for every 12 hours; namely, for noon and midnight, at Greenwich, when the Moon's semidiameter is required for any intermediate time, a proportional part of the difference or variation in 12 hours is to be applied to the semidiameter for noon or midnight: this gives the horizontal semidiameter, which is to be farther corrected by the augmentation from Table IV. (see the explanation of that Table.) The fixed stars having no sensible magnitudes, as seen from the Earth, are esteemed as mere lucid points; hence, no allowance is to be made for semidiameter in observations of the Fixed Stars.

9. The apparent distance between any two objects means the apparent distance of their centres, and is found by applying the semidiameters of those objects to the observed distance, by addition, or subtraction, according as the nearest or farthest limbs have been observed.

10. The true altitude of the Sun, or a Star, is found by subtracting the correction in altitudet from the apparent altitude. The true altitude of the Moon is found by adding the correction from Table XXI. to her apparent altitude, the true altitude of a Planet by deducting the parallax in altitude (Table P.), from the refraction (Table VI.), previously to subtracting the refraction from the apparent altitude of the Planet. In correcting a lunar distance, by the first method given in this work, the apparent altitudes only are used. In finding the time, the true altitude of the object is always used.

11. The Moon's horizontal parallax which is required to find her true altitude, and to clear the lunar distance, is found on page III. for the Month in the Nautical Almanac, and should be taken out at the same time with her semidiameter, and corrected in the same manner. The Planets' horizontal parallaxes and semidiameters are found on the right-hand pages of the month for each Planet in the Nautical Almanac.

12. The polar distance of an object is its distance from the elevated Pole of the observer. Hence, when the latitude of the place of observation, and the declination of the observed object, are both of the same name (that is, both North or both South), the difference between 90° and the declination is the polar distance; but when the latitude of the place, and the declination of the object, are of contrary names, the sum of 90° and the declination of the object is its polar distance.

13. The Sun's declination is found in pages I. and II. for the month in the Nautical Almanac for apparent and mean noons. The Moon's declination is given for every hour, in pages V. to XII. for the month. The Planet's declinations for mean noon every day are found under their respective names in the same work, also, the declinations of 100 Fixed Stars for the 1st January of each year, in

pages 432 to 434, with their annual variations. The following method of reducing (see Art. 2,) the Sun's declination will be found short and simple. Take out the declination for the noon nearest to the Greenwich date, from page I. or II. for the month, according as the Greenwich date is in apparent or mean time. If the Greenwich date be before the nearest noon, take the difference for one hour annexed to the preceding noon, to reduce the declination to the given intermediate interval. If the Greenwich date be after the nearest noon, take the difference for one hour annexed to the declination taken out to effect the reduction. The difference for one hour multiplied by the hours and parts† of an hour from the nearest noon, will be the quantity requisite to reduce the declination to the given interval from noon, to be added to or subtracted from the declination taken out, according as the declination is increasing or decreasing, or as the Greenwich date precedes or follows the nearest noon. The differences for one hour on page I. Nautical Almanac are to be used for their respective articles on page II.

EXAMPLE.

In Long. 15° W. required the O's decl. date M. T., (Astronomical.)
d. h. m.

1844, December . 3 3 10
Long. in time W.
· + 1 0

Gwh. D. in M. T.. 3 4 10

In Long. 15° W. required the O's decl. date in M. T. (Astronomical.)

d. h. m.

14. The equation of time, which is required to convert apparent time into mean time, and the contrary, is found on pages I. or II.

4m.

12m.

of an hour.

for the month, in the Nautical Almanac, and should be taken out with the Sun's declination and semidiameter. The same method of reduction to other dates and meridians is applicable to both equation of time and declination, as the differences for one hour of equation of time are given.

15. The sidereal time at mean noon, which is required to find the mean time, from observations of the Moon, Planets, or Fixed Stars, is found on page II. of the month in the Nautical Almanac. It is to be taken out for the given astronomical day at the place of observation, and to be reduced to that meridian by applying the acceleration for longitude, as directed in Table A, which contains the acceleration.

16. The right ascensions of the Moon, Planets, and Stars, also requisite for finding the mean or apparent time, by either of these bodies, are found at the same places in the Nautical Almanac, with their respective declinations, and should be taken out at the same time. The right ascension of the Moon being given for every hour, the proportional part for any intermediate portion of time is easily obtained, by finding the difference between the right ascension for the hour given, and the right ascension following, and taking onethird, or one-fourth, or one-sixth, &c., of the difference as may be required. Ample explanations are given at pages 582-3-4 in the Nautical Almanac for reducing the right ascensions and declinations of the Planets and Fixed Stars, to which refer.

17. When the Sun's right ascension is required, it is to be taken from pages I. or II. in the Nautical Almanac, according as the Greenwich date is in apparent or mean time; and as the differences for one hour are given, it can be reduced in the same manner as the declination.

18. The fraction of the year which is requisite for taking a proportional part for any date during the year, of the annual variations of the right ascensions and declinations of the Fixed Stars, given in the Nautical Almanac, is found on page XXII. for the month therein.

It is strongly recommended to young Navigators to make themselves well acquainted with all the explanations given in the Nautical Almanac, as well as the explanations of the Tables given in this work.

PROBLEM I.

Given the Latitude of a Place, together with the Sun's True Altitude and Declination; to find the Apparent Time of Observation; the Sun's True Azimuth, and the Mean Time.

RULE.

To find the Apparent Time.

1. Add together the Sun's altitude, the polar distance, and the latitude of the place of observation; find the half sum, and the difference between the half sum and the Sun's altitude.

2. To the logarithm of the polar distance, add the logarithm of the latitude and the logarithms of the half sum and difference, the sum of these 4 logarithms will be the logarithm of the apparent time.†

To find the Azimuth.

3. Take the difference between the half sum (already found), and the polar distance, call it Az.

4. Add together the secant of the altitude, the logarithms of the latitude and half sum (already found), and the cosine of Az. The sum of these 4 logarithms will be the logarithm of the Sun's true azimuth in time to be taken from the top of Table XIII., which convert into arc, by Table I.

5. This is to be reckoned from the South in North latitude, from the North in South latitude. Towards the East in the morning; the West in the afternoon.

EXAMPLE I.

Suppose the Sun's true altitude, west of the meridian, is 34° 0', his declination 10° 0′ N. and the latitude of the place of observation 42° 0' N. required the apparent time of observation and the Sun's true azimuth?

+ Table XI. contains the logarithms of the latitude and polar distance, being a table of secants and co-secants; Table XII. the logarithms of the half sum and difference, being a table of cosines and sines; and Table XIII. the logarithms of

In latitude 33° 56′ S. the Sun's true altitude, observed East of the meridian, was 24° 58', and his declination at the same time was 2° 44' N.; required the apparent time of observation and the Sun's true azimuth?

Apparent time - 20h 11m. 8s.

Log. 9-36037
True azimuth N. 67° 52′ E. = 4h. 31m. 29s. =

9.49370

If civil, or nautical time be required, when the Sun is observed east of the meridian, reject 12 hours from the astronomical time. Thus, the civil or nautical time in the last Example, would be 8h. 11m. 8s. A. M., observing that when the date of the astronomical day is given, the date of the civil or nautical day will be one day farther advanced. In the practice of the lunar observations, the mean astronomical time at the place of observation is always to be found.

To find the Mean Time.

Having obtained the apparent time, apply to it the reduced equation of time, as directed in page I. for the month in the Nautical Almanac. The result will be the mean time required. (See Art. 2 and 14.)