A Strange Wilderness

by Amir D. Aczel

  Gauss was always fascinated by numbers and their properties—studied in a field we now call number theory, which was in large part launched by his Disquisitiones Arithmeticae. He asked himself about the properties of what we call Gaussian integers—that is, complex numbers of the form a + bi, where a and b are integers. These numbers have interesting properties. For example, the number 5, which is prime in the “normal” system of numbers, is no longer a prime number in the integral domain of the Gaussian integers. Why? Because it is now factorable into two “prime numbers,” the Gaussian integers 1 + 2i and 1 – 2i: (1 + 2i)(1 – 2i) = 1 + 2i – 2i – 4i2 = 1 – 4(–1) = 5.

  THE DENSITY OF PRIMES

  Returning to “normal” integers, one of the main concerns of number theory is the study of prime numbers. We have seen earlier that Euclid proved that the prime numbers are unending—there is no possible “largest prime number,” no matter how far out in the sequence of positive whole numbers you go. Gauss knew this, but he wondered what happened to the frequency of prime numbers as he surveyed the numbers further and further. Did they become more frequent, or less so?

  Recall the Sieve of Eratosthenes we saw earlier in the book. As we go from 1 to 100, do the primes become denser (more primes per length of a sequence of numbers) or less dense (fewer prime numbers per length of a sequence of numbers)? Let’s take a look (the prime numbers appear against a gray background):

  Between 1 and 20 there are eight primes; in the next set of twenty positive integers, there are only four primes; in the next set of twenty, there are five primes; the next set of twenty has five primes; and the last set, from 80 to 100, has only three prime numbers. In the first 100 numbers, we can see that there is a moderate decrease in density: but does this trend continue? Might there be 60 primes between 1,900 and 2,000, as opposed to the mere 25 within the first 100 numbers? And if the density does continue to decrease, how fast does it decrease?

  Gauss made the first attempt at an answer to this question, one of the most important questions in number theory. On the back of the page of a table of logarithms—he studied Napier’s invention with much interest—Gauss wrote in German:

  Primzahlen unter a (= ∞) a / la

  Recognizing that “l” stands for the natural logarithm, and hence that la is the natural logarithm of a, Gauss’s statement is what we call the prime number theorem: The number of prime numbers (primzahlen) less than any given number, a, approaches the limit a / lna as the number a approaches infinity. This is a relatively slow rate of decrease, but it suggests a constant overall decrease in the number of primes as we approach infinity. It implies that for a large number, a, the probability that it is prime is roughly 1 / lna. The celebrated Riemann hypothesis, named after the German mathematician G. F. B. Riemann, implies bounds on the errors of the estimates provided by the prime number theorem, but to this day the conjecture is still unproved.

  In 1805 Gauss married a young woman from Brunswick named Johanna Osthoff. They had three children, but she died giving birth to the third in 1809. Gauss was heartbroken, but for the sake of his children, he remarried the following year and had three children with his second wife, Minna Waldeck.

  Gauss didn’t get along very well with his children—he was, perhaps, too busy doing mathematics. The exception was Joseph, his son from his first wife, who was fairly gifted in mathematics. Two sons ran away from home as adolescents, immigrated to the United States, and became farmers in Missouri. Later one of them became a wealthy merchant in Saint Louis, which was a major trade port during the colorful days of the Mississippi riverboats. The sons had several children each, and it is believed that Gauss has many descendants living in America today (perhaps excelling as mathematicians).20

  Meanwhile, when he was seventy years old, the Duke of Brunswick, who had generously supported Gauss, was called to help Germany fight against Napoleon. He was sent to Saint Petersburg to try to enlist Russia’s help, but the tsar refused, and the duke was put at the head of a large German force opposing Napoleon’s progress at Jena. Gauss was living in a house by the main highway going into Brunswick, and one fall day he saw a wagon entering town carrying the dying duke, who had been critically wounded in battle. Gauss’s anguish was palpable—the kind duke had been like a father to him—but after the duke’s death another German philanthropist, Alexander von Humboldt, stepped in to support Germany’s greatest mathematician. Through Humboldt’s patronage Gauss was made the director of the Göttingen Observatory, a position that also allowed him to teach and continue research in mathematics.

  Gauss was a great facilitator, encouraging and supporting the work of younger mathematicians. His last student was the brilliant German mathematician Peter Gustav Lejeune Dirichlet (1805–59), who always carried Gauss’s Disquisitiones with him, even sleeping with the book under his pillow. The great French mathematicians Lagrange and Laplace, who did important work in mathematics as applied to physics and astronomy, were also big fans of Gauss when he was still a young man.

  Thus the geniuses that followed the invention of the calculus achieved great breakthroughs in the history of mathematics and, in many ways, transformed the field into a cohesive and rigorous discipline. Collectively, their work affected many areas, including analysis, probability theory, topology, number theory, and geometry. They laid an important foundation for the future of mathematics.

  PART V

  UPHEAVAL IN FRANCE

  Under the reign of Napoleon Bonaparte, many mathematicians flourished—including Adrien-Marie Legendre, Joseph Lagrange, and Joseph Fourier. Napoleon is depicted in this 1805 portrait by Andrea Appiani.

  ELEVEN

  NAPOLEON’S

  MATHEMATICIANS

  France has an unusual history in general, and mathematics history in France is no exception. At first the monarchs—members of the so-called ancien régime—supported science and mathematics at least as much as the absolute rulers of countries such as Russia and Germany (Prussia). The French Royal Academy of Sciences was founded by a group of scientists and scholars who first met in King Louis XIV’s own library in 1666. Six years earlier, in 1660, the Royal Society had been founded in London under a royal charter from Charles II. With Louis XIV’s blessing and encouragement, the French scholars and scientists continued to meet in his official library for many years until the construction of the Institut de France, with its gilded dome, just across the Seine from the Louvre.

  But the French revolution changed all that. In 1793 the National Convention—the political body the revolutionists had elected to rule France instead of its toppled (and guillotined) monarchs—abolished the Royal Academy of Sciences. It was reinstituted under a different, non-Royal guise a couple of years later, but science was not a major concern for the revolutionists. They were more interested in overthrowing everything that had come before them, and the idea of “natural law” was clearly anathema to them because it would have supported the old status quo, even though the legitimacy of the monarchy was based on religious belief (God anoints the kings of France through a cleric, such as Saint Rémy, who baptized Clovis, the first king of France) rather than science. Even the names of the months were changed during the Revolution, based upon natural weather patterns in temperate France at different times of the year. For example, the months of Brumaire and Thermidor, roughly covering the same range of dates as November and August in the Gregorian calendar, took their names, respectively, from the Latin words for “fog” and “summer heat.”

  This 1671 engraving depicts King Louis XIV touring the French Royal Academy of Sciences, which was founded in Paris by a group of scholars in 1666.

  There is another important consideration that affects our story. In the eighteenth century, the study of science was sometimes considered a luxury. One of the founders of quantum mechanics, Louis de Broglie, was a prince. The French naturalist and mathematician who partially developed the theory of probability, Georges-Louis Buffon, was a count. And we’ve already met the Marquis de l’Hôpital, ano
ther nobleman. In fact, there were many nobles engaged in scientific pursuits—not only in France but throughout Europe. During the Revolution thousands of members of the French nobility were executed, and science and mathematics therefore suffered during this tumultuous and brutal period. The mathematician M. J. Condorcet, a marquis, lost his life despite the fact that his ideas were very much in the direction of the abolition of the monarchy and support for the rights of man. And for those who remained unmolested, how easy could it have been to find the tranquillity and garner the powers of concentration required for conducting mathematical or scientific research while food is scarce and one lives in constant fear of the dreaded knock on the door?

  But the Revolution did accomplish one very important achievement in mathematics as applied to society: it established the metric system. This is today a very widely used system around the world—and it is certainly important in the sciences. Ironically, the metric system owed its birth to the revolutionaries’ insistence on breaking from the past, as they were doing with the calendar.

  THE BIRTH OF THE METER

  Adrien-Marie Legendre (1752–1833) was one of the French mathematicians of the revolutionary period, advancing a number of areas of mathematics, including the least-squares method in statistics that was later completed by Gauss. He also helped measure the Paris Meridian, which passes through the center of the Paris Observatory, using triangulation methods. Legendre polynomials are polynomials used in mathematical applications such as physics; they result as solutions to a differential equation. Nevertheless, Legendre was left out of the committee of mathematicians who were charged with creating the new system of measures.

  In 1790 the French diplomat Talleyrand put forward the idea that as part of the overthrow of everything that came before them, the revolutionaries must choose a new system of weights and measures. The Academy of Sciences was still in existence at the time—it would be abolished three years later (and would be reinstituted still later)—so the problem of setting the new system was referred to a committee of the academy that included the two mathematicians Joseph-Louis Lagrange (1736–1813) and Lazare Carnot (1753–1823). The committee entertained two main proposals after various ideas had been whittled down. One system of measures was to be based on the decimal (base-10) number system, and the other on the duodecimal (base-12) number system.

  Lagrange was greatly opposed to the duodecimal system, which was gaining strong support in the committee. The arguments for a base-12 system were rooted in the fact that 12 is divisible by more numbers than 10, so presumably it would be easier to calculate in this system. (The divisors of 12 are 1, 2, 3, 4, 6, and 12, while 10 is divisible only by 1, 2, 5, and 10.) Lagrange reportedly led a charge in the opposite direction: “Why should the base of the system of measurements we adopt be a number divisible by many numbers?” he asked. “The system should be based on a prime number, such as 11,” he argued. However, it has been conjectured that Lagrange led with this argument only to derail the supporters of the duodecimal system.1 Eventually, Legendre’s celebrated success in measuring the meridian was brought into account, and since that measurement was based on the decimal system, the decimalists won. The meter was thus defined, based on the meridian, as one ten-millionth of the distance between the equator and the poles.

  Recall Eratosthenes’ estimation of the circumference of the earth as 250,000 stades. A stade is about a tenth of a mile, making that circumference 25,000 miles. This measurement is equivalent to about 40,000 kilometers, and a quarter of this length is the distance from the equator to the pole. One ten-millionth of 10,000 kilometers is 1 meter. Lagrange and Carnot—a mathematician and military leader who worked in geometry and calculus—had won! Years later it would be none other than Carnot who would promote a young colonel named Napoleon Bonaparte to the rank of general, launching a career that would change the world. But Lagrange was a far more important mathematician than Carnot.

  JOSEPH LAGRANGE

  In the words of Napoleon, “Lagrange is the lofty pyramid of the mathematical sciences.” The French emperor would eventually appoint the brilliant mathematician to the positions of senator, Count of the Empire, and Grand Officer of the Legion of Honor, the honorary body established by Napoleon.2

  Joseph-Louis Lagrange was born in Turin, Italy, to French-Italian parents. His grandfather was a captain of the French cavalry and offered his service to the king of Sardinia before settling in Turin and marrying a daughter of the influential Conti family. Their son became the treasurer of war for Sardinia and married Marie-Therese Gros, the daughter of a wealthy physician. The couple had eleven children, but only the youngest—Joseph-Louis, born in 1736—survived to adulthood.

  Joseph’s father was a speculator, and although he had inherited money from his own parents and had married into a family of means, he lost most of his assets. As an old man, Joseph once commented, “If I had inherited a fortune I should probably not have cast my lot with mathematics.”3 Mathematics would have been the poorer, for Lagrange contributed much during his mathematical career.

  At school Lagrange was interested in the classics, and only through classical studies became familiar with the works of the ancient Greek geometers. He was not impressed with geometry, however, and would have become a classical scholar had he not come upon a book by Edmond Halley, Newton’s friend, which explained Newton’s calculus with great enthusiasm. Lagrange studied Halley’s work on his own and became enamored with the new mathematics of the time. It is said that he was only sixteen years old when he became a professor of mathematics at the Royal School of Artillery in Turin, although that claim has been disputed.4

  At any rate, when he was nineteen, Lagrange wrote a book called Mécanique Analytique (Analytical Mechanics), which was recognized as one of the world’s greatest books on mathematics when it was published thirty-three years later. His dislike of geometry—ancient Greek or otherwise—was evident in the book. The preface stated, “No diagrams will be found in this work.”5 Instead, Lagrange claimed that all of mechanics could be understood as a generalized geometry that is four-dimensional—including three spatial dimensions, plus the dimension of time—and therefore cannot benefit from diagrams or drawings of any kind. With this deep insight, he anticipated Einstein, whose theories of relativity wed space and time to create space-time, and freed mechanics from its Greek origins. Instead, Lagrange introduced purely analytical methods, which utilized algebraic tools instead of geometrical ones.

  The teenage professor in Turin was the main force behind the founding of the Turin Academy of Sciences, and he contributed to its proceedings, encouraging his students—who were older than himself—to conduct research in mathematics, which would subsequently be published by the academy. One of his own contributions was the development of the calculus of variations. His work elaborated on the achievements of Newton and founded mechanics on a solid mathematical base.

  In his early twenties, Lagrange contributed to the theory of probability by applying calculus to statistical problems. He also worked on the mathematical theory of sound by considering the movement of air particles in elastic settings and studied the mathematical theory of vibrating strings. At age twenty-three he was recognized all over the world as a mathematician as great as Euler and the Bernoullis.6

  Lagrange sent some of his work to Euler, who was then in Berlin. Euler, who had been working on similar problems with little success, quickly recognized that the young French-Italian mathematician’s work would enable him to solve some of these problems. Being kind and fair, he encouraged Lagrange to publish his work first and only afterward arranged for publication of his own papers based on Lagrange’s breakthrough in the calculus of variation. He didn’t want there to be any doubt about who was first to develop the new methodology.

  Euler then devised a plan to get Lagrange to join him at the Royal Prussian Academy. At the insistence of Euler and other mathematicians, Frederick the Great invited Lagrange to Berlin. Before going to Berlin, Lagrange won a prize
from the French Academy of Sciences for solving the problem of the moon’s libration, which explains why, with small variations, we see only one face of the moon. Lagrange made strides in our understanding of the celebrated “three-body problem,” in which the sun, earth, and moon exert gravitational forces on each other. He later won several more prizes from the academy for partial solutions to this problem, which is very complicated. In 1766, when he was thirty, Lagrange was finally welcomed to the Berlin Academy and, after a short while, took Euler’s position as director of mathematics. He worked for years at the academy revising his Mécanique Analytique masterpiece with the help of Adrien-Marie Legendre.

  Lagrange made contributions to number theory, but he did even more important work in algebra, in the theory of equations. The Babylonians, Egyptians, and Greeks, followed by the Arabs, Italians, and others, had done extensive work on solving equations of increasing order: linear (where the unknown, x, is in the first power), quadratic (second power), cubic (third power), and quartic (fourth power). But Lagrange sought a general method of solution for equations of any order. He began to understand that the number of different orders in which solutions, expressed in terms of the coefficients of an equation, can be arranged had something to do with the solvability of an equation. Nevertheless, he did not find a general method, although his pioneering investigations would be carried to a far greater extent by the young Frenchman Évariste Galois in the 1800s.

  In mechanics Lagrange invented a function we now call the Lagrangian, in his honor. This function lists and ties together all the relevant parameters of a physical situation. Today it is of immense importance in modern physics—particularly quantum mechanics and modern particle theory, in addition to the Newtonian mechanics for which Lagrange had invented it in the first place. A brilliant twentieth-century mathematician, Emmy Noether, described later in this book, used the Lagrangian function—in addition to the work of Galois and the Norwegian mathematician Lie, who extended Lagrange’s work on equations—in a study of symmetry that established conservation laws in physics. These laws govern such physical processes as the conservation of energy, momentum, electric charge, and other parameters used in modern physics.