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a revolution of the sun; and a revolution of the sun is called a year. The year from equinox to equinox is called the equinoctial year, or sometimes the tropical year; for his time of returning from tropic to tropic, they being situations always holding the same relation to the equinox for the time being, is obviously the same as that from equinox to equinox. The year from any point in the ecliptic to the same point again is called the sidereal year, for the sun is then in the same position as before, with relation to the stars. The sun's angular distance from the apogee is called the true anomaly, and the period between his leaving and returning to a given situation with respect to the apogee is therefore called the anomalistic year.

§ 425. It is evident that the equinoctial is the shortest, and the anomalistic the longest of these years. When the sun starts from the equinox, it is a given point of his orbit; before he returns to it, the equinox has receded on the ecliptic, and he therefore meets it again sooner than he returns to the same spot in his orbit. The effect therefore of the retrograde motion of the equinoctial point on the ecliptic is to bring forward the time of the equinox, (or the instant at which the sun is at the equator); and hence, as we have already mentioned, the phenomenon is known by the name of the precession of the equinoxes. In the mean time, however, the apogee has moved forward on the ecliptic; and the sun, therefore, after returning to the same point in his orbit where he was at the former equinox, has still a further arc to describe before he arrives at his original position with respect to the apogee, and the time of his doing so is of course later.

The mean length of the equinoctial year is 365d. 5h. 48m. 51.6s., (or decimally 365d.242264) of mean solar time. After this, the sun has to describe 50."1 to return to the same point of his orbit at which he was at the commencement of the year, or to complete the sidereal year; and the mean length of the sidereal year is thus made 365d. 6h. 9m. 11.5s. or 365d.256383. He then has to describe a further arc of 11."8 to arrive at his original position with respect to the apogee, and the length of the anomalistic year is thus made 365d. 6h. 13m. 58s.8, or 365d. 259708.

$ 426. The lengths assigned to the equinoctial and sidereal years are only mean lengths; that given to the anomalistic year is a true one. We shall hereafter shew, from other considerations, that the length of the anomalistic year does not vary. For the present, we will assume that fact, and then it is obvious that the length of the equinoctial and sidereal years must continually vary; for each of these years is shorter than the anomalistic year by the time which the sun takes to describe a given angle of his orbit; in one case 62", in the other 11."8. Now the rate of the sun's motion is different in different parts of his orbit, faster as he is further from the apogee, slower as he approaches it; and, consequently, his times of describing these spaces of 62" and 11."8 continually vary as they are differently situated with respect to the apogee. The times therefore which are to be subtracted from the uniform length of the anomalistic year, to ascertain those of the equinoctial and sidereal years respectively, are themselves of variable duration; and the lengths of the equinoctial and sidereal year are necessarily so too. The variation, however, is very small, and the mean differs from the true length at any period by a very inconsiderable quantity.

§ 427. It is obviously necessary, for many purposes, not only of chronology and history, but even of personal and domestic convenience, that we should have the means of dividing time into definite periods of considerable length; and the most obvious and natural period to adopt, is that which includes all the various operations and appearances which succeed each other in regular order, which comprehends seed-time and harvest, summer and winter.

The tropical or civil year, of 365d. 5h. 48m. 49s.7, is the time elapsed between the consecutive returns of the sun to the mean equinoxes or solstices, including all the changes of the seasons, is a natural cycle peculiarly suited for a measure of duration. It is estimated from the winter solstice, the middle of the long annual night under the north pole. But although the length of the civil year is pointed out by nature as a measure of long periods, the incommensurability that exists between the length of the day

and the revolution of the sun, renders it difficult to adjust the estimation of both in whole numbers.

§ 428. If the revolution of the sun were accomplished in 365 days, all the years would be of precisely the same number of days, and would begin and end with the sun at the same point of the ecliptic. But as the sun's revolution includes the fraction of a day, a civil year and a revolution of the sun have not the same duration. Since the fraction is nearly the fourth of a day, in four years it is nearly equal to a revolution of the sun, so that the addition of a supernumerary day every fourth year nearly compensates the difference. But in process of time further correction will be necessary, because the fraction is less than the fourth of a day. In fact, if a bissextile be suppressed at the end of three out of four centuries, the year so determined will only exceed the true year by an extremely small fraction of a day; and if, in addition to this, a bissextile be suppressed every 4,000 years, the length of the year will be nearly equal to that given by observation. Were the fraction neglected, the beginning of the year would precede that of the tropical year, so that it would retrograde through the different seasons in a period of 1,507 years.

The division of the year into months is very old and almost universal. But the period of seven days, by far the most permanent division of time, and the most ancient monument of the common origin of the human race, was used by the Brahmins in India with the same denominations employed by us, and was alike found in the calendars of the Jews, Egyptians, Arabs and Assyrians. It has survived the fall of empires, and has existed among all successive generations.

CHAPTER XIX.

PARALLAX.

Parallax defined. Horizontal Parallax. Methods of correcting for Parallax. Determination of the Moon's Parallax. Transits of Mercury and Venus. Methods of computing the Solar Parallax. Distances of the Sun and Planets. Parallax of the Fixed Stars.

§ 429. We are now prepared to enter more particularly into the effects of parallax, both diurnal and annual, and to show the knowledge obtained by its means.

The centre of the earth is really the place to which all motions in the solar system should be referred. The centre of the sun is the point to which all out of it should be referred. Yet we are never in either of those places; we are always 4,000 miles from the one, and upwards of 95,000,000 from the other. Observations made on dif ferent parts of the surface of the earth must therefore be corrected and referred to the centre of the earth. We must always know on what part of the earth, at what period of her rotation and of her revolution, a given observation was made.

But this eccentric position, which seems so disadvantageous, is the only foundation of an accurate knowledge of the absolute dimensions of the solar system. Without it we could not ascertain the distances, and consequently the real magnitudes of the heavenly bodies.

§ 430. The sun, moon and planets assume different positions among the fixed stars which are at an incalculable distance, when viewed by two observers in different parts of the globe. This difference of position is termed the parallax of the heavenly body; and when it is in the horizon of one of the observers such difference is called its horizontal parallax.

To an observer at A, (Fig. 7, Plate II.) a heavenly body B, will appear in the horizon, either rising or setting, and he would refer its position among the fixed stars to the

point G in a circle immeasurably distant, although the limits of the diagram compel us to contract its dimensions. Another observer at A', will have the same body, which we will suppose to be the moon, in the zenith; a line joining C, the centre of the earth, and his position, will pass through B. This observer, then, views the object as it would appear from the earth's centre, and refers it to the point E; a position distant from the former by the arc EG. Now if two observers remark the position of the moon at the same instant of time, and afterwards compare notes, the measure of this arc will be known, and consequently the value of the angle E B G, which is equal to the angle A B C, or the angle which the earth's semi-diameter would subtend when seen from B, the moon.

§ 431. Since the triangle C A B, is right angled at A, we know A C, the earth's semi-diameter, the angles CB A and CA B; whence may be found the side CB, by the following proportion :

Sine ABC: rad. :: CA:: CB.

In the case of the moon, her mean horizontal parallax is 57 12"; therefore,

As sine 57' 12": rad. : : 3,956 miles, the earth's semidiameter: 237765, the moon's distance.

Her diameter may be easily found when her distance is ascertained, (Fig. 8, Plate II.): the angle ECG is that subtended by the diameter of the moon; half of this will be ECB, which is one angle of the right-angled triangle EC B, of which the base C B is known, whence the perpendicular E B may be easily found by trigonometry: this multiplied by two will give E G the diameter of the moon, which is about 2,000 miles her mean distance is about 240,000 miles.

§ 432. As the heavenly body rises above the horizon, its parallax will become less and less; thus, at H, (Fig. 7, Plate II.), its place, as seen from the centre, will be S; from A, it will be W; its arc of displacement SW being less than E G; when at N, QS will be less than SW; while at M, in the zenith, its parallactic angle will vanish, and its position as seen from C, and also from A, will be I.

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