appears also, that the Algebraists of this period knew only how to solve numerical problems with one unknown quantity, and that they had no signs for either the quantities or operations, except abbreviations of the words themselves.

The publication of De Burgo's book seems to have been the means by which the knowledge of Algebra was first diffused through Italy, and other neighbouring countries, as we find many learned men, both there and in various parts of Germany, now began to study the science with so much success, that a very few years produced many learned Algebraists.

About the year 1505, Scipio Ferreus, Professor of Mathematics at Bononia, discovered the method of solving one case of Cubic Equations; but his method was most probably (agreeably to the fashion of those dark ages) retained as a secret. Thirty years after, Nicholas Tartalea, a Mathematician of Brescia, made the same discovery, with the addition of the rules for the two remaining cases; proposing, in like manner, to conceal the methods of investigation. The secret was nevertheless, after much difficulty, drawn from him by the address of Jerome Cardan', a celebrated Physician, Astrologer,

ception: Leonard De Pisa's tract on Cubics, &c. (if such a work really existed,) was probably a single manuscript, and totally unknown to the few Algebraists of that period. Every circumstance attending the discovery of the rules for cubics, which happened a few years after, adds weight to this conclusion.

e Tartalea was born at Brescia in Italy, towards the close of the 15th century he was a very respectable teacher of the Mathematics, and published various works of merit on that subject, the chief of which was, Trattato di Numeri et Misure, fol. 1556. being a treatise on Arithmetic, Algebra, Geometry, &c. here are many of the curious particulars of the dispute between our author and Cardan. He published in 1543, at Venice, all the books of Euclid, with curious notes. He was the first author who treated of the flight and path of balls and shells, in a work entitled, Nova Scientia Inventa, 4to. published at Venice in 1537. an English translation of which, with notes and additions by Lucar, came out at London in 1588. Tartalea died about the year 1558.

f Hieronymus Cardanus was born at Paria in Italy, in 1501. At twenty years

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 : he frequently used the literal notation, expressing quantities by letters; treated fully on the transformation of Equations; 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 Maurolicus was born at Messina in 1494: he was a great proficient 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, &c. 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 fore told, were witnesses only of the mortification and disappointment of the unfor tunate 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 algebraic 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