CHAPTER VI. MATHEMATICS. THE seventeenth century was the " golden age" of mathematical science. Never, since the revival of learning, has this branch of knowledge been cultivated with such brilliant success as during that period. The grand inventions of Logarithms, by Napier, and of Fluxions, by Newton, together with the numerous discoveries and improvements of des Cartes, Huygens, Kepler, Gregory, Leibnitz, and many others, must ever render the age of those great men a distinguished æra in the annals of mathematics. It is even possible that the grand discoveries of these philosophers, and the unusual lustre of their characters, may have contributed, by an influence far from being unnatural, to repress the ambition and discourage the exertions of some who came after them. But, although the eighteenth century can boast of no discoveries so splendid, nor of any advances so honourable, as belong to the preceding, yet it produced both, in a sufficient degree to secure a reputable place in the history of this sublime science. Though the Fluxionary Analysis had been invented by Newton thirty years before, yet that great mathematician first published his new doctrine on this subject in 1704. The controversy in which he became involved with Leibnitz, in con sequence of this publication, is well known to have been one of the most curious and interesting of the age *. It seems to have been long and generally agreed, that the credit of this celebrated invention is due to the illustrious British philosopher; and, of course, that the claim of his German rival was unfounded t. Within the period under consideration, several new and valuable branches of mathematics, now in use, have been either wholly discovered, or placed on a footing, in a great measure, if not entirely, * Soon after Newton published his doctrine of Fluxions, his book was reviewed in the Acta Eruditorum of Leipsic. In the course of this review, an intimation was given that he had borrowed from Leibnitz, and that the honour of the invention properly belonged to the latter. Dr. Keill, professor of astronomy in the University of Oxford, undertook the defence of his countryman. After a number of controversial papers had been exchanged. on the subject, Leibnitz complained to the Royal Society of injustice on the part of Newton and his friends. The Society appointed a committee of its members to investigate the questions in dispute; who, after examining all the letters and other papers relating to it, decided in favour of Newton and Keill. These papers were published in 1712, under the title of Commercium Epistolicum. Svo. In the eloquent and comprehensive Eulogium upon Dr. David Rittenhouse, the late president of the American Philosophical Society, pronounced by Dr. Rush, at the request of the Society, there is the following passage: "It was during the residence of our ingenious philosopher with his father in the country, that he became acquainted with the science of Fluxions, of which sublime invention he believed himself for a while to be the author; nor did he know, for some years afterward, that a contest had been carried on between sir Isaac Newton and Leibnitz, for the honour of that great and useful discovery.. What a mind was here! Without literary friends or society, and but two or three books, he became, before he had reached his four and twentieth year, the rival of the two greatest mathematicians in Europe." among the earliest cultivators of this department of mathematical science. It was afterwards much improved and extended by the successive labours of Simpson, Price, Dodson, Morgan, and Maseres, of Great Britain; by Deparcieux, of France; and by many others, in various parts of Europe. About the middle of the 'century under review, and for some years afterwards, flourished the celebrated Euler, a native of Switzerland, and one of the greatest mathematicians, and most excellent men of the age in which he lived. He invented many new formulæ for the sines, cosines, &c., and carried to a greater degree of perfection the Integral Calculus; he also did much to elucidate the theory of the more remarkable Curves; and contributed greatly to simplify and extend the whole system of Analytical operations. In short, he may be said to have thrown new light upon almost every part of mathematical science *. Beside those branches of mathematics which are * Leonard Euler was born at Basil, in 1707, and died in 1783, in the 76th year of his age. The mathematical genius and erudition of this man were truly wonderful. No individual of the eighteenth century can be compared to him for the number and value of the discoveries which he made in this branch of science, and for the improvements of which he was the author. His publications are numerous; and there is scarcely a department of mathematics on which he has not thrown some new light, or to which he has not made some important additions. On every subject which he undertook to investigate, he displayed a vigour, a penetration, and a comprehensiveness of mind, which entitle him to a place in the first rank of philosophers. Euler was not less distinguished for the excellence of his moral and religious, than for the greatness of his intellectual character. To singular probity, and great social amiableness, he added the piety of an eminent christian. He was a warm and active friend to religion, fervent in his devotions, and exemplary in his attention to all pub entirely the growth of the last age, almost every part of this science has been extended and improved within the same period. Of a few of these some transient notice will be attempted. Since Newton published an account of his celebrated method of Fluxions, this curious part of mathematical science has received new light, and been carried to new degrees of extent, simplicity, and refinement. For these improvements we are indebted to Taylor, Craig, Maclaurin, Emerson, Landen, Simpson, and Waring, of Great Britain; to Clairaut, d'Alembert, Condorcet, de la Croix, and de la Grange*, of France; to Manfredi, of Italy; to Hindenbourg and Arbogast, of Germany; and to none, perhaps, more than to the great Euler, whose work on the Integral Calculus, or the inverse method of Fluxions, may be considered as holding the first rank on the subject of which it treats. The principles of Algebra have received important additions, and been more satisfactorily displayed during this period, than by the mathematicians of former times. Of this department of mathematical science the most distinguished cultivators were lic and private duties. If ever he felt indignation against any particular class of men, it was against the enemies of christianity, especially against the apostles of infidelity. He published a valuable work in defence of revelation, at Berlin, in 1747. * M. la Grange has lately presented to the world a very important work, entitled, the Theory of Analytical Functions, in which he is supposed to have shown, that every thing hitherto called Fluxions, or the Differential Calculus (the phrase chiefly used on the Continent of Europe to express Fluxions), whether according to the method of Newton or Leibnitz, may be reduced to the ordinary calculations of fine quantities. and many subtleties and refinements suggested by the mathematicians of the preceding age, but not sufficiently developed by them, have been clearly and satisfactorily unfolded. It is also worthy of notice, that in addition to all the improvements which have taken place in mathematical science, as such, it has been applied to many objects, during the last age, to the illustration and accomplishment of which it had never before been directed. A great number of difficult and very interesting problems in astronomy have been resolved by the Analytic Method, first applied to this subject by Euler. His calculations, by this method, of the perturbations of the earth's orbit, and of the theory of the moon, may be regarded as models of simplicity and beauty. The same illustrious mathematician also first introduced analysis into the doctrines of the motion of fluids; and by this means threw great light on the principles and laws of hydraulics. Mr. pinus, of Petersburg, before mentioned, has made an ingenious attempt to reduce the mysterious phenomena of Electricity and Magnetism to the regularity of algebraical calculation. M. de Lisle, of France, has endeavoured, with no small degree of success, to form a new system of Mineralogical Characters, on the principles of geometry; and M. Hauy, of the same country, has given a very elaborate and plausible system of doctrines on Crystallisation, which all proceed upon fixed mathematical rules. To this chapter belongs some notice of the attempts which were made, during the period under |