VIÈTE, FRANÇOIS (1540–1603)

VIÈTE, FRANÇOIS (1540–1603), French mathematician. Viète is widely viewed as the founder of modern algebra. Born in Fontenay-le-Comte in the province of Poitou, he studied law at the University of Poitiers and received his degree in 1560. Shortly thereafter he entered the service of the noblewoman Antoinette d'Aubeterre and served as legal adviser as well as educator of her daughter, Catherine of Parthenay (later Rohan). His position in the household of this leading Huguenot family involved him with increasing prominence in the tense religious rivalries of the time. In 1573, following several years in Paris, he was appointed counselor to the Parlement of Brittany in Rennes by King Charles IX, and in 1580 he became a member of Henry III's privy council. Following a period of political eclipse in the late 1580s, he was recalled to court in 1589 and served as counselor to Henry III and Henry IV until his death on 23 February 1603. During his years as royal counselor Viète specialized in cryptanalysis, becoming one of the leading code breakers in Europe. His success in decoding secret Spanish communications famously brought upon him the accusation of being in league with the devil.

Despite his active career at court, Viète found time to research and publish an impressive number of mathematical works in a range of different fields. His most influential work, however, was undoubtedly in algebra. The field known as "algebra," he contended, was not, in fact, an achievement of Arab mathematicians, but was a corruption of the ancient "Art of Analysis" which was known in classical times. Unlike synthesis, which begins with self-evident assumptions and proceeds deductively to necessary conclusions, analysis proceeds in the reverse direction. In analysis, one assumes that the desired conclusion is true and then proceeds to deduce the implications of this assumption. If this leads to a known true relationship, it is a good indication (although no proof) that the original assumption was true. The mathematician can then reverse course and use the analysis as a guide for a synthetic proof of the theorem. If, on the other hand, the assumption leads to a falsehood, it is also necessarily false.

Classical mathematicians, Viète believed, used analysis extensively in their research. Unfortunately, as they only considered synthetic proof to be proper and incontrovertible, they proceeded to suppress the analytic part of their research in their published works. This left their modern-day successors with beautiful and elaborate synthetic constructions, such as can be found in the writings of Euclid and Archimedes. The method used by the ancients to discover their theorems—namely analysis—appeared to be lost. Viète set out to correct this unfortunate state of affairs by recovering the ancient "Art of Analysis." Beginning with his Introduction to the Analytic Art of 1591, and continuing in a series of subsequent works, he laid down the basic outlines of the ancient method as he perceived it.

Viète's fundamental insight was that the "Art of Analysis" was none other than the algebra. In algebra, he pointed out, one proceeds analytically: when presented with a mathematical problem, one assumes that the solution has already been found, and sets up a mathematical relationship accordingly. One then proceeds to analyze this relationship, arriving ultimately at a true solution if such exists. This, he claimed, was precisely the approach used in ancient analysis.

Viète realized, however, that the algebra of his time was inadequate to the task. It consisted of a long and increasing list of solutions to specific problems and practical rule-of-thumb methods to help with the solution of others. This, for Viète, was evidence of the corrupt state of algebra and the need for restoration. He therefore sought to replace the haphazard algebraic practices with general rules of analysis that would guide the solution of all problems.

To accomplish this, Viète proposed a novel system of notation. For the first time, he distinguished between the given magnitudes of a problem and the unknown ones, which must be sought out. The given magnitudes, he proposed, should be signified by consonants (B, C, D, F . . . ) and the unknown ones by vowels (A, E, I, O, U, Y). This simple innovation enabled Viète to write down not just specific linear, quadratic, and cubic problems, but general types of linear, quadratic, and cubic equations. Consequently, once a general type of equation was analyzed and solved, any particular instance of this type could be solved as well. With considerable justification, Viète referred to his "recovered" Art of Analysis as "the doctrine of discovering well in mathematics" (doctrina bene inveniendi in mathematicis).

In addition to algebra, Viète contributed to numerous other mathematical fields including trigonometry, conic sections, and astronomy. His enduring reputation, however, rests firmly on his algebraic work. Despite his claim that he was merely recovering an ancient method, his approach was in fact very different from the geometrical analysis practiced in antiquity. It is ironic, but telling, that Viète, who sought to replace the corrupt "algebra" with pure "analysis," has become known to subsequent generations as the father of modern algebra.

VIÈTE, FRANÇOIS (1540–1603)

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