An observation may he made upon Polaris at any of these four points, or at any other point of its orbit, but this latter case becomes too complicated for ordinary practice, and is therefore not considered.
If the observation were made upon the star at the time of its upper or lower culmination, it would give the true meridian at once, but this involves a knowledge of the true local time of transit, or the longitude of the place of observation, which is generally an unknown quantity; and moreover, as the star is then moving east or west, or at right angles to the place of the meridian, at the rate of 15° of arc in about one hour, an error of so slight a quantity as only four seconds of time would introduce an error of one minute of arc. If the observation be made, however, upon either elongation, when the star is moving up or down, that is, in the direction of the vertical wire of the instrument, the error of observation in the angle between it and the pole will be inappreciable. This is, therefore, the best position upon which to make the observation, as the precise time of the elongation need not be given. It can be determined with sufficient accuracy by a glance at the relative positions of the star Alioth, in the handle of the Dipper, and Polaris (see Fig. 1). When the line joining these two stars is horizontal or nearly so, and Alioth is to the west of Polaris, the latter is at its eastern elongation, and vice versa, thus:
But since the star at either elongation is off the meridian, it will be necessary to determine the angle at the place of observation to be turned off on the instrument to bring it into the meridian. This angle, called the azimuth of the pole star, varies with the latitude of the observer, as will appear from Fig 2, and hence its value must be computed for different latitudes, and the surveyor must know his latitude before he can apply it. Let N be the north pole of the celestial sphere; S, the position of Polaris at its eastern elongation; then N S=1° 19' 13", a constant quantity. The azimuth of Polaris at the latitude 40° north is represented by the angle N O S, and that at 60° north, by the angle N O' S, which is greater, being an exterior angle of the triangle, O S O. From this we see that the azimuth varies at the latitude.
We have first, then, to find the latitude of the place of observation.
Of the several methods for doing this, we shall select the simplest, preceding it by a few definitions.
A normal line is the one joining the point directly overhead, called the zenith, with the one under foot called the nadir.
The celestial horizon is the intersection of the celestial sphere by a plane passing through the center of the earth and perpendicular to the normal.
A vertical circle is one whose plane is perpendicular to the horizon, hence all such circles must pass through the normal and have the zenith and nadir points for their poles. The altitude of a celestial object is its distance above the horizon measured on the arc of a vertical circle. As the distance from the horizon to the zenith is 90°, the difference, or complement of the altitude, is called the zenith distance, or co-altitude.
The azimuth of an object is the angle between the vertical plane through the object and the plane of the meridian, measured on the horizon, and usually read from the south point, as 0°, through west, at 90, north 180°, etc., closing on south at 0° or 360°.