Françoise Viète: Mathematical Biography

Françoise Viète was born in 1540 in Fontenay-le-Comte, in Western France. The son of a lawyer, he himself also became a lawyer, graduating from the University of Poitiers. Viète gave up the lawyers life in 1570 and moved to Paris and, 3 years later, was appointed by King Charles IX to the Government of Brittany at Rennes. Mathematics was always just a hobby to Viète but he was wealthy so was able to publish some papers and send them to scholars all over Europe. In 1579 he pressed for the replacement of sexagesimal fractions (those with denominators in powers of 60) with decimal fractions. In 1580 he was appointed by King Henry III to be Royal Privy Counselor in Paris but, just 4 years later, religion and politics forced him to move away to Beavoir-sur-Mer where he spent 5 years accomplishing some of his most influential mathematical work.

In 1589, the exiled King Henry III recalled him back to his parliament in Tours. After the King was killed later that year, he began working for Henry IV and became famous for decoding secret Spanish messages. In 1593, he famously and quickly solved a problem of an equation of 45 degrees set by the mathematician Adriaan Van Roonen. He recognized the relationship between Sin(x) and Sin(^x/₄₅) finding all 23 positive roots. He died in Paris in 1603 and is considered the most important mathematician in bridging the gap between Renaissance math to modern world math.

Viète is most famous for the Viète formula which is an exact expression for the value of π. The formula states:

Which goes to infinity terms on the right. Viète made this discovery as part of some research into the perimeter of polygons inscribed in a circle.

When Adriaan Van Roonen posed his famous problem,

Where

The ambassador taunted him that French mathematicians were poor and couldn’t solve the problem. To King Henry’s delight, Viète reportedly recognized the problem and solved it in a matter of minutes. He already knew all the sin and cosine formulae for nα where n is any number and α is an angle. He recognized that the left hand side of the equation is equivalent to 2Sin(45 α) if expressed merely in terms of 2Sin( α) so he found the value of  that made 2Sin(45 α) = C . The solution he got was x = 2Sin(2 α) .

Probably Viète’s most significant contributions to the world of mathematics were in terms of algebraic expressions where letters were used to denote quantities. Although already an established system for unknown quantities, it was he who popularized its use for constants in terms. While previous mathematicians would restrict themselves to specific equations such as x²+4x=13, Viète would make a more general expression like x²+bx=c.

While investigating plane and spherical triangles, Viète developed three sum-to-product formulas such as,

for example, and was the first to state the law of triangles in the form used most today:

where a and b are two sides of a triangle and α and β are the opposite angles.

Its importance underestimated, the Principle of Homogeneity was stated by Viète that the quantities in an equation, should be representative of lines, surfaces, or solids, more generally as homogeneous. Although used by some of the old Greek mathematicians, it was never laid down as a principle until Viète. The thinking has even evolved where many modern mathematicians will take an equation and make it homogeneous in order to simplify its solution.

Viète certainly gained some fame from his code-breaking exploits. Many of the codes he broke used substitutions, but not simply just a straight swap between numerals and letters, often words or syllables denoted by a numeral or combination of such. His first observation was that often, the enemies sent the same letter coded in different ways. If you can decode one of these, you can then compare the codes and get a feel for how all the codes are constructed. Similarly, by noticing that letters often began in a similar manner, equivalent to how we may begin a letter with “Dear Sir”, that gives some free clues. Certainly such work for the French King had great effects on regional politics and history.

Boyer, Carl B. A History of Mathematics

Livio, Mario The Equation that Couldn’t be Solved

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Panza, Marco “François Viète: between analysis and cryptanalysis” (hosted: www.sciencedirect.com)