Doctor here first introduces the method of registering the steps of an Equation, by characters placed for reference in the margin : he likewise invented the signs ÷ for division, for involution, and us for evolution.

Dr. Barrow's Method of Tangents was published in

county. About this time he took his degree of Doctor in Divinity, and was made Chaplain to Archbishop Sheldon ; but his thoughts being too much employed in the cultivation of the sciences, to allow him time for the management of his pecuniary concerns, he was so cheated by his tenants and dependants, that he wanted money to purchase the common necessaries, and he found difficulty to procure even pens, ink, and paper; this brought him to the King's Bench Prison, and to a state of dependance after his release: he died at the house of a friend, in 1685, and his funeral expences were defrayed jointly by Mr. Sharp, Rector of St. Giles's, and Dr. Busby. Dr. Pell published a number of mathematical books, and left behind him a large collection of papers and letters on mathematical and other subjects, which are in the bands of the Royal Society; others be left at the seat of Lord Brereton, at Brereton, in Cheshire.

d Isaac Barrow was born at London in 1630. He first went to the Charter House, where his behaviour was so bad, that little hopes of him were enter. tained by his friends; and his father has been heard to say, that, "if it pleased God to take either of his children, he hoped it would be Isaac." He was entered at Trinity College, Cambridge, in 1645, and now began to apply himself in earnest to learning. Mathematics, Natural Philosophy, Chronology, Physic, Anatomy, Chemistry, Botany, and Divinity, were the objects of his unremitting attention. Having studied these with great success, and meeting with a disappointment at the University, he quitted it, and in 1655 set out on his travels on the Continent; he returned about four years after, and having been ordained, was chosen in 1660 Greek Professor at Cambridge; two years after he was made Professor of Geometry at Gresham College, where he discharged the duties both of the geometrical and astronomical departments. The Royal Society elected him a Fellow in 1663, and the same year he was appointed the first Lucasian Professor of Mathematics at Cambridge. At length, being determined to direct his studies wholly to divinity, he resigned the mathematical chair (which he had filled with much credit) to his friend, Mr. Isaac Newton. In 1670 he was created D. D. by Mandate, and in 1672 King Charles I. appointed him Master of Trinity College by patent, observing, that "he had given it to the best scholar in England." Dr. Barrow died in 1677, and was buried in Westminster Abbey: his numerous writings on a great variety of subjects, do an honour to his country. "He was unrivalled," says Mr. Grainger, "in mathematical learning, especially in the sublime Geometry, in which he has been excelled only by his successor, Newton; he at length gave himself up entirely to divinity, and particularly to the most useful part of it, that which has a tendency to make men wiser and better."

e

1669; and about 1677 Leibnitz discovered his Methodus Differentialis, being either a variation of Newton's Fluxions, or else an extension of Barrow's Method of Tangents above-mentioned.

In 1693 appeared M. Ozanam's Course of Mathematics, in 5 volumes octavo, containing a treatise on Algebra; nine years after he sent forth a work exclusively on Algebra, in which the problems of Diophantus are skilfully handled. He was the author of several other mathematical works, among which may be mentioned his Mathematical and Philosophical Recreations, translated and published by Dr. Hutton, which afford both ainusement and instruction. A book on the same subject had been published many years before by Bachet de Meziriac, under the title of Problêmes plaisans et délectables sur les Nombres &.

The invention of Fluxions opened the way for new discoveries and improvements. Mathematicians in every

e Godfrey William Leibnitz was born at Leipsige in 1646, where, and at Jena, he received his mathematical education; his writings, which are numerous and valuable, were published separately, or among the memoirs of different academies: as a mathematician, he is chiefly remarkable for laying claim to the invention of Fluxions, in opposition to Sir Isaac Newton, and for his system of Philosophy, intended as an amendment of the Cartesian, in opposition to the Newtonian Philosophy. In his system are retained the subtile matter, the vortices, and the universal plenum of Des Cartes; and the universe is considered as a vast machine, which by the laws of mechanism will, by absolute necessity, continue in motion for ever in the most perfect state. He died in 1716.

f Jaques Ozanam was born at Boligneux in France in 1630: he was a respectable teacher of the Mathematics, first at Lyons, then at Paris, and published a great number of very useful books on various branches of the Mathematics, with some pieces in the Journal des Scavans, the Memoirs of the Academy of Sciences, the Mémoires de Trevoux, &c. Ozanam died in 1717.

8 The Récréation Mathématique of H. Van Etten is a well known work, and principally remarkable for its absurdities; Mydorgius undertook to correct them, but was not very successful, as appears by the later editions of the book, which are of little value.

The Greck Anthology is the earliest example we have of this kind of composition.

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part of Europe, aided by the light which this sublime discovery had thrown on the new Geometry of Des Cartes, were intent on extending the bounds of the science, and their labours were attended with astonishing sucThe Method of Increments, which naturally arises out of the doctrine of Fluxions, was discovered by Dr. Brook Taylor, who published, in 1715, his Methodus Incrementorum; this ingenious and learned work appearing too difficult for the generality of readers, Mr. Emerson undertook to render the subject more easily attainable; and succeeded probably beyond his own expectation, in a work which appeared in 1763.

The calculation of the probabilities in the theory of games of chance, owes its origin to M. Pascal, resulting from one of the numerous applications of the doctrine of his celebrated arithmetical triangle. Some of the greatest mathematicians of the time directed their attention to the subject, but Huygens was the first who wrote expressly on it, in his treatise De Ratiociniis in

i

Blaise Pascal was born at Clermont, in Auvergne, in 1623, and was one of the greatest and best writers France has ever produced. He was a very eminent mathematician: "his writings," says Voltaire, "may be considered as models of eloquence and humour;" his Provincial Letters, which do equal honour to his head and heart, have been published in every language and country in Europe. M. Bayle says, that a hundred volumes of sermons are not of so much avail as the history of this great man; he calls him a paragon in the human species; and adds, "when we consider his character, we are almost inclined to doubt whether he was born of a woman." Pascal died at Paris in 1662, and his works were collected by L'Abbé Bossu, and published both at Paris, and at the Hague, in 1779, in 5 volumes octavo.

i Christopher Huygens was born at the Hague in 1629: he discovered an early love for the Mathematics, and studied them with uncommon ardour under the celebrated Schooten at Leyden: in 1655 he went to France, where he had the degree of Doctor of Laws conferred on him; and five years after, visiting London, he was made a Fellow of the Royal Society. The talents of this great man were employed, not merely in promoting the interests of speculative science, but in reducing his knowledge to practical uses for the benefit of He was the first who discovered Saturn's ring, and also the third satel lite of that planet; "he first applied the pendulum to clocks, and equalized its

man.

Ludo Alea. The posthumous work of James Bernoulli *, entitled De Arte Conjectandi, is well known; Nicholas Bernoulli, the nephew of that excellent man, made an important application of the principles contained in his book to the probabilities of the continuance of human life. The Analysis of Games of Chance, by Remond de Montmort, appeared in 1711, is a work of great merit, and has for its object the subjecting of probabilities of every kind to calculation.

The importance of the theory which had employed the talents of these illustrious foreigners, soon appeared from the useful applications made of it, especially in England, to the uses and purposes of life; such as the

vibrations by means of the cycloid; and perfected the telescope then in use. In 1681, Huygens was admitted a member of the Academy of Sciences at Paris, which was the last public mark of honour he received; for his intense application having impaired his health, he retired as often as he could into the country. He died at the Hague in 1695, and his principal works being collected, two editions of them were published in 4to, under the inspection of Professor S'Gravesande, the first at Leyden, 1682, the other at Amsterdam, 1728.

k James Bernoulli was born at Basil in 1654, and died in 1705: he was Professor of Mathematics at Basil, and member of the Academies of Sciences of Paris and Berlin. John Bernoulli, brother of the above, and equally famous, was born at Basil in 1667, where he obtained the degree of Doctor of Physic; he was first, Professor of Mathematics at Groningen, but on his brother's death, was appointed to succeed him in the mathematical chair at Basil; he was member of most of the Academies of Europe, and both he and his brother cultivated the new analysis with the greatest application and success. He died in 1748. Daniel Bernoulli, the son of John, was born at Groningen in 1700; he became Professor of Physic and Philosophy at Basil, and on his father's death, succeeded him in the Academy of Sciences. It is remarkable that this Academy, from its first institution in 1699, had never been without a Bernoulli; after a long, useful, and honourable life, Daniel Bernoulli died, much lamented, in the year 1782. John Bernoulli, grandson of the above John, is well known as an industrious and skilful astronomer at Berlin. His brother, James Bernoulli, was born at Basil in 1759; he read lectures on Philosophy in the University of Basil for his uncle Daniel : afterwards he was secretary to Count Brenner, the Emperor's envoy at Venice, and in 1786 became a member of the Academy of Sciences at Petersburg, where he died in 1789. Of Nicholas Bernoulli I have not been able to obtain any satisfactory account

valuation of annuities, "assurances, reversions, &c. Mr. Abraham Demoivre' appears to have been the first in this country, who distinctly explained the doctrine and mode of application, in his Doctrine of Chances, &c. 1718, and his Annuities on Lives, 1724: indeed he was particularly famous for calculations of this kind, spending most of his time upon them. Mr. Demoivre was ably followed by Mr. Thomas Simpson ", whose two treatises On the

I Abraham Demoivre was born at Vitri in Champagne, in 1667; he was an eminent teacher of the Mathematics at London, where he (being a Protestant) had been driven by the revocation of the edict of Nantz; he was a Fellow of the Royal Society, and member of the Academies of Sciences at Paris and Berlin, and was, in consequence of his consummate abilities as a mathematician, appointed by the Royal Society to determine the dispute between Newton and Leibnitz, concerning their respective rights to the invention of Fluxions. He died at London in 1754.

Thomas Simpson was born at Market Bosworth in Leicestershire, in 1710; he was bred a weaver, but gave early indications of a thirst for knowledge of a superior kind; this not being agreeable to the views of his father, a separation soon took place, and young Simpson went to reside at Nuneaton. Here, by means of the instructions given him by an itinerant pedlar and fortune-teller, during his periodical visits, and our author's own endeavours, he acquired considerable acquaintance with the principles of Arithmetic and Algebra, and likewise sufficient skill in the art of conjuring to enable him, in the technical phrase, to answer all fair questions about futurity. We now find him working at his trade all day, teaching a school in the evening, and occasionally telling fortunes; this continued till a curious circumstance, the account of which is too long to relate, obliged him to quit both his home, and the trade of astrology: he then went to Derby, where he worked at his trade, and kept an evening school; but his emoluments being. insufficient for the support of himself and family, he was induced to repair to London about 1736; he settled in Spitalfields, working at his trade of weaving, and occasionally teaching the Mathematics. He now became better known, and the number of his pupils increasing, he was encouraged to make proposals for publishing his work on Fluxions by subscription, which proved the precursor of many other learned and valuable works; and he now, for the first time, began to discover that the pretended art of astrology is nothing more than a system of fallacy and deceit. In 1743, he was appointed Professor of Mathematics at the Royal Military Academy, Woolwich; and two years after was elected Fellow of the Royal Society: he died in 1761, at Bosworth. A more complete and circumstantial account of Mr. Simpson is to be found at the beginning of his Select Exercises,