equator e q is extended to the celestial concave, and therefore e c q is the plane of the celestial equator.

By means of the figure we may define the zenith, reduced zenith, latitude, and reduced latitude, as follows:—

The zenith is that point in the celestial equator which is the extremity of the line drawn perpendicular to the place of the spectator, as z.

The reduced zenith is that point in the celestial concavewhich is the extremity of a straight line drawn from the centre of the earth, through the place of the spectator, as z'.

The latitude of a place a on the surface of the earth, is the inclination of the perpendicular AG to the plane of the equator: thus the angle A G Q is the latitude of a. The arc

z q in the celestial concave measures the angle A & Q; hence z Q, or the distance of the zenith from the celestial equator, is equal to the latitude of the spectator.

The reduced latitude of the place a, is the inclination of z'c or Ac to the plane of the equator: or it is the angle ACQ or arc z'Q, which measures the angle. Since the curvature of the earth diminishes from the equator to the poles, the reduced latitude z' Q must be always less than the true latitude z Q, and therefore the difference z z' must be subtracted from the true latitude to get the reduced latitude.

P

H

R

The visible horizon is that circle in the celestial concave which touches the earth where the spectator stands, as har; and a circle parallel to the visible horizon, and passing through the centre of the earth, is called the rational horizon thus H C B is the

:

rational horizon. These two

circles, however, form one and the same great circle in the celestial concave: thus R and r in the figure must be sup

posed to coincide. This may be readily conceived, when we consider that the distance of any two points on the surface of the earth will make no sensible angle at the celestial concave therefore either of these two circles is to be understood by the word horizon. The poles of the horizon of any place are manifestly the zenith and nadir.

Great circles passing through the zenith are called circles of altitude or vertical circles. Thus, let z be the zenith of a spectator, where the horizon is represented by the circle. N W 8 E, then N z S, w z E, and

z o are circles of altitude. That circle of altitude which passes through the poles of the heavens is called the celestial meridian. Thus, suppose the point P to be that pole of the heavens which is above the horizon (and therefore called the elevated pole), then the circle N P Z s is the celestial

W

E

meridian of a spectator supposed to be on the earth below z. The points of the horizon through which the celestial meridian passes are called the north and south points. A circle of altitude at right angles to the meridian is called the prime vertical: thus w z E is the prime vertical. This last circle cuts the horizon in two points called the east and west points. The east and west points are manifestly the poles of the celestial meridian.

Since the horizon and celestial equator are both perpendicular to the celestial meridian, the points where the horizon and celestial equator intersect each other, must be 90° distant from every part of the meridian (Jeans' Trig., P. II, art. 65); that is, the celestial equator must cut the horizon in the east and west points. If, therefore, P is the pole of the heavens, take P Q = 90°, then the celestial equator must pass through Q, and as we see it must also pass through

the east and west points, the curve w Q E, in the figure, will represent the celestial equator.

The preceding terms are frequently explained by means of a figure projected on the plane of the celestial meridian, as thus: Let the circle P z Q represent the celestial me

H

P

R

ridian, the pole c is either the east or west point; let H R be the horizon and p the pole of the heavens above the horizon; the line P P' may represent the axis of the heavens, and q o, drawn at right angles to it, the celestial equator; the poles of the horizon will be z and z', the zenith and the nadir, and z cz', the circle

passing through the east and west points, is the prime vertical. The ecliptic is divided into twelve parts, called signs, which receive their names from constellations lying near them. These divisions or signs are supposed to begin at that intersection of the celestial equator and ecliptic which is near the constellation Aries.

Great circles passing through the poles of the heavens are called circles of declination; and great circles passing

X

C

through the poles of the ecliptic are called circles of latitude. Thus, let A Q represent a part of the celestial equator, A C a part of the ecliptic, A the first point of Aries, and therefore angle c A Q the obliquity of the ecliptic: let P be the pole of the heavens, or of the celestial equator, and p' the pole of the ecliptic, then P X R is a circle of declination, and P′ x м is a circle of latitude.

Parallels of declination and of latitude are small circles parallel respectively to the celestial equator and ecliptic.

The declination of a heavenly body is the arc of a circle of declination passing through its place in the celestial concave, intercepted between that place and the celestial equator: thus let x be the place of a heavenly body, then X R is its declination.

The right ascension of a heavenly body is the arc of the equator, intercepted between the first point of Aries and the circle of declination, passing through the place of the heavenly body in the celestial concave, measuring from the first point of Aries, eastward, from 0° to 360°; thus the arc A R is the right ascension of the heavenly body x.

In like manner, if a circle of latitude be drawn through any point x, in the celestial concave, the part of it between the point and the ecliptic is called the latitude of the point; and the arc of the ecliptic, extending eastward from the first point of Aries to the circle of latitude, is called the longitude of the point: thus the latitude of x is x M, and the longitude A M.

The altitude of a heavenly body is the arc of a circle of altitude passing through the

place of the body intercepted between the place and the horizon. Thus, if z o be a circle of altitude, and N w S E the horizon, then arc x o is the altitude of x.

The azimuth, or bearing of a heavenly body, is the arc of the horizon intercepted between the north and south points

W

X

R

T

S

D

E

and the circle of altitude passing through the place of the body; or it is the corresponding angle at the zenith between the celestial meridian and the circle of altitude passing through the body: thus the arc so or No, or the angle N zo, or s zo, is the azimuth of x.

The amplitude of a heavenly body is the distance from the

east point at which it rises, or the distance from the west point at which it sets, the arcs or distances being measured on the horizon; thus suppose the heavenly body x to rise at D, and after describing the arc D X D', to set at D', then the amplitude of x is either D E or D'w.

The hour angle of a heavenly body, is the angle at the pole between the celestial meridian and the circle of declination passing through the place of the body; thus, z PX is the hour angle of x.

CHAPTER II.

ON TIME.

The Solar year, and Sidereal year.

15. A solar year is the interval between the sun's leaving the first point of Aries, and returning to it again.

A sidereal year is the interval between the sun's leaving a fixed point, as a star, and returning to that point again.

The equinoctial points have an annual motion of 50′′1, by which they are carried back to meet the sun in its apparent motion among the fixed stars, from west to east.

On this account a solar year is shorter than a sidereal year by the time the sun takes to describe 50"·1.

The length of the solar years is found to differ a little from each other, on account of certain irregularities in the sun's apparent motion, and that of the first point of Aries. The mean length of several solar years is therefore the one made use of in the common division of time, and called the mean solar year.

To find the length of the mean solar year.

16. By comparing observations made at distant periods, it was found that the sun had described 36000° 45′ 45′′ of