metaphysics, as we may call it, of that calculus, displays none of his usual talent and accuracy of thought. He contends, that fluxions or differentials have no magnitude whatever, and are truly and literally equal to nothing; which is a harsh and inaccurate way of stating what is much better expressed by Newton in his doctrine of prime and ultimate ratios; or by Maclaurin, where he considers fluxions as the measures of velocity. There were, however, some exceptions to the generality of the observations which we are now making; and D'Alembert, in particular, though he has not written professedly on the subject of the principles of the Calculus, yet, whenever he has occasion to state any thing relative to it, never fails to do so in the luminous manner that we might expect from a geometer who was a metaphysician and a philosopher. Carnot, whose name is so well known, was one of the first among the French mathematicians who treated professedly of the metaphysics of the differential calculus. The little tract which he wrote on this subject is full of ingenious and sound views; but such as, though presented in a new form, and one that appeared quite original both to the author and his countrymen, are in reality very little removed from the method of prime and ultimate ratios. Carnot, however, had the merit of accommodating that method to the form and language of the calculus, better than we were accustomed to do, by stating that a differential equation is not an exact, but only an approximate equation, which comes continually nearer to the truth the less the fluxions or differentials are that are involved in it. Lagrange, however, has placed the matter on the true foundation; and has shown that, in delivering the general rules of the differential and integral calculus, there is no need for introducing evanescent quantities, or quantities less than any thing that is assigned. Thus, the differential calculus is reduced to the algebra of variable but finite quantities; and it is only when the application of the general formulas is made to geometric magnitudes, that the ultimate ratios of evanescent quantities come to be considered; and they do so in a manner that admits of strict demonstration. This step is undoubtedly one of the greatest that has been made in the new analysis since the period of its invention; and we have often wondered that the works of Lagrange, which contain the developement of this idea, have not produced a greater sensation among the mathematicians of this island, who have always aimed so much at accuracy in their manner of treat. ing this subject. We will not allow ourselves to suppose that this proceeds from any illiberal jealousy, or any unwillingness to acknowledge the superior success of a foreigner in a pursuit in which they themselves had been engaged. We must rather ascribe this apparent indifference to the ge neral agitation of Europe, and the interruption of all intercourse, even that of letters, between France and England. On the Continent, these works seem to meet with the reception they deserve. The Théorie des Fonctions was published in the year 5 of the French Revolution. The first edition of the Calcul des Fonctions was in 1805; and the second edition, which is now before us, in 1806. Another treatise of Lagrange is noticed in this report, Traité de la résolution des équations numériques de tous les dégrés. This is also a work of great merit, and yet it is but little known in this country, though the memoir which is the foundation of it was published by Lagrange in the Berlin Memoirs so long ago as the year 1767. It deserves to be particularly studied; and nothing more useful could be done in an elementary treatise of algebra, than to give to this method of approximating to the roots of equations the simplest form which it admits of. The last article under this head is the Mécanique Céleste of Laplace, on which, as is well known, too much praise cannot be bestowed. We have already considered this work with a minuteness that renders any further observations on it unnecessary in this place. The Report mentions three articles in practical mechanics; the timekeepers for finding the longitude, constructed by Berthoud, which gained the prize of the Institute; the hydraulic ram of Montgolfier; and, lastly, a machine approved by the Class of the Sciences, the Pyreolophorus of Messrs Lenieps, a new invention, in which, if we understand the very short notice concerning it which the editor has given in a note, the force of air suddenly expanded by heat, is made to raise a weight, or overcome a resistance. In an experiment made with this machine, it is said, that a boat, loaded with five quintals, and presenting to the water a prow of the area of six square feet, was carried up the Soane with a velocity double that of the stream. In another experiment, the pressure exerted on a piston of three square inches was in equilibrio with 21 ounces, and the fuel consumed weighed only six grains. We want here a necessary element, the time in which these six grains were consumed. This omission may perhaps be supplied from another part of the account, where it appears that each stroke of the piston takes up five seconds. The six grains were the fuel consumed in five seconds. 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 five 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 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 |