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. 33:52:0 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 rhumb-lines 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 161:160 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.

Whence it is evident that Captain Gambier saved a distance of 7454 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 apparent 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.

Introductory Problems to the Science of Nautical Astronomy.

PROBLEM I.

To convert Longitude or Parts of the Equator into Time.

RULE.

Multiply the given degrees by 4, and the product will be the corresponding time-observing that seconds multiplied by 4 produce thirds; minutes multiplied by 4 produce seconds, and degrees multiplied by 4 produce minutes, which, divided by 60, give hours, &c.

To convert Time into Longitude, or Parts of the Equator.

RULE.

Reduce the hours to minutes, to which add the odd minutes, if any; then, the minutes divided by 4 give degrees; the seconds divided by 4 give minutes, and the thirds divided by 4 give seconds.

Note. The two preceding Problems are readily solved by means of Table I-see explanation, pages 1 and 2.

PROBLEM III.

Given the Time under any known Meridian, to find the corresponding Time at Greenwich.

RULE.

Let the given time be reckoned from the preceding noon, to which apply the longitude of the place in time (reduced by Problem I., as above,) by addition if it be west, or subtraction if east; and the sum, or difference will be the corresponding time at Greenwich.

Given the Time at Greenwich, to find the corresponding Time under a

known Meridian.

RULE.

Let the given time be reckoned from the preceding noon, to which apply the longitude of the place in time (reduced by Problem I. as above,) by addition if it be east, or subtraction if west; and the sum, or difference will be the corresponding time under the given meridian.

Example 1.

When may the emersion of the first satellite of Jupiter be observed at Trincomalee, in longitude 81:22. E., which, by the Nautical Almanac, happens at Greenwich, March 4th, 1825, at 9:928: ?

Example 2.

When may the immersion of the first satellite of Jupiter be observed at Port Royal, Jamaica, in longitude 76:52:30" W., which, by the Nautical Almanac, happens at Greenwich Nov. 1st. 1825, at 18:1745 ?

Apparent time of immersion at Greenwich =

18:17:45:

Longitude of Port Royal 76:52:30" W., in time = 5. 7.30

To reduce the Sun's Longitude, Right Ascension, and Declination; and, also, the Equation of Time, as given in the Nautical Almanac, to any other Meridian, and to any time under that Meridian.

RULE.

Let the given apparent time at ship, or place, be always reckoned from the preceding noon; to which apply the longitude in time (reduced by Problem I., page 296,) by addition if it be west, or subtraction if east, and the sum or difference will be the corresponding time at Greenwich.

Take, from page II. of the month in the Nautical Almanac, the sun's longitude, right ascension and declination, or the equation of time, as the case may be, for the noons immediately preceding and following the Greenwich time, and find their difference; then,

To the proportional log. of this difference, add the proportional log. of the Greenwich time (reckoning the hours as minutes, and the minutes as seconds), and the constant log. 9. 1249; * the sum of these three logs. rejecting 10 from the index, will be the proportional log. of a correction which is always to be added to the sun's longitude, or right ascension, at the noon preceding the Greenwich time; but to be applied by addition or subtraction to the sun's declination, or the equation of time at that noon, according as these elements may be increasing or decreasing.

Remark. Since the daily difference of the equation of time is expressed, in the Nautical Almanac, in seconds and tenths of a second; if, therefore, these tenths be multiplied by 6 they will be reduced to thirds: hence, the daily difference will be obtained in seconds and thirds.-Now, if those seconds and thirds be esteemed as minutes and seconds, the operation of reducing the equation of time will become as simple as that of the sun's declination ;-observing, however, that the minutes and seconds, corresponding to the sum of the three logs., are to be considered as seconds and thirds.

* This is the arithmetical complement of the proportional log. of 24 hours esteemed as minutes.