of which comets appear only for a short time, ano frequently in circumstances where their apparent motion is rendered very complicated, by the real motion of the earth, which always carries us in a contrary direction.

Notwithstanding all these difficulties, it is possible to determine the elements of the orbits of comets by different methods. Three complete observations are sufficient for this object; others only serve to confirm the accuracy of these elements, and the truth of the theory which has been just explained. Above four and twenty comets, the numerous observations of which are exactly represented by this theory, have confirmed it beyond all doubt. It appears, therefore, that comets which have been considered as meteors, for many years, are of the same nature as planets; their motions and their returns are regulated by the same laws as planetary motions.

4. Comets do not always move in the same direction like the planets. The real, or heliocentric motion of some is direct, or according to the order of the signs; and of others, retrograde. But the geocentric motion of the same comet may be either retrograde or direct according to the position of the earth with respect to the comet, and their relative velocities.

The heliocentric motion of half the comets, whose elements have been computed, is retrograde, and of the others, direct. The inclination of their orbits is not confined within a narrow zone like that of the planetary orbits; they present every variety of inclination from an orbit nearly coincident with the plane of the ecliptic, to that perpendicular to it.

A comet is recognised when it re-appears by the identity of the elements of its orbit with those of the orbit of a comet already observed. If its perihelion distance, the position of its perihelion, its nodes, and the inclination of its orbit are very nearly the same, it is probable that the comet which appears is that which has been observed before, and which, having receded to such a distance as to be invisible. returns to that part of its orbit nearest to the sun. The duration of the revolution of comets being very long, and having been observed with very little care, till within about two centuries; the period of the revolution of one comet only, is known with certainty, that of 1682, which had been already observed in 1607 and 1531, and which has re-appeared in 1759. This comet takes about 76 years to return to its perihelion; therefore, taking the mean distance of the sun from the earth as unity, the greater axis of its orbit is 35.9, and as its perihelion distance is only 0.58, it recedes from the sun at least 35 times more than the earth, describing a very eccentric ellipse. Its return to the periheion nas been longer by thirteen months from 1531 to 1607, than from 1607 to 1682; it has been 18 months shorter from 1607 to 1682, than from 1682 to 1759.

The real or heliocentric motion of this comet was retrograde, and the elements of the orbit deduced by Dr. Halley from the observations of Apian in 1531, of Kepler in 1607, and of himself in 1682, also the elements deduced from the observations in 1759, were as follows:

Per. dist.

Passage through Earth's per Place of Place of Inclination Perihelion. dist. unity. Perihelion Node. to ecliptic.

This comet was retarded by the action of Jupiter, as Dr. Halley had foretold. This retardation was more exactly computed by Clairaut, who also calculated the retardation by Saturn. The result of his computation published before the return of the comet, fixed April 15, for the time of the passage through perihelion: it happened on March 12. Dr. Halley's computation appears also very exact, when it is considered that he did not allow for the retardation by Saturn. We may be nearly certain that this comet will reappear again in 1834.

The return of some other comets has been suspected: the most probable of these returns was that of the comet of 1532, which has been believed to be the same with that of 1661, and the revolution of which was fixed at 129 years; but this comet not having re-appeared in 1790, as was expected, there is great reason to believe that these two comets were not the same.

The preceding part of the present Chapter has been principally extracted from Laplace's System of the World.

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An ingenious computation has been made by Laplace, from the doctrine of chances, to show the probability of two comets being the same, from a near agreement of the elements. It is unnecessary to detail at length the method here. It supposes that the number of different comets does not exceed one million, a limit probably sufficiently extensive. The chance that two of these, differing in their periodic times, agree in each of the five elements within certain limits, may be computed by which it was found to be as 1200: 1. that the comets of 1607 and 1682 were not different, and thus Halley was justly almost confident of its reappearance in 1759. As it did appear then, we may expect, with a degree of probability, approaching almost without limit to certainty, that it will re-appear again at the completion of its period.

But with respect to the comet predicted for 1789, from the supposition that those of 1661 and 1532 were the same, the case is widely different. From the discrepancy of the elements of these comets, the probability that they were the same is only 3 to 2, and we cease to be surprised that we did not see one in 1789. See Dr. Brinkley's Elements of Astronomy.

Comets that appeared in 1264 and 1556 are supposed to have been the same, whence this comet may again be expected in 1848.

A comet appeared in 1770 very remarkable from the resuit of the computations of Lexell, which indicated a period of only 5 years; it has not been observed since. There can be no doubt that the periodic time of the orbit which it described in 1770, was justly determined; for M. Burcknardt has since, with great care, re-computed the observations, and his result gives a period periodic time of 5' years.

Lexell has remarked, that this comet, moving in the orbit he had investigated, must have been near Jupiter in 1767, and would also be very near it again in 1779, from whence he concluded that the former approach changed the perihelion distance of the orbit, by which the comet became visible to us, and that in consequence of the latter approach, the perihelion distance was again increased, and so the comet again became invisible, even when near its perihelion. This explanation has been in a manner confirmed by the calculations of Burckhardt, from formulas of Laplace. He has found, that before the approach of Jupiter, in 1767, the perihelion distance might have been 5.08, and that after the approach in 1779, it may have become 3.33, the earth's dis