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It may frequently happen that the objects are If the object be the sun, the apparent time is too near the meridian to deduce the true time at the meridian distance, if less than twelve hours, the place of observation from either of the alti- but if it is more than twelve hours its complitudes with required exactness ;: or, though the ment to twenty-four hours is the meridian disaltitudes may be exact enough to use in clearing tance. the distance, they may not be sufficiently so for For any other celestial object, to the apparent deducing the time from them. In either case time add the sun's right ascension, reduced to the error of the watch must be found from some the given instant of Greenwich time, and from other observation, and this error, being applied the sum subtract the object's right ascension, and to the time at which the lunar distance is taken, the remainder will be the meridian distance of will give the time at the meridian at which the the object. If it exceed twelve hours take its observation for the error of the watch was taken, complement to twenty-four hours as the meriand the longitude thence deduced will be the dian distance. longitude not of the place at which the distance With the latitude of the place, the polar disis observed, but of that for and at which the error tance of the object, and its meridian distance in of the watch was found. In this manner may degrees, proceed to compute the altitude as folthe results of a great many lunar observations belows:all referred to one place, and the situation of the Add together the cosine of the meridian disship at that time determined with a certainty tance and the cotangent of the latitude, and the to which the result of one observation can have sum rejecting ten from the index will be the tanno claim. The situation of the ship being thus gent of arc first. determined by lunars, her longitude may be kept If the meridian distance is greater than 909 by the chronometer, till another opportunity is take the sum of arc first and the polar distance, afforded to determine her place with precision otherwise their difference, for arc second. by another series of lunars.

Add together the secant of arc first, the The latitude used in computing the time is cosine of arc second, and the sine of the latitude, understood to be obtained from the course and and the sum rejecting the tens from the index distance run since the last observation for the will be the sine of the true altitude. latitude. An error of a few degrees in the lon- If the apparent altitude is required, take the gitude by account, or of half an hour in the correction corresponding to the true altitude, and time by account can be of very little importance, apply it with a contrary sign to the true latitude, as the Greenwich time by account, which they and the result will be the approximate apparent are used in finding, is only employed in taking altitude. From this approximate apparent altitude out the semidiameter and horizontal parallax of take out the correction, and, applying it with a the moon from the Nautical Almanac, and these contrary sign to the computed true altitude, the vary in general but a few seconds in twelve corrected apparent altitude will be obtained. hours; the parallax sometimes twenty-four se- In fig. 9, plate II., if A, S, and x, repreconds; and this is about its maximum variation sent the places of the first point of Aries the in twelve hours.

sun and the star respectively, P the pole, and If the time is not computed from either of the M the meridian of the plane; then LLM the altitudes taken with the distances, the time of apparent time is the sun's meridian distance, if observation by a watch must be carefully noted, it exceed twenty-four hours, the time deducted and the error of that watch found from some from twenty-four hours leaves the meridian disother observation ; but, if the time is computed tance on the other side of PM. In the case of from one of the altitudes used in the lunar, no a star the sun's right ascension APS, added to great care is required in noting the time. the apparent time SPM, gives APM the siderial

When speaking of altitudes and distances, it time; and from A PM, the star's right ascension must always be understood that the means of A Pr being taken, the remainder x PM is the several altitudes and distances are meant, when star's meridian distance. Let x P M, in fig. 8, it is possible to obtain them.

plate II., be represented by A Pz, fig. 15, plate An observer may himself take both the altitudes ÎI., : being the zenith, A B the altitude, and and distances, first taking an altitude of each ob- AP the polar distance of the object A. Now ject, then a series of distances, and again the

tan. P 2. cos. P' cot. lat. cos. P. altitude of each object, carefully noting the tan. PC =

rad.

R tines, and then finding by proportion what the altitudes must have been at the time of the whence PC is arc first. When P is acute the mean distance.

difference of PC and PA is CA arc second, It may however frequently happen that the otherwise their sum is CA; and cos. PC? distances may be observed when from darkness

cos. AC:: or Pz (or sin. lat.; cos. A ), or or fog the horizon cannot be seen, and conse

sin. A B the altitude; whence sin. AB quently the altitudes cannot be obtained. The cos. AC · sin. lat. sect. PC:cos. AC sin. lat. altitudes must then be computed, and the method

cos. PC

rad. 2. of computing them is shown in the following problem.

Erumple 1.-What are the true and apparent The latitude of a place and the time being altitudes of the sun October 3d, 1828, at 3h. 4m. known, and the longitude by account, to compute 12s. mean time, in lat. 40° 12' N., long. by acthe altitude of any known celestial object. count 18° W.

=

S.

1

h. m. S.

h. Mean time at place

3 4 12
Equa. time

11

1 sub. + 16. Equa. time with contrary sign + 11 4

Correc. for given time S App. time at place 3 15 16 = 48° 49' True equa.

11 국 Long. in time w.

1 12 0

's declin.

4° 3' 7" $. + 23 13 4 27 16 Correction

4 18

.

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True alt. 27° 4'
Correction with contrary sign

+ 1

0" sin. 9•658149 43

App. alt. 27 5 43 Erample 2.-On March 9th, 1828, in lat. 36° 27' N., long. 35o W., at 7h. 2m. 10s. P. M., niek lime, required the true and apparent altitudes of Procyon?

h. m. s. Mean time, at place.

7 2 10 The sun's R. A. at 9h. 11m. 32s. is Equa. time with contrary sign

10 38

23h. 21' 7s., the *'s R. A. 7h.

30m 19s. and polar dist. 84° 21'. App. time at place

6 51 32 Long. in time, w.

2 20 0

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App. alt.

54 25 41 In computing altitudes it is absolutely neces- Example.—On May 4th, at 3 P. M., by obsary that the apparent time for the meridian of servation I found my watch 3m. 10s. fast, after the place of observation should be known; there- sailing westerly, and making thirty-five miles dif fore, when the error of the watch is found for ference of longitude, I wanted at gh: 6m. 105, by some other meridian, the difference of longitude the watch to compute the altitude of a star

, what made from that meridian to that of the place of was the true time at the place of observation ? observation must be applied to the time shownı by the watch, adding it if east, but subtracting it if west, to obtain the time at the place of obser- Time by watch vation. Let t = the time shown by the watch, Error fast at 3 P. M. e its error as found by observation, l the difference of long ude made subseqnently, and T Time at the meridian where error the true time. Then T=ttet til being

was found reduced to time, and e + when the watch is Long. made since, W. slow and — when fast, and ľ + when the difference of longitude is east, and — when west. Time at place of calculation

h. m. $

9 6 10

3 10

9 30

2 20

9 0 40

If the error was in mean time, this must be when on the meridian, which being 1} point reduced to apparent time by applying the equa- to the left of N., the variation is 1} point east. tion of time with a contrary sign.

The variation of the compass may also be To find the variation of the compass.—If the found by the amplitudes of celestial objects. bearing of any object by the compass be com- But, as from the effect of refraction they appear pared with its known bearing, the variation or in the horizon when they are about 33' below it, deviation of the points of the compass from their the centre of the sun, or a star, ought to be corresponding points in the horizon becomes of about 33' + W. dip above the horizon, when course immediately known; the difference of the their amplitude is observed to compare with true and observed bearing being the variation. their true computed amplitude to find the varia

Now when an object is on the meridian its tion. Or the lower limb of the sun ought to be true bearing is either due north or due south; about 17' + the dip above the horizon. hence the deviation of an object from the meri- To compute the true amplitude, add together dian as observed by the compass, when the ob- the secant of the latitude, and the sine of the ject is known to be on the meridian, is the vari- object's declination, and the sum rejecting ten ation of the compass; west when the object from the index will be the sum of the true ambears to the right; and east when it bears to plitude, east when the object is rising, and west the left of the meridian.

when setting; north when the declination is Again, the middle time between equal alti- north, and south when it is south. Then if the tudes of a celestial object being the time of its true and observed amplitude, be both porth or being on the meridian, the middle point between both south, their difference, otherwise their sum, those on which it bears when it has equal' alti- is the variation; westerly when the true amplitudes is its bearing when on the meridian; hence tude is to the left of the observed, and easterly if this middle point be to the right of the meri- when the true altitude is to the right of the obdian the variation is west ; if to the left the va- served. riation is east.

Let P, fig. 16, plate II., be the pole, A the Erample 1.- The sun at noon was observed east or west points of the horizon, CB or C'B, to bear Š. by W. } W., what was the variation ? the declination of the object at rising or setting,

S. by W.; W.is 14 point to the right of S. then A C or A C' is the amplitude, B C or BẮC therefore the variation is 14 point W.

the declination, and BAC or B’AC the colatiErample 2.—The sun, when he had equal al- tude; and rad. sin. BC= sin. A C • sin. BAC, titudes on the same day, bore N.N. E. and N.W.

rsin. BC r. sin. BC by W.; what was the variation ?

whence sin. AC=

sin. BAC

cos. lat. The middle point between N. N. E. and N.W. sect. lat. sin. declin by W. is N. by W., W., the bearing of the sun

rad. Erample. If on Oct. 11th, 1828, in lat. 50° 46' N., long. 17° W., the sun rise E. 20° S. by compass at 6h. 32m. A. M., required the variation ?

h. Time Oct. 11th 32

O's declination at this time 7° 3' S. sin. 9•088970 Long. 1 8

Lat. 50 46 sect, 10:198953

E. 11 11 S.
Greenw. time 19 40

Obs. do.

E. 20 O S. sin. 9.287923

m.

18

Fine amp.

Variation

8 49, W, the true bearing being

to the left of the observed. To compute the true azimuth o) any celestial rithms will be the sine of half the azimuth, to object from its altitude, polar distance, and the be reckoned from the north in south latitud, latitude of the place of observation.

and from the south in north latitude, eastward Add together the altitude, latitude, and polar when the latitude is increasing, and westward distance, and take the difference between half when it is decreasing. the sum and the polar distance. Then add to- Then, when the true and observed azimuths gether the secani of the altitude, the secant of are both east or both west, their difference is the the latitude, rejecting ten from the index of each, variatiori, otherwise their sum is the variation, the cosine of the half sum, and the cosine of the westward when the true is to the left, and east. remainder, and half the sum of these four loga- ward when it is to the right of the observed.

For, adopting the notation employed in investigating the rule for computing the meridian distance of a celestial object, we have (fig. 10, plate II.)

a +1+p

-0. sect. I. sect. a
A 2 P
2 AZC

rad. 2
AZC
a +1+P a +1+p

-p: sect. 1 . sect. a.

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COS

2

2

or sin.

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- sin.

2

2

EV cos.

COS.

2

2

rad. 2 Erample. On Feb. 16th, 1828, in lat. 30° 14' N., long. by account 31° W. at 4h. 30m. P. M. the alt. of was 12° 56' bearing S. W. W. per compass, height of the eye fifteen feet, required the variation ?

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From the observed altitudes and the distance of are given to find the angles P MS, Z MS, 22. the sun und moon, and the compass bearing of ei- the difference of those angles is the ar: ther object, to find the latitude, longitude, and Z MP; and in the triangle Z M P are given ?! variation of the compass.

M P, and the included angle Z M P, to find Z. With the observed distance enter the Nautical the co-latitude. Hence the time at the place e Almanac for the time of the month in which the observation, and the true azimuth of eit! observation is made, and take by inspection the object, may be found ; and the time compani day and hour of Greenwich time, corresponding with the Greenwich time, previously found insi most nearly with that distance. To that time the distance, will give the longitude; and the take the moon's semi-diameter and horizontal true azimuth compared with the observed cei parallax, and, clearing the distance, find the will give the variation of the compass. Greenwich time from it. If this time differ much Example.-On September 2d, in the mornine from that before taken out by inspection, take the altitude of 0 was 15° 45' + of) 58° 4 out the moon's semi-diameter and parallax again, distance of their nearest limbs 77o O 40, lat and, again clearing the distance, find from it the by account 48° N., height of the eye twelve fett true Greenwich time. For this time take the bearing of the sun S. É. & E., required the lat

. polar distance of both objects, and proceed with tude, longitude, and variation of the compass! ihe computation thus :

By inspection in the Nautical Almanac it i Let M (fig. 17, plate II) be the true place of readily seen that the Greenwich time of this aó the moon; S that of the sun; MP, SP, their po- servation must have been about 19h. of Septerlar distances; and MZ, S2, their true zenith ber 1st. To this time the moon's semidiameter is distances; and MS their true distance. Then in 15'0", and hor. par. 55' 2" the triangles PMS, Z MS, all the sides in each

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Then in the triangle ZMP we have ZM = 30° 40' 20", MP 71° 28' 31", and angle ZMP 5° 53' 24", to find ZP. From Z on M P let PX be a perpendicular

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