of determining the masses of Planets that have Satellites, close the volume.

If the Planets were acted on by the Sun alone, they would describe orbits nearly elliptical round that luminary: But as they are affected, at the same time, by a mutual action among themselves, certain inequalities and perturbations arise, which must be computed, in order to construct accurate tables of their motions. In this, as in the problem of the Three Bodies, the present state of Analysis will not suffice for an exact solution. We must therefore be satisfied with an approximation. The difficulties are lessened, however, from this, that the masses of the Planets are exceeding small, compared with the Sun's magnitude; and that the eccentricities and planes of their orbits, vary within narrow limits. Those perturbations, which can only be detected in the lapse of ages, are known by the name of Secular Inequalities. The other inequalities, discovered in their own motions, and in the motions of their nodes and perihelions, are called Periodic. The inequalities of Jupiter and Saturn are considered by Mr Woodhouse at some length. The important results which followed, when these were first successfully investigated by La Grange and La Place, deserve some notice.

It was observed by Flamstead, in the year 1682, that there was a retardation of the mean motion of Saturn, and an acceleration of that of Jupiter. This retardation was inconsistent with any thing founded on the principles of gravitation; and it was thence concluded by mathematicians, that the disturbances of these two planets proceeded from a cause that in some essential point differed from their attraction on each other. Euler, who first endeavoured to reconcile these phenomena, altogether failed in the attempt. La Grange and La Place, who also turned their inquiries to the same object, met at first with no better success. Soon after this, La Grange, by the calculus of variations, a highly refined analytical artifice, invented by him to facilitate the solution of Isopereinetrical problems, succeeded in demonstrating the remarkable fact, that the mean periodic revolutions of the planets, and the major axes of the orbits they describe round the sun, remain for ever the same. Consequently the inequalities in the mean motions and distances must be periodical, and not subject to secular acceleration or retardation. But this discovery, so far as regarded Jupiter and Saturn, enly made the matter worse it made the acceleration of the , and retardation of the other, more anomalous than everlike Clairaut's investigation of the progression of the lunar e, it threatened to form a striking exception to motions

which were referred to gravitation, if not to overthrow the theory itself:-But, like Clairaut's, it ended by being one of its strongest confirmations. Struck with the contradictory results which seemed now almost established as true phenomena, La Place examined the steps of his process with unwearied attention; and at last succeeded in discovering the true cause of the secular inequalities in the mean motions of these planets. He conceived, that if the phenomena were to be explained by the mutual action of the two plancts; that if Jupiter caused a retardation in the mean motion of Saturn; Saturn, by his action, must reciprocally produce an acceleration in the mean motion of Jupiter. Revising his operations, he found that those terms which involved the third power of the eccentricity, (which Euler had neglected, and hence obtained no results), were very mi nute in the differential, but became, in the particular case of Jupiter and Saturn, of such value in the integral, that they could not be rejected, without vitiating the results. Retaining them, therefore, the solution turned out as he expected; it showed that Jupiter and Saturn had corresponding inequalities, which were of different affections, but of equal duration, the whole period of each being somewhat greater than 900 years. The one inequality retards Saturn's motion during half this timewhile the other accelerates Jupiter's,-and, during the other. half of the period, the contrary effect takes place. These corresponding accelerations and retardations were thus explained by the mutual attraction of the two bodies, and referred for their cause to the general principle of gravitation.

There are no phenomena, in the whole system of the universe, which more powerfully call forth our wonder and admiration, than these;-that the major axes of the planets' orbits are invariable, and that their mean periodic revolutions are always the same. While the other elements of the orbit are perpetually changing-their elliptic forms at one time approaching to circles, at another differing from that form by a sensible eccentricity-their apsides, and the lines of their nodes, constantly varying in position-the inclinations of the planes of their orbits oscillating between their greatest and least variation ;—we are yet assured that the stability of the system is secure from decay-that, by the wise dispensation of the Great Author of its formation, it is everlasting, permanent, immutable,

In concluding, we cannot but congratulate the country on the appearance of the first work calculated to convey an accurate notion of the methods used in the later and more profound researches of Physical Astronomy; the only methods, for what

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has yet appeared, by which those researches can be successfully prosecuted. The exposition of them now submitted to the pubfic, is clear and distinct, and as elementary as it can easily be rendered. The principal obstacle to their being generally received amongst us, and used to supersede the older methods, which, however elegant, are so much less powerful, seems to be the want of a text in the higher Geometry, or in the Fluxionary Calculus, to which reference might be made. We have no book in English that is in the least calculated to answer that purpose; none where the Integration of Fluxionary Equations is delivered with the necessary extension; none in which the method of Partial Differences is explained, or where the method of Variations is so much as mentioned. These great defects must be supplied, before the Geometer can penetrate into the depths of Mechanicks, Hydrodymicks, or Astronomy. A translation, or a judicious abridgment of Euler's Differential and Integral Calculus, would be the best present that could be made to the English mathematician. It is by such means only that we can hope to vie with our continental neighbours in the pursuits of Philosophy.

While we rejoice at the rapid progress that has been made in the natural sciences, during the last century, and exult in the idea that this has been effected by the increased intellectual attainments of man, we feel with regret how little Great Britain has contributed to this great work. Since the time of Newton, almost every thing that has been done in Physical Astronomy, has been done by the Mathematicians of the Continent. To account for this, it has been said, that a consciousness of inferiority in the knowledge of the higher Mathematics, has kept us back: And, as a proof of this inferiority, is adduced the fact, that the more extensive methods of analytical calculation have not been introduced into England till within these few years; and that the higher parts of the Calculus, as the method of partial differences, the calculus of variations, are even now unknown in the mathematical institutions of this Island, To this argument we allow its full weight;-that it is a proof of our present inferiority in mathematical learning, cannot be denied. But in a country which has done more for chemistry, than all Europe put together, which has given greater encouragement to the arts--to agriculture and manufactures—to literature and general science-than any nation on the face of the globe, a charge of intellectual inferiority cannot be well founded. Our deficiency in Mathematics, the obstacles that have perpetually opposed the introduction of the Calculus into Great Britain, must be accounted for on different grounds.

An extreme reverence for that immortal work, the Principia of Newton, contributed to strengthen, instead of weakening the attachment which the older geometers evinced to the synthetical methods; and operates, even to the present day, to the exclusion of those which are purely analytical. What that book has done for mathematical science in general, but more especially for Physical Astronomy, has appeared in the preceding pages; and we are sure no mathematician, grown gray in the service of Geometry, would lament more than we should, were it suffered to fall into oblivion:-But if we wish to share in the discoveries of our neighbours, we must now resort to the use of a more powerful instrument, by introducing the means which they have employed, even though our pride may suffer by the change. In addition to this feeling, which encouraged Geometry rather than Algebra, another cause may be mentioned, which we believe to have no inconsiderable effect against the advancement of the higher branches of the Mathematics in this country, namely, the very extensive dissemination of general knowledge, which is so much the case over the whole of this kingdom. Literature and the Arts give abundant occupation to the mind of a man of liberal curiosity, and leave less inducement to attach himself to abstract studies. Formerly, Theology and School Logic were essentially necessary to a man of education; and the different branches of Mathematics, and comparatively lighter studies, were pursued as a relief to the mind. Now-a-days, a man must be conversant in chemistry, mineralogy, entomology, modern languages, history, politics, and fifty hard-worded studies beside-so that, in fact, unless he chuse to devote himself almost exclusively to Mathematics, he has little chance of aspiring to discovery, or even to eminence, in that pursuit. Besides that the peculiar formation of society now, makes a man ashamed to appear ignorant upon such topics as we have mentioned, there are really so many opportunities of acquiring them perpetually thrown in his way, that it would need no small portion of ingenuity to escape picking up a little here and there. There never was a period in which museums, lectures, picture-galleries-for the arts; abstracts, reviews, encyclopædias—for literature; newspapers, pamphlets, and public meetings-for politics; were so abundant, and supported with so much spirit as the present. Our very ladies will discourse you most eloquently upon transition rocks, hydrogen and oxygen; and would consider it the worst of scandal to be thought illiterate. Now, in France, while every encouragement is given to this sort of general knowledge, yet no opportunity is neglected to promote mathematical knowledge in particular. That admirable establishment the

Royal Academy of Paris, where pensions and honours are heaped upon those who devote themselves exclusively to works of science, keeps alive the spirit of discovery; and, from the effect it has already produced in promoting the interests of science, we may safely look forward to its advancing them to a still higher state of perfection. In England, the only inducement to invention or discovery, is the hope that, by making sufficient interest, the Royal Society may be prevailed upon to allow the paper to be read before them. It is indeed much to be lamented that this Institution, intended for the advancement of science, should hold out so little encouragement to Mathematical learning.

There is another establishment in France, which is of the utmost use in this way-we mean the Polytechnic School, founded by Bonaparte. It is composed of boys chosen out of every department, in each of which there is an examination, chiefly in Mathematics, once a year. Two or three of the best proficients are selected for this school, at which they are instructed by professors in every branch of science, and excited to emulation by a judicious distribution of honours and rewards. We have lately learnt, with much satisfaction, that something on this plan has been begun at Liverpool, promoted by the subscriptions of the merchants of that place, with that judgment and liberality which characterizes all their public proceedings.

Once more we return Mr Woodhouse our thanks for his work upon Astronomy. We trust he will go on with it, and add another volume upon those important parts of the subject which yet remain ;-The Nutation of the Earth's axis-the Precession of the Equinoxes-the Tides-Figure of the Earth,will afford ample food for further speculation. We hope he will take advantage of them. We omitted to notice, in our remarks on the volume on Plane Astronomy, that no explanation is given of the method of Interpolation, which is of such extensive use in astronomical observations. The places of the heavenly bodies can be known by observation, only for instantsseparated from one another by certain finite intervals of time. By the method above mentioned, their places can be determined for any time we choose, or, if necessary, for every moment. No man has done so much to improve the studies of Cambridge as Mr Woodhouse. His Trigonometry may be said to have introduced the New Calculus into that university. We hope the present work will serve still further to recommend it,—and to Snake its value known, not to the Student only, but also to the Master,