François Viète facts for kids

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Quick facts for kids François Viète

Born	1540 Fontenay-le-Comte, Kingdom of France
Died	23 February 1603 (aged 62–63) Paris, Kingdom of France
Nationality	French
Other names	Franciscus Vieta
Education	University of Poitiers (LL.B., 1559)
Known for	New algebra (the first symbolic algebra) Vieta's formulas Viète's formula
Scientific career
Fields	Astronomy, mathematics (algebra and trigonometry)
Notable students	Alexander Anderson
Influences	Peter Ramus Gerolamo Cardano
Influenced	Pierre de Fermat René Descartes

Signature

François Viète, Seigneur de la Bigotière (Latin: Franciscus Vieta; 1540 – 23 February 1603), commonly known by his mononym, Vieta, 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 of France.

Biography

Early life and education

Viète was born at Fontenay-le-Comte in present-day Vendée. His grandfather was a merchant from La Rochelle. His father, Etienne Viète, was an attorney in Fontenay-le-Comte and a notary in Le Busseau. His mother was the aunt of Barnabé Brisson, a magistrate and the first president of parliament during the ascendancy of the Catholic League of France.

Viète went to a Franciscan school and in 1558 studied law at Poitiers, graduating as a Bachelor of Laws 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.

Serving Parthenay

In 1564, Viète entered the service of Antoinette d’Aubeterre, Lady Soubise, wife of Jean V de Parthenay-Soubise, one of the main Huguenot military leaders and accompanied him to Lyon to collect documents about his heroic defence of that city against the troops of Jacques of Savoy, 2nd Duke of Nemours just the year before.

The same year, at Parc-Soubise, in the commune of Mouchamps in present-day Vendée, Viète became the tutor of Catherine de Parthenay, Soubise's twelve-year-old daughter. He taught her science and mathematics and wrote for her numerous treatises on astronomy and trigonometry, some of which have survived. In these treatises, Viète used decimal numbers (twenty years before Stevin's paper) and he also noted the elliptic orbit of the planets, forty years before Kepler and twenty years before Giordano Bruno's death.

John V de Parthenay presented him to King Charles IX of France. Viète wrote a genealogy of the Parthenay family and following the death of Jean V de Parthenay-Soubise in 1566 his biography.

In 1568, Antoinette, Lady Soubise, married her daughter Catherine to Baron Charles de Quellenec and Viète went with Lady Soubise to La Rochelle, where he mixed with the highest Calvinist aristocracy, leaders like Coligny and Condé and Queen Jeanne d’Albret of Navarre and her son, Henry of Navarre, the future Henry IV of France.

In 1570, he refused to represent the Soubise ladies in their infamous lawsuit against the Baron De Quellenec, where they claimed the Baron was unable (or unwilling) to provide an heir.

First steps in Paris

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 1572, Viète was in Paris during the St. Bartholomew's Day massacre. That night, Baron De Quellenec was killed after having tried to save Admiral Coligny the previous night. The same year, Viète met Françoise de Rohan, Lady of Garnache, and became her adviser against Jacques, Duke of Nemours.

In 1573, he became a councillor of the Parliament of Brittany, at Rennes, and two years later, he obtained the agreement of Antoinette d'Aubeterre for the marriage of Catherine of Parthenay to Duke René de Rohan, Françoise's brother.

In 1576, Henri, duc de Rohan took him under his special protection, recommending him in 1580 as "maître des requêtes". In 1579, Viète finished the printing of his Canonem mathematicum (Mettayer publisher). A year later, he was appointed maître des requêtes to the parliament of Paris, committed to serving the king. That same year, his success in the trial between the Duke of Nemours and Françoise de Rohan, to the benefit of the latter, earned him the resentment of the tenacious Catholic League.

Exile in Fontenay

Between 1583 and 1585, the League persuaded Henry III to release Viète, Viète having been accused of sympathy with the Protestant cause. Henry of Navarre, at Rohan's instigation, addressed two letters to King Henry III of France on March 3 and April 26, 1585, in an attempt to obtain Viète's restoration to his former office, but he failed.

Viète retired to Fontenay and Beauvoir-sur-Mer, with François de Rohan. He spent four years devoted to mathematics, writing his New Algebra (1591).

Code-breaker to two kings

In 1589, Henry III took refuge in Blois. He commanded the royal officials to be at Tours before 15 April 1589. Viète was one of the first who came back to Tours. He deciphered the secret letters of the Catholic League and other enemies of the king. Later, he had arguments with the classical scholar Joseph Juste Scaliger. Viète triumphed against him in 1590.

After the death of Henry III, Viète became a privy councillor to Henry of Navarre, now Henry IV. He was appreciated by the king, who admired his mathematical talents. Viète was given the position of councillor of the parlement at Tours. In 1590, Viète discovered the key to a Spanish cipher, consisting of more than 500 characters, and this meant that all dispatches in that language which fell into the hands of the French could be easily read.

Henry IV published a letter from Commander Moreo to the King of Spain. The contents of this letter, read by Viète, revealed that the head of the League in France, Charles, Duke of Mayenne, planned to become king in place of Henry IV. This publication led to the settlement of the Wars of Religion. The King of Spain accused Viète of having used magical powers. In 1593, Viète published his arguments against Scaliger. Beginning in 1594, he was appointed exclusively deciphering the enemy's secret codes.

Gregorian calendar

In 1582, Pope Gregory XIII published his bull Inter gravissimas and ordered Catholic kings to comply with the change from the Julian calendar, based on the calculations of the Calabrian doctor Aloysius Lilius, aka Luigi Lilio or Luigi Giglio. His work was resumed, after his death, by the scientific adviser to the Pope, Christopher Clavius.

Viète 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. Viète gave a new timetable, which Clavius cleverly refuted, after Viète's death, in his Explicatio (1603).

It is said that Viète was wrong. Without doubt, he believed himself to be a kind of "King of Times" as the historian of mathematics, Dhombres, claimed.

The Adriaan van Roomen problem

In 1596, Scaliger resumed his attacks from the University of Leyden. Viète replied definitively the following year. In March that same year, Adriaan van Roomen sought the resolution, by any of Europe's top mathematicians, to a polynomial equation of degree 45. King Henri IV received a snub from the Dutch ambassador, who claimed that there was no mathematician in France. He said it was simply because some Dutch mathematician, Adriaan van Roomen, had not asked any Frenchman to solve his problem.

Viète came, saw the problem, and, after leaning on a window for a few minutes, solved it. It was the equation between sin(x) and sin(x/45). He resolved this at once, and said he was able to give at the same time (actually the next day) the solution to the other 22 problems to the ambassador. "Ut legit, ut solvit," he later said. Further, he sent a new problem back to Van Roomen, for resolution by Euclidean tools (rule and compass) of the lost answer to the problem first set by Apollonius of Perga. Van Roomen could not overcome that problem without resorting to a trick (see detail below).

Final years

In 1598, Viète 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 écu, 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 De Thou wrote, leaving two daughters, Jeanne, whose mother was Barbe Cottereau, and Suzanne, whose mother was Julienne Leclerc. Jeanne, the eldest, died in 1628, having married Jean Gabriau, a councillor of the parliament of Brittany. Suzanne died in January 1618 in Paris.

The cause of Viète's death is unknown. Alexander Anderson, student of Viète and publisher of his scientific writings, speaks of a "praeceps et immaturum autoris fatum." (meeting an untimely end).

Work and thought

Opera, 1646

New algebra

Background

At the end of the 16th century, mathematics was placed under the dual aegis of Greek geometry and the Arabic procedures for resolution. At the time of Viète, 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, Niccolò 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, from the German school of Coss, the Welsh 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. Descartes, almost a century after their invention, used them as imaginary numbers. Only positive solutions were considered and using geometrical proof was common.

The mathematician's task was in fact twofold. It was necessary to produce algebra in a more geometrical way (i.e. to give it a rigorous foundation), and it was also necessary to make geometry more algebraic, allowing for analytical calculation in the plane. Viète and Descartes solved this dual task in a double revolution.

Viète's symbolic algebra

Firstly, Viète gave algebra a foundation as strong as that of geometry. He then ended the algebra of procedures (al-Jabr and al-Muqabala), creating the first symbolic algebra, and claiming that with it, all problems could be solved (nullum non problema solvere).

Viète 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.

Viète 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 unpopular with future mathematicians and Descartes, among others, preferred the first letters of the alphabet to designate the parameters and the latter for the unknowns.

Viète also remained a 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, Viète created many innovations: the binomial formula, which would be taken by Pascal and Newton, and the coefficients of a polynomial to sums and products of its roots, called Viète's formula.

Geometric algebra

Viète 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 Viète, 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 Viète to lay down the principle that quantities occurring in an equation ought to be homogeneous, all of them lines, or surfaces, or solids, or supersolids — an equation between mere numbers being inadmissible. During the centuries that have elapsed between Viète'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. Viète 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, Viète 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 computational 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, Viète marked the end of medieval algebra (from Al-Khwarizmi to Stevin) and opened the modern period.

The logic of species

Being wealthy, Viète began to publish at his own expense, for a few friends and scholars in almost every country of Europe, the systematic presentation of his mathematic theory, which he called "species logistic" (from species: symbol) or art of calculation on symbols (1591).

He described in three stages how to proceed for solving a problem:

As a first step, he summarized the problem in the form of an equation. Viète called this stage the Zetetic. It denotes the known quantities by consonants (B, D, etc.) and the unknown quantities by the vowels (A, E, etc.)
In a second step, he made an analysis. He called this stage the Poristic. Here mathematicians must discuss the equation and solve it. It gives the characteristic of the problem, porisma (corrollary), from which we can move to the next step.
In the last step, the exegetical analysis, he returned to the initial problem which presents a solution through a geometrical or numerical construction based on porisma.

Among the problems addressed by Viète with this method is the complete resolution of the quadratic equations of the form $X^2 + X b = c$ and third-degree equations of the form $X^3 + aX = b$ (Viète 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 Viète'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 sines. This formula must have been known to Viète in 1593.

Viète's formula

In 1593, based on geometrical considerations and through trigonometric calculations perfectly mastered, he discovered the first infinite product in the history of mathematics by giving an expression of $π$ , now known as Viète's formula:

\pi= 2\times\frac{2}{\sqrt{2}}\times \frac{2}{\sqrt{2+\sqrt{2}}}\times \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\times\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\times\cdots

He provides 10 decimal places of $π$ by applying the Archimedes method to a polygon with 6 × 2¹⁶ = 393,216 sides.

Adriaan van Roomen's problem

This famous controversy is told by Tallemant des Réaux in these terms (46th story from the first volume of Les Historiettes.

This suggests that the Adrien van Roomen problem is an equation of 45°, which Viète recognized immediately as a chord of an arc of 8° ( $\frac{2\pi}{45}$ radians). It was then easy to determine the following 22 positive alternatives, the only valid ones at the time.

When, in 1595, Viète published his response to the problem set by Adriaan van Roomen, he proposed finding the resolution of the old problem of Apollonius, namely to find a circle tangent to three given circles. Van Roomen proposed a solution using a hyperbola, with which Viète did not agree, as he was hoping for a solution using Euclidean tools.

Viète published his own solution in 1600 in his work Apollonius Gallus. In this paper, Viète made use of the center of similitude of two circles. His friend De Thou said that Adriaan van Roomen immediately left the University of Würzburg, saddled his horse and went to Fontenay-le-Comte, where Viète lived. According to De Thou, he stayed a month with him, and learned the methods of the new algebra. The two men became friends and Viète paid all van Roomen's expenses before his return to Würzburg.

This resolution had an almost immediate impact in Europe and Viète earned the admiration of many mathematicians over the centuries. Viète did not deal with cases (circles together, these tangents, etc.), but recognized that the number of solutions depends on the relative position of the three circles and outlined the ten resulting situations. Descartes completed (in 1643) the theorem of the three circles of Apollonius, leading to a quadratic equation in 87 terms, each of which is a product of six factors (which, with this method, makes the actual construction humanly impossible).

Religious and political beliefs

Viète was accused of Protestantism by the Catholic League, but he was not a Huguenot. His father was, according to Dhombres. Indifferent in religious matters, he did not adopt the Calvinist faith of Parthenay, nor that of his other protectors, the Rohan family. His call to the parliament of Rennes proved the opposite. At the reception as a member of the court of Brittany, on 6 April 1574, he read in public a statement of Catholic faith.

Nevertheless, Viète defended and protected Protestants his whole life, and suffered, in turn, the wrath of the League. It seems that for him, the stability of the state was to be preserved and that under this requirement, the King's religion did not matter. At that time, such people were called "Politicals."

Furthermore, at his death, he did not want to confess his sins. A friend had to convince him that his own daughter would not find a husband, were he to refuse the sacraments of the Catholic Church. Whether Viète was an atheist or not is a matter of debate.

Publications

Between 1564 and 1568, Viète prepared for his student, Catherine de Parthenay, some textbooks of astronomy and trigonometry and a treatise that was never published: Harmonicon coeleste.
In 1571, Francisci Vietaei Universalium inspectionum ad Canonem mathematicum liber singularis (a book of trigonometry, often abbreviated Canonem mathematicum), which he published at his own expense and with great printing difficulties. This text contains many formulas on the sine and cosine and is unusual in using decimal numbers. The trigonometric tables here exceeded those of Regiomontanus (Triangulate Omnimodis, 1533) and Rheticus (1543, annexed to De revolutionibus of Copernicus).
In 1589, Deschiffrement d'une lettre escripte par le Commandeur Moreo au Roy d'Espaigne son maître.
In 1590, Deschiffrement description of a letter by the Commander Moreo at Roy Espaigne of his master, Tours: Mettayer.
In 1591:
- In artem analyticem isagoge (Introduction to the art of analysis), also known as Algebra Nova (New Algebra) Tours: Mettayer, in 9 folio; the first edition of the Isagoge.
- Zeteticorum libri quinque. Tours: Mettayer, in 24 folio; which are the five books of Zetetics, a collection of problems from Diophantus solved using the analytical art.
Between 1591 and 1593, Effectionum geometricarum canonica recensio. Tours: Mettayer, in 7 folio.
In 1593:
- Vietae Supplementum geometriae. Tours: Francisci, in 21 folio.
- Francisci Vietae Variorum de rebus responsorum mathematics liber VIII. Tours: Mettaye, in 49 folio; about the challenges of Scaliger.
- Variorum de rebus mathematicis responsorum liber VIII; the "Eighth Book of Varied Responses" in which 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.
In 1594, Munimen adversus nova cyclometrica. Paris: Mettayer, in quarto, 8 folio; again, a response against Scaliger.
In 1595, Ad problema quod omnibus mathematicis totius orbis construendum proposuit Adrianus Romanus, Francisci Vietae responsum. Paris: Mettayer, in quarto, 16 folio; about the Adriaan van Roomen problem.
In 1600:
- De numerosa potestatum ad exegesim resolutione. Paris: Le Clerc, in 36 folio; work that provided the means for extracting roots and solutions of equations of degree at most 6.
- Francisci Vietae Apollonius Gallus. Paris: Le Clerc, in quarto, 13 folio; where he referred to himself as the French Apollonius.
Between 1600 and 1602:
- Fontenaeensis libellorum supplicum in Regia magistri relatio Kalendarii vere Gregoriani ad ecclesiasticos doctores exhibita Pontifici Maximi Clementi VIII. Paris: Mettayer, in quarto, 40 folio.
- Francisci Vietae adversus Christophorum Clavium expostulatio. Paris: Mettayer, in quarto, 8 folio; his theses against Clavius.

1612:
- Supplementum Apollonii Galli edited by Marin Ghetaldi.
- Supplementum Apollonii Redivivi sive analysis problematis bactenus desiderati ad Apollonii Pergaei doctrinam a Marino Ghetaldo Patritio Regusino hujusque non ita pridem institutam edited by Alexander Anderson.
1615:
- Ad Angularum Sectionem Analytica Theoremata F. Vieta primum excogitata at absque ulla demonstratione ad nos transmissa, iam tandem demonstrationibus confirmata edited by Alexander Anderson.
- Pro Zetetico Apolloniani problematis a se jam pridem edito in supplemento Apollonii Redivivi Zetetico Apolloniani problematis a se jam pridem edito; in qua ad ea quae obiter inibi perstrinxit Ghetaldus respondetur edited by Alexander Anderson
- Francisci Vietae Fontenaeensis, De aequationum — recognitione et emendatione tractatus duo per Alexandrum Andersonum edited by Alexander Anderson
1617: Animadversionis in Franciscum Vietam, a Clemente Cyriaco nuper editae brevis diakrisis edited by Alexander Anderson
1619: Exercitationum Mathematicarum Decas Prima edited by Alexander Anderson
1631: In artem analyticem isagoge. Eiusdem ad logisticem speciosam notae priores, nunc primum in lucem editae. Paris: Baudry, in 12 folio; the second edition of the Isagoge, including the posthumously published Ad logisticem speciosam notae priores.