The Story of the Heavens Part 31

The copious outflow of heat from the sun corresponds with its enormous temperature. We can express the amount of heat in various ways, but it must be remembered that considerable uncertainty still attaches to such measurements. The old method of measuring heat by the quant.i.ty of ice melted may be used as an ill.u.s.tration. It is computed that a sh.e.l.l of ice 43-1/2 feet thick surrounding the whole sun would in one minute be melted by the sun's heat underneath. A somewhat more elegant ill.u.s.tration was also given by Sir John Herschel, who showed that if a cylindrical glacier 45 miles in diameter were to be continually flowing into the sun with the velocity of light, the end of that glacier would be melted as quickly as it advanced. From each square foot in the surface of the sun emerges a quant.i.ty of heat as great as could be produced by the daily combustion of sixteen tons of coal. This is, indeed, an amount of heat which, properly transformed into work, would keep an engine of many hundreds of horse-power running from one year's end to the other. The heat radiated from a few acres on the sun would be adequate to drive all the steam engines in the world. When we reflect on the vast intensity of the radiation from each square foot of the sun's surface, and when we combine with this the stupendous dimensions of the sun, imagination fails to realise how vast must be the actual expenditure of heat.

In presence of the prodigal expenditure of the sun's heat, we are tempted to ask a question which has the most vital interest for the earth and its inhabitants. We live from hour to hour by the sun's splendid generosity; and, therefore, it is important for us to know what security we possess for the continuance of his favours. When we witness the terrific disburs.e.m.e.nt of the sun's heat each hour, we are compelled to ask whether our great luminary may not be exhausting its resources; and if so, what are the prospects of the future? This question we can partly answer. The whole subject is indeed of surpa.s.sing interest, and redolent with the spirit of modern scientific thought.

Our first attempt to examine this question must lie in an appeal to the facts which are attainable. We want to know whether the sun is showing any symptoms of decay. Are the days as warm and as bright now as they were last year, ten years ago, one hundred years ago? We can find no evidence of any change since the beginning of authentic records. If the sun's heat had perceptibly changed within the last two thousand years, we should expect to find corresponding changes in the distribution of plants and of animals; but no such changes have been detected. There is no reason to think that the climate of ancient Greece or of ancient Rome was appreciably different from the climates of the Greece and the Rome that we know at this day. The vine and the olive grow now where they grew two thousand years ago.

We must not, however, lay too much stress on this argument; for the effects of slight changes in the sun's heat may have been neutralised by corresponding adaptations in the pliable organisms of cultivated plants.

All we can certainly conclude is that no marked change has taken place in the heat of the sun during historical time. But when we come to look back into much earlier ages, we find copious evidence that the earth has undergone great changes in climate. Geological records can on this question hardly be misinterpreted. Yet it is curious to note that these changes are hardly such as could arise from the gradual exhaustion of the sun's radiation. No doubt, in very early times we have evidence that the earth's climate must have been much warmer than at present. We had the great carboniferous period, when the temperature must almost have been tropical in Arctic lat.i.tudes. Yet it is hardly possible to cite this as evidence that the sun was then much more powerful; for we are immediately reminded of the glacial period, when our temperate zones were overlaid by sheets of solid ice, as Northern Greenland is at present. If we suppose the sun to have been hotter than it is at present to account for the vegetation which produced coal, then we ought to a.s.sume the sun to be colder than it is now to account for the glacial period. It is not reasonable to attribute such phenomena to fluctuations in the radiation from the sun. The glacial periods prove that we cannot appeal to geology in aid of the doctrine that a secular cooling of the sun is now in progress. The geological variations of climate may have been caused by changes in the earth itself, or by changes in its actual orbit; but however they have been caused, they hardly tell us much with regard to the past history of our sun.

The heat of the sun has lasted countless ages; yet we cannot credit the sun with the power of actually creating heat. We must apply to the tremendous ma.s.s of the sun the same laws which we have found by our experiments on the earth. We must ask, whence comes the heat sufficient to supply this lavish outgoing? Let us briefly recount the various suppositions that have been made.

Place two red-hot spheres of iron side by side, a large one and a small one. They have been taken from the same fire; they were both equally hot; they are both cooling, but the small sphere cools more rapidly. It speedily becomes dark, while the large sphere is still glowing, and would continue to do so for some minutes. The larger the sphere, the longer it will take to cool; and hence it has been supposed that a mighty sphere of the prodigious dimensions of our sun would, if once heated, cool gradually, but the duration of the cooling would be so long that for thousands and for millions of years it could continue to be a source of light and heat to the revolving system of planets. This suggestion will not bear the test of arithmetic. If the sun had no source of heat beyond that indicated by its high temperature, we can show that radiation would cool the sun a few degrees every year. Two thousand years would then witness a very great decrease in the sun's heat. We are certain that no such decrease can have taken place. The source of the sun's radiation cannot be found in the mere cooling of an incandescent ma.s.s.

Can the fires in the sun be maintained by combustion, a.n.a.logous to that which goes on in our furnaces? Here we would seem to have a source of gigantic heat; but arithmetic also disposes of this supposition. We know that if the sun were made of even solid coal itself, and if that coal were burning in pure oxygen, the heat that could be produced would only suffice for 6,000 years. If the sun which shone upon the builders of the great Pyramid had been solid coal from surface to centre, it must by this time have been in great part burned away in the attempt to maintain its present rate of expenditure. We are thus forced to look to other sources for the supply of the sun's heat, since neither the heat of incandescence nor the heat of combustion will suffice.

There is probably--indeed, we may say certainly--one external source from which the heat of the sun is recruited. It will be necessary for us to consider this source with some care, though I think we shall find it to be merely an auxiliary of comparatively trifling moment. According to this view, the solar heat receives occasional accessions from the fall upon the sun's surface of ma.s.ses of meteoric matter. There can be hardly a doubt that such ma.s.ses do fall upon the sun; there is certainly no doubt that if they do, the sun must gain some heat thereby. We have experience on the earth of a very interesting kind, which ill.u.s.trates the development of heat by meteoric matter. There lies a world of philosophy in a shooting star. Some of these myriad objects rush into our atmosphere and are lost; others, no doubt, rush into the sun with the same result. We also admit that the descent of a shooting star into the atmosphere of the sun must be attended with a flash of light and of heat. The heat acquired by the earth from the flashing of the shooting stars through our air is quite insensible. It has been supposed, however, that the heat accruing to the sun from the same cause may be quite sensible--nay, it has been even supposed that the sun may be re-invigorated from this source.

Here, again, we must apply the cold principles of weights and measures to estimate the plausibility of this suggestion. We first calculate the actual weight of meteoric indraught to the sun which would be adequate to sustain the fires of the sun at their present vigour. The ma.s.s of matter that would be required is so enormous that we cannot usefully express it by imperial weights; we must deal with ma.s.ses of imposing magnitude. It fortunately happens that the weight of our moon is a convenient unit. Conceive that our moon--a huge globe, 2,000 miles in diameter--were crushed into a myriad of fragments, and that these fragments were allowed to rain in on the sun; there can be no doubt that this tremendous meteoric shower would contribute to the sun rather more heat than would be required to supply his radiation for a whole year. If we take our earth itself, conceive it comminuted into dust, and allow that dust to fall on the sun as a mighty shower, each fragment would instantly give out a quant.i.ty of heat, and the whole would add to the sun a supply of heat adequate to sustain the present rate of radiation for nearly one hundred years. The mighty ma.s.s of Jupiter treated in the same way would generate a meteoric display greater in the ratio in which the ma.s.s of Jupiter exceeds the ma.s.s of earth. Were Jupiter to fall into the sun, enough heat would be thereby produced to scorch the whole solar system; while all the planets together would be capable of producing heat which, if properly economised, would supply the radiation of the sun for 45,000 years.

It must be remembered that though the moon could supply one year's heat, and Jupiter 30,000 years' heat, yet the practical question is not whether the solar system could supply the sun's heat, but whether it does. Is it likely that meteors equal in ma.s.s to the moon fall into the sun every year? This is the real question, and I think we are bound to reply to it in the negative. It can be shown that the quant.i.ty of meteors which could be caught by the sun in any one year can be only an excessively minute fraction of the total amount. If, therefore, a moon-weight of meteors were caught every year, there must be an incredible ma.s.s of meteoric matter roaming at large through the system.

There must be so many meteors that the earth would be incessantly pelted with them, and heated to such a degree as to be rendered uninhabitable.

There are also other reasons which preclude the supposition that a stupendous quant.i.ty of meteoric matter exists in the vicinity of the sun. Such matter would produce an appreciable effect on the movement of the planet Mercury. There are, no doubt, some irregularities in the movements of Mercury not yet fully explained, but these irregularities are very much less than would be the case if meteoric matter existed in quant.i.ty adequate to the sustentation of the sun. Astronomers, then, believe that though meteors may provide a rate in aid of the sun's current expenditure, yet that the greater portion of that expenditure must be defrayed from other resources.

It is one of the achievements of modern science to have effected the solution of the problem--to have shown how it is that, notwithstanding the stupendous radiation, the sun still maintains its temperature. The question is not free from difficulty in its exposition, but the matter is one of such very great importance that we are compelled to make the attempt.

Let us imagine a vast globe of heated gas in s.p.a.ce. This is not an entirely gratuitous supposition, inasmuch as there are globes apparently of this character; they have been already alluded to as planetary nebulae. This globe will radiate heat, and we shall suppose that it emits more heat than it receives from the radiation of other bodies. The globe will accordingly lose heat, or what is equivalent thereto, but it will be incorrect to a.s.sume that the globe will necessarily fall in temperature. That the contrary is, indeed, the case is a result almost paradoxical at the first glance; but yet it can be shown to be a necessary consequence of the laws of heat and of gases.

Let us fix our attention on a portion of the gas lying on the surface of the globe. This is, of course, attracted by all the rest of the globe, and thus tends in towards the centre of the globe. If equilibrium subsists, this tendency must be neutralised by the pressure of the gas beneath; so that the greater the gravitation, the greater is the pressure. When the globe of gas loses heat by radiation, let us suppose that it grows colder--that its temperature accordingly falls; then, since the pressure of a gas decreases when the temperature falls, the pressure beneath the superficial layer of the gas will decrease, while the gravitation is unaltered. The consequence will inevitably be that the gravitation will now conquer the pressure, and the globe of gas will accordingly contract. There is, however, another way in which we can look at the matter. We know that heat is equivalent to energy, so that when the globe radiates forth heat, it must expend energy. A part of the energy of the globe will be due to its temperature; but another, and in some respects a more important, part is that due to the separation of its particles. If we allow the particles to come closer together we shall diminish the energy due to separation, and the energy thus set free can take the form of heat. But this drawing in of the particles necessarily involves a shrinking of the globe.

And now for the remarkable consequence, which seems to have a very important application in astronomy. As the globe contracts, a part of its energy of separation is changed into heat; that heat is partly radiated away, but not so rapidly as it is produced by the contraction.

The consequence is, that although the globe is really losing heat and really contracting, yet that its temperature is actually rising.[43] A simple case will suffice to demonstrate this result, paradoxical as it may at first seem. Let us suppose that by contraction of the sphere it had diminished to one-half its diameter; and let us fix our attention on a cubic inch of the gaseous matter in any point of the ma.s.s. After the contraction has taken place each edge of the cube would be reduced to half an inch, and the volume would therefore be reduced to one-eighth part of its original amount. The law of gases tells us that if the temperature be unaltered the pressure varies inversely as the volume, and consequently the internal pressure in the cube would in that case be increased eightfold. As, however, in the case before us, the distance between every two particles is reduced to one-half, it will follow that the gravitation between every two particles is increased fourfold, and as the area is also reduced to one-fourth, it will follow that the pressure inside the reduced cube is increased sixteenfold; but we have already seen that with a constant temperature it only increases eightfold, and hence the temperature cannot be constant, but must rise with the contraction.

We thus have the somewhat astonishing result that a gaseous globe in s.p.a.ce radiating heat, and thereby growing smaller, is all the time actually increasing in temperature. But, it may be said, surely this cannot go on for ever. Are we to suppose that the gaseous ma.s.s will go on contracting and contracting with a temperature ever fiercer and fiercer, and actually radiating out more and more heat the more it loses? Where lies the limit to such a prospect? As the body contracts, its density must increase, until it either becomes a liquid, or a solid, or, at any rate, until it ceases to obey the laws of a purely gaseous body which we have supposed. Once these laws cease to be observed the argument disappears; the loss of heat may then really be attended with a loss of temperature, until in the course of time the body has sunk to the temperature of s.p.a.ce itself.

It is not a.s.sumed that this reasoning can be applied in all its completeness to the present state of the sun. The sun's density is now so great that the laws of gases cannot be there strictly followed. There is, however, good reason to believe that the sun was once more gaseous than at present; possibly at one time he may have been quite gaseous enough to admit of this reasoning in all its fulness. At present the sun appears to be in some intermediate stage of its progress from the gaseous condition to the solid condition. We cannot, therefore, say that the temperature of the sun is now increasing in correspondence with the process of contraction. This may be true or it may not be true; we have no means of deciding the point. We may, however, feel certain that the sun is still sufficiently gaseous to experience in some degree the rise of temperature a.s.sociated with the contraction. That rise in temperature may be partly or wholly obscured by the fall in temperature which would be the more obvious consequence of the radiation of heat from the partially solid body. It will, however, be manifest that the cooling of the sun may be enormously protracted if the fall of temperature from the one cause be nearly compensated by the rise of temperature from the other. It can hardly be doubted that in this we find the real explanation of the fact that we have no historical evidence of any appreciable alteration in the radiation of heat from the sun.

This question is one of such interest that it may be worth while to look at it from a slightly different point of view. The sun contains a certain store of energy, part of which is continually disappearing in the form of radiant heat. The energy remaining in the sun is partly transformed in character; some of it is transformed into heat, which goes wholly or partly to supply the loss by radiation. The total energy of the sun must, however, be decreasing; and hence it would seem the sun must at some time or other have its energy exhausted, and cease to be a source of light and of heat. It is true that the rate at which the sun contracts is very slow. We are, indeed, not able to measure with certainty the decrease in the sun's bulk. It is a quant.i.ty so minute, that the contraction since the birth of accurate astronomy is not large enough to be perceptible in our telescopes. It is, however, possible to compute what the contraction of the sun's bulk must be, on the supposition that the energy lost by that contraction just suffices to supply the daily radiation of heat. The change is very small when we consider the present size of the sun. At the present time the sun's diameter is about 860,000 miles. If each year this diameter decreases by about 300 feet, sufficient energy will be yielded to account for the entire radiation. This gradual decrease is always in progress.

These considerations are of considerable interest when we apply them retrospectively. If it be true that the sun is at this moment shrinking, then in past times his globe must have been greater than it is at present. a.s.suming the figures already given, it follows that one hundred years ago the diameter of the sun must have been nearly six miles greater than it is now; one thousand years ago the diameter was fifty-seven miles greater; ten thousand years ago the diameter of the sun was five hundred and seventy miles greater than it is to-day. When man first trod this earth it would seem that the sun must have been many hundreds, perhaps many thousands, of miles greater than it is at this time.

We must not, however, over-estimate the significance of this statement.

The diameter of the sun is so great, that a diminution of 10,000 miles would be but little more than the hundredth part of its diameter. If it were suddenly to shrink to the extent of 10,000 miles, the change would not be appreciable to ordinary observation, though a much smaller change would not elude delicate astronomical measurement. It does not necessarily follow that the climates on our earth in these early times must have been very different from those which we find at this day, for the question of climate depends upon other matters besides sunbeams.

Yet we need not abruptly stop our retrospect at any epoch, however remote. We may go back earlier and earlier, through the long ages which geologists claim for the deposition of the stratified rocks; and back again still further, to those very earliest epochs when life began to dawn on the earth. Still we can find no reason to suppose that the law of the sun's decreasing heat is not maintained; and thus we would seem bound by our present knowledge to suppose that the sun grows larger and larger the further our retrospect extends. We cannot a.s.sume that the rate of that growth is always the same. No such a.s.sumption is required; it is sufficient for our purpose that we find the sun growing larger and larger the further we peer back into the remote abyss of time past.

If the present order of things in our universe has lasted long enough, then it would seem that there was a time when the sun must have been twice as large as it is at present; it must once have been ten times as large. How long ago that was no one can venture to say. But we cannot stop at the stage when the sun was even ten times as large as it is at present; the arguments will still apply in earlier ages. We see the sun swelling and swelling, with a corresponding decrease in its density, until at length we find, instead of our sun as we know it, a mighty nebula filling a gigantic region of s.p.a.ce.

Such is, in fact, the doctrine of the origin of our system which has been advanced in that celebrated speculation known as the nebular theory of Laplace. Nor can it be ever more than a speculation; it cannot be established by observation, nor can it be proved by calculation. It is merely a conjecture, more or less plausible, but perhaps in some degree necessarily true, if our present laws of heat, as we understand them, admit of the extreme application here required, and if also the present order of things has reigned for sufficient time without the intervention of any influence at present unknown to us. This nebular theory is not confined to the history of our sun. Precisely similar reasoning may be extended to the individual planets: the farther we look back, the hotter and the hotter does the whole system become. It has been thought that if we could look far enough back, we should see the earth too hot for life; back further still, we should find the earth and all the planets red-hot; and back further still, to an exceedingly remote epoch, when the planets would be heated just as much as our sun is now. In a still earlier stage the whole solar system is thought to have been one vast ma.s.s of glowing gas, from which the present forms of the sun, with the planets and their satellites, have been gradually evolved. We cannot be sure that the course of events has been what is here indicated; but there are sufficient grounds for thinking that this doctrine substantially represents what has actually occurred.

Many of the features in the solar system harmonise with the supposition that the origin of the system has been that suggested by the nebular theory. We have already had occasion in an earlier chapter to allude to the fact that all the planets perform their revolutions around the sun in the same direction. It is also to be observed that the rotation of the planets on their axes, as well as the movements of the satellites around their primaries, all follow the same law, with two slight exceptions in the case of the Uranian and Neptunian systems. A coincidence so remarkable naturally suggests the necessity for some physical explanation. Such an explanation is offered by the nebular theory. Suppose that countless ages ago a mighty nebula was slowly rotating and slowly contracting. In the process of contraction, portions of the condensed matter of the nebula would be left behind. These portions would still revolve around the central ma.s.s, and each portion would rotate on its axis in the same direction. As the process of contraction proceeded, it would follow from dynamical principles that the velocity of rotation would increase; and thus at length these portions would consolidate into planets, while the central ma.s.s would gradually contract to form the sun. By a similar process on a smaller scale the systems of satellites were evolved from the contracting primary. These satellites would also revolve in the same direction, and thus the characteristic features of the solar system could be accounted for.

The nebular origin of the solar system receives considerable countenance from the study of the sidereal heavens. We have already dwelt upon the resemblance between the sun and the stars. If, then, our sun has pa.s.sed through such changes as the nebular theory requires, may we not antic.i.p.ate that similar phenomena should be met with in other stars? If this be so, it is reasonable to suppose that the evolution of some of the stars may not have progressed so far as has that of the sun, and thus we may be able actually to witness stars in the earlier phases of their development. Let us see how far the telescope responds to these antic.i.p.ations.

The field of view of a large telescope usually discloses a number of stars scattered over a black background of sky; but the blackness of the background is not uniform: the practised eye of the skilled observer will detect in some parts of the heavens a faint luminosity. This will sometimes be visible over the whole extent of the field, or it may even occupy several fields. Years may pa.s.s on, and still there is no perceptible change. There can be no illusion, and the conclusion is irresistible that the object is a stupendous ma.s.s of faintly luminous glowing gas or vapour. This is the simplest type of nebula; it is characterised by extreme faintness, and seems composed of matter of the utmost tenuity. On the other hand we are occasionally presented with the beautiful and striking phenomenon of a definite and brilliant star surrounded by a luminous atmosphere. Between these two extreme types of a faint diffused ma.s.s on the one hand, and a bright star with a nebula surrounding it on the other, a graduated series of various other nebulae can be arranged. We thus have a series of links pa.s.sing by imperceptible gradations from the most faintly diffused nebulae on the one side, into stars on the other.

The nebulae seemed to Herschel to be vast ma.s.ses of phosph.o.r.escent vapour. This vapour gradually cools down, and ultimately condenses into a star, or a cl.u.s.ter of stars. When the varied forms of nebulae were cla.s.sified, it almost seemed as if the different links in the process could be actually witnessed. In the vast faint nebulae the process of condensation had just begun; in the smaller and brighter nebulae the condensation had advanced farther; while in others, the star, or stars, arising from the condensation had already become visible.

But, it may be asked, how did Herschel know this? what is his evidence?

Let us answer this question by an ill.u.s.tration. Go into a forest, and look at a n.o.ble old oak which has weathered the storm for centuries; have we any doubt that the oak-tree was once a young small plant, and that it grew stage by stage until it reached maturity? Yet no one has ever followed an oak-tree through its various stages; the brief span of human life has not been long enough to do so. The reason why we believe the oak-tree to have pa.s.sed through all these stages is, because we are familiar with oak-trees of every gradation in size, from the seedling up to the n.o.ble veteran. Having seen this gradation in a vast mult.i.tude of trees, we are convinced that each individual pa.s.ses through all these stages.

It was by a similar train of reasoning that Herschel was led to adopt the view of the origin of the stars which we have endeavoured to describe. The astronomer's life is not long enough, the life of the human race might not be long enough, to watch the process by which a nebula condenses down so as to form a solid body. But by looking at one nebula after another, the astronomer thinks he is able to detect the various stages which connect the nebula in its original form with the final form. He is thus led to believe that each of the nebulae pa.s.ses, in the course of ages, through these stages. And thus Herschel adopted the opinion that stars--some, many, or all--have each originated from what was once a glowing nebula.

Such a speculation may captivate the imagination, but it must be carefully distinguished from the truths of astronomy, properly so called. Remote posterity may perhaps obtain evidence on the subject which to us is inaccessible: our knowledge of nebulae is too recent.

There has not yet been time enough to detect any appreciable changes: for the study of nebulae can only be said to date from Messier's Catalogue in 1771.

Since Herschel's time, no doubt, many careful drawings and observations of the nebulae have been obtained; but still the interval has been much too short, and the earlier observations are too imperfect, to enable any changes in the nebulae to be investigated with sufficient accuracy. If the human race lasts for very many centuries, and if our present observations are preserved during that time for comparison, then Herschel's theory may perhaps be satisfactorily tested.

A hundred years have pa.s.sed since Laplace, with some diffidence, set forth his hypothesis as to the mode of formation of the solar system. On the whole it must be said that this "nebular hypothesis" has stood the test of advancing science well, though some slight modifications have become necessary in the light of more recent discoveries. Laplace (and Herschel also) seems to have considered a primitive nebula to consist of a "fiery mist" or glowing gas at a very high temperature. But this is by no means necessary, as we have seen that the gradual contraction of the vast ma.s.s supplies energy which may be converted into heat, and the spectroscopic evidence seems also to point to the existence of a moderate temperature in the gaseous nebulae, which must be considered to be representatives of the hypothetical primitive chaos out of which our sun and planets have been evolved. Another point which has been reconsidered is the formation of the various planets. It was formerly thought that the rotation of the original ma.s.s had by degrees caused a number of rings of different dimensions to be separated from the central part, the material of which rings in time collected into single planets.

The ring of Saturn was held to be a proof of this process, since we here have a ring, the condensation of which into one or more satellites has somehow been arrested. But while it is not impossible that matter in the shape of rings may have been left behind during the contraction of the nebulous ma.s.s (indeed, the minor planets between Mars and Jupiter have perhaps originated in this way), it seems likely that the larger planets were formed from the agglomeration of matter at a point on the equator of the rotating nebula.

The actual steps of the process by which the primeval nebula became transformed into the solar system seem to lie beyond reach of discovery.

CHAPTER XXVII.

THE TIDES.[44]

Mathematical Astronomy--Lagrange's Theories: how far they are really True--The Solar System not Made of Rigid Bodies--Kepler's Laws True to Observation, but not Absolutely True when the Bodies are not Rigid--The Errors of Observation--The Tides--How the Tides were Observed--Discovery of the Connection between the Tides and the Moon--Solar and Lunar Tides--Work done by the Tides--Whence do the Tides obtain the Power to do the Work?--Tides are Increasing the Length of the Day--Limit to the Shortness of the Day--Early History of the Earth-Moon System--Unstable Equilibrium--Ratio of the Month to the Day--The Future Course of the System--Equality of the Month and the Day--The Future Critical Epoch--The Constant Face of the Moon accounted for--The other Side of the Moon--The Satellites of Mars--Their Remarkable Motions--Have the Tides Possessed Influence in Moulding the Solar System generally?--Moment of Momentum--Tides have had little or no Appreciable Effect on the Orbit of Jupiter--Conclusion.

That the great discoveries of Lagrange on the stability of the planetary system are correct is in one sense strictly true. No one has ever ventured to impugn the mathematics of Lagrange. Given the planetary system in the form which Lagrange a.s.sumed and the stability of that system is a.s.sured for all time. There is, however, one a.s.sumption which Lagrange makes, and on which his whole theory was founded: his a.s.sumption is that the planets are _rigid_ bodies.

No doubt our earth seems a rigid body. What can be more solid and unyielding than the ma.s.s of rocks and metals which form the earth, so far as it is accessible to us? In the wide realms of s.p.a.ce the earth is but as a particle; it surely was a natural and a legitimate a.s.sumption to suppose that that particle was a rigid body. If the earth were absolutely rigid--if every particle of the earth were absolutely at a fixed distance from every other particle--if under no stress of forces, and in no conceivable circ.u.mstance, the earth experienced even the minutest change of form--if the same could be said of the sun and of all the other planets--then Lagrange's prediction of the eternal duration of our system must be fulfilled.

But what are the facts of the case? Is the earth really rigid? We know from experiment that a rigid body in the mathematical sense of the word does not exist. Rocks are not rigid; steel is not rigid; even a diamond is not perfectly rigid. The whole earth is far from being rigid even on the surface, while part of the interior is still, perhaps, more or less fluid. The earth cannot be called a perfectly rigid body; still less can the larger bodies of our system be called rigid. Jupiter and Saturn are perhaps hardly even what could be called solid bodies. The solar system of Lagrange consisted of a rigid sun and a number of minute rigid planets; the actual solar system consists of a sun which is in no sense rigid, and planets which are only partially so.

The question then arises as to whether the discoveries of the great mathematicians of the last century will apply, not only to the ideal solar system which they conceived, but to the actual solar system in which our lot has been cast. There can be no doubt that these discoveries are approximately true: they are, indeed, so near the absolute truth, that observation has not yet satisfactorily shown any departure from them.

But in the present state of science we can no longer overlook the important questions which arise when we deal with bodies not rigid in the mathematical sense of the word. Let us, for instance, take the simplest of the laws to which we have referred, the great law of Kepler, which a.s.serts that a planet will revolve for ever in an elliptic path of which the sun is one focus. This is seen to be verified by actual observation; indeed, it was established by observation before any theoretical explanation of that movement was propounded. If, however, we state the matter with a little more precision, we shall find that what Newton really demonstrated was, that if two _rigid_ particles attract each other by a law of force which varies with the inverse square of the distance between the particles, then each of the particles will describe an ellipse with the common centre of gravity in the focus. The earth is, to some extent, rigid, and hence it was natural to suppose that the relative behaviour of the earth and the sun would, to a corresponding extent, observe the simple elliptic law of Kepler; as a matter of fact, they do observe it with such fidelity that, if we make allowance for other causes of disturbance, we cannot, even by most careful observation, detect the slightest variation in the motion of the earth arising from its want of rigidity.

There is, however, a subtlety in the investigations of mathematics which, in this instance at all events, transcends the most delicate observations which our instruments enable us to make. The principles of mathematics tell us that though Kepler's laws may be true for bodies which are absolutely and mathematically rigid, yet that if the sun or the planets be either wholly, or even in their minutest part, devoid of perfect rigidity, then Kepler's laws can be no longer true. Do we not seem here to be in the presence of a contradiction? Observation tells us that Kepler's laws are true in the planetary system; theory tells us that these laws cannot be true in the planetary system, because the bodies in that system are not perfectly rigid. How is this discrepancy to be removed? Or is there really a discrepancy at all? There is not.

When we say that Kepler's laws have been proved to be true by observation, we must reflect on the nature of the proofs which are attainable. We observe the places of the planets with the instruments in our observatories; these places are measured by the help of our clocks and of the graduated circles on the instruments. These observations are no doubt wonderfully accurate; but they do not, they cannot, possess absolute accuracy in the mathematical sense of the word. We can, for instance, determine the place of a planet with such precision that it is certainly not one second of arc wrong; and one second is an extremely small quant.i.ty. A foot-rule placed at a distance of about forty miles subtends an angle of a second, and it is surely a delicate achievement to measure the place of a planet, and feel confident that no error greater than this can have intruded into our result.

When we compare the results of observation with the calculations conducted on the a.s.sumption of the truth of Kepler's laws, and when we p.r.o.nounce on the agreement of the observations with the calculations, there is always a reference, more or less explicit, to the inevitable errors of the observations. If the calculations and observations agree so closely that the differences between the two are minute enough to have arisen in the errors inseparable from the observations, then we are satisfied with the accordance; for, in fact, no closer agreement is attainable, or even conceivable. The influence which the want of rigidity exercises on the fulfilment of the laws of Kepler can be estimated by calculation; it is found, as might be expected, to be extremely small--so small, in fact, as to be contained within that slender margin of error by which observations are liable to be affected.

We are thus not able to discriminate by actual measurement the effects due to the absence of rigidity; they are inextricably hid among the small errors of observation.

The argument on which we are to base our researches is really founded on a very familiar phenomenon. There is no one who has ever visited the sea-side who is not familiar with that rise and fall of the sea which we call the tide. Twice every twenty-four hours the sea advances on the beach to produce high tide; twice every day the sea again retreats to produce low tide. These tides are not merely confined to the coasts; they penetrate for miles up the courses of rivers; they periodically inundate great estuaries. In a maritime country the tides are of the most profound practical importance; they also possess a significance of a far less obvious character, which it is our object now to investigate.