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nearly the same, B falls between AS and A'S. In this case it will easily be seen that ABA' = BA'S + BAS. Hence ABA' the difference or sum of the known angles BA'S and BAS, is known.

From the latitudes Zdq and Z'd'q of the places A and A', the geocentric latitudes zCq and z'Cq may be found (76). The difference between Zdq and zCq gives the angle ZAz, and this angle taken from the zenith distance ZAB leaves the geocentric zenith distance zAB. In like manner we find the geocentric zenith distance z'A'B. Put, the app. geocen. zen. distances zAB and z'A'B, the horizontal parallaxes at A and A',

N, N'

P, P'

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the parallaxes ABC and A'BC.

the radii CA and CA',

and let R and be as in the last article.

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The values of r and r' may be found from the latitudes of the places A and A' (App. 52). Hence the quantities in the expression for, are all known.

It is not essential that the two observers should be on exactly the same meridian; for if the meridian zenith distances of the body be observed on several consecutive days, its change of meridian zenith distance in a given time will become known. Then if the difference of longitude of the two places is known, the zenith distance of the body, as observed at one of the meridians, may be reduced to what it would have been found to be if the observations had been made in the same latitude at the other meridian.

95. Moon's parallax and distance. In the year 1751, La Caille and La Lande, two French Astronomers, made corresponding observations on the moon; the former at the Cape of Good Hope

and the latter at Berlin.

From these observations, others of a similar kind which have since been made, and from other methods, the moon's parallax has been ascertained with much greater precision than it was previously known. The parallax and consequently the distance (93) are found to vary considerably during a revolution of the moon round the earth. It is also ascertained that the least and greatest parallaxes, or greatest and least distances, in one revolution of the moon, differ materially from those in another. There is, however, a mean distance, a mean of the average greatest and least distances, that is not subject to this change. The parallax corresponding to this mean distance is called the constant of the parallax. The constant of the moon's equatorial parallax is found to be 57′ 4′′. The equatorial parallax when least, is about 53′ 54′′, and when greatest, 61′ 32′′.

From tables that will be hereafter noticed, called lunar tables, the equatorial parallax of the moon may be obtained for any given time. The parallax computed from these is given in the Nautical Almanac for every 12 hours throughout the year; whence it may easily be obtained for any intermediate time. From the equatorial parallax the horizontal parallax at a given place may be found by (93 F), or by a table computed for the purpose.

Taking the moon's parallax 57' 4", we have, (93 E),

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Hence the moon's mean distance from the earth is about 60 times the equatorial radius of the earth or 239,000 miles nearly. The least distance is about 56 times the equatorial radius, and the greatest 64 times that radius.

96. Sun's parallax and distance. By the preceding method (94), the sun's parallax may be ascertained to be about 9". By a

*The Nautical Almanac is an astronomical ephemeris, published annually at London and republished at New York. It contains a large amount of data of great importance to the mariner and also to the practical astronomer. It is usually published about three years prior to the year for which it is computed. The Connaissance des Tems, published at Paris, the Astronomisches Jahrbuch, published at Berlin, and the Effemeridi Astronomiche, published at Milan, are ephemerides of a similar character. The American Ephemeris and Nautical Almanac, a work

method that will be noticed in a subsequent chapter, his mean equatorial parallax has been found to be 8".6. The parallax when least is about 8.5, and when greatest about 8".7. From the mean parallax the mean distance is found to be 23,984 times the equatorial radius of the earth, or 95,000,000 miles nearly (93 E).

97. The apparent semidiameter or diameter of a body, seen at different distances, is inversely proportional to the distance.

Let A and A', Fig. 17, be two positions from which a body, whose centre is C, is viewed. Then AB and A'B' being tangents to the body at B and B', the angle CAB is the semidiameter of the body as seen from A, and CA'B as seen from A'. Put s CAB,

♪'= CA'B, D = AC, and D' A'C. D' sin d', or D×8=D'×8'.


Then D sin d = CB = CB' =

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Regarding CB, Fig. 15, the distance of a body from the centre of the earth, as constant, the distance AB from a place on the surface must diminish as the altitude increases; and consequently the apparent semidiameter of the body, as seen from A, must increase. The apparent semidiameter, when the body is in the horizon, is sometimes called the horizontal semidiameter, and when it is elevated, the augmented semidiameter. When the expression, apparent semidiameter, is used without reference to the altitude of the body, it implies that of the body when in the horizon.

98. The sine of the apparent zenith distance of a body is to the sine of the true zenith distance, as the apparent diameter of the body at that zenith distance is to the horizontal diameter.

Let & be the horizontal semidiameter of the body, and s' the apparent diameter at B, Fig. 15. Then (97), AB': AB: : d' : d, or, since AB' may be regarded as sensibly equal to CB' or CB, we have CB: AB:: 8' : §. But CB AB:: sin ZAB: sin ZCB. Hence,

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recently commenced by our government, is equal in value and in some particulars superior to any one published in Europe. The number for the year 1855 has already been published, and that for 1856 will shortly appear. It is under the superintendence of Commander Charles Henry Davis, of the U. S. Navy.

If the apparent diameter of a body be measured with a micrometer at any observed zenith distance, and the apparent and true zenith distances be obtained (80 and 91), the above proportion gives the horizontal diameter.

For the moon, the difference between the apparent diameters in the horizon and zenith, amounts to about half a minute. For other bodies, the difference is nearly or quite insensible.

99. The sine of the equatorial parallax of a body is to the sine of the apparent semidiameter in a constant ratio.

ratio of sin
certained to be, sin я: sin 8::1: 0.27304.

For if R = equatorial radius of the earth, R' = radius of the body, and D = distance of the body from the earth, we have (93 E) R = D sin л and (97) R' = D sin d. Hence sin я : sin ♪ &:: R : R'. Therefore, since R and R' are constant quantities, the sin 8, is constant. sin d, is constant. For the moon this ratio is as

Cor. From the proportion we have R' = R.

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=2R. Hence, putting d = equatorial diameter of the earth and


d' = diameter of the body, we have d'

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100. Apparent and real diameters of the Sun and Moon. The apparent diameter of the sun at his mean distance from the earth is 32'3".6. When least, it is 31' 32".0, and when greatest, 32′ 36′′.5.

The apparent diameter of the moon at her mean distance is 31′ 39′′.6. When least, it is about 29′ 26′′, and when greatest, 33' 37".

Taking the sun's apparent semidiameter at his mean distance, and the corresponding parallax (96), we find (99 H) the sun's real diameter to be nearly 112 times the equatorial diameter of the earth, or more than 880,000 miles. His bulk is therefore about fourteen hundred thousand times that of the earth.

In like manner we find the moon's diameter to be about

the equatorial diameter of the earth, or 2160 miles.



The moon's surface is therefore about of that of the earth, and her volume or bulk about of the earth's volume.



101. The polar distance of a body, when on the meridian, is equal to the sum or difference of the complement of the latitude of the place and the zenith distance of the body, according as it culminates to the south or north of the zenith.

Let M, Fig. 1, be the point at which a body is when on the meridian of the place A. Then PM = PZ + ZM. But PM is the polar distance of the body, ZM its zenith distance, and PZ the complement of the latitude of the place. If the body be on the meridian at I, to the north of the zenith, we have PI = PZ — IZ; if at F, we have PF = FZ - PZ.

102. To find the polar distance or declination of a body. Let the meridian zenith distance of the body be observed at a place whose latitude is known, and be corrected for refraction and parallax. Then, by the last article, the polar distance becomes known. If the body is a fixed star, the zenith distance only requires correction for refraction, as the star has no sensible parallax. When the body has a sensible diameter, the apparent semidiameter added to, or subtracted from, the observed zenith distance of the upper or lower limb, when corrected for refraction and parallax, gives the true zenith distance of the centre.

The declination is evidently equal to the difference between the polar distance and 90°, and is north or south, according as the polar distance is less or greater than 90°. It therefore becomes known when the polar distance is known.

The polar distances or declinations of the heavenly bodies, are found to vary more or less from day to day, except those of the fixed stars, which continue sensibly the same for several days in succession; but after a longer interval, changes become also perceptible in them.

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