During this time it must therefore cross the meridian twice, once above the pole and once below; the former is called the _upper_, and the latter the _lower meridian transit or culmination_. It must also pass through the points farthest east and west from the meridian. The former is called the _eastern elongation_, the latter the _western_. 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: [Illustration] 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_.