François Viète

Who Gave The Known Quantities Letters · Mathematician & Royal Councillor · 1540–1603

The lawyer who gave letters to the KNOWN quantities as well as the unknown, and so turned the solution of a problem into the solution of every problem shaped like it.

Who was Viète?

François Viète was born at Fontenay-le-Comte in 1540, trained as a lawyer, and spent his working life in royal service — councillor to Henri III and afterwards to Henri IV, with an interval of exile when the Catholic League drove him from court. Mathematics was the business of his private hours, and in 1591 he published at Tours the Introduction to the Analytic Art, a short book that changed what algebra could be. Its central move looks small and is not. Algebra before him had a letter, or a word, for the quantity being sought; every other number in a problem was an actual number, so a method could be demonstrated only on the particular case in front of you. Viète gave letters to the given quantities too — vowels for what is sought, consonants for what is supplied — and called the result the reckoning with species, with kinds of quantity, as against the reckoning with numbers that everyone already had. A problem could now be stated, solved and the solution kept, in a form that served every set of numbers that would ever be put into it. That is what a formula is. He held to the ancient law of homogeneity, under which only magnitudes of the same dimension may be added or set equal, so a given quantity had to be declared a length, a plane or a solid and written as such; the constraint is why his pages look nothing like a modern one. He had no raised numeral for a power, writing A quadratum and A cubum in words, and no equals sign. He also established relations between the roots of an equation and its coefficients, gave a solution to a challenge problem of the forty-fifth degree set by Adriaan van Roomen, produced an early infinite product for the ratio of a circle to its diameter, and broke a Spanish cipher of several hundred characters during the war with Spain.

Major achievements

• Published the Introduction to the Analytic Art at Tours in 1591
• Gave letters to the KNOWN quantities as well as the unknown, making a general formula expressible
• Separated reckoning with species — with kinds of quantity — from reckoning with numbers
• Established relations between the roots of an equation and its coefficients
• Solved Adriaan van Roomen's challenge problem of the forty-fifth degree

Life in brief

• 1540 — Born: At Fontenay-le-Comte in the Vendée, an attorney's son.
• 1564 — In service: Enters the household of Catherine de Parthenay, who becomes his lifelong patron.
• 1580s — Royal councillor: Serves Henri III; later exiled for a time by the Catholic League.
• 1591 — The analytic art: The Introduction published at Tours: letters for the given quantities, and the reckoning with species.
• 1590s — The Spanish cipher: Breaks a despatch cipher of several hundred characters for Henri IV.
• 1593 — The challenge: Answers Adriaan van Roomen's problem of the forty-fifth degree.
• 1603 — Died: At Paris, with much of his mathematics still unpublished.

Explore the life of Viète — five chapters

1. Fontenay and the Court — A lawyer who kept mathematics for the evenings
2. The Analytic Art — Giving letters to what you already know
3. The Law He Would Not Drop — Why a length may not be added to an area
4. The Spanish Despatches — A cipher of several hundred characters
5. What Descartes Tidied — The idea, and the notation that got the credit

Read the five-chapter life of François Viète →

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