These two examples will serve for specimens of the corrections to be applied to an observed altitude, in order to deduce from it the true altitude of the body's centre. In the case of the moon, the corrections, when the utmost accuracy is sought, are rather numerous, as the last example shows.

But in finding the latitude at sea, it is usual to dispense with some of these, more especially with the corrections for temperature, for the contraction of the moon's semidiameter, and for the spheroidal figure of the earth; because an error of a few seconds in the true altitude will introduce no error worth noticing in the resulting latitude. When, however, the object of the observer is to deduce the longitude of the ship, all the data, furnished by observation, should be as accurate as possible; for the problem is one of such delicacy that by neglecting to allow for the influence of temperature would alone introduce in some cases an error of from 30 to 40 miles in the longitude.

When the object observed is a star, several of the foregoing corrections vanish; the only corrections in this case requisite are those for dip and refraction, modified as usual for the temperature.

111. To determine the latitude at sea from the meridian altitude of any celestial object whose declination is known.

The determination of the latitude, by a meridian altitude, is the most

* See p. 275.

easy and safe method of finding that element; the observations and subsequent calculations being few, are readily performed, and with but little liability to error in the result; this method, therefore, is always to be preferred at sea, unless clouds obscure the meridian whilst other portions of the heavens are left visible.

The declination of the object observed is supposed to be given in the Nautical Almanac, when it culminates or makes its meridian transit at Greenwich; its declination when it culminates at the meridian of a ship, may be found by means of the longitude by account,* which will always be sufficiently accurate for this purpose, although it should differ very considerably from the true longitude, because declination changes so slowly that even an error of an hour in the longitude would cause an error in the declination too small to deserve notice.

The declination being the distance of the object from the equator, and the observed altitude, properly corrected, being the distance of the same object from the ship's zenith, the distance of the zenith from the equator, that is, the latitude, immediately becomes known.

Let the full circle in the diagram be the meridian.

1. Let s be the object observed, the zenith z being to the north of it, and the object itself north of the equator, EQ, then the latitude Ez is equal to the zenith distance, or co-altitude zs + the declination Es, and it is north.

N

2. Let s' be the object, still north of the equator, but so posited that the zenith is south of it, then the latitude Ez is equal to the difference between the zenith distance s'z, and declination s'E, and is still north. 3. Let now the object be at s", south of the equator, and the zenith to the north of the object, then the latitude Ez is equal to the difference between the zenith distance s'z and declination s'E, and it is north.

We have here assumed the north to be the elevated pole, but if the south be the elevated pole, then we must write south for north, and north for south. Hence the following rule for all cases.

S

E

S

*For this purpose the variation of the declination in 1 hour, which is given in the Nautical Almanac for the sun, must be multiplied by the longitude in hours and fractions of an hour, and the product added or subtracted will produce the declination at the time of meridian transit at the ship..

Call the zenith distance north or south, according as the zenith is north or south of the object.

If the zenith distance and declination be of the same name, that is, both north or both south, their sum will be the latitude; but if of different names, their difference will be the latitude, of the same name as the greater.

EXAMPLE.

1. Ship Admiral, from New York to Havre, at sea Jan. 4th, 1850. Longitude 25° W.

12 29

Allowing for semidiameter (Dip 3' 26") parallax, &c. +
Required the latitude.

Ans. 49° 57′ 29′′ N.

3. On the 1st of January, 1850, the meridian altitude of Capella was 27° 35', the zenith being south of the star, and the height of the eye 22 feet; required the latitude.

* This is obtained by taking out from the Nautical Almanac, the declination for apparent noon, p. I., which is 22° 44' 5'2. Then computing the change in declination for 13 hours, the time in which the sun is passing from the meridian of Greenwich to that of the ship in long. 25° W., by multiplying the number 16" 38, found in the column in the Almanac entitled Diff. for 1 hour by 13. The product 27"-3, subtracted from 22° 44' 5''•2, because the declination is decreasing will give the declination at the meridian transit of the sun at the ship.

4. Suppose that the altitude of the moon, as given in Example 2, p. 277, was observed when the moon was upon the meridian, required the latitude of the place of observation.

The true altitude of the moon's centre being known, after applying the corrections as at p. 278, it remains to find her declination at the instant of observation. The Nautical Almanac gives the moon's declination for every even hour of the day of every day, on pages V. to XII. of each month, and the variation in declination for 10" of time. The required declination would therefore be computed as follows:,

D's dec. June 23d, at 16' (Nautical Almanac),

Diff. dec. for 41",*

19° 36' 48' 8

1 35

If the time of observation were not known, it could be computed from

the fact that the moon is on the meridian.

That is in 24 48 7° the moon is retarded in coming to the meridian, by her proper motion from W. to E. 48m 7

.. 24 48 7: 48m 7:: 4 56m 4o† : 9′′ : 39′

*The time of obs. was 40m 59 past 6 or nearly 41m. The Naut. Alm. gives 23" 17 diff. of dec. for 10m .. 10m: 2317 :: 41: 1′ 34′′ the change in dec. in 41m, which as dec. is increasing must be added. Second differences are not used.

+ This is the true alt. of the D's centre from the centre of the earth, p. 279. This is the longitude of the place of observation.

This last number is the retardation of the moon in passing from the meridian of Greenwich to that of the place of observation. The moon having crossed the meridian at Greenwich at 11" 33" on the 23d, will cross that of the station 9" 39 later, so that the time of meridian transit at the station will be 11" 36" 39*.*

It saves trouble to note the time of meridian transit by a watch, or still better by a chronometer, keeping Greenwich time.

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. Bowditch's Navigator is the most complete work of the kind.

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.f

112. 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 shown 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

This differs slightly from the time of observation given. The moon changes sc rapidly in declination that her greatest altitude is not always the meridian altitude.

That the elevation of the pole above the horizon is equal to the latitude of the place is evident from the fact that the zenith is 90° from the horizon, and the pole 90 from the equator.