Measuring the positions of  and the celestial equator
 
Use is made of the fact that the Sun’s annual path against the stellar background is the ecliptic. We know that the value of the Sun’s maximum northerly declination is the obliquity ε. Measurements of the Sun’s meridian zenith distance z at transit on a number of days around the summer solstice and a knowledge of the observer’s latitude φ give a set of values for δ by means of the equation
δ = ∅ − z. (1)
The maximum in the graph of δ against time is the value of the obliquity.
If a second set of observations of the Sun’s meridian zenith distance were carried out near an equinox, its declination will again be calculated from equation (1).
From figure1, using the four-parts formula, we obtain
sin α = tan δ cot ε
from which the Sun’s right ascension α at the time of the transit can be found.
At transit, the Sun’s hour angle is zero so that the value of α is the local sidereal time of the hour angle of . This enables the observatory sidereal clock error to be determined. The sidereal times of transits of stars may then be noted, giving their right ascensions. Their declinations can also be deduced from their meridian zenith distances by equation (1). In this way the equatorial coordinates of the stars (i.e. their positions with respect to the celestial equator and the First Point of Aries) can be found or, using the opposite viewpoint, the positions of the equator and against the stellar background can be determined at any time.

Figure 1. Measurement of the position of .

Figure 2. The celestial sphere illustrating the precessional movement of the celestial equator.