François Viète (1540 - 1603)

Viète lived during a particularly difficult time in French history, sometimes caused by the religious conflict between the Catholics and Protestants and sometimes by political dimensions.

Viète, son of a lawyer, went to college and tried to keep out of the troubles. He had an interest in mathematics and became teacher to Catherine de Parthenay, an aristocrat. She remained his benefactress throughout his life, and he seems to have enjoyed teaching her.

He developed lectures on astronomy, mathematics and cosmology although he still believed in the system of Ptolemy as he didn’t recognise the geometric breakthrough that Copernicus’ ideas represented.

Despite this he was never to become a professional mathematician - it was always a refuge from the world at constant war around him, never his single pursuit.

When Catherine de Parthenay got married, Viète went to Paris and became friendly with the heir to the French throne. Because of his close ties to the king, he became a royal privy counsellor attached to parliament.

By 1584 however his Huguenot connections had became too much of a liability and he retired to the countryside for five years. It was during this period that he turned to mathematics. He was not particularly interested in the work of Arab mathematicians, which is unfortunate as he devoted much time to rediscovering many of their results.

After this mathematical interlude, he re-entered the political scene of France in the service of the Protestant King Henry of Navarre. Working for him he famously broke the Spanish king’s letter code. This 500 symbol code was used for communicating with Catholic forces sponsored by the Spanish king. They refused to believe that the code was breakable and wrote to the Pope complaining that supernatural abilities were being used. It wasn’t supernatural abilities - just the mathematical ingenuity of Viète.

Viète’s mathematics

Viète was also interested in the work of Cardan and other previous mathematicians. He contributed to their results and tried to reconstruct the lost work of the Greek mathematician Apollonius. He is rightly called the father of algebra because of the many symbols he introduced which are still in common use today. Two examples are the ‘+’ and ‘-’ symbols (‘plus’ and ‘minus’ - both French words) and he also showed the value of using letters to present unknown quantities.

At the same time he made remarkable progress in geometry, especially in trigonometry. He developed formulae in trigonometry for multiple angles and used these to find the first infinite expansion for \pi. Viète had already found \pi correct to 9 places of decimals by the classic method - using a regular polygon with 6.2^{16}=393,216 sides. With a little more ingenuity, he adapted the same basic approach, with profoundly different results - an infinite expression for the reciprocal of \pi. This is his method:

Consider a square, inscribed in a unit circle. The length of side of the square is just,

{1\over\cos\theta}\ \mbox{where}\ \theta=45^\circ

Now consider an octagon inscribed in a unit circle; the sum of two sides of this is given by,

1\over\cos\theta{\cos{\theta\over{2}}}

the sum of four sides of a regular 16-gon is given by,

1\over\cos\theta{\cos{\theta\over{2}}}{\cos{\theta\over{4}}}

As the number of sides increases, the perimeter of the inscribed figure approaches the circumference of the circle. Viète concluded that,

{1\over{\cos\theta{\cos{\theta\over{2}}}{\cos{\theta\over{4}}}\ldots}} \to {\pi\over{2}}

which he then wrote in the form,

{2\over{\pi}}={\cos\theta{\cos{\theta\over{2}}}{\cos{\theta\over{4}}}\ldots}

Now Viète used the fact that,

\cos\theta={\sqrt{2}\over{2}},

and then from trigonometric formula, \cos{2}\theta=2\cos^2\theta-1 he concluded that,

{\cos{\theta\over{2}}=\sqrt{{1+\cos\theta}\over{2}}=\sqrt{{2+\sqrt{2}}}\over{2}},\quad\cos{\theta\over{4}}=\sqrt{{1+\cos{\theta\over{2}}}\over{2}}={{\sqrt{2+{\sqrt{2+\sqrt{2}}}}}\over{2}}

and hence that,

{2\over{\pi}}={\sqrt{2}\over{2}}.{{\sqrt{2+\sqrt{2}}}\over{2}}.{\sqrt{2+\sqrt{2+\sqrt{2}}}\over{2}}\ldots

Viète was also interested in classic problems. For instance in 1592 he lectured at Tours and discussed recent claims that the circle could be squared, an angle trisected, and the cube doubled using only ruler and compasss. He showed in these lectures that the “proofs” which had been published earlier in the year were fallacious.