Folha: “A Tale of Many Kinds of Math”
Reproduction of Marcelo Viana’s column in Folha de S. Paulo
What I like most about this story are the connections between different scientific breakthroughs. I was already familiar with most of them, but I had no idea they were related!
The story begins in 1730, with the publication of the book *Miscellanea Analytica* by Abraham de Moivre (1667–1754). In a somewhat obscure appendix, De Moivre asks: What can we say about the number of heads when we flip a coin a large number (N) of times?
The expectation is that there will be N/2 heads, of course. But it’s also no surprise if there are a few more or a few fewer. On the other hand, it’s unlikely that the number of heads will be very different from N/2. De Moivre plotted the probability curve associated with each possible outcome (percentage of heads): it’s a bell-shaped curve, high near the mean of N/2 and dropping sharply as we move away from that value, either up or down.
I mentioned here last week that, to arrive at this line of reasoning, he had to discover a formula for the factorial N! of an integer N, and I explained why that formula ended up going down in history under the name of his colleague James Stirling (1692–1770). That wasn’t the only loss for De Moivre.
Gauss’s motivation for studying these questions surprised me. Around 1800, a group of 24 experienced astronomers had been formed to find an unknown planet believed to exist between Mars and Jupiter. They became known as the “Celestial Police,” and in early 1801, they succeeded in identifying a new celestial body in that region. It was named Ceres and is currently considered a dwarf planet.
But after several observations, the Celestial Police “lost” Ceres during a transit behind the Sun. That’s when Gauss stepped in: he developed a new, highly efficient computational method and used it to successfully rediscover Ceres’ exact position in the sky.
A crucial component of this approach was the so-called least-squares method, which makes it possible to replace complex experimental data (in this case, astronomical data) with simple formulas and is now a fundamental tool in any numerical analysis.
It was to provide a foundation for this method that Gauss needed to develop the theory of the Gaussian distribution (perhaps independently of De Moivre and Laplace). But his paper was not published until 1809, as I mentioned. However, the method of least squares had already been published in 1805 by André-Marie Legendre (1752–1833), who was unaware of Gauss’s work. The motivation in that case was the calculation of comet orbits.