A philosopher, whose name is a.s.sociated with immortal discoveries, said to his audience who had allowed themselves to be influenced by ancient and consecrated authorities, "Bear in mind, Gentlemen, that in questions of science the authority of a thousand is not worth the humble reasoning of a single individual." Two centuries have pa.s.sed over these words of Galileo without depreciating their value, or obliterating their truthful character. Thus, instead of displaying a long list of ill.u.s.trious admirers of the three beautiful works of Laplace, we have preferred glancing briefly at some of the sublime truths which geometry has there deposited. Let us not, however, apply this principle in its utmost rigour, and since chance has put into our hands some unpublished letters of one of those men of genius, whom nature has endowed with the rare faculty of seizing at a glance the salient points of an object, we may be permitted to extract from them two or three brief and characteristic appreciations of the _Mecanique Celeste_ and the _Traite des Probabilites_.

On the 27th Vendemiaire in the year X., General Bonaparte, after having received a volume of the _Mecanique Celeste_, wrote to Laplace in the following terms:--"The first _six months_ which I shall have at my disposal will be employed in reading your beautiful work." It would appear that the words, the first _six months_, deprive the phrase of the character of a common-place expression of thanks, and convey a just appreciation of the importance and difficulty of the subject-matter.

On the 5th Frimaire in the year XI., the reading of some chapters of the volume, which Laplace had dedicated to him, was to the general "a new occasion for regretting, that the force of circ.u.mstances had directed him into a career which removed him from the pursuit of science."

"At all events," added he, "I have a strong desire that future generations, upon reading the _Mecanique Celeste_, shall not forget the esteem and friendship which I have entertained towards its author."

On the 17th Prairial in the year XIII., the general, now become emperor, wrote from Milan: "The _Mecanique Celeste_ appears to me destined to shed new l.u.s.tre on the age in which we live."

Finally, on the 12th of August, 1812, Napoleon, who had just received the _Traite du Calcul des Probabilites_, wrote from Witepsk the letter which we transcribe textually:--

"There was a time when I would have read with interest your _Traite du Calcul des Probabilites_. For the present I must confine myself to expressing to you the satisfaction which I experience every time that I see you give to the world new works which serve to improve and extend the most important of the sciences, and contribute to the glory of the nation. The advancement and the improvement of mathematical science are connected with the prosperity of the state."

I have now arrived at the conclusion of the task which I had imposed upon myself. I shall be pardoned for having given so detailed an exposition of the princ.i.p.al discoveries for which philosophy, astronomy, and navigation are indebted to our geometers.

It has appeared to me that in thus tracing the glorious past I have shown our contemporaries the full extent of their duty towards the country. In fact, it is for nations especially to bear in remembrance the ancient adage: _n.o.blesse oblige_!

FOOTNOTES:

[22] The author here refers to the series of biographies contained in tome III. of the _Notices Biographiques_.--_Translator_.

[23] These celebrated laws, known in astronomy as the laws of Kepler, are three in number. The first law is, that the planets describe ellipses around the sun in their common focus; the second, that a line joining the planet and the sun sweeps over equal areas in equal times; the third, that the squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun. The first two laws were discovered by Kepler in the course of a laborious examination of the theory of the planet Mars; a full account of this inquiry is contained in his famous work _De Stella Martis_, published in 1609. The discovery of the third law was not effected until, several years afterwards, Kepler announced it to the world in his treatise on Harmonics (1628). The pa.s.sage quoted below is extracted from that work.--_Translator_.

[24] The spheroidal figure of the earth was established by the comparison of an arc of the meridian that had been measured in France, with a similar arc measured in Lapland, from which it appeared that the length of a degree of the meridian increases from the equator towards the poles, conformably to what ought to result upon the supposition of the earth having the figure of an oblate spheroid. The length of the Lapland arc was determined by means of an expedition which the French Government had despatched to the North of Europe for that purpose. A similar expedition had been despatched from France about the same time to Peru in South America, for the purpose of measuring an arc of the meridian under the equator, but the results had not been ascertained at the time to which the author alludes in the text. The variation of gravity at the surface of the earth was established by Richer's experiments with the pendulum at Cayenne, in South America (1673-4), from which it appeared that the pendulum oscillates more slowly--and consequently the force of gravity is less intense--under the equator than in the lat.i.tude of Paris.--_Translator_.

[25] It may perhaps be asked why we place Lagrange among the French geometers? This is our reply: It appears to us that the individual who was named Lagrange Tournier, two of the most characteristic French names which it is possible to imagine, whose maternal grandfather was M. Gros, whose paternal great-grandfather was a French officer, a native of Paris, who never wrote except in French, and who was invested in our country with high honours during a period of nearly thirty years;--ought to be regarded as a Frenchman although born at Turin.--_Author_.

[26] The problem of three bodies was solved independently about the same time by Euler, D'Alembert, and Clairaut. The two last-mentioned geometers communicated their solutions to the Academy of Sciences on the same day, November 15, 1747. Euler had already in 1746 published tables of the moon, founded on his solution of the same problem, the details of which he subsequently published in 1753.--_Translator_.

[27] It must be admitted that M. Arago has here imperfectly represented Newton's labours on the great problem of the precession of the equinoxes. The immortal author of the Principia did not merely _conjecture_ that the conical motion of the earth's axis is due to the disturbing action of the sun and moon upon the matter acc.u.mulated around the earth's equator: he _demonstrated_ by a very beautiful and satisfactory process that the movement must necessarily arise from that cause; and although the means of investigation, in his time, were inadequate to a rigorous computation of the quant.i.tative effect, still, his researches on the subject have been always regarded as affording one of the most striking proofs of sagacity which is to be found in all his works.--_Translator_.

[28] It would appear that Hooke had conjectured that the figure of the earth might be spheroidal before Newton or Huyghens turned their attention to the subject. At a meeting of the Royal Society on the 28th of February, 1678, a discussion arose respecting the figure of Mercury which M. Gallet of Avignon had remarked to be oval on the occasion of the planet's transit across the sun's disk on the 7th of November, 1677.

Hooke was inclined to suppose that the phenomenon was real, and that it was due to the whirling of the planet on an axis "which made it somewhat of the shape of a turnip, or of a solid made by an ellipsis turned round upon its shorter diameter." At the meeting of the Society on the 7th of March, the subject was again discussed. In reply to the objection offered to his hypothesis on the ground of the planet being a solid body, Hooke remarked that "although it might now be solid, yet that at the beginning it might have been fluid enough to receive that shape; and that although this supposition should not be granted, it would be probable enough that it would really run into that shape and make the same appearance; _and that it is not improbable but that the water here upon the earth might do it in some measure by the influence of the diurnal motion, which, compounded with that of the moon, he conceived to be the cause of the Tides_." (Journal Book of the Royal Society, vol.

vi. p. 60.) Richer returned from Cayenne in the year 1674, but the account of his observations with the pendulum during his residence there, was not published until 1679, nor is there to be found any allusion to them during the intermediate interval, either in the volumes of the Academy of Sciences or any other publication. We have no means of ascertaining how Newton was first induced to suppose that the figure of the earth is spheroidal, but we know, upon his own authority, that as early as the year 1667, or 1668, he was led to consider the effects of the centrifugal force in diminishing the weight of bodies at the equator. With respect to Huyghens, he appears to have formed a conjecture respecting the spheroidal figure of the earth independently of Newton; but his method for computing the ellipticity is founded upon that given in the Principia.--_Translator_.

[29] Newton a.s.sumed that a h.o.m.ogeneous fluid ma.s.s of a spheroidal form would be in equilibrium if it were endued with an adequate rotatory motion and its const.i.tuent particles attracted each other in the inverse proportion of the square of the distance. Maclaurin first demonstrated the truth of this theorem by a rigorous application of the ancient geometry.--_Translator_.

[30] The results of Clairaut's researches on the figure of the earth are mainly embodied in a remarkable theorem discovered by that geometer, and which may be enunciated thus:--_The sum of the fractions expressing the ellipticity and the increase of gravity at the pole is equal to two and a half times the fraction expressing the centrifugal force at the equator, the unit of force being represented by the force of gravity at the equator._ This theorem is independent of any hypothesis with respect to the law of the densities of the successive strata of the earth. Now the increase of gravity at the pole may be ascertained by means of observations with the pendulum in different lat.i.tudes. Hence it is plain that Clairaut's theorem furnishes a practical method for determining the value of the earth's ellipticity.--_Translator_.

[31] The researches on the secular variations of the eccentricities and inclinations of the planetary orbits depend upon the solution of an algebraic equation equal in degree to the number of planets whose mutual action is considered, and the coefficients of which involve the values of the ma.s.ses of those bodies. It may be shown that if the roots of this equation be equal or imaginary, the corresponding element, whether the eccentricity or the inclination, will increase indefinitely with the time in the case of each planet; but that if the roots, on the other hand, be real and unequal, the value of the element will oscillate in every instance within fixed limits. Laplace proved by a general a.n.a.lysis, that the roots of the equation are real and unequal, whence it followed that neither the eccentricity nor the inclination will vary in any case to an indefinite extent. But it still remained uncertain, whether the limits of oscillation were not in any instance so far apart that the variation of the element (whether the eccentricity or the inclination) might lead to a complete destruction of the existing physical condition of the planet. Laplace, indeed, attempted to prove, by means of two well-known theorems relative to the eccentricities and inclinations of the planetary orbits, that if those elements were once small, they would always remain so, provided the planets all revolved around the sun in one common direction and their ma.s.ses were inconsiderable. It is to these theorems that M. Arago manifestly alludes in the text. Le Verrier and others have, however, remarked that they are inadequate to a.s.sure the permanence of the existing physical condition of several of the planets. In order to arrive at a definitive conclusion on this subject, it is indispensable to have recourse to the actual solution of the algebraic equation above referred to. This was the course adopted by the ill.u.s.trious Lagrange in his researches on the secular variations of the planetary orbits. (_Mem. Acad. Berlin_, 1783-4.) Having investigated the values of the ma.s.ses of the planets, he then determined, by an approximate solution, the values of the several roots of the algebraic equation upon which the variations of the eccentricities and inclinations of the orbits depended. In this way, he found the limiting values of the eccentricity and inclination for the orbit of each of the princ.i.p.al planets of the system. The results obtained by that great geometer have been mainly confirmed by the recent researches of Le Verrier on the same subject. (_Connaissance des Temps_, 1843.)--_Translator_.

[32] Laplace was originally led to consider the subject of the perturbations of the mean motions of the planets by his researches on the theory of Jupiter and Saturn. Having computed the numerical value of the secular inequality affecting the mean motion of each of those planets, neglecting the terms of the fourth and higher orders relative to the eccentricities and inclinations, he found it to be so small that it might be regarded as totally insensible. Justly suspecting that this circ.u.mstance was not attributable to the particular values of the elements of Jupiter and Saturn, he investigated the expression for the secular perturbation of the mean motion by a general a.n.a.lysis, neglecting, as before, the fourth and higher powers of the eccentricities and inclinations, and he found in this case, that the terms which were retained in the investigation absolutely destroyed each other, so that the expression was reduced to zero. In a memoir which he communicated to the Berlin Academy of Sciences, in 1776, Lagrange first showed that the mean distance (and consequently the mean motion) was not affected by any secular inequalities, no matter what were the eccentricities or inclinations of the disturbing and disturbed planets.--_Translator_.

[33] Mr. Adams has recently detected a remarkable oversight committed by Laplace and his successors in the a.n.a.lytical investigation of the expression for this inequality. The effect of the rectification rendered necessary by the researches of Mr. Adams will be to diminish by about one sixth the coefficient of the princ.i.p.al term of the secular inequality. This coefficient has for its multiplier the square of the number of centuries which have elapsed from a given epoch; its value was found by Laplace to be 10".18. Mr. Adams has ascertained that it must be diminished by 1".66. This result has recently been verified by the researches of M. Plana. Its effect will be to alter in some degree the calculations of ancient eclipses. The Astronomer Royal has stated in his last Annual Report, to the Board of Visitors of the Royal Observatory, (June 7, 1856,) that steps have recently been taken at the Observatory, for calculating the various circ.u.mstances of those phenomena, upon the basis of the more correct data furnished by the researches of Mr.

Adams.--_Translator_.

[34] [Ill.u.s.tration]

The origin of this famous inequality may be best understood by reference to the mode in which the disturbing forces operate. Let P Q R, P' Q' R'

represent the orbits of Jupiter and Saturn, and let us suppose, for the sake of ill.u.s.tration, that they are both situate in the same plane. Let the planets be in conjunction at P, P', and let them both be revolving around the sun S, in the direction represented by the arrows. a.s.suming that the mean motion of Jupiter is to that of Saturn exactly in the proportion of five to two, it follows that when Jupiter has completed one revolution, Saturn will have advanced through two fifths of a revolution. Similarly, when Jupiter has completed a revolution and a half, Saturn will have effected three fifths of a revolution. Hence when Jupiter arrives at T, Saturn will be a little in advance of T'. Let us suppose that the two planets come again into conjunction at Q, Q'. It is plain that while Jupiter has completed one revolution, and, advanced through the angle P S Q (measured in the direction of the arrow), Saturn has simply described around S the angle P' S' Q'. Hence the _excess_ of the angle described around S, by Jupiter, over the angle similarly described by Saturn, will amount to one complete revolution, or, 360.

But since the mean motions of the two planets are in the proportion of five to two, the angles described by them around S in any given time will be in the same proportion, and therefore the _excess_ of the angle described by Jupiter over that described by Saturn will be to the angle described by Saturn in the proportion of three to two. But we have just found that the excess of these two angles in the present case amounts to 360, and the angle described by Saturn is represented by P' S' Q'; consequently 360 is to the angle P' S' Q' in the proportion of three to two, in other words P' S' Q' is equal to two thirds of the circ.u.mference or 240. In the same way it may be shown that the two planets will come into conjunction again at R, when Saturn has described another arc of 240. Finally, when Saturn has advanced through a third arc of 240, the two planets will come into conjunction at P, P', the points whence they originally set out; and the two succeeding conjunctions will also manifestly occur at Q, Q' and R, R'. Thus we see, that the conjunctions will always occur in three given points of the orbit of each planet situate at angular distances of 120 from each other. It is also obvious, that during the interval which elapses between the occurrence of two conjunctions in the same points of the orbits, and which includes three synodic revolutions of the planets, Jupiter will have accomplished five revolutions around the sun, and Saturn will have accomplished two revolutions. Now if the orbits of both planets were perfectly circular, the r.e.t.a.r.ding and accelerating effects of the disturbing force of either planet would neutralize each other in the course of a synodic revolution, and therefore both planets would return to the same condition at each successive conjunction. But in consequence of the ellipticity of the orbits, the r.e.t.a.r.ding effect of the disturbing force is manifestly no longer exactly compensated by the accelerative effect, and hence at the close of each synodic revolution, there remains a minute outstanding alteration in the movement of each planet. A similar effect will he produced at each of the three points of conjunction; and as the perturbations which thus ensue do not generally compensate each other, there will remain a minute outstanding perturbation as the result of every three conjunctions. The effect produced being of the same kind (whether tending to accelerate or r.e.t.a.r.d the movement of the planet) for every such triple conjunction, it is plain that the action of the disturbing forces would ultimately lead to a serious derangement of the movements of both planets. All this is founded on the supposition that the mean motions of the two planets are to each other as two to five; but in reality, this relation does not exactly hold. In fact while Jupiter requires 21,663 days to accomplish five revolutions, Saturn effects two revolutions in 21,518 days. Hence when Jupiter, after completing his fifth revolution, arrives at P, Saturn will have advanced a little beyond P', and the conjunction of the two planets will occur at P, P' when they have both described around S an additional arc of about 8. In the same way it may be shown that the two succeeding conjunctions will take place at the points _q, q', r, r'_ respectively 8 in advance of Q, Q', R, R'. Thus we see that the points of conjunction will travel with extreme slowness in the same direction as that in which the planets revolve. Now since the angular distance between P and R is 120, and since in a period of three synodic revolutions or 21,758 days, the line of conjunction travels through an arc of 8, it follows that in 892 years the conjunction of the two planets will have advanced from P, P'

to R, R'. In reality, the time of travelling from P, P' to R, R' is somewhat longer from the indirect effects of planetary perturbation, amounting to 920 years. In an equal period of time the conjunction of the two planets will advance from Q, Q' to R, R' and from R, R' to P, P'. During the half of this period the perturbative effect resulting from every triple conjunction will lie constantly in one direction, and during the other half it will lie in the contrary direction; that is to say, during a period of 460 years the mean motion of the disturbed planet will be continually accelerated, and, in like manner, during an equal period it will be continually r.e.t.a.r.ded. In the case of Jupiter disturbed by Saturn, the inequality in longitude amounts at its maximum to 21'; in the converse case of Saturn disturbed by Jupiter, the inequality is more considerable in consequence of the greater ma.s.s of the disturbing planet, amounting at its maximum to 49'. In accordance with the mechanical principle of the equality of action and reaction, it happens that while the mean motion of one planet is increasing, that of the other is diminishing, and _vice versa_. We have supposed that the orbits of both planets are situate in the same plane. In reality, however, they are inclined to each other, and this circ.u.mstance will produce an effect exactly a.n.a.logous to that depending on the eccentricities of the orbits. It is plain that the more nearly the mean motions of the two planets approach a relation of commensurability, the smaller will be the displacement of every third conjunction, and consequently the longer will be the duration, and the greater the ultimate acc.u.mulation, of the inequality.--_Translator_.

[35] The utility of observations of the transits of the inferior planets for determining the solar parallax, was first pointed out by James Gregory (_Optica Promota_, 1663).--_Translator_.

[36] Mayer, from the principles of gravitation (_Theoria Lunae_, 1767), computed the value of the solar parallax to be 7".8. He remarked that the error of this determination did not amount to one twentieth of the whole, whence it followed that the true value of the parallax could not exceed 8".2. Laplace, by an a.n.a.logous process, determined the parallax to be 8".45. Encke, by a profound discussion of the observations of the transits of Venus in 1761 and 1769, found the value of the same element to be 8".5776.--_Translator_.

[37] The theoretical researches of Laplace formed the basis of Burckhardt's Lunar Tables, which are chiefly employed in computing the places of the moon for the Nautical Almanac and other Ephemerides. These tables were defaced by an empiric equation, suggested for the purpose of representing an inequality of long period which seemed to affect the mean longitude of the moon. No satisfactory explanation of the origin of this inequality could be discovered by any geometer, although it formed the subject of much toilsome investigation throughout the present century, until at length M. Hansen found it to arise from a combination of two inequalities due to the disturbing action of Venus. The period of one of these inequalities is 273 years, and that of the other is 239 years. The maximum value of the former is 27".4, and that of the latter is 23".2.--_Translator_.

[38] This law is necessarily included in the law already enunciated by the author relative to the mean longitudes. The following is the most usual mode of expressing these curious relations: 1st, the mean motion of the first satellite, plus twice the mean motion of the third, minus three times the mean motion of the second, is rigorously equal to zero; 2d, the mean longitude of the first satellite, plus twice the mean longitude of the third, minus three times the mean longitude of the second, is equal to 180. It is plain that if we only consider the mean longitude here to refer to a _given epoch_, the combination of the two laws will a.s.sure the existence of an a.n.a.logous relation between the mean longitudes _for any instant of time whatever_, whether past or future.

Laplace has shown, as the author has stated in the text, that if these relations had only been approximately true at the origin, the mutual attraction of the three satellites would have ultimately rendered them rigorously so; under such circ.u.mstances, the mean longitude of the first satellite, plus twice the mean longitude of the third, minus three times the mean longitude of the second, would continually oscillate about 180 as a mean value. The three satellites would partic.i.p.ate in this libratory movement, the extent of oscillation depending in each case on the ma.s.s of the satellite and its distance from the primary, but the period of libration is the same for all the satellites, amounting to 2,270 days 18 hours, or rather more than six years. Observations of the eclipses of the satellites have not afforded any indications of the actual existence of such a libratory motion, so that the relations between the mean motions and mean longitudes may be presumed to be always rigorously true.--_Translator_.

[39] Laplace has explained this theory in his _Exposition du Systeme du Monde_ (liv. iv. note vii.).--_Translator_.

APPENDIX.

(A.)

THE FOLLOWING IS A BRIEF NOTICE OF SOME OTHER INTERESTING RESULTS OF THE RESEARCHES OF LAPLACE WHICH HAVE NOT BEEN MENTIONED IN THE TEXT.

_Method for determining the orbits of comets._--Since comets are generally visible only during a few days or weeks at the utmost, the determination of their orbits is attended with peculiar difficulties.

The method devised by Newton for effecting this object was in every respect worthy of his genius. Its practical value was ill.u.s.trated by the brilliant researches of Halley on cometary orbits. It necessitated, however, a long train of tedious calculations, and, in consequence, was not much used, astronomers generally preferring to attain the same end by a tentative process. In the year 1780, Laplace communicated to the Academy of Sciences an a.n.a.lytical method for determining the elements of a comet's...o...b..t. This method has been extensively employed in France.

Indeed, previously to the appearance of Olber's method, about the close of the last century, it furnished the easiest and most expeditious process. .h.i.therto devised, for calculating the parabolic elements of a comet's...o...b..t.

_Invariable plane of the solar system._--In consequence of the mutual perturbations of the different bodies of the planetary system, the planes of the orbits in which they revolve are perpetually varying in position. It becomes therefore desirable to ascertain some fixed plane to which the movements of the planets in all ages may be referred, so that the observations of one epoch might be rendered readily comparable with those of another. This object was accomplished by Laplace, who discovered that notwithstanding the perpetual fluctuations of the planetary orbits, there exists a fixed plane, to which the positions of the various bodies may at any instant be easily referred. This plane pa.s.ses through the centre of gravity of the solar system, and its position is such, that if the movements of the planets be projected upon it, and if the ma.s.s of each planet be multiplied by the area which it describes in a given time, the sum of such products will be a maximum.

The position of the plane for the year 1750 has been calculated by referring it to the ecliptic of that year. In this way it has been found that the inclination of the plane is 1 35' 31", and that the longitude of the ascending node is 102 57' 30". The position of the plane when calculated for the year 1950, with respect to the ecliptic of 1750, gives 1 35' 31" for the inclination, and 102 57' 15" for the longitude of the ascending node. It will be seen that a very satisfactory accordance exists between the elements of the position of the invariable plane for the two epochs.

_Diminution of the obliquity of the ecliptic._--The astronomers of the eighteenth century had found, by a comparison of ancient with modern observations, that the obliquity of the ecliptic is slowly diminishing from century to century. The researches of geometers on the theory of gravitation had shown that an effect of this kind must be produced by the disturbing action of the planets on the earth. Laplace determined the secular displacement of the plane of the earth's...o...b..t due to each of the planets, and in this way ascertained the whole effect of perturbation upon the obliquity of the ecliptic. A comparison which he inst.i.tuted between the results of his formula and an ancient observation recorded in the Chinese Annals exhibited a most satisfactory accordance.

The observation in question indicated the obliquity of the ecliptic for the year 1100 before the Christian era, to be 23 54' 2".5. According to the principles of the theory of gravitation, the obliquity for the same epoch would be 23 51' 30".

_Limits of the obliquity of the ecliptic modified by the action of the sun and moon upon the terrestrial spheroid._--The ecliptic will not continue indefinitely to approach the equator. After attaining a certain limit it will then vary in the opposite direction, and the obliquity will continually increase in like manner as it previously diminished.

Finally, the inclination of the equator and the ecliptic will attain a certain maximum value, and then the obliquity will again diminish. Thus the angle contained between the two planes will perpetually oscillate within certain limits. The extent of variation is inconsiderable.

Laplace found that, in consequence of the spheroidal figure of the earth, it is even less than it would otherwise have been. This will be readily understood, when we state that the disturbing action of the sun and moon upon the terrestrial spheroid produces an oscillation of the earth's axis which occasions a periodic variation of the obliquity of the ecliptic. Now, as the plane of the ecliptic approaches the equator, the mean disturbing action of the sun and moon upon the redundant matter acc.u.mulated around the latter will undergo a corresponding variation, and hence will arise an inconceivably slow movement of the plane of the equator, which will necessarily affect the obliquity of the ecliptic.

Laplace found that if it were not for this cause, the obliquity of the ecliptic would oscillate to the extent of 4 53' 33" on each side of a mean value, but that when the movements of both planes are taken into account, the extent of oscillation is reduced to 1 33' 45".

_Variation of the length of the tropical year._--The disturbing action of the sun and moon upon the terrestrial spheroid occasions a continual _regression_ of the equinoctial points, and hence arises the distinction between the sidereal and tropical year. The effect is modified in a small degree by the variation of the plane of the ecliptic, which tends to produce a _progression_ of the equinoxes. If the movement of the equinoctial points arising from these combined causes was uniform, the length of the tropical year would be manifestly invariable. Theory, however, indicates that for ages past the rate of regression has been slowly increasing, and, consequently, the length of the tropical year has been gradually diminishing. The rate of diminution is exceedingly small. Laplace found that it amounts to somewhat less than half a second in a century. Consequently, the length of the tropical year is now about ten seconds less than it was in the time of Hipparchus.

_Limits of variation of the tropical year modified by the disturbing action of the sun and moon upon the terrestrial spheroid._--The tropical year will not continue indefinitely to diminish in length. When it has once attained a certain minimum value, it will then increase until finally having attained an extreme value in the opposite direction, it will again begin to diminish, and thus it will perpetually oscillate between certain fixed limits. Laplace found that the extent to which the tropical year is liable to vary from this cause, amounts to thirty-eight seconds. If it were not for the effect produced upon the inclination of the equator to the ecliptic by the mean disturbing action of the sun and moon upon the terrestrial spheroid, the extent of variation would amount to 162 seconds.

_Motion of the perihelion of the terrestrial orbit._--The major axis of the orbit of each planet is in a state of continual movement from the disturbing action of the other planets. In some cases, it makes the complete tour of the heavens; in others, it merely oscillates around a mean position. In the case of the earth's...o...b..t, the perihelion is slowly advancing in the same direction as that in which all the planets are revolving around the sun. The alteration of its position with respect to the stars amounts to about 11" in a year, but since the equinox is regressing in the opposite direction at the rate of 50" in a year, the whole annual variation of the longitude of the terrestrial perihelion amounts to 61". Laplace has considered two remarkable epochs in connection with this fact; viz: the epoch at which the major axis of the earth's...o...b..t coincided with the line of the equinoxes, and the epoch at which it stood perpendicular to that line. By calculation, he found the former of these epochs to be referable to the year 4107, B.C., and the latter to the year 1245, A.D. He accordingly suggested that the latter should be used as a universal epoch for the regulation of chronological occurrences.