As comets are subject to great changes of appearance, one can never be identified by any description of its magnitude, brilliancy, etc., at the time of a previous return. This can be done only by a comparison of orbits. If, for example, we find the elements of an orbit very nearly corresponding in every particular with those of a former comet, there is a degree of probability, amounting almost to certainty, that the two are identical. Sir Isaac Newton, in his _Principia_, published shortly after the appearance of the comet of 1682, explained how the periods of those mysterious visitors might thus be ascertained, thus directing the attention of astronomers to the subject. Dr. Halley soon after undertook a thorough discussion of all the recorded cometary observations within his reach. In the course of his investigations he discovered that the path of the comet observed by Kepler in 1607 coincided almost exactly with that of the one which passed its perihelion in 1682. Hence he concluded that they were the same. He found also that the comet of 1531, whose course had been particularly observed by Apian, moved in the same path. The interval between the consecutive appearances being nearly 76 years, Halley announced this as the time of the comet's revolution, and boldly predicted its return in 1758 or 1759. The law of universal gravitation had at this time just been discovered and announced. But although its application to the determination of planetary and cometary perturbations had not been developed, Halley was 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.