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°. These two co-ordinates, the altitude and azimuth, will determine the position of any object with reference to the observer's place. The latter's position is usually given by his latitude and longitude referred to the equator and some standard meridian as co-ordinates. The _latitude_ being the angular distance north or south of the equator, and the _longitude_ east or west of the assumed meridian. We are now prepared to prove that _the altitude of the pole is equal to the latitude of the place of observation_. Let H P Z Q¹, etc., Fig. 2, represent a meridian section of the sphere, in which P is the north pole and Z the place of observation, then H H¹ will be the horizon, Q Q¹ the equator, H P will be the altitude of P, and Q¹ Z the latitude of Z. These two arcs are equal, for H C Z = P C Q¹ = 90°, and if from these equal quadrants the common angle P C Z be subtracted, the remainders H C P and Z C Q¹, will be equal. To _determine the altitude of the pole_, or, in other words, _the latitude of the place_. Observe the altitude of the pole star _when on the meridian_, either above or below the pole, and from this observed altitude corrected for refraction, subtract the distance of the star from the pole, or its _polar distance_, if it was an upper transit, or add it if a lower. The result will be the required latitude with sufficient accuracy for ordinary purposes. The time of the star's being on the meridian can be determined with sufficient accuracy by a mere inspection of the heavens. The refraction is _always negative_, and may be taken from the table appended by looking up the amount set opposite the observed altitude. Thus, if the observer's altitude should be 40° 39' the nearest refraction 01' 07", should be subtracted from 40° 37' 00", leaving 40° 37' 53" for the latitude. TO FIND THE AZIMUTH OF POLARIS.