The Human Side of Science: Edison and Tesla, Watson and Crick, and Other Personal Stories behind Science's Big Ideas (2016)

Chapter 4: The Battling Bernoullis and Bernoulli's Principle

All birds need to fly are the right-shaped wings, the right pressure and the right angle.

—Daniel Bernoulli1

Antwerp was a bustling city in the Spanish Netherlands and a cosmopolitan center of world trade. Perhaps Antwerp was becoming too international, for in 1567, King Philip II of Spain sent the Duke of Alba to the Netherlands with a sizable army to reassert the king's authority and compel adherence to the Roman Catholic religion. The duke set up a court called the Council of Troubles. This court condemned over twelve thousand people for either treason or heresy. Many fled, including the Bernoulli family, who were spice merchants. Eventually, they wound up in Basel, Switzerland. Even though their family dynamics were often stormy, several Bernoullis made significant contributions to science and mathematics.

Since many of them have similar first names, the Bernoullis will be referred to by first name and, if necessary, their birth year in parentheses. The intention here is clarity, not inappropriate familiarity.

Nicolaus (1623) was a spice merchant in Basel who inherited the business originally located in Amsterdam. He fathered ten children, including (number 5) Jacob (1654) and (number 10) Johann (1667). Jacob studied philosophy and theology at his parents’ insistence but resented it greatly. Although he completed degrees in both, he also studied mathematics and astronomy, which were far more to his liking. This disagreement with authority was the start of a recurring pattern in the family. For the next five years after receiving his theology licentiate in 1676, Jacob studied mathematics and traveled throughout Europe, meeting many of the leading mathematicians. He began correspondences with several of them that lasted many years. His travels took him to England in 1681, where he met Robert Boyle and Robert Hooke. In 1687, Jacob was appointed professor of mathematics at the University of Basel.

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THE SIBLING PLOT THICKENS

Jacob's youngest brother Johann was their parents’ last child, so he was directed by them to study business. But Johann protested that business didn't interest him. His father, Nicolaus, was displeased but finally allowed Johann to study medicine as a compromise. Johann entered the University of Basel, where his older brother Jacob taught mathematics. Ostensibly studying medicine, Johann got his brother Jacob to teach him the latest mathematics—calculus—based on the recent (1684) and quite obscure work of Gottfried Leibniz. Johann was a quick study and soon bragged about how he knew more mathematics than his brother, the professor. Jacob wasn't pleased, and the two quarreled, often in public or in print.

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Johann Bernoulli (1667–1748). By Johann Rudolf Huber (1668–1748) and Johann Jakob Haid (1704–1767). From Wikimedia Commons, user Roybb95~commonswiki.

After a several-year tour of the Continent, including more than a year tutoring Guillaume Marquis de l'Hôpital in Leibniz's calculus, Johann decided to finish his doctorate and start a family. This meant that he needed a steady job. Since Jacob blocked him from any faculty position in Basel, he took a teaching job in Groningen, the Netherlands. Johann's first son, Nicolaus (1695), was born several months before he and his wife traveled to Groningen, which turned out to be a difficult journey. Things didn't improve much after they arrived. In France, de L'Hôpital published the first calculus textbook in 1696. Johann received scant mention in the book but swore it was based on notes from his tutorial sessions in France. Since he had been paid (handsomely) for his tutorial work, L'Hôpital believed the notes belonged to him. The book was well received, much to Johann's chagrin.

A NEWTON TRAP?

In 1696, Johann posed a challenge called the brachistochrone problem, which was originally considered by Galileo in 1638: “Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity which starts at A and reaches B in the shortest time?”2

The problem was published in the scholarly journal Acta Eruditorum with a six-month deadline for readers to submit their solutions. Of course, Johann had a solution, so this perhaps seems to have been another way for him to show up his brother in their ongoing feud. But a curious thing happened. No solutions were received by the deadline. Gottfried Leibniz persuaded Johann to extend the deadline to allow foreign competitors to have enough time to work on the problem. Foreign competitors? Who might that be?

Eventually, several solutions were obtained. Jacob Bernoulli solved it (so much for the sibling feud), along with Leibniz and L'Hôpital, plus an anonymous solution published in Transactions of the Royal Society in England. Hmmm. Who could that anonymous person be? Perhaps some of the additional wording of the challenge might be relevant here: “There are fewer who are likely to solve our excellent problems, aye, fewer even among the very mathematicians who boast that [they]…have wonderfully extended its bounds by means of the golden theorems which (they thought) were known to no one, but which in fact had long previously been published by others.”3

It appears that Johann Bernoulli (and perhaps Leibniz) was using this problem as a way to test Isaac Newton. After all, Newton had his difficulties with de Duillier and his depression in 1693, and had recently been made Warden of the Mint (recall from chapter 3). Perhaps this was a good time to see if Newton had lost interest in scientific matters and thereby establish Leibniz's priority claim on inventing calculus. Just to make sure Newton saw the challenge, Johann sent two copies of the challenge directly to Newton's home in London in January 1697. According to the later recollection of Newton's niece, here's what happened when the problem arrived: “ In the midst of the hurry of the great recoinage, [Newton] did not come home till four (in the afternoon) from the Tower very much tired, but did not sleep till he had solved it, which was by four in the morning.”4

Newton sent his solution to the president of the Royal Society, where it was published anonymously. Johann wasn't confused by the anonymity of the Royal Society publication. He said, “Tanquam ex ungue leonem” (We recognize the lion by his claw).5

Newton also told the Royal Society president, “I do not love to be dunned [pestered] and teased by foreigners about mathematical things.”6

The calculus feud was reignited seven years later, after Newton had resigned his professorship and devoted himself full time to the Royal Mint.

DIFFICULTIES MULTIPLY

In Groningen, Johann continued his independent streak by introducing experiments into his physics lectures. This managed to offend both Calvinists—who thought sensory data should be used to ascertain God's plan—and Cartesians—who saw sensory information as greatly inferior to reason.

Johann's next child, a daughter, then arrived but lived only a few weeks. This saddened him greatly. Several student disputes of a religious or philosophical nature then swirled around him, in addition to the ongoing controversy with his brother Jacob.

In 1700, Johann's second son, Daniel (1700), arrived and joined this unhappy brood. It wasn't long before the Bernoullis left Groningen. Johann, his wife, their sons Nicolaus and Daniel, and Johann's nephew Nicolaus Bernoulli, who had come to study math with his uncle, all set out for Basel. Johann had finally been offered an academic position—teaching Greek.

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Daniel Bernoulli (1700–1782). Used with permission from Sidney Harris.

During the journey, the family learned that Jacob had died, so when Johann arrived, he lobbied hard and was rewarded with his brother's teaching position in mathematics. You might think this would be a good time to start playing the “happily ever after” music, but it didn't turn out that way.

After a few years, Johann's third son, also named Johann (1710), arrived. Johann the elder busied himself with his mathematics and spent a lot of time writing letters supporting the Leibniz side of the Newton/Leibniz controversy (see chapter 3). He was so active that he was referred to as “Leibniz's bulldog.”7

Soon it became time for Johann to guide his sons’ choices for their life's work. The first and favorite son, Nicolaus, was already studying math, and Johann knew there was little financial reward in math. As a result, Daniel was encouraged to study business. At first, he refused but then gave in. Daniel was such a strong student that his father changed his mind and asked him to study medicine. Daniel agreed, but only if his father would teach him mathematics privately. Daniel obtained his PhD in anatomy and botany in 1721. His thesis dealt with the mechanics of breathing, and his work was based partially on his father's ideas about energy conservation. (Does this sound familiar? Genetics strikes again.)

After failing to obtain an academic post at the University of Basel, Daniel made his way to Padua to further his medical training. While traveling through Venice, he became ill and stayed there until he recuperated, working on mathematics and publishing a mathematics book. He also designed an hourglass that would function on ships during heavy seas. For this invention, he won the Paris Academy Prize in 1725. Having acquired a measure of fame, he was offered an academic job. Both Daniel and his older brother Nicolaus were offered and accepted faculty positions in mathematics at a new academy at St. Petersburg. Unfortunately, Nicolaus died within a year after their arrival. Sadness at his brother's death and the harshness of the climate made life difficult for Daniel, but a very interesting thing happened that was based on earlier developments back home. Several years prior, Daniel's father, Johann, did something entirely out of character. His old college roommate, Paul Euler (pronounced oiler) had a son, Leonhard, who wanted to study mathematics rather than become a minister like his father. Johann convinced Euler to let his son study mathematics. (Amazing flip-flop?) Just prior to Nicolaus's death, Leonhard Euler finished his doctorate at Basel. When he learned of his son's death, Johann recommended that Leonhard should fill Nicolaus's post at St. Petersburg. Leonhard took the job. Daniel was delighted and offered him a place to live in St. Petersburg. In exchange, Leonhard had to bring brandy, tea, coffee, and other delicacies from home. Thus began an incredibly fruitful collaboration for both of them.

Leonhard Euler's great analytical skills combined with Daniel's superior physical insights produced top-quality work in a wide variety of scientific and purely mathematical areas. These included the harmonic vibrations of a stretched string and the odd harmonics of an open/closed air column, hydrodynamics, and a remarkable anticipation (by one hundred years) of the kinetic theory of gases as the interaction of independent particles.

In spite of the productive collaboration with Leonhard Euler, Daniel was homesick and ready to leave St. Petersburg. In 1733, Daniel and his brother Johann, who had joined him in St. Petersburg, toured the Continent as they journeyed back to Basel. Daniel finally landed an academic position there—teaching botany. (Well, better botany in Basel than sleet in St. Petersburg, right?) Before leaving St. Petersburg, Daniel had submitted an entry to the Paris Academy competition that applied some of his ideas to astronomy. He arrived in Basel to find that he had won the prize. The only problem was that it was awarded jointly to his father, Johann, who had also entered. Johann was furious that his son could be considered his academic equal, so he wouldn't even let Daniel enter the family home.

THE WORST IS YET TO COME

Daniel had left a copy of his major work, Hydrodynamica, at the printer's in St. Petersburg, but he continued to polish it for publication. It was eventually issued in 1738. His father also had a book published, Hydraulica. Although it wasn't published until 1739, Johann had it predated to 1732. He then claimed Daniel had stolen the work from him. Ironically, the frontispiece of Daniel's work is signed “Daniel Bernoulli, son of Johann.” Johann's plagiarism was noticed before long, and Daniel received appropriate credit for the work. After sixteen years of teaching botany and physiology, Daniel was finally appointed to the chair of physics at St. Petersburg in 1750. He was extremely popular and incorporated many experiments in his lectures. He taught until 1776 and received many honors from eminent scientific societies.

In view of all the contention, jealousy, and dirty tricks of the Bernoullis, one must wonder if the controversies spurred them to greater efforts or whether they would have accomplished more without the distractions. Bernoulli family genes must be quite strong. In 2004, the chairman of the Earth Sciences Department of the University of Basel was Professor Dr. Daniel Bernoulli. In 2015, Facebook lists more than a thousand Bernoullis, and the name “Daniel Bernoulli” has almost seven times as many Facebook “likes” as “Johann Bernoulli.”

BONUS EXAMPLES

To illustrate the value of Daniel Bernoulli's work contained in Hydrodynamica, let's look at one of his major ideas, the energy conservation principle, and see some examples of it from modern life. Here is Bernoulli's principle of energy conservation from a purely conceptual (nonmathematical and somewhat simplified) viewpoint.

A flowing fluid has three forms of energy: random kinetic energy, represented by pressure; directed kinetic energy, represented by the speed of the flow; and gravitational potential energy, which depends on the height of the fluid. The sum of these three energies is called the total energy, which remains constant under many conditions. Thus, if one form of energy decreases, another must increase, and vice versa. Here are a few examples of this principle in action, as fluid flows or is raised to some height.

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Curveball

In the diagram, a ball moving from left to right has air flowing across it. If the ball spins counterclockwise as viewed from the top, the ball drags air with it on the left side, increasing its velocity. According to the Bernoulli principle, an increase in velocity must be accompanied by a decrease in pressure, so the pressure on the left side of the ball is smaller. The right side slows the air's velocity, so the pressure on the right side is greater. This imbalance in pressures leads to a net force from right to left, so the ball's path curves to the left. Baseball fans would note that this corresponds to the normal curveball thrown by a right-handed pitcher. If the batter is also right-handed, the ball curves away from him, possibly causing him to swing where the ball isn't. Strike. It's easy enough to figure what would happen for left-handers, and you could check TV slow-motion camera shots for examples. Curveballs also show up in other ball sports such as a soccer, tennis, and Wiffle ball.

Backspin

Look back at the previous diagram and suppose it was a view from the right side of the gallery watching a golfer hit a shot toward the green. The counterclockwise spin causes the golf ball to rise higher than normal. When it lands, the ball would roll forward very little and might even roll backward.

Topspin

In tennis, the idea is often to bring the ball down quickly so that it stays inbounds. To accomplish this goal, the spin direction must be reversed from the previous example, so the path of the ball would dip downward. To apply topspin, tennis players “brush up” the back of the ball as they hit it.

Airplane Wing

In normal flight, an airplane wing's orientation looks like it is going slightly uphill. This is called an angle of attack. Since the air must go farther across the top of the wing than the bottom, it must also go faster across the top. From the Bernoulli principle, this means less pressure above the wing and more pressure below. The net force is known as lift. Actually, the real situation is much more complicated because the flow is not always smooth (laminar) and turbulence alters things, as does air at the wing ends flowing from the bottom of the wing to the top, creating wingtip vortices.

Sailboat

The sail on a sailboat functions like a wing, and the force generated helps to propel the boat forward. Another component of the force from the sail pushes sideways. Since the sail is high above the boat deck, this other component creates a torque that tends to rotate the boat. To counteract this torque, boats have daggerboards, centerboards, or keels, which generate a torque in the opposite direction. The other possibility for counteracting this torque is called hiking out, as when sailboat passengers or crew hang their bodies out of the boat in the opposite direction to the boat's tilt.

Water Tower

Many communities have water towers, where water is pumped to a high storage location so that it can be distributed later with minimal pumping action. From the standpoint of the Bernoulli principle, this is an example of trading gravitational potential energy for kinetic energy. Pumping the water into the tower often occurs late at night, when there is excess electrical energy available because of minimal demand.

Beach Ball in Blower Stream

You may have seen a beach ball suspended in a vertical air stream from a blower. This demonstration is designed to show the power of the blower, but it also reveals the Bernoulli principle at work. Under these conditions, the ball is remarkably stable. If you try to push it out of the air stream, it is just pushed back toward the center. According to the Bernoulli principle, air within the stream is moving faster than air outside the stream, so the higher pressure outside pushes the ball back.

Chimney

When winter winds howl, smoke from the fire in the fireplace moves up the chimney and out into the air quite smartly. The wind outside causes reduced pressure there, so the higher pressure inside pushes the smoke out.

Here's a handy little experiment you can do to demonstrate the Bernoulli principle at work:

Hold two sheets of paper vertically in front of your face. Blow between the sheets and watch to see if they move toward or away from each other. As the air velocity increases between the sheets, the pressure decreases, so they should move together.

Next time you fly somewhere in an airplane or ride a sailboat, please think kindly about the battling Bernoullis. Their battles won something—for the rest of us.

The Bernoullis generated their own form of heat in terms of family dynamics. The next chapter will deal with heat in the physical realm—different in a sense but still capable of generating hot human interactions.