Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/212

Rh 198 no component of rotation there, the pendulum would C011- tinue to move in one and the same plane. At interinedia.te stations the rate of rotation is easily calculated ; and obser- vations conﬁrm tlie calculations, and have made the earth’s 1‘0tatlo11 actually visible. The poles of the earth are the points in which the axis of rotation, or of ﬁgure, meet the surface ; and the equator is the circle in which the surface is intersected by a plane through the earth’s centre, perpendicular to the axis of rotation. Every point of the equator is therefore equidis- tant from the poles. To determine the position of a point in space three co-ordiiiates or ineasureinents are necessary ; they may be three lines, or two lines and one angle, or two angles and one line. Thus, to deﬁne the precise position of a point on the earth’s surface, we express it by latitude, longitude, and altitude; the first two are angular measures, the third a linear magnitude, namely the height above the surface of the sea. The line in which the surface of the earth is inteisected by a plane through the axis of rotation is called a meridian, and all meridians are evidently similar curves. A line perpendicular to the surface at any point is called a vertical line; it corresponds with the direction of gravity there; being produced outwards, that is, away from the earth’s centre it meets the heavens in the zeizitk; and produced downwards it intersects the axis of revolution; it would of course pass through the earth’s centre were it a sphere ; as it is, it passes near the eartl1’s centre. The angle between the meridian planes of two stations as A and B is called the difference of longitude of A and B, or the longitude of B with reference to A. In British maps the longitudes of all places are expressed with reference to the Royal Observatory of Greenwich. The latitude of any point is the angle made by the vertical line there with the plane of the equator, or the co-latitude is the angle between the vertical line and the axis of rotation. The surface of the earth being one of revolution, any intersecting plane parallel to the equator cuts it in a circle. If we imagine the vertical lines drawn at any two points, as P and Q, in such a circle it is evident from the symmetry of the surface that these verticals make the same angle with the equator; in other words, the latitudes of all points on this circle are equal. Such circles are called parallels ; they intersect meridians at right angles. If we suppose that at any point Q of the surface the meridian, or a small bit of it, is actually traced on the surface, and also a portion of the parallel through the same point, then these lines, crossing at right angles in Q, mark there the directions which we call north and south, east and west—the meridian lying north and south, the parallel east and west. Planes containing the vertical line at Q are vertical planes there. A vertical plane is deﬁned by its azimuth, which is the angle it makes with the meridian plane ; the azimuth at Q of any object (or point) celestial or terrestrial is the angle which the vertical plane passing through the object makes with the meridian. The south meridian is generally taken as the zero of azimuth. The plane touching the surface at Q is the visible horizon there—a plane parallel to this through the centre of the earth being called the rational horizon. The altitude at Q of a heavenly body, as a star, is the angle which the line drawn from Q to the star makes with the plane of the horizon,—the zenith distance of the same star being the angle between its direction and the vertical at Q. By a degree of the meridian is meant this: if E, F are points on the same meridian such that the directions of their verticals make with each other an angle of one degree-— a ninetieth part of a right aiigle—then the distance between E and F measured along the meridian is a degree of the (i‘rEOGlt.-l’l{Y [:i.u‘iii:.iAT1c.iL. meridian. ..s the radius of curvature of an ellipse is variable, increasing from the extremity of the major axis to the extremity of the minor axis, so on the earth’s surface a degree of the meridian is found by geodetic measurement to increase from the equator to the poles. The actual length of a. degree of the meridian at the equator is 3627464 feet; at either pole it is 3664798 feet. The length of one degree of the equatorial circle is 3652311 feet. With regard to the ﬁgure of the earth as a whole, the polar radius is 394979 miles, and the radius of the equator 396336 miles; the difference of these, called the ellip- ticity, is 3,1,3 of the mean radius. A splieroid with these semiaxes is equivalent in volume to a sphere having a radius of 395879 miles. Without referring further here to the spheroidal ﬁgure, we shall now, having given the precise dimensions, regard the earth as a sphere whose radius is 3959 miles. On such a sphere one degree is 6909 miles. From the deﬁnitions given above it appears that the radius of the parallel which corresponds to all points whose latitude is <1» is 3959cos<,t; and that one degree of this circle, 23.0., one degree of longitude in the latitude <1» is 6909 cosqb expressed in miles. In the representation of the spherical earth (ﬁg. 2) P is the pole, QQ the equator, E,F any two points on the surface, Plﬂe, PF/' the meridians of those ’ /73. points intersectingthe equator in e and f I Join EF by a great circle; then in the ‘ F spherical triangle PEF the angle at P , is the difference of longitude of E and ‘ Q F, PE is the co-latitude of E, and PF f I the co-latitude of F, the latitudes being J eE and fF respectively. The angle at E, being that contained between the Fig» 2 meridian there and a vertical plane passing through F, is the azimuth of F (measured in this case from the north), while the angle at F is the azimuth of E. If, then, there be given the latitudes and longitudes of two places, to ﬁnd their distance apart, and their relative bearings, it becomes necessary to calculate a spherical triangle (PEF) in which two sides and the included angle are given,—the calculation bringing out the third side, which is the required distance, with the adjacent azimuthal angles. The latitudes and longitudes of places on the earth’s surface are determined by observations of the stars, of the sun, and of the moon. As the earth rotates, the zenith of any place (not being on the equator) traces out among the stars a small circle having for centre that point in which the axis of rotation meets the heavens. If there were a star at this last point it would be apparently motionless, having always the same altitude and azimuth. The pole star, though very conveniently near the north pole of‘ the heavens, and without perceptible motion to the unaided eye, is in reality moving in a very small circle. The zenith of a point on the equator traces out in the heavens a great circle, namely, the celestial equator. _ As the positions of points on the earth are deﬁned with reference to the equator and a certain ﬁxed meridian, so the positions of stars are deﬁned by their angular distance from the celestial equator, called in this case declination, and by their right ascension, which corresponds to terrestrial longitude. Stars which are on the same meridian plane (extended to the heavens) have the same right ascension. Right ascension is expressed in time from O” to 24-“. A sidereal clock, going truly, indicates 24” for every revolu- tion of the earth : at every observatory, the sidereal clock there shows, at each moment, the right ascension of the stars which at that moment are on the meridian. Thus the right ascension of the zenith is the sidereal time. In the left hand circle of the diagram (ﬁg. 3) two