The apparent dip is the true depression (Table 8), diminished by about of itself. As this correction is subject to great variations, the dip may be employed to the nearest minute only.

TABLE 31. MEAN ASTRONOMICAL REFRACTION.

The Refraction is given for the barometer at 30 inches, and Fahrenheit's thermometer at 50°, according to Ivory. The diff. to 10' of alt. is inserted. Ex. 1. The refraction at 20° is 2' 39'.

Ex. 2. The refr. to the alt. 38° 35' is 1′ 13′′3, deducting 2, or 1′ 13′′1.

The tenths of seconds are omitted at altitudes below 35°, on account of the uncertainty at low altitudes.

With the alt. as course and dep.
For the refr. is pro-

To find the Refraction approximately. 58, find the D. Lat. ; this is the refraction in seconds. port. nearly to the tang. of the zen. dist., and is 58"-2 at zen. dist. 45°. Ex. Alt. 10°, as course, and Dep. 58, give 329", or 5′ 29′′, the refr. required.

TABLE 32. CORRECTION OF THE MEAN REFRACTION FOR THE HEIGHT OF THE THERMOMETER.†

The Table is entered with the Alt. at the top, and the degree of Fahrenheit's therm. at the side. When the therm. is below 50°, the correction is added to the mean refr.; when above 50°, it is subtracted.

Ex. Alt. 17° 10′, therm. 72°; the corr. is 8", which, subtracted from the mean refi., 3'7", gives the true refraction 2′ 59′′.

To find the Correction, nearly. Multiply the mean refraction in seconds by 2, and by the difference between the height of the therm. and 50°, and divide the product by 1000.

Ex. Alt. 5o, therin. 38'. The mean refr. 9′ 54′′, or 594′′, mult. by 2 and by 12, is 14256, and this divided by 1000 gives 14".

TABLE 33. CORRECTION OF THE MEAN REFRACTION FOR THE HEIGHT OF THE BAROMETER.

The Correction is given to each tenth of an inch. The Table is entered like Table 32. When the barom. is above 30 inches, the correction is to be added; when below, subtracted.

Ex. Alt. 17° 10', barom. 29*2 in.; the corr. is 5", and true refr. 3′ 2′′.

To find the Correction. Multiply the mean refr. in seconds by the difference between the height of the barom. and 30 inches, and divide the product by 30.

Ex. (Above.) 3′ 7′′, or 187′′, mult. by 8, and divid. by 30, gives 5′′.

Ivory's

* The refractions now used by astronomers are those according to Bessel. exceeds these by 0.9" at alt. 45°, by 2" at alt. 20°, and 5" at alt. 10°. The difference of the tables is scarcely worth a more extended notice.

†This correction involves the term (-50°). The term

dde
dr

as insensible.-Phil. Trans. 1823, p. 476.

TABLE 34. THE SUN'S PARALLAX IN ALTITUDE AND SEMIDIA METER.

These are given for convenience on some occasions, but not for extreme precision.

To compute the Sun's Parallax in Altitude. Take the hor. par. in the Naut. Alm. as dist., and find the D. Lat. to the app. alt. as course.

TABLE 35. DIP OF A SHORE-HORIZON.

The Table shews the Apparent Dip to be used instead of the dip in Table 30, when the distant sea-horizon cannot be seen, and the altitude is observed from the water-line on the beach. The distance of this line may either be estimated nearly, as it is always less than the true dip due to the height of the eye (Table 8), or it may be found by the method No. 550, p. 180.

To compute a Term.

Take the diff. between the depr. to the eye (Table 8) and the dist. of the beach-line, and divide by twice this last; add the quotient to the app. dip in Table 30.

TABLE 36.*

This Table contains the scales of the Centigrade and Réaumur therJometers, corresponding (approximately) with that of Fahrenheit.

The zero of the two former, or the freezing point of water, being 32° of of Fahr., and their boiling points 100° and 80° respectively, while that of Fahr. is 212; the following rules are derived for the conversion of the scales.

To convert the Centigrade into Fahrenheit. Multiply the degrees of the Centigrade by 9, and divide the product by 5. When the Centigrade degrees are above 0, add 32° to the quotient; when below 0 (or marked —), subtract it from 32°.

To convert Réaumur into Fahrenheit. Multiply the degrees by 9, and divide the product by 4. Apply the quotient as directed above.

Ex. Centig. 117, find Fahr. 11.7 × 9 = 1053, this ÷ 5 = 211, which subtracted from 32 gives 10°.9.

To extend the Table. For the Centigrade add 0-555, &c., and for Réaumur 0-444, &c., for each 1° of Fahr.

TABLE 37.

This Table contains the English measures corresponding to the Mètre, Kilomètre, Décimètre, and Millimètre. † See p. 368. Thus 30 centim. are 11.81 inches; 3 kilom. are 1.618 nautical miles.

The barometer scale, in English inches, and millimètres (approximately), is annexed.

To reduce the French to the English barometer scale. Divide the millimètres by 25-4, the quotient is the number of English inches required.

* As numerous valuable works relating to Navigation are published by the French, and as other Continental nations frequently employ the language of that country in hydrographic documents, Tables 36 and 37 are added, for the ready reduction of such French measures as most frequently occur.

†The quantities are taken from the Annuaire, for 1846. The mètre is the 10-millionth part of the quadrant of a meridian.

When the French scale is given in inches and lines (or 12ths of an inch), multiply the inches by 1.065, the product is English inches.

inch.

To extend the barometer scale, add 2.54 millimètres for each 0.1 of an

TABLE 38.

CORRECTIONS OF ALTITUDE OF THE SUN AND STARS.

The Table contains the gross corr. of alt., or the corrections enumerated in No. 644, exclusive of index error, to the nearest min.

For examples, see No. 648, and p. 215.

TABLE 39. THE MOON'S CORRECTION OF ALTITUDE.

The Table contains the Correction to each minute of horizontal parallax and every 10' of alt.; for the barom. 30 inches, and Fahrenheit's therm. 50°. Ex. The corr. to app. alt. 15° 30' and hor. par. 56′, is 50′ 31′′.

For seconds of parallax. Look among the columns on the right side of the page, and against the alt., and take out the seconds, which add to the correction.

the

For minutes of altitude.

page,

Take the seconds from the extreme right of

and apply them as there directed.

Ex. Moon's App. Alt. 35° 37′, Hor. Par. 57′ 32"; find the Correction of Altitude.

To correct for the Barom. and Therm.

Take the corrections from Tables 32 and 33, but apply them to the correction of alt. the contrary way to that directed. Ex., No. 655, p. 216.

To compute a Term. Correct the app. alt. (of the centre) for refraction. To the log. sec. of this alt. add the prop. log. of the horizontal parallax; the sum is the prop. log. of the parallax in alt. From this subtract the refraction; the rem. is the correction of alt.

The Table does not give the correction with precision at low alts.*

TABLE 40. CORRESPONDING HORIZONTAL PARALLAX AND SEMIDIAMETER OF THE MOON.

As these two elements are generally required together, the Table renders it necessary to reduce the parallax alone to the Greenwich Date.

TABLE 41. DIMINUTION OF THE MOON'S HORIZONTAL PARALLAX FOR THE SPHEROIDAL FIGURE OF THE EARTH.

The Table is entered with the Horizontal Parallax at the top and the Latitude at the side; the seconds corresponding are to be subtracted from the equatorial hor. par.

The compression employed is bo

3

* In all these tables of refraction the eye is supposed at the level of the sea; when the observer is at very great elevations, low altitudes cannot be corrected with precision by the tables in common use. The refraction is in such cases too great.

TABLE 42.

AUGMENTATION OF THE MOON'S SEMIDIAMETER.

The Table is entered with the Moon's Semidiameter at the top and her Altitude at the side; the seconds corresponding are the excess by which her apparent semidiameter at her actual altitude exceeds that at which it would appear if seen from the centre of the earth. See Nos. 439 and 440, p. 147

TABLES 43 AND 44. FOR CONVERTING TRUE INTO APPARENT

ALTITUDES.

These contain the further correction necessary in reducing a true to an apparent altitude, after adding the refraction and subtracting the parallax. See Nos. 657 and 658, p. 216.

TABLE 45.

PARALLAX OF THE PLANETS IN ALTITUDE.

The Table is entered with the Planet's Horizontal Parallax at the top, and its Altitude at the side; and the corresponding seconds taken out. To compute a Term. Enter the Traverse Table with the alt. as course and the hor. par. as dist., and take out the D. Lat.

TABLE 46. AZIMUTH CORRESPONDING TO THE CHANGE OF ALTITUDE IN 1 OF TIME.

The Table shews the Change of Altitude in 1m of Time at any Azimuth in Latitudes below 66°. The azimuth is reckoned either from N. or S.

Ex. In lat. 50°, at the azim. 40°, reckoned either from N. or S., the change of alt. in Im is 6' and some seconds.

The Table shews also, roughly, the true bearing when the change of alt. in 1m is given. See also No. 677, p. 224.

The column of 6' limits the azimuth for finding the time, No. 778, p. 261.

LATITUDE.

THESE Tables are employed in the rules in Chap. V., p. 225. TABLE 47. LIMITS OF THE REDUCTION TO THE MERIDIAN AT SEA

This Table shews how long before or after noon the sun's altitude may be observed, so that the Reduction shall not be in error more than 2′ when the time is 1m in error. The Table, therefore, shews the Limits of this method for common practice at sea.

If the time be in error, or doubtful, 2m, 3m, &c., the Reduction will, at the limits, be in error, or doubtful, 4', 6', &c. In like manner, if the error of time be less than 1m, that of the Reduction will be less than 2′, in the same proportion.

If the time is doubtful 2m, 3m, &c., and we require that the error of the Reduction shall not exceed 2', we must take for the limit,, &c., that set down; thus, if in lat. 48° N., decl. 10° N., the time be doubtful 3o, we must take the alt. within of 28m, and that is, 9m from noon.

When the ame from noon, of observation, exceeds the limits set down, the error of the Reduction (caused by 1m error in the time) will exceed 2' in the same proportion; thus, in the above case, if the alt. be observed 56 from noon, the error of 1m in the time will cause 4′ error in the Reduction.

The time in the Table is that hour-angle, nearly, at which the number of minutes (of time) is equal to the number of minutes (of arc) in the Reduction.

To find this Hour-Angle. To the constant 0.4771, add the log. from Table 70; the sum is the prop. log. of the hour-angle required, in time.*

TABLE 48. VALUE OF THE REDUCTION AT WHICH THE SECOND
REDUCTION AMOUNTS TO l'.

The Table contains, against each Mer. Alt. under 85°, that value of the Reduction at which the 2d Reduction amounts to l'; and therefore shews whether it is necessary or not to compute the latter.

Ex. Suppose the mer. alt. 68° and the (first) Red. computed to be 47', then the error of omitting the 2d Red. cannot amount to 1'; but if the 1st Red. were 54', the omission of the 2d Red. would cause an error of more than 1'.

One eighth of the quantity in this Table is that (1st) Reduction at which the 2d Red. amounts to 1".

Thus, in Ex. No. 707, p. 234, the mer. alt. is 60°, the value of the 1st Red. in the Table is 1° 3', 1-8th of which is 8'; hence, if the Red. exceed 8', the 2d Red. will exceed 1".

To compute a Term.

To the constant 6.7648 (the sin. of 2′), add the log. cot. of the mer. alt.; half the sum (preserving 10 in the index) is the log. sine of the reduction required.

To find the time from noon, or the hour-angle. to which this (1st) Reduction corresponds: from the log. sine of the Red. subtract the log. in Table 70, the remainder is the log. sine square of the time or hour-angle required.

Ex. 1. Lat. 60° N., decl. 14° N. (mer. alt. 44°), Red. 1° 24′; 8.388-0130 the sin. sq. of 1h 1m 53, the hour-angle required.

8.258,

Ex. 2. Lat. 29° N., decl. 17° S. (mer. alt. 44°), Red. 1° 24′, gives oh 47m 3o.

These precepts concerning the Reductions are, of course, merely approximations near enough in practice.

TABLES 49 AND 50. FOR COMPUTING THE REDUCTION TO THE
MERIDIAN IN SECONDS.

The seconds forming part of the 1st Reduction (Table 49) are taken out to the min. and sec. of the hour-angle. When the sun is observed in the forenoon, the Table is entered with the time from midnight, for convenience.

* Mr. Towson has constructed convenient tables for reducing an alt. observed near the merid. to the mer. alt., which are published by the Hydrographic Office.