# Chapter XIII

## CHAPTER XIII.

THE NEWTONIAN PHILOSOPHY STATIONARY FOR HALF A CENTURY, OWING TO THE IMPERFECT STATE OF MECHANICS, OPTICS, AND ANALYSIS — DEVELOPED AND EXTENDED BY THE FRENCH MATHEMATICIANS — INFLUENCE OF THE ACADEMY OF SCIENCES — IMPROVEMENTS IN THE INFINITESIMAL CALCULUS — CHRISTIAN MAYER ON THE ARITHMETIC OF SINES — D'ALEMBERT'S CALCULUS OF PARTIAL DIFFERENCES — LAGRANGE'S CALCULUS OF VARIATIONS — THE PROBLEM OF THREE BODIES — IMPORTANCE OF THE LUNAR THEORY — LUNAR TABLES OF CLAIRAUT, D'ALEMBERT, AND EULER — THE SUPERIOR TABLES OF TOBIAS MAYER GAINS THE PRIZE OFFERED BY THE ENGLISH BOARD OF LONGITUDE — EULER RECEIVES PART OF THE ENGLISH REWARD, AND ALSO A REWARD FROM THE FRENCH BOARD — LAPLACE DISCOVERS THE CAUSE OF THE MOON'S ACCELERATION, AND COMPLETES THE LUNAR THEORY — LAGRANGE'S SOLUTION OF THE PROBLEM OF THREE BODIES AS APPLIED TO THE PLANETS — INEQUALITIES OF JUPITER AND SATURN EXPLAINED BY LAPLACE — STABILITY OF THE SOLAR SYSTEM THE PROOF OF DESIGN — MACLAURIN, LAPLACE, AND OTHERS, ON THE FIGURE OF THE EARTH — RESEARCHES OF LAPLACE ON THE TIDES, AND THE STABLE EQUILIBRIUM OF THE OCEAN — THEORETICAL DISCOVERY OF NEPTUNE BY ADAMS AND LEVERRIER — NEW SATELLITES OF SATURN AND NEPTUNE — EXTENSION OF SATURN'S RING AND ITS PARTIAL FLUIDITY — TWENTY-SEVEN ASTEROIDS DISCOVERED — LEVERRIER'S THEORY OF THEM — COMETS WITH ELLIPTIC ORBITS WITHIN OUR SYSTEM — LAW OF GRAVITY APPLIED TO DOUBLE STARS — SPIRAL NEBULÆ — MOTION OF THE SOLAR SYSTEM IN SPACE.

When Halley remarked that the author of the *Principia* "seemed to have exhausted his argument, and left little to be done by those who should succeed him," he committed a mistake which, though it had a tendency to check the progress of inquiry, was yet one into which philosophers are apt to fall when their science has made a great start by the discovery of some general and comprehensive law. Had Halley ventured to make this remark at the close of his life, rather than in 1687, he might have found some justification of it in the long interval which elapsed before any brilliant addition had been made to physical astronomy. During the half century which had passed away since the discovery of universal gravitation, no application of it of any importance had been made, and, as Laplace has observed, "all this interval was required for this great truth to be generally comprehended, and for surmounting the opposition which it encountered from the system of vortices, and from the prejudices of contemporaneous mathematicians." The infinitesimal analysis, as it was left by Newton and Leibnitz, was incapable of conducting the physical astronomer to any higher results than those which were consigned in the Principia; and it is a remarkable fact in the history of science, that the very men who spurned the new philosophy of gravitation, were strenuously engaged in improving that very calculus which was destined to establish and extend those great truths which they had so rashly denounced.

It has been remarked by Laplace, that "with the exception of his researches on the elliptical motion of the planets and comets, of the attraction of spherical bodies, and of the intensity of gravity at the surface of the sun, and of the planets that are accompanied by satellites, all the other discoveries which we have described were only blocked out by Newton. His theory of the figures of the planets was limited by the supposition of their homogeneity. His solution of the problem of the precession of the equinoxes, though very ingenious and accordant with ^{[1]}

In thus completing the great work of which Newton laid the foundation, it was necessary, as Laplace observes, "to bring to perfection at once the sciences of mechanics, optics, and analysis; and though physical astronomy may still be improved and simplified, yet posterity will gratefully acknowledge that the geometers of the eighteenth century have not transmitted to us a single astronomical phenomenon of which they have not determined the cause and the law. We owe to France the justice of observing, that if England had the advantage of giving birth to the discovery of universal gravitation, it is principally to the French geometers, and to the encouragement held out by the Academy of Sciences, that we owe the numerous developments of this discovery, and the revolution which it has produced in astronomy.^{[2]}

In submitting to our readers a brief history of these developments, and of that revolution, we shall gather fresh laurels for the author of the Principia. It is from what he left undone, and what he enabled others to do, that we can rightly estimate the magnitude and appreciate the value of his achievement. The importance of a great discovery does not lie in its intrinsic novelty and beauty : It is the number of its applications, and the ^{[3]} May we not make the question more special by asking why the University which he instructed and adorned, which possessed such noble endowments, and which claims the honour of having first adopted and taught his philosophy, did not rear a younger son, or even a sickly child, that could be ranked in the great family we have named ? Scotland contributed her Maclaurin, but England no European name; and a century and a half passed away till Airy and Adams adorned the birth-place of Newton's genius. In the same spirit in which we have asked these questions, M. Arago, equally jealous of the glory of his country, has freely confessed, "that no Frenchman can reflect, without an aching heart, on the small participation of his own country in the memorable achievement of the discovery of universal gravitation ;" and Mr. Grant, the latest historian of physical science,^{[4]} in responding to this liberal sentiment, has added, in the language of just severity, that "*if an Englishman could be supposed to be equally sensitive,* he has ample reason to regret the inglorious part his country played during the long period which marked the development of the Newtonian theory."^{[5]}

In the imperfect state in which the differential calculus was left by Newton and Leibnitz, its inventors, it was not fitted to grapple with the higher problems in physical astronomy which still remained to be solved; and it was fortunate for the future progress of the science, that distinguished mathematicians directed themselves to the improvement of the infinitesimal calculus, and to the discovery of new mechanical principles, or extended applications of those already known.

In 1727, the very year in which Newton died, Christian Mayer published in the Petersburg Commentaries a valuable memoir on the application of algebra to geometry; *Arithmetic of Sines,* for which Euler provided a notation and an algorithm, which have rendered it one of the most simple and valuable instruments of astronomical research. The invention of the calculus of *Partial Differences* by D'Alembert, which he first made known in 1747, was particularly applicable to the more difficult problems in physical astronomy, and when improved and extended by Euler, it became an invaluable instrument in every inquiry which demanded the aid of the pure or mixed mathematics.

But however valuable were these instruments of analysis, the *calculus of variations* discovered by Lagrange in 1760, was the greatest step in the improvement of the infinitesimal calculus which was made in the last century. It not only afforded the most complete solution of the problems that gave rise to it, but had an application of the most extensive kind, exceeding even the expectations of its inventor. Euler, who had made some progress in the same subject, at once acknowledged the superiority of his youthful rival, and with a nobility of mind not frequently displayed even by the greatest men, he renounced his own less perfect methods, and devoted himself to the study and extension of the new calculus.^{[6]}

Nearly twenty years after the death of Newton, Euler, Clairaut, and D'Alembert were engaged in solving what has been called the *problem of three bodies*, — that is, the determination of the motion of *one* body revolving round a *second* body, such as the *moon* round the *earth,* and disturbed by the attractions of a *third* body, such as the *sun.* The rigorous solution of this problem is beyond the reach of human genius, and the imperfect solution which has been obtained is only an approximate one depending for its accuracy on the more or less advanced state of the infinitesimal calculus. But even if the problem of three bodies had been susceptible of an accurate solution, it would not have diminished the difficulty of solving the more general problem of finding the motion of a planet, when simultaneously acted upon by all the other planets of the system. In this case the disturbances are very small, and when the separate action of each planet upon the disturbed body is determined, the sum of the perturbations, when applied to the place of the planet in its elliptic orbit, will give its true place in the heavens as seen from the centre of the sun.

When the three bodies are the sun, the moon, and the earth, the disturbance of the moon's motions by the action of the sun are very considerable, and hence the theory of the moon was the first subject to which the continental mathematicians directed their attention. The determination of the longitude at sea by observing the distance of the moon from the stars, had given a peculiar interest to the construction of accurate tables for computing the moon's place, and the Board of Longitude in England had offered a high reward. Mathematicians were urged to the inquiry by the united motives of wealth and fame. Newton had explained only five of the principal equations of the moon's orbit, and it was obvious that there were many other irregularities which observation alone was incapable of detecting. Clairaut seems to have been the first of the three mathematicians who undertook this inquiry; but however this may be, the competitors arrived at the same goal with nearly equal success. Clairaut had at first endeavoured to compute the lunar inequalities

by the method of Newton, but he was obliged to abandon it, and appeal to the higher powers of analysis. In 1746, Euler drew up a set of lunar tables, founded on the results of his researches, but they were not found to be very superior to those in common use. In 1754, Clairaut and D'Alembert published lunar tables, embodying the results of their theory. Those of Clairaut were far superior to any that had hitherto been published, while those of D'Alembert were very inferior in accuracy. Encouraged by the failure of his rivals, Euler resumed his investigations in 1755, and published a more complete set of lunar tables, along with his researches on the lunar theory; but though more conformable with observation than his former set, they had not that degree of accuracy which was required for the determination of the longitude.While the mathematicians, trusting too much to theory, were thus baffled in the useful application of their own results, a sagacious practical astronomer directed his attention to the improvement of the lunar tables, and carried off the prize. Tobias Mayer of Göttingen, comparing the results obtained by Euler with a number of accurate observations made by himself and others, produced a set of tables which, when compared with the observations of Bradley, gave the moon's place within *thirty seconds.* These tables were sent to the English Board of Longitude in 1755, in competition for the prize; but they did not possess that degree of precision which was required. Mayer, however, continued till the day of his death to add to their accuracy, and he left behind him a complete set of solar and lunar tables, for which the Board of Longitude awarded his widow the sum of *three thousand pounds,* a portion of the reward which they had offered for the discovery of the longitude. These tables were first published in 1770, and their greatest error was found never to exceed *one minute and a quarter.* As these tables were founded on Euler's theorems, the Board presented this distinguished mathematician with the sum of *three hundred pounds.* Though advanced in years, Euler was full of intellectual life, and having continued to labour at the lunar theory, he constructed a new set of tables, which were published in 1771, and were rewarded by the Board of Longitude in France.

Notwithstanding the accuracy of Mayer's tables, an irregularity had been discovered by observation which was not indicated by the theory of gravity. Halley and other astronomers had placed it beyond a doubt that the moon performed her monthly revolution round the earth in a shorter time than formerly. This acceleration of the moon, as it was called, amounted to nearly *ten seconds* in a century, and various hypotheses were framed to account for it. The most plausible of them was, that all space was filled with an ethereal medium which opposed such a resistance to the motions of the planets, that the force which kept them in their orbit would gradually overpower their diminished velocity, and thus shorten their period round the central body. This hypothesis was supported by Euler, and by the abettors of the undulatory theory, who required the existence of a medium for the propagation of light, and it was adopted with equal eagerness by another class of theorists, who saw in the acceleration of the celestial motions the process by which the Almighty was to destroy the solar system, by precipitating the secondary planets upon their primaries, and the primary planets upon the sun. Laplace admitted the sufficiency of the hypothesis, but as he saw no reason for admitting the existence of a resisting medium, he did not consider himself warranted in adopting such an hypothesis till it was found that gravitation was incapable of accounting for the fact. Another theory of the moon's acceleration was founded on the supposition that the daily motion of the earth was retarded by the continued blowing of the easterly winds of the tropics against the mountain ranges which extend from the equator to the poles; but Laplace satisfied himself from a rigorous examination of this supposition, that no retardation of the earth's motion could be thus produced. Another hypothesis still remained to which astronomers might appeal not only for the explanation of the moon's acceleration, but also of some considerable inequalities in the motions of Jupiter and Saturn, which appeared not to have a periodical character, and therefore to be in the same category with the moon's acceleration. Newton and every other philosopher had taken it for granted that the force of gravity was propagated instantaneously from bodies, and not in time like the rays of light; but it occurred to Laplace that if time was required for the transmission of gravity, it would affect the intensity of the force. He therefore computed the velocity of gravity that would be required to produce the observed acceleration, and he found it to be *eight millions* of times greater than the velocity of light, that is 192,500 miles multiplied by 8,000,000, or 1,540,000,000,000 miles in a second — a velocity which no language can express. After arriving at this result, Laplace found that if the acceleration is produced by another cause, then the effect of the successive transmissions would be insensible, and consequently the velocity of gravity, if it is not instantaneous, must at least be *fifty millions* of times greater than that of light, that is, must be at least 9,625,000,000,000 miles in a second.

In the course of these investigations it had been placed beyond a doubt that every inequality in the motion of the planets, and in the form of their orbits produced by their mutual gravitation, must be periodical, that is, that the inequality, after reaching its maximum, will diminish according to the same law by which it increased, and hence it became doubly interesting to discover the cause of phenomena which had this character. Although foiled in so many attempts to refer the moon's acceleration to the action of gravity, Laplace returned to the inquiry with fresh zeal, and about the end of 1787 his labours were crowned with success. It was well known to Lagrange and to himself, that the eccentricities of the planetary orbits underwent extremely slow changes, which had a very long period. To such a change the eccentricity of the earth's orbit is subject from the action of the planets. The mean action of the sun must therefore vary with the earth's eccentricity, and the earth, thus exerting a greater or a less force over the moon, will accelerate or retard her, and thus produce the secular inequality which has been observed in her mean motion. When the eccentricity is diminishing, which it has been doing since the date of the earliest astronomical observations, the moon's mean motion will be accelerated; but when the diminution ceases, and the orbit returns to its former ellipticity, the sun's action will increase, and the moon's mean motion will be retarded.^{[7]} Laplace found the acceleration to be *ten seconds* during a century, a rate which, notwithstanding its variable character, may be considered as uniform for two thousand years.

Although Halley suspected the existence of this inequa lity so early as 1693, yet it is to Mr. Dunthorne that we owe the first accurate determination of its magnitude. By means of lunar eclipses observed at Babylon in 721 B.C., and at Alexandria in 201 B.C., — a solar eclipse observed by Theon, A.D. 364, and other two by Ibyn Jounis at Cairo, about the end of the tenth century, he found the acceleration to be *ten* seconds in a hundred years. The consequence of this inequality is, that the moon is about *two hours* later in coming to the meridian than she would have been had she performed her monthly revolution in the same time that she did when the earliest Chaldean observations were made. "It is indeed a wonderful fact in the history of science," as Mr. Grant remarks, "that these rude notes of the priests of Babylon should escape the ruins of successive empires, and finally, after the lapse of three thousand years, should become subservient in establishing a phenomenon of so refined and complicated a character as the inequality we have just been considering."^{[8]} And in referring to the long period of the same inequality, Professor Playfair remarks, that "two thousand years are little more than an infinitesimal in this reckoning ; and as an astronomer thinks that he commits no error when he considers the rate of the sun's motion as uniform for twenty-four hours, so he commits none when he regards the rate of this equation as continuing the same for twenty centuries. That man, whose life, nay, the history of whose species occupies such a mere point in the duration of the world, should come to the knowledge of laws that embrace myriads of ages in their revolution, is perhaps the most astonishing fact that the history of science exhibits."^{[9]}

By this great discovery, which had eluded the grasp of Euler and Lagrange,^{[10]} Laplace may be regarded as having completed the lunar theory exactly one hundred years after it bad been sketched out in the first edition of the Principia.

The theory of the lunar motions being thus completed, the three rival mathematicians directed all the powers of their mind, and all the refinements of analysis, to the determination of the mutual action of the primary planets. In this case the three bodies were the sun, the disturbed and the disturbing planet. In 1748, the Academy of Sciences proposed the Inequalities of Jupiter and Saturn as the subject of their prize. In his Memoir, which gained the prize, Euler proved that both Jupiter and Saturn were subject to considerable inequalities, arising from their mutual action, but all of them periodical, and returning nearly in the same order after short intervals of not much more than twenty or thirty years. But though these results accorded with observation, they afforded no explanation of the great secular inequalities which in twenty centuries had produced in Jupiter an acceleration of 3° 33′, and in Saturn a retardation of 5° 13′. The Academy, therefore, again offered their prize of 1752 for the best Memoir on the same subject. Euler a second time carried off the prize ; but though he found two inequalities of long periods depending on the angle formed by the line of the *three* seconds, and for the latter a subtractive one of *fourteen* seconds ; but though this result was in its general character superior to that Euler, it yet afforded no explanation of the great inequalities we have mentioned. Having observed that the calculus had never given any inequalities but periodical ones, Lagrange now set himself to inquire, whether in the planetary system, continually increasing or continually diminishing inequalities, affecting the mean motions, could be produced by the mutual action of the two planets. Independently of any approximation, and by a method peculiarly his own, he found that all the inequalities produced by gravity must be periodical, and that amid all the changes arising from the mutual action of the planets, two elements are unchangeable, — the length of the major axis of the planet's elliptical orbit, and the time in which that orbit is described. The inclination of the orbit to the ecliptic changes, the ellipse and its eccentricity change, but its greater axis and the time of the planet's revolution are unalterable. This grand discovery, excluding every source of disorder, and securing the stability of the system, is doubtless one of the noblest in physical astronomy, and more than any other displays the wisdom of the Creator.

But though Lagrange had made this great step in celestial physics, he failed in discovering the cause of the inequalities of Jupiter and Saturn, and left to Laplace the honour of solving this perplexing problem. By a more rigorous inquiry into the effects of their mutual action, Laplace found that the mean motion of Jupiter would be accelerated, while that of Saturn would be retarded, and that the relative derangement of the two planets would be as *five* to *ten,* the ratio of their mean motion, or as 3° 58′ to 5° 16′, the result for Jupiter differing only *nine* minutes from that given by Halley. In continuing the inquiry, he found that each planet was subject to an inequality whose period was 969 years; — that of Saturn, when a maximum, being 48° 44″, and that of Jupiter 20′ 49″, with an opposite sign. These inequalities were a maximum in 1560, and from that epoch the apparent mean motions of the two planets have been approaching to their true mean motions, and became the same in 1790. By a comparison of these results with forty-three observed oppositions of Saturn, Laplace found them generally correct, and the error never exceeding *two minutes* of a degree. This difference he afterwards reduced in the case of both planets to *twelve seconds,* although the best tables of Saturn often erred *twenty minutes.* By these brilliant researches theory and observation were reconciled, — the last difficulty which beset the Newtonian theory was removed, — every inequality in the Solar system was explained, — and the law of gravitation established as a law of the universe.

In concluding this brief notice of the progress of phy sical astronomy since the time of Newton in a few of its leading features, we are naturally led to ponder on the great truth of the stability and permanence of the solar system as demonstrated by the discoveries of Lagrange and Laplace. In the present day, when worlds and systems of worlds, when life physical and life intellectual are supposed to be the result of general law, it is interesting to study those conditions of the planetary system which are necessary to its stability, and to consider whether they appear to be the result of necessity or design. It follows, from the discoveries of Laplace, that there are three conditions essential to the stability and permanence of the Solar system, namely, the motion of all the planets in the same direction, — their motion in orbits slightly elliptical, or nearly circular, — and the commensurability of their periods of revolution. That these conditions are not necessary is very obvious. Any one of them may be supposed different from what it is, while the rest remained the same. The planets, like the comets, might have been launched in different directions, and moved in planes of various and great inclinations to the ecliptic. They might have been propelled with such varieties of tangential force as to have moved in orbits of great ellipticity ; and no reason, even of the most hypothetical nature, can be assigned why their annual periods might not have been incommensurable. The arrangements, therefore, upon which the stability of the system depends, must have been the result of design, the contrivance of that omniscience which foresaw all that was future, and of that infinite skill which knew how to provide for the permanence of His work. How far the comets, whose motions are not regulated by such laws, and which move in so many directions, may in the future interfere with the order of our system, can only be conjectured. They have not interfered with it in the past, owing no doubt to the smallness of their density; and we cannot doubt that the same wisdom which has established so great a harmony in the movements of the planetary system, that the inequalities which necessarily arise from their mutual action arrive at a maximum, and then disappear, will also have made provision for the future stability of the system.

Although it is only a general view that we can take of the important discoveries in physical astronomy which have sprung from those of Newton, yet we should scarcely be justified in omitting those which relate to the figure of our earth and the tides of its ocean. Newton inferred that the figure of the earth was an oblate spheroid, whose equatorial diameter was to its polar axis in the ratio of 231 to 230, but it was reserved for Maclaurin to demonstrate, *a priori*, that the earth, if homogeneous, might assume such a form. The method which he employed, though synthetical, was remarkable for its accuracy and elegance. In 1743, Clairaut published his Treatise on the Figure of the Earth, in which he investigated the form it would assume on the supposition of its density being heterogeneous. He found that the earth would have the form of an elliptic spheroid, if its mass was arranged in homogeneous concentric strata of the same form ; and he investigated the beautiful theorem which bears his name, by which we can determine the ellipticity of the earth from measures of the force of gravity, taken in two different latitudes by the aid of the pendulum. D'Alembert, Lagrange, Legendre, Laplace, Ivory, Plana, Gauss, Poisson,, and Airy, have directed their attention to the subject of the earth's figure, but without adding much to the results obtained by Clairaut. In his *Mécanique Céleste,* Laplace has applied the deductions of his calculus to the determination of the figure of the earth, from the measurement of degrees on its surface, and the observations made in different latitudes On the length of a pendulum vibrating seconds, and he finds that the result cannot be reconciled with the hypothesis of an elliptic spheroid, unless a greater error than is probable be admitted in some of the measurements.^{[11]} Upon discussing, however, all the more recent measurements of a degree, and all the observations with the pendulum, the ellipticity of the earth in the former case has been found to be $\frac{1}{299}$, and, in the latter, $\frac{1}{289}$, the ellipticity indicated by the lunar perturbation being $\frac{1}{306}$, an agreement which is very remarkable, when we consider the local causes which necessarily affect the observations with the pendulum, as first noticed by Colonel Sabine, and the measurement of an arc of the meridian.

The theory of the tides of our ocean, though treated by a master mind in the Principia, was nevertheless susceptible of extension and improvement. The Academy of Sciences proposed it as the subject of their prize for 1740. Four dissertations competed for the prize, three of them of great merit, by Euler, Daniel Bernouilli, and Maclaurin, and a fourth by Father Cavalleri, a Jesuit, who founded his investigation on the system of Vortices. The prize was divided among all the four competitors, — a proof, doubtless, that the Cartesian doctrines were not entirely exploded. These dissertations, and others on the same subject, are founded on what is called the equilibrium theory, which supposes that the sun and moon ^{[12]}

As the ocean is often agitated by several irregular causes, such as storms and earthquakes, which raise it to great heights, and sometimes make it overstep its limits, Laplace has endeavoured to ascertain the "stability of the equilibrium of our seas." Although we find that the sea falls into its hollow bed after the ordinary commotions to which it is subject, yet we may reasonably fear that some extraordinary cause may communicate to it such a disturbance, that, though inconsiderable in its origin, may go on increasing till it raises it above the highest mountains. As such a result would afford an explanation of several phenomena of natural history, it becomes interesting to determine the conditions necessary to the absolute stability of the equilibrium of our seas, and to see if these conditions exist in nature. In submitting this question to analysis, Laplace has found that *the equilibrium of the ocean is stable if its density is less than the mean density of the earth,* and that its equilibrium cannot be subverted unless these two densities are equal, or that of the earth less than that of its waters. The experiments on the attraction of Schehallien and Mount Cenis, and those made by Mr. Cavendish, Reich, and Baily, with balls of lead, demonstrate that the mean density of the earth is at least *five* times that of water, and hence the stability of the ocean is placed beyond a doubt. As the seas therefore have at one time covered continents which are now raised above their level, we must seek for some other cause of it than any want of stability in the equilibrium of the ocean.^{[13]}

We have already seen how Newton deduced the precession of the equinoxes from the action of the sun and moon upon the excess of matter accumulated at the equator of the terrestrial spheroid. This investigation, however, was founded on principles not rigorously correct, and therefore the complete solution of the problem was left to his successors. The discovery, too, of the nutation and of its cause, by Bradley, gave a new character to the investigation, which now required the aid of the calculus of partial differences. It fell to the lot of D'Alembert to give a complete solution of the problem, whatever were the figure and the density of the strata of the terrestrial spheroid. The results which he obtained agreed accurately with observations on the precession, and he obtained also the true measure of the nutation, or the dimensions of the small ellipse described by the pole of the equator, which the observations of Bradley had left in some uncertainty.

In viewing the subject under a more general aspect than D'Alembert, Laplace was led to some very interesting results. From his researches on the oscillations of the ocean, he was led to the remarkable theorem, "that whatever be the law of the depth of the sea, and the form of the spheroid which it covers, the phenomena of the precession and the nutation are the same as if the sea formed a solid mass with this spheroid." Laplace has also shown that the rotation of the earth upon its axis, or the length of the day, cannot be affected either by currents on the ocean, rivers, trade-winds, or even earthquakes, or in general any force which can shake the earth either in its interior or upon its surface. It might have been expected that the trade winds blowing between the tropics would, by their action upon the sea, and upon the continents ^{[14]} After discussing the consequences respecting the constitution of the earth, which are accordant with his theory of the precession and nutation, Laplace states that though it does not enable us to determine the ellipticity of the earth, it fixes its limit between $\frac{1}{304}$ and $\frac{1}{578}$ part of the radius of the equator. The same theory indicates as the most probable constitution of the earth, that the density of its strata increases from its surface to its centre.^{[15]}

Such is a brief and general view of the important discoveries in physical astronomy which have illustrated the century that followed the publication of the Principia. Brilliant as they are, and evincing as they do the highest genius, yet the century in which we live has been rendered remarkable by a discovery which, whether we view it in its theoretical relations, or in its practical results, is the most remarkable in the history of physical astronomy. In the motions of the planet Uranus, discovered since the time of Newton, astronomers had been for a long time perplexed with certain irregularities, which could not be deduced from the action of the other planets. M. Bouvard, who constructed tables of this planet, seeing the impossibility of reconciling the ancient with the modern observations, threw out the idea that the irregularities from which this discrepancy arose might be owing to the action of an unknown planet. Our countryman, the Rev. Mr. Hussey, conceived "the possibility of some disturbing body beyond Uranus ;" and Hansen, with whom Bouvard corresponded on the subject, was of opinion that there must be *two new planets beyond Uranus* to account for the irregularities. In 1834, Mr. Hussey was anxious that the Astronomer-Royal should assist him in detecting the invisible planet, and other astronomers expressed the same desire, to have so important a question examined and settled. On his return to Berlin from the meeting of the British Association in 1846, the celebrated astronomer, M. Bessel, commenced the task of determining the actual position of the planet ; but in consequence of the death of M. Flemming, the young German astronomer, to whom he had entrusted some of the preliminary calculations, and of his own death not long afterwards, the inquiry was stopped.

While the leading astronomers in Europe were thus thinking and talking about the possible existence of a new planet beyond the orbit of Uranus, two young astronomers, Mr. Adams of St. John's College, Cambridge, and M. Leverrier of Paris, were diligently engaged in attempting to deduce from the irregularities which it produced in the motions of Uranus, the elements of the planet's orbit, and its actual position in the heavens. In October 1845, Mr. Adams had solved this intricate problem — the *inverse problem of perturbations,* as it has been called, placing beyond a doubt the theoretical existence of the planet, and assigning to it a place in the heavens, which was afterwards found to be little more than a single degree from its exact place! Anxious for the discovery of the planet in the heavens, Mr. Adams communicated his results to the Astronomer-Royal and Professor Challis ; but more than *nine* months were allowed to pass away before a single telescope was directed in search of it to the heavens. On the 29th July, Professor Challis began his observations, and on the 4th and 12th of August, when he directed his telescope to the theoretical place of the planet as given him by Mr. Adams, *he saw the planet, and obtained two positions of it.*

While Mr. Adams was engaged in this important inquiry, M. Leverrier, who had distinguished himself by a series of valuable memoirs on the great inequality of Pallas, — on the perturbations of Mercury, — and on the rectification of the orbits of comets, was busily occupied with the same problem. In the summer of 1845, M. Arago represented to Leverrier the importance of study ing the perturbations of Uranus. Abandoning his researches on comets, he devoted himself to the task suggested by his friend, and on the 10th November 1845, he communicated to the Academy of Sciences his *First Memoir on the Theory of Uranus.* In the following June he submitted to the Academy his Second Memoir, entitled, *Researches on the Motions of Uranus,* in which, after examining the different hypotheses that had been adduced to explain the irregularities of that planet, he is driven to the conclusion that *they are due to the action of a planet situated in the ecliptic at a mean distance double that of Uranus.* He then proceeds to determine where this planet is actually situated, what is its mass, and what are the elements of the orbit which it describes : After giving a rigorous solution of this problem, and shewing that there are not two quarters of the heavens in which we can place the planet at a given epoch, he computes its heliocentric place on the 1st January 1847, which he finds to be in the 325th degree of longitude, and he boldly asserts that in assigning to it this place, he does not commit an error of more than 10°. The position thus given to it is within a degree of that found by Mr. Adams. Anxious, like Mr. Adams, for the actual discovery of the planet, M. Leverrier naturally expected that practical astronomers would exert themselves in searching for it. The place which he assigned to it was published on the 1st of June, and yet no attempt seems to have been made to find it for nearly five months. The *exact position* of the planet was published on the 31st August, and on the 18th September was communicated to M. Galle, of the Royal Observatory of Berlin, who discovered it as a star of the eighth magnitude the very evening on which he received the request to look for it. Professor Challis had *secured* the discovery of this remarkable body six weeks before, but the honour of having actually found it belongs to the Prussian astronomer. With the universal concurrence of the astronomical world, the new planet received the name of Neptune. It revolves round the Sun in about 172 years, at a mean distance of thirty, that of Uranus being nineteen, and that of the Earth one ; and by its discovery the Solar system has been extended *one thousand millions of miles* beyond its former limits. The honour of having made this discovery belongs equally to Adams and Leverrier. It is the greatest intellectual achievement in the annals of astronomy, and the noblest triumph of the Newtonian Philosophy. To detect a planet by the eye, or to track it to its place by the mind, are acts as incommensurable as those of muscular and intellectual power. Recumbent on his easy chair, the practical astronomer has but to look through the cleft in his revolving cupola, in order to trace the pilgrim star in its course ; or by the application of magnifying power, to expand its tiny disc, and thus transfer it from among its sidereal companions to the planetary domains. The physical astronomer, on the contrary, has no such auxiliaries : he calculates at noon, when the stars disappear under a meridian sun: he computes at midnight, when clouds and darkness shroud the heavens; and from within that cerebral dome, which has no opening heavenward, and no instrument but the Eye of Reason, he sees in the disturbing agencies of an unseen planet, upon a planet by him equally unseen, the existence of the disturbing agent, and from the nature and amount of its action, he computes its magnitude and indicates its place. If man has ever been permitted to see otherwise than by the eye, it is when the clairvoyance of reason, piercing through screens of epidermis and walls of bone, grasps amid the abstractions of number and of quantity, those sublime realities which have eluded the keenest touch, and evaded the sharpest eye.

Although the philosophy of Newton has since his day enjoyed such signal triumphs, it has yet other strongholds to storm, and other conquests to achieve. In his survey of the sidereal and planetary domains, the practical astronomer has in the present century laid open new fields of research ripe for the intellectual sickle, and fitted to yield to the accomplished analyst the richest harvest of discovery.

Within our own system the detection of a satellite to Neptune, by Mr. Lassels, — of an eighth satellite to Saturn, by Mr. Lassels and Mr. Bond, between the orbits of the 4th and 5th of these bodies, — and of a new fluid ring gradually advancing to the body of the planet, will furnish interesting materials to the physical astronomer. This new and remarkable feature in the system of Saturn, has been recently studied by Mr. Bond of the United States, and M. Otto Struve, at the observatory of Pulkova, with the great Munich telescope. With that fine instrument they saw distinctly the dark interval which separates this new ring from the two old ones, and the boundaries of this interval were so well marked, that they succeeded in measuring its dimensions. They perceived, also, at the inner margin of the new ring, an edge or border feebly illuminated, which they conceived might be the commencement of another similar appendage, though the line of separation had not yet become visible. The following are the principal results which these two able astronomers have obtained: — "1. The new ring is not subject to very rapid changes. 2. It is not of very recent *That the inner border of the annular system of Saturn has, since the time of Huygens, been gradually approaching to the body of the planet,* and therefore it follows, that there has been a successive enlargement of this system. 4. That it is at least very probable that the approach of the rings towards the planet is caused particularly by the successive extension of the inner or middle ring. Hence it follows, that Saturn's system of rings does not exist, as has been generally supposed,^{[16]} in a state of stable equilibrium, and that *we may expect sooner or later, perhaps in some dozen of years, to see the rings united with the body of the planet."*

Of all the celestial phenomena which have been discovered since the time of Newton, the most remarkable are the *twenty-seven* small planets which have been discovered between the orbits of Mars and Jupiter. Dr. Olbers of Bremen, who discovered two of them, hazarded the idea that a large planet which had once occupied the same place, had been burst in pieces by some internal force. This opinion, which has been long considered as a very probable one, has only recently been called in question. M. Leverrier, the first mathematician who has directed his attention to the theory of this remarkable group of bodies, considers the opinion of Olbers as contradicted by the great inclination of the orbit of Pallas; and in place of explaining the existence of these planets by an alteration of the primitive system of the universe, he believes "that they have been regularly formed like all the other planets, and in virtue of the same laws." In a very interesting communication on this subject, lately made to the Academy of Sciences,^{[17]} M. Leverrier has endeavoured to ascertain the limit of the sum of the magnitudes of the whole group, known and unknown, by the disturbing action which they exercise on the motion of the perihelion of Mars and the Earth. If the perihelions of these small planets were distributed uniformly in all the regions of the zodiac, the action of these masses of matter, situated in one half or semi-circumference of the heavens, would be destroyed by the action of the equal masses situated in the opposite half or semi-circumference. But M. Leverrier finds that *twenty* out of *twenty-six* of the planets have the longitudes of their perihelion between 4° and 184°, a semi-circumference of the heavens, and hence their action as one mass on Mars and the Earth is not destroyed by the action of the other six planets. It is possible that the small planets, which may yet be discovered, may have more of their perihelions in the latter of their semi-circumferences than in the former; but the possibility is, that there will be more of them conjoined with the larger than the smaller group, or, at least, that they will be equally diffused over the zodiac in reference to their perihelion points.

Having shewn that the perihelion of Mars is placed much more advantageously than that of the Earth, in *eleven seconds,* a quantity which could not have escaped the notice of astronomers. Considering, therefore, that this inequality would become particularly sensible at the oppositions of Mars, M. Leverrier is led to believe, that though the orbit of Mars has not received its final improvements, yet it will not admit of an error in longitude greater than *one-fourth* of the above quantity, and hence he concludes, *that the sum total of the matter constituting the small planets situated between the mean distances 2.20 and 3.16, cannot exceed about the fourth part of the mass of the Earth*.

In examining the place of the nodes of the twenty-six planets, M. Leverrier finds that *twenty-two* of the ascending nodes of their orbits have their longitudes between 36° and 216°, that is, within a semi-circumference of the heavens,^{[18]} a result almost the same as that which takes place in their perihelions. From this fact he observes, that in considering the motion of the plane of the ecliptic, we may arrive at conclusions of the same kind respecting the magnitude of the mass of the small planets, though the limit would be less strict than in that which is derived from the grouping of their perihelions.

In his theory of the motion of comets, Sir Isaac Newton did not anticipate that bodies of this kind would be discovered moving in elliptical orbits, contained within the limits of our own system, and thus affording a new application of the law of gravity, and remarkable examples of the action of the planets upon this new class of wandering stars. It had long been the universal belief among astronomers, that every comet strayed far beyond the limits of our system, the shortest period being about seventy years. In 1818, however, M. Pons announced the discovery of a very faint comet, without a tail, the motions of which could not be reconciled with a parabolic orbit. After its fourth appearance, Professor M. Encke of Berlin, whose name is now attached to the comet, found that it moved in an elliptic orbit with a period of about 1211 days, or three years and a third, and that its orbit was included within our system, extending inward as far as Mercury, and outward only a little beyond the orbit of Pallas. He computed the perturbations produced by the action of Venus, the Earth, Mars, Jupiter, and Saturn, and he found that its periods had been diminishing between 1786 and 1838, at the rate of about $2\frac{1}{2}$ hours in each revolution — an effect which he ascribed to the resistance of an ethereal medium.

A still more remarkable comet, supposed to be the same as that of 1772, 1805, 1839, &c., was discovered in 1826 by M. Biela. Its period was found to be about 2410 days, or $6\frac{3}{4}$ years, and its orbit did not reach so far as that of Saturn. M. Damoiseau found that its arrival at its perihelion would be retarded nine days and sixteen hours by the action of Saturn, Jupiter, and the Earth ; and that on the 29th October 1832, about a month before its perihelion passage, it would cross the plane of the ecliptic, within 18,000 miles of a point in the Earth's orbit. The announcement of this fact excited such an alarm in Paris, that M. Arago was summoned to allay the fears of the community. According to prediction, the comet returned in 1839 and 1846 ; but, strange to say, it was on this last occasion *separated into two distinct comets,* the one a little fainter than the other. Their tails were parallel, and their distance, which was the same till the comet became single by the gradual disappearance of the smaller one, was found by M. Plantamour to be equal to about two-thirds of the radius of the moon's orbit, that is, about 160,000 miles!^{[19]}

Another cornet belonging to our system was discovered by M. Faye in November 1843. Dr. Goldsmicht found that its period was about 2718 days, or $7\frac{1}{2}$ years, and M. Leverrier computed that its arrival would be delayed 7 days and 16 hours by the action of the planets. Its orbit is more circular than that of any other comet, and is included between the orbits of Mars and Saturn. It had been suggested by M. Valz, that this comet might be Lexell's comet of 1770,^{[20]} which had been rendered visible by the action of Jupiter in 1767, and which was afterwards thrown into a larger orbit and rendered invisible in 1779 by the action of the same planet. M. Leverrier, however, has shown that the two bodies cannot be identical.

Before another year had expired, a fourth comet be longing to our system was discovered by M. De Vico of Rome. He first saw it on the 29th August 1844, and M. Faye found that it revolved in an elliptic orbit with a period of about 2000 days, or $5\frac{1}{2}$ years. It was supposed by some astronomers that this comet was the same as that of 1585, observed by Tycho ; but M. Leverrier has shown that they are not identical, and that the comet of De Vico is not the same as that of Lexell. He discovered, however, such a striking similarity between it and the comet observed by De la Hire in 1678, that he considers them clearly identical. It is strange, however, that this comet should only have been seen once previous to 1844, although it has frequently come very near the Earth.

Another comet of the Solar system was discovered by M. Brorsen on the 26th February 1846. Its period is 2042 days, or about $5\frac{4}{7}$ years. It is very faint, and is almost identical in its elements with the comet of 1532.

A seventh comet, discovered by M. Peters on the 26th June 1846, has been placed by the calculations of M. Arrest among those having elliptic elements and a short period, and therefore belonging to our system. Its period is 5804 days, or about 16 years.

Such are some of the important celestial phenomena within the limits of our own Solar system, to which the Newtonian theory is applicable, and to which it has been to some extent successfully applied. The sidereal phenomena which have been discovered beyond our system, in which movements of long periods, round visible and invisible centres, have been traced and measured, possess a higher interest, and to some of them also the Newtonian law of gravitation has been actually extended.

The most important of this class of phenomena are those of binary and multiple systems of stars. Among *γ* Virginis 708 years, and *γ* Leonis 1200 years. Although the list of double stars has been greatly extended, yet those whose orbits and periods have been determined with any accuracy, amount only to twenty-one. Nine of these have been computed by Mr. Madler of Dorpat, five by Sir John Herschel, four by Mr. Hind, and one by M. Savary.^{[21]} The first calculation of the orbit of a double star was made in 1830 by M. Savary, (in the case of *ξ* of the Great Bear,) who showed that the changes of place in one of the stars could be explained by an elliptic orbit, and a period of $58\frac{1}{4}$ years. The periods of the other twenty double stars vary from $31\frac{1}{2}$ to 737 years, eleven having their periods below 100 years, three below 200, two below 300, and three between 600 and 700 years. These orbits are calculated on the supposition that the force exerted by the stars varies inversely as the square of the distance, and the accuracy with which the observations are represented allows us to conclude that the Newtonian law of gravity extends to the distant region of the double stars.^{[22]}

Another sidereal phenomenon, in which we have the appearance of motion round a centre, is displayed in the spiral nebulæ discovered by Lord Rosse; that the stars which compose these spirals have been placed there in virtue of some movement related to the central mass, cannot be doubted, although it is vain for man to attempt the solution of such a problem. To suppose these spirals to be nothing more than vaporous matter, like the tail of a comet, whirled round into spiral branches, because we cannot find any explanation compatible with the almost universally admitted fact, that every nebula is composed of stars, is to renounce all faith in the great truths of astronomy, and seek for some resting-place to the mind, when reason stands aghast amid the infinite and the incomprehensible.

Beside the motions of one of the bodies which compose a binary system, a proper motion of a very peculiar kind has been observed in the stars. In one region of the heavens the distance between the stars is increasing, and in the opposite region diminishing, while in intermediate localities little or no change of place is observed. It is obvious that such changes indicate a motion of our earth, and consequently of the whole Solar system, to a point in the heavens where the increasing distance of the stars is a maximum. Before the proper motion of the fixed stars had been measured, various speculators, among whom Hooke was the earnest, hazarded the supposition that the whole Solar system was in continual motion. Tobias Mayer, in 177l, attempted in vain to deduce such a movement of the system from the proper motions of eighty stars; but a few years afterwards, in 1783, when better observations were accessible, Sir W. Herschel and M. Prévost came to the conclusion that the ^{[23]} and north declination 25°. Although both Biot and Bessel came to the same conclusion as Tobias Mayer, that no such motion existed, yet the existence of a proper motion has been more recently placed beyond a doubt by the observations made at the observatories of Dorpat, Abo, and Pulkova : And it has been shewn by the united studies of Argelander, Otto Struve, and Peters, that the point to which the Solar system is advancing at the epoch of 1840, is situated in, —

Right ascension, | 259° 35′, | with a probable error of 2° 57′ |

North declination, | 34° 33′, | with a probable error of 3° 24′. |

Not content with determining the direction of the solar motion, Otto Struve has computed the angular value of this motion, as seen at a right angle to the Sun's path, and at the mean distance of the stars of the first magnitude. His results are as follows : —

From the right ascension of the stars, | 0″.32122, | with a probable error of | 0″.03684 |

From their declination, | 0 .35719, | with a probable error of | 0 .03562 |

Or, taking the mean of these results, | 0″.33920 | 0″.03623 |

But as the parallax of stars of the first magnitude is 0″.209, we can change the angular motion of the Sun into a linear motion in space ; and hence taking the radius of the Earth's orbit as unity, M. Struve finds that the annual motion of the Sun in space is $\frac{\mathrm{0\u2033.3392}}{\mathrm{0\; .209}}=1.623$ radii of the Earth's orbit, with a probable error of 0.229.

In his interesting work on Stellar Astronomy,^{[24]} he has expressed these results in the following manner : — " *The motion of the solar system in space is directed to a point* *of the celestial vault situated on the right line which joins the two stars π and μ Herculis at a quarter of the apparent distance of these stars, reckoning from π Herculis. The velocity of this motion is such, that the Sun, with all the bodies which depend upon it, advances annually in the above direction* 1.623 *times the radius of the Earth's orbit, or* 33,550,000 *geographical miles. The possible error of this last number amounts to* 1,733,000 *geographical miles, or to a seventh part of the whole. We may then wager* 400,000 *to* 1 *that the Sun has a proper progressive motion, and* 1 *to* 1 *that it is comprised between the limits of thirty-eight and twenty-nine millions of geographical miles."*

If we take 95 millions of English miles as the mean radius of the Earth's orbit, we have $95\times 1.623=154.185$ millions of miles, and, consequently,

The velocity of the Solar system is | 154,185,000 miles in the year. |

" " | 422,424 miles in a day. |

" " | 17,601 miles in an hour. |

" " | 293 miles in a minute. |

" " | 57 miles in a second. |

As none of the celestial motions are rectilineal, the advance of the system in space must be round some distant centre, which M. Madler, without much reason, supposes to be *Alcyone,* the brightest star in the Pleiades. In the course of time, however, the point to which the system is advancing must change its place, and from the nature and magnitude of that change, its curvilineal motion, and perhaps the form of its orbit, may be established. But even if so grand a result were obtained, we may never be able to ascertain whether our Sun and planets revolve like a multiple star round a single centre, or, as in our planetary system, they form only one of a number of systems revolving round the same centre. On such a subject speculation is vain. We must rest satisfied with the simple truth, that since the earliest observation of the stars, our system has described so small a portion of its curvilineal orbit, that it cannot be distinguished from a straight line. If the buried relics of primeval life have taught us how brief has been our tenure of this terrestrial paradise, compared with its occupancy by the brutes that perish, the great sidereal truth which we have been expounding, impresses upon us the no less humbling lesson, that from the birth of man to the extinction of his race, the system to which he belongs will have described but an infinitesimal are of that immeasurable circle in which it is destined to revolve.

Such are the great sidereal movements to some of which the law of gravitation has been already applied, and nobody has ventured to doubt that all of them will, in due time, come under its rule. Every new satellite, every new asteroid, every new comet, every new planet, every new star circulating round its fellow, proclaims the universality of Newton's philosophy, and adds fresh lustre to his name. It is otherwise, however, in the general history of science. The reputation achieved by a great invention is often transferred to another which supersedes it, and a discovery which is the glory of one age is eclipsed b the extension of it in another. The fame of having invented the steam-engine has disappeared beside the reputation of the philosophers who have improved it; and the laurels which the discoverer of Ceres has worn for half a century, have been almost withered by the discovery of twenty-six similar bodies. It is the peculiar glory of Newton, however, that every discovery in the heavens attests the universality of his laws, and adds a greener leaf to the laurel chaplet which he wears.

^{[1]} *Systéme du Monde*, p. 336.

^{[2]} *Ibid.,* p. 340.

^{[3]} Professor Playfair adds, that this was "the more remarkable, as the interests of navigation were deeply involved in the question of the lunar theory, so that no motive which a regard to reputation or to interest could create was wanting to engage the mathematicians of England in the inquiry." — *Edinburgh Review*, vol. xi., p. 280. Jan. 1808.

^{[4]} *History of Physical Astronomy*, &c., p. 108. London, 1852. Mr. Grant also remarks, "that with the exception of Maclaurin and Thomas Simpson, hardly any individual of these islands deserves even to be mentioned in connexion with the history of physical astronomy during that period ;" and that, at the beginning of the present century, "there was hardly an individual in this country who possessed an intimate acquaintance with the methods of investigation which had conducted the foreign mathematicians to so many sublime results."

^{[5]}

Referring our readers to the statement at the end of Chapter IV., as showing the probable cause of the success of the French mathematicians, and of the inglorious failure of our own, we beg their attention to the following confirmation of our views by one of the wisest and most eminent of our Scottish mathematicians. In a review of Laplace's *Systéme du Monde*, Professor Playfair makes the following observations.

"The literary institution which has most completely produced its effect of any in modern times, and that has been most successful in promoting the interests of science, is that of the Royal Academy of Sciences of Paris, where *small pensions and great honours,* bestowed on a few men for devoting themselves exclusively to works of invention and discovery, have been the means of advancing the mathematical sciences in France to a state of unexampled prosperity.

"In England, *where such an institution as that just mentioned was wanting,* and where the public is perpetually prepared, with the question *cui bono,* to repress what seems the luxury of science, the same progress has not been made ; and our mercantile prejudices have so far defeated our own purpose, that if the matter had been left to us, the theory of the moon's motion would still have been extremely imperfect, and the great nautical problem of finding the longitude could have received nothing like an accurate solution." — *Edinburgh Review*, vol. xv. p. 39. Jan. 1810.

^{[6]} See the Article Mathematics in the *Edinburgh Encyclopædia*, vol. xiii. p. 380, where Sir John Herschel pronounces a beautiful eulogium on the conduct of Euler.

^{[7]} M. Leverrier has recently shewn that the earth's eccentricity will diminish during the period of *twenty-four thousand years!*

^{[8]} *History of Physical Astronomy*, pp. 63, 64.

^{[9]} *Edinburgh Review,* vol. xi. p. 261.

^{[10]} The Academy of Sciences proposed the moon's acceleration as the subject of their prize for 1770. Euler gained it, but came to the conclusion that it was not produced by the force of gravity. The same subject was again proposed in 1772, and the prize was divided between Euler and Lagrange. Euler ascribed the acceleration to a resisting medium, and Lagrange evaded the difficulty. The prize was again offered in 1774, and was gained by Lagrange, and he now doubted the existence of the inequality. It was under these circumstances that Laplace took up the ect, and obtained the results which we have mentioned.

^{[11]} *Mécanique Céleste*, tom. ii., liv. iii., chap. v.; and *Système du Monde*, liv. iv., chap. vii.

^{[12]} *Mécanique Céleste*, part i., liv. iv., chap. i., tom. ii., p. 171; and *Système du Monde*, liv. iv., chap. x., p. 248.

^{[13]} See *Mécanique Céleste*, part i., liv. iv., chap. ii., tom. ii., p. 204; and *Système du Monde,* liv. iv., chap. xi., p. 265.

^{[14]} *Système du Monde*, liv. iv., chap. xiii., pp. 276, 277. See also *Mécanique Céleste*, part i., liv. v., chap. i., tom. ii., p. 347.

^{[15]} *Mécanique Céleste*, tom. ii., pp. 354, 355.

^{[16]} Laplace has shewn that the stability of the equilibrium of the rings requires that they be irregular solids, unequally wide in different parts of their circumference, so that their centres of gravity do not coincide with their centres of figure. — See *Mécanique Céleste,* part i., liv. iii., chap. vi., tom. ii., p. 155; *Système du Monde*, liv. iv., chap. viii., p. 242.

^{[17]} *Considérations sur l'ensemble du Système des petites Planètes situées entre Mars et Jupiter par* M. U. J. Leverrier. Lu 28 Nov. 1853. *Comptes Rendus*, &c., tom. xxxvii. pp. 793-798.

^{[18]} M. Leverrier takes occasion to remark, "that we might perhaps find some systematic difference between the mean direction of the ascending nodes of the planets near the Sun, and that of the ascending nodes of the more distant planets, and that we may thus conjecture that these planets belong in reality to three distinct groups." — *Comptes Rendus*, &c., tom. xxxvii. p. 795.

^{[19]} Sir John Herschel has ventured to say, "that the orbit of Biela's comet so nearly intersects that of the Earth, that an actual collision is not impossible, and indeed (supposing neither orbit variable) must, *in all likelihood,* happen in the lapse of some millions of years." — *Outlines of Astronomy,* § 585.

^{[20]} This comet ought to have appeared *thirteen* times since 1770, and, as it has not been since seen, it must be lost. Burckhardt supposed that it might have become a satellite to Jupiter, from its aphelion being near that planet!

^{[21]} A table of the elements of their orbits is given by Sir John Herschel in his *Outlines of Astronomy*, § 843.

^{[22]} M. Madler has adduced an instance, (p Ophiuchi,) where he regards the deviations from an elliptic orbit too considerable to be accounted for by an error of observation ; but we cannot view a single fact of this kind as affecting the generality of the law of gravity.

^{[23]} M. Prévost, who used Mayer's proper motions, made the right ascension only 230°.

^{[24]} *Etudes d'Astronomie Stellaire,* of which we have given a copious abstract in the *North British Review*, vol. viii. pp. 523-534.