Knowing the latitude and longitude of the point of departure, and noting carefully the time that elapses between each separate observation of the ground speed and of the course, the air navigator, with the aid of a specially prepared set of "traverse tables" (as used by mariners), can easily plot on his chart the distance covered and the direction in which it has been covered. Hence the position of the aircraft at any time is either known definitely, or can be forecast with a fair degree of accuracy. For aërial navigation by means of "Dead Reckoning," frequent observations of ground speed and drift are necessary. If aircraft are cut off by clouds or fog from all possibility of sighting the surface of the earth, grave errors may occur, since in long distance flights the wind's velocity and direction often change without the pilot's knowledge. NAVIGATION BY ASTRONOMICAL OBSERVATION In navigation by astronomical observation, the position of the aëroplane or airship is found by observing the height above the horizon of either the sun or another heavenly body, such as a star that is easy of recognition. The method depends upon the known fact that at any given instant the sun is vertically above some definite point on the earth's surface. This point can be calculated from the time of the observation and the declination and equation of time, as tabulated in the nautical almanac. In the case of stars, the right ascension of the sun and of the star also enter into the calculation. The method of carrying out such calculations is too involved for the scope of this volume, and the reader is referred to many of the excellent text books published on the subject of navigation. Since the navigator knows, from the time of his observation, the point on the earth's surface over which is the heavenly body in question, it is clear that around this point circles on the surface of the earth may be described. From any point in any one circle the heavenly body will appear to have the same altitude or elevation above the horizon. A single observation of the altitude of any one heavenly body shows, therefore, only that the observer may be at any point on such a circle of equal altitude--otherwise known as a Sumner circle. But it does not fix that point. A second simultaneous observation, of a different heavenly body, will give a different circle, corresponding to the position of the second body. The intersection of these two circles determines the point of observation. This fact constitutes a reliable basis for fixing one's position during a clear night, when many stars are visible and choice of suitable heavenly bodies may be made. During the day, however, the light of the sun prevents other heavenly bodies from being seen, so that only a single observation is possible.