Читать книгу The Story of the Heavens - Robert S. Ball - Страница 18

Table of Contents

Gravitation—The Falling of a Stone to the Ground—All Bodies fall equally, Sixteen Feet in a Second—Is this true at Great Heights?—Fall of a Body at a Height of a Quarter of a Million Miles—How Newton obtained an Answer from the Moon—His Great Discovery—Statement of the Law of Gravitation—Illustrations of the Law—How is it that all the Bodies in the Universe do not rush Together?—The Effect of Motion—How a Circular Path can be produced by Attraction—General Account of the Moon's Motion—Is Gravitation a Force of Great Intensity?—Two Weights of 50 lbs.—Two Iron Globes, 53 Yards in Diameter, and a Mile apart, attract with a Force of 1 lb.—Characteristics of Gravitation—Orbits of the Planets not strictly Circles—The Discoveries of Kepler—Construction of an Ellipse—Kepler's First Law—Does a Planet move Uniformly?—Law of the Changes of Velocity—Kepler's Second Law—The Relation between the Distances and the Periodic Times—Kepler's Third Law—Kepler's Laws and the Law of Gravitation—Movement in a Straight Line—A Body unacted on by Disturbing Forces would move in a Straight Line with Constant Velocity—Application to the Earth and the Planets—The Law of Gravitation deduced from Kepler's Laws—Universal Gravitation.

Our description of the heavenly bodies must undergo a slight interruption, while we illustrate with appropriate detail an important principle, known as the law of gravitation, which underlies the whole of astronomy. By this law we can explain the movements of the moon around the earth, and of the planets around the sun. It is accordingly incumbent upon us to discuss this subject before we proceed to the more particular account of the separate planets. We shall find, too, that the law of gravitation sheds some much-needed light on the nature of the stars situated at the remotest distances in space. It also enables us to cast a glance through the vistas of time past, and to trace with plausibility, if not with certainty, certain early phases in the history of our system. The sun and the moon, the planets and the comets, the stars and the nebulæ, all alike are subject to this universal law, which is now to engage our attention.

What is more familiar than the fact that when a stone is dropped it will fall to the ground? No one at first thinks the matter even worthy of remark. People are often surprised at seeing a piece of iron drawn to a magnet. Yet the fall of a stone to the ground is the manifestation of a force quite as interesting as the force of magnetism. It is the earth which draws the stone, just as the magnet draws the iron. In each case the force is one of attraction; but while the magnetic attraction is confined to a few substances, and is of comparatively limited importance, the attraction of gravitation is significant throughout the universe.

Let us commence with a few very simple experiments upon the force of gravitation. Hold in the hand a small piece of lead, and then allow it to drop upon a cushion. The lead requires a certain time to move from the fingers to the cushion, but that time is always the same when the height is the same. Take now a larger piece of lead, and hold one piece in each hand at the same height. If both are released at the same moment, they will both reach the cushion simultaneously. It might have been thought that the heavy body would fall more quickly than the light body; but when the experiment is tried, it is seen that this is not the case. Repeat the experiment with various other substances. An ordinary marble will be found to fall in the same time as the piece of lead. With a piece of cork we again try the experiment, and again obtain the same result. At first it seems to fail when we compare a feather with the piece of lead; but that is solely on account of the air, which resists the feather more than it resists the lead. If, however, the feather be placed upon the top of a penny, and the penny be horizontal when dropped, it will clear the air out of the way of the feather in its descent, and then the feather will fall as quickly as the penny, as quickly as the marble, or as quickly as the lead.

If the observer were in a gallery when trying these experiments, and if the cushion were sixteen feet below his hands, then the time the marble would take to fall through the sixteen feet would be one second. The time occupied by the cork or by the lead would be the same; and even the feather itself would fall through sixteen feet in one second, if it could be screened from the interference of the air. Try this experiment where we like, in London, or in any other city, in any island or continent, on board a ship at sea, at the North Pole, or the South Pole, or the equator, it will always be found that any body, of any size or any material, will fall about sixteen feet in one second of time.

Lest any erroneous impression should arise, we may just mention that the distance traversed in one second does vary slightly at different parts of the earth, but from causes which need not at this moment detain us. We shall for the present regard sixteen feet as the distance through which any body, free from interference, would fall in one second at any part of the earth's surface. But now let us extend our view above the earth's surface, and enquire how far this law of sixteen feet in a second may find obedience elsewhere. Let us, for instance, ascend to the top of a mountain and try the experiment there. It would be found that at the top of the mountain a marble would take a little longer to fall through sixteen feet than the same marble would if let fall at its base. The difference would be very small; but yet it would be measurable, and would suffice to show that the power of the earth to pull the marble to the ground becomes somewhat weakened at a point high above the earth's surface. Whatever be the elevation to which we ascend, be it either the top of a high mountain, or the still greater altitudes that have been reached in balloon ascents, we shall never find that the tendency of bodies to fall to the ground ceases, though no doubt the higher we go the more is that tendency weakened. It would be of great interest to find how far this power of the earth to draw bodies towards it can really extend. We cannot attain more than about five or six miles above the earth's surface in a balloon; yet we want to know what would happen if we could ascend 500 miles, or 5,000 miles, or still further, into the regions of space.

Conceive that a traveller were endowed with some means of soaring aloft for miles and thousands of miles, still up and up, until at length he had attained the awful height of nearly a quarter of a million of miles above the ground. Glancing down at the surface of that earth, which is at such a stupendous depth beneath, he would be able to see a wonderful bird's-eye view. He would lose, no doubt, the details of towns and villages; the features in such a landscape would be whole continents and whole oceans, in so far as the openings between the clouds would permit the earth's surface to be exposed.

At this stupendous elevation he could try one of the most interesting experiments that was ever in the power of a philosopher. He could test whether the earth's attraction was felt at such a height, and he could measure the amount of that attraction. Take for the experiment a cork, a marble, or any other object, large or small; hold it between the fingers, and let it go. Everyone knows what would happen in such a case down here; but it required Sir Isaac Newton to tell what would happen in such a case up there. Newton asserts that the power of the earth to attract bodies extends even to this great height, and that the marble would fall. This is the doctrine that we can now test. We are ready for the experiment. The marble is released, and, lo! our first exclamation is one of wonder. Instead of dropping instantly, the little object appears to remain suspended. We are on the point of exclaiming that we must have gone beyond the earth's attraction, and that Newton is wrong, when our attention is arrested; the marble is beginning to move, so slowly that at first we have to watch it carefully. But the pace gradually improves, so that the attraction is beyond all doubt, until, gradually acquiring more and more velocity, the marble speeds on its long journey of a quarter of a million of miles to the earth.

But surely, it will be said, such an experiment must be entirely impossible; and no doubt it cannot be performed in the way described. The bold idea occurred to Newton of making use of the moon itself, which is almost a quarter of a million of miles above the earth, for the purpose of answering the question. Never was our satellite put to such noble use before. It is actually at each moment falling in towards the earth. We can calculate how much it is deflected towards the earth in each second, and thus obtain a measure of the earth's attractive power. From such enquiries Newton was able to learn that a body released at the distance of 240,000 miles above the surface of the earth would still be attracted by the earth, that in virtue of the attraction the body would commence to move off towards the earth—not, indeed, with the velocity with which a body falls in experiments on the surface, but with a very much lesser speed. A body dropped down from the distance of the moon would commence its long journey so slowly that a minute, instead of a second, would have elapsed before the distance of sixteen feet had been accomplished.[11]

It was by pondering on information thus won from the moon that Newton made his immortal discovery. The gravitation of the earth is a force which extends far and wide through space. The more distant the body, the weaker the gravitation becomes; here Newton found the means of determining the great problem as to the law according to which the intensity of the gravitation decreased. The information derived from the moon, that a body 240,000 miles away requires a minute to fall through a space equal to that through which it would fall in a second down here, was of paramount importance. In the first place, it shows that the attractive power of the earth, by which it draws all bodies earthwards, becomes weaker at a distance. This might, indeed, have been anticipated. It is as reasonable to suppose that as we retreated further and further into the depths of space the power of attraction should diminish, as that the lustre of light should diminish as we recede from it; and it is remarkable that the law according to which the attraction of gravitation decreases with the increase of distance is precisely the same as the law according to which the brilliancy of a light decreases as its distance increases.

The law of nature, stated in its simplest form, asserts that the intensity of gravitation varies inversely as the square of the distance. Let me endeavour to elucidate this somewhat abstract statement by one or two simple illustrations. Suppose a body were raised above the surface of the earth to a height of nearly 4,000 miles, so as to be at an altitude equal to the radius of the earth. In other words, a body so situated would be twice as far from the centre of the earth as a body which lay on the surface. The law of gravitation says that the intensity of the attraction is then to be decreased to one-fourth part, so that the pull of the earth on a body 4,000 miles high is only one quarter of the pull of the earth on that body so long as it lies on the ground. We may imagine the effect of this pull to be shown in different ways. Allow the body to fall, and in the interval of one second it will only drop through four feet, a mere quarter of the distance that gravity would cause near the earth's surface.

We may consider the matter in another way by supposing that the attraction of the earth is measured by one of those little weighing machines known as a spring balance. If a weight of four pounds be hung on such a contrivance, at the earth's surface, the index of course shows a weight of four pounds; but conceive this balance, still bearing the weight appended thereto, were to be carried up and up, the indicated strain would become less and less, until by the time the balance reached 4,000 miles high, where it was twice as far away from the earth's centre as at first, the indicated strain would be reduced to the fourth part, and the balance would only show one pound. If we could imagine the instrument to be carried still further into the depths of space, the indication of the scale would steadily continue to decline. By the time the apparatus had reached a distance of 8,000 miles high, being then three times as far from the earth's centre as at first, the law of gravitation tells us that the attraction must have decreased to one-ninth part. The strain thus shown on the balance would be only the ninth part of four pounds, or less than half a pound. But let the voyage be once again resumed, and let not a halt be made this time until the balance and its four-pound weight have retreated to that orbit which the moon traverses in its monthly course around the earth. The distance thus attained is about sixty times the radius of the earth, and consequently the attraction of gravitation is diminished in the proportion of one to the square of sixty; the spring will then only be strained by the inappreciable fraction of 1–3,600 part of four pounds. It therefore appears that a weight which on the earth weighed a ton and a half would, if raised 240,000 miles, weigh less than a pound. But even at this vast distance we are not to halt; imagine that we retreat still further and further; the strain shown by the balance will ever decrease, but it will still exist, no matter how far we go. Astronomy appears to teach us that the attraction of gravitation can extend, with suitably enfeebled intensity, across the most profound gulfs of space.

The principle of gravitation is of far wider scope than we have yet indicated. We have spoken merely of the attraction of the earth, and we have stated that this force extends throughout space. But the law of gravitation is not so limited. Not only does the earth attract every other body, and every other body attract the earth, but each of these bodies attracts the other; so that in its more complete shape the law of gravitation announces that "every body in the universe attracts every other body with a force which varies inversely as the square of the distance."

It is impossible for us to over-estimate the importance of this law. It supplies the clue by which we can unravel the complicated movements of the planets. It has led to marvellous discoveries, in which the law of gravitation has enabled us to anticipate the telescope, and to feel the existence of bodies before those bodies have even been seen.

An objection which may be raised at this point must first be dealt with. It seems to be, indeed, a plausible one. If the earth attracts the moon, why does not the moon tumble down on the earth? If the earth is attracted by the sun, why does it not tumble into the sun? If the sun is attracted by other stars, why do they not rush together with a frightful collision? It may not unreasonably be urged that if all these bodies in the heavens are attracting each other, it would seem that they must all rush together in consequence of that attraction, and thus weld the whole material universe into a single mighty mass. We know, as a matter of fact, that these collisions do not often happen, and that there is extremely little likelihood of their taking place. We see that although our earth is said to have been attracted by the sun for countless ages, yet the earth is just as far from the sun as ever it was. Is not this in conflict with the doctrine of universal gravitation? In the early days of astronomy such objections would be regarded, and doubtless were regarded, as well-nigh insuperable; even still we occasionally hear them raised, and it is therefore the more incumbent on us to explain how it happens that the solar system has been able to escape from the catastrophe by which it seems to be threatened.

There can be no doubt that if the moon and the earth had been initially placed at rest, they would have been drawn together by their mutual attraction. So, too, if the system of planets surrounding the sun had been left initially at rest they would have dashed into the sun, and the system would have been annihilated. It is the fact that the planets are moving, and that the moon is moving, which has enabled these bodies successfully to resist the attraction in so far, at least, as that they are not drawn thereby to total destruction.

It is so desirable that the student should understand clearly how a central attraction is compatible with revolution in a nearly circular path, that we give an illustration to show how the moon pursues its monthly orbit under the guidance and the control of the attracting earth.

Fig. 36.—Illustration of the Moon's Motion.

The imaginary sketch in Fig. 36 denotes a section of the earth with a high mountain thereon.[12] If a cannon were stationed on the top of the mountain at C, and if the cannonball were fired off in the direction C E with a moderate charge of powder, the ball would move down along the first curved path. If it be fired a second time with a heavier charge, the path will be along the second curved line, and the ball would again fall to the ground. But let us try next time with a charge still further increased, and, indeed, with a far stronger cannon than any piece of ordnance ever yet made. The velocity of the projectile must now be assumed to be some miles per second, but we can conceive that the speed shall be so adjusted that the ball shall move along the path C D, always at the same height above the earth, though still curving, as every projectile must curve, from the horizontal line in which it moved at the first moment. Arrived at D, the ball will still be at the same height above the surface, and its velocity must be unabated. It will therefore continue in its path and move round another quadrant of the circle without getting nearer to the surface. In this manner the projectile will travel completely round the whole globe, coming back again to C and then taking another start in the same path. If we could abolish the mountain and the cannon at the top, we should have a body revolving for ever around the earth in consequence of the attraction of gravitation.

Make now a bold stretch of the imagination. Conceive a terrific cannon capable of receiving a round bullet not less than 2,000 miles in diameter. Discharge this enormous bullet with a velocity of about 3,000 feet per second, which is two or three times as great as the velocity actually attainable in modern artillery. Let this notable bullet be fired horizontally from some station nearly a quarter of a million miles above the surface of the earth. That fearful missile would sweep right round the earth in a nearly circular orbit, and return to where it started in about four weeks. It would then commence another revolution, four weeks more would find it again at the starting point, and this motion would go on for ages.

Do not suppose that we are entirely romancing. We cannot indeed show the cannon, but we can point to a great projectile. We see it every month; it is the beautiful moon herself. No one asserts that the moon was ever shot from such a cannon; but it must be admitted that she moves as if she had been. In a later chapter we shall enquire into the history of the moon, and show how she came to revolve in this wonderful manner.

As with the moon around the earth, so with the earth around the sun. The illustration shows that a circular or nearly circular motion harmonises with the conception of the law of universal gravitation.

We are accustomed to regard gravitation as a force of stupendous magnitude. Does not gravitation control the moon in its revolution around the earth? Is not even the mighty earth itself retained in its path around the sun by the surpassing power of the sun's attraction? No doubt the actual force which keeps the earth in its path, as well as that which retains the moon in our neighbourhood, is of vast intensity, but that is because gravitation is in such cases associated with bodies of enormous mass. No one can deny that all bodies accessible to our observation appear to attract each other in accordance with the law of gravitation; but it must be confessed that, unless one or both of the attracting bodies is of gigantic dimensions, the intensity is almost immeasurably small.

Let us attempt to illustrate how feeble is the gravitation between masses of easily manageable dimensions. Take, for instance, two iron weights, each weighing about 50lb., and separated by a distance of one foot from centre to centre. There is a certain attraction of gravitation between these weights. The two weights are drawn together, yet they do not move. The attraction between them, though it certainly exists, is an extremely minute force, not at all comparable as to intensity with magnetic attraction. Everyone knows that a magnet will draw a piece of iron with considerable vigour, but the intensity of gravitation is very much less on masses of equal amount. The attraction between these two 50lb. weights is less than the ten-millionth part of a single pound. Such a force is utterly infinitesimal in comparison with the friction between the weights and the table on which they stand, and hence there is no response to the attraction by even the slightest movement. Yet, if we can conceive each of these weights mounted on wheels absolutely devoid of friction, and running on absolutely perfect horizontal rails, then there is no doubt that the bodies would slowly commence to draw together, and in the course of time would arrive in actual contact.

If we desire to conceive gravitation as a force of measurable intensity, we must employ masses immensely more ponderous than those 50lb. weights. Imagine a pair of globes, each composed of 417,000 tons of cast iron, and each, if solid, being about 53 yards in diameter. Imagine these globes placed at a distance of one mile apart. Each globe attracts the other by the force of gravitation. It does not matter that buildings and obstacles of every description intervene; gravitation will pass through such impediments as easily as light passes through glass. No screen can be devised dense enough to intercept the passage of this force. Each of these iron globes will therefore under all circumstances attract the other; but, notwithstanding their ample proportions, the intensity of that attraction is still very small, though appreciable. The attraction between these two globes is a force no greater than the pressure exerted by a single pound weight. A child could hold back one of these massive globes from its attraction by the other. Suppose that all was clear, and that friction could be so neutralised as to permit the globes to follow the impulse of their mutual attractions. The two globes will then commence to approach, but the masses are so large, while the attraction is so small, that the speed will be accelerated very slowly. A microscope would be necessary to show when the motion has actually commenced. An hour and a half must elapse before the distance is diminished by a single foot; and although the pace improves subsequently, yet three or four days must elapse before the two globes will come together.

The most remarkable characteristic of the force of gravitation must be here specially alluded to. The intensity appears to depend only on the quantity of matter in the bodies, and not at all on the nature of the substances of which these bodies are composed. We have described the two globes as made of cast iron, but if either or both were composed of lead or copper, of wood or stone, of air or water, the attractive power would still be the same, provided only that the masses remain unaltered. In this we observe a profound difference between the attraction of gravitation and magnetic attraction. In the latter case the attraction is not perceptible at all in the great majority of substances, and is only considerable in the case of iron.

In our account of the solar system we have represented the moon as revolving around the earth in a nearly circular path, and the planets as revolving around the sun in orbits which are also approximately circular. It is now our duty to give a more minute description of these remarkable paths; and, instead of dismissing them as being nearly circles, we must ascertain precisely in what respects they differ therefrom.

If a planet revolved around the sun in a truly circular path, of which the sun was always at the centre, it is then obvious that the distance from the sun to the planet, being always equal to the radius of the circle, must be of constant magnitude. Now, there can be no doubt that the distance from the sun to each planet is approximately constant; but when accurate observations are made, it becomes clear that the distance is not absolutely so. The variations in distance may amount to many millions of miles, but, even in extreme cases, the variation in the distance of the planet is only a small fraction—usually a very small fraction—of the total amount of that distance. The circumstances vary in the case of each of the planets. The orbit of the earth itself is such that the distance from the earth to the sun departs but little from its mean value. Venus makes even a closer approach to perfectly circular movement; while, on the other hand, the path of Mars, and much more the path of Mercury, show considerable relative fluctuations in the distance from the planet to the sun.

It has often been noticed that many of the great discoveries in science have their origin in the nice observation and explanation of minute departures from some law approximately true. We have in this department of astronomy an excellent illustration of this principle. The orbits of the planets are nearly circles, but they are not exactly circles. Now, why is this? There must be some natural reason. That reason has been ascertained, and it has led to several of the grandest discoveries that the mind of man has ever achieved in the realms of Nature.

In the first place, let us see the inferences to be drawn from the fact that the distance of a planet from the sun is not constant. The motion in a circle is one of such beauty and simplicity that we are reluctant to abandon it, unless the necessity for doing so be made clearly apparent. Can we not devise any way by which the circular motion might be preserved, and yet be compatible with the fluctuations in the distance from the planet to the sun? This is clearly impossible with the sun at the centre of the circle. But suppose the sun did not occupy the centre, while the planet, as before, revolved around the sun. The distance between the two bodies would then necessarily fluctuate. The more eccentric the position of the sun, the larger would be the proportionate variation in the distance of the planet when at the different parts of its orbit. It might further be supposed that by placing a series of circles around the sun the various planetary orbits could be accounted for. The centre of the circle belonging to Venus is to coincide very nearly with the centre of the sun, and the centres of the orbits of all the other planets are to be placed at such suitable distances from the sun as will render a satisfactory explanation of the gradual increase and decrease of the distance between the two bodies.

There can be no doubt that the movements of the moon and of the planets would be, to a large extent, explained by such a system of circular orbits; but the spirit of astronomical enquiry is not satisfied with approximate results. Again and again the planets are observed, and again and again the observations are compared with the places which the planets would occupy if they moved in accordance with the system here indicated. The centres of the circles are moved hither and thither, their radii are adjusted with greater care; but it is all of no avail. The observations of the planets are minutely examined to see if they can be in error; but of errors there are none at all sufficient to account for the discrepancies. The conclusion is thus inevitable—astronomers are forced to abandon the circular motion, which was thought to possess such unrivalled symmetry and beauty, and are compelled to admit that the orbits of the planets are not circular.

Then if these orbits be not circles, what are they? Such was the great problem which Kepler proposed to solve, and which, to his immortal glory, he succeeded in solving and in proving to demonstration. The great discovery of the true shape of the planetary orbits stands out as one of the most conspicuous events in the history of astronomy. It may, in fact, be doubted whether any other discovery in the whole range of science has led to results of such far-reaching interest.

We must here adventure for a while into the field of science known as geometry, and study therein the nature of that curve which the discovery of Kepler has raised to such unparalleled importance. The subject, no doubt, is a difficult one, and to pursue it with any detail would involve us in many abstruse calculations which would be out of place in this volume; but a general sketch of the subject is indispensable, and we must attempt to render it such justice as may be compatible with our limits.

The curve which represents with perfect fidelity the movements of a planet in its revolution around the sun belongs to that well-known group of curves which mathematicians describe as the conic sections. The particular form of conic section which denotes the orbit of a planet is known by the name of the ellipse: it is spoken of somewhat less accurately as an oval. The ellipse is a curve which can be readily constructed. There is no simpler method of doing so than that which is familiar to draughtsmen, and which we shall here briefly describe.

We represent on the next page (Fig. 37) two pins passing through a sheet of paper. A loop of twine passes over the two pins in the manner here indicated, and is stretched by the point of a pencil. With a little care the pencil can be guided so as to keep the string stretched, and its point will then describe a curve completely round the pins, returning to the point from which it started. We thus produce that celebrated geometrical figure which is called an ellipse.

It will be instructive to draw a number of ellipses, varying in each case the circumstances under which they are formed. If, for instance, the pins remain placed as before, while the length of the loop is increased, so that the pencil is farther away from the pins, then it will be observed that the ellipse has lost some of its elongation, and approaches more closely to a circle. On the other hand, if the length of the cord in the loop be lessened, while the pins remain as before, the ellipse will be found more oval, or, as a mathematician would say, its eccentricity is increased. It is also useful to study the changes which the form of the ellipse undergoes when one of the pins is altered, while the length of the loop remains unchanged. If the two pins be brought nearer together the eccentricity will decrease, and the ellipse will approximate more closely to the shape of a circle. If the pins be separated more widely the eccentricity of the ellipse will be increased. That the circle is an extreme form of ellipse will be evident, if we suppose the two pins to draw in so close together that they become coincident; the point will then simply trace out a circle as the pencil moves round the figure.

Fig. 37.—Drawing an Ellipse.

The points marked by the pins obviously possess very remarkable relations with respect to the curve. Each one is called a focus, and an ellipse can only have one pair of foci. In other words, there is but a single pair of positions possible for the two pins, when an ellipse of specified size, shape, and position is to be constructed.

The ellipse differs principally from a circle in the circumstance that it possesses variety of form. We can have large and small ellipses just as we can have large and small circles, but we can also have ellipses of greater or less eccentricity. If the ellipse has not the perfect simplicity of the circle it has, at least, the charm of variety which the circle has not. The oval curve has also the beauty derived from an outline of perfect grace and an association with ennobling conceptions.

The ancient geometricians had studied the ellipse: they had noticed its foci; they were acquainted with its geometrical relations; and thus Kepler was familiar with the ellipse at the time when he undertook his celebrated researches on the movements of the planets. He had found, as we have already indicated, that the movements of the planets could not be reconciled with circular orbits. What shape of orbit should next be tried? The ellipse was ready to hand, its properties were known, and the comparison could be made; memorable, indeed, was the consequence of this comparison. Kepler found that the movement of the planets could be explained, by supposing that the path in which each one revolved was an ellipse. This in itself was a discovery of the most commanding importance. On the one hand it reduced to order the movements of the great globes which circulate round the sun; while on the other, it took that beautiful class of curves which had exercised the geometrical talents of the ancients, and assigned to them the dignity of defining the highways of the universe.

But we have as yet only partly enunciated the first discovery of Kepler. We have seen that a planet revolves in an ellipse around the sun, and that the sun is, therefore, at some point in the interior of the ellipse—but at what point? Interesting, indeed, is the answer to this question. We have pointed out how the foci possess a geometrical significance which no other points enjoy. Kepler showed that the sun must be situated in one of the foci of the ellipse in which each planet revolves. We thus enunciate the first law of planetary motion in the following words:—

Each planet revolves around the sun in an elliptic path, having the sun at one of the foci.

We are now enabled to form a clear picture of the orbits of the planets, be they ever so numerous, as they revolve around the sun. In the first place, we observe that the ellipse is a plane curve; that is to say, each planet must, in the course of its long journey, confine its movements to one plane. Each planet has thus a certain plane appropriated to it. It is true that all these planes are very nearly coincident, at least in so far as the great planets are concerned; but still they are distinct, and the only feature in which they all agree is that each one of them passes through the sun. All the elliptic orbits of the planets have one focus in common, and that focus lies at the centre of the sun.

It is well to illustrate this remarkable law by considering the circumstances of two or three different planets. Take first the case of the earth, the path of which, though really an ellipse, is very nearly circular. In fact, if it were drawn accurately to scale on a sheet of paper, the difference between the elliptic orbit and the circle would hardly be detected without careful measurement. In the case of Venus the ellipse is still more nearly a circle, and the two foci of the ellipse are very nearly coincident with the centre of the circle. On the other hand, in the case of Mercury, we have an ellipse which departs from the circle to a very marked extent, while in the orbits of some of the minor planets the eccentricity is still greater. It is extremely remarkable that every planet, no matter how far from the sun, should be found to move in an ellipse of some shape or other. We shall presently show that necessity compels each planet to pursue an elliptic path, and that no other form of path is possible.

Started on its elliptic path, the planet pursues its stately course, and after a certain duration, known as the periodic time, regains the position from which its departure was taken. Again the planet traces out anew the same elliptic path, and thus, revolution after revolution, an identical track is traversed around the sun. Let us now attempt to follow the body in its course, and observe the history of its motion during the time requisite for the completion of one of its circuits. The dimensions of a planetary orbit are so stupendous that the planet must run its course very rapidly in order to finish the journey within the allotted time. The earth, as we have already seen, has to move eighteen miles a second to accomplish one of its voyages round the sun in the lapse of 365-1⁄4 days. The question then arises as to whether the rate at which a planet moves is uniform or not. Does the earth, for instance, actually move at all times with the velocity of eighteen miles a second, or does our planet sometimes move more rapidly and sometimes more slowly, so that the average of eighteen miles a second is still maintained? This is a question of very great importance, and we are able to answer it in the clearest and most emphatic manner. The velocity of a planet is not uniform, and the variations of that velocity can be explained by the adjoining figure (Fig. 38).