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seconds of time, allowing 150 to the hour, 15' to the minute, and 15" to the second of time.

The right ascension is also the difference in the time of transit of the heavenly body and of the first point of Aries over the meridian of any place.

The difference of right ascension of two stars is the difference in their times of meridian transit at the same place. Or it is the angle comprehended between the two hour circles which pass through the stars. A catalogue of stars is a list of them with the right ascension and the declination of each annexed.*

Having described the principal circle and points of the celestial sphere which are considered as permanent, or which do not alter with the situation of the observer on the earth, we come now to describe those which change with his place. The principal of these is the horizon, which has been defined already (Art. 79), and vertical circles, which are secondaries to the horizon, and on which the altitudes of celestial objects are measured. These vertical circles all meet in two points diametrically opposite, viz., the poles of the horizon; one of which is directly over the head of the observer, and called his zenith, and the opposite one his nadir. The ver

* The late catalogue of the British Association, the name of which is abbreviated B. A. C., gives the north polar distances (N. P. D.) of the stars instead of their declinations. The north polar distance of a star is its distance from the north pole of the heavens, measured on the circle of declination passing through the star. The right ascension of the star fixes the position of this circle in the heavens, and the north polar distance fixes the place of the star upon the circle, so that its position is completely determined by these two co-ordinates. In the British Catalogue is a column containing the annual variation in R. A., and four columns marked a, b, c, d, at top; also a column containing the annual variation in N. P. D., and four columns marked a', b', c', d'. The numbers whose logarithms are in these columns may be regarded as constant for a period of about ten years. In the Nautical Almanac, on p. XXII. of each month, will be found four columns marked A, B, C, D, at top, containing the logs. of numbers, which vary with the time, or are ephemeral.

To find the R. A. of a star for any given time, take out its R. A. for the epoch of the catalogue, viz., 1850, to which add the product of the annual variation in R. A., by the number of years between the given time and 1850. The result will be the mean R. A. at the beginning of the given year. Take out from the columns a, b, c, d, the logs. opposite the given star, and from the Nautical Almanac, from the columns A, B, C, D, the logs. corresponding to the given date, for which the apparent R. A. is required, and with these logs. compute the following formula:

da sa + Bb + cc + Dd

da being the correction to be applied to the result before found, to obtain the R. a. required. This will be the time at which a star ought to make its meridian transit by the siderial clock. The formula for the correction in declination is

disa' + Bb' + cc' + Dd'

tical circle which passes through the east and west points of the horizon is called the prime vertical; it necessarily intersects the meridian of the place (which passes through the north and south points) at right angles.

The azimuth of a celestial object has been already defined to be an arc of the horizon, comprised between the meridian of the observer and the vertical circle through the object, and hence vertical circles are sometimes called azimuth circles.

The amplitude of a celestial object is the arc of the horizon comprised between the east point and the point where the object rises, or between the west point and that where it sets; the one is called the rising amplitude, the other the setting amplitude.


105. The true altitude of a celestial object is always understood to mean its angular distance from the rational horizon of the observer. This is not obtained directly by observation; but is the result of certain corrections applied to the observed altitude.* These we shall now enumerate and explain.

* The observed altitude is obtained by means of an instrument called a quadrant of reflection, or simply a quadrant. This instrument is a frame of wood in the form of a sector of a circle, the arc of which is graduated to degrees and parts of a degree. This frame is suspended so that the plane of the circle shall be vertical. It has an arm, one extremity of which is attached to the centre of the circle, and which is movable about this point; upon this arm is a small mirror, and opposite to it is a plane glass, half of which is mirror, and half transparent. When a heavenly body, seen by double reflection in these two mirrors, is brought by the movement of the arm, upon which one of the mirrors is placed, to coincide with the line of the horizon at sea as seen through the transparent part of the opposite glass, the outer extremity of the arm points out upon the graduated are the number of degrees of altitude of the heavenly body above the horizon.

The construction of this instrument depends upon the optical principle that the angle of incidence is equal to the angle of reflection. The angular movement of the image of the heavenly body is double the angular movement of the arm, so that to measure the greatest altitudes, the limit of which is 90°, the graduated are need be but the eighth of a circumference; the degrees upon it are however numbered as if it were a quadrant, to save the trouble of doubling them. The instrument takes its name from the amount which it measures, instead of from the magnitude of its arc. There are colored glasses attached, which can be interposed so that the rays of light, coming from the heavenly body to the eye, can be made to pass through them when taking the altitude of the sun.

More complete instruments of this nature are the sextant and repeating circle, or circle of reflection, for full descriptions of which see p. 290, and p. 299.

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this observed altitude the angle HEH', called the Dip or Depression of the Horizon, must be subtracted to obtain the apparent altitude sEH.

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The angle HEH', or its equal c, is calculated for various elevations, AE, of the eye above the surface of the sea, by resolving the right angled triangle EBC, in which are known CB, the radius of the earth, and EC equal to the radius increased by the height of the eye. The results are registered in a table (Table XXXI.), the argument of which is the height of the eye.

The depression thus obtained must be lessened by the amount of terrestrial refraction, which is very uncertain; of the whole quantity has

been allowed in computing this table.


107. The foregoing correction for dip having been applied, the result will be the apparent altitude of the object observed, above the sensible horizon. If this be the upper or lower edge of the disc of the sun or moon, called the upper and lower limb, a further correction will be necessary to obtain the apparent altitude of the centre. The angle at the eye of the observer, subtended by the semidiameter or radius of the sun or moon, must be added to the altitude of the lower limb, and subtracted from that of the upper limb. This quantity, which is continually varying both for the sun and moon, in consequence of the variation of their distance from the


earth, is given in 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, i. e. the apparent semidiameter when the moon is in the horizon, where the distance from the observer is greater than when she is in the zenith by the semidiameter of the earth. Consequently her apparent semidiameter, which is inversely as her distance, will be least in the horizon, and greatest in the zenith; and its value between these limits will vary with the sine of the altitude, as may be easily seen by constructing a diagram.

The distance of the moon being about 60 semidiameters of the earth, the moon's horizontal semidiameter will be increased about part in the zenith. Therefore, if to the logarithm of the sine of of the D's horizontal semidiameter or the log. of the arc itself, which is small, we add


* It is given for noon of each day for the sun, and for noon and midnight for the moon, and is found for any other time of day by the proportion: As 24 or 12 hours the variation in 24 or 12 hours: : the time after noon or midnight, at Greenwich the variation in that time, which must be added to the semidiameter given in the Almanac or subtracted, according as the semidiameter is increasing or diminishing from day to day, in order to have the semidiameter at the required time.

Proportions of this kind, in which the terms contain two or three denominations, as hours and minutes, minutes and seconds, or hours, minutes, and seconds, degrees and minutes, &c., may be resolved conveniently by means of the table of proportional logarithms, Table XXII.

The following example will illustrate the mode of proceeding.

24: 16' 19' :: 8h 2m

Taking the first and third terms one grade lower, we find their proportional logarithms (P. L.) on pp. 134 and 132, writing the arith. comp. of the former, and taking from p. 133 the P. L. of 16' 19", the calculation will be as follows:

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In this as in many other problems of Nautical Astronomy, the time at Greenwich at the instant of observation is required, and may be found by adding or subtracting the difference of longitude in time, according as the place is w. or E. of Greenwich. Thus the time at Greenwich, corresponding to any given time at New York, is found by adding 44 56m 4 (the difference of longitude between the two places) to the latter.

the log. sine of the D's altitude, the result will be the log. of the apparent semidiameter at the given altitude.

In this way is formed the Table at the end, entitled Augmentation of the Moon's Semidiameter (Table XXXIII.), which contains 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.

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, which we shall notice after explaining the nature and effect of


108. The rays of light coming from a heavenly body, having to pass. through the atmosphere, are bent towards the vertical by refraction. As the atmosphere grows more and more dense in approaching the surface of the earth, the light bending continually towards the vertical pursues a curvilinear path in a vertical plane, and enters the eye in the last direction of its motion, which prolonged is a tangent to the curve, and it is in the direction of this tangent that the object emitting the light appears. The curve being convex upward, the tangent lies above it, and the effect of refraction is therefore to elevate the object, or to make the apparent place above the true place. The correction for refraction, therefore, likethe correction for dip, is always subtractive; it decreases 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) in accordance with the optical law that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant.

At the end of the volume we have given a table of refractions containing the correction for refraction to be applied to every altitude, from the horizon to the zenith,* and adapted to the mean state of the atmo

* It will be observed that there is in the table a column of differences for 1' of altitude. The number in this opposite the degrees in the given altitude must be multiplied by the given minutes and the result subtracted from the correction, or added to the altitude.

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