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manner the whole theory of caustics by reflection and by refraction; those celebrated curves, which Tschirnhausen had pointed out to geometricians, and of which James Bernoulli had contented himself with announcing the principal properties. Section VIII is occupied by the investigation of right or curve lines touching an infinite number of given right lines or curves: a subject curious in itself, and including questions applicable to the science of projectiles. In section Ix the author exhibits the celebrated rule for finding the value of a fraction, the numerator and denominator of which both vanish at the same time. The xth and last section presents the differential calculus in a new point of view, whence the marquis deduces the methods of des Cartes and Hudde for tangents. This subject, treated with the same precision and perspicuity as the rest, can be of no use in the present day, except to exercise the young geometrician.

The marquis de l'Hopital left behind him a work on the general theory and peculiar properties of the Conic Sections, which was published in 1707. Though the cartesian analysis is alone employed in this, it deserves to be distinguished, both for the intrinsic value of it's contents, and because it has opened the way for some problems, in which the analysis of infinites was necessary. It still holds a place among the small number of classical works.

This was soon followed by another book of still greater utility, at least in France; the Analysis demonstrated of farther Reyneau, which made it's first appearance in 1708. In this the author proposed to


himself two objects: first to demonstrate and eluci date several methods of pure algebra; secondly, to give in a similar manner the elements of the differential and integral calculus. The differential calculus being sufficiently made known by the marquis de l'Hopital's work; he enlarges but little on it, employing himself chiefly in unfolding the elements of the integral calculus, which was then in it's infancy. For a long time it was the the only guide among us for beginners, in which they could acquire a knowledge of the new calculus, and was called the Euclid of the higher geometry. But, while the author has retained the esteem justly his due, his book has been forgotten; other works, more learned and more complete, produced by the progress of science, having supplanted it's use.

The method of infinites, which the marquis de I'Hopital and father Reyneau had adopted, was not free from some difficulties, which these authors had wholly eluded, or not sufficiently explained. It was only by dint of bringing it forward, applying it to new uses, and pointing out, as opportunity offered, the conformity of it's results with those of the ancient methods, that it was at length universally admitted, as no less certain and exact than all the other geometrical theories. It left some doubts, however, in the mind of those who did not enter sufficiently into it's true principles. On this subject the reader will excuse a slight anecdote relating to myself. When I began to study the marquis de l'Hopital's book, I found it difficult to conceive, that a quantity infinitely small in comparison with a finite quantity

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might be absolutely neglected, without any errour. I communicated my difficulty to a celebrated geo. metrician, Fontaine, who said to me: admit infinitely small quantities as an hypothesis, study the practice of the calculus, and you will become a believer. I found his words true, and was convinced, that the metaphysics of the analysis of infinites are the same as those of the method of exhaustion of the ancient geometricians.

The same objection to the inaccuracy of the new calculus has been frequently repeated. In 1734 a letter appeared in England, entitled the Analyst, the author of which, a man of eminent merit in other respects, represented the method of fluxions as full of mysteries, and founded on false reasoning. There was no way of annihilating for ever these strange imputations, but by establishing the theory on principles so certain and evident, that no reasonable and intelligent man could refuse his assent to them. This difficult and necessary task was undertaken by Maclaurin. In 1742 this gentleman published his Trea tise on Fluxions, in which the principles of this calculus are demonstrated with the utmost rigour, and after the manner of the ancient geometricians, who have never been accused of laxity in the choice and soundness of their proofs. This synthetic method is a little prolix, and sometimes fatiguing to follow; but it affords the mind a clearness and satisfaction, which cannot be purchased too dearly. After having firmly established his foundations, Maclaurin gratifies the curiosity of the reader with a number of beautiful problems in geometry, mechanics, and astronomy,

some of which are new, and all resolved with an elegance remarkable for the choice of the means employed. These advantages place Maclaurin's book among those productions of genius, which do honour to their author, and to Scotland his country. It has been translated into french, and many of the mathematicians, who have since acquired celebrity in France, had it for the guide of their studies in the new geometry.

While bestowing on this excellent work all the praises it deserves; and acknowledging, that Maclaurin contributed more than any person to feed the sacred flame of the ancient geometry among the english, who make it a particular point of honour, to preserve it carefully; it must not be dissembled, that, even at the period when the treatise on fluxions appeared, the analytical part of it was incomplete in several respects. Analysis, however, though we ought not to entertain an exclusive predilection for it, is the true key to all the grand problems of mechanics and physical astronomy, of which we should in vain attempt a solution by synthesis. It was still to be wished therefore, that all the discoveries, with which geometricians had enriched and continued to enrich the science of analysis, should be collected into a system for general use.

The honour of doing this was reserved for Euler. He not only extended and improved all the branches of analysis, in the numerous papers of his to be found among those of the academies of Petersburg and Berlin, and in several other collections, but published separate works on the subject, particularly adapted

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to the instruction of readers of every class. the first and most important was his finding Curves possessing the Property of Maxima or Minima, of which a sufficient idea has already been given. Subjoined to this treatise we find a learned theory of the curvature of elastic lamina; and an essay, in which the author determines, by the method of maxima and minima, the motion of projectiles in an unresisting medium; the first important application of this method to the class of mechanical problems capable of being solved by the theory of final


The Introduction to the Analysis of Infinites,' a more elementary work by the same author, published in 1748, contains in two books such instructions in Analysis and pure geometry, as are necessary for those who would understand perfectly the differential and integral calculi. In the first book Euler explains every thing, that concerns algebraical or transcendent functions, their expansion into series, the theory of logarithms, that of the multiplication of angles, the summation of several very curious and useful series, the decomposition of equations into trinomial factors, &c. In the second he begins with establishing the general principles of the theory of geometrical curves, and their division into orders, classes, and genera: he applies these principles in detail to the conic sections, all the properties of which are here deduced from their general equation: and he concludes with a very elegant theory of the surfaces of geometrical bodies. He teaches us how to find the equations of these surfaces, by referring


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