260 THE PRACTICE of oblique SPHERICS, ETC. BOOK III. OR, To find the sides BC, by Rule II. 180°-LA-be, 180°- Bac, and 180° cosec ab cosec ac sin(be+ac+ab) = 48° 30' •12554 =120 44 ⚫06573 =148 52 9.71352 cab. reject indices. sin (bc+ac+ab)-bc= 20 22 9.54161 La=116 10 its suppt.=63° 50′ =BC. To find the side AB, by Rule II. tan c=29° 35′ 52′′ 2 Zc=59 11 44 its suppt.=120° 48′ 16′′=AB. PRACTICAL EXAMPLES. 1. In the oblique spherical triangle ABC. The A=126° 35' AC 57° 30' Given The B 48 32 Answer. BC 115 20 The c 61 44 Required the sides. 2. In the oblique spherical triangle ABC. The A 34° 15′ 4′′ Given The B 42 15 12 AC 50° 10′ 30′′ Answer. BC 40 0 10 AB 76 35 36 The c=121 36 20 Required the sides. 3. In the oblique spherical triangle abc. The A 36° 8' AC=30° 0' Given The B 46 18 Answer. BC 24 4 The c=104 0 Required the sides. AB 42 9 СНАР. Х. ASTRONOMICAL DEFINITIONS AND INTRODUCTORY PROBLEMS. I. Astronomical Definitions. (510) The Celestial Sphere is that apparent concave in which the sun, moon, stars, and all the heavenly bodies seem to be situated. (511) 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.* (512) The poles of the celestial sphere are the extremities of its axis, the one called the north pole, the other the south pole; and the points lying directly opposite to them, on the terrestrial sphere, are the north and south poles of the earth.+ (513) The equator or equinoctial is a great circle which divides the heavens into two hemispheres, the northern and southern; it is so called, because, when the sun appears in it, the days and nights all over the world are equal, viz. twelve hours each. This happens twice in the year, about the 20th of March and the 23rd of September; the former is called the vernal equinox, the latter the autumnal equinox. (514) The ecliptic is a great circle in which the sun makes his apparent annual progress; it cuts the equinoctial in an angle of 23° 28′ nearly ‡, called the obliquity of the ecliptic; and the points of intersection are called the equinoctial points. The ecliptic is divided into twelve equal parts, called signs : 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 pole star is not directly in the point which is the true north pole of the heavens; its mean right ascension, on January 1, 1840, being 1° 2′ 0′′ 68, and annual variation +16" 50; also its mean declination is 88° 27′ 21′′.9 N., and annual variation + 19" 32. (Nautical Almanac). Its mean polar distance is therefore 1° 32′ 38′′-1 on Jan. 1, 1840. 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 altitudes of the sun at any place, supposing the sun to have the greatest declination when on the meridian. each sign contains 30 degrees. Their names and characters m Scorpio Capricornus * Pisces. ■ Gemini my Virgo ↑ Sagittarius 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. (515) 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 greater planets are performed. (516) 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 ; the other the south or descending node, and is marked thus 8. The names and characters of the planets are as follow: ? Ceres ↳ Saturn Pallas 24 Jupiter H Herschel, or Georgian. 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 degrees distant, they are in a quartile aspect. If two signs, or 60 degrees, a sextile aspect. The astronomical marks are as follow: o Conjunction when planets are in the same point of the ecliptic. Sextile when 2 Signs dist. A Trine when 4 Signs dist. Quartile when 3 Signs dist. 8 Opposition when 6 Signs dist. The conjunctions and oppositions are also called the syzygies, and the quartile aspects the quadratures; these terms are applied chiefly to the moon. (517) 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. (518) 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′ (153), is a representation of the horizon. (519) The Zenith is a point in the celestial sphere directly over the head of the spectator, being the elevated pole of the horizon. (520) 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. (521) 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. (522) The prime vertical is that azimuth circle which passes through the east and west points of the horizon. (523) 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. (524) Circles of celestial longitude are great circles passing through the poles of the ecliptic, and cutting it at right angles. (525) 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. (526) 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. (527) The declination of any celestial object, is an arc of a meridian contained between the centre of that object and the equinoctial. (528) Parallels of declination are small circles parallel to the equinoctial. (529) 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. (530) Parallels of altitude are small circles parallel to the horizon. (531) Parallels of celestial latitude are small circles parallel to the ecliptic. (532) The tropics are small circles parallel to the equinoctial, at 23° 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. (533) 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. (534) 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. (535) 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. (536) 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. (537) The right ascension of an object, is the distance between the first point of 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 the first point of aries, and a meridian passing through the object. (538) 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. (539) The oblique descension is the distance of the first point of 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. (540) 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 declension is of the same name as the latitude, or sets before six, when the declination and latitude have contrary names. (541) The equinoctial colure is a great circle passing through the pole and the equinoctial points aries and libra. (542) The solstitial colure is a great circle passing through the pole and the points 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. (543) The arctic circle is a parallel of declination at the distance of 23° 28′ from the north pole, or 66° 32′ from the equinoctial. It is generally called the north polar circle. |