than to give to this method of approximating to the roots of equa tions the fimpleft form which it admits of.

The laft article under this head is the Mechanique Céleste of La Place, on which, as is well known, too much praife cannot be beftowed. We have already confidered this work with a minutenefs that renders any further obfervations on it unneceffary in this place.

The report mentions three articles in practical mechanics; the timekeepers for finding the longitude, conftructed by Berthoud, which gained the prize of the Inftitute; the hydraulic ram of Montgolfier; and, laftly, a machine approved by the Clafs of the Sciences, the Pyréolophorus of Meffrs Lenieps, a new invention, in which, if we understand the very fhort notice concerning it which the editor has given in a note, the force of air fuddenly expanded by heat, is made to raife a weight, or overcome a resistance. In an experiment made with this machine, it is faid that a boat, loaded with five quintals, and prefenting to the water a prow of the area of fix fquare feet, was carried up the Soane with a velocity double that of the ftream. In another experiment, the preffure exerted on a pifton of three fquare inches was in equili brio with 21 ounces, and the fuel confumed weighed only 6 grains. We want here a neceffary element, the time in which thefe 6 grains were confumed. This omiffion may perhaps be fupplied from another part of the account, where it appears that each troke of the piston takes up five feconds. The 6 grains were the fuel confumed in fix feconds.

Much more information, however, than we have at present, is necessary, in order to form any estimate of the merit of this machine, and to judge whether it has any chance of becoming a rival to the steam engine.

The next general head of the report is Astronomy; and here the new astronomical tables form the first, and indeed the most important article. This subject we have also anticipated in the review of Vince's Astronomy, or, as the title ought to have been, of Vince's edition of the Tables of Burg and Delambre.

A curious article is given with respect to the comet of 1770, which has long occupied the attention of astronomers, from the singular circumstance that the only ellipse that could be made to agree with its motions during the time it was visible, is one in which it must revolve in fiye years and a half nearly yet this comet has never been observed since 1770, and never was seen before. The singular problem to which this paradoxical phenomenon gives rise, was proposed as the subject of a prize by the National Institute, and the prize was gained by M. Burckhardt, a most skilful and laborious astronomer. From immense calculations he

has

has made it appear that the attraction of Jupiter had rendered that comet visible, from having been invisible before because of its great distance, and has also rendered it invisible again, by undoing its former effect, and reducing the comet to move in an orbit that does not admit of its coming near enough to the Sun to be visible from the Earth.

It is not one of the least remarkable circumstances in the history of a period big with novelty, that since the beginning of the present century four new planets have been discovered. These are all of them so small, that it is no wonder they escaped observation, and were even by astronomers confounded with the millions of stars of the same apparent magnitude that occupy almost every point of the heavens. From their smallness it follows, that they have no sensible effect in disturbing the motions of the planets already known. Their orbits are considerably eccentric, and the plane of one of them has an inclination to the ecliptic greater than the inclinations of all the other planetary orbits put together. This great inclination and eccentricity will render the calculation. of the disturbances produced in the motion of these bodies by the larger planets, (particularly of Jupiter and Mars, between which they are situated), a matter of considerable difficulty, and may be the occasion, as Delambre remarks, of extending the science of analysis beyond its present bounds.

The first of these planets was discovered by Piazzi at Palermo, the third by M. Harding, the two others by Mr Olbers of Bremen. The astronomer last named is of opinion, that these small bodies are the fragments of one large planet which an explosion, from some unknown cause, has burst in pieces; and hence he concludes, that all their orbits ought to cut one another in two opposite points of the heavens, which he found, by calculation, to be, one near the constellation Virgo, and the other near the Whale; and that, of course, they must pass through these points twice in every revolution; so that, in order to discover all the fragments, astronomers ought to examine these two places of the heavens very frequently. In effect, all the four have been found near these points; and the two last, after Olbers had suggested the idea just

mentioned.

Since the year 1789, seventeen comets have been discovered; and along with the names of Messrs Mechain, Olbers, &c. by whom they were observed, we are glad to see the name of Miss Herschel. The orbits of all these comets have been calculated, The comet that appeared so remarkable in the autumn of 1807, is thought by Olbers to revolve in a very eccentric ellipse, and to have a periodic time of no less than 1900 years.

Delambre concludes this article with Dr Herschel's descrip

tion of the heavens, the double, triple, quadruple, and nebulous stars, together with those that have disks like planets, in some cases round, in others irregular. The discovery of the revolution of Saturn's ring by the same excellent astronomer, is also mentioned, as also the coincidence of the time of that revolution with the theory of gravity, and the prediction of La Place. The observations of Dr Herschel on the figure of Saturn himself are not mentioned.

A rumour prevailed for some time, that Piazzi had discovered the parallax of the fixt stars; but as M. Delambre makes no mention of a discovery, which, if real, would be no doubt one of the greatest in astronomy, we must suppose that M. Piazzi's observations have not yet led to any satisfactory result. The notes mention a work, founded mostly on Dr Herschel's observations, by Schræter of Amsterdam, on the dimensions of the universe it was crowned by the Royal Society of Haerlem in 1802; it cannot fail to be highly interesting; and we very much regret that it has not yet reached this country.

The next general head is that of Physique Mathematique, or what we would call experimental philosophy. Delambre begins with remarking, that the revolution recently brought about in chemistry, could not happen without turning many experimentalists a little out of their ordinary course, when they saw in a neighbouring science a road opened that promised more numerous discoveries. We shall nevertheless find, in the mathematical branch of Physics, some curious researches and interesting inventions.'

Among these, one of the most remarkable is the Balance of Torsion; which, by the twisting and untwisting of a thread or wire, affords a measure for forces that are too small to be appreciated by any other means. It was with this that Coulomb was so successful in determining the law of electric attraction and repulsion, and afterwards in showing that the phenomena of magnetism follow a law altogether similar, namely, the inverse of the square of the distance. By help of the same instrument, he was able to measure the smallest effects of magnetism; to find the temperature (considerably elevated) at which these effects entirely disappear; and to show that magnetism is not, as has been generally supposed, a property peculiar to certain bodies, but one that exists in all, even in those that appear most devoid of it. The same balance enabled him to measure the resistance which fluids oppose to motion, the law of which resistance is expressed by two terms, of which Newton found out only the first, the second or being sensible except in motions that are extremely slow. The instrument by which Mr Cavendish determined the gravitation of balls of lead toward one another, is, as Delambre observes,

serves, no other than Coulomb's Balance, executed on a large scale. Mr Cavendish found from his experiments, that the mean density of the earth is five times and a half as great as that of

water.

Here, however, we must be permitted to observe, that though Mr Cavendish's Balance does not differ in principle from that of the excellent experimenter quoted by Delambre, it was not copied from it. The experiments of Mr. Cavendish were indeed made about the year 1798; and the first experiments of Coulomb with his balance are published as early as the year 1785, if we mistake not. The instrument that Mr. Cavendish employed had however been invented before that period by the Rev. Mr Mitchell, F. R. S., and was purchased at his sale by Mr. Cavendish. We are to consider the instrument therefore as originally an English invention, and re-invented in France by Coulomb, without any knowledge whatever of what was done in England.

We cannot help remarking too, when we reflect on the results. obtained from this instrument in the hands of the English and of the French philosopher, that the gravitation measured by the former, may have been affected by the magnetism which the latter supposed he had discovered in all bodies. The effects of the one force may have been mistaken for those of the other, and a degree of uncertainty is thrown on the determinations of both. This observation, however, we only throw out loosely perhaps an accurate comparison of the experiments might determine how much is to be ascribed to the one cause, and how much to the other: it is right, however, that this source of inaccuracy should be considered.

The application of the Barometer to the measurement of heights, or more properly the formula for determining heights by help of the Barometer, deduced by La Place, is mentioned among the discoveries of the latter. La Place used in his formula the specific gravity of mercury, as it is commonly stated. The coefficient or multiplier of the logarithmic difference which he thus obtained, was found, on comparing his Barometric measures with certain heights, trigonometrically determined by M. Ramond in the Pyrenees, to require a small correction. The coefficient, thus adjusted, was found by Biot to agree perfectly with the experiments on the specific gravity of mercury when accurately repeated; and his experiments also gave the same refraction which Delambre had deduced from astronomical observations.

In the prosecution of these experiments M. Biot found that the refracting power of different gases affords means more accurate than the ordinary processes of chemistry for inquiring into the composition of certain substances, such as the Diamond,

which he concluded to be partly composed of oxygen. The idea of inferring the chemical composition of bodies from their refracting power, as is well known, was first conceived by Newton : it seems to have been much extended and improved on by the philosopher just named.

It is not taken notice of in the report, but we think it right to remark it, that the rule for barometric measurements had been investigated on strict mathematical and mechanical principles long before it was done by La Place, and formulas brought out, which do not materially differ in their results, though they do considerably in their forms, from that of the French geometer. After De Luc made his improvements, and discovered by trial the very simple rule which he employs, leaving it however quite empirical, and not deduced from principles, a geometric demonstration of that rule was given by Dr Horseley in the Philosophical Transactions. An investigation of the same, purely analytical, was published by Professor Damen of Leyden; and a third, which considers the problem with great generality, and takes into view, several circumstances which had not hitherto been attended to, is given by Professor Playfair in the first volume of the Edinburgh Transactions. The investigation of La Place therefore was not entirely new as to its object or its principles, though we believe his method to be original, and in all respects worthy of its author. His rule, even when corrected as above mentioned, does not perfectly agree with that which we employ in this country, of which the form is agreeable to the investigations just mentioned, and the coefficients determined from the excellent experiments of General Roy and Sir G. Shuckborough. It is also less commodious in practice, than either our formula or that of M. Trembley of Geneva. We are not however perfectly prepared to state in what the difference consists, or to what extent it goes. As the question now stands, we think a comparison of the different Barometric formulas is an excellent subject for a mathematical memoir.

Under the article of Magnetism, the report mentions the series of observations published by M. Gilpin, in the Philosophical Transactions for 1806, from which some curious results may be deduced concerning the secular variations of the magnetical meridian. Another article relates to Dr Wollaston's apparatus for measuring, in a manner extremely simple and accurate, the refraction of transparent bodies, (Phil. Trans. 1802.) It is said, that a very valuable addition to this apparatus has been made in France, by M. Malus; and that an analytical consideration of the subject had enabled him to correct an error which had escaped Dr Wollaston, We do not know if any more particular account of M. Malus's improvement has yet reached England.

The