François Viete

François Viète (Latin: Franciscus Vieta) was a French lawyer and mathematician (Fontenay-le-Comte, 1540-Paris, 1603).

He is considered one of the main precursors of algebra. He was the first to represent the parameters of an equation by letters, being an outstanding precursor of the use of algebra in cryptography, which allowed him to decode the encrypted messages of the Spanish Crown.

François Viète was also known in his time as a subject of the king, renowned for his loyalty and competence. He was private adviser to the kings of France, Henry III and Henry IV.

Biography

A life in the service of the king

The son of a procurator, Viète studied law in Poitiers. In 1560, he became a lawyer at Fontenay-le-Comte. He was entrusted with important matters, notably the settlement of the lands in the Poitou region of the widow of Francis I and the interests of Mary Queen of Scots.

In 1564, he joined the house of Soubise as private secretary in charge of defending the interests of the family. He also became a tutor to Catherine de Partenay, with whom he would remain attached throughout his life. He moved in the circles of the best-known Calvinist aristocracy, in such a way that he met the main chiefs Coligny and Enrique I de Bourbon (Prince of Condé), and also Juana III of Navarre, queen of Navarre and her son., Enrique de Navarra, future Enrique IV.

In 1571, he became a lawyer in the Parliament of Paris, and was appointed counselor in the Parliament of Rennes in 1573. In 1576, he entered the service of King Henry III, who entrusted him with a special mission. In 1580, he passed into the exclusive service of the king in the Parliament of Paris.

Also in 1580, Viète was in charge of an important lawsuit that pitted the Duke of Nemours against Françoise de Rohan, and which was decided in favor of the latter. This earned him the hatred of the Catholic League, which in 1584 achieved that he be separated from his functions. Enrique de Navarra wrote several letters in favor of Viète, trying to get him to recover his position at the king's service, but he was not listened to. Viète dedicated those years, separated from political life, to mathematics.

Expelled from Paris in 1589, after the day of the barricades, on May 12, 1588, Henry III was forced to take refuge in Blois. Therefore, he appealed to the royal officers to meet him in Tours before April 15, 1589. To this, Viète responded to this appeal among the first.

After the death of Henri III, Viète became part of the private council of Henri IV, who greatly admired him for his mathematical talent. Starting in 1594, he was exclusively in charge of deciphering the secret enemy codes, a task that he had been developing since 1580.

In 1590, Enrique IV had published a letter from Commander Moreo to the King of Spain. The content of this letter, which Viète had deciphered, revealed that the head of the League in France, the Duke of Mayenne, aspired to become king instead of Henry IV. This publication put the Duke of Mayenne in a delicate situation and favored the development of the religious wars.

The memorandum he wrote in 1603, shortly before his death, on questions of cryptography made the encryption methods of his time obsolete.

Ill, he left the king's service in 1602 and died in 1603.

Mathematical work

Opera, 1646

Specious logistics

Renaissance mathematicians felt they were followers of Greek mathematics, which is fundamentally geometry. In Viète's time, algebra, derived from arithmetic, was perceived only as a catalog of rules. Some mathematicians, including Cardano in 1545, used geometric reasoning to justify algebraic methods.

Thus, geometry seemed to be a safe and powerful tool for solving algebraic questions, but the use of algebra to solve geometric problems seemed much more problematic. And yet, that was Viète's proposal.

From 1591, Viète, who was very wealthy, began to publish at his own expense the systematic exposition of his mathematical theory, which he calls specious logistics (de specis: symbol) or art of calculation through symbols, as opposed to numerous logistics, (Calculation with numbers).

Special logistics proceeds in three phases:

• In the first time, all the present magnitudes are recorded, as well as their relations, using an appropriate symbolism that Viète had developed. The problem is then summed up in the form of equation. Viète calls this stage zetética. Write the known magnitudes as consonants (B, D, etc.) and the quantities unknown as vowels (A, E, etc.).

• The Porstic analysis lets you then transform and discuss the equation. It is about finding a characteristic relationship of the problem, the porismafrom which you can pass to the next stage.

• In the last stage, rhetic analysisWe return to the initial problem of which we expose a solution through a geometric construction based on the porism.

Among the problems that Viète addresses with this method, we must cite the complete resolution of the second degree equations in form ax2+bx=c{displaystyle ax^{2}+bx=c} and third-degree equations x3+ax=b{displaystyle x^{3}+ax=b} with a{displaystyle a} and b{displaystyle b} positives (Viète puts successive variable changes: x=a3X− − X{displaystyle x={frac {a}{3X}}-X} and And=X3{displaystyle Y=X^{3}} taking it to a second-degree equation.

Posterity of specious logistics

The specious logistics had a very limited posterity. Viète was not the first to propose the notation of unknown quantities with letters. In addition, his mathematical notations are very heavy, and his algebraic development, which fails to clearly separate algebra and geometry, requires a long development in the most complex problems. His algebra was soon forgotten, pushed aside by Cartesian geometry.

However, he was the first to introduce the notation for the data of a problem (and not only for the unknowns), and he realized the relationship between the roots and the coefficients of a polynomial.

The main originality of Viète consisted in affirming the interest of algebraic methods and in trying to make a systematic exposition of said methods. He did not hesitate to affirm that thanks to algebra all problems can be solved ( Nullum non problem solvere ).

The Apollonius Gallus

Viète was involved in various scientific controversies. The most famous of them is told by Tallemant des Réaux in these terms:

« In the time of Henry IV, a Dutchman, named Adrianus Romanus, wise in mathematics, although not as much as he believed, wrote a book in which he posed a problem for all the mathematicians of Europe to try to solve it; in addition, in a part of his book he named all the mathematicians of Europe, and there was not a single Frenchman. It happened a short time later that an ambassador of States met the king at Fontainebleau. The king liked to teach him all the curiosities, and mentioned to him the remarkable nations that were in every profession in his kingdom. "But, Sire, the ambassador told him, you don't have any mathematicians, since Adrianus Romanus doesn't mention a single Frenchman in the catalogue he makes." "On the contrary, the king said, "I have an excellent man: let them go and find M. Viète." M. Viète had followed the council, and was in Fontainebleau; he came. The ambassador had sent for the book of Adrianus Romanus. The problem was taught to M. Viète, who was placed in one of the windows of the gallery where they were then, and before the king came out, he wrote two pencil solutions. At night he sent several more solutions to that ambassador, adding that he would give him as many as he wanted, as it was one of those problems whose solutions are infinite. »

Adriano Romano asked to solve an equation of degree 45 in which Viète immediately recognized the chord of an 8° arc as a solution. He then determined the other 22 positive solutions, the only ones admissible at that time.

In 1595, Viète published his response to Adriano Romano. He concluded by proposing to solve the last problem of a lost treatise by Apollonius, namely: find a circle tangent to three given circles. Adriano Romano will propose a solution making use of the intersection of two hyperbolas, which Viète did not consider appropriate to the method of the ancients (he expected a solution & # 34; with ruler and compass & # 34;).

Viète published his own solution in 1600, in Apollonius Gallus. He recognizes that the number of solutions depends on the relative position of the three circles and sets out the eleven resulting situations (although he ignores the singular cases, such as confused circles, tangent to each other, which Descartes will deal with). This resolution will have an almost immediate repercussion in Europe, and will earn Viète the admiration of numerous mathematicians through the centuries.

Later, Adriano Romano visited Viète in Fontenay-le-Comte, and a good friendship was forged between them. The complete work of him was published posthumously in 1607 by Paolo Sarpi.

Other jobs

In 1593, he published his eighth book of varied responses in which he returned to the problems of trisection of the angle (which he recognized is linked to a third-degree equation), of the quadrature of the circle, from the construction of the regular heptagon, etc.

The same year, starting from geometric considerations and by means of trigonometric calculations that he mastered, he discovers the first infinite product in the history of mathematics that gave an expression of π:

π π =2× × 22× × 22+2× × 22+2+2× × 22+2+2+2× × {displaystyle pi =2times {frac}{sqrt {2}{2}{sqrt {2}{sqrt {2}{2}{sqrt {2}}{2}{2}{2}{2}{2}{2}}{2}{2}{2}{2}}{2}}}{2}}{2}}{

It supplied 10 exact decimals of π using the method of Archimedes that, helping out a polygon of 393,216 sides (6⋅ ⋅ 216{displaystyle 6cdot 2^{16}), is clearly easier than multiple root extractions.

Eponymy

• The lunar crater Vieta carries this name in his memory.
• The asteroid (31823) Viète also commemorates its name.