of the angle formed at the centre of the earth, between the object and the observer, but in general this refraction is about of that angle.

DIP OF THE horizon.

Dip of the horizon is the angle of depression of the visible horizon below the true or sensible horizon (touching the earth at the observer) arising from the elevation of the eye of the observer above the level of the sea. Thus in Plate IX. Fig. 1. let ABC represent a section of the earth, whose plane produced passes through the observer and the object, and let AE be the height of the eye of the observer above the surface of the earth, then FEG drawn parallel to the tangent to the surface at A, will represent the true horizon, and EIH, touching the earth at I, will represent the apparent horizon;therefore the angle FEH will be the dip of the horizon. Let M be an object whose altitude is to be observed by a fore observation by bringing the image in contact with the apparent horizon at H; then will the angle MEH be the observed altitude, which is greater than the angle MEF (the altitude independent of the dip) by the quantity of the angle FEH; so that in taking a fore observation the dip must be subtracted from the observed altitude to obtain the altitude corrected for the dip. In a back observation the apparent horizon is in the direction EK, and by continuing this line in the direction EL we shall have the observed altitude MEL, and it is evident that to this the dip LEF (=KEG) must be added to obtain the altitude corrected for the dip.

In Table XIII. is given the dip for every probable height of the observer expressed in feet. In calculating this table, attention was paid to the terrestrial refraction which decreases the dip a little, because IE becomes a curve line instead of a straight one, and EH is a tangent to that curve in the point E.

What has been said concerning the dip of the horizon supposes it free from all incumbrances of land or other objects; but as it often happens when ships are sailing along shore, or are at anchor in a harbour, that an ob servation is wanted when the sun is over the land, and the shore nearer the ship than the visible horizon would be if it were unconfined; in this case the dip of the horizon will be different from what it otherwise would have been, and greater the nearer the ship is to that part of the shore to which the sun is brought down. For this reason Table XVI. has been inserted, which contains the dip of the sea at different heights of the eye, and at different distances of the ship from the land. This table is to be entered at the top with the height of the eye of the observer above the level of the sea in feet, and in the left hand side column with the distance of the ship from the land in sea miles and parts; under the former, and opposite the latter, stands the dip of the horizon, which is to be subtracted from the altitude observed by a fore observation instead of the numbers in Table XIII.

The distance of the land requisite in finding the dip from Table XVI. may be found nearly in the following manner-Let two observers, one placed as high on the main-mast as he can conveniently be, and the other on the deck immediately beneath him, observe at the same instant the altitude of the sun or other object that may be wanted, and let the height of the eye of the upper observer above that of the lower be measured in feet and multiplied by 0,56, and the product, divided by the difference of the observed altitudes of the sun in minutes, will be the distance in sea miles, nearly.

Thus, if the eye of the upper observer was 68 feet higher than that of the lower, and the two observed altitudes of the sun 20° 0′ and 20° 12′ the distance of the land in sea miles would be 3,2. For 68 × 0,56-38,08 and this divided by the difference of the two observed altitudes of the sun 12 gives 3,2 nearly. Now if the lower observer was 25 feet above the level of the sea, the dip corresponding to this height and the distance 5,2 miles would be 6', which subtracted from 20° 0' leaves 19° 54′ the altitude corrected for the dip.

The dip may be calculated in this kind of observations to a sufficient degree of accuracy without using Table XVI. in the following manner-Divide

the difference of the heights of the two observers in feet by the difference of the observed altitude in minutes, and reserve the quotient. Divide the height of the lower observer in feet by this reserved number, and to the quotient add one quarter of the reserved number, and the sum will be the dip in minutes corresponding to the lower observer, Thus in the above example 5',6 is the reserved number, and 4. 4, to this add one fourth of 5'.6 or 1'.4 and the sum will be the dip 5'.8 or nearly 6' corresponding to the lower observer, being the same as was found by the table.

TO FIND THE SUN'S DECLINATION.

THE declination of the sun is given to the nearest minute in Tab. IV. før every noon at Greenwich, from the year 1824 to 1838; and this table will answer for some years beyond that period, without any material error; if great accuracy is required, the declination may be taken from the second page of the month of the Nautical Almanac.* This declination may be reduced to any other meridian, by means of Table V. in the following

manner.

To find the sun's declination at noon, at any place.

RULE. Take out the declination at noon at Greenwich from Table IV. (or from the Nautical Almanac ;) then find the longitude from Greenwichi in the top column of Table V. and the day of the month in the side column; under the former, and opposite to the latter, will be a correction in minutes and seconds, to be applied to the declination taken from Table IV: to know whether this correction be additive or subtractive, you must look at the top of the column where you found the day of the month, and you will see it noted whether to add or subtract, according as the longitude is east or west. This correction being applied, you will have the declination at noon at the given place.

EXAMPLE I.

Required the declination of the sun at the end of the sea-day, October 10. 1824, in the longitude of 1140 E. from Greenwich?

Sun's declination Oct. 10, at Greenwich, at the end of the sea-day
or beginning of the day in the N. A. by Tab. IV.
Variation of Dec. Tab. V. Oct. 10, in 114° E. long. sub.

True dec. noon, Oct. 10. in long. 114 E.

EXAMPLE II.

6° 44' S.

0 7

6 $7 S.

Required the sun's declination at noon ending the sea-day of March 12, 1824, in the longitude of 750 W. from Greenwich?

Sun's declination March 12, by Tab. IV.
Var. Tab. V. March 12, long, 75° W.

sub.

True declination, noon, March 12, long. 750 W.

3° 13' S.

5

The preceding correction ought always to be applied to the declination used in working a meridian observation to determine the latitude, though many mariners are in the habit of neglecting it.

In finding the declination, or any other quantity, in the Nautical Almanac, you must be careful to note the difference between the civil, nautical, and astronomical account of time. The civil day begins at midnight, and ends the following midnight, the interval being divided into 24 hours, and is reckoned in numeral succession from 1 to 12, then beginning again at 1 and ending at 12. The nautical or spa day begins at noon, 12 hours before the civil day, and ends the following noon; the first 12 hours are marked P. M. the latter A. M. The astronomical day begins at noon, 12 hours after the civil day, and 24 bours after the sea day, and is divided into 24 hours, numbered in numeral succession from 1 to 24, beginning at noon, and ending the following noon. All the calculations of the Nautical Almanac are adapted to astronomical time; the declination marked in the Nautical Almanac, or in Table IV. is adapted to the beginning of the astronomical day, or to the end of the sea-day, it being at the end of the sea-day when mariners want the declination to determine their latitude. It would be much better if seamen would adopt the astronomical day, and wholly neglect the old method of counting by the sea day.

the difference of the heights of the two observers in feet by the difference of the observed altitude in minutes, and reserve the quotient. Divide the height of the lower observer in feet by this reserved number, and to the quotient add one quarter of the reserved number, and the sum will be the dip in minutes corresponding to the lower observer, Thus in the above example 5',6 is the reserved number, and 4. 4, to this add one fourth of 5'.6 or 1'.4 and the sum will be the dip 5'.8 or nearly 6' corresponding to the lower observer, being the same as was found by the table.

TO FIND THE SUN'S DECLINATION.

THE declination of the sun is given to the nearest minute in Tab. IV. for every noon at Greenwich, from the year 1824 to 1838; and this table will answer for some years beyond that period, without any material error; if great accuracy is required, the declination may be taken from the second page of the month of the Nautical Almanac.* This declination may be reduced to any other meridian, by means of Table V. in the following

manner.

To find the sun's declination at noon, at any place.

RULE. Take out the declination at noon at Greenwich from Table IV. (or from the Nautical Almanac;) then find the longitude from Greenwichi in the top column of Table V. and the day of the month in the side column; under the former, and opposite to the latter, will be a correction in minutes and seconds, to be applied to the declination taken from Table IV: to know whether this correction be additive or subtractive, you must look at the top of the column where you found the day of the month, and you will see it noted whether to add or subtract, according as the longitude is east or west. This correction being applied, you will have the declination at noon at the given place.

EXAMPLE I.

Required the declination of the sun at the end of the sea-day, October 10, 1824, in the longitude of 1140 E. from Greenwich?

Sun's declination Oct. 10, at Greenwich, at the end of the sea-day
or beginning of the day in the N. A. by Tab. IV. . .
Variation of Dec. Tab. V. Oct. 10, in 114° E. long. sub.

True dec. noon, Oct. 10. in long. 1140 E.

EXAMPLE II.

6° 44′ S.

0 7

6 87 S.

Required the sun's declination at noon ending the sea-day of March 12, 1824, in the longitude of 750 W. from Greenwich?

Sun's declination March 12, by Tab. IV.
Var. Tab. V. March 12, long. 75° W.

sub.

True declination, noon, March 12, long. 75° W.

3° 13' S. 5

3 8 S.

The preceding correction ought always to be applied to the declination used in working a meridian observation to determine the latitude, thongh many mariners are in the habit of neglecting it.

* In finding the declination, or any other quantity, in the Nautical Almanac, you must be careful to note the difference between the civil, nautical, and astronomical account of time. The civil day begins at midnight, and ends the following midnight, the interval being divided into 24 hours, and is reckoned in numeral succession from 1 to 12, then beginning again at 1 and ending at 12. The nautical or spa day begins at noon, 12 hours before the civil day, and ends the following noon; the first 12 hours are marked P. M. the latter A. M. The astronomical day begins at noon, 12 hours after the civil day, and 24 hours after the sea day, and is divided into 24 hours, numbered in numeral succession from 1 to 24, beginning at noon, and ending the following noon. All the calculations of the Nautical Almanac are adapted to astronomical time; the declination marked in the Nautical Almanac, or in Table IV, 1s adapted to the beginning of the astronomical day, or to the end of the sea-day, it being at the end of the sea-day when mariners want the declination to determine their latitude. It would be much better if seamen would adopt the astronomical day, and wholly neglect the old method of counting by the sea lax.