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not twenty years since we have begun to perceive that we were far behind all the rest of Europe in these sciences; not from want of abundance of first-rate talent, but from a misapplication of that talent to unworthy objects, or at least to such as were not of a nature calculated to lead to any great advance in the state of knowledge. Within the period named, the works and inventions of the great continental mathematicians have been introduced and studied; and it is needless to say, no sooner were they understood and appreciated, than they have called forth, in turn, an ardent spirit at least for the cultivation of these methods,-though, perhaps, that spirit has been shown rather in detailed improvements and amended treatises, than in any extensive original researches. Yet these have not been altogether wanting; and we need not fear to place in competition with the inventions of the Continent the analytical researches of Messrs. Woodhouse, Bromhead, Ivory, Babbage, and Herschel.

Dynamics.

In the application of mathematics to physical investigations, the English philosophers of a foriner age were not more happy. Here, again, with a very few exceptions, an entire devotion to the letter of the “Principia” seems to have impressed Newton's followers with the notion, that nothing further was to be effected beyond what was accomplished in that stupendous work. The "Principia," indeed, is a work which will stand as long as science shall exist, an enduring monument of the transcendent genius of its author. The truths which it establishes are unquestionably those on which the whole system of physical astronomy is securely founded; and the manner and style in which they are delivered afford the most indisputable evidence of that superiority of talent, which, by the application of such extremely simple principles, could educe such surprising conclusions. It is a work which, both for matter and style, stands forth finished and complete in itself. But it is no disparagement to its merits to admit, that when further

and more complex questions arose to be determined, the work of solving them, and reducing them into a system, would not be capable of being performed by the same means, or presented in the same form.

The methods of ultimate ratios and fluxions seem to have been, as it were, created for the express purpose of conducting Newton to the analysis of the motions of bodies acted upon by gravitating and central forces, and the solution of the great problems arising out of it; but they were not so applicable when the limits of these branches of enquiry became extended: and, to investigate more complex relations in the physical system of the world, more refined, more general and comprehensive methods were required. Hence an exclusive devotion to the methods, however excellent, invented and applied by Newton, was not likely to assist his disciples in carrying farther the discoveries he had begun; and while they confined their attention and acquirements to the knowledge of his writings and methods, they were not in a condition to enlarge the boundaries of science, and were not availing themselves of those more powerful instruments which were required for the work, in proportion as it became more difficult and extensive.

Newton had stretched the powers of geometry to an unprecedented degree, and had successfully applied it in extending the dominion of science over the system of the universe; but such a mode of proceeding would by no means be applicable in all cases: the generality of those engaged in such researches would require easier methods, and more certain and systematic rules, to assist them in their investigations and computations. Geometry, as wielded by Newton, was like the sling and the stone in the hand of David; a weapon with which no ordinary combatant would choose to attack a giant.

Adhering, then, to such methods, the British philosophers in general did but little in the way of prosecuting dynamical enquiries; while we find the continental analysts busily engaged, and rapidly extending

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their researches in proportion to the facilities supplied to their hands by the increasing powers of the new calculus. The discussion which arose in 1724 respecting the true measure of force, and which, in the end, was clearly seen to be a mere dispute about terms, had the effect of calling forth many valuable researches ; though, when the English philosophers, Maclaurin, Stirling, Clarke, and others, took a share in it, and opposed Bernoulli, Herman, S'Gravesande, and Muschenbroek, the discussion assumed too much of the angry tone of the fluxional controversy, the bitterness of which had not worked itself out till a later period. The controversy may be considered to have ended with the publication of D'Alembert's "Dynamique,” in 1743.

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The general character of the improvements effected in mechanical and hydrostatical science, about this riod, was marked chiefly by the increasing developement of the analytical formulæ. To this has been owing the great advances in the investigation of principles and laws, which, perhaps, may be dated from Bernoulli and D'Alembert, and which have been so advantageously pursued by a long train of their illustrious successors, and perfected in the " Mécanique Analytique" of La Grange. These, however, are precisely those of which it is the least in our power, within our limits, to give any adequate notion, and of which an enumeration, without such illustration, could answer no useful purpose.

Of Lagrange, however, we must add, that the distinctive mark of his genius consists in the unity and grandeur of his views. He attached himself wholly to a simple, though just and highly elevated thought. His principal work, just mentioned, refers all the laws of equilibrium and motion to a single principle: and, what is not less admirable, it submits them to a single method of calculation, of which he himself was the inventor. All his mathematical compositions are remarkable for their singular elegance, for that symmetry of form and

generality of method which constitutes the perfection of the analytical style

Physical Astronomy.

Physical astronomy was a science of purely British origin, yet, after the death of its founder, it had very few cultivators in England, and scarcely any advances of importance were made towards the investigation of the more abstruse consequences of the law of gravitation, and the more complex phenomena of the system of the world; or in providing general and comprehensive methods suited to computing the laws and consequences of such actions.

On the continent of Europe, during the same period, widely different was the condition and progress of this science. The Newtonian theory of gravitation, though at first admitted with some delay and hesitation, was at length triumphant; and no sooner were its excellencies perceived, and its conclusions assented to, than the genius of the continental mathematicians was immediately directed to the extension of its applications, and the improvement of the methods required for those applications.

To complete the theory of tides, and to investigate the lunar inequalities, the motion of comets, and the figure of the earth, were immediately among the objects which gave employment to the talents of the great ornaments of the continental school: and the success of their investigations on these subjects kept exact pace with the increased powers of the instruments of research with which they had provided themselves from the enlarged resources of the new calculus. Every year added new inventions and discoveries, and extensions of former theories, to the successes of science. The fabric of physical astronomy was rapidly extending, upɔn the plan which Newton had laid down; and the work advanced as might be expected, precisely in proportion to the more powerful means of carrying it on, which

were put into the hands of the workmen,-in proportion to the extension and generalisation of the processes of the analytical calculus. A long and brilliant catalogue of names might be adduced of those who united the extension of abstract methods with the prosecution of physical research, from the days of the Bernoullis down to the present times.

The innumerable and complex consequences of the principle of gravitation formed an immense legacy of research bequeathed by Newton to his successors.

The first who commenced working upon them was Clairaut, in his celebrated memoirs addressed to the Academy of Sciences in 1743, 1745, and 1754. These, in fact, contain the developement of the principle of the perturbations, known by the name of "the problem of the three bodies."

Nearly contemporary with him were Euler, Mayer, Thomas Simpson, and D'Alembert, whose united talents were also devoted to the further prosecution of the same difficult but highly important and interesting subjects, the lunar and planetary theories, that is, the account of all the inequalities of their motions as affected by the disturbances of their mutual attractions.

Besides a host of other philosophers (with the exception of Maclaurin, chiefly it must be confessed continental), who were engaged in carrying on and perfecting the various elaborate details of these subjects, and the computations they involved, we now find two great luminaries of science beginning to appear on the intellectual horizon, the latter of whom has become the sole but worthy competitor with Newton for the honours attending the completion of a perfect mathematical and dynamical system of the mechanism of the heavens: Lagrange and Laplace.

The former having succeeded in developing and verifying the dynamical truths which have become the basis of the whole analytical system of forces, applied them to the system of the world; and laid down the principles on which the invariability of the mean distances of all

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