where _m_ is the ma.s.s of each planet, _d_ its mean distance from the sun, _e_ the excentricity of its...o...b..t, and [theta] the inclination of its plane. However the expressions above formulated may change for individual planets, the sum of them for all the planets remains invariable.

The period of the variations in excentricity of the earth's...o...b..t is 86,000 years; the period of conical revolution of the earth's axis is 25,800 years. About 18,000 years ago the excentricity was at a maximum.

LECTURE XI

LAGRANGE AND LAPLACE--THE STABILITY OF THE SOLAR SYSTEM, AND THE NEBULAR HYPOTHESIS

Laplace was the son of a small farmer or peasant of Normandy. His extraordinary ability was noticed by some wealthy neighbours, and by them he was sent to a good school. From that time his career was one brilliant success, until in the later years of his life his prominence brought him tangibly into contact with the deteriorating influence of politics. Perhaps one ought rather to say trying than deteriorating; for they seem trying to a strong character, deteriorating to a weak one--and unfortunately, Laplace must be cla.s.sed in this latter category.

It has always been the custom in France for its high scientific men to be conspicuous also in politics. It seems to be now becoming the fashion in this country also, I regret to say.

The _life_ of Laplace is not specially interesting, and I shall not go into it. His brilliant mathematical genius is unquestionable, and almost unrivalled. He is, in fact, generally considered to come in this respect next after Newton. His talents were of a more popular order than those of Lagrange, and accordingly he acquired fame and rank, and rose to the highest dignities. Nevertheless, as a man and a politician he hardly commands our respect, and in time-serving adjustability he is comparable to the redoubtable Vicar of Bray. His scientific insight and genius were however unquestionably of the very highest order, and his work has been invaluable to astronomy.

I will give a short sketch of some of his investigations, so far as they can be made intelligible without overmuch labour. He worked very much in conjunction with Lagrange, a more solid though a less brilliant man, and it is both impossible and unnecessary for us to attempt to apportion respective shares of credit between these two scientific giants, the greatest scientific men that France ever produced.

First comes a research into the libration of the moon. This was discovered by Galileo in his old age at Arcetri, just before his blindness. The moon, as every one knows, keeps the same face to the earth as it revolves round it. In other words, it does not rotate with reference to the earth, though it does rotate with respect to outside bodies. Its libration consists in a sort of oscillation, whereby it shows us now a little more on one side, now a little more on the other, so that altogether we are cognizant of more than one-half of its surface--in fact, altogether of about three-fifths. It is a simple and unimportant matter, easily explained.

The motion of the moon may be a.n.a.lyzed into a rotation about its own axis combined with a revolution about the earth. The speed of the rotation is quite uniform, the speed of the revolution is not quite uniform, because the orbit is not circular but elliptical, and the moon has to travel faster in perigee than in apogee (in accordance with Kepler's second law). The consequence of this is that we see a little too far round the body of the moon, first on one side, then on the other. Hence it _appears_ to oscillate slightly, like a lop-sided fly-wheel whose revolutions have been allowed to die away so that they end in oscillations of small amplitude.[23] Its axis of rotation, too, is not precisely perpendicular to its plane of revolution, and therefore we sometimes see a few hundred miles beyond its north pole, sometimes a similar amount beyond its south. Lastly, there is a sort of parallax effect, owing to the fact that we see the rising moon from one point of view, and the setting moon from a point 8,000 miles distant; and this base-line of the earth's diameter gives us again some extra glimpses. This diurnal or parallactic libration is really more effective than the other two in extending our vision into the s.p.a.ce-facing hemisphere of the moon.

These simple matters may as well be understood, but there is nothing in them to dwell upon. The far side of the moon is probably but little worth seeing. Its features are likely to be more blurred with acc.u.mulations of meteoric dust than are those of our side, but otherwise they are likely to be of the same general character.

The thing of real interest is the fact that the moon does turn the same face towards us; _i.e._ has ceased to rotate with respect to the earth (if ever it did so). The stability of this state of things was shown by Lagrange to depend on the shape of the moon. It must be slightly egg-shape, or prolate--extended in the direction of the earth; its earth-pointing diameter being a few hundred feet longer than its visible diameter; a cause slight enough, but nevertheless sufficient to maintain stability, except under the action of a distinct disturbing cause. The prolate or lemon-like shape is caused by the gravitative pull of the earth, balanced by the centrifugal whirl. The two forces balance each other as regards motion, but between them they have strained the moon a trifle out of shape. The moon has yielded as if it were perfectly plastic; in all probability it once was so.

It may be interesting to note for a moment the correlative effect of this aspect of the moon, if we transfer ourselves to its surface in imagination, and look at the earth (cf. Fig. 41). The earth would be like a gigantic moon of four times our moon's diameter, and would go through its phases in regular order. But it would not rise or set: it would be fixed in the sky, and subject only to a minute oscillation to and fro once a month, by reason of the "libration" we have been speaking of. Its aspect, as seen by markings on its surface, would rapidly change, going through a cycle in twenty-four hours; but its permanent features would be usually masked by lawless acc.u.mulations of cloud, mainly aggregated in rude belts parallel to the equator. And these cloudy patches would be the most luminous, the whitest portions; for of course it would be their silver lining that we would then be looking on.[24]

Next among the investigations of Lagrange and Laplace we will mention the long inequality of Jupiter and Saturn. Halley had found that Jupiter was continually lagging behind its true place as given by the theory of gravitation; and, on the other hand, that Saturn was being accelerated.

The lag on the part of Jupiter amounted to about 34-1/2 minutes in a century. Overhauling ancient observations, however, Halley found signs of the opposite state of things, for when he got far enough back Jupiter was accelerated and Saturn was being r.e.t.a.r.ded.

Here was evidently a case of planetary perturbation, and Laplace and Lagrange undertook the working of it out. They attacked it as a case of the problem of three bodies, viz. the sun, Jupiter, and Saturn; which are so enormously the biggest of the known bodies in the system that insignificant ma.s.ses like the Earth, Mars, and the rest, may be wholly neglected. They succeeded brilliantly, after a long and complex investigation: succeeded, not in solving the problem of the three bodies, but, by considering their mutual action as perturbations superposed on each other, in explaining the most conspicuous of the observed anomalies of their motion, and in laying the foundation of a general planetary theory.

[Ill.u.s.tration: FIG. 79.--Shewing the three conjunction places in the orbits of Jupiter and Saturn. The two planets are represented as leaving one of the conjunctions where Jupiter was being pulled back and Saturn being pulled forward by their mutual attraction.]

One of the facts that plays a large part in the result was known to the old astrologers, viz. that Jupiter and Saturn come into conjunction with a certain triangular symmetry; the whole scheme being called a trigon, and being mentioned several times by Kepler.

It happens that five of Jupiter's years very nearly equal two of Saturn's,[25] so that they get very nearly into conjunction three times in every five Jupiter years, but not exactly. The result of this close approach is that periodically one pulls the other on and is itself pulled back; but since the three points progress, it is not always the same planet which gets pulled back. The complete theory shows that in the year 1560 there was no marked perturbation: before that it was in one direction, while afterwards it was in the other direction, and the period of the whole cycle of disturbances is 929 of our years. The solution of this long outstanding puzzle by the theory of gravitation was hailed with the greatest enthusiasm by astronomers, and it established the fame of the two French mathematicians.

Next they attacked the complicated problem of the motions of Jupiter's satellites. They succeeded in obtaining a theory of their motions which represented fact very nearly indeed, and they detected the following curious relationship between the satellites:--The speed of the first satellite + twice the speed of the second is equal to the speed of the third.

They found this, not empirically, after the manner of Kepler, but as a deduction from the law of gravitation; for they go on to show that even if the satellites had not started with this relation they would sooner or later, by mutual perturbation, get themselves into it. One singular consequence of this, and of another quite similar connection between their positions, is that all three satellites can never be eclipsed at once.

The motion of the fourth satellite is less tractable; it does not so readily form an easy system with the others.

After these great successes the two astronomers naturally proceeded to study the mutual perturbations of all other bodies in the solar system.

And one very remarkable discovery they made concerning the earth and moon, an account of which will be interesting, though the details and processes of calculation are quite beyond us in a course like this.

Astronomical theory had become so nearly perfect by this time, and observations so accurate, that it was possible to calculate many astronomical events forwards or backwards, over even a thousand years or more, with admirable precision.

Now, Halley had studied some records of ancient eclipses, and had calculated back by means of the lunar theory to see whether the calculation of the time they ought to occur would agree with the record of the time they did occur. To his surprise he found a discrepancy, not a large one, but still one quite noticeable. To state it as we know it now:--An eclipse a century ago happened twelve seconds later than it ought to have happened by theory; two centuries back the error amounted to forty-eight seconds, in three centuries it would be 108 seconds, and so on; the lag depending on the square of the time. By research, and help from scholars, he succeeded in obtaining the records of some very ancient eclipses indeed. One in Egypt towards the end of the tenth century A.D.; another in 201 A.D.; another a little before Christ; and one, the oldest of all of which any authentic record has been preserved, observed by the Chaldaean astronomers in Babylon in the reign of Hezekiah.

Calculating back to this splendid old record of a solar eclipse, over the intervening 2,400 years, the calculated and the observed times were found to disagree by nearly two hours. Pondering over an explanation of the discrepancy, Halley guessed that it must be because the moon's motion was not uniform, it must be going quicker and quicker, gaining twelve seconds each century on its previous gain--a discovery announced by him as "the acceleration of the moon's mean motion." The month was constantly getting shorter.

What was the physical cause of this acceleration according to the theory of gravitation? Many attacked the question, but all failed. This was the problem Laplace set himself to work out. A singular and beautiful result rewarded his efforts.

You know that the earth describes an elliptic orbit round the sun: and that an ellipse is a circle with a certain amount of flattening or "excentricity."[26] Well, Laplace found that the excentricity of the earth's...o...b..t must be changing, getting slightly less; and that this change of excentricity would have an effect upon the length of the month. It would make the moon go quicker.

One can almost see how it comes about. A decrease in excentricity means an increase in mean distance of the earth from the sun. This means to the moon a less solar perturbation. Now one effect of the solar perturbation is to keep the moon's...o...b..t extra large: if the size of its...o...b..t diminishes, its velocity must increase, according to Kepler's third law.

Laplace calculated the amount of acceleration so resulting, and found it ten seconds a century; very nearly what observation required; for, though I have quoted observation as demanding twelve seconds per century, the facts were not then so distinctly and definitely ascertained.

This calculation for a long time seemed thoroughly satisfactory, but it is not the last word on the subject. Quite lately an error has been found in the working, which diminishes the theoretical gravitation-acceleration to six seconds a century instead of ten, thus making it insufficient to agree exactly with fact. The theory of gravitation leaves an outstanding error. (The point is now almost thoroughly understood, and we shall return to it in Lecture XVIII).

But another question arises out of this discussion. I have spoken of the excentricity of the earth's...o...b..t as decreasing. Was it always decreasing? and if so, how far back was it so excentric that at perihelion the earth pa.s.sed quite near the sun? If it ever did thus pa.s.s near the sun, the inference is manifest--the earth must at one time have been thrown off, or been separated off, from the sun.

If a projectile could be fired so fast that it described an orbit round the earth--and the speed of fire to attain this lies between five and seven miles a second (not less than the one, nor more than the other)--it would ever afterwards pa.s.s through its point of projection as one point of its elliptic orbit; and its periodic return through that point would be the sign of its origin. Similarly, if a satellite does _not_ come near its central orb, and can be shown never to have been near it, the natural inference is that it has _not_ been born from it, but has originated in some other way.

The question which presented itself in connexion with the variable ellipticity of the earth's...o...b..t was the following:--Had it always been decreasing, so that once it was excentric enough just to graze the sun at perihelion as a projected body would do?

Into the problem thus presented Lagrange threw himself, and he succeeded in showing that no such explanation of the origin of the earth is possible. The excentricity of the orbit, though now decreasing, was not always decreasing; ages ago it was increasing: it pa.s.ses through periodic changes. Eighteen thousand years ago its excentricity was a maximum; since then it has been diminishing, and will continue to diminish for 25,000 years more, when it will be an almost perfect circle; it will then begin to increase again, and so on. The obliquity of the ecliptic is also changing periodically, but not greatly: the change is less than three degrees.

This research has, or ought to have, the most transcendent interest for geologists and geographers. You know that geologists find traces of extraordinary variations of temperature on the surface of the earth.

England was at one time tropical, at another time glacial. Far away north, in Spitzbergen, evidence of the luxuriant vegetation of past ages has been found; and the explanation of these great climatic changes has long been a puzzle. Does not the secular variation in excentricity of the earth's...o...b..t, combined with the precession of the equinoxes, afford a key? And if a key at all, it will be an accurate key, and enable us to calculate back with some precision to the date of the glacial epoch; and again to the time when a tropical flora flourished in what is now northern Europe, _i.e._ to the date of the Carboniferous era.

This aspect of the subject has recently been taught with vigour and success by Dr. Croll in his book "Climate and Time."

A brief and partial explanation of the matter may be given, because it is a point of some interest and is also one of fair simplicity.

Every one knows that the climatic conditions of winter and summer are inverted in the two hemispheres, and that at present the sun is nearest to us in our (northern) winter. In other words, the earth's axis is inclined so as to tilt its north pole away from the sun at perihelion, or when the earth is at the part of its elliptic orbit nearest the sun's focus; and to tilt it towards the sun at aphelion. The result of this present state of things is to diminish the intensity of the average northern winter and of the average northern summer, and on the other hand to aggravate the extremes of temperature in the southern hemisphere; all other things being equal. Of course other things are not equal, and the distribution of land and sea is a still more powerful climatic agent than is the three million miles or so extra nearness of the sun. But it is supposed that the Antarctic ice-cap is larger than the northern, and increased summer radiation with increased winter cold would account for this.

But the present state of things did not always obtain. The conical movement of the earth's axis (now known by a curious perversion of phrase as "precession") will in the course of 13,000 years or so cause the tilt to be precisely opposite, and then we shall have the more extreme winters and summers instead of the southern hemisphere.

If the change were to occur now, it might not be overpowering, because now the excentricity is moderate. But if it happened some time back, when the excentricity was much greater, a decidedly different arrangement of climate may have resulted. There is no need to say _if_ it happened some time back: it did happen, and accordingly an agent for affecting the distribution of mean temperature on the earth is to hand; though whether it is sufficient to achieve all that has been observed by geologists is a matter of opinion.

Once more, the whole diversity of the seasons depends on the tilt of the earth's axis, the 23 by which it is inclined to a perpendicular to the orbital plane; and this obliquity or tilt is subject to slow fluctuations. Hence there will come eras when all causes combine to produce a maximum extremity of seasons in the northern hemisphere, and other eras when it is the southern hemisphere which is subject to extremes.

But a grander problem still awaited solution--nothing less than the fate of the whole solar system. Here are a number of bodies of various sizes circulating at various rates round one central body, all attracted by it, and all attracting each other, the whole abandoned to the free play of the force of gravitation: what will be the end of it all? Will they ultimately approach and fall into the sun, or will they recede further and further from him, into the cold of s.p.a.ce? There is a third possible alternative: may they not alternately approach and recede from him, so as on the whole to maintain a fair approximation to their present distances, without great and violent extremes of temperature either way?

If any one planet of the system were to fall into the sun, more especially if it were a big one like Jupiter or Saturn, the heat produced would be so terrific that life on this earth would be destroyed, even at its present distance; so that we are personally interested in the behaviour of the other planets as well as in the behaviour of our own.

The result of the portentously difficult and profoundly interesting investigation, here sketched in barest outline, is that the solar system is stable: that is to say, that if disturbed a little it will oscillate and return to its old state; whereas if it were unstable the slightest disturbance would tend to acc.u.mulate, and would sooner or later bring about a catastrophe. A hanging pendulum is stable, and oscillates about a mean position; its motion is periodic. A top-heavy load balanced on a point is unstable. All the changes of the solar system are periodic, _i.e._ they repeat themselves at regular intervals, and they never exceed a certain moderate amount.

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 ma.s.ses of the several planets by m_1, m_2, &c.; their mean distances from the sun (or radii vectores) by r_1, r_2, &c.; the excentricities of their orbits by e_1, e_2, &c.; and the obliquity of the planes of these orbits, reckoned from a single plane of reference or "invariable plane," by [theta]_1, [theta]_2, &c.; then all these quant.i.ties (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, m_1e_1^2 [square root]r_1 + m_2e_2^2 [square root]r_2 + &c., will remain always the same. This is summed up briefly in the following statement: