Chap. 3. gives the application of the integral calculus to the theory of series, and to their summation and interpolation; an investigation of the values of definite integrals, and of series proper to give the value of integrals that are functions of great numbers; an exposition of the uses of definite integrals to express functions given by differential equations, &c. Chap. 4. on equations of mixed differences; here we have their analytical theory, and their application to geometrical questions. Of all the information which books could afford, we have already observed, M. LACROIX has availed himself; and it is certainly pro per for an author to be acquainted with the labors of preceding writers on the subjects which he designs to discuss. In abstract science, however, this rule may be less strictly observed: indeed, it certainly is less strictly observed; and D'Alembert said that ma thematicians were very little acquainted with mathematical works. As it is vain to expect perfection, very extensive reading, and much original invention, we wish that M. LA CROIX had devoted part of the time which he has spent in learning what others had thought and invented, in simplifying those theories which are perplexed and involved; and in rendering many demonstrations more direct and rigorous than they are found to be in his treatise. At the conclusion of our observations on his former production, we entered into a short discussion of the respective advantages of the fluxionary and the differential calculus; and by a note to the present work, we find that the author's opinion coincides with our own. He animadverts on the change of notation introduced by La Grange in his analytic functions, and in his treatise on the resolution of numerical equations; and certainly such a change appears to be unnecessary and incommodious. It is true that no alteration is made in the principles of a method by a difference in the notation: but it is desirable to relieve the mind in calculation; and there does not seem to be a more effectual way than that of simplifying the system of signs, by which the process of deduction is carried on. Whether the notations in the differential calculus be more commodious than in the fluxionary, can only be known by observing them in their application to different cases: such an application. we have already made: but, although M. LA CROIX supports us in our opinion, yet we find not, in what he has said, any such confirmation as deserves to be particularly noticed. The present treatise does not abound with very many acute reflections, nor with comprehensive views of the nature, progress, and possible future advancement of analytic science: yet it contains some which are worthy of notice, and which, amid the maze of analytical operations, tend to invigorate the tedious Kk 2 ness ness of calculation; as in pages 174. 460. 483, &c. We may therefore conclude, not that such reflections and views dis. pleased M. LA CROIX when they occurred, but that they did not frequently present themselves *. If we may conjecture the turn of the author's mind from some passages in his former work, and from the general style of his composition, it is not such as would make him linger over demonstration, in order to render it more luminous or more exact: "If it were done, when 'tis done, then 'twere well it were done quickly +." The author appears to us to belong to that class of mathematicians, whom the facility of modern calculation has engendered: who are desirous" to operate rather than to know," and who look more to truth of result than to justness of inference. The ease by which modern analysis obtains its results is certainly, in our opinion, one of the causes which tended to the neglect of evidence and rigour of demonstration; and another cause is the disgust created by the foolish subtleties and vain refinements, into which certain metaphysical mathematicians have been betrayed: but why rush from one extreme to another? All examination of principle does not necessarily degenerate into unsatisfactory subtlety; nor does all discussion inevitably terminate in vain refinement. Because men striving to reason well have reasoned ill, are the rules of an exact logic to be thrown aside? Is there to be no weighing in the scrupulous scales of accurate metaphysics, because things are judged equal by the gross de-cision of sense? The cautions of those who tell us to beware of metaphysical subtleties are to be heard, indeed, but heard with suspicion; and some authors who, with politic regard to their own works, have warned their readers against the nice and curious examination of principles and methods, deserve no higher rank than that of "rude mechanicals" in science;-mere men of the loom and mill, who, if that which they put in comes out stuff and flour, care not by what means and contrivances the process has been conducted. To express our judgment of this work more formally and succinctly is unnecessary; the parts of that judgment are mingled with our preceding observations; and we feel no wish either to add to them, or to take away from them. In a longer discussion, indeed, we could have done more ample justice to M. LA CROIX, and have afforded a more complete view of his treatise: but the frowns of a numerous class of readers check and forbid the frequency of extensive mathematical discussions; and, in an *«Non illi displicuisse jocos, sed non contigisse." other other part of this Appendix, we must pay our respects at some length to an important publication on the Calculus of Derivations, by M. ARBOGAST. ART. VII. Histoire du Général Moreau, &c. i. e. The History of General Moreau, to the Peace of Luneville; containing an Account of his Life, of his Campaigns on the Rhine and in Italy, with Anecdotes and Traits of that Greatness of Mind, Genius, and Bravery, by which he is characterized. 12mo. pp. 250. Paris. 1801. Imported by De Boffe, London. IT is probable that the hero, who is the subject of these pages, is great only as a soldier, and that nothing relating to him in any other view deserves to be consigned to the press. Be this as it may, however, it is certain that the work before us contains little that respects him but as a soldier, though the title page led us to expect something more. To speak of him. in this light; it is the opinion of the best judges that he is the ablest General who has appeared in the late wars. He is said also to be modest, and a stranger to factions; and he manifested conduct interesting to humanity, and to the first personages in this country, when, in the face of the infernal decree of those monsters who composed the Committee of Public Safety, he ventured to save the garrison of Nieuport, consisting almost wholly of Hanoverians: for which indulgence he would have answered with his life, had not those wretches, in the mean time, paid the forfeit of their crimes. Victor Moreau, we are here told, was born at Morlaix in 1763, and studied law at Rennes, where he was admitted to the degree of advocate. The writer says that he will not stop to detail the particulars of Moreau's infancy, nor those of his youth; because, as they possess no superior interest, they cannot enhance his merit, nor impart any new lustre to his history. When his department raised a battalion, the interest of Petiet, since minister of war, and now counsellor of state, gained for him the appointment of its commander. His rise was afterward rapid; till, in the year iv. he was made General in chief of the army of the Rhine and Moselle, at which period the present history commences its details. It would have been interesting to have given us some insight into the mode by which Moreau, after the age of manhood, so speedily made himself master of a new art, wholly distinct from his former pursuits : but no such information here occurs. We say master of a new art, because of all the French Generals of the present day, he is allowed to possess the most military science; and not to have succeeded solely by the powers of genius, and the assistance of Kk 3 adventitious adventitious circumstances. The work forms, however, a neat, and, we believe, an accurate recapitulation of the operations conducted by this celebrated officer; and it will be read with pleasure and instruction by persons who are skilled in military science, and in the geography of the countries which were the theatres of action. Having detailed Moreau's atchievements in his march from Manheim to Munich, the author adds; we are now to see him run a career more difficult and more perilous. Every where surrounded by enemies, and exposed to the greatest dangers, he commenced his famous retreat; which is not less admirable on account of the firmness of the soldiers, than the genius of the chief who directed it, and which will be quoted in all ages as a real prodigy.' The attention of Europe was fixed on this brave army, and its General; and the enemy reckoned on its certain capitulation: while Moreau, by scientific combinations, by his address in drawing the utmost advantage from the excellent dispositions of his troops, and their undaunt ed courage, disappointed their expectations, and maintained and augmented his own fame. After a march of a hundred leagues, amid a thousand difficulties, covered with glorious trophies and brilliant victories, which rank this retreat among the most brilliant operations of which history makes mention, he conducted his army to the frontiers.' A very interesting detail of the siege of Kehl is next inserted. It is here said that the brave resistance made by Moreau and Dessaix, in this fortress, occasioned the surrender of Mantua: certainly, Dessaix's address in obtaining terms is not less admirable than his bravery in defending the place. The French were allowed a certain time to remove whatever they chose; and their alacrity and activity on this occasion, considering the hardships: of the siege which they had undergone, appear wonderful, They left nothing behind them except the bare walls, but carried away every thing; even the palisades, and the balls which the enemy had thrown; and the Austrians who observed them were lost in astonishment, while the French openly said to them, in German, we will not leave thee a nail. The author informs us that Bonaparte and Moreau had never met, till after the return of the former from Egypt; that they received each other with frankness, and testified great reciprocal esteem; that they were above all' jealousy; and that Moreats entered fully into the measures of the dix-huit Brumaire. The writer says nothing respecting the coolness which is reported (perhaps falsely) to have since taken place, between these two conspicuous characters. A por 1 A portrait of Moreau is prefixed to this little work. It presents a countenance not altogether very prepossessing, but indicative of thought and of talents. We know not whether it can claim the merit of accurate resemblance. ART. VIII. Pouvoir Legislatif, &c. i. e. On Legislative Power under Charlemagne. By M. BONNAIRE DE PRONVILLE. 8vo. 2 Vols. pp. 200 in each. Brunswick. 1800. Imported by Dulau and Co. London. Price 10s. 6d. sewed. THE HE authors of the Helvetic Revolutions of the fourteenth century, as we learn from Mr. Planta's excellent History*, founded their claims on prescription: the champions of the people in England, at a later period, proceeded on similar grounds; and even the more early of the French assertors of the popular cause did not disdain to press antiquity into their service, to set up a sort of hereditary right to liberty, and to de- ' duce a kind of legal title to certain political prerogatives, from the records and memorials of remote times. Hence they made themselves the descendants of those Germans, whose liberties the matchless pencil of Tacitus has sketched. The courts and synods of the Merovingian and Carlovingian kings were described as legislative senates; the commonalty was represented as raised to a share in the legislation, by the House of Pepin; and the monarchs were said to be absolute at the head of the army, but under the controul of law as civil chiefs. M. DE PRONVILLE (who, we think, appears to have been bred a lawyer, and was perhaps a member of parliament under the ancien regime,) regards all this as "the baseless fabric of a vision," conjured up for the purposes of innovation, and originating rather in malâ fide, than in ignorance and self delusion. The professed object of the present work, therefore, is to strip off the mask from these artificers of revolutions, by detecting their misrepresentations, exposing their errors, refuting their conclusions, and putting the world in possession of the real facts. He disclaims, on the part of the Franks, any affinity to the free tribes of the Roman historian; and he thinks that the assemblies, with which we see the kings of the first and second race so often act, were merely called together to render an account of their administrations, to give information to the momarch respecting the state of his dominions, to listen to his reprehensions and admonitions, to learn his future pleasure, to be made sensible of their subject condition, to repledge their. fealty, and to serve as pageants to reflect lustre on the court of See M. R. Vol. xxxII. N. S. Kk 4 their |