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 họnourable, 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 consequence 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."

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. 8vo.

m In the eloquent and comprehensive Eulogium upon Dr. DAVID RITTENHOUSE, the late Fresident of the American Philosophical Society, pronounced by Dr. Rusu, 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 afterwards, 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 mathe maticians in Europe."

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, new. It will be proper briefly to mention some of the more important of these.

In 1717 Dr. BROOKE TAYLOR invented a new branch of analysis, which he called the Method of Increments, in which a calculus is founded on the properties of the successive values of variable quantities, and their differences or increments. This method is nearly allied to NEWTON's doctrine of Fluxions, and arises out of it; insomuch, that many of the rules formed for one serve also, with little variation, for the other. By means of the Method of Increments many curious and useful problems are easily solved, which scarcely admit of a solution in any other way. It is, particularly, of great use in finding any term of a series proposed, and also in finding the sums of a series given. In 1763 an ingenious and instructive treatise on this new method was published by Mr. EMERSON, Who threw further light upon it. The Differential Method of Mr. STIRLING, which he applied to the summation and interpolation of series, is of the same nature with the Method of Increments, but not so general and extensive.

In 1724 M. LAGNY, of France, discovered a new mode of measuring angles, which he denominated Goniometry. By means of this method he was enabled to ascertain the measure of angles, without the use of either scales or tables, and with great exactness; a method which exceedingly abbreviated, or rendered wholly unnecessary, many tedious calculations.

In 1746 the Rev. Dr. STEWART, of Scotland, published new and elegant Theorems, of great value to the mathematician, by which he extended

the application of geometry to many problems, to the solution of which the Algebraic Calculus had been alone supposed adequate.

About the year 1758 the invention of a new branch of the analytic art, under the name of the Residual Analysis, was published by Mr. LANDen, of Great-Britain. By means of this new operation he enabled the mathematician to solve a variety of problems, to which the method of fluxions had usually been applied, in a way entirely original, and by a process more simple, natural, and elegant, than formerly. He applied this method to drawing tangents, and finding the properties of curve lines, and to the solution of many curious and difficult problems, both in mechanics and physics.

The invention of the Antecedental Calculus, a new method of geometrical reasoning, first published in 1793, by JAMES GLENIE, Esq. of NorthBritain, also deserves some notice. This is a branch of general geometrical proportion, or universal comparison, derived from an examination of the antecedents of ratios, having consequents, and a standard of comparison given, in the various degrees of augmentation and diminution which they undergo by composition and decomposition. This method proceeds without any consideration of motion or of time, but is, notwithstanding, in the opinion of the inventor, applicable to every purpose to which the celebrated doctrine of fluxions has been or can be applied.

The doctrines of Tontines, Annuities, and Reversionary Payments, were first reduced to system, and brought into use in the eighteenth century. Dr. HALLEY, of Great-Britain, and DE MOIVRE, of France, were among the earliest cultivators of this department of mathematical science. It was afterwards much improved and extended by the successive labours of SIMPSON, PRICE, WEBSTER,

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 the calculation by Sines; he carried to new degrees of perfection the Integral Calculus; he did. much to elucidate the theory of the more remarkable Curves; he contributed greatly to simplify and extend the whole system of Analytical operations; and may be said to have thrown new light upon almost every part of mathematical science."

Besides those branches of mathematics which are 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

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 public 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.

and refinement. For these improvements we are indebted to TAYLOR, CRAIG, MACLAURIN, EmMERSON, LANDEN, SIMPSON, and WARING, of Great-Britain; to CLAIRAUT, NICOLE, D'ALEMBERT, CONDORCET, DE LA CROIX, and DE LA GRANGE, of France; to MANFREDI, of Italy; to PACASSI, a nobleman 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 STIRLING, SIMPSON, and WARING, of Great-Britain; the BERNOULLIS, CRAMER, and EULER, of Switzerland; and CLAIRAUT, BEZout, LAGNY, DE LA GRANGE, and DE LA PLACE, of France.

It may be asserted that in almost every branch of what is called Modern Analysis, much new light, and many curious refinements have been introduced by the mathematicians of the eighteenth century. In the doctrines of Series, of Increments, of Differences, of Infinitesimals, &c. great ingenuity has been successfully employed in modern times. And the application of these to astronomy, and other branches of philosophy, may be considered as forming a grand æra in the history of science. For many of these improvements the public is indebted to several of the mathematicians men

M. LA GRANGE has lately presented to the world a very important work, entitled, the Theory of the 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 OF LEIBNITZ, may be reduced to the ordinary calculations of fine quantities.