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After the appearance of the Principia, no discovery of importance in geometry had been published by the english, if we except the solution of the problem of the line of swiftest descent. Toward the end of the year 1704, Newton gave to the World in one volume his Optics in english, an enumeration of lines of the third order, and a treatise on the quadratures of curves, both in latin. His optics are foreign to the purpose here. The enumeration of lines of the third order is a profound and original work, though it rests simply on the common analysis, and the theory of series, which Newton had carried to a great length. It contains little more than enunciations and results, and has since been commented upon by several learned geometricians, to whom it has afforded an ample harvest of very curious researches. His other work, the treatise on quadratures, belongs to the new geometry.
The particular object of this treatise is the resolution of differential formulæ of the first order, or of a single variable quantity; on which depends the precise, or at least the approximate, quadrature of curves. With great address Newton forms series, by means of which he refers the resolution of certain complicated formulæ to those of more simple ones; and these series, suffering an interruption in certain cases, then give the fluents in finite terms. The development of this theory affords a long chain of very elegant propositions, where among other curious problems we remark the method of resolving rational fractions, which was at that time difficult, particularly when the roots are equal, Such an important
and happy beginning makes us regret, that the author has given only the first principles of the analysis of differential equations. It is true he teaches us to take the fluxions, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus: but he does not inform us, how to solve the inverse problem; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem. Newton's opponents have argued from his treatise on quadratures, that, when this work appeared, the author was perfectly acquainted only with that branch of the inverse method of fluxions which relates to quadratures, and not with the resolution of differential equations.
Newton almost entirely melted down the treatise of Quadratures into another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinites, that is to say, the methods of determining the tangents of curve-lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several lustroque times
times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher geometry. In 1736, nine years after Newton's death, Dr. Pemberton gave it to the World in english. In 1740 it was translated into french, and a preface was prefixed to it, in which the merits of Leibnitz are depreciated so excessively, and in such a decided tone as might impose on some readers, if the writer of this preface [Buffon] had not sufficiently blunted his own criticisms, by betraying how little knowledge of the subject he possessed. Notwithstanding his public and frequently repeated efforts, he was never able to penetrate to any depth in the higher geometry; and the anecdote of the strange meaning he had affixed to the latin words de testitudine quadrabili of Viviani, on which he had written a little dissertation, that was fortunately left out of that preface by the advice of one of his friends, is still remembered. To posterity he is already known only by his Natural History, in which the philosopher, while he condemns some wanderings of the imagina tion, cannot avoid admiring several grand and just ideas, as well as the loftiness and elegance of the diction.
In 1711 another work of Newton's appeared, his Method of Fluxions; the foundation of which he had already laid in his Principia under a different form. The object of this Method is, to find the linear coefficients of an equation that satisfies as many conditions as there are coefficients, or to con
struct a curve of the parabolic kind passing through any number of given points. From this arises an easy and commodious method of finding by approximation the quadratures of curves, of which a certain number of ordinates can be determined. In this work, however, Newton has employed only common simple algebra; and it is by mistake, that some of his admirers, actuated by a little too much zeal, have imagined they find in it the first clements of the integral calculus with finite differences, so celebrated in our days.
In the beginning of the last century considerable progress in the new geometry was made in Italy. This was principally owing to the work, which Gabriel Manfredi published in 1707 under the following title: De Constructione Equationum differentialium primi Gradus. The author displays much address in subjecting certain differential equations to conditions which render them integral. In his method of sepaFating the indeterminate quantities in homogeneous differential equations of the first order he has shown a similarity of genius and doctrine with John Ber
The loss Germany had sustained in geometry by the death of James Bernoulli was repaired in some measure by the scholars of that celebrated man, as James Hermann, his countryman, Nicholas Bernoulli, his nephew, and others, whom it would be too tedious to enumerate.
Hermann first made himself known by a method of finding the osculatory radii in polar curves; and soon after he published an elegant solution of the
problem, of the indefinite section of circular arcs,' at that time agitated between the two Bernoullis. He afterward distinguished himself still more by various works, of which I shall have occasion to speak.
Nicholas Bernoulli rendered himself celebrated at an early age in the doctrine of chances, by following the steps of his uncle James, whose excellent work, Ars Conjectandi, is well known. In 1709 he also made an important application of the principles of this work to the probabilities of the continuance of human life. We are likewise indebted to him for several other profound geometrical researches, of which we shall take particular notice, when speaking of the subjects to which they belong.
In France the marquis de l'Hopital had neither contemporary, nor immediate successor, of his rank in geometry. Yet we possessed several learned geometricians at the time, who, though they did not enlarge the boundaries of the science, at least in any striking degree, surmounted difficulties at that time attached to the methods of applying it. The chief were Parent, Varignon, and Saurin.
To Parent we are indebted for the solution of a very elegant and useful problem de maximis et minimis. Having remarked generally, that, if the disposition of the parts of a machine be such, as to occasion the velocity of the moving power to become greater or less, according as the weight moved becomes on the contrary less or greater, there exists a ratio between the two velocities for rendering the effect of the machine a maximum or minimum; he demonstrates, that, in hydraulic wheels moved by