principles, and it must be admitted that they formed an important step towards a rigorous solution of the problem.

When Halley learned the extreme pretensions of Hooke, he deemed it his duty to acquaint Newton with the charge preferred against him. This called forth a long and interesting letter from Newton, dated June 26th, 1686, in which he mentions a variety of particulars connected with the progress of his researches. He asserts that he had discovered the law of the inverse square of the distance (for circular orbits) even previous to the publication of Huygen's treatise "De Horologio Oscillatorio."* He admits that he was led to consider the law of the force in an elliptic orbit by Hooke's letter to him in 1679, but he positively denies being indebted to him in any other way for the results at which he arrived. This letter contains some interesting information relative to the progress of his labours in composing his great work. "I designed," says he, "the whole to consist of three books; the second was finished last summer, being short, and only wants transcribing, and drawing the cuts fairly. Some new propositions I have since thought of, which I can as well let alone. The third wants the theory of comets." Thus it appears that, about fifteen months after he returned from Lincolnshire to Cambridge, he had almost completed the three books of the Principia. This fully corroborates the statement of Pemberton, that Newton was engaged only about eighteen months in the composition of his immortal work. When we contemplate, in connexion with this fact, the prodigious mass of original discoveries announced in the Principia, the mind is lost in amazement at the power of thought which could have reared into existence so stupendous a monument in such a brief space of time.

Newton seems to have been so much disgusted with Hooke's violent conduct, that, in the letter above referred to, he intimated his resolution to suppress the third book altogether, containing the application of his dynamical discoveries to the system of the world. On the occasion of announcing his splendid discoveries in Optics at an earlier period, he had experienced much annoyance from the ignorance and jealousy of rival claimants, and he now feared that his peace of mind might be disturbed again by a similar cause. "Philosophy," says he, "is such an impertinently litigious lady, that a man had as good be engaged in lawsuits, as have to do with her. I found it so formerly, and now I am no sooner come near her again but she gives me warning. The two first books without the third will not bear so well the title of 'Philosophia Naturalis Principia Mathematica;' and therefore I had altered it to this, De Motu Corporum libri duo;' but upon second thoughts I retain the former title, 'twill help the sale of the book, which I ought not to diminish now 'tis yours." Halley wrote a soothing reply to Newton, declaring his belief in the groundlessness of Hooke's charges, and imploring him not to persevere in his resolution of suppressing the third book of his work. Newton seems to have listened favourably to the advice of his friend, and he gave a proof of his conciliatory disposition by adding a scholium to the fourth proposition of the first book, in which he mentions that Wren, Hooke, and Halley, had all found, by means of the relation between the periodic times and the distances, that the force which retains

In a subsequent letter to Halley, dated July 14th, 1686, he mentions having arrived at the law of the inverse square of the distance, by means of Kepler's theorem, about twenty years previously. This would carry back his original speculations to about the time assigned to them by Pemberton. The original of this letter is in the guard-book of the Royal Society.

the celestial bodies in their orbits (supposed circular) varies according to the inverse square of the distance.

It is impossible too much to admire the conduct of Halley in regard to the part he took in the publication of the Principia. Indeed we may reasonably doubt whether that immortal work would ever have been written at all, if it had not been for his enlightened zeal in the cause of science; for Newton himself appears to have been imbued much more strongly with the love of pondering in secret over his discoveries, than he was urged by the equally natural feeling of communicating them to others. This disposition of mind was fostered by a lively recollection of the annoyance he had suffered from the publication of his researches in Optics, and the consequent dread he entertained of having his tranquillity again disturbed by a controversy with envious rivals. Halley, therefore, besides discovering the only individual living who could unfold the physical theory of the celestial motions, is entitled to the credit of having persuaded him to communicate his discoveries to the world. Nor was this all; for, as has been already hinted, he defrayed the expense of publishing the Principia, at a time too when his finances could ill afford such an outlay ; and also undertook the revision of it in its progress through the press. Posterity has retained a grateful recollection of those princes who at different periods of history have distinguished their reign by a munificent patronage of learning and science; but, among all those who have thus contributed indirectly to the progress of knowledge, there is none who exhibits such a bright example of disinterestedness and selfsacrificing zeal as the illustrious superintendent of the first edition of the Principia. It is pleasing to reflect that Halley received such a noble reward for his exertions in the splendid discovery with which his name is immortally associated, and to which he was mainly conducted by Newton's researches on comets.

The Principia was published in 1687, and was dedicated to the Royal Society. At the beginning of it was inserted a Latin poem in hexameter verse by Halley, in honour of Newton's discoveries. The con cluding line runs thus :—

"Nec fas est propius mortali attingere divos; "+

"an eulogium," says the severe Delambre, "which no one has charged with exaggeration."§

The whole work is divided into three books. The first book treats of motion in free space; the second is occupied chiefly with questions relating to resisted motion; the third is upon the system of the world.

The first book is divided into fourteen sections, and contains ninetyeight propositions, besides a number of corollaries, lemmas, and scholia. In the first section, Newton explains the geometry which he employs in his subsequent investigations. It is termed by him the method of prime and ultimate ratios, and is essentially the same as the differential calculus. In the second section he enters upon the subject of centripetal forces, demonstrating Kepler's theorem of areas, and investigating the law of the

It must be understood that Halley was subsequently reimbursed for the expenses connected with the publication of the Principia by the sale of the copies of the work. + He was brought up in affluent circumstances, but in 1684 his father died, after completely wasting his fortune.

‡ Nor is it lawful for mortals to approach nearer the Deity, Histoire de l'Astronomie de Dixhuitième Siècle, p. 2,

force in various curves. In the third section, he considers the motion of a body compelled to revolve in any of the conic sections by a force directed continually to the focus. The fourth and fifth sections are purely geometrical, relating to methods of drawing conic sections through given points and touching given straight lines. The sixth section treats of the motion of a body in a given orbit. The seventh treats of the motion of a body ascending or descending in a straight line relative to the centre of force. The eighth contains the investigation of the orbit described by a body when the law of the centripetal force is given. The ninth relates to the motion of bodies in moveable orbits. This section contains the famous investigation of the motion of the apsides. The tenth treats of bodies moving on given surfaces, and of the motion of pendulums.

Hitherto Newton has been considering only the motion of material points. In the eleventh section he investigates the motion of bodies exposed to their mutual attraction. The twelfth treats of the attraction of spheres. The thirteenth of the attraction of bodies not spherical. The fourteenth relates to the motion of small particles passing from one medium into another.

The second book is divided into nine sections, and contains fifty-three propositions. It treats of bodies moving in resisting media upon different hypotheses of the resistance; and, whether moving in straight lines, or curves, or vibrating like pendulums. It also takes cognizance of the more recondite parts of several other branches of the Physico-mathematical sciences. The second lemma to the eighth proposition contains an exposition of the method of Fluxions, which is rendered necessary in most of the investigations of this and the following book.

The third book contains forty-two propositions. From the first to the eighteenth inclusive, Newton demonstrates various general theorems relative to the attraction of the sun, moon, and planets. In the nineteenth and twentieth he investigates the ratio of the earth's axes, and compares the weights of bodies at the surface in different latitudes. In the four following propositions, he shows that the precession of the equinoxes, the irregularities of the moon and the other satellites, and the phenomena of the tides, are all explicable by the principle of gravitation. From the twenty-fifth to the thirty-fifth inclusive, he computes the various inequalities of the moon's motion. The thirty-sixth and thirty-seventh treat of the tides. The thirty-eighth, of the figure of the moon. The thirty-ninth, of the precession of the equinoxes. The remaining three propositions are devoted to the theory of comets. At the conclusion is a scholium to the whole work, containing general reflections on the constitution of the material universe, and on the eternal and omnipotent Being who presides over it *.

The publication of the Principia marks by far the most important epoch in the history of physical science. Previous to its appearance the researches of philosophers may be said to have resembled the voyages of the early navigators, who continued creeping timidly along the coasts, without daring to launch their barks into the boundless ocean. Newton, like another Columbus, disdained to confine himself within the common

Besides the original edition of the Principia, two others were published during the life of the author. The second edition was published at Cambridge in 1713, under the superintendence of Cotes. The third edition was published at London in 1726, by Pemberton.

place conventionalities of ordinary minds; and, guided by the eagle eye of genius, explored the secret springs which animate a whole system of worlds. We cannot convey to the general reader a more adequate idea of the merits of the incomparable work just mentioned, than by citing the judgment pronounced upon it by the most illustrious of Newton's followers. Laplace, after enumerating the various astronomical discoveries first announced in the Principia, concludes in the following terms: The imperfection of the Infinitesimal Calculus, when first discovered, did not allow Newton to resolve completely the difficult problems which the system of the world offers, and he was often compelled to give mere hints, which are always uncertain until they are confirmed by a rigorous analysis. Notwithstanding these unavoidable defects, the number and generality of his discoveries relative to this system, and many of the most interesting points of the Physico-mathematical sciences, the multitude of original and profound views, which have been the germ of the most brilliant theories of the geometers of the last century, all of which were presented with much elegance, will assure to the Principia a preeminence above all the other productions of the human intellect." *

CHAPTER II.

Newton's Intellectual Character considered in connexion with his Scientific Researches.His Inductive Ascent to the Principle of Gravitation. -Motion of a Body in an Orbit of Variable Curvature.-Attraction of a Spherical Mass of Particles.-Developement of the Theory of Gravitation.-General Effects of Perturbation.-Inequalities of the Moon computed.-Aid afforded by the Infinitesimal Calculus.-Figure of the Earth. Attraction of Spheroids.-Precession of the Equinoxes.-General accuracy of Newton's Results.-Anecdotes illustrative of his Natural Disposition.-His Death and Interment.

NEWTON was singularly endowed with all those qualities which enable the mind to unfold the laws of the material world. He could detect with a glance the distinctive features of natural phenomena, and with marvellous sagacity divine the principles on which they depended. With these valuable qualities he combined a proneness to generalization, which constantly led him to connect together the facts he was contemplating, and advance from them to more comprehensive views of the operations of nature. He possessed also powers of mathematical invention adequate on all occasions to surmount the difficulties he might encounter, either in ascending by induction to general laws, or in subsequently redescending from them to the explanation of their various consequences. When we consider, moreover, that he was imbued with an extreme love of truth, which induced him to reject all speculations, however ingenious and beautiful, that were not reconcileable with facts-that his whole soul was wrapped up in the study of nature and her works, and that he possessed in an extraordinary degree the power of concentrating the whole energies of his intellect upon the object of his researches, we may form some conception of the advantages under which he approached the examination of physical questions. It is, in fact, in consequence of his possession of • Exposition du Système du Monde, liv. v. chap. v.

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all these qualities in so high a degree, that he stands without a rival among ancient or modern philosophers. His discovery of Universal Gravitation, beyond all comparison the greatest achievement that the human mind can boast of, affords abundant illustration of the truth of this remark. Throughout the magnificent train of investigations which that discovery suggested to his mind, we see him constantly uniting the sagacious and comprehensive views of the genuine interrogator of nature with the fertility of invention, the skilful research, the profundity and elegance, of the consummate mathematician. We have, in fact, presented to us the unexampled combination in one individual of all those attributes of genius which ennoble the human intellect, and which have thrown the halo of immortality around the names of Kepler and Leibnitz-of Galileo and Descartes-of Bradley and Laplace.

The transcendent powers of Newton's intellect are equally discernible in his inductive ascent to the principle of gravitation, and in his subsequent developement of its numberless consequences. Notwithstanding the sagacity he exhibited in connecting the fall of a stone at the surface of the earth with the motion of the moon in her orbit, and both of these phenomena with the motions of the planets round the sun, he would inevitably have failed in establishing this sublime conception as a physical truth, if he had not also possessed sufficient mathematical genius to solve the problem of central forces for an orbit of variable curvature. To those who are acquainted with the state of mechanical science in Newton's time it would be superfluous to mention that the highest powers of invention were indispensable for this purpose. When we reflect on the fact that Kepler spent a considerable part of his life in vain efforts to establish a connexion between the motions of the planets and the continual agency of some physical principle, that the question entirely escaped the sagacity of Galileo, and that Huygens, although in complete possession of the laws of motion, was unable to advance in its solution beyond the case of a circular orbit, we may well imagine the obscurity in which it was enveloped, and the mathematical difficulties which the investigation must have offered. Even when Newton had succeeded in this research, he merely established the mutual gravitation of the planets, according to the law of the inverse square of the distance, but he was not also enabled to extend the same principle to the ultimate particles of which the masses of the planets are composed. In order to effect this object, and thereby to establish the law of gravitation in its widest generality, he was compelled to determine the effect of the attraction of a spherical agglomeration of particles. This problem is of a totally opposite nature to the one already referred to; for here we have an infinite number of particles in juxtaposition, all attracting the body with unequal intensities and in different directions. Its intricacy is manifest at first sight; nor was this circumstance compensated by any preliminary hints calculated to facilitate its solution, for the mere conception of such a problem had not yet occurred to any mathematician. Newton, however, again triumphed over opposing difficulties, and thus succeeded in riveting, with the bonds of demonstrative reasoning, all the links of his magnificent generalization.

In redescending from the principle of universal gravitation, and pursuing it into its remoter consequences, he displays even more astonishing force of genius than he does in the course of his inductive ascent. It might be supposed that when once the highest step of generalization was