The extraordinary length and brilliancy which was ascribed to the tail, upon this occasion, have led astronomers to investigate the circumstances under which its brightness and magnitude would be the greatest possible; and upon tracing back the motion of the comet to the year 1456, it has been found that it was then actually in the position, with respect to the earth and sun, most favorable to magnitude and splendor. So far, therefore, the result of astronomical calculation corroborates the records of history. The next return took place in 1531. Pierre Appian, who first ascertained the fact that the tails of comets are usually turned from the sun, examined this comet with a view to verify his statement, and to ascertain the true direction of its tail. He made, accordingly, numerous observations upon its position, which, although rude, compared with the present standard of accuracy, were still sufficiently exact to enable Halley to identify this comet with that observed by himself. The next return took place in 1607, when the comet was observed by Kepler. This astronomer first saw it on the evening of the twenty-sixth of September, when it had the appearance of a star of the first magnitude, and, to his vision, was without a tail; but the friends who accompanied him had better sight, and distinguished the tail. Before three o'clock the following morning the tail had become clearly visible, and had acquired great magnitude. Two days afterwards, the comet was observed by Longomontanus, a distinguished philosopher of the time. He describes its appearance, to the naked eye, to be like Jupiter, but of a paler and more obscured light; that its tail was of considerable length, of a paler light than that of the head, and more dense than the tails of ordinary comets. The next appearance, and that which was observed by Halley himself, took place in 1682, a little before the publication of the '_Principia_.' In the interval between 1607 and 1682, practical astronomy had made great advances; instruments of observation had been brought to a state of comparative perfection; numerous observatories had been established, and the management of them had been confided to the most eminent men in Europe. In 1682, the scientific world was therefore prepared to examine the visitor of our system with a degree of care and accuracy before unknown.