of fx by the increment: but, since f'x is independent of h, h must disappear by this division, and may be represented by any. symbol at pleasure. Hence, for the sake of uniformity in the dfx x= Hence to find dfx, signs, let be represented by dx. f'x dx or the differential of fx, write in fx, x+dx, for x, develope f(x+dx) as far as the terms affected with the first power of dx, and subtract fx. Since f(x+dx)-fx=ƒ'xdx+, &c. = , f11xdx=df'x &c. &c. f'xdx=dfx df x dfx therefore fx dx = d(ax dx But since dx is invariable, f'(x+dx) dx-f'xdx=f11xdx' but df'xdx=ddfx=d'fx (d' not meaning a symbol d squared, but denoting the second differential of fx to be taken): hence it appears, since ƒ"_d(fx) and = d2fx dx2 ; and thus the derived functions f'x, ƒ`x, ƒ'''*, &c, may be represented by the quantities. dfx difx d3fx &c. dx dx2 dx3' so that the developement of f(x + b) takes this form, the celebrated theorem of Taylor. M. LA CROIX next proceeds to demonstrate the known forms for the differentials of rectangles, fractions, hyperbolic logarithms, circular arcs, &c. He also deduces the differentials of equations, and of a function of two variable quantites, as ƒ (x. y); and his methods, although they want the neatness and brevity which are displayed in those of M. La Grange, yet do not essentially differ from them. The latter part of the first chapter is occupied in shewing the affinity of the method of differential coefficients, to that of the limits of ratios which the increments of variable quantities have to each. The affinity is thus shewn: Let u be a func * The expressions dx, d'x, d'x, dx2, dx3, &c. in the English notation are equivalent to x, x, x', ¤ ̈1, * ̈3, &c. tion of x: when x becomes th, u becomes u+ph+qb2 + sh* +, &c. p, q, r, &c, being derived from %, and independent of h. k Let k=ph+gb2+, &c.—then =p+qh+rb2+, &c. k b. k Let b=0; then, or the limit of is p, the first differential coefficient derived from u: In like manner, confidering as a new function, when x becomes + h, p becomes p+p'b+q'h2, &c. &c, When bo, limit of is p', and so on. Chapter II. is on the principal analytical Uses to which the dif ferential Calculus is applied. These uses are the conversion of multinomials, raised to any power, into series ascending to the powers of the unknown quantities; the expression of the logarithm of quantities, and the arcs of circles, by infinite series in terms of sine, cosine, tangent, &c; the determination of the values of fractions, when the numerators and denominators vanish; the methods of ascertaining the maxima and minimą of quantities, &c. To these uses, it is well known, the differential calculus has long been applied. The application is of no great difficulty, and the method followed by M, LA CROIX does not materially differ from the methods pointed out in former treatises, We pass on, therefore, to Chapter III. which is called a Digression on Algebraic Equations. The problems first solved by M. LA CROIX are the assigning the sum of the mth powers of the roots of an equation, the sum of quantities of this form, amß"y", &c.+a"ßmy", &c. a, ß, y, being the roots of an algebraic equation: these two problems are those which were solved by Waring in the beginning of his Meditationes Algebraica. The author next goes on to shew, the possibility of decomposing, into real factors of second degree, any equation of an even degree. He proves that every equation of a degreep will have a real factor of second degree, if every equation of the degree P.(-1) has a real factor, whether of 1a or 2a degree; p.(p 2 xmamo is resolved into its factors. In detecting the number of impossible roots of an equation, he employs a rule of M, La Grange, which is the same as the rule given by Waring in page 82 of his Meditationes. In speaking of the controversy Between Leibnitz, Bernouilli, Euler, and D'Alembert, concerning the logarithm of negative quantities, M. LA CROIX inclines to the opinion of Euler. Chap. IV. treats on Curve Lines :-but we must now suspend the farther consideration of this work till a future opportunity. [To be continued.] ART. VI. M, Millin's National Antiquities of France, Vol. V. AFTER long delay and much diligent inquiry, we are at length able to lay before our readers some account of another part of this publication. We must refer them to Vol. X. of our New Series, p. 545, for the preceding volume; and, they will there see that we bestowed considerable commendation on the performance, which we have no reason for recalling on a perusal of the pages before us. The author is not a commonplace retailer of objects which fall under every person's notice, nor indeed a mere describer of those which are more neglected, or concealed from general attention; but he appears to be a scientific describer, disposed to entertain, and qualified to improve the reader. Were we to enter into a minute account of the contents of this work, we should transgress those boundaries which justice and necessity impose: we shall therefore only offer a general view, with remarks on two or three of the subjects which present themselves. The articles in this volume are not numerous; because some of them are replete with topics. which demand particular attention, and are therefore extended to greater length than others. Their distinct titles are as follow: Collegiate Church of St. Nicholas at Amiens; Carthusian Monastery of Paris; Antient Palace of Lisle; Collegiate Church of St. Peter, Lisle; Hôpital Comptesse at Lisle; Dominicans at Lisle; Recollects at Lisle; Castle; de Comines; Saint Landry, Paris; Abbey of St. Geneviève, Paris; Library of St. Peter, Lisle. The Chartreuse de Paris, or the Carthusian Monastery, makes a distinguished figure; its domain was extensive, its buildings were large and various, its different departments and ornaments were curious, and often well executed, and it is besides connected with history and events which cannot fail to interest an exact and intelligent examiner. In one part, we remark an inscription dictated, as the author properly expresses it, by fanaticism and adulation, congratulating Lewis XIII on his triumphant entry on the extermination of heretics at Rochelle: that murderous exploit was honored by the monks with the erection of a triumphal arch for the sanguinary monarch. --In another part, we observe the tomb of Hugh, or in the catholie 5 style, 'style, St. Hugues, an early and zealous Carthusian, invited by Henry II. to England; where, as Bishop of Lincoln, he was distinguished during that reign, and those of Richard and John. He died in the year 1200; and his funeral was pompous, it is said, beyond any thing that England had seen; besides multitudes of other ranks, barons and kings assisted at the interment at Lincoln; and several years afterward, the body, found entire, was removed and deposited with the utmost magnificence in the monastery in France. In connection with this-country, we may take notice of another saint, Saint-Hervé; whose father, a native of England, came very young to the court of France. Hervey was born blind; yet, under this impediment, he made considerable proficiency in the sort of knowlege which marked his time, and (as it is said) wrought several miracles, two or three of which, rather of a humourous and diverting kind, are here related; sufficient to discover the base assurance of those who could forge and impose these legendary tales, and the simplicity or sottishness of those who could bestow on them any regard. In one of the numerous chapels belonging to this edifice, an altar appears, with this inscription; Autel Privilégié. Several of our readers probably know that these altars have been held in high estimation. We are sometimes carried back, for their origin, to the year 820; while others bring it down so low as the council of Trent, or about the year 1563, when a brief for the purpose was obtained from Pope Gregory XIII, and these privileged altars no doubt proved a source of rich consolation. The regular monks, as well as the mendicants, readily adopted the practice, and even added to the golden inscription, here by every mass a soul is released from purgatory. Some are said to have promoted the delusion so far as, by means of artificial sparks or flames, during the impious service, to persuade the people that the released soul then flew into paradise. monk, it is related, passing through an usual examination, being interrogated concerning his opinion of privileged altars, replied, "I think them very productive;" or that "they yield great profit," (sont d'un grand produit). One jesuit amused the people with an ingenious, and to himself, no doubt, a lucrative conceit, when he persuaded them that he was himself an autel privilégié; that is, that he imparted benefits of the same kind to any and every altar at which he officiated. A In one of the cloisters, are twenty-two paintings, which we are informed have attracted the attention and admiration of artists throughout Europe; and which were executed in the year 1648, by the celebrated Lesueur, mentioned as holding the first rank in the French school for painting. They are on wood, wood, and placed in handsome arcades opposite to the windows of painted glass. The subject is the life of Saint Bruno, the founder of the monastery, whose history had been painted twice before; once, in 1350, on the walls; and a second time, in 1510, on linen cloth. These paintings are particularly described, with remarks on their beauties or defects. Each of these pictures was accompanied with lines of poetry. At the commencement of the revolution, some care was adopted for their removal: but, unfortunately, the intendant, or overseer of buildings, began to take them down before a place was provided for their reception; and they were hurried into granaries, where they have perished, with many others of the superb gallery of Rubens. This article is terminated by a brief notice of Dom Felix Nonante, the last prior of this house; who is said to have been equally eminent for virtue and for the, love of letters: but nothing could escape the persecution, from internal or external causes, which has recently deluged France with blood; and he perished, says this writer, by the punishment at that time reserved for whatever was useful and meritorious.-In such terms, do those, who approved the principles of the first revolution in France, speak of the horrible consequences which art, fraud, and wickedness afterward effected. The collegiate church of St. Peter at Lisle occupies a great number of pages; accompanied by a short account of the city, in which we particularly remark the use made of dogs for carriage and for draught. They are harnessed as horses are; and three of them draw a little loaded cart, of which a view is given in the print of the antient churches of St. Etienne and St. Peter. We are told that this practice has been introduced at Paris, since the failure of horses.-Numerous are the tombs and monuments here brought under review; all of which we must pass over, and proceed to the chapels. Cybele and Diana had not in antient times, says M. MILLIN, so many churches and chapels as are dedicated to Mary. The chapel of Notre Dame immediately followed the building of the church of St. Peter. The stone-statue of the virgin, sometimes exhibited in public procession, is surrounded by an ornamented and gilded lattice; on which account it is distinguished as Notre Dame de la Treille; and we may imagine, remarks this writer, that Notre Dame de la Treille does not yield for the gift of miracles to all or any of the other Notre Dames in the world. It yields, however, in decoration and magnificence, to that which next occurs, Notre Dame des sept douleurs, (of the seven sorrows) raised at the expence of Philip the Good, who signalized his devotion by a representation of Mary, exerting herself to re-animate the body of |