How Bernoullis made their mark

Once upon a time, there lived a family in Switzerland—by the name of Bernoulli. I mention the name because most once-students of science know of it from physics in high-school or early-college years. Students like me, for example.

I refer here to Bernoulli’s Principle. It goes something like this: when you have a fluid that’s flowing—water through a pipe, air around the wing of an aircraft—there’s a relationship between the speed at which it flows and the pressure in the fluid. As the speed increases, the pressure decreases.

This is a physics nugget that does not stay abstract long. For example, it explains why topspin imparted to a tennis ball—think Rafa Nadal—makes it descend faster than a ball that’s hit without spin. It also helps explain how an aircraft’s wing generates the force needed to lift a plane into the air. Well actually, there is some nuance and argument about how exactly the Principle applies to that lift. Still, because it’s been connected to phenomena like these, the name is known more widely than just in physics circles.

In any case, the Principle is named after Daniel Bernoulli, who discovered this relationship in the 1730s. Daniel studied business and mathematics and medicine, finishing a PhD in anatomy and botany and an MD in medicine by the age of 21 years. His first love seems to have been mathematics, though, and particularly its applications to fluid mechanics. When he happened upon the Principle, he found ways to use it in real life, devising an instrument to measure blood pressure, for example, that worked because of his Principle. While we no longer measure blood pressure that way, you can find essentially the same instrument on the outside of planes. It uses pressure to measure the speed of the plane.

Because Bernoulli’s Principle became so well-known, students like me tended to assume without much thought that there was just one Bernoulli. Or at any rate, that Daniel was the only Bernoulli to make a mark in science. Both are serious misconceptions. In fact, there can’t be too many families in mathematics or science, or really in any field, quite like the Bernoullis. In the 17th and 18th Centuries, this remarkable Swiss family produced at least eight—maybe as many as a dozen—scientists and mathematicians who found widespread recognition for their work. Daniel was only one of them.

To be sure, he was one of the brightest stars in the family. But there’s Daniel’s younger brother, Johann II. He was a professor of mathematics at the University of Basel. His particular interest was the way light moves, but there were other subjects he dabbled in, too, with much success: he won prizes for his work from the Academy of Sciences in Paris. Johann II had four sons. The eldest was Johann III. This Bernoulli was considered a child prodigy: he completed his PhD at just 13, and was named Berlin’s “Astronomer Royal” when he was 19. His youngest brother was Jakob II, who loved geometry and learned it from his uncle Daniel. He became a professor of mathematics in St Petersburg. Sadly, he drowned soon after, short of his 30th birthday.

The earlier Bernoullis, though, were even more eminent. Daniel’s cousin Nicolaus I had a deep interest in probability. He dreamed up what mathematicians know as the St Petersburg paradox (though it was Daniel who actually gave it that name). This arises from a game of flipping coins that, in theory, offers unlimited returns, but potential players don’t see that, and are unwilling to pay more than a token amount to enter the game.

Daniel’s father Johann and uncle Jacob were early pioneers and users of calculus as we know it today, that uses the idea of infinitesimal quantities. Unfortunately, the two grew professionally jealous of each other, a feeling that, after Jacob died, Johann transferred to his son Daniel. In both cases, the relationships eventually broke down altogether.

Jacob is known for deriving the well-known law of large numbers in probability theory: if an experiment has an expected result, and if you perform it a large number of times, the average of the results you get will tend to approach the expected result. For example, if you toss a coin 10 times and count how many tails you get, you expect five. But the first time, you may get seven. Then maybe six, four, five, eight—but in the long run, the average of all your 10-toss experiments will get closer and closer to five.

We can also thank Jacob for discovering “e” (2.71828...)—a number just as ubiquitous and important in mathematics as TT is. For just one example, it is fundamental to the idea of compound interest—which is actually how Jacob found it.

In the early 18th Century, he also stumbled upon the so-called Seki-Bernoulli numbers, independently of but almost simultaneously with the Japanese mathematician Seki Takakazu. They ran into them by chance, while looking for a formula for the sums of powers of integers. This is a prize that mathematicians since ancient times—Archimedes, Aryabhata, Fermat and Pascal, among many others—have chased.

The Seki-Bernoulli numbers turn up in the branch of mathematics called “analysis”. I will not say more about that, but here are some of these numbers, and some remarks about them, taken from a 1911 paper by the great Srinivasa Ramanujan (http://ramanujan.sirinudi.org/Volumes/published/ram01.pdf):

#2: 1/6

#4 and #8: 1/30

#12: 691/2730

#30: 8615841276005/14322

and more.

(They are infinite in number, and every odd-numbered Seki-Bernoulli number is 0.)

In that paper, Ramanujan noted some intriguing things about these numbers. Two examples:

* All the denominators have 2 and 3 as prime factors, but only once each. Thus, all denominators are divisible by 6, but not by 4 (2 x 2) or 9 (3 x 3).

* They are fractions, and if you divide one by its serial number, the numerator of the result is prime. This works for #12 above, but - and I can hardly believe this - not for #30. This is a known mistake in Ramanujan’s work, though I am flummoxed that the great man made such a simple mistake.

That conundrum aside, Ramanujan has plenty more to say about these numbers. Just as I have plenty more to say about the Bernoullis. Sadly, space only allows this much. Still, I’m honoured to have at least that in common with Ramanujan. Thank you, Bernoulli family.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.