From these, the Greenwich mean time is to be deduced in the same manner as before. It will be proper, however, to find the mean time at ship by a set of altitudes, taken as soon as possible before or after those altitudes, which are to be employed in correcting the distance. It seldom happens but the altitudes of at least one of the objects, may be observed at the same time as the distances; in this case, it is generally proper to observe the altitudes of the Sun or Star along with the distances, and then deduce the altitude of the Moon, as in the foregoing Example. (See Art. 6, p. 18).

To form an estimate of the respective values of two sets of observations of objects on opposite sides of the Moon, or of different objects on the same side, compare the relative intervals of mean time obtained by observation, with the interval estimated as in the following example, taken from pp. 20 and 21.

As the interval of time measured by the hour angles, and that estimated, agree nearly, the hour angles may be esteemed as correct. The interval by distances is too small, being 1m. 9s., or 17' of longitude less than the estimated interval; but this difference need not be regarded, and the mean of the two sets may be taken for the longitude. Had the difference been double that amount, or more, the second set would have been entitled to the preference, the pro log. from the N. A. being of the first distance 3429=1h. 21m. 44s., and the second 2974-1h. 30m. 45s. (See N. A. p. 574, 2d paragraph.)

ON FINDING THE

LONGITUDE BY CHRONOMETERS.

THIS method of finding the longitude depends on the same principle as the lunar method, that is, on being able to find the respective times at two meridians, for the same instant of absolute time, when the difference of these times will give the difference of longitude between the two meridians. For example:

Suppose a Chronometer that keeps mean time exactly, be set to mean time at Greenwich, and then taken to another meridian, where the mean time is found, by observation, to be 4 hours at the instant that the time by the Chronometer is only 2 hours; we know that the place of observation is 30° E. of Greenwich, because the time at that place is 2 hours farther advanced than the time at Greenwich: but if the time shewn by this Chronometer were 4 hours at the instant, the mean time found, by observation, is only 2 hours, then the longitude of the place is 30° W. of Greenwich, because the time at Greenwich is 2 hours farther advanced, than the time at the place of observation.

A Chronometer generally deviates something from mean time in its rate of going; the portion of time which it gains or loses on mean time, during 24 hours, is called its Daily Rate, or simply the Rate, and what a Chronometer is fast or slow, for mean time at a given meridian, is called its Error for that meridian. Those who reckon the longitude from the meridian of Greenwich, should always have the errors of their Chronometers for that meridian. If the rate of a Chronometer and its error, for any particular time be known, the error for any other time is found by multiplying the rate by the number of days between the times. Thus, let the rate of a Chronometer be 5.4s. gaining, and it is found to be fast for mean time at Greenwich, at noon, on the 5th of June, Oh. 11m. 31s.; the error on the 2d of July, at noon, would be Oh. 13m. 57s.; for here the number of days elapsed is 27, and 5.4s. × 27 = 145 8s. or 2m. 26s., and Oh. 11m. 31s. + 2m. 26s. Oh. 13m. 57s.

It is evident, that when the error is wanted for any other time than noon at Greenwich, a proportional part of the daily rate must be

When only one or two Chronometers are used, the daily rate or rates should be carried on in the book in which the sights for time, &c. are worked. When there are two, or three, or more, the longitude may be deduced from the mean of their Greenwich dates, or the preference will be given to either, according to its performance, when in the maker's hands, and his reputation for skill in manufacture and adjustment.

When there are two, three, or more Chronometers, irregularities in their rates may be detected by comparison of each with the others, unless they should all alter their rates to the same amount. The difference of the daily comparison between any two watches should be the same as the difference of their daily rates given, otherwise one or the other has altered its rate, possibly both.

When two watches maintain the proper difference of daily comparison, and a third watch varies from both in an equal amount of difference of comparison, it may be presumed that the two have maintained their given rates, and that the third alone has altered; for it appears to be scarcely possible that two watches should be so exactly alike, as to alter their rates to the same amount from the same cause; for instance, a change of temperature may make three watches gain; but it has not been proved that three watches will each gain 2s., or any other amount; neither does it appear that they alter their rates, from the same cause, in a ratio to their respective given rates.

When a comparison book is kept, the Barometer and Thermometer (See Art. 6, p. 52), and state of the weather should be noted in it. The uneasy motion of a ship will, at times, have an effect upon the rates of Chronometers.

The following form for a comparison book may be extended for any number of watches:

As the above is only inserted for the sake of the form, it is unnecessary to make any

PROBLEM VII.

Having a Chronometer, of which the error for Mean Time at Greenwich is known, and also the Latitude of a place, to find the Longitude of that place.

RULE.

1. When the Sun, Moon, a Planet, or a known Star, is at a proper distance from the meridian, take a set of altitudes and note the corresponding times by the Chronometer; to the mean of the times by the Chronometer, apply its error for mean time at Greenwich, by addition or subtraction, according as it is too slow or too fast, the sum or difference will be the Mean Time at Greenwich when the observation is made.

2. The Mean Time at Greenwich obtained by Chronometer, is from midnight to noon following, according to civil reckoning, therefore to obtain the Astronomical Greenwich Date at any moment during that interval, 12 hours must be added to the time shewn by Chronometer. In the following example, the astronomical Greenwich date is 1d. 19h. 28m. 56s. If the Sun's declination and the equation of time were to be reduced by logarithms the Astronomical Time would be requisite. The Greenwich Date by Chronometer should always be employed to reduce the elements from the Nautical Almanac, (See Art. 2, Introduction.)

3. From the mean observed altitude, deduce the true altitude.

4. With the Latitude of the place, the true altitude, and the declination of the object, find the mean time by Problem I. or II. according as the altitude of the Sun, the Moon, a Planet, or a Star has been observed. The difference between this Mean Time and the Mean Time at Greenwich, is the Longitude of the Ship in time, and the Longitude will be east or west, according as the time at the Ship is greater or less than the time at Greenwich.

EXAMPLE I.

1844. January 2d., 9h., A. M., civil time, the following observations were taken to obtain the Long. by Chronometer, and the variation of the compass. Bar. 29.75, Ther. 68°, Sun's magnetic Azimuth, N. 108° E. Error of Chronometer 18m. 45s. slow for Greenwich mean time. The true course and distance run from the time of observation till apparent Noon, was E. S. 15', when the Lat. by observation was found to be 37° 0′ 30′′ S.

+ The seconds are rejected above as they would only make a difference of about 2s. in the horary angle, and for general purposes at sea, working to the nearest minute is sufficient.

Twelve hours are rejected from the apparent astronomical time by observation to compare it with the time shewn by Chronometer.

§ In South Lat. read the Sun's Azimuth by compass from the north point as above, and in North Lat. from the South, because the true Azimuths are calculated by this method, from the North in South Lat., and from the South in