amount of the diurnal variations with sufficient frequency; sometimes intervals of fifty days would be short enough, at others it might be necessary to resort to intervals of twelve, eight, or even four days, or less, according as the attraction of the disturbing body varies in a lesser or greater degree in each interval; because it is assumed that the changes go on uniformly, or nearly so, and consequently that the diurnal variation for the middle day of each interval, if multiplied by the number of days included in it, shall give very nearly the true variation in that interval. By another method, instead of using equal periods of time, the calculations are conducted for equal intervals of the comet's excentric anomaly. These processes, to be fully understood in theory, require an acquaintance with some of the highest branches of mathematical science, while their practical application is so tedious and intricate, that few but the most zealous and expert computers will attempt to manage them.

We have just alluded to the use of a table of the elements of cometary orbits in identifying returns of the same body. By this comparison of elements the periodicity of at least one comet has been established, and we have strong reasons to suspect it in other cases. In the following chapter we shall briefly describe the circumstances which led our countryman Halley to the first discovery of the periodical return of a comet, while he was occupied in applying Newton's law of gravitation to the apparently capricious and intricate movements of these bodies; by the application of the Newtonian principles at a vast expense of time and labour, Halley laid the founda

tion upon which cometary astronomy has since risen. We shall endeavour so to arrange our descriptions of this and other comets, that the reader may gain an insight into the kind of interest attaching to this department of the science, and the difficulties with which astronomers have to contend in prosecuting their inquiries respecting these extraordinary bodies.

CHAPTER IV.

THE COMET OF HALLEY.

AMONGST the comets which particularly attracted the attention of Dr. Halley during his arduous investigations, were those of 1531, 1607, and 1682.

For the most important observations of the comet of 1531 we are indebted to Peter Apian, astronomer to the Emperors Charles V. and Ferdinand I. of Austria, who observed at Ingoldstadt in Bohemia. His results were published in 1540 in the Astronomicum Cæsarium, an extremely rare work, which was found with difficulty even so long since as the time of Halley. A fine copy is now preserved in the library of the Royal Astronomical Society. The observations were made each evening at the time of transit of the bright star Arcturus over the prime vertical westward, and extend from August 13th to the 23rd, Julian style. The comet was first seen at the end of July, and was visible in some parts until the beginning of September, its motion in the interval being from Leo into Libra. Lavather says it was observed in Italy, Germany, and France, and was of a reddish or yellow colour. Camerarius says it belonged to the class Pogonia. The Chinese astronomers found the comet in Gemini on August 5th, and it was last perceived on the 8th of September in the constellation

Virgo; they describe its course amongst the stars, agreeing in every essential point with the relation of Peter Apian.

On the 11th of September, 1607, the Chinese discovered a comet in Gemini, which they finally lost sight of in the sign Scorpio at the beginning of October. In Europe it was observed by the celebrated Kepler at Prague, by Longomontanus at Copenhagen, and at Malmoge, in Scania, by William Lower at Ilfracombe, and by our distinguished countryman Harriot. Kepler's observations are found in a work published at Augsburg in 1619, and entitled, De Cometis libelli tres. Professor Rigaud has printed in detail the observations of Lower and Harriot, which had been fortunately discovered amongst the papers of the Earl of Egremont. According to Longomontanus, the comet had a pretty long and dense tail in a direction opposite to that of the sun. The head appeared of about the size of Jupiter, its colour being livid, and resembling that of Saturn; the tail was visible from the time the comet was first perceived by Kepler, on September 16th, until the 12th of the following month. The head was not quite round; it had a pale watery light; the tail long and glittering. Gottfried Wendelin saw the comet till November 5th, and says its form was like that of "a burning lamp,' or "a flaming sword," 7° long. Coruscations of the tail were remarked by Kepler and others. The apparent path was through Ursa Major, Bootes, Serpens, and Ophiuchus.

On the evening of the 15th of August, 1682, Flamsteed's assistant saw a comet at the Royal Observatory,

Greenwich. A few days later it was more visible; the tail 5° long, the diameter of the head about two minutes of arc. On the 21st the tail was 10° long, and bent towards the right. Flamsteed's observations extend to September 9th, when the head was dull, and scarcely visible in the twilight. Halley saw it a day later. Picard found the comet at Paris on August 26th; the head then appeared as large as a star of the second magnitude. On the 29th the tail was curved, the concavity on the eastern side. On September 11th the head was so confused that it was not without difficulty a luminous point could be perceived. Picard's observations extend to September 12th. Hevelius, who observed the comet at Dantzic, says it was bright at the end of August, and could be seen all night with a tail from 12° to 16° long. In large telescopes a nucleus of an oval or gibbous form was constantly remarked. On many occasions the tail was not directed exactly from the sun. About September 8th a kind of luminous ray or sector was thrown out from the nucleus into the tail, a phenomenon which appears to have struck Hevelius as very remarkable, and induced him to give a figure, which conveys a better idea of its nature than could have been obtained from verbal description. The same phenomenon was witnessed at a subsequent return of this comet. Besides the observers we have named, Kirch of Leipsic, Zimmerman of Nuremberg, Bäert of Toulon, and Montanari of Padua, are amongst those who have left descriptions or observations of this body.

Dr. Halley calculated the parabolic elements of the comet of 1682 from Flamsteed's observations, agree