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sphere (Table XXX.).* When the temperature of the atmosphere is raised, which is indicated by the thermometer, the refraction decreases; and when the density of the atmosphere is increased (indicated by the rising of the mercury in the barometer), its refractive power increases. The change in refraction for a difference of 1° of Fahrenheit, and of 1 inch in the barometer from the mean state, is given in separate columns, and must be multiplied the one by the number of degrees which the thermometer differs from 50°, and the other by the number of inches and fractions of an inch which the barometer differs from 30°, and the result added or subtracted, as the case may require. It should be observed that below 4° the refraction is very variable and uncertain, and such low altitudes should be avoided as much as possible at sea.

It will be unnecessary to use the correction for the state of the barometer and thermometer, when the latitude of the ship is the only object of the observation, as this could seldom make a difference so great as half a mile in the resulting latitude; but, in determining the longitude by the Lunar Observations, the neglect of these small corrections would sometimes introduce an error in the resulting longitude of more than thirty miles.

When the foregoing corrections have been applied to the observed altitude, the result will be the true altitude of the centre above the sensible horizon, and it now remains to apply the correction necessary to reduce this to the true altitude of the centre above the rational horizon; that is, to the altitude which the body would have if the observer were situated at the centre of the earth instead of on its surface. This last correction is called

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* Such a table might be formed by comparing the observed altitude of a star with

its altitude computed from the declination or N.P.D. hour angle and latitude.

in reference to the rational horizon CR; and the difference of these angles is the parallax called parallax in altitude when the object is above the horizon as at s, and horizontal parallax when it is in the horizon as at H.

Since the angle SE'H is equal to the angle SCR, EH and CR being parallel by definition, we have for the parallax in altitude SE'H- SEH ESC (Geom. Th. 15), that is, the parallax is the angle which the semidiameter of the earth subtends at the object;* it is obviously greatest in the horizon, and nothing in the zenith, and is the quantity which must be added to the true altitude above the sensible horizon to obtain the true altitude above the rational horizon.

The sun's parallax in altitude is given in a Table at the end (Table XXXIV.), his horizontal parallax being nearly constant; and the moon's horizontal parallax is given for the noon and midnight at Greenwich, of every day of the year, in the Nautical Almanac, and from the horizontal parallax thus obtained, parallax in altitude must be calculated. This is easy; for since in the triangle SEC we have the proportion

SC EC sin SEC sin SEZ COS SEH : sin ESC;

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it follows (since sc, the distance of the heavenly body, as well as Ec, the semidiameter of the earth, may be regarded as constant for a single day), that the sine of the parallax in altitude varies as the cosine of the altitude; but when the altitude 0, as in the case of horizontal parallax, cos. altitude - 1, and the constant ratio sc to EC, the above proportion shows to be equal to the sine of the horizontal parallax. But from the proportion itself we see that it is necessary to multiply this ratio by the cosine of the altitude, to have the sine of the parallax in altitude; but as the parallax is always a very small angle, it is usual to substitute the arc for its sine, or par. in alt. = hor. par. X cos. alt., so that

log. hor. par. in seconds + log. cos. alt. 10

log. par. in alt. in seconds.

We must observe here that the horizontal parallax, given in the Nautical Almanac, is calculated to the equatorial radius of the earth;

* This result might be arrived at much more simply by means of our definition of an angle (Geometry, def. 10), viz. " the difference of direction of two lines," and a definition of parallax, viz. the difference of direction in which an object is seen from the centre and surface of the earth, or in a more enlarged sense of the term, from any two points. This in the diagram will be the difference of direction of the two lines cs and Es, i. e. the angle CSE, or the angle subtended by the line joining the two points of observation.

and, therefore, except at the equator, a small subtractive correction of the horizontal parallax will be necessary, on account of the spheroidal figure of the earth, in consequence of which the radius of the earth is smaller everywhere else than at the equator, and consequently subtends a smaller parallax. A table of such corrections is given at the end. (See Table XXXV.) It must evidently be a table of double entry, the two arguments being the equatorial horizontal parallax and the latitude, upon which two quantities the correction depends.

110. Such are the corrections necessary to be applied to the observed altitudes of celestial objects, in order to obtain their true altitudes. A few other preliminary, but very simple and obvious operations, must also be performed upon the several quantities taken out of the Nautical Almanac, in order to reduce them to their proper value at the time and place of observation; for the elements furnished by the Nautical Almanac are computed for certain stated epochs, and their values for any intermediate epoch must be found by proportion. But ample directions for these preparatory operations are contained in the "Explanation of the Articles in the Nautical Almanac,' "* to be found in the last pages of that work.

* It may be well, however, to give here some general account of the arrangement of the Nautical Almanac. The first twenty-two pages contain the right ascension, declination, semidiameter, and a variety of other elements relating to the sun and moon for every day of the month of January, the right ascension of the sun at mean noon and at apparent noon, that of the moon at the beginning of every hour of mean time throughout the day at Greenwich. The next twenty-two pages contain the same elements for the month of February, and so on, each month occupying twenty-two pages, marked with the Roman numerals, I. II., &c. The year thus being gone through, after a few pages containing the sun's co-ordinates, follows the ephemeris of the planets, beginning with Mercury, the one nearest the sun. This contains the semidiameter and declination, apparent right ascension, as affected by aberration of light, and some other elements of the planet for every day in the year of mean noon at Greenwich, and also at the time of the planet's meridian transit at Greenwich, each month occupying two pages. This Ephemeris extends from p. 275 to p. 455, in the almanac of 1850. The next three pages contain the mean places or right ascension and declination on the 1st of Jan. of 100 principal fixed stars, with their annual variations in right ascension and declination, marked + or - .

The latter multiplied by the fraction of the year which has elapsed, which is given in the last column of p. XXII. of each month, will be the quantity to be added or subtracted, in order to have the mean R. A. and Dec. at the time. To obtain the true places, corrected for nutation, &c., recourse must be had to formulas and tables given in the next three pages of the Almanac, except that of a number of the principal stars, the true R. A. and Dec. are given for every ten days from p. 468 to p. 501, in the edition of 1850. The remaining matters contained in the Nautical Almanac will be noticed as occasion requires.

EXAMPLES OF THE CORRECTIONS.

1. On the 14th of July, 1833, suppose the observed altitude of the sun's lower limb* to be 16° 36' 4", the observer's eye to be 18 feet above the level of the sea, the barometer to stand at 29 inches, and the thermometer at 58°; required the true altitude of the sun's centre.

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2. On the 23d of June, 1850, in longitude 4 56′′ 4′ W., latitude about 40° 43′ N., at 11' 44TM 55′ mean time, the double altitude of the moon's upper limb was observed by reflection from Mercury to be 58° 14'; the index error of the sextant was 15" subtractive; the barometer stood at 30.74 in., and the thermometer at 76°, required the true altitude of the moon's centre.

The object in this example being the moon, it is necessary to compute her semidiameter and parallax in altitude at the instant of obser

* The limb of the sun or moon is the edge or border of the disc.

Take out the refractions for 16° 30' of altitude from the table, then the diff. for 1' of altitude in the column adjoining, multiplying the latter by 2, and subtracting the product from the refraction for 160 30'; the result will be that for 16° 32', when the barometer is at 30 in. and the thermometer at 50°. The correction for refraction is always subtractive.

The barometer standing at 29 in. the number taken from the column entitled cor. for 1 must be subtracted from the refraction or added to the altitude, the atmosphere being less dense than in its medium state.

§ The thermometer standing at 80 above its medium state, the atmosphere is more rare, and the number taken from the column Diff. for 10 Fah., after being multiplied by 8, must be subtracted from the refraction, or added to the altitude.

|| Table XXXIV., the parallax in alt. for 100 is 9', and for 200 is 8". Therefore for 160 by proportion it is 8'4. This correction for par. in alt. is always additive.

vation since these elements, for the moon changes sensibly in a very short time. The semidiameter of the moon at noon and midnight is given in the Nautical Almanac for every day in the year, at page III. of each month, and the difference between these will be the variation of the semidiameter in 12 hours. Therefore we must say as 12: the variation in 12:: the interval between the preceding noon or midnight and the instant of observation the variation of the semidiameter in that interval; the fourth term of this proportion added to or subtracted from the semidiameter at the preceding noon or midnight, according as the semidiameter is observed from the numbers in the almanac to be increasing or decreasing, will give the semidiameter at the instant of observation.

In a similar manner must the moon's horizontal parallax, which is given for every noon and midnight on the same page of the Nautical Almanac, be reduced by proportion to the time of observation. The computation of these elements is as follows:

Mean time of observation at the station
Add longitude of the place of observation
Corresponding mean time at Greenwich
Time after midnight Gr. June 23d

Semidiameter previous mid

night, June 23d (Naut. Alm.)

14' 50"-3

Semidiam. noon (June 24th), 14 48 1

Variat. in 124.

... 124 : 2''•2 : : 4h 40m 59* :
Semidiam. at midnight (23d), 14 50 3

11' 44" 55'

4 56

4

16 40 59
4 40 59

Horizontal parallax preceding

midnight, June 23d (Naut.
Alm.)

Hor. par. noon (24th),

54' 27''-3

54 19 1

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* This is the interval from midnight at Greenwich to the instant of observation. + Table XXXIII. This augmentation is in consequence of the moon being nearer to the observer, as it approaches the zenith. See p. 272.

This is occasioned by the effect of refraction, whic.. is to make every vertical arc, such as the vertical semidiameter of the sun or moon, appear shorter in the heavens than it really is. This will obviously be the case, because the lower extremity of the arc is more elevated by refraction than the higher, and consequently the two extremities are brought nearer together, and thus the arc is shortened. The contraction is obtained from Tab. XXXII.

§ Table XXXV., see p. 275, last paragraph of Art. 109.

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