Extraordinary as these conjectures must have appeared at the time, they were soon strictly realized. Halley, who was then a young man, but possessed one of the best minds in England, undertook the labor of examining the circumstances attending all the comets previously recorded, with a view to discover whether any, and which of them, appeared to follow the same path. Antecedently to the year 1700, 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 twenty-four bodies; he found the exact points at which each one of them crossed the ecliptic, or their nodes; also the angle which the direction of their motion made with that plane,—that is, the inclination of their orbits; he also calculated the nearest distance at which each of them approached the sun, or their perihelion distance; and the exact place of the body when at that nearest point,—that is, the longitude of the perihelion. These particulars are called the elements of a comet, because, when ascertained, they afford sufficient data for determining a comet's path. On comparing these 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 one hundred and twenty-nine 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 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 seventy-five and a half years. The probability of this conclusion is strikingly exhibited to the eye, by presenting the elements in a tabular form, from which it will at once be seen how nearly they correspond at these regular intervals.
| Time. | Inclination of | Long. of the | Long. Per. | Per. Dist. | Course. |
| the orbit. | node. | ||||
| 1456 | 17°56´ | 48°30´ | 301°00´ | 0°58´ | Retrograde. |
| 1531 | 17 56 | 49 25 | 301 39 | 0 57 | " |
| 1607 | 17 02 | 50 21 | 302 16 | 0 58 | " |
| 1682 | 17 42 | 50 48 | 301 36 | 0 58 | " |
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 which, at least in point of time, agreed with the period assigned to the comet of 1682. Collecting confidence from these circumstances, he announced his discovery as the result of observation and calculation combined, and entitled to as much confidence as any other consequence of an established physical law.
There were, nevertheless, two circumstances which might be supposed to offer some difficulty. First, the intervals between the supposed successive returns were not precisely equal; and, secondly, 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 for 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. The theory of gravitation, which was in its infancy in 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 inquiry: the differential and integral calculus, or the science of fluxions, as it is sometimes called,—a branch of the mathematics, expressed by algebraic symbols, but capable of a much higher reach, as an instrument of investigation, than either algebra or geometry,—was its first and greatest offspring. This branch of science was cultivated with an ardor and success by which it was enabled to answer all the demands of physics, and it contributed largely to the advancement of mechanical science itself, building upon the laws of motion a structure which has since been denominated 'Celestial Mechanics.' Newton's discoveries having obtained reception throughout the scientific world, his inquiries and his theories were followed up; and the consequences of the great principle of universal gravitation were rapidly developed. Since, according to this doctrine, every body in nature attracts and is attracted by every other body, it follows, that the comet was liable to be acted on by each of the planets, as well as by the sun,—a circumstance which rendered its movements much more difficult to follow, than would be the case were it subject merely to the projectile force and to the solar attraction. To estimate the time it would take for a ship to cross the Atlantic would be an easy task, were she subject to only one constant wind; but to estimate, beforehand, the exact influence which all other winds and the tides might have upon her passage, some accelerating and some retarding her course, would present a problem of the greatest difficulty. Clairaut, however, a celebrated French mathematician, undertook to estimate the effects that would be produced on Halley's comet by the attractions of all the planets. His aim was to investigate general rules, by which the computation could be made arithmetically, and hand them over to the practical calculator, to make the actual computations. Lalande, a practical astronomer, no less eminent in his own department, and who indeed first urged Clairaut to this inquiry, undertook the management of the astronomical and arithmetical part of the calculation. In this prodigious labor (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 one who is not conversant with such investigations, an adequate notion of the labor which such an inquiry 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 characterize 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 belong to the orbits of comets, which, on the contrary, are highly eccentric, and make all possible angles with the ecliptic. The consequence of this is, that the calculation of the disturbances produced in the cometary orbits by the action of the planets must be conducted not like the planets, in one general calculation applicable to the whole orbits, 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 the whole orbit of the planet. Now, when it is considered that the period of Halley's comet is about seventy-five 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 labor 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 labor, in which it was necessary to calculate the distance of each of the two planets, Jupiter and Saturn, from the comet, and their attraction upon that body, separately, for every successive degree, and for one hundred and fifty years."
The attraction of a body is proportioned to its quantity of matter. Therefore, before the attraction exerted upon the comet by the several planets within whose influence it might fall, could be correctly estimated, it was necessary to know the mass of each planet; and though the planets had severally been weighed by methods supplied by Newton's 'Principia,' yet the estimate had not then attained the same measure of accuracy as it has now reached; nor was it certain that there was not (as it has since appeared that there actually was) one or more planets beyond Saturn, whose attractions might likewise influence the motions of the comet. Clairaut, making the best estimate he was able, under all these disadvantages, of the disturbing influence of the planets, fixed the return of the comet to the place of its nearest distance from the sun on the fourth of April, 1759.
In the successive appearances of the comet, subsequently to 1456, it was found to have gradually decreased in magnitude and splendor. While in 1456 it reached across one third part of the firmament, and spread terror over Europe, in 1607, its appearance, when observed by Kepler and Longomontanus, was that of a star of the first magnitude; and so trifling was its tail that, Kepler himself, when he first saw it, doubted whether it had any. In 1682, it excited little attention, except among astronomers. Supposing this decrease of magnitude and brilliancy to be progressive, Lalande entertained serious apprehensions that on its expected return it might be so inconsiderable, as to escape the observation even of astronomers; and thus, that this splendid example of the power of science, and unanswerable proof of the principle of gravitation, would be lost to the world.
It is not uninteresting to observe the misgivings of this distinguished astronomer with respect to the appearance of the body, mixed up with his unshaken faith in the result of the astronomical inquiry. "We cannot doubt," says he, "that it will return; and even if astronomers cannot see it, they will not therefore be the less convinced of its presence. They know that the faintness of its light, its great distance, and perhaps even bad weather, may keep it from our view. But the world will find it difficult to believe us; they will place this discovery, which has done so much honor to modern philosophy, among the number of chance predictions. We shall see discussions spring up again in colleges, contempt among the ignorant, terror among the people; and seventy-six years will roll away, before there will be another opportunity of removing all doubt."
Fortunately for science, the arrival of the expected visitor did not take place under such untoward circumstances. As the commencement of the year 1759 approached, "astronomers," says Voltaire, "hardly went to bed at all." The honor, however, of the first glimpse of the stranger was not reserved for the possessors of scientific rank, nor for the members of academies or universities. On the night of Christmas-day, 1758, George Palitzch, of Politz, near Dresden,—"a peasant," says Sir John Herchel, "by station, an astronomer by nature," first saw the comet.