The Calculus of Dysfunction

Every family has its tensions: the uncle who brings up politics over the turkey, the board game that went badly one Christmas and is never mentioned again. So if you have ever sat through a holiday dinner and thought your relatives were impossible, you may find it a comfort to know that you were not born into the Bernoulli family of Basel.

Over three generations the Bernoullis produced eight mathematicians of real distinction, which is about seven more than most families manage and a quite extraordinary number to come out of one household of Swiss spice merchants. They gave the world probability theory, the calculus of variations, the mathematics of fluids, and the number e. They also spent a great deal of their time loathing one another, sometimes publicly, often in Latin, and occasionally in ways that we would now expect to see in a courtroom.

It started, as these things usually do, with a father who had other plans. Nicolaus Bernoulli was a prosperous merchant who wanted his eldest son, Jacob, to become a minister. Jacob studied theology as he was told, then went quietly off and taught himself mathematics and astronomy. He took as his personal motto Invito patre sidera verso ("Against my father's will I study the stars"), which is a wonderfully dramatic thing to put on your crest and suggests that rebellion in seventeenth-century Basel was a fairly low-key affair.

Jacob was, by all accounts, brilliant, methodical and touchy. He became professor of mathematics at the University of Basel in 1687, and around the same time he did something he would regret for the rest of his life: he began teaching mathematics to his younger brother, Johann.

Johann had been meant for business, which he hated, and then for medicine, which he tolerated. But he took to mathematics at once, and within a few years he was at least as good at it as his brother. He was also, it soon became clear, vain, competitive and incapable of letting anything go, which is an unfortunate combination in a sibling and a disastrous one in a colleague. The two brothers had just been introduced to the new calculus of Gottfried Leibniz, which nearly nobody else in Europe yet understood, and together they did more than anyone except Leibniz himself to show what it could do. They did it, however, mostly while trying to make each other look like fools.

Their chosen battlefield was the learned journal. In June 1696, Johann published a challenge in the Acta Eruditorum addressed to "the most acute mathematicians of the whole world." It concerned the brachistochrone, from the Greek for "shortest time": if a bead slides without friction from one point to another lower point, which curve gets it there fastest? The obvious answer is a straight line, which is wrong. The correct answer is a curve called a cycloid, the path traced by a spot on the rim of a rolling wheel. It is not obvious, which is exactly why it makes a good challenge. And although the invitation was addressed to the whole world, everyone understood that it was mainly aimed at one man in Basel.

Several people solved it, Jacob included. But the solution that has become famous came from England. Isaac Newton, by then in his fifties and working at the Royal Mint, received the problem one afternoon in January 1697. According to his niece, he came home tired from the Mint, sat down with it, and had it solved by four o'clock the next morning. He sent the answer anonymously to the Royal Society. Johann looked at it and is said to have remarked that he recognised the author tanquam ex ungue leonem, "as the lion by his claw." It is one of the finest compliments in the history of science, and it is somewhat spoiled by the knowledge that Johann almost never paid one to anyone else.

Jacob's solution, as it happens, was the more profound. Johann's was ingeniously clever. He worked it out by pretending the bead was a ray of light passing through layers of glass, but Jacob's method could be generalised, and it became the foundation of a whole new branch of mathematics, the calculus of variations. Jacob then struck back with a problem of his own, concerning the shapes that enclose the greatest area for a given perimeter. Johann rushed out a solution, which was wrong. Jacob said so, at length and in print, and the two men spent the following years exchanging insults that grew less mathematical and more personal with each round. Johann never let anyone forget that when Jacob had been stuck on the problem of the curve formed by a hanging chain, he, Johann, had solved it in a single night. Jacob, for his part, liked to point out that Johann had learned everything he knew from him. Both, on different occasions, had a point.

Johann also found time for a strange little side arrangement. In 1694 a wealthy French nobleman, the Marquis de l'Hôpital, offered him a regular salary in return for private lessons and the right to use Johann's mathematical discoveries more or less as he pleased. Johann, who needed the money, agreed. In 1696 l'Hôpital published the first calculus textbook ever written, and in it was a useful result that has been known ever since as L'Hôpital's rule. It was Johann's. Generations of students have learned it under the wrong name, which, given everything else we know about Johann, may be the one form of justice the universe was able to arrange.

Jacob died in 1705, at the age of fifty, with the feud unresolved. He left behind a great unfinished book on probability, the Ars Conjectandi, which contains the first proof of what we now call the law of large numbers and which, by itself, would have been enough to make him famous. He also left instructions for his tombstone. He had long loved the logarithmic spiral, a curve that keeps its shape however much it grows, and he asked that one be carved on his grave with the words Eadem mutata resurgo: "Though changed, I shall arise the same." The stonemason, who was evidently not a mathematician, carved an Archimedean spiral instead, which is a quite different curve. You can still see it in the cloisters of Basel Minster. It seems a fittingly Bernoullian ending: even in death, Jacob was being corrected.

Johann promptly took his brother's chair at the university, and then turned his considerable talent for resentment on his own children.

His second son, Daniel, was by most accounts the nicest Bernoulli there ever was, which is admittedly not a hard competition to win. Johann did exactly what his own father had done to him. He told Daniel there was no money in mathematics and steered him first into business and then into medicine. Daniel duly became a doctor and then, with the family's characteristic disregard for parental advice, became one of the great mathematical physicists of the century anyway.

His masterpiece was Hydrodynamica, published in 1738, which contains the idea now known as Bernoulli's principle: that where a fluid moves faster, its pressure drops. It is often offered as the reason aeroplanes stay up and shower curtains billow inward to cling to your legs, though I should warn you that physicists have been arguing for years about exactly how much of either it explains. They are, it turns out, quite a lot like Bernoullis.

The real trouble had begun a few years earlier. In 1734 the Paris Academy of Sciences awarded its grand prize jointly to Daniel and Johann, which most fathers would have regarded as a lovely thing to share with their son. Johann regarded it as an outrage. He was so incensed at being ranked level with his own child that he threw Daniel out of the family house. Then, a few years after Hydrodynamica appeared, Johann produced a book of his own called Hydraulica, which covered much of the same ground and which he dated 1732, six years earlier than his son's book. It did not actually appear until around 1742. Daniel wrote to a friend in some anguish that he had been robbed of his whole Hydrodynamica all at once. He went on to win the Paris prize about ten times in all, which may or may not have been meant as a reply.

The friend he wrote to was Leonhard Euler, and with him our story becomes much more pleasant.

Euler (pronounced "Oiler," a fact that seems to come as a surprise to nearly everyone who first meets the name) was also from Basel. His father was a pastor who had studied mathematics under Jacob Bernoulli. As a boy, Euler went to Johann for lessons. Johann was too busy to teach him properly, but he let him come by on Saturday afternoons to ask about whatever he had been unable to work out during the week. Euler made such good use of these sessions that Johann, of all people, talked Euler's father out of making him a minister. It was very probably the most generous thing Johann ever did, and it was well worth it.

In 1727, at the age of nineteen, Euler followed Daniel Bernoulli to the new Academy of Sciences in St Petersburg. There was no opening in mathematics, so he accepted a post in physiology, and he arrived in Russia on the very day the Empress Catherine I died, which threw the whole academy into confusion. Happily, he drifted over to physics and mathematics before anyone could ask him to dissect anything. He and Daniel became close friends and worked side by side for some years, the one explaining how the world moved and the other building the mathematics to describe it.

It is hard to describe how productive Euler was without sounding as if you are exaggerating. He wrote something like 850 papers and books, on everything from number theory to ship design to the tuning of musical instruments. A collected edition of his works was begun in 1911 and has since run to more than seventy large volumes, and the editors are still not quite done. Most of the notation we take for granted is his, or became standard because he used it: f(x) for a function, e for the base of natural logarithms, i for the square root of −1, Σ for a sum. The symbol π had been introduced by a Welshman named William Jones in 1706, but it was only after Euler took it up that everyone else did.

Unlike the Bernoullis, Euler seems to have been a thoroughly decent and even-tempered man. He married, and had thirteen children, of whom, in the way of the times, only five survived to adulthood. A visitor reported that he did some of his best work with a baby on his lap and the older children playing around his feet. He had his disagreements, as everyone in eighteenth-century science did, but he never did anything to anyone as unpleasant as what the Bernoullis did to one another as a matter of routine.

He also went blind, by degrees. He lost most of the sight in his right eye in his early thirties, possibly from a fever or possibly, as he himself thought, from overwork on maps. He is often said to have shrugged that he would now have fewer distractions, and it is a lovely line, though there's no very good evidence that he ever said it, and it is so much the sort of thing a great man ought to say that it should probably be treated with suspicion. Frederick the Great, who employed him in Berlin for twenty-five years and never warmed to him, called him a "mathematical cyclops," a remark that tells you rather more about Frederick than about Euler. In 1766 Euler returned to St Petersburg, and soon afterward a cataract took the sight of his other eye.

It made no difference. If anything, he got faster. With the help of his sons and a small team of assistants to whom he dictated, Euler produced about half of his entire life's work after he had gone blind. He did immensely complicated calculations in his head. When his house burned down in 1771, a servant from Basel carried him out of the flames, and most of his manuscripts were saved. He went straight back to work.

He died on 18 September 1783 in a manner that would have been hard to improve upon. He had spent the morning giving a mathematics lesson to one of his grandchildren, and the afternoon working out the mathematics of hot-air balloons, which had just been invented, and of the newly discovered planet Uranus. Over tea, he suffered a stroke. In the words of the French mathematician Condorcet, he "ceased to calculate and to live."

Johann had died thirty-five years earlier, aged eighty, still a professor, still feuding and, as far as anyone could tell, wholly unrepentant. Daniel lived on peacefully in Basel until 1782. Between them, the Bernoullis and Euler built much of the mathematics that the modern world runs on, and I find it oddly comforting that so much of it was done by people who couldn't get through a family dinner.