four hundred and twenty-five of these bodies had been recorded in history; but those which had appeared before the fourteenth century had not been submitted to any observations by which their paths could be ascertained, at least not with a sufficient degree of precision to afford any hope of identifying them with those of other comets. Subsequently to the year 1300, however, Halley found twenty-four comets on which observations had been made and recorded, with a degree of precision sufficient to enable him to calculate the actual paths which these bodies followed while they were visible. He examined with the most elaborate care the courses of each of these twentyfour bodies; he found the exact points at which each of them penetrated the plane of the earth's orbit; also the angle which the direction of their motion made with that plane; he also calculated the nearest distance at which each of them approached the sun, and the exact place of the body when at that nearest distance. In a word, he determined all the circumstances which were necessary to enable him to lay down with sufficient precision the path which these comets must have followed while they continued to be visible.

On comparing their paths, Halley found that one which had appeared in 1661, followed nearly the same path as one which had appeared in 1532. Supposing then these to be two successive appearances of the same comet, it would follow that its period would be 129 years; and Halley accordingly conjectured that its next appearance might be expected after the lapse of 129 years, reckoning from 1661. Had this conjecture been well founded, the comet must have appeared about the year 1790. No comet, however, appeared at or near that time following a similar path.

In his second conjecture, Halley was more fortunate, as indeed might be expected, since it was formed upon more conclusive grounds. He found that the paths of comets which had appeared in 1531 and 1607, were very nearly identical, and that they were in fact the same as the path followed by the comet observed by himself in 1682. He suspected, therefore, that the appearances at these three epochs were produced by three successive returns of the same comet, and that consequently its period in its orbit must be about 75 years.

So little was the scientific world at this time prepared for such an announcement, that Halley himself only ventured at first to express his opinion in the form of conjecture; but after some further investigation of the circumstances of the recorded comets, he found three others which at least in point of time agreed with the period assigned to the comet of 1682, viz. those of 1305,

1380, and 1456.* Collecting confidence from these circumstances, he announced his discovery as the result of combined observation and calculation, and entitled to as much confidence as any other consequence of an established physical law.

There were nevertheless two circumstances which to the fastidious sceptic might be supposed to offer some difficulty. These were, first, that the intervals between the supposed successive returns to perihelion were not precisely equal; and, secondly, that the inclination of the comet's path to the plane of the earth's orbit was not exactly the same in each case. Halley, however, with a degree of sagacity which, considering the state of knowledge at the time, cannot fail to excite unqualified admiration, observed that it was natural to suppose that the same causes which disturbed the planetary motions must likewise act upon comets; and that their influence would be so much the more sensible upon these bodies because of their great distances from the sun. Thus, as the attraction of Jupiter upon Saturn was known to affect the velocity of the latter planet, sometimes retarding, and sometimes accelerating it, according to their relative position, so as to affect its period to the extent of thirteen days, it might well be supposed, that the comet might suffer by a similar attraction an effect sufficiently great to account for the inequality observed in the interval between its successive returns; and also for the variation to which the direction of its path upon the plane of the ecliptic was found to be subject. He observed, in fine, that as in the interval between 1607 and 1682 the comet passed so near Jupiter that its velocity must have been augmented, and consequently its period shortened by the action of that planet, this period, therefore, having been only seventy-five years, he inferred that the following period would probably be seventy-six years or upwards; and consequently that the comet ought not to be expected to appear until the end of 1758, or the beginning of 1759. It is impossible to imagine any quality of mind more enviable than that which, in the existing state of mathematical physics, could have led to such a prediction. The imperfect state of mathematical science rendered it impossible for Halley to offer to the world a demonstration of the event which he foretold. He, therefore,' says M. de Pontecoulant, could only announce these felicitous conceptions of a sagacious mind as mere intuitive perceptions, which must be received as uncer

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* The path of the comet of 1456 was afterwards fully identified with that of 1682.

tain by the world, however he might have felt them himself, until they could be verified by the process of a rigorous analysis.' The theory of gravitation, which was in its cradle at the time of Halley's investigations, had grown to comparative maturity before the period at which his prediction could be fulfilled. The exigencies of that theory gave birth to new and more powerful instruments of mathematical enquiry: the differential and integral calculus was its first and greatest offspring. This branch of science was cultivated with an ardour and success by which it was enabled to answer all the demands of physics, and consequently mechanical science advanced, pari passu. Newton's discoveries having obtained reception throughout the scientific world, his enquiries and his theories were followed up; and the consequences of the great principle of universal gravitation were rapidly developed. Among these enquiries one problem was eminently conspicuous for the order of minds whose powers were brought to bear upon it. One of the first and simplest results of the theory of gravitation was, that if a single planet attended the sun (its mass being insignificant compared with that of the sun), that planet must revolve in an ellipse, the focus of which must be occupied by the centre of the sun; but, if a second planet be admitted into the system, then the elliptic form of their paths round the sun can be preserved only by the supposition, that the two planets have no attraction for each other, and that no physical influence is in operation, except the attraction of the solar mass for each of them. But the law of universal gravitation is founded upon the principle, that every body in nature must attract and be attracted by every other body. Thus, the elliptic character of the orbit is effaced the moment a second planet is introduced. But let us remember, that in this case each of the two supposed planets are in mass absolutely insignificant compared with the sun. The amount of attraction depending on the greatness of the attracting body, the intensity of the solar attraction of each of the planets must predominate enormously over the comparatively feeble influence of their diminutive masses on each other. The tendency of the solar attraction to impress the elliptie form on the paths of those planets, must therefore prevail in the main; and although their mutual attraction, however feeble, cannot be wholly ineffective, their orbits will, in obedience to the solar mandate, preserve a general elliptic form, subject to those very slight deviations, or disturbances, due to their reciprocal attraction. The problem to discover the nature and amount of these disturbances is one of paramount importance in astronomy, and has been called the 'problem of three bodies;' and its extension embraces the effects of the mutual gravitation of all the planets of

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the system upon each other. This celebrated problem presented enormous mathematical difficulties. A particular case of it, which, from the comparative smallness of the third body considered, was attended with greater facility, was solved by Euler, D'Alembert, and Clairaut. This was the case in which the single planet, revolving round the sun, was the earth, and the third body the moon.

Clairaut undertook the difficult application of this problem to the case of the Comet of 1682, with a view to calculate the effects which would be produced upon it by the attractions of the different planets of the system; and by such means to convert the conjecture of Halley into a distinct astronomical prediction, attended with all the circumstances of time and place. The exact verification of the prediction would, it was obvious, furnish the most complete demonstration of the principle of universal gravitation; which, though generally received, was not yet considered so completely demonstrated as to be independent of so remarkable a body of evidence as the fulfilment of such a calculation would afford.

To attain completely the end proposed, it was necessary to solve two very different classes of problems, requiring different powers of mind, and different habits of thought and application. The mathematical part of the enquiry, strictly speaking, consisted in the discovery of certain general analytical formulæ, applicable to the case in question; which, when translated into ordinary language, would become a set of rules expressing certain arithmetical processes, to be effected upon certain given numbers; and when so effected would give as the final results, numbers which would determine the place of the Comet, under all the circumstances influencing it from hour to hour. The actual place of the body being thus determined, it became a simple question of practical astronomy, to ascertain its apparent place in the firmament, at corresponding times. table, exhibiting its apparent place thus determined in the firmament for stated intervals of time, is called its Ephemeris; it is the final result to which the whole investigation must tend, and is that whose verification by observation would ultimately decide the validity of the reasoning, and the accuracy of the computations. Clairaut, a mathematician and natural philosopher, was eminently qualified to conduct such an investigation, as far as the attainment of those general analytical formula which embodied the rules by which the practical astronomer and arithmetician might work out the final results; but beyond this point, neither his habits nor his powers would conduct him. Lalande, a practical astronomer, no less eminent in his own department, and

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who indeed first urged Clairaut to this enquiry, accordingly undertook the management of the astronomical and arithmetical part of the calculation. In this prodigious labour (for it was one of most appalling magnitude) he was assisted by the wife of an eminent watchmaker in Paris, named Lepaute, whose exertions on this occasion have deservedly registered her name in astronomical history.

It is difficult to convey to the reader who is not conversant with such investigations an adequate notion of the labour which such an enquiry involved. The calculation of the influence of any one planet of the system upon any other, is itself a problem of some complexity and difficulty; but still, one general computation, depending upon the calculation of the terms of a certain series, is sufficient for its solution. This comparative simplicity arises entirely from two circumstances which characterise the planetary orbits. These are, that though they are ellipses, they differ very slightly from circles; and though the planets do not move in the plane of the ecliptic, yet none of them deviate considerably from that plane. But these characters do not, as we have already stated, belong to the orbits of comets, which, on the contrary, are highly eccentric, and depart from the ecliptic at all possible angles. The consequence of this is, that the calcu lation of the disturbances produced in the cometary orbit by the action of the planets must be conducted, not like the planets, in one general calculation applicable to the whole orbit, but in a vast number of separate calculations; in which the orbit is considered, as it were, bit by bit, each bit requiring a calculation similar to that of the whole orbit of the planet. In fact, for a very small part of its course, we treat the comet as we would a planet ; making our calculations, and completing them nearly in the same manner; but for the next part we are obliged to enter upon a new calculation, starting with a different set of numbers, but performing over again similar arithmetical operations upon them. When it is considered that the period of Halley's comet is about seventyfive years, and that every portion of its course, for two successive periods, was necessary to be calculated separately in this way, some notion may be formed of the labour encountered by Lalande and Madame Lepaute. During six months,' says Lalande, we calculated from morning till night, sometimes even at meals, the consequence of which was, that I contracted an illness which 'changed my constitution for the remainder of my life. The 'assistance rendered by Madame Lepaute was such, that without her, we never could have dared to undertake this enormous labour, in which it was necessary to calculate the distance of each of the two planets, Jupiter and Saturn, from the come