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time, will be the interval between noon and sunset. Supposing the declination not to change during the day, it is evident the interval between sunrise and noon will be just the same. The angle APM or its measure, the arc DQ, is called the semi-diurnal arc, and the arc ED, the semi-nocturnal arc. The arc DW, contained between a declination circle, passing through the point of setting or rising, and the western or eastern point of the horizon, is called the ascensional difference.

In the right angled spherical triangle ADW, we have (App. 48), tang AD sin DW tang EWH sin DW cot PH,

or, sin DW tang PH tang AD,

or, sin ascen. diff. tang lat. x tang declin.

Now, as the angle WPQ, or its measure WQ, is 90° or 6 hours, it is evident that when the declination is of the same name with the latitude, the sun will set at some point A, between H and W, and the ascensional difference DW, converted into time and added to 6 hours, will give the semi-diurnal arc in time. When the declination is of a different name from the latitude, the sun will set at some point A', between W and R, and the ascensional difference subtracted from 6 hours will give the semi-diurnal arc.

The semi-diurnal arc in time, evidently, expresses the time of sunset, and, subtracted from 12 hours, it gives the time of sunrise. These are in apparent time, which may be converted into mean, by applying the equation of time.

We may find an expression for the semi-diurnal are without first. obtaining the ascensional difference. For cos DQ = cos (WQ + DW) = cos (90° + DW) sin DW. Hence, cos DQ tang PH tang AD,

or, cos semi-diur. arc

tang lat. × tang declin.

184. To find the time of the sun's apparent rising or setting. When the sun's centre appears to be in the horizon at a point a, Fig. 31, we have (80 and 91), his zenith distance Zb 90° + re fraction — parallax.

Put, Z = Zb = sun's zenith distance,

L= PZ complement of the latitude,


D= Pb sun's polar distance, and

P = angle ZPb = dQ semi-diurnal arc.

Then in the spherical triangle ZPb, we have (App. 38),

sin P = ✓

sin 1 (Z + D — L) sin (Z + L — D)

sin L sin D

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If we take Z = 90° + refraction — parallax - semi-diameter, the above formula gives the semi-diurnal arc for the apparent rising or setting of the sun's upper limb.


185. To find the time of beginning or end of twilight.

At the beginning or end of twilight the sun is 18° below the horizon (88). Let B be the position of the sun when at this distance below the horizon. Then, in the triangle ZBP, we have L PZ and D PB as in the last article, and Z Hence,


sin P


ZB 90°

sin (108° + DL) sin (108° + L −
sin L sin D


186. Given the latitude of a place and the sun's declination ană altitude, to find the time of day.

Let S be the position of the sun. Taking D and L as above, and Z ZS 90° SK, and P = the hour angle ZPS; the value of P may be found by the formula in article 184.




187. Conjunction, &c. A body is said to be in conjunction with the sun, or simply in conjunction, when its position is such, that its longitude and that of the sun are the same;* to be in opposition, when their longitudes differ 180°; and to be in quadrature, when their longitudes differ 90° or 270°. The term syzygy is used to denote either conjunction or opposition.

* When any two of the heavenly bodies have the same longitude, they are said to be in conjunction.

When the body is in any of the four positions, midway between the syzygies and quadratures, it is said to be in octant.

As each of the planets Mercury and Venus revolves at a less distance from the sun than that of the earth (18), either of them may be in conjunction both on the same side of the sun with the earth and on the opposite side. The former is called inferior, and the latter, superior, conjunction.

Some of the preceding terms are frequently designated by characters, as follow:

Conjunction d

Opposition 8

Quadrature .

188. Nodes. The two points in which the orbit of the moon or a planet is cut by the plane of the ecliptic, are called nodes. That node in which the body is, when passing from the south to the north side of the ecliptic, is called the ascending node, and the other the descending node. The nodes are frequently designated by the following characters:

Ascending node &

Descending node 8.

189. Different revolutions of a body. The sidereal or periodic revolution of a body is the time during which it makes a real revolution round the central body; or, it is the period that elapses from the time the body and a fixed star have equal longitudes, supposing them to be observed at the central body, till their longitudes are again equal.

The tropical revolution of a body is the period that elapses from the time it is at the vernal equinox, or any given longitude, as seen from the central body, till its return to the same.

The synodic revolution of a body is the interval between two consecutive conjunctions or oppositions of the body. In the case of Mercury or Venus, it is the interval between two consecutive conjunctions of the same kind.

The anomalistic revolution of a body is the interval between two consecutive returns of the body to the perigee or perihelion of its orbit, or to the same anomaly.

The nodical revolution of a body is the interval between two consecutive returns of the body to the same node.


189.a Moon revolves round the earth. The moon appears to make a complete circuit of the heavens in rather less than a month (6), Hence, either the moon really revolves round the earth, or the latter revolves round the former. That the moon's apparent motion is not, like that of the sun, produced by a motion of the earth, but that it is a real motion, follows from the relative sizes of the bodies: the bulk of the earth being nearly fifty times that of the moon (100).

Strictly speaking, the earth and moon both revolve about the centre of gravity of the two, which is a point in the line joining the centres, situated a small distance within the earth's surface. This follows from the principles of mechanics, and is in accordance with their motions as deduced from observations.

The distance of the moon from the earth is only about the 400th part of that of the sun (95, 96). While, therefore, the moon revolves round the earth, she at the same time revolves with the earth round the sun.

190. Moon's tropical revolution. From observations made when the moon is on the meridian, her right ascension and declination are easily obtained (142, 102). With these, and the known obliquity of the ecliptic, her longitude may be computed (119). By daily, or at least frequent, observations and computations of this kind, the interval, from the time at which the moon has any given longitude, till her return to the same, may be determined. This interval, which is the tropical revolution of the moon, is found to be subject to considerable variation. Its mean length is about 27.32 mean solar days.*

191. Moon's path. The moon's observed right ascension and declination serve to determine her latitude as well as longitude (119). Frequent determinations of both, during her revolution round the earth, show that her path does not coincide with the

* For more exact expressions of this and other periods, see tables at end of Part I.

ecliptic, but is inclined to it in a small angle, intersecting it in two opposite points or nodes.

192. Moon's mean daily motion in longitude. As the moon moves through 360° of longitude during a tropical revolution, we have the following proportion :-As moon's mean tropical revolution: 1 day :: 360° : moon's mean daily motion in longitude. The mean daily motion is thus found to be 13° 10′ 35′′.

193. Acceleration of moon's mean motion. A comparison of modern observations with those of former periods proves that the moon's mean motion is subject to a small, but, thus far, a continual acceleration. The mean daily motion given in the preceding article is that which had place at the beginning of the present century. Investigations in physical astronomy, by Laplace, have made known the cause of the acceleration, and have shown that it is a periodical inequality in the moon's mean motion, requiring a great length of time to go through its different values. It is occasioned by the change in the eccentricity of the earth's orbit (162).

As the acceleration is a periodical inequality, the mean length of the moon's revolution which is now diminishing must, in process of time, cease to do so, and afterwards increase.

194. Moon's sidereal revolution. In consequence of the westerly or retrograde motion of the equinoxes (124), the moon returns to the vernal equinox, or to the same longitude, before she has quite completed a real or sidereal revolution. From the known precession of the equinoxes, which is 50′′.2 in a year (125), their motion during a tropical revolution of the moon is found to be 3".75 nearly. This, subtracted from 360°, leaves the arc of the ecliptic moved through by the moon during a tropical revolution. Hence, as this arc : 360° :: the tropical revolution of the moon: the sidereal revolution. It is thus found that the sidereal revolution exceeds the tropical by about 7 seconds.

195. Moon's synodic revolution. It is evident, that during the interval between two consecutive conjunctions of the moon with the sun, that is, during a synodic revolution of the moon, the excess of the moon's motion in longitude above that of the sun must be 360°. From the mean daily motions of the sun and moon in

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