The Story of the Heavens by Sir Robert Stawell Ball (fantasy books to read .txt) 📕

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 assumed and the stability of that system is assured for all time. There is, however, one assumption which Lagrange makes, and on which his whole theory was founded: his assumption is that the planets are _rigid_ bodies.

No doubt our earth seems a rigid body. What can be more solid and unyielding than the mass of rocks and metals which form the earth, so far as it is accessible to us? In the wide realms of space the earth is but as a particle; it surely was a natural and a legitimate assumption 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 circumstance, 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 asserts 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 quantity. 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 assumption of the truth of Kepler's laws, and when we pronounce 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.

These daily pulses of the ocean have long ceased to be a mystery. It was in the earliest times perceived that there was a connection between the tides and the moon. Ancient writers, such as Pliny and Aristotle, have referred to the alliance between the times of high water and the age of the moon. I think we sometimes do not give the ancient astronomers as much credit as their shrewdness really entitles them to. We have all read--we have all been taught--that the moon and the tides are connected together; but how many of us are in a position to say that we have actually noticed that connection by direct personal observation? The first man who studied this matter with sufficient attention to convince himself and to convince others of its reality must have been a great philosopher. We know not his name, we know not his nation, we know not the age in which he lived; but our admiration of his discovery must be increased by the reflection that he had not the theory of gravitation to guide him. A philosopher of the present day who had never seen the sea could still predict the necessity of tides as a consequence of the law of universal gravitation; but the primitive astronomer, who knew not of the invisible bond by which all bodies in the universe are drawn together, made a splendid--indeed, a typical--inductive discovery, when he ascertained the relation between the moon and the tides.

We can surmise that this discovery, in all probability, first arose from the observations of experienced navigators. In all matters of entering port or of leaving port, the state of the tide is of the utmost concern to the sailor. Even in the open sea he has sometimes to shape his course in accordance with the currents produced by the tides; or, in guiding his course by taking soundings, he has always to bear in mind that the depth varies with the tide. All matters relating to the tide would thus come under his daily observation. His daily work, the success of his occupation, the security of his life, depend often on the tides; and hence he would be solicitous to learn from his observation all that would be useful to him in the future. To the coasting sailor the question of the day is the time of high water. That time varies from day to day; it is an hour or more later to-morrow than to-day, and there is no very simple rule which can be enunciated. The sailor would therefore welcome gladly any rule which would guide him in a matter of such importance. We can make a conjecture as to the manner in which such a rule was first discovered. Let us suppose that a sailor at Calais, for example, is making for harbour. He has a beautiful night--the moon is full; it guides him on his way; he gets safely into harbour; and the next morning he finds the tide high between 11 and 12.[45] He often repeats the same voyage, but he finds sometimes a low and inconvenient tide in the morning. At length, however, it occurs to him that _when he has a moonlight night_ he has a high tide at 11. This occurs once or twice: he thinks it but a chance coincidence. It occurs again and again. At length he finds it always occurs. He tells the rule to other sailors; they try it too. It is invariably found that when the moon is full, the high tide always recurs at the same hour at the