When the scale readings of a sextant are suspected of error, or if an assurance of scale accuracy is needed, optical equipment for establishing angles is rarely available. Comparison with another sextant is possible, but if the arc between two stars were measured by bringing them into coincidence this could be compared with the calculated angular distance. To calculate the distance from a spherical triangle, knowing the stars' declinations and right ascensions, would be laborious and the refraction corrections would be difficult to apply—as they were in the old lunar distance methods. It is far simpler, in the northern hemisphere, to take Polaris as one of the two stars. If the other is within an hour of its meridian passage, so that the measured arc is approximately vertical, the usual corrections for refraction are sufficient. Also the calculated angular distance will then be simply 90° – δ, where δ is the declination of the star selected. This calculated distance is therefore subject to two corrections:

(i) refraction corrections for both stars,

(ii) a correction for Polaris not being exactly at the pole.