ficial, and we trust that the publication of them will not be without effect.

In his last lecture, M. VOLNEY proposed to resume the subject after some respite, and to illustrate his maxims by examples: but the Normal School being shortly afterward dissolved, the design was relinquished.

We must not omit to mention that, in one of his lectures, the author has introduced a short dissertation on the construction of halls for deliberative assemblies. The object to which he has principally attended is that of enabling a speaker, with ease, to make himself distinctly heard. Where it is otherwise, he observes, a debate will be frequently carried by strength of lungs. The plan recommended is accompanied with a drawing. In a note to the same lecture, M. VOLNEY has made a comparison between Voltaire and J. J. Rousseau, which we submit to the judgment of our readers.

There is this characteristic difference between Rousseau and Vol taire, considered as leaders of opinions; that if you attack Voltaire before his partizans, they defend him without heat, by reasoning, or by pleasantry, and at most only regard you as a man of bad taste: but, if you attack Rousseau before his admirers, you excite in them a species of religious horror, and they consider you as flagitious.Voltaire, speaking more to the understanding than to the heart, to thought rather than to sentiment, excites no passion; by having been occupied more in combating the opinions of others than in esta blishing his own, he has produced a habit of doubt, more favourable to tolerance than is that of affirmation. Rousseau, on the contrary, addresses the heart and the affections, rather than the understanding; he exalts the love of virtue and of truth (without defining them) by the love of women, so capable of creating illusion; and from a strong persuasion that his own principles are perfect, he suspects in others first the opinion, and afterward the intention:-a situation of mind whence proceed aversion and intolerance.'

Traité du Calcul Differentiel, &c.; i. e. A Treatise on the Differential and Integral Calculus. By S. F. LA CROIX. 4to. 2 Vols. pp. 520 and 730. Paris.-Imported by De Boffe, London. Price 21. 25.

HE reasons assigned for the publication of the present work, by M. M. La Place and Legendre, who were ap pointed to examine it, are the scarcity of the works of Euler on the differential and integral calculus, the improvements which have been made in all. branches of analysis since the time of that mathematician, and the difficulty of access to the -volumes of the several academies, among which so many important memoirs are dispersed.

In the plan of this work, it is proposed to comprehend and systematize all that has been written on the differential and integral calculus; an undertaking of great utility, labour, and difficulty: since, from the time of Newton to the present day, there has not been produced a single treatise which was at once clear and profound. With deep and intricate disquisitions on the differential and integral calculus, the analytic art has indeed been most abundantly enriched :-but the real source of complaint arises from obscurity and want of evidence in its principles. Although, to remove this obscurity, and to vindieate the stability on which the doctrine of fluxions is built, many elaborate treatises have been written, the greater part of them abound with absurdity and sophistry, and excite a doubt whether the praise which Bacon gives to the mathematics, of habituating the mind to just reasoning, be really due to them. Who would direct his ridicule against the refinements, subtleties, and trifling of the schoolmen, if he read what has been written by some men who were presumed to be the greatest masters of reason, and whose employment and peculiar priviJege consisted in deducing truth by the justest inferences from the most evident principles? The history of the differential calculus, indeed, shews that even mathematicians sometimes bend to authority and a name, are influenced by other motives than a love of truth, and occasionally use (like other men) false metaphysics and false logic. No one can doubt this, who reads the controversial writings to which the invention of fluxions gave rise: he will there find most exquisite reasonings concerning quantities which survived their grave, and, when they ceased to exist, did not cease to operate; concerning an infinite derivation of velocities, and a progeny of infinitesimals smaller than the "moonshine's wat'ry beams," and more numerous than

"Autumnal leaves that strow the brooks

In Vallombrosa." (Milton, Par. Lost. I. 302.)

The contemporaries and partizans of Newton were men infinitely inferior to him in genius: but they had zeal, and were resolved to defend his opinions and judgments. Hence they undertook the vindication of fluxions, according to the principles and method of its author; although it may be fairly inferred, from the different explanations given of that doctrine by New

• The following passage, from the masterly preface of Torelli, was aimed at the partizans of Newton: "Qui vero, quod unus aliquis affirmaverit, id ita esse sine ulla probatione credit, præpostere agit; dum id homini tribuit, quod unice rationi tribuendum est.”

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ton in different parts of his works, that Newton himself was not perfectly satisfied of the stability of the grounds on which he had established it. It was not to be expected that pigmies should effect that which had baffled a giant but they acted as men always act in similar circumstances. Bigoted to their opinion, yet, unable to establish it by evident and just reasoning, they employed the arts of sophistry; and hence arose elaborate and refined disquisitions concerning nothings, and perplexing paradoxes concerning the infinite divisibility of matter, abstract extension, and velocities that were to be conceived independently of time and space. The "ghosts of departed quantities," a mighty host of " shadowy entities," were conjured up to enthrall the common sense of mankind. That men, and even those who assume the title of philosophers, should fall into error and absurdity, is no great matter for wonder; history has its thousand similar instances: but it may justly create surprize that Newton, who had invented the method of séries, and who to solve a problem in the second book of his Principia had employed first the method of series and then that of fluxions, should not have given to the latter a more natural and scientific origin; and that, in the collision of opposite arguments, more truth should not have been elicited;—because ali men, however they might admire, did not believe Newton to be infallible, and some freely and frequently arraigned the obscurity of his method. We may wonder, also, that, having beyond all controversy obtained truth, mathematicians should have been unable to make it science; for the method was simple and easy in its application, and exact and rigorous in its conclusions. Viewed as a whole, it appeared to possess the greatest stability; though its foundations, seen through a mist, seemed uncertain, and of discordant and unsuitable materials;

" Minimas rerum discordia vexat,

Pacem summa tenent.”

The treatise of M. LA CROIX is introduced by an excellent preface, containing a history of the differential and integral calculus: which history is properly dated from the times of

To find the law of resistance necessary for a heavy body to describe freely a given curve, Newton employed, in the first edition of his Principia, the method of series. Here, by neglecting some terms of the series which ought to have been considered, he fell into an error. John Bernouilli detected this error, but not the real cause; and Newton himself, not discerning it, acandoned the method, and gave, in the following edition of his Principia, a solution of the same problem according to the method of fluxigns.

Euclid

Euclid and Archimedes, because the methods of exhaustions and limits rest on the same foundation.

• The discovery of the differential and integral calculus (says the writer,) is to be assigned to no higher a period than that of the last age: but the questions which led to its discovery have been discussed since the earliest times of geometry. When the antient geometricians sought to compare curvilinear figures, either one with another, or with rectilinear figures, they were necessitated to give a new turn to their demonstrations. The 12th proposition of the 12th book of Euclid's Elements offers the first essay of this kind of problems, that has come down to us :-its object is to prove that the surfaces of circles are to one another as the squares of their diameters. In this proposition we pass from finite to infinite; since, in the foregoing proposition, Euclid proves that this relation is the same as that of similar polygons described in two different circles; and it seems evident to me that the geometrician, whoever he was, who discovered this truth, perceiving it to be independent of the number of the sides of the polygons, and at the same time that they differed the less from circles as their sides were more numerous, hence necessarily concluded, by virtue of the law of continuity, that the property of the first belonged to the second. By such reasonings, the proposition (their object) is now considered as sufficiently proved; and the generality of elementary books do not give the reasonings equally complete. The antients were more difficult in this respect than we are; they never indulged themselves in the privilege of confounding two quantities which had a difference, however minute that difference might be. To place beyond the reach of doubt and cavil the proposition of which they may be said to have divined the existence, by the considerations which I have mentioned, they sought to prove that the relation of circles to one another could be neither greater nor less than that of the squares of their dia meters; and to obtain such proof, they began by shewing that an inscribed polygon might always be found which only differed from the corresponding circumscribed polygon, and à fortiori from the circle itself, by a quantity less than any given magnitude.

Archimedes, by nearly similar methods, advanced to the solution of much more difficult problems; such as the relations between the surfaces and solid contents of the cylinder and sphere, the quadrature of the parabola, and the properties of spirals; but let us not imagine that he arrived at these discoveries by the methods transmitted to

us.'

M. LA CROIX then proceeds in his history of the differential calculus, as connected with the several discoveries made by Cavalleri, Roberval, Descartes, Fermat, Huygens, Gregoire de St. Vincent, Pascal, Wallis, Barrow, Leibnitz, and Newton. In our critique on M. Lagrange's Theory of Analytical Functions, (App. Rev. vol. xxviii. p. 481.) we entered into this history; and we refrain from quoting the observations of M. LA CROIX, because they differ so little from those which we have already given to the public. We must confess, how

ever, that we feel no inconsiderable satisfaction in finding that our opinions on many subjects agree with those of the author of the present treatise. In the article just quoted, we consi dered particularly the nature of the principle which Newton gave to his doctrine of fluxions; and we shewed that such principle was foreign to the subject, unnecessary, and fictitious; since, when analysed, it was not the real principle from which the doctrine was deduced. M. LA CROIX says:

Newton supposed lines to be generated by the motion of a point, and surfaces by the motion of a line; and he gave the name of fluxions to the velocities which regulated the motions. These notions, although rigorous, are foreign to geometry, and their application is difficult. It is very true that, by imagining a point which moves on a line while the line is carried parallel to itself with an uniform velocity, we may represent any curve whatever: but the velocity of the de scribing point being variable at each instant, we can only determine it by recurring to the method of the antients, to that of exhaustions, or to that of prime and ultimate ratios. It is of this last method that Newton almost always avails himself; so that, properly speaking, fluxions were to him only a means of giving a sensible existence to the quantities on which he operated. By the method of prime and ultimate ratios, he understood the investigation of the relation of quantitics at the first and last instant of their existence, when the quantities were generated or vanished together; and he found, in the prime ratio of spaces described by the ordinate on the line of the abscissas, and by the describing point on the ordinate, (spaces which he called moments,) the ratio of the fluxion of the abscissa to that of the ordinate; whence he determined the direction of the tangent. The calculus was merely that used by Barrow in his method of tangents; which Newton, by means of his formula for the binomial, and by his reduction into series, had extended to irrational expressions. The advantage of the method of fluxions over the differential calculus, in point of metaphysics, consists in this; that, fluxions being finite quantities, their moments are only infinitely small quantities of the first order, and their fluxions are finite; by these means, the consideration of infinitely small quantities of superior orders is avoided.'

The controversy between Newton and Leibnitz, concerning the discovery of the fluxionary calculus, is here examined and decided; and here also we agree with M. LA CROIX, as well as in his opinion concerning the real merit of Leibnitz. The decision of the Committee of the Royal Society is not hastily and inconsiderately to be accused of unjust partiality but we strongly insist that those men must have had minds deplorably weak, or miserably perverted by party-zeal and national prejudice, who denied the praise of genius to Leibnitz; who, amid a thousand other pursuits, instructed Bernouilli and rivalled Newton, not on subjects of small concern and easy APP. REV. VOL. XXXI.

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