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Francois Viete (1540 – 1603) was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV.
Viete was born at Fontenay-le-Comte, Vendee. His grandfather was a merchant from La Rochelle. His father, Etienne Viete, was an attorney in Fontenay-le-Comte and a notary in Le Busseau. His mother was the aunt of Barnabe Brisson, a magistrate and the first president of parliament during the ascendancy of the Catholic League of France.
Viete went to a Franciscan school and in 1558 studied law at Poitiers, graduating as a Bachelor of Law in 1559. A year later, he began his career as an attorney in his native town. From the outset, he was entrusted with some major cases, including the settlement of rent in Poitou for the widow of King Francis I of France and looking after the interests of Mary, Queen of Scots.
In 1571, he enrolled as an attorney in Paris, and continued to visit his student Catherine. He regularly lived in Fontenay-le-Comte, where he took on some municipal functions. He began publishing his Universalium inspectionum ad canonem mathematicum liber singularis and wrote new mathematical research by night or during periods of leisure. He was known to dwell on any one question for up to three days, his elbow on the desk, feeding himself without changing position (according to his friend, Jacques de Thou).
In 1579, Viete printed his canonem mathematicum (Metayer publisher). A year later, he was appointed maitre des requetes to the parliament of Paris, committed to serving the king. That same year, his success in the trial between the Duke of Nemours and Francoise de Rohan, to the benefit of the latter, earned him the resentment of the tenacious Catholic League.
In 1582, Pope Gregory XIII published his bull Inter gravissimas and ordered the Catholic kings to comply with the change from the Julian calendar, based on the calculations of the Calabrian doctor Aloysius Lilius or Giglio. His work was resumed, after his death, by the scientific adviser to the Pope, Christopher Clavius.
Viete accused Clavius, in a series of pamphlets (1600), of introducing corrections and intermediate days in an arbitrary manner, and misunderstanding the meaning of the works of his predecessor, particularly in the calculation of the lunar cycle. Viete gave a new timetable, which Clavius cleverly refuted, after Viete's death, in his Explication (1603).
He said that Viete was wrong. Without doubt, he believed himself be a kind of "King of Times" as the historian of mathematics, Dhombres, claimed. It is true that Viete held Clavius in low esteem, as evidenced by De Thou: "He said that Clavius was very clever to explain the principles of mathematics that he heard with great clarity what the authors had invented, and wrote various treaties compelling what had been written before him without quoting its referencies. So, his works were in a better order which was scattered and confused in early writings..."
In 1598, Viete was granted special leave. Henry IV, however, charged him to end the revolt of the Notaries, whom the King had ordered to pay back their fees. Sick and exhausted by work, he left the King's service in December 1602 and received 20,000 ecu, which were found at his bedside after his death.
A few weeks before his death, he wrote a final thesis on issues of cryptography, whose memory made obsolete all encryption methods of the time. He died on 23 February 1603, as wrote De Thou. The cause of Viete's death is unknown.
At the end of 16th century, mathematics was placed under the dual aegis of the Greeks, from whom they borrowed the tools of geometry, and the Arabs, who provided procedures for the resolution. At the time of Viete, algebra therefore oscillated between arithmetic, which gave the appearance of a list of rules, and geometry which seemed more rigorous. Meanwhile, Italian mathematicians Luca Pacioli, Scipione del Ferro, Niccolo Fontana Tartaglia, Ludovico Ferrari, and especially Raphael Bombelli (1560) all developed techniques for solving equations of the third degree, which heralded a new era.
On the other hand, the German school of the Coss, the English mathematician Robert Recorde (1550) and the Dutchman Simon Stevin (1581) brought an early algebraic notation, the use of decimals and exponents. However, complex numbers remained at best a philosophical way of thinking and Descartes, almost a century after their invention, used them as imaginary numbers. Only positive solutions were considered and using geometrical proof was common.
The task of the mathematicians was in fact twofold. It was necessary to produce algebra in a more geometrical way, i.e. to give it a rigorous foundation and, on the other hand, it was necessary to give geometry a more algebraic sense, allowing the analytical calculation in the plane. Viete and Descartes solved this dual task in a double revolution. Firstly, Viete gave algebra a foundation as strong as in geometry. He then ended the algebra of procedures (al-Jabr and Muqabala), creating the first symbolic algebra. In doing so, he did not hesitate to say that with this new algebra, all problems could be solved. twofold In his dedication of "the Isagoge to Catherine de Parthenay", Viete wrote, "These things which are new are wont in the beginning to be set forth rudely and formlessly and must then be polished and perfected in succeeding centuries. Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbarians, that I considered it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudo-technical terms…".
Viete did not know "multiplied" notation (given by William Oughtred in 1631) or the symbol of equality, =, an absence which is more striking because Robert Recorde had used the present symbol for this purpose since 1557 and Guilielmus Xylander had used parallel vertical lines since 1575.
Viete had neither much time, nor students able to brilliantly illustrate his method. He took years in publishing his work, (he was very meticulous) and most importantly, he made a very specific choice to separate the unknown variables, using consonants for parameters and vowels for unknowns. In this notation he perhaps followed some older contemporaries, such as Petrus Ramus, who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, etc., only when these were exhausted. This choice proved disastrous for readability and Descartes, in preferring the first letters to designate the parameters, the latter for the unknowns, showed a greater knowledge of the human heart.
Viete also remained prisoner of his time in several respects: first, he was heir of Ramus and did not address the lengths as numbers. His writing kept track of homogeneity, which did not simplify their reading. He failed to recognize the complex numbers of Bombelli and needed to double check his algebraic answers through geometrical construction. Although he was fully aware that his new algebra was sufficient to give a solution, this concession tainted his reputation.
However, Viete created many innovations: the binomial formula, which would be taken by Pascal and Newton, and the link between the roots and coefficients of a polynomial, called Viete's formula.
Viete was well skilled in most modern artifices, aiming at the simplification of equations by the substitution of new quantities having a certain connection with the primitive unknown quantities. Another of his works, "Recensio canonica effectionum geometricarum", bears a modern stamp, being what was later called an algebraic geometry — a collection of precepts how to construct algebraic expressions with the use of ruler and compass only. While these writings were generally intelligible, and therefore of the greatest didactic importance, the principle of homogeneity, first enunciated by Viete, was so far in advance of his times that most readers seem to have passed it over. That principle had been made use of by the Greek authors of the classic age; but of later mathematicians only Hero, Diophantus, etc., ventured to regard lines and surfaces as mere numbers that could be joined to give a new number, their sum.
The study of such sums, found in the works of Diophantus, may have prompted Viete to lay down the principle that quantities occurring in an equation ought to be homogeneous, all of them lines, or surfaces, or solids, or super solids – an equation between mere numbers being in admissible. During the centuries that have elapsed between Viete's day and the present, several changes of opinion have taken place on this subject. Modern mathematicians like to make homogeneous such equations as are not so from the beginning, in order to get values of a symmetrical shape. Viete himself did not see that far; nevertheless, he indirectly suggested the thought. He also conceived methods for the general resolution of equations of the second, third and fourth degrees different from those of Scipione dal Ferro and Lodovico Ferrari, with which he had not been acquainted. He devised an approximate numerical solution of equations of the second and third degrees, wherein Leonardo of Pisa must have preceded him, but by a method which was completely lost.
Above all, Viete was the first mathematician who introduced notations for the problem (and not just for the unknowns). As a result, his algebra was no longer limited to the statement of rules, but relied on an efficient computer algebra, in which the operations act on the letters and the results can be obtained at the end of the calculations by a simple replacement. This approach, which is the heart of contemporary algebraic method, was a fundamental step in the development of mathematics. With this, Viete marked the end of medieval algebra (from Al-Khwarizmi to Stevin) and opened the modern period.
Among the problems addressed by Viete with this method is the complete resolution of the quadratic equations of the form and third-degree equations of the form (Viete reduced it to quadratic equations). He knew the connection between the positive roots of an equation (which, in his day, were alone thought of as roots) and the coefficients of the different powers of the unknown quantity (see Viete's formulas and their application on quadratic equations). He discovered the formula for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sine. This formula must have been known to Viete in 1593.
In the Eighth Book of the varied responses, he talks about the problems of the trisection of the angle (which he acknowledges that it is bound to an equation of third degree) of squaring the circle, building the regular heptagon, etc.
The same year, based on geometrical considerations and through trigonometric calculations perfectly mastered, he discovered the first infinite product in history of mathematics by giving an expression of :

He provides 10 decimal places of by applying the Archimedes method to a polygon with 6 x 216 = 393 216 sides.
In 1600, he created a work that provided the means for extracting roots and solutions of equations of degree at most 6.
Thirty four years after the death of Viete, the philosopher Rene Descartes published his method and a book of geometry that changed the landscape of algebra and built on Viete's work, applying it to the geometry by removing its requirements of homogeneity. Descartes, accused by Jean Baptiste Chauveau, a former classmate of La Fleche, explained in a letter to Mersenne (1639 February) that he never read those works. "I have no knowledge of this surveyor and I wonder what he said, that we studied Viete's work together in Paris, because it is a book which I cannot remember having seen the cover, while I was in France."
Elsewhere, Descartes said that Viete's notations were confusing and used unnecessary geometric justifications. In some letters, he showed he understands the program of the Artem Analyticem Isagoge; in others, he shamelessly caricatured Viete's proposals. One of his biographers, Charles Adam, noted this contradiction: "These words are surprising, by the way, for he (Descartes) had just said a few lines earlier that he had tried to put in his geometry only what he believed "was known neither by Viete nor by anyone else". So he was informed of what Viete knew; and he must have read his works previously".
Although Viete was not the first to propose notation of unknown quantities by letters (Jordanus Nemorarius had done it in the past), we can reasonably estimate that it would be simplistic to summarize his innovations for that discovery and place him at the junction of algebraic transformations made during the late sixteenth – early 17th century.

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