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of right lines be drawn from it, forming, with each other, the angles that are described daily. Then, as these lines would pass through the centres of the Earth and Sun at those respective times (each noon for instance), from S on each of these lines set off the corresponding radii vectores, SG, SE, SE', &c. and through all these points G, E, E', &c. describe a curve line, which will be the ecliptic, or the orbit described by the Earth in its annual revolution round the Sun. This curve is an ellipse, in which CA or CP is the mean distance; and GH, perpendicular to AP at the centre C, is the less axis: and since GS is equal GF, and their sum is AP, SG is therefore equal to CP, that is, to the mean distance. Finally, CS or CF is what is called the excentricity, which is the difference between the mean radius CP and the greatest distance AS, or the least, SP.
It is this elliptic orbit which the Earth describes in the course of a year; and, in order to determine the actual duration of that period, astronomers have carefully observed the secular movement of the Sun, or the arc described by his apparent motion during 100 Julian years, which M. Delambre has determined to be equal to 36000°.7625 very nearly. Then the tropical year is readily obtained by a simple proportion; for the arc described by the apparent movement of the Sun in 100 years, is to 360 degrees, as the number of days in 100 years is to the time of making one revolution or describing 360 degrees; viz.
as 36000°.7625 : 360° :: 36525d : 3654.242264; hence the exact length of the tropical year, according to this statement, is 365 days 5 hours 48 minutes and 51.6 seconds.
It is by observing the appearances which the Moon presents, and submitting them to analysis, that we ascertain the laws of her movement, and the
numerous irregularities to which it is subject. These methods we shall now endeavour briefly to explain.
The variations to which the apparent diameter of the Moon is subject, indicate that her distance from the Earth is considerably greater at one time than another. These variations may either be ascertained by observation, and directly measured by means of the micrometer, or found from the times which elapse between the immersions and emersions in occultations of the stars, by the Moon in her passage round the Earth. The greatest apparent diameter of the Moon has been determined by various methods, and found to be 2011".068 or 33'.5178; and the least equal 1761.912 or 29'.3652. The analogous values for the sum are 1955".567 or 32'.5928, and 1890".961 or 31'.516. Hence the variations in the distance of the Moon from the Earth are greater than those in the distance of the Earth from the Sun; since the apparent diameter of the Moon is sometimes greater than the greatest, and at others less than the least, apparent diameter of the Sun.
Observations of the parallaxes also confirm this result. By comparing the parallaxes of the Moon observed at the same place, but at different times, considerable variations have been found. The greatest horizontal parallax of the Moon for 48° or 50° of latitude, is 1°.0248, and the least 0°8975; and it takes successively all possible values between these
Now, calculating the greatest and least distance of the Moon from the Earth, according to these given quantities, we obtain the following results in terrestrial radii :
The arithmetical mean between these two extremes 59.87915, that is about 60 times the radius of the
Earth in the same latitude. If the greatest distance. be divided by the least, the quotient 1.1417 will give the greatest variations to which the distance between the Earth and the Moon is subject. The same result may likewise be obtained by dividing the extreme parallaxes, the one by the other; for these are inversely as the distances. The ratio found in the same manner for the Sun is 1.034; and the union of these two results leads to two important consequences, viz.-The Moon is very near the Earth in comparison with the Sun, and the variations it experiences in its distance are proportionally much greater than those of the Sun.
The lunar parallax not only varies for different epochs, in consequence of these inequalities, but its value is not the same at the same time, when observed at different places of the Earth's surface. Modern observations give sensible differences in this respect. In order to comprehend these consequences, it must be recollected that the horizontal parallax of the Moon is the angle which a radius of the Earth, drawn from the centre to the observer, subtends at the Moon. Since this parallax is not the same at different places at the same epochs, it proves that the radii of the Earth are not all equal to each other, and consequently that the surface of the Earth is not spherical. Hence the Moon, by the variations in its parallax, indicates the inequalities of the Earth's surface, as well as proves its general sphericity by its eclipses. The inequalities of these radii have therefore an influence upon the apparent positions of the Moon when observed from different latitudes. These inequalities must be taken into the account when the apparent position of the Moon is required to be calculated with great accuracy; as in the occultations of the stars.
By dividing the apparent semidiameter of the Moon, reduced to the centre of the Earth, by its horizontal parallax for any given place, the result S
will give a constant ratio. This ratio is that which subsists between the radius of the Moon and the radius of the Earth, considered as spherical. According to the best observations, the value of this ratio for the equator is 0.24564, by taking 0°.27293 for the semidiameter reduced to the centre of the Earth, when the parallax is 0°.9999, or very nearly 10. This result being once obtained, it will serve for calculating the apparent diameter when the horizontal parallax is known; or, reciprocally, for determining the parallax when the diameter is given.
In the diurnal revolution of the Moon about the Earth, she approaches nearer to an observer situated on its surface when she is in his zenith than in the horizon. This difference, it has already been observed, produces a sensible effect upon her apparent diameter, which increases with her altitude. This augmentation is easily calculated, and its total value from the horizon to the zenith is about th, since the distance between the observer and the Moon is diminished by a quantity nearly equal to the radius of the Earth, which is very nearly a 60th part of the mean distance.
The Moon moves in an elliptical orbit, of which the Earth, occupies one of the foci, and she accompanies the Earth in its annual orbit round the Sun. In the preceding figure, E represents the Earth, M the Moon, mMo the lunar orbit, and EM the radius vector of that orbit. This radius vector, in its motion round the point E, describes areas which are nearly proportional to the times of description. The mean distance of this body being taken for unity, the excentricity of its orbit in 1800 was 0.0548553; and the method by which this result is deduced from observation is the same as employed for the solar orbit. The revolution of the Moon with respect to her apsides, called the anomalistic revolution, is performed in 27.5546 days.
The orbit of the Moon is oblique to the equator
and to the ecliptic; its inclination to the latter plane varies within small limits, which may easily be determined; and its mean value is about 5°.14998. The differences of these values at each revolution may be immediately found from observation, by determining the greatest latitude of the Moon, in the same manner as the obliquity of the ecliptic is found from the greatest declination of the Sun.
The mean arc which the Moon describes parallel to the ecliptic in a hundred Julian years, constitutes what is called its secular tropical movement. In 1800, this arc was very nearly 481267°.8793; and which divided by the number of days in 100 years, or 36525, gives the tropical motion equal to 130.17636; and hence this motion of the Moon is about thirteen times as great as that of the Sun.
From these given quantities, the time in which the Moon returns to the same longitude, or her periodic revolution, may be easily found by proportion; for as 4812679.8793: 360°:: 36525 days: 27.32158 days, which is denominated the lunar month.
If from the secular tropical motion the precession of the equinoxes for a century be subtracted, the remainder will be the secular sidereal movement, or that of the Moon with respect to the stars; for the motion of the stars and that of the Moon, with respect to the equinoxes, are both in the same direction, and, therefore, their relative motion is equal to the difference of the arcs described in the same time. Now, as the precession of the equinoxes in a century is equal 5010".012 or 1°.39167, and which taken from the secular tropical movement, gives 481266°.4876 for this movement in 100 years; and, therefore, by a proportion in all. respects similar to that above by which the lunar month is obtained, we have 27.32166 days for the sidereal revolution. These two revolutions are therefore connected together by the equinoxes, and consequently may readily be deduced the one from the other. They have not always had the