pied by her in falling through a given space was exactly 60 times greater than that occupied by a body at the earth's surface in falling through an equal space. It thus appeared that the force which retained the moon in her orbit, as deduced from her actual motion, was less than the force of gravity at the earth's surface, in the exact ratio of the inverse square of the distance from the centre of the earth.

When Newton had thus satisfied himself by indisputable evidence that he had discovered the true law of gravitation, he proceeded to investigate more profoundly its real character. He had found that the planets gravitate towards the centre of the sun, and the satellites towards the centres of their respective primaries, but it did not escape his sagacity that these points could not of themselves exert any physical influence; and that the attractive force was directed towards them solely in consequence of the mass of material particles which in each case surround them. He was thus led to regard the principle of attraction as residing in the constituent particles of the attracting body, and to conclude that the tendency of the force to the centre was no other than the resultant of all the molecular forces acting with unequal intensities and in different directions. In order to establish this important fact, it was necessary for him to investigate the nature of the attraction exercised by a mass of particles agglomerated in the form of a sphere; for observation shewed that all the heavenly bodies were spherical, or very nearly so. In the course of these researches he was conducted to the remarkable conclusion that, if the sphere were of uniform density, or even if it consisted of concentric strata of uniform density throughout each stratum, but differing in density from one stratum to another, the combined effect of the attraction of all the molecules would be the same, both in intensity and direction, as if the whole mass had been collected at the centre. This result afforded a most satisfactory explanation of the fact that, in accounting for the motion of the planets by a solar force, varying according to the inverse square of the distance, it was in all cases found necessary to measure the distance from the centre of the sun; and the same explanation applied to the motions of the satellites round their respective primaries.

Having thus assured himself that the tendency towards the central body was due to a quality inherent in the constituent particles, and not to any virtue residing in the centre, he naturally was led to suppose that this tendency must be mutual for all the parts of matter, and that as the sun attracts the planets, and the planets the satellites, so, in like manner, the planets attract the sun, and the satellites the planets, and even objects at the surface of the earth attract the earth. The equality of action and reaction, which was strikingly illustrated in all the other relations of the material world, rendered this proposition self-evident; nor did his sagacity fail to discover sensible manifestations of this principle in the irregular movements of the celestial bodies, especially in those of the moon. He

* The force which retains the moon in her orbit is here supposed to act in the same direction during a very short space of time. This supposition is not strictly true, but for a very small arc of the lunar orbit it cannot sensibly affect the final result.

It is said that Newton became so much agitated as soon as he began to suspect the probable result of his calculation, that he was compelled to assign to a friend the task of bringing it to a conclusion.

Cotes, in his admirable preface to the second edition of the Principia, demonstrates in the following simple and convincing manner that the action of gravity is equal on both sides:-"Let the mass of the earth be divided into any two parts whatever, either equal or anyhow unequal; now, if the weights of the parts towards each other were not mutually

therefore finally arrived at the conclusion, that every particle of matter in the universe attracts every other particle, with a force varying inversely as the square of their mutual distances, and directly as the mass of the attracting particle.

When Newton had thus ascended to the principle of gravitation in its most comprehensive form, he devoted the whole energies of his vast intellect to the unfolding of its consequences; and, with a sagacity and power of investigation unexampled in ancient or modern times, he succeeded in tracing all the grand phenomena of the universe to its agency. Considering generally a body projected in free space, and exposed to the action of a central force, varying according to the inverse square, of the distance, he demonstrated, by means of a beautiful geometry which he had specially invented for such researches, that the body would revolve in a curvilinear orbit which would be some one of the conic sections. It might be a circle, an ellipse, a parabola, or an hyperbola, but it must necessarily be one of them-the question as to the particular species of curve depending entirely on the primitive position of the body, and the velocity of the impulse. He showed that, when once the initial distance and the velocity and direction of the impulse were given, not only the conic section in which the body would move was readily assignable, but also the magnitude, position, and form of the orbit. Applying these principles to the motions of comets, he discovered that these bodies, like the planets, are retained in their orbits by the attraction of the sun; and he invented a method for determining the elements of a comet's orbit, by means of three distinct observations.

He perceived that, while the planets and satellites are mainly influenced by the attraction of the central bodies round which they revolve, they are also liable to be disturbed in their motions by their mutual attraction. Considering the moon as disturbed by the sun in her orbit round the earth, he found that the action of that body would account for the numerous inequalities which astronomers had from time to time detected in her motion. He demonstrated that the mean effect of such a disturbing force would be to cause the apsides to advance in the direction of the moon's motion, and the nodes to regress in the opposite direction, both of which results are conformable to observation; nor did he stop here, but actually computed the exact quantity of many of the most important of the lunar inequalities. He discovered that the mutual gravitation of the molecules composing the earth's mass, combined with the centrifugal force generated by her motion round her axis, would cause her to be flattened at the poles. Assuming the actual figure to be an oblate spheroid, he assigned the ratio between the polar and equatorial axes, and determined the law of gravity at the surface. With a sagacity almost divine, he perceived that the action of the sun and moon upon the redundant matter accumulated at the equator, would produce the slow conical motion of the earth's axis which occasions the Precession of the Equinoxes, and he indicated the quantity of the motion due to each of the two disturbing bodies. He shewed, also, that the attraction of the sun and moon, by elevating the waters of the ocean, would continually disturb their equilibrium, and would thereby give rise to the phenomenon of the Tides. Finally, what is

equal, the lesser weight would give way to the greater, and the two parts joined together would continue moving in a right line ad infinitum, towards the part to which the greater weight tends; a result which is entirely contrary to experience."

perhaps the most astonishing of all the results to which he was conducted by his theory, he found that the quantities of matter contained in the heavenly bodies might be ascertained by observing the effects of their mutual attraction. By means of this principle, he was enabled to compare the mass of the sun with the masses of those planets that are accompanied by satellites, and also to compare the mass of the moon with that of the earth *.

Newton has given a full exposition of these sublime discoveries in his immortal work, the Principia. As the appearance of this work was destined to introduce a new era in science, it may not be uninteresting to mention briefly the circumstances connected with its publication. Newton does not appear to have contemplated communicating to the world his researches on the subject of gravitation until the occasion of a visit paid him by Dr. Halley in 1684. About the beginning of that year, Halley had discovered, by means of Kepler's third law, that the centripetal force for circular orbits varied according to the inverse square of the distance. This result gave him the law of the solar force from one orbit to another, on the supposition that the planets move in circles, with the sun in the centre; but, as in reality, they move in elliptic orbits, with the sun in the focus, the distance, in the same orbit, was subject to continual variation; and hence it became necessary to ascertain the corresponding variation of the force. Finding his mathematical powers inadequate to the task of successfully grappling with this more difficult problem of dynamics, he applied to Wren and Hooke, in hopes of receiving from either of them a solution of it. Wren, according to Newton's statement, had deduced the law of the inverse square of the distance (for circular orbits) several years previous to Halley's present communication with him. When Halley proposed to him the problem of the law of the force in an elliptic orbit, he replied, that he had bestowed much thought on it, but was compelled to give it up from inability to make any impression on it. Hooke asserted that he had solved it, and had found that the force varied according to the inverse square of the distance. When pressed to produce his solution, he refused to do so, declaring that he would conceal it, until others trying and failing, might know how to value it when he should make it public. It is quite clear, however, that he was unable to support his assertion by any mathematical proof, for if such had been the case he would have given it forth to the world as the surest means of vindicating his claims, when he attempted, a year or two afterwards, to appropriate to himself the credit of Newton's discoveries.

Unable to obtain a solution of this interesting problem from any of his acquaintance in London, Halley proceeded to Cambridge, in the month of August, 1684, for the express purpose of conferring with Newton on the subject. To his inexpressible delight, he learned the good news that his friend had already brought the demonstration to perfection. So little was Newton's mind occupied at this time with such researches, that he was unable to lay his hand on the papers relating to them when Halley visited him, but he promised to send them to him soon after his return to London. It appears that Newton subsequently worked out the propositions afresh, and transmitted them to Halley, in the month of November of the same year. Halley immediately set out upon a second visit to

⚫ For a concise but very luminous exposition of the mode by which Newton established the principle of gravitation, see the "History of Astronomy," Library of Useful Knowledge, p. 83, et seq.

Cambridge, to procure more information, and to encourage Newton to pursue his researches. In December, of the same year, we learn the progress of Newton's labours, from Halley's announcement to the Society, on the 10th of that month, " that he had lately seen Mr. Newton, at Cambridge, and that he had shown him a curious treatise de Motu,' which, upon his desire, he said was promised to be sent to the Society, to be entered upon their register."* In fulfilment of his promise, Newton transmitted to the Society, about the middle of February, 1685, a paper containing his early researches on centripetal forces. This communication consisted of eleven propositions, the greater number of which were similar to those which subsequently formed the second and third sections of the Principia. Newton, in acknowledging the registration of his paper by the Society, thus writes to Mr. Aston, the secretary, on the 23rd of February. "I thank you for entering in your register my notions about motion. I designed them for you before now; but the examining several things has taken greater part of my time than I expected, and a great deal of it to no purpose. And now I am to go into Lincolnshire for a month or six weeks. Afterwards I intend to finish it as soon as I can conveniently."† It is quite clear from the above letter that, although Newton was already in possession of the groundwork of all his discoveries in Physical Astronomy, he had not at this time developed his thoughts beyond the substance of the brief essay transmitted to the Society. Indeed, he can hardly be said to have entered seriously upon the composition of the Principia until his return to Cambridge, in April, 1685. Mr. Rigaud has justly remarked, in reference to this fact, that the Principia was not a protracted compilation from memoranda which might have been written down under the impression of different trains of thought. It had the incalculable advantage of being composed by one continued effort, during which the mutual bearing of all the several parts was vividly presented to the author's mind .

On the 21st of April, 1686, Halley read before the Royal Society a paper on gravity; in which, after alluding to the labours of Galileo, Toricelli, and Huygens, he mentions the truths "now lately discovered by our worthy countryman, Mr. Isaac Newton, who has an incomparable Treatise of Motion' almost ready for the press."§ The prospect held out by Halley was very soon realised; for, on the 25th of the same month, Dr. Vincent presented to the Society a manuscript treatise of Mr. Isaac Newton's, entitled, "Philosophiæ Naturalis Principia Mathematica." This was the first book of the Principia. The Society directed that a letter of thanks should be addressed to the author: they also referred the question of printing it to the consideration of the Council, and the drawing up of a report on it to Dr. Halley. On the 19th of May, the Society ordered that the book should be printed forthwith: whence an impression has been generally formed that the Principia was printed at the expense of that body. This conclusion, however, is not borne out by the words on the title page of the work, which are, "Jussu Societatis Regiæ," not "Jussu et Sumptibus," as was usual in those cases where the expense of printing was defrayed out of the funds of the Society. But a decisive

* Journal Book of the Royal Society; see also Rigaud's Historical Essay on the first publication of the Principia. Oxford, 1838.

See also Rigaud's Essay. Ap

+ Letter Book of the Royal Society, vol. x. p. 28. pendix, page 24. The original letter has not been discovered. Rigaud's Essay, page 25.

Phil. Trans., vol. xvi. p. 6.

refutation of the current opinion is furnished by a resolution passed at the meeting of the Council on the 2nd of June, to the effect that Mr. Newton's book be printed, and "that E. Halley shall undertake the business of looking after it and printing it at his own charge, which he engaged to do." The fact is, that when the Council, which took cognizance of all the pecuniary affairs of this Society, came to consider the resolution adopted at the general meeting of May the 19th, they found that the state of their finances could not admit of their carrying it into effect. A work, "De Historia Piscium," by Fr. Willughby, had been published in 1686, “Jussu et Sumptibus," and the outlay incurred by this publication appears to have completely exhausted the funds of the Society. To such extremities, indeed, were they reduced by this act of imprudent liberality, that they were compelled to pay their officers in copies of this work on fishes, in consequence of their inability to procure purchasers for it.

Meanwhile a violent reclamation was raised by Hooke relative to the discovery of the law of gravitation. This individual, who would be well entitled by his genius to occupy a high place in the history of physical science, if he had displayed more uprightness and moderation in his relations with contemporary philosophers, had no sooner heard of the manuscript which Dr. Vincent had presented to the society in Newton's name, than he asserted that it was he who first communicated to the author the law of the inverse square of the distance, as well as various other discoveries announced in the manuscript. We have mentioned that, as early as 1666, Hooke had arrived at very accurate notions on the subject of centripetal forces. In 1674 he published a work, entitled "An Attempt to prove the Motion of the Earth from Observations," in which he describes the general nature of gravitation with remarkable clearness and accuracy. Although, however, he remarked that the attractive forces acting between bodies are more powerful as the distances from the centres are less," it is quite clear that the idea of computing by a mathematical investigation the intensity of the force in any case at different distances from the centre, and thereby ascertaining the law of its variation, did not at all occur to him; for, after referring to the varying intensity of the force, he then goes on to say: now what these several degrees are I have not yet experimentally verified." It would appear, however, that, guided by the analogy of other emanations from centres, he had subsequently adopted the inverse square of the distance as the law of the force which retains the planets in their orbits; and then, extending the same law to the earth, he concluded, by an inversion of the question, that the path of a projectile was an ellipse, with the centre of the earth in the focus. We have mentioned already that Hooke was unable to produce a demonstration of the law of the inverse square of the distance, although he boasted repeatedly that he had arrived by legitimate reasoning at that result. The fact is that, although a man of extraordinary acuteness in physical matters, he had no talents for mathematical science; and this defect constituted an effectual bar towards his establishing, upon a satisfactory basis, any of the great truths relating to the theory of gravitation.

But although Hooke's powers were inadequate to the complete investigation of the problem of centripetal forces, there was much merit in the clearness with which he pointed out the mode in which a body is retained in a curvilinear orbit by a force continually directed towards a fixed centre. His views on this subject were in strict accordance with mechanical