The true amplitude of any celestial object is, an arch of the horizon intercepted between the true east or west point thereof, and the object's centre at the time of its rising or setting.

The magnetic amplitude of an object is, the arch of the horizon that is intercepted between its centre, and the east or west point of the compass, at the time of its rising or setting; or, it is the compass bearing of the object when in the horizon of the eastern or western hemisphere.

The true amplitude of a celestial object is found by calculation; and the magnetic amplitude is found by an azimuth compass.

The true azimuth of a celestial object is, the angle contained between the true meridian and the vertical circle passing through the object's

centre.

The magnetic azimuth is, the angle contained between the magnetic meridian and the azimuth, or vertical circle passing through the centre of the object; or, in other words, it is the compass bearing of the object, at any given elevation above the horizon.

The true azimuth of a celestial object is found by calculation; and the magnetic azimuth by an azimuth compass.

PROBLEM I.

Given the Latitude of a Place, and the Sun's magnetic Amplitude; to find the Variation of the Compass.

RULE.

Reduce the apparent time of the sun's rising or setting to the meridian of Greenwich, by Problem III., page 297; to which time let the sun's declination at noon of the given day be reduced, by Problem V., page 298. Then, to the logarithmic secant of the latitude, add the logarithmic sine of the sun's reduced declination; and the sum, abating 10 in the index, will be the logarithmic sine of the true amplitude,-to be reckoned north or south of the true east or west point of the horizon, according to the name of the declination. Now, if the true amplitude, thus found, and the magnetic amplitude, observed per azimuth compass, be both north or both south, their difference is the variation; but if one be north and the other south, their sum is the variation:-and to know whether it be east or west, let the observer look directly towards that point of the compass representing the true amplitude; then, if the magnetic amplitude be to the left hand of this, the variation is easterly; but if to the right hand, it is westerly.

Example 1.

May 20th, 1825, in latitude 48:50 N., and longitude 6:30. W., at about 7:40", the sun was observed to set W. 56:42. N.; required the variation of the compass?.

Latitude of the place of observ.= 48°50'N.

Sun's reduced declination = . 20: 2:54′′N. Log. sine = 9.535057

20: 2:54 N.

Log. secant

10.181608

25:18:52"; which is west, because the

magnetic amplitude is to the right hand of the true amplitude.

Example 2.

July 10th, 1825, in latitude 18:40 N., and longitude 73:45: W., at about 17:29", the sun was observed to rise E. 30:12. N.; required the variation of the compass?

magnetic amplitude is to the left hand of the true amplitude.

Example 3.

October 17th, 1825, in latitude 42°10′ N., and longitude 14:30? W., at about 5:27", the sun was observed to set W. 7:33 N.; required the variation of the compass?

Variation =

=

42:10: ON. Log. secant 10. 130067 9.21.11 S. Log, sine =

9.210901

W. 12:39:57" S. Log. sine = 9.340968
W. 7.33. 0 N.

20:12:57"; which is west, because the

magnetic amplitude is to the right hand of the true amplitude.

Remarks.

In finding the variation of the compass by this method, the sun's amplitude should be taken, with an azimuth compass, when the altitude of his lower limb is equal to the sum of his semi-diameter and the dip of the horizon. Thus, if the sun's semi-diameter be 16:5, and the dip of the horizon 4'17" (for 20 feet), the sum = 20:22" is the height which the lower limb of that object should be above the horizon, at the time of observing its amplitude.

If the index of the quadrant be set to the altitude, thus determined, the sun's magnetic amplitude may be taken when his lower limb attains that altitude, either at rising or setting; for, although the sun is apparently so elevated, yet, on account of the atmospherical refraction, his centre is actually then in the horizon of the place of observation.

Note. For the principles of finding the variation of the compass by the amplitude of a celestial object, see "The Young Navigator's Guide to the Sidereal and Planetary Parts of Nautical Astronomy," page 261.

PROBLEM II.

Given the Latitude of a Place, the Sun's Altitude, and his magnetic Azimuth; to find the Variation of the Compass.

RULE.

Reduce the apparent time of observation to the meridian of Greenwich, by Problem III., page 297; to which time let the sun's declination, at noon of the given day, be reduced, by Problem V., page 298.

Find the true central altitude of the sun, by Problem XIV., page 320;

now,

To the sun's polar distance, add its true central altitude and the latitude of the place of observation; take half their sum, and call the difference between it and the polar distance the remainder.

:

Then, to the logarithmic secants, less radius, of the true central altitude and the latitude, add the logarithmic co-sines of the half-sum and the remainder half the sum of these four logarithms will be the logarithmic co-sine of an arch; which, being doubled, will be the true azimuth, to be reckoned from the north in north latitude, but from the south in south latitude; towards the east in the forenoon, and towards the west in the afternoon.

Now, if the true azimuth, thus found, and the magnetic azimuth, observed per azimuth compass, are on the same side of the meridian, their difference is the variation; but if on different sides, their sum is the variation :—and to know whether it be east or west, let the observer look directly towards that point of the compass which represents the true azimuth; then, if the magnetic azimuth be to the left hand of this, the variation is easterly; but if to the right hand, it is westerly.

Example 1.

April 15th, 1825, in latitude 39:40 N., and longitude 14:0: W., at. 410 per watch, the observed altitude of the sun's lower limb was 27:11, and the bearing of his centre, by azimuth compass, N. 80:37:30% W.; the height of the eye above the level of the sea was 24 feet; required the variation of the compass?

magnetic azimuth is to the right hand of the true azimuth.