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Hank: If you’ve ever taken a science or math class, you’re probably used to lots of things being named after the same people. There are hundreds of principles and theorems and equations named after superstars like Euler and Laplace and Gauss. So it might not seem all that weird to learn about Bernoulli’s brachistochrone solution, Bernoulli’s equation in fluid dynamics, and the Bernoulli probability distribution. And you would assume that they’re all named after the same person. But they’re not!

In the seventeenth and eighteenth centuries, there were actually eight mathematically gifted Bernoullis, from three generations of the same family. But three of the Bernoullis really stood out: Jacob, Johann, and Daniel. They were all descendants of Nikolaus Bernoulli, who was born in 1623.

Nikolaus had three sons, who he named Jacob, Nikolaus — after himself — and Johann. The younger Nikolaus became a painter and a city official, but his brothers were famous mathematicians. Jacob, the oldest, solved a number of early calculus problems that are still taught in schools today. But Jacob’s most important discoveries were in probability.

He was the first person to prove something called the law of large numbers, and he came up with what’s now known as the Bernoulli distribution. Both are ways of predicting a series of random events.

Say you roll a six-sided die a bunch of times in a row. If you roll the die, say, 3 times, you might get a 1, a 4, and a 6. You also might roll three 2s. It’s random. But the law of large numbers says that as you roll the die more times, the average of all the numbers you rolled will get closer and closer to 3.5 — the average of all the possible rolls.

That’s because after a lot of rolls — say, a thousand — you’ll probably have rolled each number roughly the same amount of times. So the average of all your rolls will be very close to 3.5.

Jacob Bernoulli was the first person to officially prove this, and it became one of the early fundamental concepts of probability theory. The Bernoulli distribution is a similar idea that applies the law of large numbers to something like a coin flip, where there are only two possible outcomes. Jacob also discovered the famous mathematical constant e, which is used all the time in math and the sciences. It can describe anything that grows continuously — from bacteria to a bank account that’s accumulating interest.

Jacob’s younger brother Johann was also interested in math. In 1696, he posed a fun problem for the world’s mathematicians: What’s the fastest path for a ball to follow if it rolls down a track between two points? He called it the brachistochrone problem, from the Greek words for “shortest” and “time.” You might assume that the fastest path would be a straight line. And a straight line is the shortest path, but it’s not the fastest one.

The fastest path for a ball to roll between two points is actually a kind of stretched out piece of a circle called a cycloid, because of the way gravity makes the ball accelerate. Galileo and a few others had figured this out on a conceptual level, but Johann used calculus to prove it. Jacob then found his own solution, laying the foundation for a whole new branch of calculus while he was at it.

Johann also discovered ways of finding an answer to things like 0 divided by 0 and infinity divided by infinity. His method is now known as l’Hôpital’s rule, but all Guillaume de l’Hôpital did was publish his notes from Johann’s calculus lectures.

Johann’s three sons were all mathematicians, as well, but the probably most influential one was Daniel. One of the things that’s named after him is Bernoulli’s principle, a concept in fluid dynamics that describes the relationship between pressure and speed in a moving fluid. Bernoulli’s principle is part of why airplanes fly: the air on the top of the wings is going faster than the air on the bottom. In physics, Daniel worked with the famous Euler to develop Euler-Bernoulli beam theory, which describes the way forces make strong rods bend. Euler-Bernoulli beam theory and the physics based on it are still super useful for engineers these days — there are, of course, beams in our bridges and buildings. So that’s-- we should understand them. Daniel also developed new mathematical methods of measuring risk, and he was one of the first physicists to study the behavior of gasses.

Part of the reason the Bernoullis were so successful was that they were in the right place at the right time. Newton and Leibniz had just invented calculus, so there was this powerful new technique for describing the universe. But the Bernoullis were also talented mathematicians and scientists who used the tools they had to uncover all kinds of new things about the universe. Which is why their names are now all over our science and math textbooks. Even if it’s sometimes hard to keep track of which Bernoulli did what.

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