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PRACTICAL EXAMPLES. 1. In the oblique spherical triangle ABC. The LA=1269.37'.
ACE 570.30'. Given The LB= 480.30' Answer.BC=1150.20'. 61°.41'
Sac=50°.10.30": . Given The LB= 42°.15'.13" Answer. BC=40°. 0'.10". The Lo=1219.36'.20"
AB=760.35'.36". Required the sides. 3. In the oblique spherical triangle ABC. The LĀ= 36o. 8'
AC=30°. Given The LB= 46°.19 Answer.BC=240.4'. The Lc=104°
AB=420.9'. Required the sides.
ASTRONOMICAL DEFINITIONS AND INTRODUCTORY
1. Astronomical Definitions. (A) The Celestial Sphere, is that apparent concave in which the sun, moon, stars, and all the heavenly bodies seem to be situated.
(B) The axis of the celestial sphere is an imaginary line passing through the centre of the earth, about which all the heavenly bodies appear to have a diurnal revolution.*
(C) The poles of the celestial sphere are the extremities of its axis, the one called the north pole, the other the south pole.t
* Although the earth's real motion on its axis from west to east, is the cause of day and night; and its motion in its orbit, or path round the sun, is the cause of the variation of the seasons of the year : yet as all appearances and places of the celestial bodies will be the same, whether the earth moves and the celestial sphere is at rest; or the earth is at rest, and the celestial sphere in motion; astronomers, for the ease of calculation, assume the earth as a point at rest in the centre of the celestial sphere, and ascribe to the heavenly bodies that motion which they appear to have to a spectator on the earth.
+ The polar star is a star of the second magnitude, near the north pole, in the tail of the little bear. Its mean right ascension, for the beginning of the year 1820, was 119.13.7", and its declination 880.20.55" north. Connaissance des Tems for 1820, page 168.
In the Nautical Almanac for 1820 the north polar distance of the polar star is stated to be 10.39'.44”.50 for the year 1818, and its annual variation-19":45.
(D) The equinoctial is a great circle which divides the heavens into two hemispheres, the northern and southern; it is called the equinoctial, because, when the sun appears in it, the days and nights all over the world are equal, viz. 12 hours each. This happens twice in the year, about the 21st of March and the 23d of September; the former is called the vernal equinox, the latter the autumnal equinox.
-(E) The ecliptic is a great circle in which the sun makes his apparent annual progress; it cuts the equinoctial in an angle of 23o.28'*, called the obliquity of the ecliptic; and the points of intersection are called the equinoctial points.
Thé ecliptic is divided into twelve equal parts called signs, each sign contains 30 degrees. Their names and characters are as follow :
vs Capricornus 8 Taurus SL Leo
m Scorpio Aquarius I Gemini
The Virgo † Sagittarius * Pisces. The first six signs lie on the north of the equinoctial, and are called northern signs, the six following lie on the south side of the equinoctial, and are called southern signs. The sun continues about a month in one of these signs, and goes through nearly a degree in a day.
(F) The Zodiac is a space which extends about 8 degrees on each side of the ecliptic, like a belt or girdle, within which the motions of all the planets are performed.
(G) The Nodes are the points where the orbits or paths of the planets round the sun intersect the ecliptic. That where the planet ascends from the south towards the north of the ecliptic is called the north or ascending node, and is marked thus 8; the other the south or descending node, and is marked thus 8. The names and characters of the planets are as follow : o The Sun ) The Moon
$ Pallas H Herschel, Venus
Vesta 21 Jupiter or Georgian. • The Earth Juno
When two planets are referred to the same point of the ecliptic, they are said to be in conjunction; and those that are referred to opposite points of the ecliptic are said to be in opposition, or 180 degrees apart. If they be three signs or 90° distant, they are in a quartile aspect. If two sines or 60 degrees, a sextile aspect. The astronomical marks are as follow:
* The angle which the ecliptic makes with the equinoctial is a variable quantity, and is equal to half the difference between the greatest and least meridian altitude of the sun at any place, supposing the sun to have the greatest declination when on the meridian.
ó Conjunction when planets are in the same point of the ecliptic. * Sextile when 2 Signs dist. A Trine when 4 Signs dist. o Quartile when 3 Signs dist. 8 Opposition when 6 Signs dist.
The conjunction and opposition are called the syzygies, and the quartile aspects the quadratures ; these terms are applied chiefly to the moon.
(H) The horizon is a great circle which separates the visible half of the heavens from the invisible.
This horizon is distinguished by the sensible and rational horizon, when applied to the earth. The sensible horizon is the boundary of the spectator's view at sea or land; and a plane parallel to this circle, passing through the earth's centre, is called the rational horizon.
(1) The cardinal points are the east, west, north and south, points of the horizon. The mariner's compass, which is divided into 32 points, each 11°.15', (F. 74.) is a representation of the horizon.
(K) The Zenith is a point in the celestial sphere directly over the head of the spectator, being the elevated pole of the horizon.
(L) The Nadir is a point in the celestial sphere directly under the feet of the spectator, and is diametrically opposite to the zenith ; being the depressed pole of the horizon.
(M) Azimuth, or vertical circles, are great circles passing through the zenith and nadir. They cut
the horizon at right angles. The altitudes of the heavenly bodies are measured on these circles.
(N) The prime vertical is that azimuth circle which passes through the east and west points in the horizon.
(O) Meridians are great circles passing through the poles of the world, and cutting the equinoctial at right angles. They are also called hour circles; and upon the terrestrial sphere, circles of longitude.
(P) Circles of celestial longitude are great circles passing through the poles of the ecliptic, and cutting it at right angles.
(Q) The latitude of any object in the heavens, is an arc of a circle of longitude contained between the centre of that object and the ecliptic.
(R) The latitude of any place on the earth, is the elevation of the pole above the horizon, and the complement of the latitude, is the distance of the pole from the zenith. Or the latitude is the distance of the zenith of the place from the equinoctial, on the celestial sphere.
(S) The declination of any celestial object, is an arc of a meridian contained between the centre of that object and the equinoctial.
(T) Parallels of declination are small circles parallel to the equinoctial.
(U) The altitude of any object in the heavens, is an arc of an azimuth or vertical circle, contained between the centre of the object and the horizon.
(W) Parallels of altitude are small circles parallel to the horizon.
(X) Parallels of celestial latitude are small circles parallel to the ecliptic.
(Y) The tropics are small circles parallel to the equinoctial, at 230.28' from it, and touch the ecliptic in the points of cancer and capricorn; they are the limits of the sun's progress to the north and south of the equinoctial.
(Z) The zenith distance of any celestial object is the arc of a vertical circle, contained between the centre of that object and the zenith, being the complement of the altitude.
(A) The polar distance of any object in the heavens, is an arc of a meridian contained between the centre of that object and the pole of the equinoctial.
(B) The amplitude of any celestial object is an arc of the horizon, contained between the centre of the object when rising or setting, and the east or west point of the horizon.
(C) The azimuth of any object in the heavens, is an arc of the horizon, contained between an azimuth or vertical circle, (passing through the object,) and the north or south point of the horizon.
(D) The right ascension of an object, is the distance between the point aries and a meridian passing through the object, reckoned on the equinoctial. It is so called, because, in a right sphere, this meridian will coincide with the horizon when the object is rising. Or, we may define it to be the angle at the pole, formed between a meridian passing through aries, and a meridian passing through the object.
(E) The oblique ascension of an object, is the distance of the equinoctial point aries from the horizon when the object is rising. Or, it is that degree of the equinoctial which rises with the object in an oblique sphere.
(F) The oblique descension is the distance of the point aries from the horizon when the object is setting. Or, it is that degree of the equinoctial which sets with the object in an oblique sphere.
(G) The ascensional, or descensional difference, is the difference between the right and oblique ascension or descension, and with respect to the sun, it is the time he rises before six, when his declination is of the same name as the latitude, or sets
before six, when the declination and latitude have contrary names.
(H) The equinoctial colure is a great circle passing through the pole and the equinoctial points aries and libra.
(1) The solstitial colure is a great circle passing through the pole and the points B and w; called solstitial points, because when the sun is near these points he seems to have nearly the same altitude at noon, for several days, and therefore apparently stops or stands still.
(K) The arctic circle is a parallel of declination at the distance of 230.28' from the north pole, or 66°.32' from the equinoctial. It is generally called the north polar circle.
(L) The antarctic circle, called likewise the south polar circle, is the same distance from the south pole as the arctic circle is from the north pole.
(M) Apparent noon, the time when the sun comes to the meridian, or 12 o'clock, as shewn by a sun-dial.
(N) True, or mean noon, twelve o'clock as shewn by a well regulated chronometer, so adjusted as to go 24 hours in a mean solar day.*
(O) The equation of time at noon, is the interval between the true and apparent noon.
(P) A sidereal year is the interval of time from the sun's leaving any fixed star till he returns to it again, and consists of 365d. 6h. 9m. 12sec. of mean solar time.
(Q) A tropical or solar jear is the interval of time from
A mean solar day is a period not marked out by any observable phenomena, but an artificial interval of time. The tiine elapsed from the sun's leaving the meridian on any day till it returns to the same meridian the next day is called a true so'ar day, and is subject to a continual variation, arising from the obliquity of the ecliptic, and the unequal motion of the earth in its orbit,
A clock or chronometer, therefore, which measures time by equal motion, can. not be so adjusted as to keep time exactly with the sun, or always to shew 12 o'clock when the sun is on the meridian ; to correct these irregularities, the year is divided into as many imaginary days, each of 24 hours in length, as there are real days in the year measured by the sun's return to the meridian ; one of these imaginary days is called a mean solar day, and a clock adjusted so as to go 24 hours in one of these days, is said to be regulated to mean solar time.
The year thus consists of as many mean solar days as true solar days; the clock being just as much before the sun, on some days of the year, as the sun is before the clock on others. The difference is given in page II. of the Nautical Almanac for every day in the year. The time shewn by the clock is called true or mean time, and the time shewn by the sun is called apparent time.
If a clock be adjusted to go 24 hours, from the passage of any fixed star over the meridian till it returns to it again, its rate of going at any time may be determined by comparing it with the transit of that fixed star. A clock thus regulated is said to be adjusted to sidereal time. Here nature affords a standard exceeding in exactness any imitation that can be produced by art, there is no irregularity in the earth's diurnal motion, its diurnal revolution on its axis being uniformly per. formed in 24 hours of sidereal time=23h. 56m. 4sec. of mean solar time.