* The paraila, being small, is here neglected, and the sun's semidiameter is supposed to be 16'. ↑ The declinations of these bright stars are given for every 10 days in the Nautical Almanac. When great accuracy is required, these declinations should be used instead of the numbers in Table VIII

We have observed, in the directions for finding the meridian altitude of an object, hat an error will arise if the ship be in motion, or the sun's declination vary. The amount of this correction may be estimated in the following manner:

Find the number of miles and tenths of a mile northing or southing made by the ship in one hour, and also the variation of the sun's declination in an hour, expressed also in miles and tenths. Add these together, if they both conspire to elevate or depress the sun; otherwise take their difference, which call the arc A. Find, in Table XXXII., the arc B, expressed in seconds, corresponding to the latitude and declination; then the arc A, divided by twice the arc B, will express the time in minutes from noon, when the greatest (or least) altitude is observed. Moreover, the square of the arc A, divided by four times the arc B, will be the number of seconds to be applied to the observed altitude to obtain the true altitude, which would have been observed if the ship had been at rest.

Thus, if the ship sail towards the sun south 11 miles per hour, and the declination increases northerly 1' per hour, we shall have A=11+1=12. If the latitude is 42° N., and the declination 2° S., we shall have by Table XXXII. B=2′′. In this case, the time from noon is 12=3 minutes, and the correction of altitude 144 — 18 seconds only.

The declination of this star is given for every day in the Nautical Almanac; when great accuracy is required, this declination should be used instead of that in Table VIII.

The refraction, being small, is neglected.

22

TO FIND THE LATITUDE BY A MERIDIAN ALTITUDE OF THE MOON.

THE latitude may be found at sea, by the moon's meridian altitude, more accurately than by any other method, except by the meridian altitude of the sun; but to do this, it is necessary to find the time of her passing the meridian, and her declination at that time. To facilitate these calculations, we have given the Tables XXVIII. and XXIX., the uses of which will evidently appear from the following rules and examples.

To find the mean time of the moon's passing the meridian.

Find, in the Nautical Almanac, the time of the moon's coming to the meridian of Greenwich for one day earlier than the sea account,* and also the time of her coming to the meridian of Greenwich the next day, when you are in west longitude, but the preceding day when in east longitude; take the difference between these times, with which you must enter the top column of Table XXVIII., and against the ship's longitude in the side column will be a number of minutes to be applied to the time taken from the Nautical Almanac, for the day immediately preceding the sea account, by adding when in west longitude, but subtracting when in east longitude; the sum or difference will be the mean time of passing the meridian of the given place.

EXAMPLE.

Required the time of the moon's passing the meridian of Philadelphia, April 19 1836, sea account.

The day preceding the sea account is April 18; on this day, the moon passed the meridian of Greenwich at 1h 55m.6, and, being in west longitude, we find the time of her passing the meridian the next day 2 43.0. The difference between these two times is 47.4, which is to be found at the top of Table XXVIII.; the nearest tabular number is 48m; under this, and opposite 75°, (the longitude of Philadelphia,, is the correction 10m, nearly, to be added to 1h 55m.6, to obtain the time of passing the meridian at Philadelphia, April 19a 2h 05m.6, sea account, or April 184 2 05 6, P. M., civil account.

To find the moon's declination when on the meridian.

Find the time of the moon's coming to the meridian as above; turn the ship's longitude into time by Table XXI., and add it thereto if in west longitude, but subtract it in east; the sum or difference will be the time at Greenwich. Take out the moon's declination from the Nautical Almanac, for the nearest hour preceding the Greenwich time, and also the variation for 10 minutes in the next column.

Taking the time one day earlier than the sea account, reduces it to astronomical time used in the Nautical Almanac. We may observe that the time of the moon's coming to the meridian, is given in the Nautical Almanac to tenths of a minute, instead of seconds of time. This is done to facilitate the calculation of the right ascension and declination, by using common decimal fractions instead of sexagesimals.

=

Longitude may be turned into time, without the help of Table XXI., by multiplying the degrees and minutes of the longitude by 4, and considering the product as minutes and seconds of time respectively; and, by the inverse process of dividing by 4, we may turn time into degrees, &c. Thus. 80° X 4 = 320m 5h 20m; and 15° 16' x 4 = 61m 04• — 1a [m 4•. In like manner, 1h 20m or 80m, being divided by 4, ges 20°, and 196m, being divided by 4, gives 49°, which agree with the table If the ship be furnishe with a chronometer, regulated for mean time at Greenwich, we may avoid the labor of this part of the operation by taking the time at Greenwich as shown by the chronometer, at the very moment when the meridian altitude of the moon is observed.

If the time at Greenwich fall exactly upon any hour, the declination can then be taken from the Nautical Almanac, by mere inspection, without any reduction. We may also remark, that the reduc tion of the declination for the minutes and tenths of a minute of time, can be found by means of Table

This variation is to be multiplied by the minutes and tenths of a minute which occut in the time at Greenwich; the product, being divided by 10, gives the correction of the declination taken from the Nautical Almanac, additive if that declination be increasing, subtractive if decreasing; the sum or difference will be the true declina tion at the time of passing the meridian.

NOTES.

1. By the above rule, the day of the month on which the moon passes the meridian must be taken one less than the sea account. When the longitude, turned into time, is added to the time of passing the meridian, and the hours of the same exceed 24h, you must subtract 24h, and add one to the day of the month; if the longitude be subtractive, and greater than the time of passing the meridian, you must, before the subtraction, add 24 hours to the time of passing the meridian, and subtract one from the day of the month; the sum or difference will be the time at Greenwich.

2. When the declination, taken from the Nautical Almanac for the nearest hour preceding the time at Greenwich, is decreasing, and the correction to be subtracted exceeds this declination, the difference of the two quantities will be the required declination, with a different name from that of the declination taken from the Nautical Almanac.

3. In the same manner we may find the declination for any other time of the day, by making use of the given time instead of the time of the moon's passing the meridian. In all these rules, the second differences of the moon's motion are neglected.

EXAMPLE.

Required the moon's declination at the time of her passing the meridian of Philadelphia, April 19, 1836, sea account.

The time of passing the meridian of Philadelphia was found, in the preceding example, to be April 191 2h 5m.6 sea account, or April 18 2h 5m.6 by astronomical account; adding this to the longitude of Philadelphia, in time 5h 1m nearly, we obtain the time at Greenwich, April 184 7h 6m.6. The declination in the Nautical Almanac for April 18d 7h is 21° 13′ 52" N., and the variation 89" for 10 minutes of time nearly; multiplying this by 6m.6, and dividing by 10m, we get 59, to be added to 21° 13′ 52", because the declination is increasing, and we obtain 21° 14′ 51′′ N. for the required declination at the time of the moon's passing the meridian of Philadelphia.

To find the latitude by the mon's meridian altitude, obtained by a fore observation.

At the time of the moon's passing the meridian, the altitude of her round limb must be observed, whether it be the upper or lower limb. This altitude must be corrected for the semidiameter, dip, parallax, and refraction, in order to obtain the central altitude; with which, and the declination, we may find the latitude by the same rules as we have used in finding the latitude from the sun's meridian altitude. In making these calculations, we must find, from the Nautical Almanac, the moon's semidiameter and horizontal parallax, corresponding to the time of observation, reduced to the meridian of Greenwich, which was used in computing the declination. The moon's semidiameter is to be increased by the correction in Table XV., and this augmented semidiameter is to be added to the observed altitude, if the moon's lower limb be observed; but if the upper limb be observed, we must subtract this augmented semidiameter from the moon's observed altitude, to obtain the central altitude. From this central altitude you must subtract the dip of the horizon, found in Table XIII., to obtain the apparent altitude. The correction for parallax and refraction is likewise to be added; this correction is easily found by means of Table XIX., by subtracting the tabular number corresponding to the moon's altitude and horizontal parallax from 59′ 42′′; the remainder will be the correction for parallax and refraction, which is to be added to the apparent central altitude, to obtain the true altitude; and, by subtracting this true altitude from 90°, we obtain the true zenith distance. With this and the declination, we deduce the latitude by the usual rules, similar to those given for the sun in pages 166, 167.

* In computing this table, the mean refraction is used; but, when very great accuracy is required, the true refraction ought to be used. The corrections arising from this cause may be obtained from Table XXXVI., and are to be applied to the above-found zenith distance, with the same signs as in

this table.

EXAMPLE I.

Suppose that, on the 27th of June, 1836, sea account, in the longitude of &f W. from Greenwich, the meridian altitude of the moon's upper limb was observed to be 40° 0', bearing south, the eye of the observer being elevated nineteen feet above the surface of the sea; required the true latitude.

June 27th, sea account, is June 26th by the Nautical Almanac; on this day the moon passes the meridian of Greenwich at 9h 55.9, mean time, and the next day at 10 59.8, the daily difference being 639. In Table XXV. II., under m, (which is the nearest number in the table to 63.9,) and opposite to the longit le 80°, stand 14; adding this to 9h 55m.9, we get 10h 09.9 for the time of passing the meridian at the place of observation.

D passes the merid.....June 26d 10h 10m
Ship's long. 80° W., in time, 5 20
Time at Greenwich....June 26 15 30
D's decli. June 26d 15h 23° 37′ 43.2 S.
Cor. for 30m is 30 x 8.798 + 4 23.9
Required declination.... 23 42 07 .1 S.

Here the variation of the declination for 10m is, by the Nautical Almanac, 87".98, or 8.798 for 1m. Multiplying this by 30, we get the correction for 30m, equal to 263.94, or 4' 23.9, as above. This is additive, because the declination is increasing. For the same time at Greenwich, we find D's hor. par. 60′ 58', and

EXAMPLE II.

49 35 14 N

23 42 07 S.

25 53 07 N

Suppose that, on the 27th September, 1836, sea account, in the longitude of 90° E., the meridian altitude of the moon's lower limb was observed to be 50° 0′, bearing south, the eye of the observer being seventeen feet above the surface of the sea, required the true latitude.

Sept. 27th, sea account, is Sept. 26th, astronomical account; on this day the moon passed the meridian of Greenwich at 13h 28.0, and the preceding day at 12" 42" .8, differing 452. In Table XXVIII., under 46m, (which is the nearest tabular number,) and opposite to 90°, are 11m, which, being subtracted from 13" 28", leaves 13h 17 for the time of passing the meridian of the place of observation. Subtracting the longitude 6, gives the corresponding time at Greenwich Sept. 26 7h 17m.

The latitude may also be obtained from the mocn's meridian altitude, by the following approximative method, which will vary but very little from the truth, except when the horizontal parallax and semidiameter are very large or very small :

Abridged approximative method of finding the latitude by the moon's meridiar altitude, obtained by a fore observation.

To the observed altitude of the moon's lower limb add 12'; but if her upper limb