RA and DEC of Jupiter can be calculated. (He is actually working in ecliptic coordinates, but the idea is the same.)
He uses four nearby stars and thus gets four estimates of the separation angle from Jupiter to stars in his catalog. This allows him to detect errant observations and to select the one he feels has the least error. Tycho's Latin sometimes substitutes 'f' for 's'.
