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X = A & tan. L + B & tan. D, where
T = the interval of time expressed in hours;
L= the latitude of the place, minus when south;
D = the declination at noon, also minus when south;
8 = the double daily variation in declination in seconds, deduced from the noon of the preceding day to that of the following, minus when the sun is proceeding to the south; and
the required correction in seconds, A being minus when the time of noon is required.
The result is of course apparent noon, to which must be applied the equation of time, in order to compare a chronometer with
If the rate only of a chronometer is required, it can be obtained by observing the transits of a star on successive days, or by equal altitudes of the same star, on the same side of the meridian, on different evenings, as a star attains the same altitude after each interval of a sidereal day, which is 3m 56.91s less than a mean solar day; but if the refraction is not alike on the days of observation, a correction will be required.
By reading the azimuths, when the sun or a star has equal altitudes, we obtain the true meridian line, which will be again alluded to. Very frequently the afternoon altitude cannot be observed on account of intervening clouds, but the time can be calculated from the observed single altitude, as in the last problem.
TO FIND THE LONGITUDE.
THE usual method of finding the longitude at sea, is by comparing the local time found by observation, with that shown by a chronometer set to Greenwich time, and whose error and rate are
* See table 14, page 194.
known. The accuracy of the result depends of course upon the chronometer keeping its rate correctly, which cannot always be relied upon, and various methods have been resorted to, to render the solution of this most important problem independent of such uncertain data. Any celestial phenomenon which should be visible at the same predicted instant of time in different parts of the globe, would of course furnish the necessary standard of comparison, and all the methods in use for determining the longitude are based upon this foundation, but they are not generally practicable at sea, with the exception of that derived from the observed angular distances between the moon and the sun, or certain stars, which are calculated for every three hours of Greenwich time, and which lunar distance is measured with a sextant, or other reflecting instrument. Artificial signals have been resorted to as a means of ascertaining the difference of longitude between two observatories with considerable success.
In the philosophical transactions for 1826, is an account drawn up by Sir J. Herschel, of a series of observations made in the summer of 1825, for the purpose of connecting the royal observatories of Greenwich and Paris, and undertaken by the Board of Longitude, in conjunction with the French Minister of War. The signals were made by the explosion of small portions of gunpowder fired at a great elevation by means of rockets, from three stations, two on the French, and one on the English side of the Channel, and were observed at Greenwich and Paris, as well as at two intermediate places, Legnieres, and Fairlight-Down, near Hastings. The difference of longitude thus obtained 9′ 21.6", is supposed by Sir J. Herschel to be correct within one tenth of a second, and the observations were taken with such care, that those of the French and English observers at the intermediate stations only differed one-hundreth part of a second.
At page 198 also, of "Francœur's Godesie," is a description of similar operations for the purpose of ascertaining the difference of longitude between Paris and Strasburg. In operations of this nature all that is necessary is, that the rates of the chronometers used
* It is usual to have several chronometers on board, and to take the mean of those most to be depended upon. If one varies considerably from the others it is rejected.
should be uniform for the short period of time occupied by the transmission of the signals, which can in no case exceed one hour. Suppose A and B are two places, whose difference of longitude is required, and that they are too far distant to allow of one signal being seen from each
C and D are taken as intermediate stations, and the first signal, made at S, is observed from A and C, and the times noted; the second at S', is observed from C and D, some fixed number of minutes after, and then that at S" from D and B. Suppose these two intervals to have been five minutes each, then the difference of longitude = the time at A+ ten minutes - time at B, if A is to the eastward of B.
Everything in this operation depends upon the correct observation of the times, which should be kept in sidereal intervals, or reduced to such if observed with a chronometer regulated to mean time. Chronometers conveyed from one place to another for a comparison of the respective local time at each, have likewise been used for determining differences of longitude; but the irregular variation of their rates from being moved any considerable distance; renders this method one upon which much dependance cannot be placed.
The eclipses of Jupiter's satellites are phenomena of very frequent occurrence, the precise instants of which can be calculated with certainty for Greenwich time.*
* The time occupied by light in travelling from the earth to the sun, is also ascertained by means of the eclipses of Jupiter's satellites.
E and J
represent the places of the earth and Jupiter, this planet and the sun being in opposition.
E' and J P, when in conjunction.
The difference of distance the light has to travel from Jupiter to the earth, on the occasion of an eclipse of one of the Satellites L, happening when they are in opposition or conjunction, is evidently EE' on the major axis of the earth's orbit. This difference has been ascertained to be 16′ 26′′.4, which divided by two, gives 8' 13".2 for the time occupied by light in passing from the sun to the earth.
The distance of the sun from the earth again, ha, been determined by means of the transit of Venus over the sun's disc.
A telescope magnifying at least forty times is required for their observation, and those of different powers are found to give such different results as to the moment of immersion or emersion, that the method is not susceptible of the accuracy it would appear to promise, it is moreover almost impracticable at sea. In determining the longitude by this method, the local time must be found by observations of one or more fixed stars, unless a chronometer whose error or rate is at hand, be regulated to this time.
The eclipses of the sun and moon also enable us to determine the longitude, the former with considerable accuracy; but their rare occurrence render them of little or no practical benefit; and the results obtained by the eclipses of the moon are generally unsatisfactory, owing to the indistinct outlines of the shadow of the earth's border.
The three methods upon which the most dependence can be placed, are 1st, by a "lunar observation," which as before stated, possesses the great advantage of being easily taken at sea; 2ndly, by the meridional transits of the moon, compared with those of certain stars previously agreed on, and which are given in the "Nautical Almanac" under the head of " Moon Culminating Stars," and 3rdly, by occultations of the fixed stars by the moon. The two latter methods are the most accurate of any, but the first requires the use of a transit instrument, and the latter a good telescope; both involve also, long and intricate calculations, which will be found fully detailed in the works of Dr. Pearson, and in chapter 37, of "Woodhouse's Astronomy." The methods given in the following pages, considerably shorten the labour of the more accurate computations, and are the same as those in Mr. Riddle's "Navigation."
Method 1st-By a Lunar Observation.-The observations for this method of ascertaining the longitude of any place, can be taken by one individual; but as there are three angles required as data, which, if not obtained simultaneously, must be reduced to what they would have been if taken at the same moment of time, it is better, if possible, to have that number of observers.
The lunar distance, which is of the first importance, is measured by bringing the enlightened edge of the moon and the star, or either limb of the sun, in perfect contact. The other observations
required are, the altitudes of the moon, and that of the other object, whether the sun, a fixed star, or a planet, and as these are only taken for the purpose of correcting the angular distance, by clearing it from the effects of parallax and refraction, they do not require the same accuracy, or an equal degree of dexterity in observing. When the observations are made consecutively by one person, the two altitudes are first taken (noticing of course the times), then the lunar distance repeated any number of times, from whence a mean of the times and distance is deduced, and afterwards the altitudes again in reverse order, which altitudes are to be reduced to the same time as that of the mean of the lunar distances.
It being of great moment to simplify and render easy the solution of this problem, which is of the most vital importance at sea, a number of celebrated practical astronomers have turned their attention to the subject, and tables for “clearing the lunar distance" are to be found in all works on Nautical Astronomy, by the use of which the operation is undoubtedly much shortened; † but as none of these methods show the steps by which this object is attained, the example given below is worked out by spherical trigonometry, and the process will be rendered perfectly easy and intelligible by the following description :—
In the accompany
ing figure Z represents the zenith, P the Pole, M the observed place of the moon, and S
that of the sun or star. The data given are MS, the measured angular distance; and ZM and ZS the two zenith distances(or coaltitudes) from whence the angleMZSis found, the value of which is
*These altitudes, if not observed, can be calculated when the latitude is known. Dr. Pearson enumerates no less than twenty-four astronomers who have published different methods of facilitating the " Clearing the Lunar Distance."