47. In the north latitude of 51° 30′ I observed the sun 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 declination?

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 ?

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 3h 55m 16s 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?

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?

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?

,,

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′ 53′′, time of year Apr 17, or Aug. 26, time of day 9h 25m 56′ A. M. 56. In what north latitude is the shortest day equal to 1 of the longest at London?

8

5

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?

Ans. N. latitude 68° 17′ 12′′.

59. To find the declination of that star whose change in azimuth is a maximum 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 declination 20° 25' south.

60. What are of a circle is equal to its tangent?

Ans. Arcs of 257° 27′ 12′′, 442° 37′ 28′′, 621° 45′ 38′′, 805° 56′ 1′′, 986° 40′ 36′′, 1167° 11′ 23′′, 1347° 33′ 55′′,

1527° 51′9′′, 1708° 5′ 44′′, or 1888° 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 sine of triple the same arc?

62. What arc is that whose sine is half the sine of its triple arc?

n

63. What arc is that whose sine is the th part of the sine of its triple arc? Ans. The arc whose sine is√(3

ན ། ན།།་

n).

64. What arc is that whose sine is equal to the sine of its quadruple arc?

65. What arc is that whose sine is equal to the sine of its quintuple are?

66. What arc is that whose sine is equal to n times its cosine?

Ans. The arc whose sine is

n + 1

67. What arc is that whose secant is equal to n times its tangent?

Ans. The arc whose tangent is n2

68. In a right angled triangle the right lines drawn from the acute angles to the middle points of the opposite sides are equal to a and b respectively: required the acute angles.

Ans. The acute angles are those whose tangents are

Thus if a = 4, 6 = 3, then the tangents are

31° 5′ 28′′.

and the angles are 58° 54' 32" and

69. Demonstrate the truth of Dr. Maskelyne's four

M

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 longitude 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.

B

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 в and c 20°; the difference of the azimuths of a and c was found to be 90°; and the middlemost в was the least distance possible from the zenith. Required the altitudes of the three stars?

Ans. Alt. of A, 72° 18′ 14′′, of в, 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 sunrise and noon?

Ans. In N. lat. 64° 35′ 48."

73. Let ABC be a plane triangle of which

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.

75. If in a plane triangle tan B = n tan A, then de

monstrate that

76. Let E be the spherical excess in seconds of a spherical triangle ABC, then it is to be demonstrated

that

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

79. If a person could approach so near to the moon as to see one third of her convex surface, what angle