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alone, but must be shared, in various proportions, among the philosophers and mathematicians of all ages. Their efforts, from the age of Euclid and Archimedes, to the time of Newton and La Place, have all been required to the accomplishment of this great object; they have been all necessary to form one man for the author, and a few for the readers, of the work before us. Every mathematician who has extended the bounds of his seience; every astronomer who has added to the number of facts, and the accuracy of observation ; every artist who has improved the construction of the instruments of astronomy-all have cooperated in preparing a state of knowledge in which such a book could exist, and in which its merit could be appreciated. They have collected the materials, sharpened the tools, or constructed the engines employed in the great edifice, founded by Newton, and completed by La Place.
In this estimate we detract nothing from the merit of the author himself; his originality, his invention, and comprehensive views, are above all praise ; nor can any man boast of a higher honour than that the Genius of the human race is the only rival of his fame.
This review naturally gives rise to a great variety of reflections. We shall state only one or two of those that most obviously occur.
When we consider the provision made by nature for the stability and permanence of the planetary system, a question arises, which was before hinted at, - whether is this stability necessary or contingent, the effect of an unavoidable or an arbitrary arrangement? If it is the necessary consequence of conditions which are themselves necessary, we cannot infer from them the existence of design, but must content ourselves with admiring them as simple and beautiful truths, having a necessary and independent existence. If, on the other hand, the conditions from which this stability arises necessarily, are not necessary themselves, but the consequences of an arrangement that might have been different, we are then entitled to conclude, that it is the effect of wise de sign exercised in the construction of the universe.
Now, the investigations of La Place enable us to give a very satisfactory reply to these questions; viz. that the conditions essential to the stability of a system of bodies gravitating mutually to one another, are by no means necessary, insomuch that systems can easily be supposed in which no such stability exists. The conditions essential to it, are the movement of the bodies all in one direction, their having orbits of small eccentricity, or not far different from circles, and having periods of revolution not commensurable with one another. Now, these conditions are not necessary; they may casily be supposed different ; any of
them might be changed, while the others remained the same. The appointment of such conditions therefore as would necessarily give a stable and permanent character to the system, is not the work of necessity; and no one will be so absurd as to argue, that it is the work of chance : It is therefore the work of design, or of intention, conducted by wisdom and foresight of the most perfect kind. Thus the discoveries of La Grange and La Place lead to a very beautiful extension of the doctrine of final causes, the more interesting the greater the objects are to which they relate. This is not taken notice of by La Place; and that it is not, is the only blemish we have to remark in his admirable work. He may have thought that it was going out of his proper province, for a geometer or a mechanician to occupy himself in such speculations. Perhaps, in strictness, it is so; but the digression is natural: and when, in any system, we find certain conditions established that are not necessary in themselves, we may be indulged so far as to inquire, whether any explanation of them can be given, and whether, if not referable to a mechanical cause, they may not be ascribed to intelligence.
When we mention that the small eccentricity of the planetary orbits, and the motion of the planets in the same direction, are essential to the stability of the system, it may naturally occur, that the comets which obey neither of these laws in their motion may be supposed to affect that stability, and to occasion irregularities which will not compensate one another. This would, no doubt, be the effect of the comets that pass through our system, were they bodies of great mass, or of great quantity of matter. There are many reasons, however, for supposing them to have very little density; so that their effect in producing any disturbance of the planets is wholly inconsiderable.
An observation somewhat of the same kind is applicable to the planets lately discovered. They are very small; and therefore the effect they can have in disturbing the motions of the larger planets is so inconsiderable, that, had they been known to La Place (Ceres only was known), they could have given rise to no change in his conclusions. The circumstance of two of these planets having nearly, if not accurately, the same periodic time, and the same mean distance, may give rise to some curious applications of his theorems. Both these planets may be consisiderably disturbed by Jupiter, and perhaps by Mars,
Another reflection, of a very different kind from the preceding, must present itself, when we consider the historical details concerning the progress of physical astronomy that have occurred in the foregoing pages. In the list of the mathematicians and philosophers, to whom that science, for the last sixty or seventy years,
has been indebted for its improvements, hardly a name from Great Britain falls to be mentioned. What is the reason of this ? and how comes it, when such objects were in view, and when so much reputation was to be gained, that the country of Bacon and Newton looked silently on, without taking any share in so noble a contest? In the short view given above, we have hardly mentioned any but the five principal performers; but we might have quoted several others, Fontaine, Lambert, Frisi, Condorcet, Bailly, &c. who contributed their share to bring about the conclusion of the piece. In the list, even so extended,
there is no British name. It is true, indeed, that before the period to which we now refer, Maclaurin had pointed out an improvement in the method of treating central forces, that has been of great use in all the investigations that have a reference to that subject.. This was the resolution of the forces into others parallel to two or to three axes given in position and at right angles to one another. In the controversy that arose about the motion of the apsides in consequence of Clairaut's deducing from theory only half the quantity that observation had established, as already stated, Simpson and Walmesley took a part; and their essays are allowed to have great merit. The late Dr Mathew Stewart also treated the same subject with singular skill and success, in his Essay on the Sun's distance. The same excellent geometer, in his Physical Tracts, has laid down several propositions that had for their object the determination of the moon's irregularities. His demonstrations, however, are all geometrical; and leave us to regrete, that a mathematician of so much originality preferred the elegant methods of the ancient geometry, to the more powerful analysis of modern algebra. Beside these, we recollect no other names of our countrymen distinguished in the researches of phy, sical astronomy during this period; and of these none made any attempt toward the solution of the great problenis that then occupied the philosophers and mathematicians of the continent. This is the more remarkable, that the interests of navigation were deeply involved in the question of the lunar theory; so that no motive, which a regard to reputation or to interest could create, was wanting to engage the mathematicians of England in the inquiry. Nothing, therefore, certainly prevented them from eng ging in it, but consciousness that, in the knowledge of the higher geometry, they were not on a footing, with their brethren on the Continent. This is the conclusion which unavoidably forces itself upon us, and which will be but too well confirmed by locking back to the particulars which we stated in the beginning of this review, as either essential or highly conducive to the improvements in physical astronomy.
The calculus of the sines was not known in England till within these few years. Of the method of partial differences, no mention, We believe, is yet to be found in any English author, much less the application of it to any investigation. The general methods of integrating differential or fluxionary equations, the criterion of integrability, the properties of homogeneous equations, &c. were all of them unknown, and it could hardly be said, that, in the more difficult parts of the doctrine of Fluxions, any improvement had been made beyond those of the inventor. At the moment when we now write, the treatises of Maclaurin and Simpson, are the best which we have on the fluxionary calculus, though such a vast multitude of improvements have been made by the fo. reign mathematicians, since the time of their first publication. These are facts, which it is impossible to disguise ; and they are of such extent, that a man may be perfectly acquainted with every thing on mathematical learning that has been written in this country, and may yet find himself stopped at the first page of the works of Euler or D'Alembert. He will be stopped, not from the difference of the fluxionary notation, (a difficulty easily overcome), nor from the obscurity of these authors, who are both very clear writers, especially the first of them, but from want of knowing the principles and the methods which they take for granted as known to every mathematical reader. If we come to works of still greater difficulty, such as the Méchanique Céleste, we will venture to say, that the number of those in this island, who can read that work with any tolerable facility, is small indeed. If we reckon two or three in London and the military schools in its vicinity, the same number at each of the two Eng lish Universities, and perhaps four in Scotland, we shall not hardly exceed a dozen ; and yet we are fully persuaded that our reckoning is beyond the truth.
If any further proof of our inattention to the higher mathematics, and our unconcern about the discoveries of our neighbours were required, we would find it in the commentary on the works of Newton, that so lately appeared. Though that commentary was the work of a man of talents, and one who, in this country, was accounted a geometer, it contains no information about the recent discoveries to which the Newtonian system has given rise; not a word of the problem of the Three Bodies, of the disturbances of the planetary motions, or of the great contrivance by which these disturbances are rendered periodical, and the regularity of the system preserved. The same silence is observed as to all the improvements in the integral calculus, which it was the duty of a commentator on Newton to have traced to their origin, and to have connected with the discoveries of his master. If Dr FOL. XI. NO. 22.
Horseley has not done so, it could only be because he was unacquainted with these improvements, and had never studied the methods by which they have been investigated, or the language in which they are explained.
At the same time that we state these facts as incontrovertible proofs of the inferiority of the English mathematicians to those of the Continent, in the higher departments; it is but fair to acknowledge, that a certain degree of mathematical science, and indeed no inconsiderable degree, is perhaps more widely diffused in England, than in any other country of the world. The Ladies Diary, with several other periodical and popular publications of the same kind, are the best proofs of this assertion. In these, many curious problems, not of the highest order indeed, but still having a considerable degree of difficulty, and far beyond the mere elements of science, are often to be niet with; and the great number of ingenious men who take a share in proposing and answering these questions, whom one has never heard of any where else, is not a little surprising. Nothing of the same kind, we believe, is to be found in any other country. The Ladies' Diary has now been continued for more than a century; the poetry, enigmas, &c. which it contains, are in the worst taste possible; and the scraps of literature and philosophy are so childish or so old-fashioned, that one is very much at a loss to form a notion of the class of readers to whom they are addressed. The geometrical part, however, has always been conducted in a superior style ; the problems proposed have tended to awaken curiosity, and the solutions to convey instruction in a much better manner than is always to be found in more splendid publications. If there is a decline, therefore, or a deficiency in inathematical knowledge ir this country, it is not to the genius of the people, but to some other cause that it must be attributed.
An attachment to the synthetical methods of the old geometers, in preference to those that are purely analytical, has often been assigned as the cause of this inferiority of the English mathematicians since the time of Newton. This cause is hinted at by several foreign writers, and we must say that we think it has had 110 inconsiderable effect. The example of Newton himself may have been hurtful in this respect. That great man, influenced by the prejudices of the times, seems to have thought that algebra and Muxions might be very properly used in the investigation of truth, but that they were to be laid aside when truth was to be communicated, and synthetical demonstrations, if possible, substituted in their room. This was to embarrass scientific method with a clumsy and ponderous apparatus, and to render its progress indirect and slow in an incalculable degree. The controversy