The eye of the spectator is at z, and his zenith-line is The smaller circle is a section of the earth, and the larger of the sphere of the heavens. The figure is drawn of dimensions so false, that the sphere of the heavens is represented about as well as a common orrery represents the solar system. The Hoerzos ie the small circle drawn perpendicular to o z through ran; the altitude of the pole of the and horizon make an angle equal to the colatitude of tho place of observation ; a star which is distant from the north pole by less than the latitude of the place of observation can never set nor go below the horizon (it is called a circumpolar star).
heavens (r' being that of the earth) is the angle w E P. Now suppose the earth and the spectator to diminish until they cannot be distiu guished from the point o, the sphere of the heavens remaining the same. All angles at o remain unaltered : the altitude of the pole of the heavens becomes Q o p, equal to the angle A 0 E, the latitude of the spectator, and the horizon of the latter coincides with the great circle drawn through R Q perpendicular to o z. The great circle, Q P z R, passing through the pole and the zenith, is the meridian ; the second ary to the horizon perpendicular to the meridian is the prime vertical. We here exhibit a skeleton of the sphere, showing nu z P N, half the meridian ; N En, the horizon (N, E, its north, east, and south points); z E, the prime vertical; a portion of p o, the axis ; E M, the equator, perpendicular to the axis.
We now give three positions of the sphere, differing only in the manner of projecting the figure. Each one represents the state of the heavens some two or three hours before noon in an October morning, in a latitude somewhat greater than our own. The first figure is pro jected on the plane of the meridian ; that is, the meridian is the circle which bounds the view of the sphere. The second is projected on the prime vertical; the third, on the horizon.
The diagrams have many letters and numerals which are useless, except in tracing the affinities of the figures. The meridian, n PZN; the prime vertical, Z E z; the horizon, n E N, and its poles, the zenith and nadir, z and z ; the equator, II E in, and its poles p and p (which are called the poles, from their importance), are supposed to be well known. The reader who is new to the subject should learn to see the following propositions in each of the figures, namely poles of the meridian are the cast and west points of the horizon; the poles of the prime vertical are the north and south points of the horizon ; the equator and prime vertical make an angle equal to the latitude of the place of observation (which is r N, or the angle of r 11); the equator The diurnal motion carries the sphere round the axis in the direction of the arrows marked upon the equator. The meridian, horizon, and prime vertical, must be considered as detached from the sphere, and not moving with it. Every point of the sphere describes It small circle parallel to the equator : and all stars which are at the same dis tance from the pole describe the same small circle. The whole revolu tion takes place in what is called a sidereal day [TIME], about four minutes less than the mean solar day shown by a good clock. A secondary to the equator describes angles uniformly about the pole at the rate of to 24 sidereal hours, or to 1 sidereal hour: [ANGLE.] Hence if we would know how long it will bs before the diurnal motion will bring a star at x into the position s, we must turn the angle SP IC, which is measured by the arc Q 5, into sidereal time at the rate of 15' to and then turn the sidereal time so obtained into common clock time, at the rate of about 23h 56" of clock time to 24" of sidereal time. For purposes of general explanation, the two species of time may be confounded. The sidereal day is always made to begin when a certain point of the equator, presently to he noticed (the vernal equinox), comes on the south side of the meridian, and the hours are measured on to We shall now explain the systems of co-ordinates which are use of in describing the positions of stars.
1. Horizontal System. Altitude and this case the horizon is the primitive circle employed ; its north point, N, is the origin, and the position of a point w is determined by its azimuth, w I, and its altitude, Ltv; zwE being a secondary to the horizon. Sines the altitude and azimuth are reckoned by means of a fixed circle, both are perpetually changing their values for any one star. The followius assertions will serve to try the reader's comprehension of these terms : pointy on the north side of the meridian are in azimuth 0°, on the south side in azimuth ISO° •, the zenith has all azimuths, and every other point of the cast side of the prime vertical 90' of azimuth ; the altitude of • star which sets is greatest when it. is on the meridian; the meridian altitudes of a circumpolar star are the greatest and least of all its altitudes, and their half sum is always the latitude of the place of observation. [ALTITUDE; AZIMUTH.) 2. Equatorial System. Right Ascension and Declination.—Tho pri mitive circle here is the equator • the poiut of the equator called the vernal equinox (presently described) Is the origin, and the direction of the sun's motion from west to cast is the direction in which right ascension is measured. In the diagrams T is not the vernal, but the autumnal equinox, the point opposite to the vernal equiuox, conse quently T has 180' of right ascension, and so have w, v, and all points on the same half of the secondary rwT. The other co-ordinate, declination, is measured on the secondary to the equator north or south according to its direction : thus s has for its right ascension 180' + T Q, and Q s of south declination ; while n has the same right ascension, and Q n of north declination. The secondaries to the equator are called hour-circles, and the difference of the right ascensions of two stars is the angle made by their hour-circles: thus the angle qr5, measered by the arc q 5, is obviously the difference of the right ascensions of the point B and 1. The equator moves with the sphere, so that the right ascension and declination of a star remain the same, as long as it moves only with the diurnal motion. The right ascension is generally ex
pressed in time, as before described ; and the following assertions will serve for exercise in the meaning of these terms :—the sidereal day beginning when the vernal equinox is on the meridian, the right ascension of any star, turned into time, expresses the moment of the sidereal day at which that star will be on the meridian; when the venial equinox is on the meridian of Greenwich, the longitude of any place, measured eastwards, is the same as the right ascension of a star which is on the meridian of that place; the meridian altitude of any star, diminished by its declination (if north), or increased by its declination (if south), is the co-latitude of the place of observation ; every star which has the same declination as the place of observation has latitude, passes directly over the head of the spectator at that place ; the time of rising of a star, and the time during which it remains above the horizon, depend solely upon the declination, and not at all upon the right ascension. [RIGHT ASCENSION; DECLLYATION.) 1 Ecliptic System. Celestial Longitude and Latilude.—The ecliptic (B T S b) is the circle which the sun appears to describe in the course of a year, the direction of this orbital motion being from west to east. One half of it is north, the other half south, of the equator ; and the point of the equator in which the ecliptib cuts it, and through which the sun passes when it leaves the southern and enters the northern part of the ecliptic, is the vernal equinox, the opposite point being the autumnal equinox. Consequently, T, as drawn, is the autumnal equinox, for motion from west to east, or in the direction B T s, makes the sun peas from the northern to the southern side of the equator. In this system of co-ordinates the ecliptic is the primitive circle, the Vernal equinox is the origin, longitude is measured from west to east on the ecliptic, and latitude north or south, as the case may be, is measured on a secondary to the ecliptic drawn through the star. In fact, celestial longitude and latitude arc to the ecliptic precisely what right ascension and declination are to the equator. The obliquity of the ecliptic is the angle made by the equator and the ecliptic ; and the secondaries to the ecliptic, drawn through the vernal and autumnal equinoxes, are the equinoctial and solsticial colures. [LONGITUDE AND LATITUDE.] A complete understanding of all these terms makes the comprehen sion of the globe easy,end also the application of spherical trigonometry to those who know the latter science. We now describe the diagrams, in order to point out how such applications are made. The point s is the sun, of course in the ecliptic ; its right ascension is 180° + T Q, its declination Q s south, its longitude 180° + T s, its latitude 0°, its azimuth N L, its altitude L s, its hour-angle ( a name given to the angle made by the hour-circle of a star with the meridian) s r ac, measured by x Q. The parallel to the equator csee would be the diurnal path of the sun, if it continued at the point s of the ecliptic ; but as the sun has a slow motion of its own towards K, it is not strictly (though very nearly) correct to say that, for the day in question, the sun continues in the parallel. Hence we may say, without sensible error, that the sun moves over c x during half the night, and through x c during half the day. It rises when at the point K, and the angle K r s, turned into time, shows the sidereal time elapsed since the rising, while the angle r • shows the time which is yet to elapse before noon. As to the time of the year, observe that the sun was at the autumnal equinox T on the 21st of September, since which tune it has moved over T 8, independently of the diurnal rotation of the sphere. We see then what is meant by saying that the diagram represents some morning in October. The use of the globe is thus explained, as far as setting it for any hour and day is concerned. The polo r must first be elevated until the elevation is equal to the latitude of the place, the sun must then be put in its proper place In the ecliptic for the time of the year, and its hour-angle must then be made to represent the time which is waited of noon, or has elapsed since noon. All this ou the globe is done without attending to the distinction of sidereal and solar time, which need hardly be attended to when no greater degree of accuracy is wanted than can be obtained on a globe. We now refer the reader to works on the use of the globes, and shall conclude this article by a few indications of the mode of applying spherical trigonometry.
To find the time of sunrise, observe that in the spherical triangle K N, right-angled at N, we have P K given, being 90° + the sun's declination, and also P N, the latitude of the place of obecrvation. Hence the angle K P N can be found, which being turned into sidereal time, gives a good approximation to the time of sunrise, refraction and the sun's proper motion being neglected.
Given s L the sun's altitude, and the latitude of the place ; required the time of day. In the triangle s z P, we now know z s the sun's co altitude, a P which is 90' + declination, and z P the co-latitude of the place. Hence the angle s 3'z can be found, and thence the time from noon. If s, instead of the sun, were a known star, the question would be solved in the same way, except that the sun's hour-angle is no longer s r z, but that angle increased or diminished by the difference of the right ascensions of the sun and star.
Two known stars, w and s, are observed to be iu the same circle of altitude s w L at a given place ; required the time of day. Ilere P W and r 5, the co-declinations of the stars, are known, and also the angle w P 8, which is the difference of their right ascensions ; hence in the triangle s w r the angle s w r can be found, and thence its supplement, the angle z w P. Then, in the triangle w z r, we know the angle z w r w the co-declination of the star iv, and z r the co-latitude of the place : whence the angle iv r z can be found; and thence, by com parison of w with the sun, the time of day.
For the actual applications we must refer to mathematical works on astronomy.