Popular Science Monthly/Volume 59/September 1901/The Discovery of the Law of Gravitation
|THE DISCOVERY OF THE LAW OF GRAVITATION.|
By the late Professor JOHN T. DUFFIELD,
THE law of gravitation, that all matter attracts all matter, directly as the mass and inversely as the square of the distance, whether we consider the extent of its reach, or the number and variety and peculiar interest of the problems of which it furnishes the solution, or the grandeur of many of those problems by reason of the magnitude of the elements involved; whether we consider the power which it gives us to anticipate nature, and predict with the minutest accuracy and certainty of a mathematical demonstration celestial phenomena for ages to come; we cannot but regard it as the most important truth in the whole book of nature and its discovery as the most interesting event in the history of physical science. As there is but one material universe and the law of gravitation solves the enigma of its structure, no other problem of equal interest and importance can ever occupy the attention of the student of nature.
Kepler has remarked that: "The occasions by which men have acquired a knowledge of celestial phenomena, are not less admirable than the discoveries themselves." If this be so, the history of the discovery of that great law of nature by which all celestial phenomena are determined can never cease to be a matter of peculiar interest.
In the account which we propose to give of the discovery we shall select as our chronological starting point the beginning of the seventeenth century. At that period the theory in regard to the structure of the material universe which, with few exceptions had been held from time immemorial, still prevailed. The earth was regarded as the center of the universe, about which the sun, moon, planets and stars performed their ceaseless revolutions. More than half a century before (in 1543) Copernicus, in his memorable work, 'De Orbium Cœlestium Revolutionibus,' had, indeed, announced the true system of the universe, yet as he was led to the adoption of the theory he proposed, not so much by positive evidence in its favor as by the difficulty of reconciling certain phenomena with the Ptolemaic theory; moreover, as the objections to this theory were from their very nature such that few could appreciate their force, whilst in the apparent motions of the heavenly bodies every one could see what seemed to be an ocular demonstration of its truth, it is not strange that the doctrine of Copernicus should have been for so long a time generally regarded as an interesting yet fanciful speculation. It remained for a subsequent age to furnish proof of the truth of the Copernican system which could not be gainsaid or resisted.
At the beginning of the seventeenth century, the dicta of Aristotle, in regard to matters of science as well as philosophy, were still accepted, as they had been for many centuries preceding, as of infallible authority. In regard to the subject of our inquiry, he taught that bodies at the surface of the earth fell or tended to fall toward the center of the earth, not in virtue of any attraction of the earth, but in virtue of the fact that the center of the earth was the center of the material universe—that if the earth itself should be moved out of its place and then left free to move, it would return to its place by the same law of nature which controlled all terrestrial bodies. He taught moreover that celestial bodies were different in kind from bodies terrestrial—that whilst the latter were imperfect, corruptible and changeable, the former were perfect, (and therefore, according to his fancy, perfectly spherical in form) incorruptible, unchangeable and self-luminous. Being different in kind, he held that they were subject to entirely different physical laws; that whereas the motion of terrestrial bodies when free to move was rectilinear, by a necessity of their nature, the motion of celestial bodies was circular by a like necessity of their nature. His language on this point is worth quoting as an illustration of the contrast between the ancient and modern method of philosophizing in regard to natural phenomena. He says: "All simple motion must be rectilinear or circular; either to a center or from a center, each of which is rectilinear, or about a center. It is natural for two of the elements—earth and water—which are heavy, to tend to a center; two—air and fire—which are light, to tend from a center. As the motion of all terrestrial elements is therefore rectilinear, it seems reasonable that celestial bodies, which are of a different nature, should have only the other simple motion possible, namely, circular motion."
The year 1609 marks a new era in the history of astronomy. In this year two events occurred, independent, yet alike memorable as contributing to the overthrow of the theory in regard to the structure of the material universe which had previously prevailed and establishing the doctrine of Copernicus upon an immovable foundation. The invention of the telescope by Galileo, and the immediate discovery by means of it of the inequalities of the moon's surface, the phases of Venus, the satellites of Jupiter and the rings of Saturn, at once annihilated the fancies of Aristotle as to the perfectly spherical form of the planets, their self-luminosity, and their difference in kind from bodies terrestrial. The other memorable event referred to was the publication of Kepler's great work on 'The Motions of Mars,' in which, with much that was fanciful, two of the three laws of planetary motion were for All the first time announced. Some twelve years later, in his work, entitled 'Harmonies' he announced the third law of planetary motion, fully establishing his right to the title, by which he has since been distinguished 'The Legislator of the Heavens.'
These laws of Kepler are: First, that the orbits of the planets are elliptical, the sun being at one of the foci; second, that the radius vector, that is, a line drawn from a planet to the sun, passes over equal spaces in equal times; third, that the squares of the times of revolution of the different planets, are to each other as the cubes of the mean distances from the sun.
Together with these laws of planetary motion, two of the three axioms of the science of mechanics, known as the Laws of Motion, were about this time discovered, or rather, were now for the first time distinctly apprehended and enunciated. The first of these was given by Kepler—the law of inertia, namely, that a body will persevere in the state in which it is, whether of rest or motion, until it is acted on by some force; or more precisely, a body at rest will continue at rest until acted on by some force, and when acted on by any single force, if free to move, its motion will be rectilinear, uniform and continuous until the body is acted on by some other force. The second law of motion was announced by Galileo, and is known as the law of the coexistence of motions, or independence of forces. It may be expressed as follows: If a body be acted on by several forces simultaneously, it will obey the impulse of each force, just as it would if the others were not acting. [The simplest illustration of this law is what is known as the parallelogram of forces. If the direction and intensity of two forces acting simultaneously on a body be represented by the sides of a parallelogram, the body will describe the diagonal of the parallelogram; that is, at the end of a unit of time the body will be just where it would have been if the forces had, each for a unit of time, acted consecutively.]
The true system of the universe, the laws of planetary motion and the fundamental principles of mechanics having become known, for the first time in the history of the race any intelligent inquiry as to the physical causes of the motions of the heavenly bodies became possible. With earnestness and assiduity proportioned to the interest and grandeur of the problem, men of science at once applied themselves to its solution, and yet half a century of gradual progress elapsed before the desired result was reached. From the facts which we shall have occasion to mention it will appear how much, or rather how little, foundation there is for the common belief that the idea of the law of gravitation was wholly original with Newton—suggested to him for the first time by observing the fall of an apple, and then suddenly coming forth from his brain like Minerva from the head of Jove, unheralded and complete. The ordinary method of transition from widespread and plausible error to the truth is by slow and gradual progress, and the discovery of the law of gravitation, so far from being an exception to this rule, is but one of its most striking illustrations. Such an accident as that which the discovery of the law of gravitation is supposed to have been is of the kind which only happens to men of large knowledge, profound thought, and after intense and protracted mental effort. Simple as this law is known to be, and easily apprehended and even demonstrated by ordinary minds, it needed one endowed with the most gigantic intellect probably ever given to mortal—availing himself of the suggestions and the results of the labors of those who had preceded him in the same field of inquiry—to make the discovery.
In tracing the history of this discovery, from the epoch when by the previous discovery of all the necessary data it for the first time became possible, the first place in the order of time, and next to Newton in the order of merit, is undoubtedly due to Kepler. Possessing a singularly lively imagination—we might say, volatile fancy—combined with a love for the truth that amounted to a ruling passion, and a breadth of knowledge in his favorite science far in advance of any other man of his age, he was eminently fitted for the work which he so successfully performed of scientific discovery. Fertile in hypotheses—sometimes the most extravagant—he was indefatigable in his labors to test his hypotheses by the facts. Without the slightest pride of opinion, he seemed to take a satisfaction in exploding his own theories when they were false, that was only exceeded by his delight when successful in demonstrating their truth. Of the men who have contributed to the advancement of science, there are few to whom we are under greater obligation, or whose character as an investigator of nature is more worthy of admiration, than 'The Legislator of the Heavens'—the father of modern astronomy.
In the introduction to his memorable work on 'The Motions of Mars,' referred to above, he opposed the doctrine of Aristotle on the subject of terrestrial gravity, and in the course of the discussion uses the following remarkable language:
A mathematical point, whether the center of the universe, or not, has no power to move heavy bodies to approach it. Let philosophers prove, if they can, that natural things have any sympathy with that which is nothing.
If the earth were not round, heavy bodies would not tend from every side toward its center, but to different points from different sides.
If two stones were placed in any part of the universe, near each other, and beyond the sphere of the influence of a third cognate body, these stones would come together at an intermediate point, each approaching the other at a distance proportional to the comparative mass of the other.
If the moon and the earth were not retained in their orbits by their annual force, or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance from each other, and the moon would fall toward the earth through the other fifty-three parts, that is, assuming that the substance of the earth is of the same density.
The sphere of the attractive virtue which is in the moon, extends to the earth and entices up the waters, but as the moon flies rapidly across the zenith and the waters cannot follow so quickly, a flow of the ocean is occasioned toward the westward.If the attractive virtue of the moon extends to the earth, it follows, with greater reason, that the attractive virtue of the earth extends to the moon and much farther, and in short, nothing which consists of earthy substance, however constituted, although thrown up to any height, can ever escape the powerful operation of this attractive virtue.
These views of Kepler—so novel at the time they were announced by him and yet which we now know to be in the main so correct—were published more than thirty years before Newton was born. As we read them, our first feeling is one of surprise that any subsequent investigator of the phenomena of gravitation should be able, by his discoveries, to achieve for himself a fame which should not only render his name immortal but should almost wholly hide from view the merit of the great pioneer in this field of inquiry. To appreciate, however, the importance of the work which yet remained to be performed, we should bear in mind that whilst Kepler's views in regard to terrestrial gravity were so remarkably just, he at the same time, in common with the age in which he lived, and, we may say, with all preceding ages—regarded the tendency of bodies near the earth to fall toward its center, and the motions of heavenly bodies, as entirely different phenomena and not at all referable to the same physical cause. He indeed speculated on the possibility of referring the motions of the planets to an attractive force emanating from the sun, similar to that which caused bodies near the earth to tend toward its center, and concluded that such a hypothesis was untenable, inasmuch as the motion in one case was rectilinear, and in the other curvilinear. Again, not to overestimate the merit of Kepler in connection with the discovery of the law of gravitation, we should remember that a theory as to the physical cause of natural phenomena, even if it be in the main correct, will furnish no complete solution of the problems which phenomena present, unless it express accurately and precisely the measure as well as the mode of the action of the assigned cause. For example, to know merely that all matter attracts all matter, would not enable us to explain the phenomena of gravitation; we need to know precisely how the intensity of this attraction is affected by the comparative magnitude of the masses and by the distance of the masses from each other. Now the theory of Kepler in regard to gravity was correct as to the first of these points, namely that the intensity of this attraction was directly as the mass, but he was in error in regard to the second point, as he supposed that the intensity of the attraction was inversely as the distance, instead of what was subsequently found to be the fact, the square of the distance.
Once more, to estimate at its just value the part which Kepler performed in the discovery of the laws of gravitation, we should bear in mind, that an hypothesis, even if subsequently it be found to be correct, is of no authority until its truth be demonstrated. It may be of great importance, by way of suggestion, in directing the labors of subsequent inquirers, but the chief merit of the discovery of the truth is due to the individual who furnishes its demonstration. When this is done, and not before, that which was previously but an hypothesis takes its place among the recognized laws of nature.
As in Kepler's day, the tendency of bodies near the earth to fall toward its center and the motions of the heavenly bodies were regarded as phenomena of entirely different laws of nature, his views as to the physical cause of planetary motion next claim our attention. He supposed that the motion of the planets in their orbits was due to an influence emanating from the sun, but assuming that if this influence were an attractive force, similar to terrestrial gravity, its effect would be to cause the planets to fall toward the sun in straight lines, instead of the actual motion of revolution about the sun; he supposed that the emanation was of a corporeal nature, somewhat analogous to light; that as the sun revolved on its axis, this emanation revolved with it just as the spokes of a wheel when the hub revolves, and that the planets were swept along in their orbits by the revolution of this emanation—the force which caused them to move being a propulsion and not an attraction. As the hypothesis would seem to require that the times of revolution of all the planets should be the same, whereas they are different, the nearer performing their annual revolution in a time less than the more remote—he supposed that the density of the emanation diminished as its distance from the sun increased; that consequently its virtue, or propulsive energy, diminished in like manner, just as the intensity of light diminishes with the increase of the distance from the luminous center. This would account in a general way for the fact that the times of revolution of the planets nearer the sun are shorter than the times of revolutions of those more remote, but the precise difference in the observed times of revolution was not exactly that which would be required by the hypothesis. Moreover, he had observed that the orbits of the planets were not circular, as would seem to be required by his hypothesis, but elliptical, the sun being at one of the foci; also, the ever-varying radius vector always passed over equal spaces in equal times, hence the motion of the planet in its orbit was not uniform, as his hypothesis would require, but ever-varying; and this variation too was evidently not fortuitous or uncertain but increased or diminished in the exact ratio to the varying distance of the planet from the sun required by the law just mentioned, of equal spaces in equal times. These facts, apparently so inconsistent with his hypothesis, Kepler accounted for by supposing that each of the planets was animated by an intelligent spirit, by whose agency the motion of the planet was, in part at least, determined. We have seen an allusion to this theory in the quotation above given, on the subject of gravity. He regarded each of the heavenly bodies, and the earth as one of them, as literally a huge animal, and in one of his works describes with some minuteness the habits of that particular animal on whose body it is our lot to live.
Kepler's hypothesis of an emanation from the sun of a corporeal nature by whose revolution the planets were propelled in their orbits was received with more or less favor for a time, but was soon superseded by another memorable hypothesis no more reasonable or plausible and yet from the time of its announcement until the publication of 'The Principia' demonstrated its fallacy, it was adopted by most men of science and may be said to have been the accepted theory on the subject. We refer to the Vortices of Descartes. This distinguished philosopher, born 1596, rose to eminence about the time of Kepler's death, which occurred in 1630. By the force of his genius, illustrated not only by that achievement for which his name will ever be held in honored remembrance—the invention of analytical geometry—but by the abundance and ability of his labors in every department of science and philosophy, Descartes, for more than half a century, occupied a position in the learned world scarcely inferior to that which for ages preceding had been held by Aristotle.
As to the cause of planetary motion, Descartes assumed the existence, throughout the limits of our system, of a subtle transparent fluid in ceaseless revolution about the sun as its center, and that the planets floated in this fluid and were consequently carried round by the sun in its motion, just as in a whirlpool a cork or floating body is carried round by the motion of the water. To account for the difference in the times of revolution of different planets, he supposed that the velocity of the revolution of the fluid, at different distances from the sun was different. To account for the revolution of the satellites of the planets, he assumed that in the hypothesis there are several fatal objections, as was subsequently demonstrated by D'Alembert, of which it will be sufficient to mention that the very existence of a spherical vortex is a mechanical impossibility. And yet such was the weight of the authority of its author, and the ingenuity with which it was defended by himself and his followers, that, as was mentioned above, it not only was received with general favor but for more than half a century it was accepted by most men of science without questioning and continued to be maintained by some, even after Newton had announced and demonstrated the law of gravitation. It is a notable illustration of the tenacity of error when once it becomes firmly fixed and widespread, that for some years after the publication of 'The Principia,' a Latin translation from the French of 'The Physics of Rohault'—a work entirely Cartesian—continued to be the text-book in Philosophy at the University of Cambridge—Newton himself being at the time Lucasian Professor of Mathematics. We have the authority of Play fair for the statement (which, indeed, has been called in question by Sir David Brewster, in his 'Life of Newton,' though so far as we have been able to see, without any sufficient reason) that the doctrines of 'The Principia' were introduced into the regular course of instruction at Cambridge by strategem. Dr. Samuel Clarke, a zealous advocate of the Newtonian Philosophy, prepared a new and more elegant translation of Rohault, with copious notes, in which the doctrines of 'The Principia' were explained and defended, and it was by this work, more directly than by the lectures of Newton himself, that Cartesianism was finally driven from the University.of each planet this fluid revolved about the planet as a center. To this purely fanciful
Whilst Kepler's speculations as to the cause of the motions of heavenly bodies were soon supplanted by the hypothesis of Descartes, his more just views in regard to terrestrial gravity commended themselves to the scientific world and speedily passed into universal and abiding favor. In the memorable work of Galileo on the true system of the universe—completed the very year after Kepler's death, and published two years after; a work which, aside from its own merit, 'The Holy Inquisition,' by the persecution of its author, has made immortal—we find the doctrine of Kepler, on the subject of gravity, distinctly stated and elaborately defended. The Inquisition had power to imprison Galileo and commit copies of his work to the flames, but the truth it contained could not be burnt or bound. The earth 'still moved,' and matter continued to attract matter, unawed by the terrors of the Inquisition. The truth, once distinctly apprehended and announced, was never again to be lost, but was destined to grow in importance and be extended in its application far beyond the conceptions even of the great prophets of nature who were the first to proclaim it. The doctrine of Kepler on the subject of gravity may be regarded as. historically, the foundation of that sublime superstructure which in a subsequent age was reared by Newton, and which, by reason of the magnitude of its proportions and the multiplicity of its details, all pervaded and determined by the most admirable unity, now stands and in all probability will ever stand, as the most imposing monument ever erected by the human intellect.
Although Kepler's theory, that bodies terrestrial mutually attracted each other, met with ready reception, more than thirty years elapsed after the publication of this work before the idea was entertained, at least favorably, of accounting for the revolutions of the heavenly bodies on the theory of the universality of the attraction of gravitation. Kepler indeed, as we have remarked above, alludes to such an hypothesis only however to expose, as he imagined, its fallacy. The motions of the heavenly bodies being curvilinear, whilst the motions of bodies under the influence of gravity were rectilinear, it was taken for granted as a thing self-evident that the two phenomena must be due to entirely different physical causes. Familiar as we are with the fact, that by the two laws of motion above mentioned, the hypothesis of an attractive force of the sun, combined with the hypothesis of a tendency of the planets to move in a straight line in virtue of an original impulse communicated to them, would satisfactorily and readily account for their curvilinear motion, it cannot but be a matter of surprise that the truth should have remained so long unrecognized.
The credit of having been the first to generalize the idea of gravity, and refer the revolutions of the heavenly bodies to the attraction of matter for matter, appears to be due to Borelli, an Italian philosopher, a pupil of Galileo. It is announced in a work which he published "On the Satellites of Jupiter,' in 1666, although, as we shall have occasion to notice subsequently, Newton had conceived the same idea at least as early as 1665. Both Newton and Huyghens, however, attributed to Borelli the honor of having been the first to announce the important truth.
The idea, having been suggested, was at once accepted by many with favor and immediately led to the investigation of a hitherto unexplored field in the department of mechanical philosophy. Whilst the labors of others in this field were not unimportant, particularly those of Wallis, the name which is especially deserving of honorable mention in this connection is that of Huyghens. In a work published in 1672, we meet for the first time with a scientific discussion of the doctrine of Central Forces. His investigations were remarkably satisfactory and complete as to the phenomena of circular motion, the attractive force being at the center and contributing largely to the success of the labors of subsequent inquirers.
A great step had been taken toward the solution of the problem of planetary motion, but a formidable difficulty yet remained to be overcome. The orbits of the planets were not circular, but elliptical, and the sun—the center of the attractive force—was not at the center of the ellipse, but at one of the foci. For the complete solution of the actual problem which the phenomena presented, a calculus was needed which neither Borelli nor Huyghens possessed, and the preeminent genius of Newton was illustrated, probably more by the invention of the needed calculus than by his successful application of it to the solution of the important problem in question.
The general fact having been established that the curvilinear motion of the heavenly bodies was explicable on the hypothesis of a central attractive force, it was soon surmised that the particular character of the planetary orbits—involving as it did a continual variation in the distance of each planet from the sun, as well as a continual variation in the velocity of the planet's motion—could be due to no other cause than a difference in the intensity of the sun's attractive force at different distances. The query was: What was the precise law of this variation in intensity, which would account for the phenomena? Was the attraction inversely as the distance? or, as the square of the distance? or, as the cube? or, was it such as admitted of any precise expression? Guided probably by the best known fact as to the distribution of light, of heat, indeed of any emanation radiating in all directions from a center, several individuals, independently as it would seem, adopted the conclusion which was afterwards demonstrated to be correct, namely: That the attractive force of matter for matter varied inversely as the square of the distance, that is, at double the distance the attraction is one-fourth, at treble the distance one-ninth, and so on. The first to announce the true law of variation in the intensity of attraction was a French philosopher, Bouilland, or as his name ordinarily appears in the Latinized form, Bullialdus. About the same time. Sir Christopher Wren, the distinguished architect of St. Paul's, Dr. Hooke, for a long time secretary of the Royal Society, and the eminent mathematician astronomer, Halley, had arrived at the same conclusion. It was still however but a conjecture. In spite of the most earnest and persevering effort no one was able to furnish a demonstration.
As contributing to the discovery of the demonstration, the place of merit next to that of Newton, though of course far inferior, is doubtless due to Hooke. His labors were probably of aid to Newton by way of suggestion, without however affording any just ground for the charge which Hooke subsequently made that Newton was wearing the laurels to which he himself was justly entitled. As early as 1666 Hooke exhibited in the presence of the Royal Society an experiment now quite familiar but at this time new and of exceeding interest. He suspended from the ceiling a long wire to the end of which a ball of wood was attached—a simple pendulum on a large scale. On removing the pendulum from the vertical position and then giving it a lateral impulse at right angles to the plane in which it tended to oscillate, the ball described an ellipse—the eccentricity of the ellipse varying with a variation of the intensity of the lateral impulse. An ocular demonstration was thus given of the important fact that elliptical motion could be produced by the combined action of two forces—one impulsive and the other central—and that the particular form of the ellipse depended upon the relative intensities of the two forces. Although in the experiment the attractive force was at the center of the ellipse, whilst in the case of planetary motion it was at one of the foci, still the fact exhibited must have been highly suggestive to any subsequent inquirer as to the cause of planetary motion.
In 1674 Hooke published a dissertation entitled 'An Attempt to prove the motion of the earth by observations,' in which he says: "I shall hereafter explain a system to the world, differing in many particulars from any yet known, depending upon three suppositions." The first—which he gave at some length—is a distinct statement of the universality of the attraction of gravitation. The second is substantially Kepler's law of inertia. The third is "that the attractive powers of the heavenly bodies are so much the more powerful, by how much nearer the body wrought upon is to their own centers." And, he adds, "Now what these several degrees are, I have not yet experimentally verified, but it is a notion which, if fully prosecuted, as it ought to be, will mightily assist the astronomers to reduce all celestial motions to a certain rule, which, I doubt not, will never be done without it." From this declaration it is evident, first, that at this time he was still in doubt as to the true law of gravitation; and second, that he was endeavoring to discover it by experiment—a method by which he could never have arrived at the truth. A few years later, as appears from his correspondence with Newton, Wren and Halley, he was fully convinced that the intensity of the attraction of gravitation was inversely as the square of the distance, and he even professed to be able to furnish a demonstration. In this he was either insincere at the time or discovered subsequently that his supposed demonstration was defective, as he never presented it, though repeatedly urged by Wren and Halley to do so.
We are now prepared to understand and appreciate aright the precise work which Newton performed in connection with the discovery of the law of gravitation. Born on Christmas day of the year 1642, the year in which Galileo died, in 1665 we find Newton a student of Trinity College, Cambridge, which he had entered in 1660. But twenty-three years of age, he had already not only mastered all of value that had previously been written on Mathematics, Astronomy and Natural Philosophy, but he had discovered the Binomial Theorem, and had conceived and to an extent developed the Differential Calculus—an achievement with which few other events in the history of science deserve to be compared, after we except his own subsequent brilliant discoveries in Optics, and his successful application of the calculus to the discovery of the law and explanation of many of the most interesting phenomena of gravitation. In the summer of 1665 he left Cambridge on account of the plague which prevailed there at the time and returned to his native town of Woolsthorpe in Lancashire. It was during this visit to Woolsthorpe that the famous incident occurred which, as is generally supposed, first suggested to him the idea of gravitation and was the occasion of his great discovery. The account of it is given by his contemporary and friend Pemberton. One day as he was sitting under an apple tree in the garden an apple fell before him. This turned the currents of his thoughts and led him to reflect upon the nature of that mysterious influence which urges all terrestrial bodies toward the center of the earth, causing them, when free to move, to fall with a constantly accelerated velocity, which continues to act moreover without sensible diminution in intensity at the top of the highest towers or even the summit of the loftiest mountain. The thought was suggested to his mind, why may not this power extend to the moon? And if so, is not this the influence which retains her in her orbit round the earth? He at once applied himself to the determination if possible of the truth of this conjecture. If the moon were really retained in her orbit by terrestrial gravity, he concluded that the planets were probably retained in their orbits by a similar influence of the sun. Moreover, if the attractive influence of the earth extended to the moon and that of the sun to the farthest limits of our system, he concluded that the intensity of the attraction in each case diminished as the distance from the center of attraction increased. If this were so, it would manifest itself by a difference in the velocities of the planets, they being at different distances from the sun, and he accordingly inferred that by a comparison of the velocities of the motions of the several planets with each other, the law of variation of the intensity of the attractive force might be determined. Kepler's third law, that the squares of the times are as the cubes of the mean distances, furnished him at once with the necessary data for the calculation. He was not at the time able to solve the precise problem which the actual phenomena presented, the planetary orbits being elliptical and the attractive force at one of the foci, but assuming the orbits to be circular and the attractive force at the center, he found that Kepler's law would follow, if the variation in the intensity of the attraction were inversely as the square of the distance.
It deserves to be noticed, that to solve even this problem, Newton must at the time have been familiar with the doctrine of central forces, though Huyghens' work on that subject was not published until more than six years after.
Though the data which Newton assumed were not precisely those which the planetary systems presented, the result reached was highly interesting and calculated to encourage and direct further inquiry. The next question to be determined was, the law of the variation of the earth's attraction—Was this also inversely as the square of the distance? If so, the universality of the attraction of gravitation varying in intensity according to the law just mentioned, would be almost indubitable.
The method by which Newton undertook to determine the variation of the earth's attractive influence—so simple when once suggested—was entirely original with him, and is one, though but one, of the grounds for attributing to him preeminently the honor of the discovery of the law of gravitation. Hooke, and doubtless others, subsequently labored for years to determine whether the intensity of the earths attraction diminished with an increase of the distance from the center, and if so, according to what law, and yet all their efforts were fruitless. Newton's method was simply this, assuming the supposed distance of the moon from the earth to be correct, the length of the entire orbit of the moon may be readily determined. Moreover, the time of a complete revolution of the moon about the earth being known, the arc which she describes in one minute of time becomes known. Regarding this arc, which differs but little from a straight line, as the diagonal of a parallelogram, by the parallelogram of forces one of the sides of this parallelogram would represent the distance which the moon actually falls toward the earth under the influence of the earth's attraction in one minute of time. The arc just mentioned being known, this distance, which is the versed sine of the arc, may be readily determined. A measure is thus obtained of the intensity, at the moon, of the earth's attraction. By comparing this with the intensity of the attraction at the surface of the earth, as indicated by the distance a body near the surface will fall in one minute, the law of the variation in the intensity may be determined. Upon making the necessary computations the result was not just that which Newton anticipated, or rather hoped for. The distance which the moon ought to have fallen in one minute, according to the hypothesis, was one-sixth greater than that which, as it appeared, she actually did fall. Most men would have regarded this discrepancy as of little account, and accepting the result as, for the time at least, a sufficiently accurate demonstration of the hypothesis, would at once have given it publicity. Newton, however, though he could not but feel assured that the true law of gravitation was indicated in the result he had reached, with that singular reticence as to his labors and indifference to fame which were among the marked features of his character, not only did not publish his investigations but did not even in his correspondence with his friends allude to the subject. For more than thirteen years he does not appear to have made any further progress toward the solution of the problem of gravitation. Though his attention was doubtless at times directed to it, he was mainly occupied during this period with other scientific labors, particularly in investigating the phenomena of light, making many brilliant discoveries on this subject which, even if he had not subsequently discovered the law of gravitation, would have entitled him to a distinction among men of science scarcely inferior to that which is now awarded him.
In 1679, after Bouilland, Hooke, Wren, Halley and others had become well convinced of the true law of gravitation and yet were unable to furnish a demonstration of it, Newton was led to a renewed investigation of the subject. Hooke had for some time been investigating the motion of projectiles, and in a letter to Newton about this time asserted that a body acted on by an impulsive force and at the same time by an attractive force varying in intensity inversely as the square of the distance, would describe an ellipse. What proof Hooke had of the fact asserted does not appear. It may be regarded as certain that he was not able to give a mathematical demonstration of it. As he had become well convinced that the attraction of gravitation varied according to the law mentioned, it is altogether probable that the main if not the sole ground for his assertion, was the fact that the orbits of the planets are elliptical. However this may be, Newton at once appreciated the importance of the assertion if it could be demonstrated, and was led to attempt the solution of the problem suggested by Hooke's or rather the converse problem, namely, to determine the law of variation in intensity of a central force which would cause the body acted upon to describe an ellipse. By the aid of the calculus, which he had by this time considerably perfected, he finally succeeded, after long and laborious effort, in demonstrating in its most general form the truth of Hooke's assertion. The importance of the result cannot be over estimated. The enigma which the elliptical orbits of the planets had presented was solved, and not only the fact of the sun's attraction but the precise law of the variation in intensity of that attraction was at last established beyond the possibility of further doubt or questioning.
The demonstration of the universality of gravitation however was still incomplete. The sun indeed attracted the planets with a force varying inversely as the square of the distance, but was this a property common to all matter? Was it identical with the attraction of the earth which caused bodies near it to fall toward its center? Were the revolutions of the planets and the revolution of the moon phenomena referable to one and the same great law of nature? The result which Newton had reached in his investigations in 1665 seemed to render this doubtful, or at least presented a difficulty for the time inexplicable. Accordingly, with that characteristic reticence to which we have previously referred, Newton refrained from communicating to any one the important discovery he had made, preferring to await the solution of the difficulty which the anomalous fact of the apparent intensity of the earth's attraction at the moon presented.
Three years afterwards, in June, 1682, Newton attended a meeting of the Royal Society. Whilst in London, he accidentally learned that Picard in France had just measured the arc of the meridian with great accuracy, and that the result which he obtained for the length of a degree in that latitude differed somewhat from the measurement previously accepted as reliable. Newton at once perceived the importance of this fact in connection with the determination of the intensity of the earth's attraction. If the commonly received measure of a degree of the meridian was erroneous, the accepted estimate of the size of the earth was erroneous; moreover, if the assumed semi-diameter of the earth was incorrect, the supposed distance of the moon from the earth, in the calculation of which the earth's semi-diameter is involved, must also be incorrect. The possible explanation of the annoying result he had reached in 1665 was immediately suggested. Obtaining accurately the measurement of a degree of the meridian as given by Picard, immediately on his return to Cambridge he determined the size of the earth and the distance of the moon, on the supposition that Picard's measurement was the true one. With the data thus obtained he returned to the problem at which he had labored sixteen years before, and by the same method then pursued sought anew to determine the law of the variation of the earth's attraction. Perceiving, as he advanced in the calculation, the tendency of the numbers to produce the desired result, he became so much agitated that he was unable to finish the computation and was under the necessity of requesting a friend to do it for him. The identity of the force which causes bodies near the earth to fall toward its center and that which causes the heavenly bodies to revolve was fully established, the universality of the law of gravitation was finally and forever demonstrated, the solution of the grand problem of the universe was complete.
We might have supposed that Newton would have eagerly hastened to announce his great discovery and secure for himself the eminent honor to which he was entitled, and yet more than two years elapsed before the discovery was published to the world; and then, not of his own motion but at the instance of his friend, Halley, who subsequently boasted that he was the Ulysses who had discovered Achilles and brought him forth from his concealment. In the month of August, 1684, Halley, having become satisfied that Hooke could not furnish the demonstration of the law of gravitation which he had repeatedly promised, visited Cambridge to confer with Newton on the subject on which he had become deeply interested and been long laboring without any satisfactory result. He inquired of Newton, what would be the curve described by the planets on the supposition that the attractive influence of the sun diminished as the square of the distance? Newton at once replied, 'An ellipse.' When asked how he knew this, he replied: 'I have calculated it.' Halley, surprised and delighted at the announcement, asked to see the demonstration. Newton was unable to lay his hands on the calculation he had made two years before, nor could he at the moment reproduce it. He promised Halley however that he would send him the demonstration as soon as he was able, and in the month of November following he fulfilled his promise. Halley immediately revisited Cambridge to obtain Newton's consent to the publication of the discovery. In this he succeeded, and on the 10th of December he informed the Royal Society of the discovery and that Newton had consented to prepare a paper on the subject for the society. In February, 1685, the promised communication was received—a paper of twenty-four pages, containing four theorems and seven problems. He refers to it in the accompanying letter, as his 'Notions about Motion.' This humble yet most memorable paper ever presented to the Society was the germ of 'The Principia.'
The great discovery having been made public, Newton seems to have felt that the time had come to enter on the gigantic task he had doubtless proposed to himself when the discovery was first made but which other occupations had hitherto prevented him undertaking, namely, putting his demonstration in a complete and conclusive form and applying it to the solution of the many interesting and sublime problems which the phenomena of the material universe presented. For two years he dismissed from his mind all other occupation, and devoted himself with all the energy of his mighty intellect to the Herculean task. With untiring industry, prolonged attention, and intense thought, probably never paralleled in the history of intellectual effort, he lived but to meditate and calculate; oftentimes so wholly absorbed with the grand themes which occupied his mind as to be for the time unconscious of all the ordinary concerns of life. Frequently, on rising in the morning he would sit for hours on his bedside arrested by some new conception, and had it not been for the attention of the members of his family would often have neglected to take his daily food.
Wc cannot enter upon any detail or present even a summary of the magnificent result of these labors. They are to be found in his immortal work, the 'Philosophiæ Naturalis Principia Mathematica' given to the world in 1687 under the auspices of the Koyal Society. Of this work, the great Laplace, who, of those who have applied the highest powers of the human mind to the investigation of the phenomena of gravitation, stands second only because Newton lived before him, says: "The universality and generality of the discoveries it contains, the number of profound and original views respecting the system of the universe it presents, and all presented with so much elegance, will insure to it a lasting preeminence over all other productions of the human mind." "It is a work," says Sir David Brewster, "which will be memorable, not in the annals of one science or one country only, but which will form an epoch in the history of the world, and will ever be regarded as the brightest page in the records of human reason. It is a work which would be read with delight in every planet of our system, and in every system in the universe. There was but one earth on whose form and movements and tides the philosopher could exercise his genius; one moon whose perturbations and inequalities and action he could study; one sun whose controlling force and apparent motions he could calculate and determine; one system of comets whose eccentric paths he could explore and rectify; one universe of stars to whose binary and multiple combinations he could extend the law of gravity. To have been the chosen sage, summoned to the study of that earth, these systems and that universe, the favored lawgiver to worlds unnumbered, the high-priest in the temple of boundless space, was a privilege that could be granted to but one member of the human family; and to have executed the task, was an achievement which, in its magnitude, can be measured only by the infinite in space, and in the duration of its triumphs by the infinite in time."