Part 10 (1/2)

These two laws completely prescribe the motion of a planet round the sun. The first defines the path which the planet pursues; the second describes how the velocity of the body varies at different points along its path. But Kepler added to these a third law, which enables us to compare the movements of two different planets revolving round the same sun. Before stating this law, it is necessary to explain exactly what is meant by the _mean_ distance of a planet. In its elliptic path the distance from the sun to the planet is constantly changing; but it is nevertheless easy to attach a distinct meaning to that distance which is an average of all the distances. This average is called the mean distance. The simplest way of finding the mean distance is to add the greatest of these quant.i.ties to the least, and take half the sum. We have already defined the periodic time of the planet; it is the number of days which the planet requires for the completion of a journey round its path. Kepler's third law establishes a relation between the mean distances and the periodic times of the various planets. That relation is stated in the following words:--

”_The squares of the periodic times are proportional to the cubes of the mean distances._”

Kepler knew that the different planets had different periodic times; he also saw that the greater the mean distance of the planet the greater was its periodic time, and he was determined to find out the connection between the two. It was easily found that it would not be true to say that the periodic time is merely proportional to the mean distance. Were this the case, then if one planet had a distance twice as great as another, the periodic time of the former would have been double that of the latter; observation showed, however, that the periodic time of the more distant planet exceeded twice, and was indeed nearly three times, that of the other. By repeated trials, which would have exhausted the patience of one less confident in his own sagacity, and less a.s.sured of the accuracy of the observations which he sought to interpret, Kepler at length discovered the true law, and expressed it in the form we have stated.

To ill.u.s.trate the nature of this law, we shall take for comparison the earth and the planet Venus. If we denote the mean distance of the earth from the sun by unity then the mean distance of Venus from the sun is 07233. Omitting decimals beyond the first place, we can represent the periodic time of the earth as 3653 days, and the periodic time of Venus as 2247 days. Now the law which Kepler a.s.serts is that the square of 3653 is to the square of 2247 in the same proportion as unity is to the cube of 07233. The reader can easily verify the truth of this ident.i.ty by actual multiplication. It is, however, to be remembered that, as only four figures have been retained in the expressions of the periodic times, so only four figures are to be considered significant in making the calculations.

The most striking manner of making the verification will be to regard the time of the revolution of Venus as an unknown quant.i.ty, and deduce it from the known revolution of the earth and the mean distance of Venus. In this way, by a.s.suming Kepler's law, we deduce the cube of the periodic time by a simple proportion, and the resulting value of 2247 days can then be obtained. As a matter of fact, in the calculations of astronomy, the distances of the planets are usually ascertained from Kepler's law. The periodic time of the planet is an element which can be measured with great accuracy; and once it is known, then the square of the mean distance, and consequently the mean distance itself, is determined.

Such are the three celebrated laws of Planetary Motion, which have always been a.s.sociated with the name of their discoverer. The profound skill by which these laws were elicited from the ma.s.s of observations, the intrinsic beauty of the laws themselves, their widespread generality, and the bond of union which they have established between the various members of the solar system, have given them quite an exceptional position in astronomy.

As established by Kepler, these planetary laws were merely the results of observation. It was found, as a matter of fact, that the planets did move in ellipses, but Kepler a.s.signed no reason why they should adopt this curve rather than any other. Still less was he able to offer a reason why these bodies should sweep over equal areas in equal times, or why that third law was invariably obeyed. The laws as they came from Kepler's hands stood out as three independent truths; thoroughly established, no doubt, but unsupported by any arguments as to why these movements rather than any others should be appropriate for the revolutions of the planets.

It was the crowning triumph of the great law of universal gravitation to remove this empirical character from Kepler's laws. Newton's grand discovery bound together the three isolated laws of Kepler into one beautiful doctrine. He showed not only that those laws are true, but he showed why they must be true, and why no other laws could have been true. He proved to demonstration in his immortal work, the ”Principia,”

that the explanation of the famous planetary laws was to be sought in the attraction of gravitation. Newton set forth that a power of attraction resided in the sun, and as a necessary consequence of that attraction every planet must revolve in an elliptic orbit round the sun, having the sun as one focus; the radius of the planet's...o...b..t must sweep over equal areas in equal times; and in comparing the movements of two planets, it was necessary to have the squares of the periodic times proportional to the cubes of the mean distances.

As this is not a mathematical treatise, it will be impossible for us to discuss the proofs which Newton has given, and which have commanded the immediate and universal acquiescence of all who have taken the trouble to understand them. We must here confine ourselves only to a very brief and general survey of the subject, which will indicate the character of the reasoning employed, without introducing details of a technical character.

Let us, in the first place, endeavour to think of a globe freely poised in s.p.a.ce, and completely isolated from the influence of every other body in the universe. Let us imagine that this globe is set in motion by some impulse which starts it forward on a rapid voyage through the realms of s.p.a.ce. When the impulse ceases the globe is in motion, and continues to move onwards. But what will be the path which it pursues? We are so accustomed to see a stone thrown into the air moving in a curved path, that we might naturally think a body projected into free s.p.a.ce will also move in a curve. A little consideration will, however, show that the cases are very different. In the realms of free s.p.a.ce we find no conception of upwards or downwards; all paths are alike; there is no reason why the body should swerve to the right or to the left; and hence we are led to surmise that in these circ.u.mstances a body, once started and freed from all interference, would move in a straight line. It is true that this statement is one which can never be submitted to the test of direct experiment. Circ.u.mstanced as we are on the surface of the earth, we have no means of isolating a body from external forces. The resistance of the air, as well as friction in various other forms, no less than the gravitation towards the earth itself, interfere with our experiments. A stone thrown along a sheet of ice will be exposed to but little resistance, and in this case we see that the stone will take a straight course along the frozen surface. A stone similarly cast into empty s.p.a.ce would pursue a course absolutely rectilinear. This we demonstrate, not by any attempts at an experiment which would necessarily be futile, but by indirect reasoning. The truth of this principle can never for a moment be doubted by one who has duly weighed the arguments which have been produced in its behalf.

Admitting, then, the rectilinear path of the body, the next question which arises relates to the velocity with which that movement is performed. The stone gliding over the smooth ice on a frozen lake will, as everyone has observed, travel a long distance before it comes to rest. There is but little friction between the ice and the stone, but still even on ice friction is not altogether absent; and as that friction always tends to stop the motion, the stone will at length be brought to rest. In a voyage through the solitudes of s.p.a.ce, a body experiences no friction; there is no tendency for the velocity to be reduced, and consequently we believe that the body could journey on for ever with unabated speed. No doubt such a statement seems at variance with our ordinary experience. A sailing s.h.i.+p makes no progress on the sea when the wind dies away. A train will gradually lose its velocity when the steam has been turned off. A humming-top will slowly expend its rotation and come to rest. From such instances it might be plausibly argued that when the force has ceased to act, the motion that the force generated gradually wanes, and ultimately vanishes. But in all these cases it will be found, on reflection, that the decline of the motion is to be attributed to the action of resisting forces. The sailing s.h.i.+p is r.e.t.a.r.ded by the rubbing of the water on its sides; the train is checked by the friction of the wheels, and by the fact that it has to force its way through the air; and the atmospheric resistance is mainly the cause of the stopping of the humming-top, for if the air be withdrawn, by making the experiment in a vacuum, the top will continue to spin for a greatly lengthened period. We are thus led to admit that a body, once projected freely in s.p.a.ce and acted upon by no external resistance, will continue to move on for ever in a straight line, and will preserve unabated to the end of time the velocity with which it originally started. This principle is known as the _first law of motion_.

Let us apply this principle to the important question of the movement of the planets. Take, for instance, the case of our earth, and let us discuss the consequences of the first law of motion. We know that the earth is moving each moment with a velocity of about eighteen miles a second, and the first law of motion a.s.sures us that if this globe were submitted to no external force, it would for ever pursue a straight track through the universe, nor would it depart from the precise velocity which it possesses at the present moment. But is the earth moving in this manner? Obviously not. We have already found that our globe is moving round the sun, and the comprehensive laws of Kepler have given to that motion the most perfect distinctness and precision. The consequence is irresistible. The earth cannot be free from external force. Some potent influence on our globe must be in ceaseless action.

That influence, whatever it may be, constantly deflects the earth from the rectilinear path which it tends to pursue, and constrains it to trace out an ellipse instead of a straight line.

The great problem to be solved is now easily stated. There must be some external agent constantly influencing the earth. What is that agent, whence does it proceed, and to what laws is it submitted? Nor is the question confined to the earth. Mercury and Venus, Mars, Jupiter, and Saturn, unmistakably show that, as they are not moving in rectilinear paths, they must be exposed to some force. What is this force which guides the planets in their paths? Before the time of Newton this question might have been asked in vain. It was the splendid genius of Newton which supplied the answer, and thus revolutionised the whole of modern science.

The data from which the question is to be answered must be obtained from observation. We have here no problem which can be solved by mere mathematical meditation. Mathematics is no doubt a useful, indeed, an indispensable, instrument in the enquiry; but we must not attribute to mathematics a potency which it does not possess. In a case of this kind, all that mathematics can do is to interpret the results obtained by observation. The data from which Newton proceeded were the observed phenomena in the movement of the earth and the other planets. Those facts had found a succinct expression by the aid of Kepler's laws. It was, accordingly, the laws of Kepler which Newton took as the basis of his labours, and it was for the interpretation of Kepler's laws that Newton invoked the aid of that celebrated mathematical reasoning which he created.

The question is then to be approached in this way: A planet being subject to _some_ external influence, we have to determine what that influence is, from our knowledge that the path of each planet is an ellipse, and that each planet sweeps round the sun over equal areas in equal times. The influence on each planet is what a mathematician would call a force, and a force must have a line of direction. The most simple conception of a force is that of a pull communicated along a rope, and the direction of the rope is in this case the direction of the force.

Let us imagine that the force exerted on each planet is imparted by an invisible rope. Kepler's laws will inform us with regard to the direction of this rope and the intensity of the strain transmitted through it.

The mathematical a.n.a.lysis of Kepler's laws would be beyond the scope of this volume. We must, therefore, confine ourselves to the results to which they lead, and omit the details of the reasoning. Newton first took the law which a.s.serted that the planet moved over equal areas in equal times, and he showed by unimpeachable logic that this at once gave the direction in which the force acted on the planet. He showed that the imaginary rope by which the planet is controlled must be invariably directed towards the sun. In other words, the force exerted on each planet was at all times pointed from the planet towards the sun.

It still remained to explain the intensity of the force, and to show how the intensity of that force varied when the planet was at different points of its path. Kepler's first law enables this question to be answered. If the planet's path be elliptic, and if the force be always directed towards the sun at one focus of that ellipse, then mathematical a.n.a.lysis obliges us to say that the intensity of the force must vary inversely as the square of the distance from the planet to the sun.

The movements of the planets, in conformity with Kepler's laws, would thus be accounted for even in their minutest details, if we admit that an attractive power draws the planet towards the sun, and that the intensity of this attraction varies inversely as the square of the distance. Can we hesitate to say that such an attraction does exist? We have seen how the earth attracts a falling body; we have seen how the earth's attraction extends to the moon, and explains the revolution of the moon around the earth. We have now learned that the movement of the planets round the sun can also be explained as a consequence of this law of attraction. But the evidence in support of the law of universal gravitation is, in truth, much stronger than any we have yet presented.

We shall have occasion to dwell on this matter further on. We shall show not only how the sun attracts the planets, but how the planets attract each other; and we shall find how this mutual attraction of the planets has led to remarkable discoveries, which have elevated the law of gravitation beyond the possibility of doubt.

Admitting the existence of this law, we can show that the planets must revolve around the sun in elliptic paths with the sun in the common focus. We can show that they must sweep over equal areas in equal times.

We can prove that the squares of the periodic times must be proportional to the cubes of their mean distances. Still further, we can show how the mysterious movements of comets can be accounted for. By the same great law we can explain the revolutions of the satellites. We can account for the tides, and for other phenomena throughout the Solar System. Finally, we shall show that when we extend our view beyond the limits of our Solar System to the beautiful starry systems scattered through s.p.a.ce, we find even there evidence of the great law of universal gravitation.

CHAPTER VI.

THE PLANET OF ROMANCE.