François Viète: A Life in Five Chapters

Chapter 1 · Fontenay and the Court

A lawyer who kept mathematics for the evenings

1540 – 1589 · Fontenay-le-Comte · Paris · Tours

François Viète was born at Fontenay-le-Comte, in the Vendée, in 1540. His father was an attorney and Viète followed him into the law, taking his degree at Poitiers and beginning to practise in his early twenties. Mathematics was never his profession. It was the thing he did with the hours a legal career left over, and he paid for the printing of his own books.

In 1564 he entered the household of a noble family as tutor to a girl of eleven, Catherine de Parthenay. It turned out to be the most important relationship of his working life. She became a formidably educated woman, remained his patron for four decades, and several of his works are dedicated to her. Teaching her astronomy and mathematics is also, plausibly, where some of his own thinking began.

The rest of his career was royal service in a country tearing itself apart. France in these decades was in the middle of the Wars of Religion, and Viète — a Huguenot sympathiser though not straightforwardly a Protestant — served Henri III as a councillor, was driven from court for several years when the Catholic League gained the upper hand, and returned under Henri IV. The years of exile were not wasted. It was during them, largely, that the mathematics got written.

Why this matters

The decisive work in Renaissance algebra was done by somebody for whom mathematics was never the day job — which is worth knowing when the subject is presented as the property of professionals.

A lawyer, a tutor and a councillor, with the mathematics squeezed in between. What would you ask him?

Ask Viète

• “How did you find time for mathematics inside a legal career?”
• “What did you teach Catherine de Parthenay, and what did she teach you?”
• “Why were you driven from court, and where did you go?”
• “Did the years of exile help the mathematics or hinder it?”
• “Why did you pay to print your own books?”

Chapter 2 · The Analytic Art

Giving letters to what you already know

1591 · Tours

In 1591 Viète published at Tours a short book with a long title, usually shortened to the Introduction to the Analytic Art. It contains the move that made modern algebra possible, and the move is easy to state and easy to underrate.

Algebra before him already had a symbol for the quantity you were hunting. What it did not have was any way of writing the quantities you had been given. Those had to be actual numbers, which meant a method could only ever be demonstrated on a particular case: here is how to solve this equation, and you may reasonably hope the same steps work on the next one. Viète gave letters to the given quantities as well as the sought ones — vowels for the unknown, consonants for the known — and called the result the reckoning with species, with KINDS of quantity, as opposed to the reckoning with numbers that everybody already had.

The consequence is the formula. Once the given quantities have names, a problem can be solved once and the solution kept, in a form that will serve every set of numbers anybody ever puts into it. The difference between solving this quadratic and solving all quadratics is exactly the difference between a number and a letter standing where the number was.

He did not claim to have invented this. He presented it as the restoration of an analytic method he believed the Greeks — Pappus and Diophantus in particular — had possessed and lost.

Why this matters

This is the origin of the thing a student uses every day without noticing: a letter that stands for a quantity which is given but not yet chosen.

One book, one idea, and everything after it changed. What would you ask him?

Ask Viète

• “Why does giving a letter to a KNOWN quantity change anything?”
• “What is the difference between reckoning with numbers and reckoning with species?”
• “Why vowels for the unknown and consonants for the given?”
• “Why do you say you restored the analytic art rather than invented it?”
• “What could you solve after 1591 that you could not solve before?”

Chapter 3 · The Law He Would Not Drop

Why a length may not be added to an area

1591 – 1603 · Tours · Paris

Open a page of Viète and it does not look like algebra. There are no raised numerals: a squared quantity is written A quadratum, a cubed one A cubum, spelled out. There is no equals sign — he writes aequatur, is equal to. And every given quantity carries a declaration of what KIND it is: B planum, B taken as a plane; D solidum, D taken as a solid.

That last is the law of homogeneity, inherited from Greek geometry, and he kept it absolutely. Under it, only magnitudes of the same dimension may be added together or set equal to one another. A length is not an area and an area is not a volume, so a length may not be added to an area any more than three apples may be added to two hours. If an equation has a squared term in it, every other term must be brought up to that same dimension before the equation can even be written down.

It made his notation cumbersome and his successors abandoned it. Descartes, writing in 1637, simply allowed quantities of different dimensions to be combined, and algebra became far easier to write.

Something went with it, though, and it is worth naming. Viète could not commit the error of adding unlike things, because his notation would not let him form the expression. A modern student adds quantities of unlike kinds without noticing — which is why physics teachers spend so much time on units.

Why this matters

The constraint that made his pages hard to read was also a guard against a category error that modern notation permits silently.

A rule that cost him convenience and bought him safety. What would you ask him?

Ask Viète

• “What is the law of homogeneity, and why did you keep it?”
• “How did you write a power without a raised numeral?”
• “Was it a mistake for your successors to abandon homogeneity?”
• “What could you not write down at all, because of the law?”
• “Why would you not accept a negative quantity as a root?”

Chapter 4 · The Spanish Despatches

A cipher of several hundred characters

c. 1589 – 1594 · Tours · Paris

France was at war with Spain, and Spanish commanders in France corresponded with Philip II in cipher. Intercepted despatches were useless until somebody could read them. Viète could.

The cipher he broke was large — several hundred distinct characters, a nomenclator in which symbols stood not only for letters but for whole words, names and phrases, with nulls scattered in to defeat counting. Breaking a system of that size by hand, from intercepted material alone, is patient statistical work: counting, guessing at likely words, testing a guess across several messages, and building the key up piece by piece. He worked at it for a considerable time and succeeded, and the French crown read Spanish correspondence for years afterwards.

The story usually told alongside this is that Philip II, on discovering that his despatches had been read, complained to the Pope that the French were employing black magic against him. It is repeated in many accounts. It should be labelled as a report rather than a documented fact — it is not something Viète recorded, and it has the shape of a good story.

What the episode does show plainly is the same cast of mind as the mathematics. A cipher is a general problem wearing a particular disguise: solve the system, not the message.

Why this matters

Codebreaking and his algebra are the same instinct — go after the structure that generates the cases rather than the case in front of you.

Patient work on a cipher, in the middle of a war. What would you ask him?

Ask Viète

• “How do you begin to break a cipher with hundreds of characters in it?”
• “Is the story about Philip II complaining of magic actually true?”
• “Did the codebreaking and the mathematics feel like the same work to you?”
• “What did Henri IV do with what you gave him?”
• “Was there anything troubling about reading another king's letters?”

Chapter 5 · What Descartes Tidied

The idea, and the notation that got the credit

1603 – 1637 · Paris · Leiden

Viète died at Paris in 1603, with a good deal of his mathematics unpublished. Editors, among them the Scottish mathematician Alexander Anderson, saw further works into print over the following years.

In 1637 René Descartes published the Geometry, and with it the notation almost every reader now takes for granted: letters from the end of the alphabet for unknowns and from the beginning for knowns, the raised numeral for a power, and quantities of different dimensions freely combined. It is enormously easier to read than Viète, and because it is easier to read it is what gets remembered.

But the idea being tidied was Viète's. Descartes changed which letters do which job and how a power is printed; he did not invent the practice of lettering the given quantities, which is the thing that makes a formula a formula. Historians have argued for a long time about how much Descartes owed him and how much he acknowledged, and the answer is not entirely settled.

Viète's other results have kept their names better: the relations between the roots of an equation and its coefficients are still called Viète's formulas, and his infinite product for the ratio of a circle to its diameter — one of the earliest such expressions in European mathematics — is still called Viète's formula too.

Why this matters

A useful case of how credit actually attaches in the history of ideas: often to whoever made a thing convenient rather than to whoever thought of it.

The notation went to somebody else. The idea did not. What would you ask him?

Ask Viète

• “What did Descartes change about your algebra, and what did he keep?”
• “Do you think you were given proper credit for the analytic art?”
• “What are the relations between a root and a coefficient that carry your name?”
• “Why was so much of your work published only after you died?”
• “If you saw a modern algebra page, what would please you and what would alarm you?”

What Viète changed

The formula — a solution kept in terms of quantities nobody has yet chosen — begins with Viète's decision to letter the given as well as the sought. Everything a student does when they rearrange a subject, substitute into a relation, or use a general solution instead of starting again depends on it. His name also survives on the relations between the roots and coefficients of an equation, and on one of the earliest infinite products in European mathematics.

A debate that continues

How much Descartes owed to Viète, and how far he acknowledged it, has been argued over for centuries and is not settled. The story of Philip II complaining to the Pope about French magic is widely repeated but rests on later report rather than on documentation.

Keep exploring — ask Viète

• “Was keeping the law of homogeneity a strength or a weakness?”
• “Is inventing a notation as important as inventing an idea?”
• “What in mathematics today would you refuse to write down?”

Related lives

• René Descartes — Who Conserved The Quantity Of Motion
• Pierre de Fermat — The Margin Was Too Narrow
• Jacob Bernoulli — Against My Father's Will
• Thomas Bayes — The Question Turned Round

Related themes

Algebraic notation · Formulae and rearranging · Cryptography