The moon passed the merid. of Greenwich Feb. 19 (Naut. Alm.) 6h 56m 0s

Hence 1h 3m is the moon's retardation in 25h 3m, and, by proportion using for the longitude 40° W., its value in time 2h 40m, we have,

25h 3m: 1h 3m :: 2h 40m : 0h6m 423;

that is, the moon is retarded 6m 42s in passing from the meridian of Greenwich to that of the ship, and, therefore, instead of the apparent time at the ship being 6h 56m, as it necessarily would be if there were no retardation, it will be 6m 42o later.

Hence

Having thus got the apparent time at Greenwich when the observation was made, we may, by a reference to the Nautical Almanack and a subsequent proportion, find the moon's declination at that time: thus

Moon's declination at Greenwich, Feb. 19 at noon

26°38′ 17′′

Feb. 19 at midn.

26 54 39

16 22.

Change of declination in 12 hours

... 12h: 9h 42m 42s :: 16′ 22′′ : 13′ 15′′;

hence 13′ 15′′ is the amount of the change of declination, from noon to 9h 43m, on the supposition, however, that the motion of the moon in declination may be considered as equable during the twelve hours. But on account of the irregular motion of the moon, this supposition introduces a sensible error, which may however be corrected by means of the table of "Equation of second Differences," given in the Nautical Almanack, and explained in Dr. Maskelyne's accompanying " Explanation." The correct change of declination is thus found to be 14′ 16′′. But from the year 1833, the declination of the moon will be given in the Nautical

Almanack to every three hours, and the change for any shorter interval may then he obtained with the requisite accuracy by proportion, as above. Taking in the present case 14' 16" for the correct change, we have

Before we can find the proper correction for parallax, we must deduce

the apparent altitude of the centre,

Having thus reduced all the corrections to the time of observation, we readily obtain the true altitude, and thence the latitude as follows,

SCHOLIUM.

These examples will, no doubt, be found sufficient to put the student in possession of the method of applying the various corrections to the observed meridian altitude of a celestial object, in order to deduce from it the latitude of the ship. But it should be remarked, that in most works on Nautical Astronomy, subsidiary tables are inserted for the purpose of abridging some of the foregoing corrective operations; such tables, therefore, offer very acceptable aid to the practical navigator. The most esteemed works of this kind are Dr. Mackay's "Treatise on the Theory and Practice of finding the Longitude at Sea"; the "Nautical Tables" of J. De Mendoza Rios, and Mr. Riddle's book on Navigation and Nautical Astronomy.

It should also be observed here, that in the preceding examples the celestial object is supposed to be on the meridian above the pole; that is, to be higher than the elevated pole. But, if a meridian altitude be taken below the pole, which may be done if the object is circumpolar, or so near to the elevated pole as to perform its apparent daily revolution about it without passing below the horizon, then the latitude of the place will be equal to the sum of the true altitude, and the codeclination or polar distance of the object; for this sum will obviously measure the elevation of the pole above the horizon, which is equal to the latitude.

(79.) To determine the latitude at sea, by means of two altitudes of the sun, and the time between the observations.

In the preceding article we have shewn how to determine the latitude of the ship by the meridian altitude of the sun, or of any other heavenly body, whose declination may be found. But, as already remarked, the object we wish to observe may be obscured when it comes to the meridian, and this may happen for many days together, although it may be frequently visible at other times of the day. As therefore the opportunity for a meridian observation cannot be depended upon, it becomes an important problem to determine the latitude at sea, by observations made out of the meridian; and considerable attention has accordingly been paid, by scientific persons, to the method of finding the latitude by "double altitudes," and various tables have been computed to facilitate

the operation. But the direct method, by spherical trigonometry, though rather long, involving three spherical triangles, will be more readily remembered, and more easily applied by persons familiar with the rules and formulas of Trigonometry, than any indirect or approximative process; we shall therefore explain the direct method.

Let P be the elevated pole, Z the zenith of the ship, and S, S', the two places of the sun, when the altitudes are taken. Then, drawing the great circle arcs as in the figure, we shall have these given quantities, viz. the codeclinations PS, PS'; the coaltitudes ZS, ZS', and the hour angle SPS', which measures the interval between the observations; and the quantity sought is the colatitude ZP. Now, in the triangle PSS', we have given two sides and the included angle to find the third side SS', and one of the remaining angles, say the angle PSS'. In the triangle ZSS' we have given the three sides to find the angle S'SZ; having then the angles PSS', S'SZ, the angle ZSP becomes known, so that we have, lastly, two sides and the included angle in the triangle ZSP, to find the third side ZP.

Before the application of the trigonometrical process, the observed altitudes must, of course, be reduced to the true altitudes, as in the preceding examples. Moreover, as the ship most probably sails during the interval of the observation, an additional reduction becomes necessary; the first altitude must be reduced to what it would have been if taken at the place where the second was taken: this correction will be known if we know the number of minutes the ship has sailed directly towards or directly from the sun, upon leaving the place where the first observation was made. To find this, take the angle included between the ship's course and the sun's bearing, at the first observation; and considering this angle as a course, and the distance sailed as the corresponding distance, find by the traverse table, or by the operation of plane sailing, the difference of latitude, which will be the amount of the approach to, or departure from, the sun. This must be added to the first altitude if the angle is less than 90°, because the ship will have approached towards the sun; but it must be subtracted when the angle exceeds 90°. If the angle is 90° no correction for the ship's change of place will be necessary.

The truth of this correction will be immediately seen by considering that if the sun's centre were the elevated pole, what is in reality the coaltitude would then be the colatitude, and, therefore, that, by whatever quantity this latter is increased or diminished by the ship's motion, on the one hypothesis, by the same quantity will the former be increased or diminished on the other hypothesis.

Where great accuracy is aimed at, account should be taken of the ship's change of longitude during the interval of the observations; when converted into time it must be added to the interval of time between the observations when the ship has sailed eastward, and subtracted when she has sailed westward. This correction is very easily applied. Having thus mentioned the necessary preparative corrections, we shall now give an example of the trigonometrical operation.

EXAMPLES.

Let the two zenith distances corrected be (see last fig.) ZS=73° 54′ 13′′, ZS′ = 47° 42′ 51′′, the corresponding declinations 8° 18′ and 8° 15′ north, and the interval of time three hours; to determine the latitude.

Considering SS' to be the base of an isosceles spherical triangle, of which one of the equal sides is (PS + PS') = 81° 43′ 30′′, and the vertical angle equal to 3 or 45°, let the perpendicular PM be drawn, then we have in the triangle PMS right angled at M, PS=81° 43′ 30′′, 45°

and P =

2

22° 30′; given to find SMSS' as follows.