from the earth would suffice to prevent the identification of the two comets. A comet formerly of extreme brilliancy might reappear as a feeble nebulosity. It would have been difficult to recognise the same body in the comet of 1607, whose light appeared to Kepler pale and weak; in that of 1682, which Lahire and Picard compared to a star of the second magnitude; in that of 1759, which appeared to Messier like a star of the first magnitude; and, lastly, in the famous comet of 1456, which all historians (except two Poles),' says Pingré, agree in describing as great, terrible, and of an extraordinary size, drawing after it a long tail which covered two celestial signs, or 60 degrees.' These were, nevertheless, one and the same comet. Astronomers, it is true, mistrust, and justly, the nearly always exaggerated expressions of the ancient chroniclers; but precisely for that reason a resemblance of aspect is not to be relied upon for establishing the identity, and consequently the periodicity, of two comets. We must have more precise elements of comparison. These elements are those of the parabolic orbit, when records have been left of observations-that is to say, of positions and dates sufficient for the calculation of the orbit-when, in a word, the comet instead of having been simply seen has been observed. A catalogue of ancient comets is therefore necessary, and it was whilst consulting the table of twenty-four comets which he had calculated that Halley made the prediction, the history of which we are about to give.

If the longitudes of the ascending node and of the perihelion, the inclination of the plane of the orbit, the perihelion distance, and the direction of movement, are all the same, or nearly the same, in two cometary orbits, in all probability we have two successive, if not consecutive, apparitions of the same comet. Taking the interval between the apparitions for the period itself, we are enabled by the third law of Kepler to

calculate the dimensions of the major axis of the corresponding elliptic orbit, and to assure ourselves that the new orbit is in accordance with the whole of the known observations. If this be so, we can calculate more or less exactly the comet's next return; that is to say, its perihelion passage, and all the circumstances of its future apparition.

The second method consists in the direct calculation of the elliptic elements. It requires, as a rule, exact observations, especially if the orbit be greatly elongated, since there is then but little difference between the apparent path followed by a comet, whether it be a parabola, a very long ellipse, or an hyperbola slightly flattened. The first attempts by this method--a very legitimate one in theory-prove that it is subject to many difficulties and uncertainties. Euler, on first applying it to the comet of 1744, obtained a hyperbolic orbit from the observations made at Berlin. But afterwards, having received the observations made by Cassini, he found the orbit to be a very long ellipse, with a period of many centuries.

The first example of an elliptic orbit calculated with precision by this second method is, we believe, that of Lexell's comet (or comet of 1770), a comet of short period (five years and a half), and having an orbit of comparatively slight elongation, but which, unfortunately—we shall come to its history further on has undergone enormous perturbations, and has not again been seen. Since then the direct calculation of the elliptic movement, without reference to previous observations, has been employed for various comets, and with success in several instances, as the return of the periodical comets of Faye, Brorsen, d'Arrest, and Winnecke (1819) has been rendered certain by numerous and careful observations.

The above two methods both require observed positions of the comet, whose periodicity is to be discovered, and also that these observations should possess a certain degree of accuracy.

In the absence of these conditions the end may, however, be attained, but the result is, in that case, as conjectural as the method itself. This third method consists in making a comparison of the different historical comets, in noting the resemblance of their aspect, and in ascertaining if the intervals of their successive apparitions agree with the hypothesis of a certain period, whose duration, in this case, must be necessarily contained nearly an exact number of times in these intervals. The elements calculated for one apparition may then suffice to render probable the identity of several comets. In this way M. Laugier is of opinion that he has identified the comets of 1299, 1468, and 1799 by assuming a period of one hundred and sixty-nine years, which is twice included between the two last dates. In the same manner the comets of 1301, 1152, 760, and several others (which we shall mention presently) have been identified as former apparitions of Halley's comet, the true period of which has long been calculated and known.

Discovery of the identity of the comets of 1682, 1607, and 1531; Halley announces the next return for the year 1758--Clairaut undertakes the calculation of the disturbing influence exercised by Jupiter and Saturn upon the comet of 1682; collaboration of Lalande and Mdlle. Hortense Lepaute-The return of the comet to its perihelion is fixed for the middle of April 1759; the comet returns on the 13th of March-Return of Halley's comet in 1835; calculation of the perturbations by Damoiseau and Pontécoulant; progress of theory-The comet will return to its perihelion in May 1910.

เ

LET us recal the memorable words of Seneca in his Quæstiones Naturales: Why should we be surprised that comets, phenomena so seldom presented to the world, are for us not yet submitted to fixed laws, and that it is still unknown from whence come and where remain these bodies, whose return takes place only at immense intervals? ... An age will come when that which is mysterious for us will have been made clear by time and by the accumulated studies of centuries. ... Some day there will arise a man who will demonstrate in what region of the heavens the comets take their way, why they journey so far apart from other planets, what their size, their nature.' Eighteen centuries have elapsed, and not one man, but the accumulated efforts of many men have raised a corner of the veil spoken of by Seneca. As far as the laws of cometary movement are concerned Newton has realised his prediction; whilst that which relates to the return of comets and their calculated periodicity has been fulfilled by Halley.

UNIV. OF

VINNQUITY.

HALLEY'S COMET.

This learned man, modest as he was laborious, published in 1705 his catalogue of twenty-four comets. On comparing their elements he remarked that three comets-namely, those of 1531, of 1607, and of 1682—had orbits nearly identical. He at once suspected the identity of the comets themselves; and more than that, he announced the next return of the comet for the year 1758. Let us subjoin the elements which Halley calculated, and leave him afterwards to speak for himself:—

The following is the passage in Halley's memoir* concerning the periodicity of the comet which at the present day bears his

name:

'Now, many things lead me to believe that the comet of the year 1531, observed by Apian, is the same as that which, in the year 1607, was described by Kepler and Longomontanus, and which I saw and observed myself, at its return, in 1682. All the elements agree, except that there is an inequality in the times of revolution; but this is not so great that it cannot be attributed to physical causes. For example, the motion of Saturn is so disturbed by the other planets, and especially by Jupiter, that his periodic time is uncertain, to the extent of several days. How much more liable to such perturbations is a comet which recedes to a distance nearly four times greater than Saturn, and a slight increase in whose velocity could change its orbit from an ellipse to a parabola? The identity of these comets is confirmed by the fact that in

[The title of Halley's memoir is Astronomia Cometica Synopsis, and it was published in the Philosophical Transactions, vol. xxiv. (1704-5), pp. 18821899.-ED.]