I am thinking that it would be possible to write a pi calculus with a volume of a sphere, but the problem is with the polyhedrons that must be ever increasingly smaller (it is impossible to have more of five regular polyhedra, so that a generalization of Viete in three dimension is not possible). If we start with a pentagon, we’ll get a formula for pi that involves the golden ratio! Since there’s no need to start with a square, we might as well start with a regular n-gon inscribed in the circle and repeatedly bisect its sides, getting better and better approximations to pi. Now let’s see in detail how Viète’s formula works.
And indeed, before Viète came along, Ludolph van Ceulen had computed pi to 35 digits using a regular polygon with sides! So Viète’s daring new idea was to give an exact formula for pi that involved an infinite process. In a more modern way of thinking, you can figure out everything you need to know by starting with the angle and using half-angle formulas 4 times to work out the sine or cosine of. Since, you can draw a regular 96-gon with ruler and compass by taking an equilateral triangle and bisecting its edges to get a hexagon, bisecting the edges of that to get a 12-gon, and so on up to 96. In modernized notation, it looks like this:īy approximating the circumference of a circle using a regular 96-gon. Among other things, it includes a formula for pi. In 1593 he came out with another book, Variorum De Rebus Mathematicis Responsorum, Liber VIII. (Later people switched to using letters near the beginning of the alphabet for known quantities and letters near the end like for unknowns.) For example, he emphasized care with the use of variables, and advocated denoting known quantities by consonants and unknown quantities by vowels. This deserves to be much better known it was very familiar to Descartes and others, and it was an important precursor to our modern notation and methods. In 1591, François Viète came out with an important book, introducing what is called the new algebra: a symbolic method for dealing with polynomial equations. The king admired his mathematical talents, and Viète soon confirmed his worth by cracking a Spanish cipher, thus allowing the French to read all the Spanish communications they were able to obtain. By 1590 he was working for King Henry IV. Nonetheless, he was highly successful in law. A friend said he could think about a single question for up to three days, his elbow on the desk, feeding himself without changing position.
But his true interest was always mathematics. He began his career as an attorney at a quite high level, with cases involving the widow of King Francis I of France and also Mary, Queen of Scots. He studied law at Poitiers, graduating in 1559. You can see the overall shape, but the really exciting stuff is hidden.įrançois Viète is a French mathematician who doesn’t show up in those simplified stories. The simplified stories we learn about the history of math and physics in school are like blurry pictures of the Mandelbrot set. It’s not exactly self-similar, but the closer you look at any incident, the more fine-grained detail you see. It’s probably not new, and it certainly wouldn’t surprise experts, but it’s still fun coming up with a formula like this. Greg Egan and I came up with this formula last weekend.
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