Suppose, for example, that the spectator is at the latitude of fifty degrees. His horizon reaches fifty degrees beyond the pole of the earth, and gives the same apparent elevation to the pole of the heavens. It cuts the equator and all the circles of daily motion, at an angle of forty degrees,--being always equal to what the altitude of the pole lacks of ninety degrees: that is, it is always equal to the co-altitude of the pole. Thus, let H O, Fig. 15, represent the horizon, E Q the equator, and P P´ the axis of the earth. Also, _l l, m m, n n_, parallels of latitude. Then the horizon of a spectator at Z, in latitude fifty degrees, reaches to fifty degrees beyond the pole; and the angle E C H, which the equator makes with the horizon, is forty degrees,--the complement of the latitude. As we advance still further north, the elevation of the diurnal circle above the horizon grows less and less, and consequently, the motions of the heavenly bodies more and more oblique to the horizon, until finally, at the pole, where the latitude is ninety degrees, the angle of elevation of the equator vanishes, and the horizon and the equator coincide with each other, as before stated. [Illustration Fig. 15.] _The circle of perpetual apparition is the boundary of that space around the elevated pole, where the stars never set._ Its distance from the pole is equal to the latitude of the place. For, since the altitude of the pole is equal to the latitude, a star, whose polar distance is just equal to the latitude, will, when at its lowest point, only just reach the horizon; and all the stars nearer the pole than this will evidently not descend so far as the horizon. Thus _m m_, Fig. 15, is the circle of perpetual apparition, between which and the north pole, the stars never set, and its distance from the pole, O P, is evidently equal to the elevation of the pole, and of course to the latitude. In the opposite hemisphere, a similar part of the sphere adjacent to the depressed pole never rises. Hence, _the circle of perpetual occultation is the boundary of that space around the depressed pole, within which the stars never rise._ Thus _m´ m´_, Fig. 15, is the circle of perpetual occultation, between which and the south pole, the stars never rise.