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Given the true distance of the moon's centre from that of the sun, or a. fixed star, as computed in the last problem, to find the apparent time at Greenwich.

For the given day, or in that preceding or following it, take from the Nautical Almanac the two distances between which the given distance falls, and write them under the given distance in the order of time in which they stand in the Almanac. Take the difference between the middle one of these three distances, and each of the others, and subtract the proportional log of the greater difference from the proportional log of the less, and the remainder will be the proportional log of the time to be added to that corresponding to the first distance taken from the Almanac, for the required Greenwich time.

Note. If the given distance be exactly found in the Almanac, the apparent time at Greenwich will be found above it.

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EXAMPLE On September 16, 1823, the true distance of the moon's centre from Antares was 65° 49' 30''; required the apparent time at Greenwich ? True distance..

65° 49' 30" Dist. per Naut. Alm.

}at 6h

at 6h 64 34 42, first diff. (see Table 32)

1° 14' 48", prop. log .... 3814 Distance at ... 9h 66 5 2, second diff. 1 30 20, prop. log 2994

2h 29m 20s prop. log

820 Time of first distance

6 0 0 Greenwich time.

8 29 20

EXAMPLES FOR EXERCISE, In each of the following examples the apparent time at Greenwich is required?

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TO FIND THE LONGITUDE BY LUNAR OBSERVATIONS.

With the time by the watch when the observations are taken, and the longitude by account, find the Greenwich time by account, as in Problem 4, page 219, and for that time take the moon's semidiameter and horizontal parallax from the Nautical Almanac, correcting the semidiameter by Table 12, and the parallax by Table 14. Take the sun's semidiameter for the given day, and correcting the observed altitudes for semidiameter and dip, call the results the apparent altitudes.

Add the sum of the sun and moon's semidiameters to the observed distance, if the sun is one of the objects observed; but if the observed distance is of the moon's limb from a star, add her semidiameter to the observed distance, or subtract it, according as the distance is measured from the nearest or farthest limb, and the result will be the apparent central distance.

The semidiameters ought, in strictness, to be corrected by Table 13 ; and, in practice, the inclination of the semidiameter to the horizon, to enter the table, may in general be estimated near enough by the eye ; but this correction, being in all cases small, and in most cases utterly insignificant, is in sea practice generally disregarded.

From the apparent altitudes, the parallaxes, and apparent distance, compute the true distance, (see page 254,) and find the apparent time at Greenwich, to which the true distance corresponds, by the last problem.

Then if the sun or the star is at a proper distance from the meridian, compute from its true altitude, &c. the apparent time at the place of observation, (see pages 243 and 245) and the difference between this time, and the apparent time at Greenwich, found from the distance, will be the longitude of the place in time; west if the Greenwich time is before, but east if the Greenwich time is behind the time at the place of observation.

If the sun or the star is too near the meridian for computing from its altitude the time with exactness, or if the altitudes have been indifferently observed, the error of the watch by which the times of taking the distances were noted must be found by altitudes taken for the purpose, when the sun or a known fixed star bears nearly east or west, or is, at any rate, at a considerable distance from the meridian. If the watch is found slow for apparent time, add its error to the time which it showed when the distances were observed ; but if fast, subtract its error, and the sum or remainder will be the apparent time at the meridian where the error of the watch was found, at the instant when the distances are measured ; the difference between which time and the Greenwich time, found as above from the distance, will be the longitude of the place where the altitudes are taken for finding the error of the watch.

But it may sometimes happen, that the lunar distance can be measured, when, from the obscurity of the horizon, the altitudes cannot be observed at all. In this case, the altitudes for clearing the distance must be computed ; but, to compute the altitudes, it is necessary that therefore the error of the watch for apparent time be found by altitudes, taken at some convenient opportunity, before or after the distances are observed. Compute, from the log, the difference of longitude made in the interval between taking the distances, and taking the altitudes for the error of the watch; and if the distances are taken to the eastward of the altitudes, add the difference of longitude in time to the time of taking the distances, corrected by the error of the watch; but subtract it if the distances are taken to the westward of the altitudes; and the sum or remainder will be the apparent time at the place where the distances were taken at the instant at which they were observed. Then with this time, the latitude at the same instant deduced from the log, the declinations, &c. of the objects, let their apparent altitudes be computed, as shown at p: 252; and with the apparent altitudes and distance compute the true distance, and thence find the Greenwich time and longitude as before.

EXAMPLE I.

On September 12, 1823, in latitude 26° 30° N, longitude by account 24° 30 W, at 5h 34m P. M. per watch, the altitude of was 7° 37' -, of 2 35° 35', distance of their nearest limbs 95° 19' 58", height of the eye 25 feet, required the longitude ?

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) 's semidiameter at noon 14' 52" - 3", )'s hor. Parallax at noon 54' 31"-9" Correction for Green. time 2

5 14 50 Equatorial parallax .... 54 26 Aug. for alt. Table 12

9

Red. for lat. Table 14.. 2 )'s true semidiameter ... 14 59

}'s reduced hor. par

54 24

+

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Min. ) 's parallax 54'.
Table 24.

35° 40' 42' 28" )'s apparent altitude

0

0 3 0 20"..

0 16 Sec. )'s parallax

0 4 0 3 Correction ) 's aititude

42 50 O's apparent alt. 7° 48', correction 6 30 • Difference of corrections

Table 25.
60° 17' 12"
0 2
0 0 7
0 0 1

36 20

0 0 2 60.17 24 aux. arc.

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Sum of apparent altitudes

43°33' 5" Auxiliary arc .......

60 17 24 Apparent distance

95 50 53 Sum of auxiliary arc and preceding one 103 50 29 vers 39098 Difference of ditto in

16 44 19 vers 42345 Sum of auxiliary arc and following one 156 8 17 vers 14490 Difference of ditto

35 33 29 vers 86392 Sum of true altitudes

44 9 25 suvers 17316

99641 Sum of parts for "

397 Vers true distance

00038 95°44' 0 vers 99899 29

139 True distance.. 95 44 29

Parts for "

137 26 33 82 119

parts for" of true distance.

To find the apparent time at Greenwich.

True distance

95°44' 29" Distance at......

6h 95 8 17 first diff. 0° 36' 12" prop. log 6966 Distance at ......9h 96 30 13 second diff. 1 21 56 prop. log 3418

1h 19m 3ls prop. log 3548 Time of first distance 6 0 0 Apparent time at Greenwich 7 19 31

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To find the apparent time at the place of observation. O's dec. at noon 4°24 71 N-22' 56' Green. time 7h 20m log., Table 30 *5149 Cor, for G.T. 7 0

Daily change O's dec. 22' 56' P.log •8948 O's true dec. 4 17 7

Cor. O's dec. for Green. T. 710" P. log 1.4097 90 O's polar dist. 85 42 53

.

O's true altitude

7041' 31" Latitude....

26 30 0 sec 048209 O's polar distance

85 42 53 cosec .001216 Sum

2)119 54 24 Half sum

59 57 12 cos 9.699582 Remainder

52 15 41 sin 9.898073

2)19.647080 Half hour angle 41° 46' 0"...... sin 9.823540

8 5h 34m 8s apparent time at place of observation,

7 19 31 apparent time at Greenwich.

EXAMPLE II. On September 27, 1823, at 5h 7m A. M. per watch, in latitude 28° 37' S, longitude by account 112° W, the altitude of 2 was 34° 20', and of Regulus 10° 40' +, distance of the star from )'s nearest limb 53° 30° 28'', height of the eye 12 feet, required the longitude ? Time per watch..

17h 7m, September 26.
Longitude 1120 W, in time.... 7 28
Greenwich time by account

0 35 September 27. )'s semidiameter at noon 16' 6'1 + 2', ) 's hor. parallax at noon 59' 711 + 61 Correction for Green. time 0 0

0 16 6

Equatorial parallax Aug. for altitude

9

Reduction for latitude .. 3 D's true semidiameter

)'s reduced hor.

par.

59 4 Altitude 2

34°20' 0" *'s observed altitude 10040' 0" Semidiameter + 16 15 Dip.

3 25 34 36 15 *'s apparent altitude 10 36 35 Dip....... 3 25 Refraction

0 4 58 ) 's apparent altitude •• 34 32 50

*'s true altitude.... 10 31 37

.... 597

.. 16 15

34°30'....

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Min. ) 's parallax 59'.
Table 24.

Table 25.
47' 10"

60°18' 221 ) 's apparent altitude 0 3 0 4

0 0 1 Sec. parallax...... 0 41.... 0 3

0 0 1 Correction ) 's altitude ... 47 17 *'s apparent altitude 10° 37', cor. 4 57

0 0 Sum of corrections

52 14

60 18 25 aux. arc. ) 's apparent altitude... 340 32 50" *'s apparent altitude. 10 36 35 Observed distance 53° 30' 28" Diff. apparent altitudes 23 56 15 ) 's semidiameter + 16 15 Sum of corrections.... 0 52 14 Apparent central dist...53 46 43 Difference of true altitudes 24 48 29

The trué distance computed by the second method.
Difference of apparent altitudes 23°56' 1511
Auxiliary arc .......

60 18 25 Apparent distance

53 46 43

Parts for 11 Sum of auxiliary arc and preceding one 84 14 40 suvers 00188

97 Difference of ditto 36 22 10 suvers 05066

145 Sum of aux. arc and following one ..114 5 8 vers 08065

35 Difference of ditto

6 31 42 vers 06461 Difference of true altitudes 24 48 29 vers 92222

58

12002 Sum of parts for "

359 Vers true distance

12361 54°0' On vers

12215 37

146 parts for 1l of true

distance. True distance.... 54 0 37

24

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