47. In the north latitude of 51°30' I observed the sum to rise on a certain point of the compass; and eleven hours after the sun had arisen, I perceived my shadow to be projected towards the same point. What was the sun’s declimation? Ans. Declination 9°54'41" N. 48. To what height must a person be raised above the city of London, on June 21st, at midnight, so that he may just see the sun’s upper limb 2 - I Ans. 155-4283 miles, the radius of the earth being supposed 3980. 49. In the latitudes of 30° and 50° north, on the same meridian, on the 21st of June in the morning, it is required to determine the exact instant, when the sun’s. altitude will be equal, if observed at both places; also the latitude at that time? - Ans. Time 3" 5.5" 16" from noon; altitude 37° 37' 17". 50. At what time of the year is the night (exclusive of twilight) longer at York (N. lat. 54°) than it is either at London (N. lat. 51°32') or at Edinburgh (N. lat. 56° 7')? Ans. Whilst the sun’s declination south is between 14° 14′ 46” and 14° 40' 34"; that is, on February 10th and November 1st. . 51. In what north latitude may an erect south declining dial be fixed, to have the declination of the plane, the distance of the substyle from the meridian, and the. height of the style, all equal 2 Ans. The latitude of 38° 2'; and the plane's declination will be 38° 13'. 52. In the spring quarter last year, day broke at 3 o'clock, and the sun’s altitude that morning when due east was 32°42'. Where and when did this happen 2 - * * Ans. Lat. 38°39'20" N.; Dec. 19°43' 18" N. answering to May the 19th. 53. At a certain place I observed the sun to rise at 10 minutes past 4 o’clock, and his altitude at noon to be 58°40'. What were the latitude and day of observation? - - - i. - Ans. Lat. 51°30' N.; Dec. 20° 10' N., answering to May 21st and July 22d. 54. On the longest day last year it was observed at a certain place, that the sun’s altitude when due east was 14°46’ more than it was at 6 o'clock the same morning: what were those altitudes, and what was the latitude of the place . . . . . - - Ans. Altitudes 32°8'5" and 17°22' 5", Latitude 48° 30'49" N. 55. What is the northern latitude, time of the year, and time of the day, in 1816, when the sun's altitude, his azimuth from the east, the arc from noon, and the co-latitude of the place are equal to each other? Ans. Lat. 51°28'33", time of year Apr 17, or Aug. 26, - time of day 9"25" 56 A.M. 56. In what north latitude is the shortest day equal to **, of the longest at London 2 - Ans. Latitude 43°28′. 57. In what latitude and time of the year does the sun, rise at half past 5 o'clock, and appear due east at 10 ” - Ans. Lat. 21° 13', Dec. 18° 35', both of the same kind. - - * . 58. Where is the sun’s altitude at 6 o'clock, on the longest day, equal to the co-latitude 2 Ans. N. latitude 68° 17' 12". 59. To find the declination of that star whose change in azimuth is awmarimum or minimum in a given time, reckoned from the time that it transits a given almucantar in a given latitude. Suppose the latitude of London, the time, one hour, and the almucantar 15° above the horizon. Ans. Star’s declimation 20° 25' south. 60. What arc of a circle is equal to its tangent?Ans. Arcs of 257°27' 12", 442° 37' 28”, 621° 45'88", 805° 56' 1", 986°40'36", 1167°11'23", 1347° 33'55", i527°51'9", 1708°5'44", or iss's 16' 12", answer the conditions of the question; and these arcs fall in the 2d, 4th, 6th, 8th, 10th, 12th, &c. quadrants, running continually round the same circle. . . . . . . . 61. What arc is that whose sine shall be equal to the so wooibreside is half the side of its - - - - I Ans. The arc whose tangent is =T. 68. In a right angled triangle the right lines drawn from the acute angles to the middle points of the oppo. site sides are equal to a and b respectively; required the acute angles. Ans. The acute angles are those whose tangents are rules for determining the log. sines and tangents of small arcs, given at pa. 55 of this volume. 70. Supposing the latitude of London to be 51°30' N., the latitude and longitude of Moscow 55° 45' N. and 38°E., and the latitude and löngitude of Constantinople 41°30' N. and 29°15' E. It is required to determine the latitude and longitude of a place which shall be equidistant from the former three. - Ans. Lat. 51° 17' N., long. 19° 13' E. 71. Three stars A, B, and c, were all observed to be in the arc of a great circle; the distance of A and B was found to be 10°, of B and c 20°; the difference of the azimuths of A and c was found to be 90°; and the middlemost B was the least distance possible from the zemith. Required the altitudes of the three stars 2 Ans. Alt. of A, 72°18'14", of B, 75°19'32", and of c,65°22' 33”. 72. In what north latitude will the sun appear due east, on the longest day, at the mid-time between sumrise and noon : Ans. In N. lat. 64° 35' 48.” 73. Let ABc be a plane triangle of which # = n : - AC it is required to demonstrate, that the value of the angle A is expressed in seconds by the first or second of the following series, according as the perpendicular from B on Ac falls within or out of the triangle: viz. m sinc m” sin 20 m3 sin 3c mo sin 4c + &c. 75. If in a plane triangle tan B = n tan A, then demonstrate that 76. Let E be the spherical excess in seconds of a spherical triangle ABC, then it is to be demonstrated that _ tan obtan ocsin A tan” obtan” #c sin2A * E - sin 1" - sin?" tano obtan? §c sin 3A + sin 3" 77. Demonstrate also, that . cot #8 cot oc. sin AT" and find E, when A = b = c = 179°. Ans. E = 35.7° 59' 58". 78. Given the latitude of the place, and the position of two hour circles, with respect to the meridian; to determine the declination of that star whose change in altitude shall be the greatest possible in passing over the interval between those hour circles. Ans. Let h’ be the greatest, and h the least hour angle from the meridian, L the latitude, and D the declination; then sin # (h' — h) tan D c tan *Tini (TW) 79. If a person could approach so near to the moon as to see one third of her convex surface, what angle |