tion Nouvelle en l'Algebre, published at Amsterdam in 1629, presents us with several useful discoveries. It appears that he was the first who understood the use of negative roots in the solution of geometrical problems, and the general doctrine of the formation of the coefficients of powers from the sums of the roots, their products, &c. or gave rules for summing the powers of the roots of an Equation. He was the first who treated of imaginary roots, and understood that every Equation might have as many roots, real and imaginary, and no more, as there are units in the index of the highest power; he introduced the parenthesis as a convenient substitute, in many cases, for the vinculum: he was the first who employed the table of sines in the solution of the irreducible case of Cubics; and who distinguished negative quantities by the ridiculous appellation of quantities less than nothing; a name which has given some very able teachers much unnecessary trouble to explain.

One of the most scientific men that this nation ever produced, was Thomas Harriot, who, for his great and valuable improvements, is justly considered as the father of modern Algebra; his work on the subject, entitled Artis Analytica Praxis, &c. which was published in 1631, after his death, by his friend Warner, will ever remain a monument of his superior skill as an

Thomas Harriot was born at Oxford in 1560: he was a commoner at St. Mary's Hall, where he took his Bachelor's degree in 1579. His uncommon proficiency in the Mathemathics attracted the favourable attention of Sir Walter Raleigh, who took him into his family in the character of a preceptor, allowing him a handsome pension. In 1584 he went to Virginia with Sir Walter's new colony, where he was employed in surveying and making maps of the country. He died in 1621 from an ulcer formed in his lip, in consequence, it is said, of holding the brass mathematical instruments, while working, in his mouth. Harriot was undoubtedly the most eminent algebraist of his time,

analyst: this work is a specimen of great and original genius, and supplies the first instance of the form of Algebra at present in use. He here, first of any, shewed the universal generation of all affected Equations by the continual multiplication of simple ones; "thereby exhibiting to the eye all the circumstances of the nature, mystery, and number of roots, with the composition and relations of the coefficients," from whence many important properties have since been deduced; "he greatly improved the numeral Exegesis, or the extraction of the roots of all Equations, by clear and explicit rules and methods, drawn from the foregoing generation of affected Equations ;" and to him we are indebted for many original and elegant solutions of quadratic, cubic, and biquadratic Equations, which have since been discovered among his manuscripts; he was likewise the inventor of the signs greater than, and less than.

The Reverend William Oughtred", rector of Aldbury in Surry, and one of the first mathematicians of the time, published in 1631 his Clavis Mathematica, which was originally written for the use of Lord William Howard, his pupil. "His style and manner were very concise, obscure, and dry; and his rules and precepts so

"William Oughtred was born at Eton in 1573, and educated at the school there; in 1592 he went to King's College, Cambridge, where he spent 12 years, and became a fellow; he was presented to the living of Aldbury in 1603, and about 1628 was appointed Tutor to Lord William Howard, by Lord Arundel, his father. The chief mathematicians of the age were indebted to Oughtred for much of their skill, his house being open to all who came for instruction: after a life of temperance, exercise, study, and zeal in performing effectually the duties of his sacred function, he died, it is said, of a sudden extasy of joy, on hearing that the Parliament had passed a vote for the restoration of Charles II. Besides the works on Arithmetic, Algebra, Geometry, Trigonometry, &c. published in his life time, such of the manuscripts which he left, and which were found proper for the press, were published in 1676 at Oxford, in 8vo. under the title of Opuscula Mathematica hactenus Inedita, &c.

involved in symbols and abbreviations, as rendered his mathematical works difficult to be understood;" nevertheless, his writings were considered as valuable, and are still held in great esteem among the learned. Besides some characters not at present in use, he introduced the sign for multiplication, :: for proportion, for continued proportion, for greater, and 1 for less, with the method of multiplying or dividing by the component parts of a number, instead of the number itself; of multiplying and dividing decimals by the contracted method usually taught in our schools at present, with other neat and convenient abbreviations.

The application of Algebra to geometrical lines and curves, began about this period to exercise the skill of the learned. Fermat, a learned and ingenious French mathematician, was the first who successfully cultivated this branch; but it was Des Cartes *, who incorporated

* Réné Des Cartes was descended from an ancient noble family in Touraine, and born in 1596. At eight years old he was placed under the tuition of Father Charlet, at the Jesuits' College of La Fléche; conceiving a dislike to philosophy, he quitted the College in 1612, proposing to himself a military life: for this purpose he speedily acquired the necessary accomplishments, but a weak constitution rendering him unfit for the duties of a martial profession, he went to Paris, where, by the advice of Father Mersenne, and others of his learned acquaintance, he was prevailed on to resume his studies; after two years he returned to the army, and was successively a volunteer in the service of the Prince of Orange, and the Duke of Bavaria. After this he travelled for improvement, and was indefatigable in the study of almost every branch of science; "he extended," says Voltaire, "the limits of Geometry as far beyond the place where he found them, as Newton did after him."" He employed this geometrical and inventive genius to Dioptries, which, when treated by him, became a new art." Voltaire acknowledges, that the rest of Des Cartes' works contain innumerable errors: indeed, his system of the universe, which is called The Cartesian Philosophy, and which prevailed until Sir Isaac Newton's system supplanted it, depends solely on hypothesis; and the theory of vortices, on which the Cartesian system is founded, is absolutely false. See Dr. Keill's Examination of Burnet's Theory, &c. Des Cartes died at Stockholm in 1650.

and improved the theories and observations of preceding writers, and first gave the doctrine a form and consistence. The Geometry of Des Cartes was published in 1637: it is properly neither Algebra purely, nor Geometry, but the application of the one to the other; nevertheless it contains improvements in both. The principal of those which relate to the present subject, are the following; viz. the geometrical construction of the higher orders of Equations, whereby the nature and properties of their roots, positive, negative, and impossible, are clearly elucidated. The rule for resolving biquadratie Equations, by means of a cubic and two quadratics, which usually goes by his name; and he is the first who denoted the unknown quantities in an Equation, by the final letters x, y, z, w, &c. and the known ones by the initials a, b, c, d, &c. according to the present mode of practice. But while we do justice to the superior talents of this distinguished philosopher, it must be acknowledged, that several of his methods and observations, which at the time of their publication were considered as new, were afterwards traced to the writings of Harriot, the English Algebraist '.

The Geometry of Des Cartes, now called the New Geometry, was soon cultivated with ardour and success. Francis Van Schooten, a Dutch mathematician, translated it out of French into Latin, adding a commentary of his

y M. Roberval, a member of the Academy of Sciences, and Professor of the Mathematics at the College Royal, was once (shortly after the above work appeared) extolling the ingenuity of Des Cartes, for his contrivance in placing all the terms of an Equation on one side, and making the whole equal to nothing; upon which Sir Charles Cavendish, who was present, hinted that the praise was due-not to Des Cartes, but to Harriot: a few days after Sir Charles produced Harriot's Algebra, and Roberval, after he had examined it, exclaimed, Oui, il l'a vu! il l'a vu! "Yes, he has seen it! he has seen it!" See Dr. Wallis's Algebra, p. 198.

own, and notes by M. De Beaune, 1649. Schooten's Principles of Universal Mathematics appeared in 1651, and Exercitationes Mathematica, six years after; in both which are much excellent matter, and a variety of curious analytical pieces. The Method of Indivisibles of Cavalerius, was published in 1635, and proved a new æra in analytics, from whence arose new modes of computation. Our learned countryman, Dr. John Wallis, was the author of

• Bonaventura Cavalieri was a native of Milan, a Friar of the order of the Jesuati of St. Jerome, a disciple of Galileo, the friend of Torricellius, and Professor of Mathematics at Bologna, where he died in 1647, leaving behind him several learned treatises on Geometry, Trigonometry, Logarithms, &c. besides the above-mentioned work on Indivisibles.

• Dr. John Wallis was born at Ashford in Kent, in 1616; after acquiring a tolerable proficiency in the Latin, Greek, Hebrew, and French languages, with the rudiments of Logic, Music, &c. he went to Emanuel College, Cambridge: he took orders, and obtained a fellowship of Queen's College; after this he was Chaplain, first to Sir Richard Darley, and then to Lady Vere. In 1642 he was appointed Savilian Professor of Geometry at Oxford, a situation which he filled with great ability; five years after he took the degree of Doctor in Divinity, and the next year his controversy with Mr. Hobbes commenced; this and his dispute with Mr. Stubbe lasted more than three years, during which several pamphlets were written on both sides, and the Doctor displayed a degree of spirit, moderation, and address, highly creditable to himself. In 1658 he was chosen Custos Archivorum of the University; King Charles II. respected him both for his talents, and for his attachment to himself and to his unfortunate father; and in consequence our author was, on the restoration, confirmed in all the places he held, and appointed one of the Chaplains in Ordinary, and likewise to assist in revising the book of Common Prayer. He was a very industrious and useful member of the Royal Society, and kept up a constant literary correspondence with most of the principal learned men of the time. Dr. Wallis died at Oxford in 1703, leaving behind him one son and two daughters; the Rev. Henry Peach, B. D. the present worthy Rector of Cheam, to whose kindness the writer acknowledges himself, with gratitude, under great and repeated obligations, is grandson of one of the latter. The publications of Dr. Wallis comprehend a great variety of subjects, and are for the most part filled with learned and useful matter; and he has contributed to nearly all the first 25 volumes of the Philosophical Transactions. His mathematical works, which had appeared separately, were published together by the University in 1699.