and Lecturer on the Mathematics at Milan; who after adding perjury to falsehood, dared to insert in a large work on the Mathematics, which he was then printing, those very rules which he had obtained from Tartalea under the most solemn promises, confirmed by an oath, of inviolable secresy. Cardan, although a bad man, was indisputably the best algebraist of the age, and the improvements he introduced into the science were very considerable; especially in the discoveries he had drawn from Tartalea, deriving from them rules for the solution of all the forms of Cubics: he was well acquainted with all the real roots of Equations, both positive and negative; shewing that the even roots of positive quantities are either positive or negative, that the odd roots of negative quantities are real and negative, and that their even roots are impossible. He knew the number and nature of the roots of an Equation, as depending on the signs of the terms, and the magnitude and relation of the coefficients; that the number of positive roots is equal to the number of changes in the signs of the terms; that the coefficient of the second term is equal to the difference between the of age he became a student at the University of Milan, and two years after explained Euclid. In 1524 he was admitted Master of Arts, and the year following Doctor of Physic; about 1533 he became Professor of Mathematics at Milan, where six years after he was received a member of the College of Physicians, and read public lectures on Medicine; he taught successively at Paria, Bologna, and Rome: at the latter he was admitted a member of the College of Physicians, and received a pension from the Pope, which he enjoyed till his death, which happened in 1575. Cardan was so great an adept in astrology, that the greatest personages in Europe had recourse to his skill; among these we find Edward VI. of England, whose nativity was calculated by our astrologer as he passed through London from Scotland, having been sent for there by the Archbishop of St. Andrews, to cure him of a dangerous disorder. Cardan was the greatest, although the most eccentric and restless genius of his time; possessing splendid talents, accompanied with a wicked and depraved heart. The Lyons edition of his works printed in 1663, consists of no less than ten volumes folio, positive and negative roots, and that consequently, when the second term is wanting, the sums of the negative and positive roots will be equal; that changing the signs of the even terms changes the signs of the roots; that the roots fail in pairs. He knew how to compose Equations with given roots, or to change them from one form to another, by taking away any intermediate term; he could extract the roots of such binomials as would admit of extraction he knew all the difficulties attending the irreducible case of Cubics, and the attempts he made to solve it led him to the discovery of rules, whereby the roots may be approximated to, in all cases whatever: hè frequently used the literal notation, expressing quantities by letters; treated fully on the transformation of Equa tions; and shewed how to apply Algebra to the solution of Geometrical Problems. : About the year 1540, Lewis Ferrari, the pupil of Cardan, discovered a rule for the solution of biquadratics, which the latter has demonstrated, explained, and exemplified, and given in his treatise on Algebra. Tartalea's Quesiti et Inventioni Diverse, printed at Venice in 1546, and dedicated to King Henry VIII. of England, is chiefly remarkable for the account it gives of the invention of the above-mentioned rules for Cubic Equations, of the artful methods employed by Cardan to obtain them, and the quarrel which ensued. Tartalea was public Lecturer on the Mathematics at Venice. About this time Franciscus Maurolicus, Abbot of Santa Maria del Porto, in Sicily, distinguished himself by his * Franciscus Maurolicns was born at Messina in 1494: he was a great profi cient in the Mathematics, which he taught with unbounded applause; to him we are indebted for the "Tabula Benefica," or Canon of Secants, and likewise an edition of the Spherics of Theodosius; Emendatio et Restitutio Conicorum Apollonii Pergæi; Archimedis Monumenta Omnia; Euclidis Phænomena, &. he died in 1575, great skill in the Mathematics; in particular, he cultivated a branch of analysis then but little known, namely, the summation of series; he gave theorems for summing the series of natural numbers, their squares, &c. for triangular and other figurate numbers, all remarkable for subtilty of invention, and simplicity of result. Algebra seems to have been in a more advanced state about this time in Germany, than it was in Italy, and to have approached nearer to the modern method; although it does not appear that the Germans knew any thing of the rules for Cubics. The earliest writer of that country was Michael Stifelius", a Protestant minister, and an eminent mathematician; his chief work, entitled Arithmetica Integra, was published at Nuremberg in 1544: it is an excellent treatise on both Arithmetic and Algebra, and contains several ingenious inventions in both. In this work he introduces the characters +, -, and, and the numeral exponents, both positive and negative, of powers, teaching the general use of exponents in the several operations on powers, as is practised at present. He understood the nature and use of Logarithms, although under another name; but it does not appear that he knew the use of fractional indices. He employed the capitals A, B, C, D, &c. to express unknown quantities, treated of quadratics in a more general manner than had been before done, and made various other improvements. John Scheubelius, Professor of Mathema h Stifelius was born at Eslingen in Germany, some time about the year 1490, and died at Jena in Thuringia in 1567; his improvements in Algebra are briefly mentioned above. Unfortunately he was not content with the credit of being a skilful mathematician, but wished to extend his fame by becoming a prophet; accordingly he predicted that the world would be at an end on a certain day in the year 1553: multitudes of his followers met him in the open air on the appointed day, but instead of being spectators of the awful event foretold, were witnesses only of the mortification and disappointment of the unfortunate prophet. tics at Tubingen, in the Duchy of Wirtemberg, wrote several treatises on Arithmetic and Algebra, about the year 1550 he is the first algebraist who makes mention of Diophantus; most probably he knew nothing of the discoveries of Ferrari and Tartalea, as he takes no notice of Cubic Equations. The first English writer whose works on Algebra were printed, was Dr. Robert Recorde', a learned physician and mathematician, who flourished under Edward VI. and Mary. He published a treatise on Arithmetic in 1552, entitled The Ground of Arts, a work much esteemed at that time, and which continued many years the standard in that branch of knowledge. In 1557 he sent abroad a second part, under the title of Cos Ingenii, or the Whetstone of Witte; this part treats of Algebra in the form of a dialogue: in his method he imitated the Germans Stifelius and Scheubelius, especially the latter, whom he sometimes quotes and copies. The first instance of the extraction of the roots of compound alge braic quantities occurs in this book, and here also are first introduced the terms binomial and residual, and the sign of equality. = After Recorde's death, it appears that Algebra was not much cultivated in England for several years, inasmuch as John Dee, in his Preface to Billingsley's Euclid, i Robert Recorde was born in Wales early in the 16th century'; and about 1525 went to Oxford, where in 1531 he became fellow of All Souls College: making physic his profession, he repaired to Cambridge, where he was honoured with the degree of M. D. in 1545. He afterwards taught the Mathematics with great applause at Oxford, and probably next at London; he was physician to both the monarchs mentioned above, and the author of several mathematical treatises. He was confined for debt in the King's Bench Prison, where he died in the year 1558. k John Dee was born in London in 1527; at fifteen years old he went to St. John's College, Cambridge: after five years close attention to the Mathematics, Astronomy, &c. he set out for the Continent; returning the next year, he was printed at London in 1570, mentions Algebra as a mystery scarcely heard of by the studious in Mathematics here. In 1558, Peletarius published at Paris a very ingenious and masterly composition, entitled Jacobi Peletarii Cenomani, de occulta parte Numerorum, quam Algebram vocant, Lib. duo; in which he ably treats of all the parts of the subject then known, excepting Cubic Equations, and teaches some curious properties of square and cube numbers, with the method of constructing a table of each, by addition only; how to reduce trinomial surds to rational quantities, and even to simple ones, by means of certain compound multipliers; and that the root of an Equation is one of the divisors of the known or absolute term. The Stratioticos of Mr. Thomas Digges', containing a elected fellow of Trinity College. His great application to Astronomy, together with some ingenious mechanical inventions with which he occasionally amused himself, gave rise to a suspicion that he was a conjurer, and he was obliged in consequence to quit the country. He went to the University of Louvain, thence to the College of Rheims, where he read lectures upon Euclid; in 1551 he returned to England, and had the rectory of Upton upon Severn: afterwards, in consequence of a correspondence he had with Elizabeth, he was accused of practising enchantment against the life of Queen Mary her sister, and suffered a tedious confinement. On the accession of Queen Elizabeth, he was introduced to her, and (agreeably to the superstitious customs of that period) consulted respecting a propitious day for the coronation. She employed him afterwards in making geographical descriptions and maps of the countries to which England might have any claim; in this he acquitted himself with credit, as he did in his labours respecting the reformation of the calendar. In 1581, meeting with Edward Kelly, a credulous alchymist, our eccentric author and he performed together divers imaginary incantations, and held a pretended intercourse with angels and spirits. Our two conjurers asserted, that they were in possession of the secret of transmuting the baser metals into gold; and meeting with one Albert Laski, a Polish Nobleman, as credulous and ridiculous as themselves, they all three set out together for the Continent: here they imposed upon such of the rich and affluent as were silly enough to believe them, and lived on the profits of their trade in great affluence. Some disputes arising, Dee returned to England, and was graciously received by the Queen, who in 1595 made him Warden of Manchester College. He died at Mortlake in 1608, leaving some valuable works behind him. 1 Thomas Digges flourished in the reign of Elizabeth, but the times of his |