# XII. NEWTON AND THE LAW OF GRAVITATION

We come now to the story of what is by common consent the greatest of scientific achievements. The law of universal gravitation is the most far-reaching principle as yet discovered. It has application equally to the minutest particle of matter and to the most distant suns in the universe, yet it is amazing in its very simplicity. As usually phrased, the law is this: That every particle of matter in the universe attracts every other particle with a force that varies directly with the mass of the particles and inversely as the squares of their mutual distance. Newton did not vault at once to the full expression of this law, though he had formulated it fully before he gave the results of his investigations to the world. We have now to follow the steps by which he reached this culminating achievement.

At the very beginning we must understand that the idea of universal gravitation was not absolutely original with Newton. Away back in the old Greek days, as we have seen, Anaxagoras conceived and clearly expressed the idea that the force which holds the heavenly bodies in their orbits may be the same that operates upon substances at the surface of the earth. With Anaxagoras this was scarcely more than a guess. After his day the idea seems not to have been expressed by any one until the seventeenth century's awakening of science. Then the consideration of Kepler's Third Law of planetary motion suggested to many minds perhaps independently the probability that the force hitherto mentioned merely as centripetal, through the operation of which the planets are held in their orbits is a force varying inversely as the square of the distance from the sun. This idea had come to Robert Hooke, to Wren, and perhaps to Halley, as well as to Newton; but as yet no one had conceived a method by which the validity of the suggestion might be tested. It was claimed later on by Hooke that he had discovered a method demonstrating the truth of the theory of inverse squares, and after the full announcement of Newton's discovery a heated controversy was precipitated in which Hooke put forward his claims with accustomed acrimony. Hooke, however, never produced his demonstration, and it may well be doubted whether he had found a method which did more than vaguely suggest the law which the observations of Kepler had partially revealed. Newton's great merit lay not so much in conceiving the law of inverse squares as in the demonstration of the law. He was led to this demonstration through considering the orbital motion of the moon. According to the familiar story, which has become one of the classic myths of science, Newton was led to take up the problem through observing the fall of an apple. Voltaire is responsible for the story, which serves as well as another; its truth or falsity need not in the least concern us. Suffice it that through pondering on the familiar fact of terrestrial gravitation, Newton was led to question whether this force which operates so tangibly here at the earth's surface may not extend its influence out into the depths of space, so as to include, for example, the moon. Obviously some force pulls the moon constantly towards the earth; otherwise that body would fly off at a tangent and never return. May not this so-called centripetal force be identical with terrestrial gravitation? Such was Newton's query. Probably many another man since Anaxagoras had asked the same question, but assuredly Newton was the first man to find an answer.

The thought that suggested itself to Newton's mind was this: If we make a diagram illustrating the orbital course of the moon for any given period, say one minute, we shall find that the course of the moon departs from a straight line during that period by a measurable distance—that: is to say, the moon has been virtually pulled towards the earth by an amount that is represented by the difference between its actual position at the end of the minute under observation and the position it would occupy had its course been tangential, as, according to the first law of motion, it must have been had not some force deflected it towards the earth. Measuring the deflection in question—which is equivalent to the so-called versed sine of the arc traversed—we have a basis for determining the strength of the deflecting force. Newton constructed such a diagram, and, measuring the amount of the moon's departure from a tangential rectilinear course in one minute, determined this to be, by his calculation, thirteen feet. Obviously, then, the force acting upon the moon is one that would cause that body to fall towards the earth to the distance of thirteen feet in the first minute of its fall. Would such be the force of gravitation acting at the distance of the moon if the power of gravitation varies inversely as the square of the distance? That was the tangible form in which the problem presented itself to Newton. The mathematical solution of the problem was simple enough. It is based on a comparison of the moon's distance with the length of the earth's radius. On making this calculation, Newton found that the pull of gravitation—if that were really the force that controls the moon—gives that body a fall of slightly over fifteen feet in the first minute, instead of thirteen feet. Here was surely a suggestive approximation, yet, on the other band, the discrepancy seemed to be too great to warrant him in the supposition that he had found the true solution. He therefore dismissed the matter from his mind for the time being, nor did he return to it definitely for some years.

{illustration caption = DIAGRAM TO ILLUSTRATE NEWTON'S LAW OF GRAVITATION (E represents the earth and A the moon. Were the earth's pull on the moon to cease, the moon's inertia would cause it to take the tangential course, AB. On the other hand, were the moon's motion to be stopped for an instant, the moon would fall directly towards the earth, along the line AD. The moon's actual orbit, resulting from these component forces, is AC. Let AC represent the actual flight of the moon in one minute. Then BC, which is obviously equal to AD, represents the distance which the moon virtually falls towards the earth in one minute. Actual computation, based on measurements of the moon's orbit, showed this distance to be about fifteen feet. Another computation showed that this is the distance that the moon would fall towards the earth under the influence of gravity, on the supposition that the force of gravity decreases inversely with the square of the distance; the basis of comparison being furnished by falling bodies at the surface of the earth. Theory and observations thus coinciding, Newton was justified in declaring that the force that pulls the moon towards the earth and keeps it in its orbit, is the familiar force of gravity, and that this varies inversely as the square of the distance.)}

It was to appear in due time that Newton's hypothesis was perfectly valid and that his method of attempted demonstration was equally so. The difficulty was that the earth's proper dimensions were not at that time known. A wrong estimate of the earth's size vitiated all the other calculations involved, since the measurement of the moon's distance depends upon the observation of the parallax, which cannot lead to a correct computation unless the length of the earth's radius is accurately known. Newton's first calculation was made as early as 1666, and it was not until 1682 that his attention was called to a new and apparently accurate measurement of a degree of the earth's meridian made by the French astronomer Picard. The new measurement made a degree of the earth's surface 69.10 miles, instead of sixty miles.

Learning of this materially altered calculation as to the earth's size, Newton was led to take up again his problem of the falling moon. As he proceeded with his computation, it became more and more certain that this time the result was to harmonize with the observed facts. As the story goes, he was so completely overwhelmed with emotion that he was forced to ask a friend to complete the simple calculation. That story may well be true, for, simple though the computation was, its result was perhaps the most wonderful demonstration hitherto achieved in the entire field of science. Now at last it was known that the force of gravitation operates at the distance of the moon, and holds that body in its elliptical orbit, and it required but a slight effort of the imagination to assume that the force which operates through such a reach of space extends its influence yet more widely. That such is really the case was demonstrated presently through calculations as to the moons of Jupiter and by similar computations regarding the orbital motions of the various planets. All results harmonizing, Newton was justified in reaching the conclusion that gravitation is a universal property of matter. It remained, as we shall see, for nineteenth-century scientists to prove that the same force actually operates upon the stars, though it should be added that this demonstration merely fortified a belief that had already found full acceptance.

Having thus epitomized Newton's discovery, we must now take up the steps of his progress somewhat in detail, and state his theories and their demonstration in his own words. Proposition IV., theorem 4, of his Principia is as follows:

"That the moon gravitates towards the earth and by the force of gravity is continually drawn off from a rectilinear motion and retained in its orbit.

"The mean distance of the moon from the earth, in the syzygies in semi-diameters of the earth, is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60 1/3; to Street, 60 2/3; and to Tycho, 56 1/2. But Tycho, and all that follow his tables of refractions, making the refractions of the sun and moon (altogether against the nature of light) to exceed the refractions of the fixed stars, and that by four or five minutes NEAR THE HORIZON, did thereby increase the moon's HORIZONTAL parallax by a like number of minutes, that is, by a twelfth or fifteenth part of the whole parallax. Correct this error and the distance will become about 60 1/2 semi-diameters of the earth, near to what others have assigned. Let us assume the mean distance of 60 diameters in the syzygies; and suppose one revolution of the moon, in respect to the fixed stars, to be completed in 27d. 7h. 43', as astronomers have determined; and the circumference of the earth to amount to 123,249,600 Paris feet, as the French have found by mensuration. And now, if we imagine the moon, deprived of all motion, to be let go, so as to descend towards the earth with the impulse of all that force by which (by Cor. Prop. iii.) it is retained in its orb, it will in the space of one minute of time describe in its fall 15 1/12 Paris feet. For the versed sine of that arc which the moon, in the space of one minute of time, would by its mean motion describe at the distance of sixty semi-diameters of the earth, is nearly 15 1/12 Paris feet, or more accurately 15 feet, 1 inch, 1 line 4/9. Wherefore, since that force, in approaching the earth, increases in the reciprocal-duplicate proportion of the distance, and upon that account, at the surface of the earth, is 60 x 60 times greater than at the moon, a body in our regions, falling with that force, ought in the space of one minute of time to describe 60 x 60 x 15 1/12 Paris feet; and in the space of one second of time, to describe 15 1/12 of those feet, or more accurately, 15 feet, 1 inch, 1 line 4/9. And with this very force we actually find that bodies here upon earth do really descend; for a pendulum oscillating seconds in the latitude of Paris will be 3 Paris feet, and 8 lines 1/2 in length, as Mr. Huygens has observed. And the space which a heavy body describes by falling in one second of time is to half the length of the pendulum in the duplicate ratio of the circumference of a circle to its diameter (as Mr. Huygens has also shown), and is therefore 15 Paris feet, 1 inch, 1 line 4/9. And therefore the force by which the moon is retained in its orbit is that very same force which we commonly call gravity; for, were gravity another force different from that, then bodies descending to the earth with the joint impulse of both forces would fall with a double velocity, and in the space of one second of time would describe 30 1/6 Paris feet; altogether against experience."[1]

All this is beautifully clear, and its validity has never in recent generations been called in question; yet it should be explained that the argument does not amount to an actually indisputable demonstration. It is at least possible that the coincidence between the observed and computed motion of the moon may be a mere coincidence and nothing more. This probability, however, is so remote that Newton is fully justified in disregarding it, and, as has been said, all subsequent generations have accepted the computation as demonstrative.

Let us produce now Newton's further computations as to the other planetary bodies, passing on to his final conclusion that gravity is a universal force.

"PROPOSITION V., THEOREM V.

"That the circumjovial planets gravitate towards Jupiter; the circumsaturnal towards Saturn; the circumsolar towards the sun; and by the forces of their gravity are drawn off from rectilinear motions, and retained in curvilinear orbits.

"For the revolutions of the circumjovial planets about Jupiter, of the circumsaturnal about Saturn, and of Mercury and Venus and the other circumsolar planets about the sun, are appearances of the same sort with the revolution of the moon about the earth; and therefore, by Rule ii., must be owing to the same sort of causes; especially since it has been demonstrated that the forces upon which those revolutions depend tend to the centres of Jupiter, of Saturn, and of the sun; and that those forces, in receding from Jupiter, from Saturn, and from the sun, decrease in the same proportion, and according to the same law, as the force of gravity does in receding from the earth.

"COR. 1.—There is, therefore, a power of gravity tending to all the planets; for doubtless Venus, Mercury, and the rest are bodies of the same sort with Jupiter and Saturn. And since all attraction (by Law iii.) is mutual, Jupiter will therefore gravitate towards all his own satellites, Saturn towards his, the earth towards the moon, and the sun towards all the primary planets.

"COR. 2.—The force of gravity which tends to any one planet is reciprocally as the square of the distance of places from the planet's centre.

"COR. 3.—All the planets do mutually gravitate towards one another, by Cor. 1 and 2, and hence it is that Jupiter and Saturn, when near their conjunction, by their mutual attractions sensibly disturb each other's motions. So the sun disturbs the motions of the moon; and both sun and moon disturb our sea, as we shall hereafter explain.

"SCHOLIUM

"The force which retains the celestial bodies in their orbits has been hitherto called centripetal force; but it being now made plain that it can be no other than a gravitating force, we shall hereafter call it gravity. For the cause of the centripetal force which retains the moon in its orbit will extend itself to all the planets by Rules i., ii., and iii.

"PROPOSITION VI., THEOREM VI.

"That all bodies gravitate towards every planet; and that the weights of the bodies towards any the same planet, at equal distances from the centre of the planet, are proportional to the quantities of matter which they severally contain.

"It has been now a long time observed by others that all sorts of heavy bodies (allowance being made for the inability of retardation which they suffer from a small power of resistance in the air) descend to the earth FROM EQUAL HEIGHTS in equal times; and that equality of times we may distinguish to a great accuracy by help of pendulums. I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes hanging by eleven feet, made a couple of pendulums exactly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations. And therefore the quantity of matter in gold was to the quantity of matter in the wood as the action of the motive force (or vis motrix) upon all the gold to the action of the same upon all the wood—that is, as the weight of the one to the weight of the other: and the like happened in the other bodies. By these experiments, in bodies of the same weight, I could manifestly have discovered a difference of matter less than the thousandth part of the whole, had any such been. But, without all doubt, the nature of gravity towards the planets is the same as towards the earth. For, should we imagine our terrestrial bodies removed to the orb of the moon, and there, together with the moon, deprived of all motion, to be let go, so as to fall together towards the earth, it is certain, from what we have demonstrated before, that, in equal times, they would describe equal spaces with the moon, and of consequence are to the moon, in quantity and matter, as their weights to its weight.

"Moreover, since the satellites of Jupiter perform their revolutions in times which observe the sesquiplicate proportion of their distances from Jupiter's centre, their accelerative gravities towards Jupiter will be reciprocally as the square of their distances from Jupiter's centre—that is, equal, at equal distances. And, therefore, these satellites, if supposed to fall TOWARDS JUPITER from equal heights, would describe equal spaces in equal times, in like manner as heavy bodies do on our earth. And, by the same argument, if the circumsolar planets were supposed to be let fall at equal distances from the sun, they would, in their descent towards the sun, describe equal spaces in equal times. But forces which equally accelerate unequal bodies must be as those bodies—that is to say, the weights of the planets (TOWARDS THE SUN must be as their quantities of matter. Further, that the weights of Jupiter and his satellites towards the sun are proportional to the several quantities of their matter, appears from the exceedingly regular motions of the satellites. For if some of these bodies were more strongly attracted to the sun in proportion to their quantity of matter than others, the motions of the satellites would be disturbed by that inequality of attraction. If at equal distances from the sun any satellite, in proportion to the quantity of its matter, did gravitate towards the sun with a force greater than Jupiter in proportion to his, according to any given proportion, suppose d to e; then the distance between the centres of the sun and of the satellite's orbit would be always greater than the distance between the centres of the sun and of Jupiter nearly in the subduplicate of that proportion: as by some computations I have found. And if the satellite did gravitate towards the sun with a force, lesser in the proportion of e to d, the distance of the centre of the satellite's orb from the sun would be less than the distance of the centre of Jupiter from the sun in the subduplicate of the same proportion. Therefore, if at equal distances from the sun, the accelerative gravity of any satellite towards the sun were greater or less than the accelerative gravity of Jupiter towards the sun by one-one-thousandth part of the whole gravity, the distance of the centre of the satellite's orbit from the sun would be greater or less than the distance of Jupiter from the sun by one one-two-thousandth part of the whole distance—that is, by a fifth part of the distance of the utmost satellite from the centre of Jupiter; an eccentricity of the orbit which would be very sensible. But the orbits of the satellites are concentric to Jupiter, and therefore the accelerative gravities of Jupiter and of all its satellites towards the sun, at equal distances from the sun, are as their several quantities of matter; and the weights of the moon and of the earth towards the sun are either none, or accurately proportional to the masses of matter which they contain.

"COR. 5.—The power of gravity is of a different nature from the power of magnetism; for the magnetic attraction is not as the matter attracted. Some bodies are attracted more by the magnet; others less; most bodies not at all. The power of magnetism in one and the same body may be increased and diminished; and is sometimes far stronger, for the quantity of matter, than the power of gravity; and in receding from the magnet decreases not in the duplicate, but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations.

"PROPOSITION VII., THEOREM VII.

"That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain.

That all the planets mutually gravitate one towards another we have proved before; as well as that the force of gravity towards every one of them considered apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence it follows, that the gravity tending towards all the planets is proportional to the matter which they contain.

"Moreover, since all the parts of any planet A gravitates towards any other planet B; and the gravity of every part is to the gravity of the whole as the matter of the part is to the matter of the whole; and to every action corresponds a reaction; therefore the planet B will, on the other hand, gravitate towards all the parts of planet A, and its gravity towards any one part will be to the gravity towards the whole as the matter of the part to the matter of the whole. Q.E.D.

"HENCE IT WOULD APPEAR THAT the force of the whole must arise from the force of the component parts."

Newton closes this remarkable Book iii. with the following words:

"Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centre of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes used to do), but according to the quantity of solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always in the duplicate proportions of the distances. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding from the sun decreases accurately in the duplicate proportion of the distances as far as the orb of Saturn, as evidently appears from the quiescence of the aphelions of the planets; nay, and even to the remotest aphelions of the comets, if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypothesis; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. . . . And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies and of our sea."[2]

The very magnitude of the importance of the theory of universal gravitation made its general acceptance a matter of considerable time after the actual discovery. This opposition had of course been foreseen by Newton, and, much as be dreaded controversy, he was prepared to face it and combat it to the bitter end. He knew that his theory was right; it remained for him to convince the world of its truth. He knew that some of his contemporary philosophers would accept it at once; others would at first doubt, question, and dispute, but finally accept; while still others would doubt and dispute until the end of their days. This had been the history of other great discoveries; and this will probably be the history of most great discoveries for all time. But in this case the discoverer lived to see his theory accepted by practically all the great minds of his time.

Delambre is authority for the following estimate of Newton by Lagrange. "The celebrated Lagrange," he says, "who frequently asserted that Newton was the greatest genius that ever existed, used to add—'and the most fortunate, for we cannot find MORE THAN ONCE a system of the world to establish.' " With pardonable exaggeration the admiring followers of the great generalizer pronounced this epitaph:

"Nature and Nature's laws lay hid in night;

God said `Let Newton be!' and all was light."

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