well aware that the attractive influence of Jupiter and Saturn might accelerate or retard the motion of the comet, so as to produce a considerable variation in its period. During the interval from 1682 to 1759, the application of the higher mathematics to problems in physical astronomy had been studied with eminent success. The disturbing effect of the two large planets, Jupiter and Saturn, was computed with almost incredible labor by Clairaut, Lalande, and Madame Lepaute. The result as announced by Clairaut to the Academy of Sciences in November, 1758, was that the period must be 618 days longer than that immediately preceding, and that the comet accordingly would pass its perihelion about the 13th of April, 1759. It was stated, however, that, being pressed for want of time, they had neglected certain quantities which might somewhat affect the result. The comet, in fact, passed its perihelion in March, within less than a month of the predicted time. When it is considered that the attraction of the earth was not taken into the account, and that Uranus, whose influence must have been sensible, had not then been discovered, this must certainly be regarded as a remarkable approximation.

But during the next interval of 76 years the theory of planetary perturbations had been more perfectly developed. The masses of Jupiter and Saturn had been determined with greater accuracy, and Uranus had been added to the known members of the planetary system. A nearer approximation to the exact time of the comet's perihelion passage in 1835 was therefore to be expected. Prizes were offered by two of the learned societies of Europe—the Academy of Sciences at Turin, and the French Institute—for the most perfect discussion of its motions. That of the former was awarded to Damoiseau,—that of the latter to Pontecoulant. The times assigned by these distinguished mathematicians for the comet's perihelion passage were very nearly the same, and differed but a few days from the true time. Had the present received mass of Jupiter been used in the calculations, Pontecoulant, it is believed, would not have been in error as much as 24 hours. It may be proper to remark that, during the entire period from 1759 to 1835, the position of Neptune was such that it could produce no considerable effect on the motion of the comet.

This interesting object will again return about 1911.

The visit of 1531 was the earliest that Halley succeeded in determining with any degree of certainty. Peter Apian, by whom it was at that time observed, was the first European to ascertain the fact that, as a general thing, the tails of comets are turned from the sun.[3] To confirm this discovery, he carefully followed the body in its progress through the constellations. By means of his recorded observations Halley was enabled to identify this comet with that of 1607 and 1682. The great comet of 1456 he conjectured to be the same, from the date of its appearance. Pingré subsequently confirmed this suspicion by a careful examination of the few trustworthy records that could be collected from the writers of that period.

From the earlier descriptions of this comet we infer that its brilliancy is gradually diminishing. In 1456 its tail, which was slightly curved like a sword or sabre, extended two-thirds of the distance from the horizon to the zenith. The appearance of such an object, in a grossly superstitious age, excited throughout Europe the utmost consternation. The Moslems had just taken Constantinople, and were threatening to advance westward into Europe. Pope Calixtus III., regarding the comet as confederate with the Turk, ordered prayers to be offered three times a day for deliverance from both. The alarm, however, was of short duration. Within ten days of its appearance the comet reached its perihelion. Receding from the sun, the sword-like form began to diminish in brilliancy and extent; and finally, to the great relief of Europe, it entirely disappeared.

The perihelion passage of 1456 was, until recently, the earliest known. It was shown by Laugier, however, in 1843, that among the notices of comets extracted by Edward Biot from the Chinese records, were observations of a body in 1378, which was undoubtedly the comet of Halley. Further researches among these annals enabled the same astronomer to recognize two ancient returns, one in 760, the other in 451. Still more recently the distinguished English astronomer, Mr. Hind, has traced back the returns to the year 11 B.C. He remarks, however, that previous to that epoch, "the Chinese descriptions of comets are too vague to aid us in tracing any more ancient appearances," and that "European writers of these remote times render us no assistance." Let us now inquire whether the comet had probably made any former approach to the sun in an orbit nearly identical with the present. It is well known that the modern period of this body is considerably less than the ancient. Thus, the mean period since A.D. 1456 has been 75.88 years; while from 11 B.C. to 1456 A.D. it was 77.27 years. In determining the approximate dates of former returns, the ancient period should evidently be employed. Now, it is a remarkable fact that of more than 70 comets,[4] or objects supposed to be comets, whose appearance was recorded during the six centuries immediately preceding the year 11 B.C., but one—that of 166 B.C.—was observed at a date corresponding nearly to that of a former return of Halley's comet. Of this object it is merely recorded that "a torch was seen in the heavens." Whether this was a comet or some other phenomenon, it is impossible to determine. But as the comet of Halley was more brilliant in ancient than in modern times, it seems highly improbable that seven consecutive returns of so conspicuous an object should have been unrecorded, especially as twelve comets per century[5] were observed during the same period. It would appear, therefore, that the perihelion passage of 11 B.C. was in fact the first ever made by the comet, or at least the first in an orbit nearly the same as the present.

The motion of Halley's comet is retrograde. The point of its nearest approach to the sun is situated within the orbit of Venus. Its greatest distance from the centre of the system is nearly twice that of Uranus, or 36 times that of the earth. The comet is, consequently, subject to great changes of temperature. When nearest the sun its light and heat are almost four times greater than the earth's; when most remote, they are 1200 times less. In the former position, the sun would appear much larger than to us; in the latter, his apparent diameter would not greatly exceed that of Jupiter, as viewed from the earth. It would be difficult to conjecture what the consequences might be, were our planet transported to either of these extremes of the cometary path. In the perihelion, the waters of the ocean would undoubtedly be reduced to a state of vapor; in the aphelion, they would be solidified by congelation.

II. Encke's Comet.

It was formerly supposed that all comets have their aphelia far beyond the limits of the planetary system. In