BY CONSTRUCTION t. (Plate V. Fig. 22.) 1. Describe the primitive circle with the chord of 60°, on which set off AB=82°.28', and from A and B through the centre P draw A Pe and BPr. 2. Set off the side Ac=57°.30′, by a scale of chords, from A to mɩ, and draw the parallel circle mcm. (Z. 157.) The construction of the figure to this example is not essentially different from the construction of the figure to the preceding example, and is introduced here to shew in what manner the general figures, Case XI, &c. of oblique spheŕics with a perpendicular, were formed.-Compare the two figures in this Case (V) with those in Case XI. referred to above. 3. Set 3. Set off the supplement of the side BC=64°.40' from r to n, and draw the parallel circle ncn. (Z. 157.) 4. Through A, and the point of intersection c, draw the oblique circle Ace; and through B, and the same point, draw вcr. 5. Then ABC is the triangle required; and cd, or CD, are perpendiculars on the base AB produced. 2. In the oblique spherical triangle ABC. Given The side BC=40°. 0'.10" Ans. Required the angles. 3. In the oblique spherical triangle ABC. The side AC=30° Given The side BC=24°.4' Answer. Required the angles. LA= 34°.15'. 3′′. 4B 42°.15'.13′′. <c=121°.36′.20". LA 36°.8'. B=46°.19'. <c=104°. (0) CASE VI. Given the three angles to find the sides. The 4A Given The B 51°.30′ 59°.16 ́ Required the sides. The c=131°.30′ BY CONSTRUCTION. (Plate V. Fig. 23.) 1. With the chord of 60° describe the primitive circle, through the centre P draw cre, and arr at right angles to it. 2. Draw the great circle cвe, making an angle of 48°.30' (the supplement of c) with the primitive (P. 155.) and find its pole p. (N. 155.) 3. With the semi-tangent of the A51°.30′ and centre P, describe the small circle oso. 4. Through e and p, draw epm; make mw=59°.16', the ▲ B; draw ew cutting ar in n; bisect nn in v; with v as a centre, and radius vn, draw the small circle nsn, cutting oso in s. 5. Throughs and P draw bs Pb, and Ard at right angles to it; through a and s draw Asx, make xy an arc of 90°, and draw Ay cutting bs pb in t. 6. Through the three points Atd draw a great circle. Then ABC is the triangle required. To measure the required parts. 7. AC=S0.19'; BC=63 ̊.50′ and AB=120°.47'. (C. 158.) BY By the same rule the other sides may be found. Cosec ab OR, To find the side вc, by Rule II. 180°— A⇒bc; 180°— ▲в=ac; and 180°- ▲c=ab. Sine(bc+ac+ab) =148°.52′ 9.71352 Cosec ac Sine (bc+ac+ab)-bc 20°.22′ 9.54161 4c 59°.11'.40" its suppt. 120°48′20′′=AB. PRACTICAL ASTRONOMICAL DEFINITIONS, &c. WITH THE APPLICATION OF RIGHT-ANGLED SPHERICAL TRIANGLES TO ASTRONOMICAL PROBLEMS. (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 root 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 1804, was 13°.14.43", and its declination 88°.15.44" north. (D) The (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 22d 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 23°28'*, 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,. each sign contains 30 degrees. Their names and characters are as follow, 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, the other the south or descending node. The names and characters of the planets and nodes are follow: Saturn | Sol (the sun)!! Mars ☀ Mercury Luna (the Moon)|| Herschel, & Moon's N. Node for Georgian. 8 her S. Node 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 signs 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 continually diminishing. But this diminution is very small, and its quantity is variously stated by different astronomers. I i ¿ Conjunction |