The period is something enormous. They will not have gone through all their changes until a period of 2,000,000 years has elapsed. This is the period of the planetary oscillation: "a great pendulum of eternity which beats ages as our pendulums beat seconds." Enormous it seems; and yet we have reason to believe that the earth has existed through many such periods.
The two laws of stability discovered and stated by Lagrange and Laplace I can state, though they may be difficult to understand:—
Represent the masses of the several planets by m1, m2, &c.; their mean distances from the sun (or radii vectores) by r1, r2, &c.; the excentricities of their orbits by e1, e2, &c.; and the obliquity of the planes of these orbits, reckoned from a single plane of reference or "invariable plane," by θ1, θ2, &c.; then all these quantities (except m) are liable to fluctuate; but, however much they change, an increase for one planet will be accompanied by a decrease for some others; so that, taking all the planets into account, the sum of a set of terms like these, m1e22√r1 + m2e22√r2 + &c., will remain always the same. This is summed up briefly in the following statement:
Σ(me2√r) = constant.
That is one law, and the other is like it, but with inclination of orbit instead of excentricity, viz.:
Σ(mθ2√r) = constant.
The value of each of these two constants can at any time be calculated. At present their values are small. Hence they always were and always will be small; being, in fact, invariable. Hence neither e nor r nor θ can ever become infinite, nor can their average value for the system ever become zero.
The planets may share the given amount of total excentricity and obliquity in various proportions between themselves; but even if it were all piled on to one planet it would not be very excessive, unless the planet were so small a one as Mercury; and it would be most improbable that one planet should ever have all the excentricity of the solar system heaped upon itself. The earth, therefore, never has been, nor ever will be, enormously nearer the sun than it is at present: nor can it ever get very much further off. Its changes are small and are periodic—an increase is followed by a decrease, like the swing of a pendulum.
The above two laws have been called the Magna Charta of the solar system, and were long supposed to guarantee its absolute permanence. So far as the theory of gravitation carries us, they do guarantee its permanence; but something more remains to be said on the subject in a future lecture ([XVIII]).
And now, finally, we come to a sublime speculation, thrown out by Laplace, not as the result of profound calculation, like the results hitherto mentioned, not following certainly from the theory of gravitation, or from any other known theory, and therefore not to be accepted as more than a brilliant hypothesis, to be confirmed or rejected as our knowledge extends. This speculation is the "Nebular hypothesis." Since the time of Laplace the nebular hypothesis has had ups and downs of credence, sometimes being largely believed in, sometimes being almost ignored. At the present time it holds the field with perhaps greater probability of ultimate triumph than has ever before seemed to belong to it—far greater than belonged to it when first propounded.