the nature of their paths. Hevelius had established the fact that their line of motion is more curved at some parts of their course than at others; and had even suggested that it resembled a parabola, having its vertex at the point where the comet came nearest to the sun.

Newton, seizing this hint, perceived in it a simple case of the law of gravitation. The nature of that force might make a body move in a parabola, provided the intensities of the central and original projectile force were properly adjusted. Or, again, they might be so adapted as to cause it to move in an ellipse of very great excentricity; such, for example, that the part of it which came within our system might hardly sensibly differ from a parabola. The reader will have no difficulty in conceiving this, by merely attending to what we observed, in an early part of our history, respecting the mode of describing the conic sections. Would then observation accurately confirm the idea of such orbits? Newton showed, by the example of precise observations made on the brilliant comet of 1680, that this agreement was very close. The observations on the position of the comet accorded with the form of a parabolic orbit, and its rate of motion with the law of the equable description of areas. Comets range

into our system from all quarters of the heavens, and from distances inconceivably great in the depths of space. Hence gravitation extended in all directions, and to unknown and inconceivably great distances from the centre of our world.

We have thus taken a rapid survey* of the main points of Newton's investigations, establishing the great system of the world, in a collective point of view, as they are presented to us in his immortal" Principia." In terminating our sketch, therefore, we must not omit to mention, that, after pursuing the elaborate

The reader will, of course, look for the further explanation of all the points here referred to, in their proper places in the TREATISE ON ASTRONOMY, Cab. Cyclo.

exposition of the mechanism of the heavens, which constitutes the third book; after that beautiful developement of the entire system of forces which guide and modify the revolutions of the planets primary and secondary, Newton concludes the whole with a sublime application, worthy of the preceding parts of the work, in which he deduces, in the direct line of argument from the laws of material phenomena, the proofs of the existence of the Great First Cause, and the evidences of the divine perfections.

In the view we have before taken of the progress of enquiry in the several departments on which the discoveries of Newton shed so new and brilliant a light, we have clearly traced at what point the labours of preceding philosophers had failed; and, on the other hand, precisely how much they had effected, and to what amount, each in their several ways, they had contributed towards the great work which it was reserved for Newton to complete.

The mathematical methods of investigation had been, in some measure, prepared by the invention of Newton's immediate precursors, and the geometrical truths which, from the immovable foundation of the whole, had been established by the ancients. But the discoveries of Plato, Euclid, and Apollonius, and the investigations of Kepler, Wallis, and Barrow, wanted their connecting principle, till it was supplied in the powerful methods created by the master mind of Newton.

The dynamical and mechanical truths contributed by Archimedes and Galileo; the laws of motion and force collected by Wren and Huyghens; the notions of attraction expounded by Hooke, had been brought together, but not as yet reduced into systematic connection.

The phenomena of the heavens, recorded from the infancy of the world, had been referred to certain laws, and order educed out of apparent confusion by the successive labours of astronomers; the magnitude of the earth had been measured, and computation had been rendered easy by logarithms.

The telescope, in the hands of Galileo, had assimilated the planets to the earth, and placed the earth in the rank of a planet. The small systems about Jupiter and Saturn had presented the most striking analogies; and, above all, the wonderful harmony pervading all the motions of our system, displayed in the constant observance of those singular proportions elicited by Kepler, and extended by subsequent observers, formed a mass of not less valuable and important than mysterious truths.

But these great facts and relations of the planetary world were unconnected with the dynamical laws; nor were the mathematical truths yet applied to combine them in their proper places and relations.

Thus from all quarters the materials were provided; but there wanted an arrangement and a connection. The substantial foundations were supplied from one quarter, the richly finished workmanship from another ; but there wanted the genius of the architect to arrange and cement the whole into a well-designed edifice. The stones had been hewn out of the mountains; the cedar had been contributed by Hiram, and David had provided brass and gold; but there wanted the Solomon to plan and rear the edifice. He erected it, and consecrated it a temple to the Lord.

Style and Method of the "Principia.”

With regard to the style and manner in which these invaluable researches are presented to us by their author, it appears unquestionable that, though he has throughout adopted the language and method of the ancient geometry, deviating from it only where the nature of the case absolutely compelled him to do so, yet the investigations were originally pursued by a very different method from that in which they are actually delivered in the "Principia." There can be no doubt that the results were first obtained by the use of the fluxional calculus, and then synthetical proofs of them invented

with that happy facility which so powerfully characterised Newton's genius.

This, indeed, appears certainly to have been the case from the unpublished correspondence of Newton with Cotes, preserved at Cambridge, relating to the preparation of the second edition of the "Principia," which Cotes superintended. In these letters the analytical method is almost always used in their mutual discussion of such points as appeared to want further explanation. It has also been supposed that Newton must have been in possession of some analytical processes of a higher class, such as "the calculus of variations," for solving some of the problems which now appear in the Principia" in a totally different form. But, perhaps, it is more probable that, without any such general systematic method, he worked out his way by some mode of investigation peculiar to himself, hardly reducible, perhaps, to fixed rules, and evincing in the highest degree the resources of his genius. "L'inspiration," says Bailly*, "de cette faculté divine lui a fait aper. cevoir les déterminations qui n'étoient pas encore accessibles; soit qu'il eût des preuves qu'il a supprimées, soit qu'il eût dans l'esprit un sorte d'estime, une espèce de balance pour approuver certaines vérités, en pesant les vérités prochaines, et jugeant les unes par les autres.'

When drawing up the first portion of his researches to send to the Royal Society, he says (in one of his letters), "I then composed a few theorems;" evidently using that word in the sense of the old geometers, and meaning, that, having already discovered the results by analysis, he now put them into a synthetical form ; without which he was unwilling that they should go forth to the world.

In some parts of the "Principia," when he could not keep entirely to the ancient model, we find him apologising to his readers, and, as it were, excusing himself for so transgressing, on the plea of desiring to be concise. "Componi possent harum assertionum

* Hist. de l'Astron. tom. ii. liv. xii. 28.

demonstrationes more magis geometrico, sed brevitati consulo." Often, probably, he satisfied his own mind as to the truth of the conclusion at which he had arrived, without thinking it necessary to give them a formal demonstration at all; or worked them out by various indirect methods of calculation, which, with singular sagacity and address, he could always apply to cases apparently the least susceptible of such solution: but it was utterly offensive to his taste to allow any investigations in such a state to appear before the public. We have already had occasion to quote his recorded sentiments on this point, with regard to mathematical theorems. In this case there was also another and more powerful motive; he was well aware that he had on all hands violent prejudices to encounter. He feared the announcement of the fluxional principle, because he foresaw it would have been misunderstood, and have laid him open to every species of cavil and objection; and he had now to announce discoveries, which would not only stir up every remnant of the Ptolemaic prejudices, to which there were still some lingering adherents, but would raise up against him the powerful phalanx of the disciples of the Cartesian philosophy, which now reigned triumphant in the universities. would then have been highly imprudent to risk the announcement of his discoveries in a form in which they would be exposed to double misrepresentation, equally on their own account, and on that of the method by which they were delivered. In the case of almost any philosopher under similar circumstances, this would surely afford a perfectly sufficient explanation for a backwardness in bringing such investigations before the public; but when to this we add the consideration of the known peculiarities of Newton's character, it seems to us every appearance of difficulty will vanish in accounting for his reluctance to publish his system, and for his preference of the ancient form of demonstration, which have afforded so much matter of surprise and speculation. Both, in fact, have been strangely

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