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horizon; and, therefore, it is numbered both ways, from those points, towards the meridian: that is, from 0 to 90%. The second space being adapted to horizontal azimuths, viz., to amplitudes reckoned from the meridian, is therefore numbered both ways, from the north and south points of the horizon towards the east and west points thereof: that is, from 0 to 90%, in a contrary order to the last. The third space is intended for the accommodation of an azimuth when the observation is reckoned from the south in north latitude, or from the south in south latitude: hence, it is numbered both ways from the south to the north point of the compass, or from 0 to 180. The fourth, or outer space, is designed for azimuths reckoned from the north in north latitude, or from the north in south latitude, according to the will of the observer; and, therefore, it is numbered both ways from the north to the south, or from 0 to 180°, &c.-See the Frontispiece to this volume.

Besides the evident uses of a compass card, graduated after this manner, in observing amplitudes and azimuths, it will also be found of the greatest utility in taking correct surveys of coasts and harbours, and in settling the true positions of places on shore from a known position at sea. It may, moreover, be applied successfully to many astronomical purposes; nay, it may even be applied to the determination of the longitude by lunar observations, as thus:-Let two observers, with two good compasses of the above description, take the azimuths of the moon and sun, or a fixed star, &c., at the same instant; then, if those two azimuths be reckoned from the same point of the horizon, their sum, subtracted from 360°, will be the angle at the zenith comprehended between the zenith distances of the objects; with which, and the true zenith distances of the objects, the true central distance may be found by oblique angled spherical trigonometry, Problem III., Remark I or 2, page 203 or 204; and, hence, the longitude of the place of observation, by Problem VIII., page 454.

An azimuth compass of this description would be of real advantage to the practical navigator; whereas, the one now in common use at sea is so very ill adapted to the important purposes for which it is designed, that it is very seldom resorted to for those purposes; and, therefore, it is scarcely ever seen upon deck, except for the simple purpose of comparing its parallelism with that of the binnacle, or steering compass.

SOLUTION OF PROBLEMS RELATIVE TO FINDING THE
APPARENT TIMES OF THE RISING AND SETTING
OF THE CELESTIAL BODIES.

PROBLEM I.

Given the Day of the Month, the Latitude of a Place, and the Height of the Eye above the Level of the Horizon: to find the apparent Times of the Sun's Rising and Setting.

RULE.

Let the sun's declination, at noon of the given day, be reduced to the meridian of the given place, by Problem V., page 298; then, to the logarithmic tangent of this reduced declination, add the logarithmic tangent of the latitude; and the sum (abating 10 in the index,) will be the logarithmic co-sine of an arch; which, being converted into time, will be the approximate time of the sun's rising, and its supplement to 12 hours will be that of the sun's setting, the latitude and the declination being of the same name; but if these elements be of contrary names, the above arch, reduced into time, will be the approximate time of the sun's setting, and its complement to 12 hours that of the sun's rising.

Reduce the approximate times of rising and setting, thus found, to the correspondent times at Greenwich, by Problem III., page 297; to which times, respectively, let the sun's declination be reduced, by Problem V., page 298; then,

To the aggregate of 90 degrees,* the horizontal refraction,† and the dip of the horizon, diminished by the sun's horizontal parallax,‡ add the sun's polar distance, and the co-latitude of the place of observation : take half the sum; the difference between which and the first term, call the remainder.

Now, to the logarithmic co-secants, less radius, of the polar distance, and the co-latitude, add the logarithmic sines of the half sum, and of the remainder half the sum of these four logarithms will be the logarithmic sine of an arch; which, being doubled, and converted into time, will be the apparent time of the sun's rising. In the same manner the apparent time of the sun's setting is to be computed; but, in this case, the half sum of the four logarithms is to be considered as a logarithmic co-sine.

Example 1.

Required the apparent times of the sun's rising and setting, July 13th, 1824, in latitude 50:48. N., and longitude 120: W., the height of the eye above the level of the sea being 30 feet?

The sun's distance from the zenith when his centre is in the horizon. + The horizontal refraction of a celestial object is 33 minutes of a degree. The sun's horizontal parallax is about 9 seconds.

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To find the apparent Time of the Sun's Rising :

Approximate time of the sun's rising =
Longitude 120: W., in time =

4.239:

+ 8. 0. 0

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Sun's dec., reduced to Greenwich time 21:49:50′′N.

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To find the apparent Time of the Sun's Setting :

Approximate time of the sun's setting = 7:57 21:
Longitude 120: W., in time =
+8. 0. 0

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Greenwich time past noon of given day = 15:57"21:

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Sun's dec., reduced to Greenwich time = 21:43:53 N.

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Required the apparent times of the sun's rising and setting, October 1st, 1824, in latitude 40:30? N., and longitude 105: E.; the height of the eye above the level of the sea being 29 feet?

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Approx, time of the sun's setting = 5:4912: Appr.time O's ris.61048:

To find the apparent Time of the Sun's Setting:

Approximate time of sun's setting

= ·

5:49 12:

Longitude 105: E., in time =

-7. 0. 0

Greenwich time past noon, September 30th, 22:49"12:

Sun's declination at noon, September 30th, = 2:52:46" S.
Correction of ditto for 22:4912: =

+22.11

Sun's declination, reduced to Greenwich time = 3:14:57" S.

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To find the apparent Time of the Sun's Rising :

Approximate time of sun's rising =
Longitude 105 E., in time =

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6:1048:

7. 0. 0

Greenwich time past noon, September 30th, 11:1048:

=

2:52:46 S.

+10.52

Sun's declination at noon, September 30th, =
Correction of ditto for 11:10:48: =

Sun's declination, reduced to Greenwich time, = 3: 3:38" S.

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See Examples 1 and 2, page 125; and, also, the Example, pages 126,127.

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