axis of the Earth in every part of its revolution about the Sun, will make an angle of 66° 32′ with the plane of the ecliptic; and this inclination occasions the successions of the four seasons, as has already been illustrated, in the preceding part of the work.

Observations, separated by a long interval, point out that the obliquity of the ecliptic is diminishing at nearly the rate of half a second in a year; that is, the ecliptic appears approaching the equinoctial by half a second in a year. The secular diminution of the obliquity of the ecliptic is at this time 50 seconds, according according to Dr. Maskelyne and M. De La Land. But later observations, and the calculations of La Place, give 52,1 for the secular diminution,

Physical astronomy shows that this arises from a change in the plane of the Earth's orbit, occasioned by the action of the planets: that this change of obliquity will never exceed a certain limit, which limit, according to La Place, is 20 42'; and that by this action of the planets, the ecliptic is progressive on the equator 14" in a century.

Besides this progressive motion of the Earth's axis towards a perpendicular direction to the plane of the ecliptic, it has a kind of libratory motion, by which the inclination is continually varying, a certain number of seconds, backwards and forwards; the period of these variations is nine years. The tremulous motion is termed the nutation of the Earth's axis. This motion is caused by the joint effect of the inequalities of the action of the Sun and Moon upon the, spheroidal figure of the Earth.

5. The equinoctial points have a retrograde motion, at the rate of about 1o in 72 years, or more accurately 50" in a year; consequently the Sun returns again to the same equinoctial point before he has completed his revolution in the ecliptic: so that the equinoxes precede continually the complete apparent revolution of the Sun in the ecliptic by 20 minutes, 20 seconds of time.

This retrograde motion, of 501" in a year, of the equinoctial points, is usually called in books of astronomy the precession of the equinoxes; but, as Delambre very pro

perly remarks, it should be called the recession or retrogradation of the equinoctial points, and reserve the term precession for the anticipation of the moment of the equinox; so that 20 minutes 20 seconds, the time which the equinoxes precede continually the complete revolution of the Sun in the ecliptic, in consequence of the recession of the equinoctial points, ought to be called the precession of the equinoxes. Although the place of the celestial pole among the fixed stars has been considered as not changed by the annual motion of the Earth, yet in a longer period of time it is observed to be changed, and also the situation of the celestial equator, while the ecliptic retains the same situation among the fixed stars. Observation shows that this change of the pole and equinoctial is nearly regular. The pole of the equinoctial appears to move with a slow and nearly uniform motion, in a small circle, round the pole of the ecliptic, while the intersections of the equinoctial and ecliptic move backward in the ecliptic, or contrary to the order of the signs, with a motion nearly uniform. In consequence of this apparent motion, all the fixed stars increase their lon gitudes by 50-" in a year, and also change their right as censions and declinations. Their latitudes remain the same. The period of the revolution of the celestial equinoctial pole about the pole of the ecliptic is nearly 25,920 years.

The north celestial pole therefore will be, about 12,960 years hence, nearly 490 from the present polar star; and about 10,000 years hence, the bright star Vega in Lyra will be within 50 of the north pole. This star, therefore, which now, in the latitude of about 54 degrees, passes the meridian within a few degrees of the zenith, and twelve hours after is near the horizon, will then remain nearly stationary with respect to the horizon. All which will readily appear, from considering the celestial concave surface as represented by a common celestial globe.

This motion of the celestial pole originates from a real motion in the Earth, whereby its axis, preserving the same inclination to its orbit, has a slow retrograde conical motion. The cause is shown, by physical astronomy, to arise from the attraction of the Sun and Moon on the excess of matter at the equatorial parts of the Earth.

6. It has already been observed that the astronomical days are not equal, and that two cause combine to produce their difference: that is, t

obliquity of the ecliptic with respect to the equa tor, and the unequal motion of the Earth in an.elliptical orbit.

7. That part of the equation of time, or the difference between the mean and apparent time, arising from the obliquity of the ecliptic is the greatest about February 5th, May 6th, August 8th, and November 8th; and is nothing about March 21st, June 21st, September 23d, and December 21st, or when the Sun is in the four cardinal points of the ecliptic.

As the Earth's axis is perpendicular to the plane of the equator, any equal portions of this circle, by means of the Earth's rotation on its axis, pass over the meridian in equal times; and so, in like manner, would any equal portion of the ecliptic, provided it were parallel to, or coincident with, the equator. But as this is not the case, the daily motion of the Earth on its axis carry unequal portions of it over the meridian in equal times; the difference being always proportional to the obliquity: and as some parts of the ecliptic are much more obliquely situated with respect to the equator than others, those differences will, therefore, be unequal among themselves.

This part of the subject may be pleasingly illustrated on the terrestrial globe, by placing patches on the ecliptic and equator at every tenth or fifteenth degree; then, by turning the globe gradually on its axis, the patches will pass under the meridian at different times, thus exhibiting the phenomena already described. And that part of the equation of time depending upon the obliquity of the ecliptic, may be found by the terrestrial globe: thus, bring the Sun's place in the ecliptic to the brazen meridian; count the number of degrees from the beginning of the sign Aries to the brazen meridian, both on the equator and on the ecliptic; the difference, reckoning 4 minutes of time to a degree, is the equation of time. If the number of degrees on the ecliptic exceed those on the equator, the Sun is faster than the clock; but if the number of degrees on the equator exceed those on the ecliptic, the Sun is slower than the clock.

8. The second part of the equation of time arising from the unequal motion of the Sun in the

ecliptic, is the greatest about March 30th, and Oсtober 3d; and least, or nothing, about July 1st, and December 31st; the Sun on the last two days being in the apsides of his orbit.

9. As the Sun moves from the apogee to the perigee, the time shown by the Sun precedes that shown by a well regulated clock, or mean solar time; but whilst the Sun moves round the perigee to the apogee, the mean time precedes the apparent time.

Illustration. Let ABCDA be the ecliptic, or the elliptical orbit, which the Sun, by an irregular motion, describes in the space of a year; the dotted circle abed, the orbit of an imaginary star, or sun, coincident with the piane of the ecliptic, and in which it moves through equal arcs in equal times.

Let HIK, also, be the Earth which revolves round its axıs every twenty-four hours, from west to east; and suppose the Sun and star to set out together from A and a, in a right line with the plane of the meridian EH; that is the Sun at A being at his greatest distance from the Earth, at which time his motion is slowest; and the star, or fictitious sun at a, whose motion is equable, and its distance from the Earth always the same. Then because the motion of the star is always uniform, and the motion of the Sun, in this part of his orbit,

the slowest; it is plain that whilst the meridian revolves from H to h, according to the order of the letters, H, I, K, L, the sun will have proceeded forward in his orbit from A to F; and the star, moving with a quicker motion, will have gone through a larger arc, from a to f'; from which it is plain, that the meridian EH will revolve sooner from H to h, under the sun at F, than from H to k under the star at f, and consequently it will be noon by the Sun sooner than by the clock.

As the Sun moves from A to C, the swiftness of his motion will continually increase, till he comes to the point C, where it will be the greatest; and the Sun C, and the star c, will be together again, and consequently it will be noon by them both at the same time; the meridian EH having revolved to EK.

From this point, the increased velocity of the Sun being now the greatest, will carry him before the star; and, therefore, the same meridian will, in this situation, come to the star sooner than to the Sun. For, whilst the star moves from c to g, the Sun will move through a greater arc, from C to G; and, consequently, the point K has its noon by the clock when it comes to k, but not its noon by the Sun till it comes to l. And though the velocity of the Sun diminishes all the way from C to A, yet they will not be in conjunction till they come to A, and then it is noon by them both at the same instant.

From this it appears that the solar noon is always later than the clock, whilst the Sun goes from C to A, and sooner whilst he goes from A to C; and at those two points, it is noon by the Sun and clock at the same time.

The obliquity of the ecliptic to the equator, which is the first mentioned cause of the equation of time, would make the Sun and clock agree on the four days of the year, which are when he enters Aries, Cancer, Libra, and Capricorn; but the other causes, which arise from his unequal motion in his orbit, would make the Sun and clocks agree only twice a year, that is, when he is in his apogee and perigee; and, consequently when these two points fall in the beginnings of Cancer and Capricorn, or of Aries and Libra, they will concur in making the clocks and Sun agree in those points. But the apogee, at present, is in the tenth degree of Cancer, and the perigee in the tenth degree of Capricorn; and, therefore, the times shown by the Sun and clocks cannot be equal about the beginning of those signs, nor at any other time of the year, except when the swiftness or slowness of equation, resulting from one of the causes, just balances owness or swiftness arising from the other. About

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