A Strange Wilderness

by Amir D. Aczel

  FROM 1689 TO 1690, Newton was a member of the Convention Parliament in London, representing Cambridge University. In 1696 he was elected Warden of the Royal Mint in London and, three years later, promoted to the very prestigious post of Master of the Royal Mint, in recognition of his service to society in the creation and dissemination of knowledge. Income from his position as the head of the mint enhanced his wealth, and while many others might have rested on their laurels, Newton worked hard at the mint, improving methods to combat counterfeiting and making the minting process more efficient as new coins were brought into circulation.

  While working in London, Newton suffered from a severe inability to sleep, and he developed poor eating habits. News of his illness spread throughout Europe, bringing wishes for a quick recovery from many. On hearing the news that Newton was well again in 1693, Leibniz wrote to a friend saying how relieved he was. But that same year, Newton heard that calculus was spreading through the continent of Europe and that it was credited to the work of Leibniz rather than himself. Thus began what historian Eric Temple Bell calls “the most shameful squabble about priority in the history of mathematics.”

  On June 20, 1688, Newton wrote to Halley that science was “such a litigious Lady, that a man had as good be engaged to lawsuits, as to have to do with her.”15 He was becoming consumed with his fight against Leibniz and was determined that he, not the German mathematician, be given full credit for inventing calculus.

  In 1703 Newton was elected to preside over the Royal Society, and thereafter he was reelected to this position of power and prestige every year for the rest of his life. In 1705 he was knighted by Queen Anne. Newton used his new powers against Leibniz: he appointed the com-mittee that investigated the priority over calculus, and (anonymously) authored its final report, finding in his own favor.

  But the dispute over calculus was never settled, and today we recognize both Newton and Leibniz jointly as the co-inventors of calculus. Leibniz’s dx and dy notation eventually won out over Newton’s and “fluxions,” however. (But Newton’s “dot” notation is still used in physics and other applications of calculus. In truth, Newton’s calculus was both cumbersome and less rigorous, while Leibniz’s notation and ease of use made his form of calculus useful.) Newton’s calculus, though, was developed before that of Leibniz, albeit not published early enough. Regardless of who is rightfully credited with its discovery, European mathematicians found in calculus fertile ground for developing and extending much beautiful and useful mathematics in the years to follow.

  Several key mathematical geniuses of the Enlightenment, including members of the Bernoulli family and Leonhard Euler, studied at the University of Basel, seen in this modern photograph.

  TEN

  GENIUSES OF

  THE ENLIGHTENMENT

  The seventeenth century brought immense advances in mathematics, which took the pioneering work of the ancient Greeks and catapulted it into the modern age, culminating in the birth of the calculus. In the century that followed, calculus was taken to higher levels of understanding, application, and abstraction, and mathematical analysis as we know it today was born. Imaginary numbers came of age, giving rise to complex analyses, and new geometries that went far beyond Euclid’s imagination unseated the ancient scholar and his predecessors.

  Sixteenth-century mathematics flourished in Italy, and mathematics of the seventeenth century found fertile ground in France and Britain. By contrast, leading mathematicians of the eighteenth century were mostly Swiss and German, though, having migrated from one part of the continent to another, they may properly be considered pan-European.

  THE BERNOULLI DYNASTY

  In Basel, Switzerland, there lived a family of gifted mathematicians called the Bernoullis. The Bernoulli family had originated in Spanish-controlled Antwerp, but in 1576 they fled religious persecution. Seven years later they moved to Basel. This most unusual family begat a dozen great mathematicians.1 Two of them—brothers who lived in the late seventeenth and early eighteenth centuries—and their descendants play a role in our story.

  Nicolas Bernoulli, the father of these two brothers, was a spice merchant and prominent citizen who sat on the Basel city council; his wife came from a wealthy family of Swiss bankers. He was not interested in his sons pursuing mathematical careers. He wanted Jacob (also known as James or Jacques), the older brother, to be a cleric and the younger, John (also known as Jean or Johann), to be a merchant or a doctor. The father piled obstacles in the way of the two brothers pursuing mathematics, but to no avail.

  Jacob was forced by his parents to study philosophy and theology at the University of Basel in preparation for a career as a minister. He received his master’s degree from the university in 1671 and five years later was awarded a license to practice theology in Switzerland. Throughout his education, however, he read mathematics—his true passion in life—on the side. Jacob then moved to Geneva, in the French part of Switzerland, and from there to France, where he concentrated his studies on the mathematics and philosophy of Descartes. On a visit to Britain, he met Newton’s associates Boyle and Hooke and thereafter maintained an extensive correspondence with them and many other mathematicians and physicists he met during his travels. Throughout this period, and to his parents’ dismay, Jacob supported himself by working as a mathematics tutor.

  After studying all the works of Leibniz from 1684 to 1686, Jacob Bernoulli became convinced of the great power of the new calculus. In fact, it is he who suggested to Leibniz that the term integral denote the area under the curve represented by a function. Jacob extended the range and depth of calculus in a number of papers he published in professional journals. Like Newton, he investigated infinite series and was able to prove that the harmonic series (1/n) is divergent—although, according to him, his bother John first made the discovery.2 He also showed that the series of the reciprocals of the Pythagorean square numbers—i.e., 1/n2—is convergent, although he did not determine what it converged to.

  Jacob also proved the interesting Bernoulli Inequality: (1 + x)n > 1 + nx, where x is a real number greater than –1 and not 0. The relationship also works for non-integers. In 1690 he published the first paper to reference the integral, substituting Leibniz’s calculus summatorius with calculus integralis. He also proved the law of large numbers used in probability theory.

  When John Bernoulli read Leibniz, he was so taken with his calculus that he spent two years writing about it. Not having a profession, he went to Paris—against his father’s better judgment—to work as a tutor for a young French nobleman, the Marquis de l’Hôpital (1661–1704). John taught the marquis about Leibniz’s calculus, but because he needed money after he left Paris, he agreed to keep sending the marquis his mathematical developments—to be disposed of as the marquis wished—for a continuing salary. Thus in 1694 Bernoulli sent l’Hôpital a rule that states that if two functions possess derivatives and are equal to 0 at the same point, and if the limit of the ratio of the two derivatives exists at that point, then the limit of the ratio of the two derivatives at the point is equal to the limit of the ratio of the two functions at that point. This is a standard rule in advanced calculus, and all students learn it under the name l’Hôpital’s rule. The Marquis de l’Hôpital used his agreement with Bernoulli to publish the rule in his book Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes (Analysis of the Infinitely Small to Understand Curved Lines) without giving its discoverer any credit.

  This Swiss stamp features a likeness of Jacob Bernoulli painted by his brother Nicolas, along with Bernoulli’s law of large numbers.

  John was a fervent supporter of Leibniz and aggressively pushed for Leibniz’s priority in his dispute with Newton. Though his acquiescence in sending his works to l’Hôpital may suggest otherwise, he was actually a very competitive and combative person. In fact, when his very gifted son Daniel Bernoulli (1700–82) won a prize for mathematics given by the French Academy of Sciences, John drove him out of the house becaus
e he had applied for the same prize and lost.

  John had two other sons, named Nicolas and John. All three were superb mathematicians, but other members of this extended family achieved renown in mathematics as well.

  THE PETERSBURG PARADOX

  Throughout his life, Leibniz lobbied for the founding of royal academies. In France the Royal Academy of Sciences—today known simply as the Academy of Sciences, having survived the fall of the monarchy during the French Revolution—is the official body to which top French (and a few foreign) scholars belong. Germany and Russia, too, had very active royal academies in those days. The monarch—Catherine I in the case of Russia, and Frederick the Great in the case of Prussia—each established a royal academy to attract the top thinkers to the employ of the crown. Not only were royal academies the most prestigious research institutions, but they were also the leading places in Europe for doing research in science and mathematics in particular. Universities were more interested in teaching than in research, and many sciences were too new to be supported strongly there. Thus for top scientists and mathematicians the royal academy was the best place to conduct research in one’s field.

  In 1725 the Bernoulli brothers Daniel and Nicolas (John’s sons) were invited to the Royal Academy of Saint Petersburg. They were both interested in the theory of probability, and together they worked on the idea of mathematical expectation, as illustrated by the following question: If an investment has a 50 percent chance of earning $1,000 and a 50 percent chance of losing $200, how much is it worth? According to the laws of the theory of probability, the answer can be determined by multiplying the values by their probabilities and adding the results:

  0.5(1,000) + 0.5(–200) = $400

  This calculation of the expected value presents a reasonable long-term outcome for a series of trials. In other words, if continuously faced with a long sequence of repetitions of this investment opportunity, one will earn an average of $400 per investment. Expanding on this theory, Daniel and Nicolas Bernoulli came up with a scenario that led to the famous Petersburg Paradox.

  The example Daniel and Nicolas developed involves Peter and Paul. Peter tosses a coin continuously until a head appears, and the number of tosses correlates to the amount he owes Paul in the following way. If a head appears on the first toss, Peter pays Paul two crowns. If the result is a tail, Peter throws again, and if a head appears, Peter pays Paul four crowns. If two tosses are tails and the third one is a head, Peter pays Paul eight crowns; and so on. The question is: For Paul, what is this game worth? First of all, we need to represent the probability of tossing a head after x number of tosses with a function. The initial probability of tossing a head is 1/2, but the probability of tossing a head after tossing a tail is only 1/4. The probability of tossing a head on the third toss is 1/8, and so on, so we can represent the probability of tossing a head after x tosses with the function y = (1/2)x. The amount of money Peter owes Paul after each toss can be represented by the function y = 2x. Further, if we represent the expected value as a summation of the factors of these two functions multiplied together, as the rule for expected values requires us to do, we find that the game has infinite worth to Paul:

  This expectation is an apparent paradox because the game won’t last to infinity. In fact, simulations have shown that the amount earned by Paul is usually less than five crowns.

  The Petersburg Paradox occupied European mathematicians for many years and, thanks to the Bernoullis, led to the distinction between actual expectation and “moral expectation”—i.e., the fact that the actual “expected value” is not always mathematical; rather, it may be affected by people’s attitudes to money and risk—that is, whether they are risk-averse or risk-seeking.

  LEONHARD EULER

  Daniel and Nicolas Bernoulli were friends of another extremely gifted Swiss mathematician from Basel who was employed at the Saint Petersburg Academy: Leonhard Euler (1707–83). Euler (pronounced “oiler”) has been called “the most prolific mathematician in history.” His contemporaries called him “analysis incarnate,” and in the words of the French scholar Francois Arago, “Euler calculated without apparent effort, as people breathe or as eagles sustain themselves in the wind.”3 Euler’s works comprise 73 volumes of collected papers and 886 books and articles.

  Leonhard Euler was the first of six children born to the family of a Calvinist minister in Basel, Switzerland. At age seventeen he earned his master’s degree from the University of Basel, where he studied mathematics, physics, astronomy, languages, and medicine. His father, Paul Euler, a minister, was also an amateur mathematician. In fact, he had studied mathematics under Jacob Bernoulli. At a young age Leonhard learned mathematics from his father, but because his father was eager to have his son succeed him as pastor of the village of Riechen, young Euler also studied theology and Hebrew. At the University of Basel, he excelled so much in mathematics that he caught the attention of John Bernoulli, who began giving Euler a private lesson once a week.4 John’s children, Nicolas and Daniel, commenced a friendship with the young genius that would last a lifetime. When Euler’s father insisted that he be groomed to take over as pastor of Riechen after graduating in 1724, the two Bernoulli brothers intervened, promising Euler’s father that his son would be a far greater mathematician than clergyman. Though saved from a life of religious service, Euler would remain a man of faith, leading his large family in prayer and giving sermons to his children.

  In 1727 the French Academy of Sciences offered a prize to the person who could design the best mast for a ship. It is believed that, other than small vessels sailing on Swiss lakes, Euler had never seen a ship—yet his design was the runner-up for the big prize and deemed the best from a mathematical-theory point of view. In later years he would win this prestigious award twelve times. In fact, throughout his life he saw the physical universe as nothing but an excuse to pursue mathematics—he was never actually interested in the applications that often instigated mathematical pursuits.

  Euler applied for a professorship at the University of Basel but didn’t get it. By then, his two friends were in Russia, working for Catherine I as researchers in her royal academy. The two wrote Euler constantly, informing him of their continuing efforts to secure him a similar appointment. In the meantime, Euler remained a student, taking more courses at Basel. At one point the brothers wrote their friend that an opening had come up in medicine. Euler then threw himself into the study of medicine and consequently was invited to take a position in medicine at the Saint Petersburg Academy in 1727. Euler rushed to Saint Petersburg, and when he arrived, there was so much confusion at the academy because of political issues, hires, and dismissals that no one noticed when the person invited to assume a medical position slipped unnoticed into the mathematics section. There, Euler thrived, and in 1733, after Daniel Bernoulli returned to his native Switzerland, he became the head of the department.

  At age twenty-six Euler decided that he would make a home in Russia. He met a Swiss woman named Katharina Gsell, the daughter of the artist Georg Gsell, who had been recruited by Peter the Great of Russia to serve as curator of the Imperial Art Gallery in Saint Petersburg. The tsar had met Gsell in Amsterdam and, hoping to improve art in Russia, asked the Swiss artist to help him acquire works of Dutch Masters. In 1734 Gsell’s daughter Katharina married Euler. The couple had thirteen children, but only five survived to adulthood. Euler was so attached to his children that he would often work on mathematics while holding a baby and watching the other children play around him.5

  Just over a year into his marriage, Euler went almost blind in his right eye. Though the problem may have been exacerbated by the extremely concentrated work Euler did over a period of several weeks, his vision problems did not slow the progress of his extensive research in mathematics. In fact, even when he lost most of the sight in his other eye many years later, in 1766, he still found ways to continue his prodigious work, relying heavily on his perfect photographic memory and imagination. At the height of his career,
he would produce a mathematical paper within a few days, although legend has it that he would finish an entire paper between two calls for dinner.6

  In light of the political turmoil in Russia, which caused difficulties for foreign residents, Euler decided in 1741 to accept an invitation for a position at another great royal academy of the time, that of Frederick the Great in Berlin. During his twenty-five-year stay in Germany, Euler undertook a massive amount of work and wrote nearly four hundred articles, but he did not get along well with the Prussian emperor. Apparently, the emperor wanted Euler to help him design a water-transport system for his summer palace of Sanssouci, outside the city, and Euler—a pure mathematician, not an engineer—failed at this task. A pious Swiss country boy who excelled in mathematics but could not match the clever rhetoric of the philosophers in the king’s employ, Euler was considered by Frederick to be ultimately too “simple” for his tastes. Meanwhile, the vision in Euler’s right eye deteriorated so much that he seemed to squint perpetually, earning him the unflattering moniker Cyclops.

  The deteriorating vision that afflicted his right eye and caused him to squint is barely noticeable in this ca. 1756 portrait of the great Swiss mathematician Leonhard Euler.