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 8 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 result 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 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 anproach in 1779, it may have become 3.33, the earth's di

tance being unity. With both these perihelion distances, the comet must have been invisible during its whole revolution. The perihelion distance in 1770 was 0.67.

'This comet was also remarkable by having approached nearer the earth than any other comet that has been observed: and by that approach having enabled us to ascertain a limit of its mass or quantity of matter. Laplace has computed, that if it had been equal to the earth, it would have shortened the length of our year by of a day. Now it has been perfectly ascertained, by the computations of Delambre on the Greenwich observations of the sun, that the length of the year has not been changed, in consequence of the approach of that comet by any perceptible quantity, and thence Laplace has concluded that the mass of that comet is less than 3600 of the mass of the earth. The smallness of its mass is also shown by its having traversed the orbits of the satellites of Jupiter without having occasioned an alteration in their motions. From those and other circumstances, it seems probable that the masses of the comets are in general very inconsiderable; and therefore, as Dr. Brinkley remarks, that astronomers need not be under apprehensions of having their tables deranged in consequence of the near approach of a comet to the earth or moon, or to any bodies of the solar system.

5. The motion of a comet, like that of a planet, is accelerated, when moving from its aphelion to its perihelion, and retarded from its perihelion to its aphelion. On account of the great eccentricity of a comet's orbit, its motion in the perihelion is prodigiously swift, and in the aphelion proportionably slow.

According to Newton, the velocity of the comet of 1680, which came nearest to the sun of any upon record, was eight hundred and eighty thousand miles an hour. On taking the perihelion distance of this comet, equal to .00603, as given by Piagré, (proportioned according to the present mean parallax of the sun deduced from the transit of Venus of 1769,) I find, says Squire in his Astronomy, by two different calculations, that the velocity of this comet in its perihelion was no less than 1,240,108 miles per hour; at which time it was only 572,850 miles from the centre of the sun, or about 130,000 miles from his surface.

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The velocity velocity of this comet in its perihelion was so great, that, if continued, would have carried it ed it through 124 degrees in an hour. But its actual hourly motion during that interval, before and after it passed its perihelion, was 81° 46′52′′.

From Dr. Halley's determination of the orbit of this comet, it cannot be less than 13,000 millions of miles from the sun when in its aphelion.

According to Pingré, the elements of the orbit of the comet of 1680, were as follows: this comet passed through its perihelion December 18th, at 1 minute, 2 seconds after 12 o'clock at noon mean time at Greenwich; place of the perihelion Ss 22° 40′ 10′′, or 22° 40′ 10′′ of Sagittarius; and its distance from the sun when in the perihelion, .00603, the mean distance of the earth from the sun being considered as unity or 1; the longitude or place of the ascending node 9 signs, 1° 57′ 13′′, or 1° 57′ 13′′ of Capricornus; and the inclination of the orbit to the plane of the ecliptic 61° 22′ 55".

It appears from the great diurnal motion of some comets, that they must have come very near the earth. For, according to Regiomontanus, the comet of 1472 moved over an arc of 1200 in one day. And the comet of 1759 described the apparent arc of 410 in the same interval of time.

The comet of 1811 was first seen at Viviers, by Flaugergues, on the 25th of March, and was visible till the end of May; it must have been very faint and near the horizon all the time, it having during that interval great southern latitude. The Earth was in about 5 degrees of Libra, on the 25th of March, and therefore the comet must be nearly in opposition to the Sun, which certainly was the most favourable position for seeing it. It was then moving towards its perihelion, but its motion being slow, and the Earth retreating from it, it was lost sight of when the Earth arrived at the beginning of Sagittarius. The comet passed the ascending node on July 11th, when the Earth was between Capricornus and Aquarius; it was then approaching its conjunction with the Sun, and was invisible from the end of May till the 31st of August, when, between 3 and 4 o'clock that morning, it was observed by Bouvard, at the imperial observatory; its right ascension was 147° 18', and declination 32° 53' north. The comet was first observed at Greenwich, on the 5th of September; its geocentric longitude at that time was 145° 3' 10", and its geocentric latitude 280 36′ 39′′. The comet was at its perihelion, at a distance of 97,128,950 miles from the Sun on the 12th September.