To compute the successive Latitudes at which the Ship should arrive :

Since the several successive polar angles, obtained as above, evidently reduce the two right angled spherical triangles A F S and B F S, into a series of right angled spherical triangles, to each of which the perpendicular FS is common; therefore, in each triangle of this series we have the perpendicular and the angle adjacent, to find the hypothenuse, or co-latitude. Thus, in the right angled spherical triangle F S 1, right angled at F, given the perpendicular F S = 29:5:51%, and the polar angle FS1= 63:4:5; to find the hypothenuse, or co-latitude S I ;-In the right angled spherical triangle F S 2, given the perpendicular F S 29:551%, and the polar angle F S 2 = 58:4.5"; to find the hypothenuse, or colatitude S 2, &c. &c. &c.

Hence, by right angled spherical trigonometry, Problem IV., page 188,

To the co-latitude

S1 = 50.51.36 Log. co-tangent = 9.910540

39: 8:24 S. at which the ship should arrive.

To find the Hypothenuse, or Co-latitude S 2 :—

As the perpendicular FS = 29? 551? Log. co-tang = 10.254507* Is to the radius = 90. 0. 0 Log. sine = So is the polar angle F S 2 = 58. 4. 5 Log. co-sine =

10.000000

9.723383

To the co-latitude S 2 =
Second latitude =

46.27.28 Log. co-tang. =

9.977890

43:32:32" S. at which the ship should arrive.

Hence, the first latitude at which the ship should arrive is 39:8:24" S.; and the second latitude 43:32:32" S.-And since it is the latitude, and not its complement that is required; therefore, if the log. tangent of the sum of the three logs. be taken, it will give the latitude direct; and, by rejecting the radius from the calculation, the work will be considerably facilitated.-Proceeding in this manner, the several successive latitudes cor

* The log. co-tangent is used, so as to save the trouble of finding the arithmetical complement of the log. tangent.

responding to the proposed alterations of longitude will be found as shown in the 3d column of the following Table.

Now, let the several successive longitudes be arranged (agreeably to the proposed change, and to the measure of the corresponding polar angles), as exhibited in the 2d column of the following Table; and find the difference between every two adjacent longitudes, as shown in the 4th column of that Table.-Find the difference between every two adjacent latitudes, and place those differences in the 5th column.-Find the meridional parts corresponding to the several successive latitudes, which place in the 6th column; and find the difference between every two adjacent meridional latitudes, as shown in the 7th column.-Then find, by Mercator's sailing, Problem I., page 238, the respective courses and distances between the several successive latitudes and longitudes; and, let the courses and distances, so found, be arranged in regular succession, as exhibited in the two last columns of the Table.-Then, will this Table be duly prepared for navigating a ship on the arc of a great circle, agreeably to the proposed alterations of longitude.—And, should the sum of the several successive differences of longitude, contained in the Table, coincide with the whole difference of longitude between the two given places ;-the sum of the several successive differences of latitude be found to agree with the whole difference of latitude comprehended under the mean, or highest latitude, and its corresponding extremes ;-the sum of the several meridional differences of latitude to be equal to the whole meridional difference of latitude corresponding to the mean, or highest latitude, and its respective extremes,—and the sum of the several successive distances to make up the whole spherical distance (or nearly so,) between the two given places; then, those several concurring equalities will be so many satisfactory proofs that the work is right.

Note. In the spherical track laid down in the following Table, it is presumed that there is not any land to intercept a ship's progress: but since this track will take the navigator into high southern latitudes, it will be indispensably necessary to keep a sharp look-out at all times, particularly during the night, so as to guard against any of the ice-bergs that may be floating to the northward of the Antarctic circle;-though, if the track be made in the months of November, December, January, or February, there will be no real night or darkness to experience; for during these months there will be a strong twilight between the latitudes of 53, and 61 degrees south; and thus the navigating at night will be attended with very little more danger than that by day.

A TABLE,

Now, the sum of the several successive differences of longitude, viz. 8212 miles, coincides exactly with the whole difference of longitude between the two given places; the sum of the successive differences of latitude = 3295.30 miles, agrees with the whole difference of latitude comprehended under the highest latitude at which the ship should arrive, and the latitudes of the two given places; viz. 33:52:07 S; 60:54:9 S, and 33:1:0 S:-and, the sum of the several meridional differences of latitude = 5011.80 miles, makes up the whole difference of latitude corresponding to the highest latitude and the latitudes of its respective extremes :-these several concurrences or agreements, form, therefore, the most satisfactory and indisputable proofs that the work has been properly conducted.

The sum of the several distances, measured on the respective rhumblines intercepted between the successive longitudes and latitudes, as given in the last column of the Table, is 6108.73 miles ;-but the true spherical distance on the arc of a great circle is 6107.87 miles; the difference, therefore, is only 0.86, or a little more than three-fourths of a mile; which is a very close approximation in the measure of so great an arc.

The distance by Mercator's sailing is 6853. 16 miles; which is 745.29, or about 745 miles more than by great circle sailing.--Hence, it is evident that the shortest and most direct route from Port Jackson to Valparaiso is by the latitude of 60:54.9" S; and that the ship must make, successively, the several longitudes and latitudes contained in the 2nd and 3rd columns of the Table, in the same manner precisely, as if they were so many ports or places of rendezvous, at which she was directed to touch.

.

The first course, therefore, from Port Jackson to Valparaiso, is S. 37:18 E. distance 398 miles; which will bring the ship to longitude 156:16:0 E. and latitude 39:8:24" S;-the second course is S. 40:26 E. distance 347 miles; which brings the ship to longitude 161:16:0 E. and latitude 43:32:32" S;-the third course is S. 43:53 E. distance 304 miles, which brings the ship to 166:16:0′′ E. and latitude 47:11:36 S. &c. &c. &c.

Whence it is evident that Captain Gambier saved a distance of 745 miles in that judicious and well-planned route: And this saving of distance should be an object of the highest consideration to every captain who wishes to recruit the strength and spirits of his ship's company by a generous supply of fresh provisions after a fatiguing and tedious voyage; the measure of which falls very little short of being equal to one-fourth of the earth's circumference as taken under the equator, or to the one-third of that circumference if taken under the given parallel of latitude.

SOLUTION OF PROBLEMS IN NAUTICAL ASTRONOMY.

NAUTICAL ASTRONOMY is the method of finding, by celestial observation, the latitude and longitude of a ship at sea; the variation of the compass; the mean time at ship; the altitudes of the heavenly bodies, &c. &c. &c.-Or, it is that branch of mathematical astronomy which shows how to solve all the important Problems in navigation by means of spherical operations, when the altitudes, or distances of the celestial objects are under consideration.

Before entering upon the Astronomical part of this work, it appears to be indispensably necessary that a few observations should be made relative to the new and most important element called "Sidercal Time," which is given in page II. of the month in the Nautical Almanac : for this essentially useful element enters so generally into all the calculations in which the moon, stars, and planets are concerned, that, without a previous knowledge of the principles upon which it is founded, and of the uses to which it may be applied, it would be in vain to attempt the solution of a Problem which has any relation to mean time.—Indeed, unless the nature of "Sidereal Time" be clearly understood, even the most simple of the preliminary problems will appear to be incomprehensible.

Although it is my intention to associate a concise explanation with every problem that may appear to demand a specific illustration, yet, with the view of elucidating the elementary expressions in the Nautical Almanac, and of giving the young navigator a competent knowledge of the above-named important element, I shall here place before his view the following

General Definitions.

1. The grateful phenomenon of Day and Night is produced by the circumrotation of the earth from west to east, round an imaginary line called the axis, the extremities of which are named Poles: the extremity which lies towards the most northern part of Europe is denominated the North Pole, and its opposite, the South Pole.

2. The Celestial Poles are two immovable points, round which the heavens seem to turn, or move apparently from east to west: they may be considered as the extremities of the earth's axis produced to the firmament, or to the sphere of the fixed stars. The apparent motion of the heavens from east to west, is occasioned by the actual diurnal motion of the earth round its axis from west to east, which gives an illusive motion, in a contrary direction, to all the heavenly bodies.

3. The Equator is a great circle on the earth, every point of which is