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the Nautical Almanac for every day in the year. But in the case of the moon the semidiameter itself requires a small correction depending upon the observed altitude. For the semidiameter, furnished by the Nautical Almanac, is the apparent horizontal semidiameter, or the angle it subtends when in the horizon ; but as the moon approaches the zenith, her distance from the observer diminishes, and therefore her semidiameter is viewed under a greater angle. As she is nearer to the observer when in the zenith than when in the horizon, by one semidiameter of the earth, and as her distance from the earth's centre is about 60 semi-diameters of the earth, the horizontal semidiameter will in the zenith become increased by about it part, and at intermediate eleva, tions the increase will he as the sine of the altitude. On this principle is formed the Table at the end, entitled Augmentation of the Moon's Semidiameter, (Table XII.) and containing the proper correction to be added to the given horizontal semidiameter to obtain the true semidiameter.
On account of the great distance of the sun, no such correction of his semidiameter is necessary.
The corrections for dip and semidiameter being thus applied, the result is called the apparent altitude of the centre. In the case of the stars the only correction for the apparent altitude is the dip. It must however, be here remarked, that if the centre of the object were visible, and its altitude, instead of that of the limb, were to be taken, we should not, after applying the correction for dip, obtain precisely the same result as that which we have just called the apparent altitude of the centre, but should get a value somewhat less. The reason of this is, that every vertical arc in the heavens is shortened by refraction, as we shall shortly explain, so that the centre would not exceed the observed altitude of the lower limb, or fall short of that of the upper, by so great a quantity as the true semidiameter. Hence, from the apparent altitude of the centre, as found from applying the true semidiameter to the apparent altitude of the limb, a small quantity should in strictness be subtracted, and this small correction becomes
necessüly when the longitude is to be determined with accuracy. This correction was first proposed by Dr. Thomas Young. A table for it is given at the end. (Table XI.)
To obtain the true altitude requires two other corrections, viz. for refraction and for parallax. The former of these has indeed an effect upon the two preceding corrections, dip and semidiameter, which require certain modifications in consequence. One of these we have adverted to above, and the other will be noticed more particularly in the following article.
Of Refraction. 108. It is a universal fact in optics, that if a ray of light pass obliquely out of one medium into another of greater density, it will be bent out of its original direction at the point where it enters the new medium, and proceed through it in a direction more nearly perpendicular to its surface at that point. Hence the rays of light, proceeding from the celestial bodies, become bent downwards as soon as they enter the atmosphere, their course being directed more nearly towards the centre of the earth, so that the rays which enter the eye of an observer, and by which any celestial object becomes visible to him, would, if not thus bent down, pass over his head ; the object is therefore seen by him above its true place; the angle between this apparent direction and the true direction of the object, measures the refraction ; and, like the correction for dip, it is always subtractive; it increases from the horizon, where it is greatest, to the zenith, where it vanishes, as the rays from objects in the zenith enter the atmosphere perpendicularly.
It is the refraction which causes the sun and moon, when near the horizon, to present sometimes an elliptical appearance, the vertical diameter (and, indeed, every oblique diameter) seeming to be shorter than the horizontal, because the lower limb, or edge, being more elevated by refraction than the upper, the two are brought, in appearance, more nearly together.
At the end of the volume we have given a table of refrac
tions for different altitudes, from the horizon to the zenith, and adapted to the mean state of the atmosphere, (Table VIII.); but, as the actual state of the atmosphere generally differs from this, it becomes necessary, where the true altitude of the body is required with the utmost accuracy, to apply a correction to the numbers in this table, so as to adapt them to the existing temperature and density of the atmosphere at the time of observation, as indicated by the thermometer and barometer. This table of corrections is annexed to the table of mean refractions. It should, however, be observed that below 4o 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 this annexed table for correcting the altitude of a celestial object when the latitude of the ship is the only object of the observation, as such a correction 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.
It should be remarked here, that the dip, as determined in article (106), is on the supposition that refraction has not elevated the apparent horizon, but as such is not the case, the dip requires a correction; the amount of this correction is very uncertain, on account of the irregularity of the horizontal refractions, although it is unquestionable that some correction is requisite. It is usual to allow about or to of the computed dip for the correction. In our table to is allowed, which is according to Dr. Maskelyne, but Lambert and Legendre make it too
When the foregoing corrections have been applied to the observed altitude, the result will be the true altitude of the centre above the visible 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.
to the observer's sensible horizon EH ; and the angle SCR will be the true altitude, in reference to the rational horizon CR; and the difference of these angles is the parallax in altitude. If the body be at H, in the sensible horizon, then the difference of which we speak is the entire angle HCR; this is called the horizontal parallax.
Since the angle SE'H is equal to the angle SCR, we have for the parallax in alt., SE'H — SEH = ESC; 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 X.) 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 the 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;
it follows 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 rad. hence as rad. is to the cosine of the altitude, so is the sine of the hori
zontal parallax, to the sine of the parallax in altitude. other words, the log. sine of the horizontal para’lax, added to the log. cosine of the altitude, abating 10 from the index, will give the log. 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, 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; and, therefore, except at the equator, a small subtractive correction wil} be necessary, on account of the spheroidal figure of the earth. A table of such corrections is given at the end. (See Table XIII.)
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
l 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,” by the late Dr. Maskelyne, which accompanies every edition of that work.
Examples of the Corrections,
1. On the 14th of January, 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.
• The limb of the sun or moon is the circumforence of the disc.