EXPLANATION OF THE TABLES.

TABLES 1, 2.

TRAVERSE TABLES, OR SOLUTIONS OF PLANE RIGHT TRIANGLES.

Tables 1 and 2 were calculated by the natural sines taken from the fourth edition of Sherwin's Logarithms, which were previously examined, by differences; when the proof-sheets of the first edition were examined the numbers were again calculated by the natural sines in the second edition of Hutton's Logarithms; and if any difference was found, the numbers were calculated a third time by Taylor's Logarithms.

The first table contains the difference of latitude and departure corresponding to distances not exceeding 300, and for courses to every quarter-point of the compass. Table 2 is of the same nature and extent, but for courses consisting of whole degrees. The manner of using these tables is particularly explained under the article of Inspection, in the different Problems of Plane, Middle Latitude, and Mercator's Sailing.

These tables may also be employed in the solution of right-angled triangles, as may be seen in Art. 112, Chap. III, Part I.

TABLE 3.

MERIDIONAL PARTS.

This table contains the meridional parts, or increased latitudes, for every degree and minute to 87°, calculated by the following formula, viz:

The results are tabulated to one decimal place, being sufficient for the ordinary problems of navigation. The practical application of this table is illustrated in Art. 66, Chap. II, Part I, and in the various problems of Mercator's Sailing, Chap. III, Part I.

TABLE 4.

This table gives the length of a degree in both latitude and longitude at each parallel of latitude on the earth's surface.

TABLE 5.

This table has been calculated to facilitate the operation of finding the distance from an object by two bearings, having the distance run and course. In the first part of the table the arguments are given in points; in the second part, in degrees.

It is illustrated in Art. 148, Chap. IV, Part I.

TABLE 6.

This table contains the distances at which any object is visible at sea calculated by the formula

d = 1.317 √x in feet,

in which is the distance in statute miles, x the height of the eye or the object in feet.
The explanations and use of this table are given in the Useful Problems of the Appendix.

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TABLE 7.

To reduce Longitude into Time, and the contrary.-In the first column of this table are contained degrees and minutes of longitude, in the second the corresponding hours and minutes, or minutes and seconds of time; the other columns are a continuation of the first and second respectively. The use of this table will evidently appear by a few examples.

To find the Time of the Sun's Rising and Setting, and the Length of the Day and Night.

RULE. Find the sun's declination at the top of the table, and the latitude in either side column; under the former, and opposite the latter, will be the time of the sun's setting if the latitude and declination are of the same name, but the time of rising if of different names. The time of rising, subtracted from 12 hours, will give the time of setting; or the time of setting, subtracted from 12 hours, will give the time of rising. The time of rising, being doubled, will give the length of the night; and the time of setting, being doubled, will give the length of the day.

For the SUN the H. A. is the app. time of rising or setting.

For the Moon or a STAR. Find the app. time (or mean time, as required) of the meridian passage. Then, for approximate time at rising, subtract the hour angle from the time of meridian passage (increased by 24h if necessary); for approximate time of setting add them together, rejecting 24h in the result if it exceeds that.

It may be noted that the numbers of Table 10 were calculated for the moment the sun's centre appears in the true horizon; allowance ought to be made for the dip, parallax, and refraction, by which the sun and stars, when near the horizon, appear in general to be elevated above half a degree above their true place, and the moon as much below her true place.

TABLE 11.

This table was calculated by proportioning the daily variation of the time of the moon's passing the meridian. The numbers taken from this table are to be added to the time at Greenwich, in West longitude, but subtracted in East.

These are tables of proportional parts;

TABLES 12, 13.

For finding the variation of the Sun's Right Ascension, of the Declination, of the Equation of Time, or of the Moon's Right Ascension, in any number of minutes of time, the Horary Motion being given at the top of the page in seconds, and the number of minutes of time in the side column ;

Also, for finding the variation of the Moon's Declination in seconds of time; the motion in one minute being given at the top, and the numbers in the side column being taken for seconds;

Also, for finding the Sun's Right Ascension for any given number of hours.

TABLE 14.

This table contains the dip of the sea horizon, calculated by the formula

D=58.8 √ F,

in which Fheight of the eye above the level of the sea in feet.

It is explained in Art. 251, Chap. V, Part II.

TABLE 15.

The table contains the dip for various distances and heights, calculated by this rule,

in which D represents the dip in miles or minutes, d the distance of the land in sea miles, and the heigh of the eye of the observer in feet.

TABLE 16.

The table contains the Sun's parallax in altitude calculated by the formula

Parallax in altitude of a planet is found by entering at the top with the planet's horizontal parallax, and at the side with the altitude.

TABLE 18.

The table gives the augmentation of the moon's semi-diameter calculated by the formula,

The table contains the augmentation of the moon's horizontal parallax, or the correction to reduce the moon's equatorial horizontal parallax to that point of the earth's axis which lies in the vertical of the observer in any given latitude, computed by the formulas

table.

Mean refraction, reduced from Bessel's tables, to barometer 30in and thermometer 50o.

TABLES 21, 22.

Corrections of the mean refraction for the height of the barometer and thermometer, deduced also from Bessel's

TABLE 23.

The table contains the correction of the moon's altitude for parallax and refraction, corresponding to the parallax 57' 30".

TABLE 24.

This table contains the correction to be applied to the moon's apparent altitude to each minute of horizontal parallax and every 10' of altitude from 50, for height of barometer 30 and Fahrenheit thermometer 50°.

For seconds of parallax, enter the table abreast the approximate correction and find the seconds of hor. parallax, viz: the tens of seconds at the side and the units at the top. Under the latter and opposite the former will be the seconds to add to the correction.

For minutes of altitude, take the seconds from the extreme right of the page, and apply them as there directed. The table was computed by correcting each app. altitude (of the centre) for refraction; then log. sec. alt. + P. L. Hor. Par. = P. L. par. in altitude. From the parallax corresponding to this proportional logarithm subtract the refraction; the remainder is the correction of altitude.

TABLE 25.

The table gives the variation of the altitude of any heavenly body arising from a change of 100" in the declination. It is useful in finding the latitude by two altitudes of a body when the declination changes during the interval of elapsed time.

If the change move the body toward the elevated pole, apply the correction to the altitude with the signs in the table; otherwise change the signs.

TABLE 26.

Table 26 contains the variation of the altitude of any heavenly body, for one minute of time from noon, for various degrees of latitude and declination. The following method was used in constructing the table: A and B were calculated for each degree of declination by these formulas:

and then the correction of the table corresponding to the zenith distance Z (

lat. dec.) was found by this formula: AX cotan ZB. To facilitate the computation of these numbers, a table of the products of A by the whole numbers from 1 to 9 was calculated.

TABLE 27.

Table 27 contains the squares of the minutes and parts of a minute of time corresponding to every second from 0 to 12m 598. This requires no explanation.

The manner of using the two preceding tables is exemplified in the body of the work in finding the latitude by reduction to the meridian, Art. 278, Chap. VII, Part II.

TABLE 28, A, B, C, D.

For finding the Latitude of a Place by Altitudes of Polaris.

The formula* on which these tables are based is

L= hp cos t + 1⁄2 på sin 1"' sin2 t tan h

in which

sin

cos t sint + p sin3 1" sin' t tan h;

Table A contains for the declination 88° 40', or fo= = 1° 20" 4800", the first correction,

A:

-

=

Po cost- po sin2 1" cos t sin2 t;
Argument, the hour angle of the star, or 24h
the hour angle.

Table B contains the second correction,

B = 1⁄2 po sin 1′′ sin2 t tan h + 1⁄2 p。 sin3 1“ sin1 t, tan3 h ;

Arguments, the true altitude of the star and the hour angle, or 24h - the hour angle. This correction is always additive.

Table C contains the third correction,

C = 1⁄2 (p2 — p2。) sin 1" sin2 t tan h;

Arguments, B and the declination of the star from 88° 39′ 20′′ to 88° 41′ 20′′. Table D contains the fourth correction,

Arguments, A and the declination of the star from 88° 39′ 20′′ to 88° 41′ 20′′.

The quantities are given to the nearest o'.1: a. placed after some of them indicates a doubt between the figure given and the next highest, or that the correct value is o".05 greater than that given. Thus, 3.7. indicates the actual value 3.75.

The refractions employed are BESSEL'S, and were taken from his table (Astronomische Untersuchungen, Vol I, p. 200), which gives directly K =rtan h.

The application of this table will be found in "Lunar Distances," Arts. 301 to 311, Chap. VIII, Part II.

TABLE 30.

Logs of A, B, C, and D, for computing the First Correction of the Lunar Distance, computed by the formulas

Η

Ah

ΔΗ

log K' =

.000126,

sun's, planet's, or star's apparent altitude (denoted in the tables by O or 's App. Alt.),
difference of 's apparent and true altitudes,

difference of O's or 's apparent and true altitudes,

*CHAUVENET'S Spherical and Practical Astronomy, Vol. I, p. 256.