Although the invariability of the mean distances of the planetary orbits has been more completely demonstrated since the appearance of the memoir above referred to, that is to say, by pushing the analytical approximations to a greater extent, it will, notwithstanding, always constitute one of the admirable discoveries of the author of the Mécanique Céleste. Dates, in the case of such subjects, are no luxury of erudition. The memoir in which Laplace communicated his results on the invariability of the mean motions or mean distances is dated 1773.* It was in 1784 only that he established the stability of the other elements of the system from the smallness of the planetary masses, the inconsiderable eccentricity of the orbits, and the revolution of the planets in one common direction around the sun.

The discovery of which I have just given an account to the reader excluded, at least from the solar system, the idea of the Newtonian attraction being a cause of disorder; but might not other forces, by combining with attraction, produce gradually-increasing perturbations, as Newton and Euler dreaded? Facts of a positive nature seemed to justify these fears.

A comparison of ancient with modern observations revealed the existence of a continued acceleration of the mean motions of the moon and the planet Jupiter, and an equally striking diminution of the mean motion of Saturn. These variations led to conclusions of the most singular nature.

In accordance with the presumed cause of these perturbations, to say that the velocity of a body increased from century to century, was equiv alent to asserting that the body continually approached the center of motion. On the other hand, when the velocity diminished the body must be receding from the center.

Thus, by a strange arrangement of nature, our planetary system seemed destined to lose Saturn, its most mysterious ornament, to see the planet, accompanied by its ring and seven satellites, plunge gradually pended. In this way he found the limiting values of the eccentricity and inclination for the orbit of each of the principal 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.

*

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 plants, 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 circumstance 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 analysis, 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.

into unknown regions, whither the eye, armed with the most powerful telescope, has never penetrated. Jupiter, on the other hand, the planet compared with which the earth is so insignificant, appeared to be moving in the opposite direction, so as to be ultimately absorbed in the incandescent matter of the sun. Finally, the moon seemed as if it would one day precipitate itself upon the earth.

There was nothing doubtful or speculative in these sinister forebodings. The precise dates of the approaching catastrophes were alone uncertain. It was known, however, that they were very distant. cordingly, neither the learned dissertations of men of science nor the animated descriptions of certain poets produced any impression upon the public mind.

It was not so with our scientific societies, the members of which regarded with regret the approaching destruction of our planetary system. The Academy of Sciences called the attention of geometers of all countries to these menacing perturbations. Euler and Lagrange descended into the arena. Never did their mathematical genius shine with a brighter luster. Still the question remained undecided. The inutility of such efforts seemed to suggest only a feeling of resignation on the subject, when from two disdained corners of the theories of analysis the author of the Mécanique Céleste caused the laws of these great phenomena clearly to emerge. The variations of velocity of Jupiter, Saturn, and the moon flowed, then, from evident physical causes, and entered into the category of ordinary periodic perturbations, depending upon the principle of attraction.

The variations in the dimensions of the orbits, which were so much dreaded, resolved themselves into simple oscillations, included within narrow limits. Finally, by the powerful instrumentality of mathemati cal analysis, the physical universe was again established on a firm foundation.

I cannot quit this subject without at least alluding to the circumstances in the solar system upon which depend the so long unexplained variations of velocity of the moon, Jupiter, and Saturn.

The motion of the earth around the sun is mainly effected in an ellipse, the form of which is liable to vary from the effects of planetary pertur bation. These alterations of form are periodic; sometimes the curve, without ceasing to be elliptic, approaches the form of a circle, while at other times it deviates more and more from that form. From the epoch of the earliest recorded observations, the eccentricity of the terrestrial orbit has been diminishing from year to year; at some future epoch the orbit, on the contrary, will begin to deviate from the form of a circle, and the eccentricity will increase to the same extent as it previously dimin ished, and according to the same laws.

Now Laplace has shown that the mean motion of the moon around the earth is connected with the form of the ellipse which the earth de

scribes around the sun; that a diminution of the eccentricity of the ellipse inevitably induces an increase in the velocity of our satellite, and vice versa; finally, that this cause suffices to explain the numerical value of the acceleration which the mean motion of the moon has experienced from the earliest ages down to the present time.*

The origin of the inequalities in the mean motions of Jupiter and Saturn will be, I hope, as easy to conceive.

Mathematical analysis has not served to represent in finite terms the values of the derangements which each planet experiences in its movement from the action of all the other planets. In the present state of science, this value is exhibited in the form of an indefinite series of terms diminishing rapidly in magnitude. In calculation it is usual to neglect such of those terms as correspond in the order of magnitude to quantities beneath the errors of observation. But there are cases in which the order of the term in the series does not decide whether it be small or great. Certain numerical relations between the primitive elements of the disturbing and disturbed planets may impart sensible values to terms which usually admit of being neglected. This case occurs in the perturbations of Saturn produced by Jupiter, and in those of Jupiter produced by Saturn. There exists between the mean motions of these two great planets a simple relation of commensurability, five times the mean motion of Saturn being, in fact, very nearly equal to twice the mean motion of Jupiter. It happens in consequence that certain terms, which would otherwise be very small, acquire from this circumstance considerable values. Hence arise, in the movements of these two planets, inequalities of long duration, which require more than 900 years for their complete development, and which represent with marvelous accuracy all the irregularities disclosed by observation. Is it not astonishing to find in the commensurability of the mean motions of two planets a cause of perturbation of so influential a nature; to discover that the definitive solution of an immense difficulty, which baffled the genius of Euler, and which even led persons to doubt whether the theory of gravitation was capable of accounting for all the phenomena of the heavens, should depend upon the fortuitous circumstance of five times the mean motion of Saturn being equal to twice the mean motion of Jupiter? The beauty

* Mr. Adams has recently detected a remarkable oversight committed by Laplace and his successors in the analytical 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 co-efficient of the principal term of the secular inequality. This co-efficient 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, 1856,) that steps have recently been taken at the observatory for calculating the various circumstances of those phenomena, upon the basis of the more correct data furnished by the researches of Mr. Adams.—TRANSLATOR.

of the conception and the ultimate result are here equally worthy of admiration. *

2

T

2

R'

R

P

S

P

*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 illustration, 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. Assuming that the mean motion of Jupiter is to that of Saturn exactly in the proportion of five to two, it follows that T 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 threefifths of a revolution. Hence, when Jupiter arrives at T, Saturn will be a little in advance of T'. Let as 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 PS 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 motious 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 de scribed 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 circum ference, or 240°. In the same way it may be shown that the two planets will come into con junction 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 1200 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 retarding 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 retarding 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 be 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 retard 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 con

We have just explained how Laplace demonstrated that the solar system can experience only small periodic oscillations around a certain mean state. Let us now see in what way he succeeded in determining the absolute dimensions of the orbits.

What is the distance of the sun from the earth? No scientific question has occupied, in a greater degree, the attention of mankind; mathematically speaking, nothing is more simple. It suffices, as in common operations of surveying, to draw visual lines from the two extremities of a known base to an inaccessible object. The remainder is a process of elementary calculation. Unfortunately, in the case of the sun, the distance is great, and the bases which can be measured upon the earth are comparatively very small. In such a case the slightest errors in the direction of the visual lines exercise an enormous influence upon the results.

In the beginning of the last century, Halley remarked that certain interpositions of Venus between the earth and the sun, or, to use an expression applied to such conjunctions, that the transits of the planet across the sun's disk, would furnish at each observatory an indirect means of fixing the position of the visual ray very superior in accuracy to the most perfect direct methods.*

Such was the object of the scientific expeditions undertaken in 1761 junction 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 80 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 1200, and since in a period of three synodic revolutions, or 21,758 days, the line of conjunction travels through an arc of 80, 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 traveling 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 retarded. 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 mass 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 circumstance will produce an effect exactly analogous 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 accumulation, of the inequality.-TRANSLATOR.

*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.)—TRANS

LATOR.