round the sun. To effect a complete restoration of the p anetary orbits to their original position, with reference to their perihelion points, will require a grand compound cycle, amounting to millions of years. Yet the time will come when all the orbits will come again to their primitive positions, to start once more on their ceaseless journeys. In the changes of the eccentricities, it will be remembered, the stability of the system was involved. Should these changes be ever progressive, no matter how slowly, a time would finally come when the original figure of the orbit would be destroyed, the planet either falling into the sun, or sweeping away into unknown regions of space. But a limit is assigned, beyond which the change can never pass.Some of the planetary orbits are becoming more cir.. cular, others growing more elliptical; but all have their limits fixed. The earth’s orbit, for example, should the present rate of decrease of eccentricity continue, in about half a million of years will become an exact circle. There the progressive motion of the changes stops, and it slowly commences to recover its ellipticity. This is not the case with the motions of the perihelia. Their positions are in no way involved in the well being of a planet, or in its capacity to sustain the life which exists on its surface; and since the stability of the system is not endangered by progressive change, it ever continues in the same direction, until the final restoration is effected, by an entire revolution about the sun. Let us now examine the inclinations of the planetary orbits. Here it is found that there is no guarantee for the stability of the system, provided the angles a under which the orbits of the planets are inclined to each other do not remain nearly the same forever.If changes are found to exist, by which the inclina. tions are made to increase, without stopping and returning to their primitive condition, then is the perpetuity of the system rendered impossible. Its fair proportions must slowly wear away, the harmony which now prevails be destroyed, and chaos must come again. Commencing again with the earth, we find that, from the earliest ages, the inclination of the earth's equator to the ecliptic has been decreasing. Since the measure of Eratosthenes, 2,078 years ago, the decrease has amounted to about 23' 44", or about half second every year. Should the decrease continue, in about 85,000 years the equator and ecliptic would coincide, and the order of nature would be entirely changed ;-perpetual spring would reign throughout the year, and the seasons would be lost forever. Of this, however, there is no danger. The diminution will reach its limit in a comparatively short time, when the decrease of inclination will change into an increase, and thus slowly rocking backwards and forwards in thousands of years, the seasons shall ever preserve their appointed places, and seed time and harvest shall never fail. These changes of inclination are principally due to the perturbations of Venus, and arising from configurations, will be ultimately entirely compensated. The angles under which the planetary orbits are inclined to each other are in a constant state of mutation. The orbit of Jupiter at this time forms an angle with the ecliptic of 4,731 seconds, and this ungle is decreasing at such a rate that, in about 20,000 years the planes would actually coincide. This would not affect the well being of the planets or the stability of the system, but should the same change now continue, the angle between the orbits might finally come to fix them even at right angles to each other, and a subversion of the present system would result. A profound investigation of the problem of the planetary inclinations, accomplished by Lagrange, resulted in the demonstration of a relation between the masses of the planets, the principal axes of their orbits, and the inclinations, such that, although the angles of inclination may vary, the limits are narrow, and they are all found slowly to oscillate about their mean positions, never passing the prescribed limits, and securing, in this particular, the perpetuity of the system. Here, again, we are presented with the remarkable fact, that whenever mutation involves stability, this mutation is of a compensatory character, always returning upon itself, and, in the long run, correcting its own effects. If all this mighty system was organized by chance, how happens it that the angular motions of the perihelia of the planetary orbits are ever progressive, while the angular inotions of the planes of the orbits are vibrating? Design, positive and conspicuous, is written all over the system, in characters from which there is no escape. We now proceed to an examination of the lines in which the planes of the planetary orbits cut each other, or the lines in which they intersect a fixed plane. These are called the lines of nodes. They all pass through the sun's centre, and, in case they ever Q were coincident, they now radiate from a common point in all directions. Here is an element in no degree involving in its value the stability of the system, and from analogy we already begin to anticipate that its changes, whatever they may be, will probably progress always in the same direction. This is actually the case. The nodes of the planetary orbits are all slowly retrograding on a fixed plane, and in vast periods, amounting to thousands of years, accomplish revolutions, which, in the end, return them to their primitive positions. Thus are we led to the following results. Of the two elements which fix the magnitude of the planetary orbits, the principal axes, and the eccentricity, the axes remain invariable, while the eccentricity oscillates between narrow and fixed limits. In the long run, therefore, the magnitudes of the orbits are preserved. Of the three elements which give position to the planetary orbits, viz: the place of the perihelion, the lines of nodes, and the inclinations, the two first ever vary in the same direction, and accomplish their restoration at the end of vast periods of revolution, while the inclinations vibrate between narrow and prescribed limits. One more point, and we close this wonderful invegtigation. The last question which presents itself is this :—May not the periodic times of the planets be 80 adjusted to each, as that the results of certain configurations may be ever repeated without any compensation, and thus, by perpetual accumulation, finally effect a destruction of the system ? If the periodic times of two neighboring planets were exact multiples of the same quantity, or if the one was double the other, or in any exact ratio, then the contingency would arise, above alluded to, and there would be perturbations which would remain uncompensated. A near approach to this condition of things actually exists in the system, and gave great trouble to geometers. It was found, on comparing observations, that the mean periods of Jupiter and Saturn were not constant—that one was on the decrease, while the other was on the increase. This discovery seemed to disprove the great demonstration which had fixed as invariable the major axes of the planetary orbits, and guaranteed the stability of the mean motions. It was not until after Laplace had instituted a long and laborious research, that the phenomenon was traced to its true origin, and was found to arise from the near commensurability of the periodic times of Jupiter and Saturn-five of Jupiter's periods being nearly equal to two of Saturn's. In case the equality were exact, it is plain that if the two planets set out from the same straight line drawn from the sun, at the end of a cycle of five of Jupiter's periods, or two of Saturn’s, they would be again found in the same relative positions, and whatever effect the one planet had exerted over the other would again be repeated under the same precise circumstances. Hence would arise derangements which would progress in the same direction, and eventually lead to permanent derangement of the system. But it happens that five of Jupiter's periods are not exactly equal to two of Saturn's, and in this want of equality safety is found. The difference is such that the point of conjunction of the planets does not |