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but the sun or star's correction must always be subtracted. The sun's correction is the difference between the refraction (Table IV.), and the parallax in altitude (Table VI.), and a star's correction is the refraction (Table IV.).

EXAMPLE I.

The apparent distance of the moon's centre from the star Regulus was 630.35'.13", when the apparent altitude of the moon's centre was 24o.29'.44", the apparent altitude of the star 450.9.12" ; and the moon's horizontal parallax 55'.2"; required the true distance ?

D's horizontal parallax=55'.2", the parallax in altitude= 50'.5" (T.96.), refraction=2.4" (Table IV.) hence the D's correction=(50'.5" — 2.4"=) 48'.1", and the *'s correction=57". (Table IV.)

Diff. app. altitudes 20°.32'.28" Nat. cos. 93570
Apparent distance 63°.35'.13" Nat. cos. 44485

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Moon's apparent altitude 24o.29'.44"log. sec. 10·04097
Star's apparent altitude | 45°. 9.12"log. sec. 10:15169
Moon's corrected altitude 250.17.45"log. cos. 9.95622
Star's corrected altitude 45o. 8'.15"log. cos. 9.84843.

Natural number

48782 4.688.26

Diff. true altitudes 19o.50'.30" Nat. cos. 94063

True distance

63°. 4.33" Nat. cos. 45281

* Instead of the logarithmical secants of the apparent altitudes, you may take the logarithmical cosines, add them together and take the sum from 20, the remainder will be the same as the sum of the logarithmical secants without the indices.

+ The sum of these four logarithms, rejecting the tens, is 9.99731. This number may be found at once from the XLIId Table of Dr. Mackay's treatise on the Longirude, 3d edition, using the ) 's horizontal parallax 55'.2", and the apparent altitude 24° 29'. 44". The operation would be rendered shorter by such a table, but it occupies 45 pages of Royal Octavo, and, therefore, could not be inserted in this work. The IXth of the Requisite Tables is not so extensive, but then it is

proportionably imperfect. The author of the article Longitude in Dr. Rees’s new Cyclopedia has been pleased to remark, that the rule given by me may be simplified by using the IXth of the Requisite Tables, or Dr. Mackay's XLIId Table; this I was fully aware of: my object was not to give a short rule for solving the problein by auxiliary tables not in common use, but to render the subject plain and easy to a learner without the help of any other tables than those at the end of this treatise.

EXAMPLE II.

The apparent distance of the moon's centre from the sun's was 106° 46'.44", when the apparent altitude of the sun's centre was 45°.32.30", and the moon's 199.43'.22"; the moon's correction 50'.3", and the sun's 50"; required the true distance of their centres ?

Diff. of app. alts. - 25o.49'. 8" Nat. cos. 90018
Apparent distance 1069.46'.44" Nat. cos. 28868

Sum, common log. 1188865 5.07513 Moon's app. altitude 190.43' 22" log. sec. 10:02625 Sun's app. altitude 450.32.30" log. sec. 10:15466 Moon's corrected alt. 20°.33.25" log. cos. 9.97143 Sun's corrected alt.

450.31'.40" log. cos. 9.84544

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Natural number

118281

5°07291

Diff. of true alts. 24°.58'.15' Nat. cos. 90652

True distance 106o. 2.19" Nat. cos.

27629

EXAMPLE III. The apparent

distance of the sun and moon's centres was 68o.42.11", when the apparent altitude of the sun's centre was 32°.0.1", and that of the moon's 249.0'.10'; the sun's correction was 1'.23", and the moon's 51'.1"; required the true distance of their centres ?

Answer. 68o.19.46".

EXAMPLE IV.

The apparent distance of the moon's centre and a star was 20.20', when the apparent altitude of the star's centre was 11o.14', and that of the moon's 90.39'; the moon's correction was 51'.30", and the star's 4'.40”; required the true distance of their centres ?

Answer. 1o.49'.

EXAMPLE V.

The apparent distance of the moon from the star Spica was observed to be 31°.17'.53", when the apparent altitude of the moon's centre was 189.56'.45”, and the apparent altitude of the star 20°.10'.56"; the moon's correction was 50'.46", and the star's 2.35"; required the true distance of their centres ? Answer. The true distance is 319.11.44".

EXAMPLE VIL The apparent distance of the moon's centre from that of the sun was 70°.8'.23" at a time when the apparent altitude of the sun's centre was 22°.14'.55", and the apparent altitude of the moon's centre 80°.52.22"; the sun's correction was 2.11", and. the moon's correction 8.47"; required the true distance of their centres ?

Answer: 700.71.56".

EXAMPLE VII., The apparent distance of the sun and moon's centre was 639.5'.46", the apparent altitude of the sun 45°.9'.12", that of the moon 24o.29'.40"; the sun's correction 57", and the moon's, 47'.47"; required the true distance of their centres ?

Answer. 620.44'.

EXAMPLE VIII;

The apparent distance of the sun and moon's centres was 72o.21.40”, the apparent altitude of the moon 19o.19', that of the sun 25o.16'; the moon's correction being 50'.40", and the sun's 1':53"; required the true distance of their centres?

Answer, 72°.31.51.".

PROBLEM XVIII.

(U) The latitude of a place and its longitude by account, the distance between the sun and the moon, or the moon andą star* in the NAUTICAL ALMANAC,. being given, to find the correct longitude.

* The principal stars used for finding the longitude at sea are the following : a Arietis in the head of Aries. This is a small star, without the zodiac, and cannot be readily found and applied by the generality of persons; it appears about 22° to the right hand of the Pleiades.,

2. Aldebaran in the Bull's eye is easily distinguished by its largeness, colour, and position to the other stars, being half way between the Pleiades and the star which forms the western shoulder of Orion.

* 3. a Pegasi or Markab, about 44o to the right band of Arietis, a line drawn in imagination from the Pleiades through Arietis will pass through a Pegasi.-The constallation Pegasus is very remarkable, the three principal stars in it, with the head of Andromeda, form a large square, of which the four corner stars are all of the second magnitude.

4. Pollux, a little north ward of Aldebaran, and about 45° towards the left hand, there are two stars nearly together : the right hand one is Castor, the left hand one Pollux.

5. Regulus, about 38° S. E. of Pollux, is easily distinguished by being the southern-most of four bright stars resembling the letter Z inverted, it lies to the north-east of Aldebaran.

I. Turn the longitude, by account, into time (X. 265 and Z. 266.) and add it to the astronomical time* where you are if west longitude, but subtract it from that time if east, and you

will have the astronomical time at Greenwich nearly, which call reduced time.

II. In page VII. of the Nautical Almanac, for the given month and day, take out the moon's semi-diameter, and horizontal parallax, for the noon and midnight between which the reduced time falls, subtract the less semi-diameter from the greater, and the less parallax from the greater.

Then, as 12 hours are to the first difference, so is the reduced time to a fourth number, which must be added to the moon's semi-diameter if the tables be increasing, but subtracted if they be decreasing; to the sum or difference add the augmentation (Table VII.), and you will have the moon's true semi-diameter at reduced time.

Again, As 12 hours are to the second difference, so is the reduced time to a fourth number, which must be added to or subtracted from the horizontal parallax at the nearest noon, or midnight, preceding the reduced time, according as the tables are increasing or decreasing, and it will give the horizontal parallax at reduced time. (See Example II. page 272.)

III. Clear the observed altitude of the moon of dip (Table V.) and semi-diameter t, and you have the apparent altitude of her centre : to the cosine of the moon's apparent altitude, add the logarithm of the horizontal parallax at reduced time in seconds; the sum, rejecting 10 from the index, will be the logarithm of

6. Spica, or a Virginis, a white sparkling star, about 54° south-east of Regulus. -7. Antares. The are of a great circle passing through Regulus and Spica Virgins east-south-east, or to the left hand of Regulus southward, will pass through Antares, which is about 45° frou Spica Virginis.

8. Fomalhaut, about 45° south of a Pegasi.
9. a Aquila, 47° westward, or to the right hand of a Pegasi.

The distance between the moon and the sun, and between the moon and the nine stars above described (being near her path, are given in the VIIIth, IXth, Xth, and XIth pages et the Nautical Almanac, to every three hours apparent time, by the meridian of Greenwieb. The observer who uses the Nautical Almanac, and certainly no observer ought to be without it, is under the necessity of taking the distance of one or other of the above stars from the moon. The distances ealculated in the Nautical Almanac afford, perhaps, the readiest method of knowing the star from which the moon's d stance ought to be observed. For the sextant being tixed to that distance, and the moon found upon the horizon glass, there is nothing more to do than to look to the east or west of the moon, according as the distance corresponds to the VIIIth and IXth, or Xth and XIth pages of the Nautical Almanae, guiding the sextant in a line with the moon's shortest axis.

* By a late order from the admiralty, the Nautical Cay begins at midnight, ia coosequence of which the Navy Logs correspond with the civii reckoning; but the tables in the Nautical Almanac are adapted to Astronomical time, see the note,

† Viz, take their difference and add it to the observed altitude of the moun's lower limb; or take their sun and subtraet it from the observed altitude ot the moon's upper limb, according as the lower or upper limb has been observed.

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the moon's parallax in altitude in seconds (T.96.), from which take the refraction of the moon in altitude (Tablo IV.), the remainder will be the moon's correction.

The moon's correction must always be added to the apparent altitude to obtain the true altitude.

IV. Additional Preparation for the Sun and Moon. Clear the observed altitude of the sun of dip and semidiameter *, and you have the apparent altitude of his centre.

From the refraction of the sun's altitude (Table IV.) take his. parallax in altitude (Table VI.), and you have the correction of the sun's altitude; which must always be subtracted from the apparent altitude to obtain the true altitude.

V. To the observed distance of the sun and moon's nearest limbs, add their semi-diameters at reduced time, and the sum will be the apparent distance of their centres.

IV. Additional Preparation for the Moon and a Star. From the star's observed altitude take the dip of the horizon (Table V.), the remainder will be its apparent altitude. The refraction of a star (Table IV.) is the correction of its altitude, and must always be subtracted from the apparent altitude to, obtain the true altitude.

V. To the observed distance of the moon from a star, add the moon's semi-diameter at reduced time, the sum will be the apparent distance; if the farthest limb was observed, subtract the semi-diameter.

VI. To find the true distance. With the apparent altitudes, their corrections, and the apparent distance, find the true distance. by the general rule. (T. 333 and 334.)

Note. If the watch has not been previously regulated, the true time must now be found with the mean altitude of the sun or star, and the latitude of the ship, as in Prob. XIV. and XV. taking care to calculate the sun's declination to the reduced time. (B. 270.)

VII. To find the longitude. In page VIII, IX, X, or XI, of the Nautical Almanac for the given month and day, look for the computed distance bea tween the moon and the other object; if you find it there exactly, the time at Greenwich stands at the top of the column; but if

you do not find it exactly, take the nearest distance to it both less and greater; take their difference, and likewise the difference between the computed distance and the earliest Ephemeris distance.

* Viz. take their difference, and add it to the observed altitude of the O's lower limb; or take their sum, and subtract it from the observed altitude of the O's upper limb, according as the lower or upper limb has been observed.

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