VI.9 François Viète

b. Fontenay-le-Comte, France, 1540; d. Paris, 1603

Trigonometry; algebraic analysis; classical problems; numerical solution of equations

Viète obtained a bachelor’s degree in law in 1560 from the University of Poitiers, but left the profession from 1564 to 1568 to oversee the education of Catherine de Parthenay, daughter of a local aristocratic family. His earliest scientific writings were his lectures to Catherine. He spent the remainder of his life in high public office, apart from a period between 1584 and 1589 when he was banished from the court in Paris for political and religious reasons. He died in Paris in 1603. Throughout his life, it was only during the time he had free from official duties that he was able to devote himself to mathematics.

The work for which Viète is best known appeared during the 1590s, beginning with In Artem Analyticem Isagoge (“Introduction to the analytic art”) in 1591. In the Isagoge Viète began to combine classical Greek geometry with algebraic methods that had originated from Islamic sources, and in doing so laid the foundations for the algebraic approach to geometry. Viète saw that the symbols in equations (traditionally variants of R, Q, and C, for the unknown, its square, and its cube) could represent either numbers or geometric quantities, and that this was potentially a powerful tool for analyzing and solving geometric problems.

Viète’s understanding of analysis was based on his reading of the Synagoge (“Collection”) of Pappus (early fourth century C.E.), where analysis was described as a method of investigating a problem by assuming that the solution is in some sense known, as we would do now by representing the solution by a symbol and carrying out mathematical manipulations involving that symbol. Algebra achieves this by regarding all quantities, known or unknown, as of equal status; equations are then formed from prestated conditions (a process Viète called zetetics), and solved to produce the unknown quantity in terms of those given (exegetics). For Viète the final step in geometric problems was to provide a specific construction for the solution: this was the geometric synthesis arising from the preceding algebraic analysis.

In several further treatises, mostly written or published around 1593, Viète taught the necessary skills of forming equations and carrying out the corresponding geometric constructions, and these books together made up his Opus Restitutae Mathematicae Analyseos, seu Algebra Nova (“The work of restored mathematical analysis, or the new algebra”), which he offered with the famous and ambitious hope of leaving no mathematical problem unsolved (nullum problema non solvere). For most of the seventeenth century, algebra continued to be known as the “analytic art,” or simply “analysis.”

Recognizing that not all equations could be solved algebraically, Viète also put forward a method of numerical solution based on successive approximations. This was the first appearance of such techniques in Europe, and was important not only for practical purposes, but also because it rapidly led to a deeper understanding of the relationships between roots and coefficients of equations.

Viète’s style of writing is wordy and often obscure, thanks in part to his liking for technical Greek terms. In his algebraic treatises, however, he devised some rudimentary notation. It had long been the case that rules for solving equations were presented through particular examples that were understood to represent a general class, but Viète took the step of replacing known quantities by consonants B, C,. . . , and unknowns by vowels A, E,. . . , so that numbers were replaced throughout by letters, or “species.” However, he had no simple or systematic way of denoting powers (for squares and cubes he used the verbal A quadratus and A cubus), and his connectives (“added to,” “equals,” and so on) were also written in words, so that his algebra was still very far from symbolic.

One of the first people to study Viète’s work in depth was Thomas Harriot in England, who, through study of Viète’s numerical method shortly after 1600, discovered that polynomials could be written as products of linear and quadratic factors, a major breakthrough in the understanding of equations. Harriot also rewrote much of Viète’s mathematics in what is essentially modern algebraic notation. In France, Viète’s work was taken up in the 1620s by FERMAT [VI.12], who was profoundly influenced by it. DESCARTES [VI.11], on the other hand, denied that he had ever read either Viète or Harriot, though in the 1630s he developed a number of very similar ideas.

Viète and his immediate successors dealt only with equations of finite degree. Only much later in the seventeenth century with the work of NEWTON [VI.14] was analysis extended to include what were thought of as infinite equations, or what we would now call infinite series, hence bringing the word “analysis” much closer to its modern meaning.

Jacqueline Stedall