CHAPTER III

THE LAW OF GRAVITATION

The fundamental thought which, recognised or not, had lain at the root of the Ptolemaic system, as indeed it lies at the root of all science, was that "like causes must always produce like effects." Upon this principle there seemed to the ancient astronomers no escape from the inference that each planet must move at a uniform speed in a circle round its centre of motion. For, if there be any force tending to alter the distance of the planet from that centre, it seemed inevitable that sooner or later it should either reach that centre or be indefinitely removed from it. If there be no such force, then the planet's distance from that centre must remain invariable, and if it move at all, it must move in a circle; move uniformly, because there is no force either to hasten or retard it. Uniform motion in a circle seemed a necessity of nature.

But all this system, logical and inevitable as it had once seemed, had gone down before the assault of observed facts. The great example of uniform circular motion had been the daily revolution of the star sphere; but this was now seen to be only apparent, the result of the rotation of the Earth. The planets revolved round the Sun, but the Sun was not in the centre of their motion; they moved, not in circles, but in ellipses; not at a uniform speed, but at a speed which diminished with the increase of their distance from {31} the Sun. There was need, therefore, for an entire revision of the principles upon which motion was supposed to take place.

The mistake of the ancients had been that they supposed that continued motion demanded fresh applications of force. They noticed that a ball, set rolling, sooner or later came to a stop; that a pendulum, set swinging, might swing for a good time, but eventually came to rest; and, as the forces that were checking the motion--that is to say, the friction exercised by the ground, the atmosphere, and the like--did not obtrude themselves, they were overlooked.

Newton brought out into clear statement the true conditions of motion. A body once moving, if acted upon by no force whatsoever, must continue to move forward in a straight line at exactly the same speed, and that for ever. It does not require any maintaining force to keep it going. If any change in its speed or in its direction takes place, that change must be due to the introduction of some further force.

This principle, that, if no force acts on a body in motion, it will continue to move uniformly in a straight line, is Newton's First Law of Motion. His Second lays it down that, if force acts on a body, it produces a change of motion proportionate to the force applied, and in the same direction. And the Third Law states that when one body exerts force upon another, that second body reacts with equal force upon the first. The problem of the motions of the planets was, therefore, not what kept them moving, but what made them deviate from motion in a straight line, and deviate by different amounts.

It was quite clear, from the work of Kepler, that the force deflecting the planets from uniform motion in a {32} straight line lay in the Sun. The facts that the Sun lay in the plane of the orbits of all the planets, that the Sun was in one of the foci of each of the planetary ellipses, that the straight line joining the Sun and planet moved for each planet over equal areas in equal periods of time, established this fact clearly. But the amount of deflection was very different for different planets. Thus the orbit of Mercury is much smaller than that of the Earth, and is travelled over in a much shorter time, so that the distance by which Mercury is deflected in a course of an hour from movement in a straight line is much greater than that by which the Earth is deflected in the same time, Mercury falling towards the Sun by about 159 miles, whilst the fall of the Earth is only about 23.9 miles. The force drawing Mercury towards the Sun is therefore 6.66 times that drawing the Earth, but 6.66 is the square of 2.58, and the Earth is 2.58 times as far from the Sun as Mercury. Similarly, the fall in an hour of Jupiter towards the Sun is about 0.88 miles, so that the force drawing the Earth is 27 times that drawing Jupiter towards the Sun. But 27 is the square of 5.2, and Jupiter is 5.2 times as far from the Sun as the Earth. Similarly with the other planets. The force, therefore, which deflects the planets from motion in a straight line, and compels them to move round the Sun, is one which varies inversely as the square of the distance.

But the Sun is not the only attracting body of which we know. The old Ptolemaic system was correct to a small extent; the Earth is the centre of motion for the Moon, which revolves round it at a mean distance of 238,800 miles, and in a period of 27 d. 7 h. 43 m. Hence the circumference of her orbit is 1,500,450 miles, and the length of the straight line which she would travel {33} in one second of time, if not deflected by the Earth, is 2828 feet. In this distance the deviation of a circle from a straight line is one inch divided by 18.66. But we know from experiment that a stone let fall from a height of 193 inches above the Earth's surface will reach the ground in exactly one second of time. The force drawing the stone to the Earth, therefore, is 193 x 18.66; _i.e._ 3601 times as great as that drawing the Moon. But the stone is only 1/330 of a mile from the Earth's surface, while the Moon is 238,800 miles away--more than 78 million times as far. The force, therefore, would seem not to be diminished in the proportion that the distance is increased--much less in the proportion of its square.

But Newton proved that a sphere of uniform density, or made up of any number of concentric shells of uniform density, attracted a body outside itself, just as if its entire mass was concentrated at its centre. The distance of the stone from the Earth must therefore be measured, not from the Earth's surface, but from its centre; in other words, we must consider the stone as being distant from the Earth, not some 16 feet, but 3963 miles. This is very nearly one-sixtieth of the Moon's distance, and the square of 60 is 3600. The Earth's pull upon the Moon, therefore, is almost exactly in the inverse square of the distance as compared with its pull on the stone.

Kepler's book had found its "reader." His three laws were but three particular aspects of Newton's great discovery that the planets moved under the influence of a force, lodged in the Sun, which varied inversely as the square of their distances from it. But Newton's work went far beyond this, for he showed that the same law governed the motion of the Moon round the {34} Earth and the motions of the satellites revolving round the different planets, and also governed the fall of bodies upon the Earth itself. It was universal throughout the solar system. The law, therefore, is stated as of universal application. "Every particle of matter in the universe attracts every other particle with a force varying inversely as the square of the distance between them, and directly as the product of the masses of the two particles." And Newton further proved that if a body, projected in free space and moving with any velocity, became subject to a central force acting, like gravitation, inversely as the square of the distance, it must revolve in an ellipse, or in a closely allied curve.

These curves are what are known as the "+conic sections+"--that is, they are the curves found when a cone is cut across in different directions. Their relation to each other may be illustrated thus. If we have a very powerful light emerging from a minute hole, then, if we place a screen in the path of the beam of light, and exactly at right angles to its axis, the light falling on the screen will fill an exact circle. If we turn the screen so as to be inclined to the axis of the beam, the circle will lengthen out in one direction, and will become an ellipse. If we turn the screen still further, the ellipse will lengthen and lengthen, until at last, when the screen has become parallel to one of the edges of the beam of light, the ellipse will only have one end; the other will be lost. For it is clear that that edge of the beam of light which is parallel to the screen can never meet it. The curve now shown on the screen is called a +parabola+, and if the screen is turned further yet, the boundaries of the light falling upon it become divergent, and we have a fourth curve, the +hyperbola+. Bodies moving under the influence of {35} gravitation can move in any of these curves, but only the circle and ellipse are closed orbits. A particle moving in a parabola or hyperbola can only make one approach to its attracting body; after such approach it continually recedes from it. As the circle and parabola are only the two extreme forms of an ellipse, the two foci being at the same point for the circle and at an infinite distance apart for the parabola, we may regard all orbits under gravitation as being ellipses of one form or another.

From his great demonstration of the law of gravitation, Newton went on to apply it in many directions. He showed that the Earth could not be truly spherical in shape, but that there must be a flattening of its poles. He showed also that the Moon, which is exposed to the attractions both of the Earth and of the Sun, and, to a sensible extent, of some of the other planets, must show irregularities in her motion, which at that time had not been noticed. The Moon's orbit is inclined to that of the Earth, cutting its plane in two opposite points, called the "+nodes+." It had long been observed that the position of the nodes travelled round the ecliptic once in about nineteen years. Newton was able to show that this was a consequence of the Sun's attraction upon the Moon. And he further made a particular application of the principle thus brought out, for, the Earth not being a true sphere, but flattened at the poles and bulging at the equator, the equatorial belt might be regarded as a compact ring of satellites revolving round the Earth's equator. This, therefore, would tend to retrograde precisely as the nodes of a single satellite would, so that the axis of the equatorial belt of the Earth--in other words, the axis of the Earth--must revolve round the pole of the ecliptic. {36} Consequently the pole of the heavens appears to move amongst the stars, and the point where the celestial equator crosses the equator necessarily moves with it. This is what we know as the "+Precession of the Equinoxes+," and it is from our knowledge of the fact and the amount of precession that we are able to determine roughly the date when the first great work of astronomical observation was accomplished, namely, the grouping of the stars into constellations by the astronomers of the prehistoric age.

The publication of Newton's great work, the _Principia_ (_The Mathematical Principles of Natural Philosophy_), in which he developed the Laws of Motion, the significance of Kepler's Three Planetary Laws, and the Law of Universal Gravitation, took place in 1687, and was due to his friend EDMUND HALLEY, to whom he had confided many of his results. That he was the means of securing the publication of the _Principia_ is Halley's highest claim to the gratitude of posterity, but his own work in the field which Newton had opened was of great importance. Newton had treated +comets+ as moving in parabolic orbits, and Halley, collecting all the observations of comets that were available to him, worked out the particulars of their orbits on this assumption, and found that the elements of three were very closely similar, and that the interval between their appearances was nearly the same, the comets having been seen in 1531, 1607, and 1682. On further consulting old records he found that comets had been observed in 1456, 1378, and 1301. He concluded that these were different appearances of the same object, and predicted that it would be seen again in 1758, or, according to a later and more careful computation, in 1759. As the time for its return drew near, CLAIRAUT {37} computed with the utmost care the retardation which would be caused to the comet by the attractions of Jupiter and Saturn. The comet made its predicted nearest approach to the Sun on March 13, 1759, just one month earlier than Clairaut had computed. But in its next return, in 1835, the computations effected by PONTÉCOULANT were only two days in error, so carefully had the comet been followed during its unseen journey to the confines of the solar system and back again, during a period of seventy-five years. Pontécoulant's exploit was outdone at the next return by Drs. COWELL and CROMMELIN, of Greenwich Observatory, who not only computed the time of its perihelion passage--that is to say, its nearest approach to the Sun--for April 16, 1910, but followed the comet back in its wanderings during all its returns to the year 240 B.C. Halley's Comet, therefore, was the first comet that was known to travel in a closed orbit and to return to the neighbourhood of the Sun. Not a few small or telescopic comets are now known to be "periodic," but Halley's is the only one which has made a figure to the naked eye. Notices of it occur not a few times in history; it was the comet "like a flaming sword" which Josephus described as having been seen over Jerusalem not very long before the destruction by Titus. It was also the comet seen in the spring of the year when William the Conqueror invaded England, and was skilfully used by that leader as an omen of his coming victory.

The law of gravitation had therefore enabled men to recognise in Halley's Comet an addition to the number of the primary bodies in the solar system--the first addition that had been made since prehistoric times. On March 13, 1781, Sir WILLIAM HERSCHEL {38} detected a new object, which he at first supposed to be a comet, but afterwards recognised as a planet far beyond the orbit of Saturn. This planet, to which the name of Uranus was finally given, had a mean distance from the Sun nineteen times that of the Earth, and a diameter four times as great. This was a second addition to the solar system, but it was a discovery by sight, not by deduction.

The first day of the nineteenth century, January 1, 1801, was signalised by the discovery of a small planet by PIAZZI. The new object was lost for a time, but it was redetected on December 31 of the same year. This planet lay between the orbits of Mars and Jupiter--a region in which many hundreds of other small bodies have since been found. The first of these "+minor planets+" was called Ceres; the next three to be discovered are known as Pallas, Juno, and Vesta. Beside these four, two others are of special interest: one, Eros, which comes nearer the Sun than the orbit of Mars--indeed at some oppositions it approaches the Earth within 13,000,000 miles, and is therefore, next to the Moon, our nearest neighbour in space; the other, Achilles, moves at a distance from the Sun equal to that of Jupiter.

Ceres is much the largest of all the minor planets; indeed is larger than all the others put together. Yet the Earth exceeds Ceres 4000 times in volume, and 7000 times in mass, and the entire swarm of minor planets, all put together, would not equal in total volume one-fiftieth part of the Moon.

The search for these small bodies rendered it necessary that much fuller and more accurate maps of the stars should be made than had hitherto been attempted, and this had an important bearing on the next great event in the development of gravitational astronomy.

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The movements of Uranus soon gave rise to difficulties. It was found impossible, satisfactorily, to reconcile the earlier and later observations, and in the tables of Uranus, published by BOUVARD in 1821, the earlier observations were rejected. But the discrepancies between the observed and calculated places for the planet soon began to reappear and quickly increase, and the suggestion was made that these discrepancies were due to an attraction exercised by some planet as yet unknown. Thus Mrs. Somerville in a little book on the connection of the physical sciences, published in 1836, wrote, "Possibly it (that is, Uranus) may be subject to disturbances from some unseen planet revolving about the Sun beyond the present boundaries of our system. If, after the lapse of years, the tables formed from a combination of numerous observations should still be inadequate to represent the motions of Uranus, the discrepancies may reveal the existence, nay, even the mass and orbit of a body placed for ever beyond the sphere of vision." In 1843 JOHN C. ADAMS, who had just graduated as Senior Wrangler at Cambridge, proceeded to attack the problem of determining the position, orbit, and mass of the unknown body by which on this assumption Uranus was disturbed, from the irregularities evident in the motion of that planet. The problem was one of extraordinary intricacy, but by September 1845 Adams had obtained a first solution, which, he submitted to AIRY, the Astronomer Royal. As, however, he neglected to reply to some inquiries made by Airy, no search for the new planet was instituted in England until the results of a new and independent worker had been published. The same problem had been attacked by a well-known and very gifted French mathematician, U. J. J. LEVERRIER, and {40} in June 1846 he published his position for the unseen planet, which proved to be in close accord with that which Adams had furnished to Airy nine months before. On this Airy stirred up Challis, the Director of the Cambridge Observatory, which then possessed the most powerful telescope in England, to search for the planet, and Challis commenced to make charts, which included more than 3000 stars, in order to make sure that the stranger should not escape his net. Leverrier, on the other hand, communicated his result to the Berlin Observatory, where they had just received some of the star charts prepared by Dr. Bremiker in connection with the search for minor planets. The Berlin observer, Dr. Galle, had therefore nothing to do but to compare the stars in the field, upon which he turned his telescope, with those shown on the chart; a star not in the chart would probably be the desired stranger. He found it, therefore, on the very first evening, September 23, 1846, within less than four diameters of the Moon of the predicted place. The same object had been observed by Challis at Cambridge on August 4 and 12, but he was deferring the reduction of his observations until he had completed his scrutiny of the zone, and hence had not recognised it as different from an ordinary star.

This discovery of the planet now known as Neptune, which had been disturbing the movement of Uranus, has rightly been regarded as the most brilliant triumph of gravitational astronomy. It was the legitimate crown of that long intellectual struggle which had commenced more than 2000 years earlier, when the first Greek astronomers set themselves to unravel the apparently aimless wanderings of the planets in the assured faith that they would find them obedient unto law. {41} But of what use was all this effort? What is the good of astronomy? The question is often asked, but it is the question of ignorance. The use of astronomy is the development which it has given to the intellectual powers of man. Directly the problem of the planetary motions was first attempted, it became necessary to initiate mathematical processes in order to deal with it, and the necessity for the continued development of mathematics has been felt in the same connection right down to the present day. When the Greek astronomers first began their inquiries into the planetary movements they hoped for no material gain, and they received none. They laboured; we have entered into their labours. But the whole of our vast advances in mechanical and engineering science--advances which more than anything else differentiate this our present age from all those which have preceded it--are built upon our command of mathematics and our knowledge of the laws of motion--a command and a knowledge which we owe directly to their persevering attempts to advance the science of astronomy, and to follow after knowledge, not for any material rewards which she had to offer, but for her own sake.

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