François Viète (; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
whose work on
new algebra
New or NEW may refer to:
Music
* New, singer of K-pop group The Boyz
* ''New'' (album), by Paul McCartney, 2013
** "New" (Paul McCartney song), 2013
* ''New'' (EP), by Regurgitator, 1995
* "New" (Daya song), 2017
* "New" (No Doubt song), 1 ...
was an important step towards modern algebra, due to his 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
Fontenay-le-Comte (; Poitevin dialect, Poitevin: ''Funtenaes'' or ''Fintenè'') is a Communes of France, commune and Subprefectures in France, subprefecture in the Vendée Departments of France, department in the Pays de la Loire Regions of France ...
in present-day
Vendée
Vendée () is a department in the Pays de la Loire region in Western France, on the Atlantic coast. In 2019, it had a population of 685,442.La Rochelle
La Rochelle (, , ; Poitevin-Saintongeais: ''La Rochéle'') is a city on the west coast of France and a seaport on the Bay of Biscay, a part of the Atlantic Ocean. It is the capital of the Charente-Maritime Departments of France, department. Wi ...
. 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
The term magistrate is used in a variety of systems of governments and laws to refer to a civilian officer who administers the law. In ancient Rome, a '' magistratus'' was one of the highest ranking government officers, and possessed both judi ...
and the first president of parliament during the ascendancy of the Catholic League of France.
Viète went to a
Franciscan
The Franciscans are a group of related organizations in the Catholic Church, founded or inspired by the Italian saint Francis of Assisi. They include three independent Religious institute, religious orders for men (the Order of Friars Minor bei ...
school and in 1558 studied law at
Poitiers
Poitiers is a city on the river Clain in west-central France. It is a commune in France, commune, the capital of the Vienne (department), Vienne department and the historical center of Poitou, Poitou Province. In 2021, it had a population of 9 ...
, graduating as a
Bachelor of Laws
A Bachelor of Laws (; LLB) is an undergraduate law degree offered in most common law countries as the primary law degree and serves as the first professional qualification for legal practitioners. This degree requires the study of core legal subje ...
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
Francis I (; ; 12 September 1494 – 31 March 1547) was King of France from 1515 until his death in 1547. He was the son of Charles, Count of Angoulême, and Louise of Savoy. He succeeded his first cousin once removed and father-in-law Louis&nbs ...
and looking after the interests of
Mary, Queen of Scots
Mary, Queen of Scots (8 December 1542 – 8 February 1587), also known as Mary Stuart or Mary I of Scotland, was List of Scottish monarchs, Queen of Scotland from 14 December 1542 until her forced abdication in 1567.
The only surviving legit ...
Huguenot
The Huguenots ( , ; ) are a Religious denomination, religious group of French people, French Protestants who held to the Reformed (Calvinist) tradition of Protestantism. The term, which may be derived from the name of a Swiss political leader, ...
military leaders, and accompanied him to
Lyon
Lyon (Franco-Provençal: ''Liyon'') is a city in France. It is located at the confluence of the rivers Rhône and Saône, to the northwest of the French Alps, southeast of Paris, north of Marseille, southwest of Geneva, Switzerland, north ...
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
Mouchamps () is a Communes of France, commune in the Vendée Departments of France, department in the Pays de la Loire Regions of France, region in western France.
The grave of the noted French prime minister, Georges Clemenceau, is located near ...
in present-day
Vendée
Vendée () is a department in the Pays de la Loire region in Western France, on the Atlantic coast. In 2019, it had a population of 685,442.Catherine de Parthenay, Soubise's twelve-year-old daughter. He taught her science and mathematics and wrote for her numerous treatises on
astronomy
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
and
trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
, 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
Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of p ...
and twenty years before
Giordano Bruno
Giordano Bruno ( , ; ; born Filippo Bruno; January or February 1548 – 17 February 1600) was an Italian philosopher, poet, alchemist, astrologer, cosmological theorist, and esotericist. He is known for his cosmological theories, which concep ...
's death.
John V de Parthenay presented him to King
Charles IX of France
Charles IX (Charles Maximilien; 27 June 1550 – 30 May 1574) was List of French monarchs, King of France from 1560 until his death in 1574. He ascended the French throne upon the death of his brother Francis II of France, Francis II in 1560, an ...
. 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
Henry IV (; 13 December 1553 – 14 May 1610), also known by the epithets Good King Henry (''le Bon Roi Henri'') or Henry the Great (''Henri le Grand''), was King of Navarre (as Henry III) from 1572 and King of France from 1589 to 16 ...
.
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
Jacques de Savoie, duc de Nemours (12 October 153115 June 1585) was a French military commander, governor and Prince étranger, Prince Étranger. Having inherited his titles at a young age, Nemours fought for king Henri II of France, Henri II duri ...
Rennes
Rennes (; ; Gallo language, Gallo: ''Resnn''; ) is a city in the east of Brittany in Northwestern France at the confluence of the rivers Ille and Vilaine. Rennes is the prefecture of the Brittany (administrative region), Brittany Regions of F ...
, 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
A Master of Requests () is a counsel of the French ''Conseil d'État'' (Council of State), a high-level judicial officer of administrative law in France. The office has existed in one form or another since the Middle Ages.
The occupational titl ...
". In 1579, Viète finished the printing of his ''Universalium inspectionum'' (Mettayer publisher), published as an appendix to a book of two trigonometric tables (''Canon mathematicus, seu ad triangula'', the "canon" referred to by the title of his ''Universalium inspectionum'', and ''Canonion triangulorum laterum rationalium''). 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 king 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
Henry III (; ; ; 19 September 1551 – 2 August 1589) was King of France from 1574 until his assassination in 1589, as well as King of Poland and Grand Duke of Lithuania from 1573 to 1575.
As the fourth son of King Henry II of France, he ...
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
New or NEW may refer to:
Music
* New, singer of K-pop group The Boyz
* ''New'' (album), by Paul McCartney, 2013
** "New" (Paul McCartney song), 2013
* ''New'' (EP), by Regurgitator, 1995
* "New" (Daya song), 2017
* "New" (No Doubt song), 1 ...
'' (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 of France. He was appreciated by the king, who admired his mathematical talents. Viète was given the position of councillor of the ''parlement'' at
Tours
Tours ( ; ) is the largest city in the region of Centre-Val de Loire, France. It is the Prefectures in France, prefecture of the Departments of France, department of Indre-et-Loire. The Communes of France, commune of Tours had 136,463 inhabita ...
. In 1590, Viète broke the key to a Spanish
cipher
In cryptography, a cipher (or cypher) is an algorithm for performing encryption or decryption—a series of well-defined steps that can be followed as a procedure. An alternative, less common term is ''encipherment''. To encipher or encode i ...
, 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
Charles de Lorraine, duc de Mayenne (26 March 1554 –3 October 1611) was a French noble, governor, military commander and rebel during the latter French Wars of Religion. Born in 1554, the second son of François, Duke of Guise, François d ...
, 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
Aloysius Lilius ( 1510 – 1576), also variously referred to as Luigi Lilio or Luigi Giglio, was an Italian physician, astronomer, philosopher and chronologist, and also the "primary author" who provided the proposal that (after modifications) ...
, 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. It is true that Viète held Clavius in low esteem, as evidenced by De Thou:
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
Adriaan van Roomen (29 September 1561 – 4 May 1615), also known as Adrianus Romanus, was a mathematician, professor of medicine and medical astrologer from the Duchy of Brabant in the Habsburg Netherlands who was active throughout Central Europ ...
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
In religious context, sin is a transgression against divine law or a law of the deities. Each culture has its own interpretation of what it means to commit a sin. While sins are generally considered actions, any thought, word, or act considered ...
(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
Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...
. 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
The term ''écu'' () may refer to one of several France, French coins. The first ''écu'' was a gold coin (the ''écu d'or'') minted during the reign of Louis IX of France, in 1266. The value of the ''écu'' varied considerably over time, and si ...
s, which were found at his bedside after his death.
A few weeks before his death, he wrote a final thesis on issues of cryptography, which essay 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
Brittany ( ) is a peninsula, historical country and cultural area in the north-west of modern France, covering the western part of what was known as Armorica in Roman Gaul. It became an Kingdom of Brittany, independent kingdom and then a Duch ...
. Suzanne died in January 1618 in Paris.
The cause of Viète's death is unknown.
Alexander Anderson Alexander Anderson may refer to:
Arts and entertainment
* Alexander Anderson (cartoonist) (1920–2010), American cartoonist, creator of "Rocky the Flying Squirrel"
* Alexander Anderson (illustrator) (1775–1870), American illustrator
* Alexander ...
, student of Viète and publisher of his scientific writings, speaks of a "praeceps et immaturum autoris fatum" (meeting an untimely end).
Work and thought
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
Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
Gerolamo Cardano
Gerolamo Cardano (; also Girolamo or Geronimo; ; ; 24 September 1501– 21 September 1576) was an Italian polymath whose interests and proficiencies ranged through those of mathematician, physician, biologist, physicist, chemist, astrologer, as ...
,
Lodovico Ferrari
Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italians, Italian mathematician best known today for solving the biquadratic equation.
Biography
Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of M ...
, 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
Robert Recorde () was a Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus (+) and minus (−) signs to English speakers in 1557.
Biography
Born around 1510, Robert Recorde was the sec ...
(1550) and the Dutchman
Simon Stevin
Simon Stevin (; 1548–1620), sometimes called Stevinus, was a County_of_Flanders, Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He a ...
(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'').H. J. M. Bos : Redefining geometrical exactness: Descartes' transformatio Google Books /ref>Jacob Klein: Greek mathematical thought and the origin of algebra Google Books /ref>
In his dedication of the '' Isagoge'' to Catherine de Parthenay, Viète wrote:
Viète did not know "multiplied" notation (given by
William Oughtred
William Oughtred (5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernard, T. Birch and J. Lockman, ''A General ...
in 1631) or the symbol of equality, =, an absence which is more striking because
Robert Recorde
Robert Recorde () was a Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus (+) and minus (−) signs to English speakers in 1557.
Biography
Born around 1510, Robert Recorde was the sec ...
had used the present symbol for this purpose since 1557, and Guilielmus Xylander had used parallel vertical lines since 1575. Note also the use of a 'u' like symbol with a number above it for an unknown to a given power by
Rafael Bombelli
Rafael Bombelli (baptised on 20 January 1526; died 1572) was an Italian mathematician. Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers.
He was the one who finally manag ...
in 1572.
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
Petrus Ramus (; Anglicized as Peter Ramus ; 1515 – 26 August 1572) was a French humanist, logician, and educational reformer. A Protestant convert, he was a victim of the St. Bartholomew's Day massacre.
Early life
He was born at the village ...
, 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
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
, which would be taken by Pascal and Newton, and the
coefficients
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a ...
of a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
to sums and products of its
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusin ...
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
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
—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
A hero (feminine: heroine) is a real person or fictional character who, in the face of danger, combats adversity through feats of ingenuity, courage, or Physical strength, strength. The original hero type of classical epics did such thin ...
,
Diophantus
Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations.
Although Jose ...
, 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
Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italians, Italian mathematician best known today for solving the biquadratic equation.
Biography
Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of M ...
, 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
Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
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 and third-degree equations of the form (Viète reduced it to quadratic equations). He knew the connection between the positive
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusin ...
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
In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
). 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 mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product
:
\prod_^ a_n = a_1 a_2 a_3 \cdots
is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound ...
in the history of mathematics by giving an expression of , now known as Viète's formula:
:
He provides 10 decimal places of by applying the
Archimedes
Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
method to a polygon with 6 × 216 = 393,216 sides.
Adriaan van Roomen's challenge and the problem of Apollonius
This famous controversy is told by Tallemant des Réaux in these terms (46th story from the first volume of ''Les Historiettes. Mémoires pour servir à l’histoire du XVIIe siècle''):
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
In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...
, 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
The Julius Maximilian University of Würzburg (also referred to as the University of Würzburg, in German ''Julius-Maximilians-Universität Würzburg'') is a public research university in Würzburg, Germany. Founded in 1402, it is one of the ol ...
, 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
New or NEW may refer to:
Music
* New, singer of K-pop group The Boyz
* ''New'' (album), by Paul McCartney, 2013
** "New" (Paul McCartney song), 2013
* ''New'' (EP), by Regurgitator, 1995
* "New" (Daya song), 2017
* "New" (No Doubt song), 1 ...
. 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.Dhombres, Jean. François Viète et la Réforme. Available a cc-parthenay.fr 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
; Chronological list
* 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 1579, the trigonometric table ''Canon mathematicus, seu ad triangula'' published together with a table of rational-sided triangles ''Canonion triangulorum laterum rationalium'', and a book of trigonometry ''Universalium inspectionum ad 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
Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrument ...
(Triangulate Omnimodis, 1533) and Rheticus (1543, annexed to ''De revolutionibus'' of
Copernicus
Nicolaus Copernicus (19 February 1473 – 24 May 1543) was a Renaissance polymath who formulated a mathematical model, model of Celestial spheres#Renaissance, the universe that placed heliocentrism, the Sun rather than Earth at its cen ...
). Alternative scan of a 1589 reprint
* 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
New or NEW may refer to:
Music
* New, singer of K-pop group The Boyz
* ''New'' (album), by Paul McCartney, 2013
** "New" (Paul McCartney song), 2013
* ''New'' (EP), by Regurgitator, 1995
* "New" (Daya song), 2017
* "New" (No Doubt song), 1 ...
'') Tours: Mettayer, in 9
folio
The term "folio" () has three interconnected but distinct meanings in the world of books and printing: first, it is a term for a common method of arranging Paper size, sheets of paper into book form, folding the sheet only once, and a term for ...
; 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
Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations.
Although Jose ...
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
Quarto (abbreviated Qto, 4to or 4º) is the format of a book or pamphlet produced from full sheets printed with eight pages of text, four to a side, then folded twice to produce four leaves. The leaves are then trimmed along the folds to produc ...
, 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
Adriaan van Roomen (29 September 1561 – 4 May 1615), also known as Adrianus Romanus, was a mathematician, professor of medicine and medical astrologer from the Duchy of Brabant in the Habsburg Netherlands who was active throughout Central Europ ...
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.
; Posthumous publications
* 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 Alexander Anderson may refer to:
Arts and entertainment
* Alexander Anderson (cartoonist) (1920–2010), American cartoonist, creator of "Rocky the Flying Squirrel"
* Alexander Anderson (illustrator) (1775–1870), American illustrator
* Alexander ...
.
* 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''.
Reception and influence
During the ascendancy of the Catholic League, Viète's secretary was Nathaniel Tarporley, perhaps one of the more interesting and enigmatic
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s of 16th-century England. When he returned to London, Tarporley became one of the trusted friends of
Thomas Harriot
Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his con ...
.
Apart from Catherine de Parthenay, Viète's other notable students were: French mathematician Jacques Aleaume, from Orleans, Marino Ghetaldi of Ragusa, Jean de Beaugrand and the Scottish mathematician
Alexander Anderson Alexander Anderson may refer to:
Arts and entertainment
* Alexander Anderson (cartoonist) (1920–2010), American cartoonist, creator of "Rocky the Flying Squirrel"
* Alexander Anderson (illustrator) (1775–1870), American illustrator
* Alexander ...
. They illustrated his theories by publishing his works and continuing his methods. At his death, his heirs gave his manuscripts to Peter Aleaume. We give here the most important posthumous editions:
* In 1612: ''Supplementum Apollonii Galli'' of Marino Ghetaldi.
* From 1615 to 1619: ''Animadversionis in Franciscum Vietam, Clemente a Cyriaco nuper'' by Alexander Anderson
* ''Francisci Vietae Fontenaeensis ab aequationum recognitione et emendatione Tractatus duo Alexandrum'' per Andersonum. Paris, Laquehay, 1615, in 4, 135 p. The death of Alexander Anderson unfortunately halted the publication.
* In 1630, an ''Introduction en l'art analytic ou nouvelle algèbre'' ('Introduction to the analytic art or modern algebra''), translated into French and commentary by mathematician J. L. Sieur de Vaulezard. Paris, Jacquin.
* The ''Five Books of François Viette's Zetetic'' (''Les cinq livres des zététiques de François Viette''), put into French, and commented increased by mathematician J. L. Sieur de Vaulezard. Paris, Jacquin, p. 219.
The same year, there appeared an ''Isagoge'' by Antoine Vasset (a pseudonym of Claude Hardy), and the following year, a translation into Latin of Beaugrand, which Descartes would have received.
In 1648, the corpus of mathematical works printed by
Frans van Schooten
Frans van Schooten Jr. also rendered as Franciscus van Schooten (15 May 1615 – 29 May 1660) was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes. He translated La Géométrie in Latin and wrote c ...
, professor at Leiden University (Elzevirs presses). He was assisted by Jacques Golius and Mersenne.
The English mathematicians
Thomas Harriot
Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his con ...
and
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, and the Dutch physicist
Willebrord Snellius
Willebrord Snellius (born Willebrord Snel van Royen) (13 June 158030 October 1626) was a Dutch astronomer and mathematician, commonly known as Snell. His name is usually associated with the law of refraction of light known as Snell's law.
The ...
, the French mathematicians
Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
and
Blaise Pascal
Blaise Pascal (19June 162319August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
Pascal was a child prodigy who was educated by his father, a tax collector in Rouen. His earliest ...
all used Viète's symbolism.
About 1770, the Italian mathematician Targioni Tozzetti, found in Florence Viète's '' Harmonicon coeleste''. Viète had written in it: ''Describat Planeta Ellipsim ad motum anomaliae ad Terram''. (That shows he adopted Copernicus' system and understood before
Kepler
Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of p ...
the elliptic form of the orbits of planets.)
In 1841, the French mathematician
Michel Chasles
Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician.
Biography
He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coal ...
was one of the first to reevaluate his role in the development of modern algebra.
In 1847, a letter from
François Arago
Dominique François Jean Arago (), known simply as François Arago (; Catalan: , ; 26 February 17862 October 1853), was a French mathematician, physicist, astronomer, freemason, supporter of the Carbonari revolutionaries and politician.
Early l ...
, perpetual secretary of the Academy of Sciences (Paris), announced his intention to write a biography of François Viète.
Between 1880 and 1890, the polytechnician Fréderic Ritter, based in Fontenay-le-Comte, was the first translator of the works of François Viète and his first contemporary biographer with Benjamin Fillon.
Descartes' views on Viète
Thirty-four years after the death of Viète, the philosopher
René Descartes
René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
published his ''method'' and a book of geometry that changed the landscape of algebra and built on Viète's work, applying it to the geometry by removing its requirements of homogeneity. Descartes, accused by Jean Baptiste Chauveau, a former classmate of La Flèche, explained in a letter to Mersenne (1639 February) that he never read those works. Descartes accepted the Viète's view of mathematics for which the study shall stress the self-evidence of the results that Descartes implemented translating the symbolic algebra in geometric reasoning. Descartes adopted the term ''
mathesis universalis
(from , "science or learning", and "universal") is a hypothetical universal science modelled on mathematics envisaged by René Descartes, Descartes and Gottfried Wilhelm Leibniz, Leibniz, among a number of other 16th- and 17th-century philosop ...
'', which he called an "already venerable term with a received usage", which originated in van Roomen's book ''Mathesis Universalis''.
Elsewhere, Descartes said that Viète'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 Viète's proposals. One of his biographers, Charles Adam, noted this contradiction:
Current research has not shown the extent of the direct influence of the works of Viète on Descartes. This influence could have been formed through the works of Adriaan van Roomen or Jacques Aleaume at the Hague, or through the book by Jean de Beaugrand.
In his letters to Mersenne, Descartes consciously minimized the originality and depth of the work of his predecessors. "I began," he says, "where Vieta finished". His views emerged in the 17th century and mathematicians won a clear algebraic language without the requirements of homogeneity. Many contemporary studies have restored the work of Parthenay's mathematician, showing he had the double merit of introducing the first elements of literal calculation and building a first axiomatic for algebra.For example:
Although Viète was not the first to propose notation of unknown quantities by letters - Jordanus Nemorarius had done this 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.
See also
*
Vieta's formulas
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (1540-1603), more commonly referred to by the Latinised form of his name, "Franciscus Vieta."
Basi ...
Rafael Bombelli
Rafael Bombelli (baptised on 20 January 1526; died 1572) was an Italian mathematician. Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers.
He was the one who finally manag ...
Notes
Bibliography
* Bailey Ogilvie, Marilyn; Harvey, Joy Dorothy. ''The Biographical Dictionary of Women in Science: L–Z'' Google Books p 985.
* Bachmakova, Izabella G., Slavutin, E.I. “ Genesis Triangulorum de François Viète et ses recherches dans l’analyse indéterminée ”, ''Archives for History of Exact Science'', 16 (4), 1977, 289-306.
* Bashmakova, Izabella Grigorievna; Smirnova Galina S; Shenitzer, Abe. ''The Beginnings and Evolution of Algebra''. Google Books pp. 75–.
* Biard, Joel; Rāshid, Rushdī. ''Descartes et le Moyen Age''. Paris: Vrin, 1998 Google Books
* Burton, David M (1985). ''The History of Mathematics: An Introduction''. Newton, Massachusetts: Allyn and Bacon, Inc.
* Cajori, F. (1919). ''A History of Mathematics'' pp. 152 and onward
* Calinger, Ronald (ed.) (1995). ''Classics of Mathematics''. Englewood Cliffs, New Jersey: Prentice–Hall, Inc.
* Calinger, Ronald. ''Vita mathematica''. Mathematical Association of America Google Books * Chabert, Jean-Luc; Barbin, Évelyne; Weeks, Chris. ''A History of Algorithms'' Google Books * Derbyshire, John (2006). ''Unknown Quantity a Real and Imaginary History of Algebra''. Scribd.com
* Eves, Howard (1980). ''Great Moments in Mathematics (Before 1650)''. The Mathematical Association of America. Google Books * Grisard, J. (1968) ''François Viète, mathématicien de la fin du seizième siècle: essai bio-bibliographique'' (Thèse de doctorat de 3ème cycle) École Pratique des Hautes Études, Centre de Recherche d'Histoire des Sciences et des Techniques, Paris.
* Godard, Gaston. ''François Viète (1540–1603), Father of Modern Algebra''. Université de Paris-VII, France, Recherches vendéennes.
* W. Hadd, Richard. ''On the shoulders of merchants'' Google Books * Hofmann, Joseph E (1957). ''The History of Mathematics'', translated by F. Graynor and H. O. Midonick. New York, New York: The Philosophical Library.
* Joseph, Anthony. ''Round tables''. European Congress of Mathematics Google Books * Michael Sean Mahoney (1994). ''The mathematical career of Pierre de Fermat (1601–1665)'' Google Books * Jacob Klein. ''Die griechische Logistik und die Entstehung der Algebra'' in: Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien, Band 3, Erstes Heft, Berlin 1934, p. 18–105 and Zweites Heft, Berlin 1936, p. 122–235; translated in English by Eva Brann as: ''Greek Mathematical Thought and the Origin of Algebra''. Cambridge, Mass. 1968,
* Mazur, Joseph (2014). ''Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers''. Princeton, New Jersey: Princeton University Press.
* Nadine Bednarz, Carolyn Kieran, Lesley Lee. ''Approaches to algebra'' Google Books * Otte, Michael; Panza, Marco. ''Analysis and Synthesis in Mathematics'' Google Books * Pycior, Helena M. ''Symbols, Impossible Numbers, and Geometric Entanglements'' Google Books * ''Francisci Vietae Opera Mathematica'', collected by F. Van Schooten. Leyde, Elzévir, 1646, p. 554 Hildesheim-New-York: Georg Olms Verlag (1970).
* The intégral corpus (excluding Harmonicon) was published by
Frans van Schooten
Frans van Schooten Jr. also rendered as Franciscus van Schooten (15 May 1615 – 29 May 1660) was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes. He translated La Géométrie in Latin and wrote c ...
, professor at Leyde as ''Francisci Vietæ. Opera mathematica, in unum volumen congesta ac recognita, opera atque studio Francisci a Schooten'', Officine de Bonaventure et Abraham Elzevier, Leyde, 1646 Gallica.bnf.fr (pdf).
* Stillwell, John. Mathematics and its history. Google Books * Varadarajan, V. S. (1998). ''Algebra in Ancient and Modern Times'' The American Mathematical Society Google Books