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therefore the difference between the time shown by such an instrument, and the mean time at any other meridian, determined by observation, or otherwise, would be the longitude of that meridian in time, twenty-four hours of time corresponding to the circumference of the equator, or to 360° of longitude.

The simplicity of this method of finding the longitude at sea, and the perfection to which the construction of chronometers has been brought, have combined to introduce it into very general practice, and its usefulness has been amply proved. But so delicate a machine as a chronometer must be peculiarly liable to be put out of order, even by causes which are difficult to detect and impossible to avoid; it is therefore desirable that, if possible, we should have some independent method of ascertaining the time at the first meridian.

Now the moon revolves round the earth, or appears to revolve among the stars, from west to east, with an angular velocity so considerable, that the instant at which she is at a given distance from a celestial object, lying in the direction of her motion, may be determined with considerable precision; and from the principles of Physical Astronomy, aided by observations, her place in the heavens can now be predicted with sufficient exactness for the practical purposes of navigation; and, in fact, in the Nautical Almanac, the distances of her centre from that of the sun, and from one or more of nine of the principal fixed stars that lie in the direction in which she moves are given, and published for several years in advance, for every three hours of apparent Greenwich time, except near the change, when she is invisible.

Hence if an observer can determine by observation the moon's distance from the sun, or any of these stars, he may easily find the apparent time at Greenwich, by comparing the observed distance with the distances given in the Nautical Almanac.

But the distances there given are those which would be seen at the centre of the earth, and therefore before any comparison can be instituted between them, and distances observed upon the surface, for the purpose of determining the Greenwich time, the distances observed on the surface must be reduced to what they would have been if seen at the centre. Now the places of celestial objects, as seen at the surface, differ from their places as seen from the centre by the effects of parallax and refraction, which vary with the altitude of the objects;

the moon's place as seen from the centre being above and that of any other celestial object below its place as seen from the surface.

Hence before the true distance can be computed, the altitudes of the objects, as well as their apparent distance must be known.

In practice, the altitudes and distances of the objects are generally measured at the same instant, by different observers, while a fourth


notes, by a watch, the times at which the observations are taken.. Several sets of observations should, if possible, be taken, and a mean of the whole used as a single observation.

Such an observation is called a lunar observation, and this method of finding the longitude, by the distance of the moon from the sun or a star, is called the method of finding the longitude by lunar observations.

In altitudes used only for computing the true distance, no great precision is necessary; but an altitude for computing the time ought to be taken as exactly as possible. Great care however is requisite in measuring the distance, as an error of 1' in it will generally produce a mistake of about 2m of time, or of about half a degree in the longitude deduced from it. The distance of the nearest limbs of the sun and moon is always measured, and their semidiameters added to obtain the apparent distance of their centres. The distance of a star is measured from the round, or enlightened edge or limb of the moon, whether that limb be the nearest to, or farthest from the

and the moon's semidiameter is added to the observed distance when it is measured from the nearest limb, but subtracted when from the farthest limb, to obtain the apparent distance of the star from the moon's centre.

A dexterous observer may himself obtain both the altitudes and distance, by taking the altitudes of the objects both before and after he measures the distance, noting the time of each observation, and then computing, by proportion, from the change of the altitudes, what they must have been at the time at which the distance was observed. The proportion may conveniently be made by proportional logarithms; see the use of Table 29.

Various other methods have been proposed for finding the longitude by observation at sea, but none of them have yet been found capable of being reduced to practice. The methods of which we have here sketched the principles, are therefore those to which the attention of the practical mariner ought chiefly to be directed. We

may re-state that, to find the longitude, we must be able to do two things which are perfectly distinct in themselves, viz. to find the time at the place at which we are, and to find the time, at the same instant, at a place whose situation we know. The former of these is found at sea from the observed altitudes of celestial objects, and the latter by the aid of a chronometer, or by the distance of the moon from the sun or a fixed star, from which her distance is computed in the Nautical Almanac.

The distance of the moon from the planets Venus, Jupiter, Mars, and Saturn, are also now annually published by Professor Schumacker, of Copenhagen.

When the apparent motion of a planet is contrary to that of the tance from it, than by one from a fixed star ; and, besides, Venus and Jupiter can often be seen when there is daylight enough to take their altitudes with every requisite degree of exactness, either for clearing the distance, or computing the time.

Altitudes can seldom be obtained at night at sea with sufficient exactness for computing the time; it will therefore be generally found preferable to find the error of the watch from altitudes of the sun during the day, and then to find the time at the ship at which a lunar distance is taken, by allowing for the error of the watch and the difference of longitude between the places where the altitude for the time, and the lunar distance are measured. It will often indeed be found difficult to take altitudes at night with sufficient precision for the purpose of clearing the distance; but as the time may be inferred from the altitudes of celestial objects, so, conversely, their altitudes may be inferred from the time, and it will often be found that at sea the altitudes of stars can be determined by computation with greater correctness than they can be observed. Having said this much by way of introduction, we proceed now to give, in order, the various practical problems connected with and subservient to the method of finding the longitude, either by chronometers or by lunar observations.


To deduce the mean and apparent time at any place from the altitude of

the sun, when at a distance from the meridian on a given day; the latitude being known, and the longitude as well as the time at the place nearly.

Rule. With the supposed time and longitude, find the corresponding Greenwich time; and for that time take from the Nautical Almanac the equation of time, and the sun's declination, and thence (Problem 9, p. 224) find the polar distance. Or the equation of time and declination may be found, with sufficient exactness for practice at sea, from Tables 19, 20, and 22. Correct the altitude, and proceed with the computation as follows.

Add together the true altitude, the latitude, and the polar distance at the time of observation, and take the difference between half the sum and the altitude. Then add together the secant of the latitude, the cosecant of the polar distance, (rejecting 10 from each of their indices,) the cosine of half the sum, and the sine of the remainder, and half the sum of these four logarithms will be the sine of half the hour angle, or of half the sun's meridian distance. Reduce the double of this angle into time, as in Problem 1, p. 218, or multiply the half hour angle by 8, which will effect the same purpose.

If the altitude is decreasing, the meridian distance is the apparent


time; but if it is increasing, subtract the meridian distance from 24 hours, and the remainder will be the apparent time past noon of the preceding day.

To the apparent time apply the equation of time, by addition or subtraction, as directed in the Nautical Almanac, or in Table 22, and the result will be the mean time at the place of observation.

Note. Any mistake in the observation will produce the least error in the computed time, when the azimuth at the time of observation is 90°; therefore the object, whose altitude is observed for the time, should be as near the east or west as possible; and, at any rate, it should be at a.considerable distance from the meridian,


If on August 12, 1859, at 4h 7m 2s P.M., the altitude of be 19° 40 28. decreasing, latitude 14° 55' S, longitude 20° W, height of the eye 16 feet, required the apparent and mean time?

4h 7m 2sP. M. time by watch. O's Dec. at noon, Green1 20

0 W longitude in time. wich time, Aug. 12, 1823 15° 10'7"N-182? 5 27 2 P.M. Green. time. Red. to 1859, (Tab. 20,).. 5 10

15 4 57 Cor. for Green, time.

4 6 True declination........ 15 0 51 N

90 0 0 Polar distance

105 0 51

Add. Equation of time at noon, Greenwich time, August 12, 1823, 4m 51s – 10s Reduction to 1859, (Table 20,)

- 3

4 48 Correction for Greenwich time

2 True equation..

4 46

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19° 49'50" Latitude

14 55 00 sec 10.014887 Polar distance 105 00 51 cosec 10.015085

2) 139 45 41

69 52 50 cos 9.536531
50 03 00 sin 9.884572

32° 6' 35" sin 9.725537

19040' 28" observed altitude.

3 56 dip. 19 36 32

2 31 correction of alt. 19 34 1 + 15 49 semidiameter. 19 49 50 alt. O's centre.

4h 16m 53s apparent time.

+ 4 46 equation of time. 4 21 39 mean time.


In each of the following examples the apparent and mean times at the place of observation are required ?

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h m 1 May 20, 1823 43 30 'N 20 W 7 43 Ở A.M. 2 August 10, 1824 54 12 N 25 W 5 33 0 P.M. 16 50– 3 June 9, 1825 15 18 N 40 W 7 42 18 A.M. 0 29 40+ 4 Jan. 20, 1828 57 30 S70 E 2 45 20 P.M. Q 41 505 May 10, 1851 56 48 N 145 E 6 43 45 A.M. 20 14+ 6 June 21, 1829 65 30 N 33 W5 26 10 P.M. 0 24 37 7 Feb. 8, 1837 40 36 S20 E 4 28 3 P.M. 0 27 468 Aug. 8, 1860 31 05 N 44 W 7 38 16 A.M. 29 15+ 9 Nov. 2, 1873 51 30 S 27 W 7 34 28 A.M., o 26 17+ 10 April 12, 1862 00 00

4 3 20 P.M. o 28 40



S 19 44 09 5 32 38 19 45 29 2 46 23 18 43 25 5 25 36 4 25 49 19 39 45 19 32 25 4 5 22

hm S 19 40 26

5 37 37 19 44 13

2 57 35 18 39 36 5 26 56 4 40 21 19 45 09 19 16 9 4 6 6

To find the time by an altitude of a fixed star, when at a distance from

the meridian on a given day, the latitude being known, and the longitude as well as the time at the place nearly.

Rule. With the given time and longitude find the apparent time at Greenwich, and for that time take the sun's right ascension, and the equation of time from the Nautical Almanac, or Tables 21 and 22.

Take the star's right ascension and declination from Table 23, and reduce them by the annual variation to the given time ; correct the altitude for dip and refraction, and find the polar distance. Then with the latitude, and the altitude, and polar distance of the star, find its meridian distance, as that of the sun was found in the last problem, and subtract it from the star's right ascension, if the altitude is increasing, but add it if the altitude is decreasing, and the sum ôr remainder will be the sidereal time, or the right ascension of the meridian, from which subtract the sun's right ascension, and the remainder will be the apparent time. Add or subtract the equation of time, as directed in the Nautical Almanac, or Table 22, and the sum or remainder will be the mean time at the place.

If in adding, the sum should exceed 24 hours, reject 24 hours from it; and if in subtracting, the time to be substracted should exceed that from which it is to be taken, conceive 24 hours to be added to it before the subtraction is made.

The time may be found in the same manner from the altitude of the moon, or the principal planets, the moon's declination and right ascension being taken from the Nautical Almanac, and reduced to the given instant, by Table 30; and further corrected by the equation of second difference, Table 18. The right ascension and declination of the planets may be taken from Shumacker's Ephemeris.

There are some small periodical corrections of the right ascena

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