best tables of Saturn exceeded 20'*. By this capital discovery Laplace banished empiricism from the tables of Jupiter and Saturn, and extricated the Newtonian theory from one of its gravest perils. "The irregularities of the two planets," says that illustrious geometer, appeared formerly to be inexplicable by the law of universal gravitation-they now form one of its most striking proofs. Such has been the fate of this brilliant discovery, that each difficulty which has arisen has become for it a new subject of triumph, a circumstance which is the surest characteristic of the true system of nature."+

We shall now give a brief account of the circumstances connected with a remarkable inequality in the moon's motion, which continued to form the subject of toilsome research until its true physical cause was at length discovered. From an extensive comparison of ancient with modern observations, it was established beyond doubt by the astronomers of the last century, that the mean motion of the moon has been becoming continually more rapid ever since the epoch of the earliest recorded observations. Halley was the first person who suspected this important fact. We may remark that, if the moon's mean motion be more rapid now than it was in ancient times, the place of that satellite, when computed for any remote epoch by means of the modern tables, will be less advanced than her actual place, and hence the time of an eclipse, when calculated in this manner, will appear to happen earlier than the recorded time. It is also obvious that, if we make a similar computation for any intermediate epoch, the moon in this case too will be thrown back in her orbit, though not to such an extent as in the previous case, and it is manifest that the error will diminish continually as we descend towards the epoch of the tables. Now this was the character of the results which Halley obtained from an examination of some ancient eclipses recorded by Ptolemy and the Arabian astronomers, and which in consequence induced him to suppose that the moon's mean motion was subject to a continual acceleration. He first alluded to this phenomenon in 1693, but no attempt was made to confirm his suspicion until the year 1749, when Dunthorne communicated a memoir to the Royal Society, in which he discussed all the observations calculated to throw light upon the subject. He computed by the modern tables an eclipse of the moon observed at Babylon in the year 721 a.c.; another, observed at Alexandria in the year 201 A.c.; a solar eclipse, observed by Theon in the year 364 A.D., and two similar phenomena, observed by Ibyn Jounis, at Cairo, in Egypt, towards the close of the tenth century. In all these cases the computed time of the phenomenon was earlier than the observed time; and the error generally was greater as the eclipse was more ancient. He therefore concluded that the several observations could only be reconciled with the tables by assuming that the mean motion was continually accelerated agreeably to the remark of Halley, and, from a comparison between the observed and computed times

* Laplace first explained these inequalities in the volume of the Academy of Sciences for the year 1784. In the volumes for the two following years he gave a complete analysis of the theory of Jupiter and Saturn, and shewed its accordance with the ancient and modern observations. An admirable exposition of the origin of the famous inequality mentioned in the text is contained in Airy's Treatise on Gravitation; a little work which should be in the hands of every person (whether a mathematician or not) who desires to obtain clear ideas of the various modes in which the planets disturb each other by their mutual attraction.

+ Mém. Cél., tome v. p. 324.

of a number of eclipses, he was induced to fix the amount of the acceleration at 10" in a century, counting from the year 1700.

A similar discussion conducted the celebrated astronomer Mayer to a secular acceleration of the mean motion. In his lunar tables, published in 1753, he fixed it at 7′′ in a century; but in those published at London, in 1770, it was raised to 9". Lalande also investigated the question in the year 1757, and deduced from his researches a secular equation of 9.886, which he ultimately fixed at 10".

Astronomers having thus demonstrated by incontestable evidence that the moon's mean motion was becoming continually more rapid, it henceforth became an interesting question to discover the physical cause of this phenomenon. The Academy of Sciences at Paris, always actuated by a zealous desire to promote the cause of science, offered their prize of 1770 for an investigation, which should have for its object to ascertain whether the theory of gravitation could render a satisfactory account of this secular inequality in the moon's motion. The prize was carried off by Euler; but that illustrious geometer was unable to discover any equations in the mean motion of a secular character. Towards the conclusion of his memoir, he uses the following remarkable words ::-"There is not one of the equations about which any uncertainty prevails, and now it appears to be established by indisputable evidence, that the secular inequality in the moon's mean motion cannot be produced by the forces of gravitation."* The future history of this inequality should teach us to accept with the utmost caution the dictum of any authority, however high, when it tends to impugn the generality of a principle supported, as in the present instance, by a multitude of phenomena of the most unequivocal character. Anxious to obtain a solution of this difficult question, the Academy of Sciences again proposed it for their prize of 1772. Euler and Lagrange were declared the successful competitors and shared the prize between them. Euler concluded his memoir by repeating the assertion he had made on the previous occasion, adding that no doubt henceforth could exist that the inequality arose from the resistance of an ethereal fluid pervading the celestial regions t. Lagrange, in his memoir, gave a new solution of the Problem of Three Bodies, which he applied to the moon, but he reserved for a future occasion a rigorous inquiry into the cause of the acceleration. Meanwhile, some persons began to entertain a suspicion that the spheroidal figures of the earth and moon, by disturbing the law of their mutual attraction, might occasion the inequality. This induced the Academy again to propose their prize of 1774 for an investigation of the subject. Lagrange was declared the successful competitor. He examined the effects of the moon's figure upon her motion, by a very skilful analysis, but he could find no equation of a secular character. By a simple process of reasoning, he extended the same conclusion to the earth, and he assured himself with equal confidence that the attraction of the planets and satellites could not be the cause of the phenomenon. He then entered upon a critical discussion of the observations upon which the alleged acceleration of the mean motion was founded, and his final conclusion was, that in general the data were of a doubtful character, and that perhaps the best course would be to reject the inequality altogether §.

Laplace, about the same time, investigated this interesting subject. Having carefully examined the ancient observations, he was induced to consider it as fully established, that the moon's mean motion was becoming more rapid in modern times. Some persons had endeavoured to explain the phenomenon by means of a continual retardation of the earth's diurnal motion. If this supposition were true, an acceleration ought to have manifested itself in the mean motions of the planets, as well as in that of the moon, but this was not borne out by observation. But, besides, no sufficient cause could be assigned why the rotatory motion of the earth should be continually retarded. It was indeed alleged, that this effect might be produced by the continual blowing of the easterly winds, generated by the heats of the torrid zone, against the great mountain chains which run from north to south in both hemispheres. Laplace, however, mentions that he examined this point with attention, and arrived at the conclusion that no retardation of the diurnal motion could possibly arise from such a cause. He considered another solution of the problem, founded on the supposition that the regions of space are occupied by an ethereal fluid, which continually resists the motions of the celestial bodies. He admits that such an hypothesis suffices to explain the phenomenon, but he contends that we have no independent proof of the existence of an ethereal fluid, and until we are assured beyond all possibility of doubt that the theory of gravitation cannot account for the moon's acceleration, we ought not to have recourse to any extraneous source of explanation *. His views on this subject are unquestionably more sagacious and philosophical than those of Euler or Lagrange

Unable to discover a secular inequality in the disturbing action of the sun, and yet reluctant to derive this result from any foreign principle, he was led to consider what effect might be produced by adopting a different conception of gravity. It had been always assumed that the effects of this principle were propagated instantaneously from bodies. Laplace, however, considered, that some time might be required for this purpose, and he readily perceived that such a supposition would have the effect of modifying the intensity of the force exerted on the moving body. He therefore computed what ought to be the velocity of gravity, in order that the gradual transmission of that principle should occasion the observed acceleration of the moon's mean motion, and he arrived at the remarkable conclusion that it must exceed the velocity of light eight millions of times. He remarked that if a satisfactory account of the origin of the phenomenon be adduced, without having recourse to this hypothesis, it would follow that the effects of the successive transmission of gravity would be insensible, and therefore the velocity must be at least fifty millions of times greater than the velocity of light!

No further progress was made in this question until the close of the year 1787, when Laplace finally announced that he had discovered the cause of the phenomenon in the gradual diminution of the mean action of the sun, arising in consequence of the secular variation of the eccentricity of the terrestrial orbit +. The mean action of the sun upon the moon tends to diminish the moon's gravity to the earth, and thereby causes a diminution of her angular velocity. This diminution being once supposed to occur, the angular velocity would afterwards remain constant, provided the mean solar action always retained the same value. This, • Mém. des Savans Étrangers, tome vii.

Mém. Acad. des Sciences, 1786.

however, is not the case, for it depends to a certain extent on the eccentricity of the terrestrial orbit, an element which we know to be in a state of continual though inconceivably slow variation, from the action of the planets on the earth. This variation of the earth's eccentricity will, therefore, produce a corresponding variation in the mean action of the sun; and the earth, in consequence, having more or less power over the moon, will either quicken or retard her angular velocity, whence will ensue a secular inequality in the mean motion conformably to observation. Now, the eccentricity of the earth's orbit has been continually diminishing from the date of the earliest recorded observations down to the present time; hence the sun's mean action must also have been diminishing, and consequently the moon's mean motion must have been continually increasing. This acceleration will continue as long as the earth's orbit is approaching towards a circular form, but as soon as this process ceases, and the orbit again begins to open out, the sun's mean action will increase, and the acceleration of the moon's mean motion will be converted into a continual retardation *.

Laplace computed the acceleration, and found it to amount to 10.1816213, t denoting the number of centuries before or after the year 1801. This result agrees as nearly as possible with that which astronomers have derived from a comparison of ancient with modern observations.

If the inequality were rigorously determinable by the preceding formula, it would obviously continue for ever to increase in the same direction, a conclusion which would be totally at variance with the explanation we have just given of its physical cause. The fact is, however, that the complete analytical expression of it is a periodic function of the time, and the quantity 10".1816213t is merely the second term in the developement of it t, the others being so small as to admit of being rejected, when the computation does not extend to more than about 2000 years. Laplace, indeed, found that when the moon's place was calculated for the time of the Chaldean observations, it would be necessary to take into account the term depending on the cube of t. The inequality would then be expressed thus: 10".1816213+0.01853844t".

The variation of the earth's eccentricity, upon which the inequality in the moon's mean motion depends, cannot be calculated from theory without a knowledge of the masses of the planets. When it is considered what uncertainty prevails respecting the masses of Mars and Venus, it is surprising how close the agreement is between theory and observation. Fortunately, the planet which exercises by far the greatest influence on the eccentricity is Jupiter, whose mass is easily derived from the elongations of his satellites. It is remarkable that the action of the planets on the moon, when transmitted to her indirectly through the medium of the sun, should be more considerable than their direct action upon her.

The moon, in the present day, is about two hours later in coming to the • It does not necessarily follow that because an inequality is secular it should increase continually in the same direction. Lagrange found that the secular inequalities in the mean motions of Jupiter and Saturn were of a recurring character, although their duration extended to the immense period of 70414 years! The secular inequality in the moon's mean motion, being a more complicated phenomenon, has a much longer period

than this.

The first term, being proportional to the time, is absorbed in the mean motion, and therefore cannot form part of the inequality as determined by observation,

meridian than she would have been if she had retained the same mean motion as in the time of the earliest Chaldean observations. It is a wonderful fact in the history of science, that those rude notes of the priests of Babylon should escape the ruin of successive empires, and, finally, after the lapse of nearly 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.

Laplace also discovered that the lunar perigee and nodes were subject to secular inequalities from the same cause. He found that the inequality in the perigee was to the corresponding inequality in the mean motion as 33 to 10, and was subtractive from the mean longitude. He also discovered that the secular inequality of the nodes amounted to seven-tenths of that of the mean motion, and was additive to the mean longitude. Thus it appeared that, while the mean motion was continually accelerated, the perigee and nodes were continually retarded, the three inequalities being as the numbers 1, 3, .7*. It hence also followed that the moon's motions, with respect to the sun, her perigee, and her nodes, continually increased in the ratios of 1, 4, 0.265. The existence and magnitude of these inequalities were confirmed in a most satisfactory manner by Bouvard, who for this purpose instituted an extensive comparison between the ancient and modern observations.

These great inequalities all depend on the secular variation of the earth's eccentricity. They will continually become more perceptible as ages roll on, but a vast number of years will elapse before they will have passed through all their values t. They will one day affect the secular motion of the moon to the extent of at least the fortieth part of the circumference, and that of the perigee to the extent of the thirteenth part.

It might be imagined that the secular variation in the position of the ecliptic would have the effect of modifying the sun's action on the moon, and would in consequence disturb the mean inclination of the lunar orbit. Laplace, however, found that the moon was constantly maintained by the sun at the same inclination towards the moveable plane of the ecliptic, so that her declinations were subject to the same secular changes as those of the sun, and were due solely to the continued diminution of the obliquity of the ecliptic.

It was not until Laplace announced his discovery of the cause of the moon's acceleration, that Lagrange became aware of the oversight he had committed, while engaged in similar researches in 1783, by neglecting to apply his analysis to the moon. He now made the required substitutions, and, computing the numerical value of the inequality, he obtained a result which almost coincided with that of Laplace §.

We shall conclude this historical notice of the secular inequalities of

Since the mean motion of the lunar perigee is direct, the effect of an inequality, which is subtractive from the mean longitude, will manifestly be to retard the perigee behind its true place. On the other hand, since the motion of the nodes is retrograde, a retardation can only take place when the inequality is additive to the mean longitude.

Leverrier has found that the eccentricity of the terrestrial orbit will continue to diminish during the period of 23,980 years. It will then attain a minimum value equal to 0.003314. Mémoire sur les variations séculaires des élémens des orbites pour les sept planètes principales. See also Connaissance des Temps, 1843. Exposition du Système du Monde, tome ii,, liv. iv, chap. v.

§ Mém. Acad. Berlin, 1792-3.