It will be noticed that meridians are circles, not straight lines, and that they all meet at the poles. But, for the purposes of ordinary surveying, where only comparatively small areas are dealt with, meridians are treated as parallel straight north-and-south lines; or, rather, all points within the area surveyed are assumed to have the same meridian. The error arising from this mode of treatment is too small to be considered in such work.

In

3. Latitude.-The latitude of a point on the earth's surface is the angle that the radius of the earth passing through that point makes with the plane of the equator. Fig. 1, the latitudes of B and G are, respectively, the angles BOR and GOG. These angles are measured by the arcs RB and G' G of the meridians passing through the two points. The latitude of a point may, therefore, be also defined as the angular distance of the point from the equator, that distance being measured on the meridian through the point, and being the number of degrees in the arc of the

meridian included between the equator and the point. When it is stated that the latitudes of the points B, G, F, are, respectively, RB, G' G, F' F, it should be understood that these arcs are to be expressed in degrees, latitude being an angular quantity, not a linear quantity.

The latitude of a point is said to be north or south according as the point is north or south of the equator.

4. Parallels of Latitude.-Any plane parallel to the equator, that is, perpendicular to the earth's axis, cuts the earth's surface in a circle called a parallel of latitude, or simply a parallel. The circle eqrt, Fig. 1, whose plane is parallel to that of EQRT, is a parallel of latitude. All points, as r, G, F, on a parallel have the same latitude, as the arcs Rr, G' G, F F are evidently equal.

5. Longitude. -The longitude of a point on the earth's surface is the angle between the meridian plane through that point and another meridian plane assumed as a plane of reference. This angle is also referred to as the angle between the meridian passing through the given point and the meridian determined by the plane of reference. The meridian of Greenwich, England, is generally taken as a reference meridian. Suppose that NG S, Fig. 1, is the meridian of Greenwich; the longitude of the point C, referred to that meridian, is the angle between the planes of the meridians NGS and NCS. This angle is the same as G' OF', or GOF, and is measured by the arc G' F' of the equator, or by the arc, as GF, of any parallel included between the meridian of Greenwich and the meridian through the point. considered. The longitude of a point may also be defined as the angular distance of the point west or east of the reference meridian.

Longitude is usually counted from the reference meridian toward the west, from 0° to 360°.

NOTE. It will be noticed that some terms, as latitude and longitude, are here used in a sense somewhat different from that given to them in previous Sections of this Course. The circumstances under which these terms are employed, however, always indicate plainly in what sense they should be taken.

THE CELESTIAL SPHERE

6. Definition.-The sun and stars seem to be and move on the inner surface of an immense sphere. This sphere is called the celestial sphere, or the heavens.

7. The Celestial Poles.-The points where the axis of the earth, produced, meets the celestial sphere are called the celestial poles. The one nearer the north pole of the earth is called the north pole; the other, the south pole.

8. The Celestial Equator and Meridians. The celestial equator is the circle in which the plane of the terrestrial equator intersects the celestial sphere.

A celestial meridian is a circle in which a meridian plane intersects the celestial sphere.

9. The Zenith.-If a vertical line at any point on the earth's surface is produced upwards, the point where it pierces the heavens is called the zenith of that point.

10. The Celestial Horizon. The celestial horizon 1 of any point is the circle in which a horizontal plane through that point intersects the celestial sphere.

11. Altitude.-The altitude of a point, with regard to a point on the horizon, is the angle that a line from the latter to the former point makes with the plane of the horizon.

12. Latitude and Longitude Measured on Celestial Circles.-In Fig. 2 are represented the earth O-nrse and the celestial sphere O-NRSE. The dimensions of the earth, as compared with those of the sphere, are shown very much exaggerated. In reality, the earth is so small, compared with the celestial sphere, that for all astronomical purposes it is treated as a point. The earth's axis sn, when produced, meets the sphere in the celestial poles N and S. The meridian planes nas, ng s, n rs, when extended, intersect the sphere in the celestial meridians NAS, NGS, and NRS, respectively. The celestial equator EQR is the intersection of the celestial sphere and the plane of the terrestrial equator e qr.

Assuming ng s, or NGS, to be the meridian of Greenwich (or any other reference meridian), and nas, or NAS, the meridian of a point a on the earth's surface, the longitude of that point is measured by the arc a'g', on the terrestrial equator, or by the arc A' G', on the celestial equator. The latitude of a is the angle a O a', or AOA', measured by either a'a or A' A. Since Oa is a radius of the earth, or a vertical

line, the point A in which that line produced meets the heavens is the zenith of the point a. Therefore, the latitude of any

point on the earth's surface is measured by the arc A' A of the celestial meridian of that point included between the equator and the zenith of that point. Hence, the latitude of a point on the earth's surface may be defined as the angular distance of the zenith of that point from the celestial equator.

TIME

13. Culmination.-All celestial bodies have an apparent motion from east to west around the axis of the celestial sphere. In this motion, they cross every meridian twice in 24 hours. The passage of a celestial body across the meridian of a place is called the culmination, or transit, of that body, with respect to that meridian; and the culmination of a celestial body is called upper culmination or lower culmination, upper transit or lower transit, according as the body crosses the meridian above or below the pole.

14. Sidereal Time.-The interval of time that elapses between two successive upper or lower transits of a star over the same meridian is called a sidereal day. This day is divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds. It begins, for any place, when a special point of the equator, called the vernal equinox, crosses the meridian above the pole. This instant is called sidereal noon. Sidereal hours, minutes, and seconds are reckoned from 0 to 24 hours, starting from sidereal noon. Time expressed in sidereal days and fractions (hours, minutes, seconds) is called sidereal time.

15. True Solar, or Apparent, Time.-The interval between two successive upper transits of the sun is called a true solar day, or an apparent day. Like the sidereal day, the apparent day is divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds. Time expressed in these units is called apparent time.

On account of the fact that the sun does not, like the stars, move in a plane perpendicular to the axis of the heavens, and that its motion is not uniform, a solar day is not equal to a sidereal day; nor are all solar days of equal duration. Apparent time is therefore not convenient to use for the ordinary affairs of life.

16. Mean Solar Time.--The earth makes one complete revolution around the sun in 366.2422 sidereal days, or 365.2422 true solar days. This motion constantly changes