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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 S 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 Zs is the celestial
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
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 repreR sent the axis of the heavens, and Q Q', 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
through the poles of the ecliptic are called circles of latitude. Thus, let A Q represent a part of the celestial equator, ▲ 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 M 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 AR 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 м, 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 zo 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
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 N o, or the angle N z 0, 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.
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
longitude in 36525 days. Now in one solar year the sun separates from the first point of Aries 360°, taking into consideration its own apparent motion from west to east, and the actual motion of the first point of Aries in the opposite direction.
Let, therefore, x = the length of a mean solar year;
then, 36000° 45′ 45′′ : 360° :: 36525a : x
.'. x=365d 5h 48m 51s∙6 = 365a·242264*
To find the length of the sidereal year.
17. Since the first point of Aries moves with a slow annual motion of about 50"-1 from east to west to meet the sun, the arc of the ecliptic described by the sun from the first point of Aries to the first point of Aries again, must be 360° — 50′′1 = 359° 59′ 9′′·9, and this is the arc described by the sun in a mean solar year; but in a sidereal year the sun describes 360°; hence a sidereal is greater than a solar year by the time the sun takes to move over the arc of 50"-1. Let the length of the sidereal year.
then sidereal year
mean solar year: 360° : 360° — 50′′1 or, sidereal year: 365-242264 :: 360° : 359° 59′ 9′′.9 ... sidereal year = 365d 6h 9m 11s.5 †
The sidereal day, the apparent solar day, and the mean solar day.
18. The sidereal day is the interval between two successive transits of the first point of Aries over the same meridian. It begins when that point of Aries is on the meridian.
The apparent solar day is the interval between two successive transits of the sun's centre over the same meridian. It begins when that point is on the meridian.
According to Bessel the formula for determining the length of the mean solar or tropical year is
where t is the number of years since 1800.
The length of a sidereal year according to Bessel is
365-256374322 = 365d 6h 9m 10s-7423 mean time.