François Viète

• L analyzes poristic then makes it possible to transform and discuss the equation. It is a question of finding a relation characteristic of the problem, the porisma from which one can pass at the following stage.

• In the last stage, the analyzes rethic , one returns to the initial problem which one exposes a solution by a geometrical construction resting on the porisma.

Among the problems that Viète approaches with this method, let us quote the complete resolution of the quadratic equations of the form $ax^2\; +\; bx\; =\; C$ and of the cubic equations of the form $x^3+\; ax\; =\; b$ with $a$ and $b$ positive (Viète poses the successive changes of variable: $x\; =\; \backslash \; frac\; \{has\}\; \{3X\}\; -\; X$ then $Y\; =\; X^3$ and is brought back thus to a quadratic equation).

Posterity of specious logistics

Specious logistics had an extremely limited posterity. Viète was not the first to propose to note unknown quantities by letters. In addition, its mathematical notations are very heavy, and its algebraic step, which does not manage to separate algebra and geometry clearly, requires of long development in the most complex problems. Its algebra was quickly forgotten, with the profit of the Cartesian Géométrie.

It is the first to however introduce notations for the facts of the case (and not only for the unknown factors), and it notices the bond between the roots and the coefficients of a polynomial.

The originality of Viète is especially to affirm the interest of the algebraic methods and to try to give a systematic talk of it. He does not hesitate to affirm that thanks to the algebra all the problems could be solved ( Nullum not problema solvere. )

The Apollonius Gallus

Viète was mingled with several scientific polemics. Most famous is told by Tallemant of Réaux in these terms (historiette 46):

Du time of Henri IV, a Dutchman, named Adrianus Romanus, scientist with mathematics, but not as long as he believed, made a book where he put a proposal which he gave to solve with all the mathematicians of Europe; however, in a place of its book it named all the mathematicians of Europe, and one did not give any to France. It arrived little from time after an ambassador of the States found the King with Fontainebleau. The King took pleasure with him of showing all curiosities of them, and told him excellent people that there was in each profession in his kingdom. “But, Lord, tells him the ambassador, you do not have mathematicians, because Adrianus Romanus does not name of them one of French in the catalog only it makes some. -- If made, if made, said the King, I have an excellent man: that one suits me to quérir Mr. Viète. ” Mr. Viète had taken the advice, and was in Fontainebleau; he comes. The ambassador had sent to seek the book of Adrianus Romanus. One shows the proposal with Mr. Viète, who puts at one windows of the gallery where they were then, and before the king left there, he wrote two solutions with pencil. The evening it sent several to this ambassador, and added of them that it him in donneroit as long as it would like him, because it was one of these proposals whose solutions are infinies.

Adrien Romain required to solve an equation of degree 45 in which Viète immediately recognizes as solution the cord of an arc of 8°. It determines then the 22 other positive solutions, the only acceptable ones at the time.

In 1595, Viète publishes its answer to Adrien Romain. He concludes while proposing to him to solve the last problem of a lost treaty of Apollonius, namely: to find a circle tangent with three circles given. Adrien Romain will propose a solution calling upon a hyperbole, which Viète does not consider as in conformity with the method of Old (it awaited a solution “with the rule and the compass”).

Viète publishes its own solution in 1600, in the Apollonius Gallus . He recognizes that the number of solutions depends on the relative position on the three circles and makes a statement on the eleven resulting situations (but are unaware of the singular cases (confused, tangent circles between them, etc) that Descartes will treat). This resolution will have an almost immediate repercussion in Europe, and will be worth in Viète the admiration of many mathematicians through the centuries.

Thereafter, Adrien Romain will return visit in Viète to Fontenay-the-Count, and the two men will become friendly.

Other work

In 1593, it publishes its eighth book of the varied answers in which it reconsiders the problems of the Trisection of the angle (of which it recognizes that it is related to a cubic equation), Quadrature of the circle, construction of the regular heptagon, etc

The same year, on the basis of geometrical considerations and by means of trigonometrical calculations that it had a perfect command of, it discovers the infinite first produced of the history of mathematics giving an expression of π:

$\backslash \; pi=\; 2\; \backslash \; times\; \backslash \; frac\; \{2\}\; \{\backslash \; sqrt\; \{2\}\}\; \backslash \; times$

It provides 10 exact decimals of π by having recourse to the method of Archimedes which, using a Polygone at 393.216 sides ($6\; \backslash \; cdot\; 2^\; \{16\}$), is definitely simpler than multiple extractions of encased square roots.

Sources

The corpus of mathematical works of Viète was printed only well after its death by Frans van Schooten, professor at the university of Leyde. This edition in a volume carries the title: “Francisci Vietæ Operated mathematica: in unum volumen congesta ac recognita, atque opera Francisci studio has Schooten…” (1646), Lugduni Batavorum, ex officina B. and A. Elzeviriorum. It is downloadable since the site of the BNF, here.

• Evelyne Barbin (to dir.) and Anne Boye (to dir.), François Viète, a mathematician under the Rebirth , Vuibert

• Chronomath
• college A. Camus of the academy of Strasbourg
• Site of Jean-Paul Guichard

Complementary bibliography

• Jean-Paul Guichard and Jean-Pierre Sicre, “François Viète. A lawyer mathematician” in scientific Adventures. Scientists in Poitou-Charentes of XVIe at the XXe century (J. Dhombres, to dir.), 1995, editions of the Topicality Poitou-Charentes (Poitiers): 222-235.