of

Newton had determined the curve described by a projectile in a medium resisting in the ratio of the simple velocity: but he had not touched upon the case, at that time more difficult, where the resistance of the medium is as the square of the velocity. This case Keill proposed to John Bernoulli, who hot only resolved it in a very short time, but extended the solution to the general hypothesis, in which the resistance of the medium should be as any power the velocity of the projectile. When he had discovered this theory, he offered repeatedly to send it to a confidential person in London, on condition, that Keill would give up his solution likewise: but Keill, though strongly urged, maintained a profound silence. The reason of this it is not difficult to conjecture: he had not resolved his own problem. When he proposed it, he expected, that no one would discover what had escaped the sagacity of Newton. In this conjecture he was cruelly mistaken: and his challenge, which was something more than indiscreet, drew on him a reprimand from the swiss geometrician, that was so much the more poignant, as the only mode of answering it satisfactorily was by a solution of the problem, which he could neither effect by his own skill, nor by the assistance of his friends. John Bernoulli's triumph was complete. In the first intoxication of victory he indulged himself in sarcasms and jests against his rivals, not commendable for their elegance, but certainly pardonable in a man of a frank and honest disposition insidiously attacked, and who had to avenge affronts, offered

offered not only to himself, but also to an illustrious friend, whose loss he still lamented.

These learned contests drew the attention of all geometricians; and notwithstanding the acrimony infused into them by the passions, they stimulated men's minds, and produced new proselytes to mathematics on all sides.

I shall now step back a little, and resume some other subjects, which I have been obliged to leave in

arrear.

In 1711 appeared the Analysis of Games of Chance,' by Remond de Montmort: a work abounding with acute and profound ideas, the object of which is, to subject probabilities to calculation; to estimate chances; to regulate wagers, &c. It does not properly belong to the new geometry, yet it con tributed to it's progress, by stimulating the spirit of combination in general, and by the extent which the author gave to the theory of series, a happy supplement to the imperfection of the rigorous methods in all the branches of mathematics.

Three years afterward de Moivre published a little treatise on the same subject, entitled, Mensura Sortis, chiefly remarkable for containing the elements of the theory of recurrent series, and some very ingenious applications of it. This Essay, gradually increased by the reflections of the author, has grown up into a considerable work, admired by all geometricians. The best edition is that of 1738, in english, under the title of the Doctrine of Chances. De Moivre was a french geometrician, whom the revocation of the edict of Nantes had obliged to quit his

country, and who had fled to London. Born with superiour talents for geometry, the narrowness of his fortune obliged him to teach mathematics for a livelihood. Newton had the highest esteem for him. It is reported, that during the last ten or twelve years of Newton's life, when any person came to ask him for an explanation of any part of his works, he used to say: Go to Mr. de Moivre; he knows all these things better than I do.'

Nicholas Bernoulli, the nephew, came to Paris in 1711. His great reputation, and mild and easy manners, gained him many illustrious friends. Among the number of these was Montmort, with whom he formed a strict intimacy, in consequence of the similarity of their dispositions, and taste for the analysis of probabilities. They spent three whole months together in the country, solely employed in resolving the most difficult problems on this subject. All these new researches, and the elucidations arising from them, produced a second edition of Montmort's book in 1714, much superiour to the first.

I have already slightly mentioned Dr. Taylor's Methodus Incrementorum, but this work, celebrated even in the present day, deserves more particular notice.

The author gives the name of increments or decrements of variable quantities to the differences, whether finite or infinitely small, of two consecutive terms in a series formed according to a given law. When these differences are infinitely small, their calculus, either direct or inverse, belongs to the leibnitzian analysis, or the method of fluxions; and Dr.

Taylor

Taylor resolves a great number of problems of this kind. But when the differences are finite, the me thod of finding the relations they bear to the quantities that produce them forms a new kind of calculus, the first principles of which were given by Dr. Taylor; and in this respect his book has the merit of originality. In this manner he has summed up some very curious series.

The extreme conciseness, or rather obscurity, with which this work is written, long retarded the success which was due to it. Nicole, however, a very dis tinguished french geometrician, was able to understand the very clearly unfolded the method for resolving finite differences, and added several new series of his own invention. The two excellent papers, which he published on this subject in the Memoirs of the Academy of Sciences for 1717 and 1728, may be considered as the first methodical and luminous elementary treatise on the integral calculus with finite differences, that ever appeared.

Several other works of the time might be mentioned, but I must be brief. I would request the reader, therefore, to consult the periodical publications of Germany, France, England, Italy, &c., with those by the different academies, where he will find a number of valuable papers on every branch of mathematics.

It has been observed, that the Royal Society of London and the Academy of sciences at Paris arose nearly at the same time, or about the year 1660. The Academy of Berlin, the establishment of which was projected in 1700, took a regular and legal form

in 1710, under the auspices of Frederic, elector of Brandenburg, and the first king of Prussia, and Leibnitz was appointed perpetual president. The Institute of Bologna was founded in 1713, through the means of the celebrated count de Marsigli, to whom natural history is so much indebted. In 1726 Catharine I, the widow of Peter the Great, founded the Academy of Petersburg. Several other learned societies have since been formed, which it would take up too much room to particularize. All these esta▾ blishments have been of extreme utility to the progress of the sciences.

CHAP.