A Strange Wilderness

by Amir D. Aczel

  Euler eventually returned to his adoptive Russia in 1766 at the warm invitation of Catherine II (Catherine the Great), who was eager to have the greatest mathematician of the time back in her country. Catherine II had become empress in 1762 after the assassination of her husband, Peter III. Four years later Euler—then fifty-nine years of age—was back at his old academy. Through an operation, doctors attempted to restore the sight in his left eye, which was clouded by a cataract, but to Euler’s despair infection soon set in and his sight deteriorated again. The empress provided a large furnished house for the mathematician and his dependents and even assigned one of her cooks to make all their meals.7

  As blindness set in, Euler had to make accommodations. He dictated equations to his grown sons, and they wrote them out on a board. His faith kept him in good spirits while his eyesight deteriorated, and his mental abilities were so immensely powerful that he performed calculations in his head that others found difficult to solve on paper. The Marquis de Condorcet, a French mathematician with whom Euler had worked, recounted that when Euler was already blind, two of his students summed up seventeen terms of a convergent series and achieved results that disagreed, according to this story, in the fiftieth decimal place. Asked to decide who was right, Euler performed the entire long calculation in his head and determined the correct answer.8

  Catherine II, a.k.a. Catherine the Great, acted as Euler’s patron during his stay in Russia from 1766 until his death in St. Petersburg in 1783.

  In 1771 a fire raged in Saint Petersburg, and Euler’s house was destroyed. Euler’s Swiss servant Grimm courageously carried him out of the house to safety and also rescued his wife; the empress then restored the furniture and the library of her favorite mathematician. Five years later Euler’s wife of forty years died. He later married her half sister, Salome Abigail Gsell.

  Despite his misfortunes, Euler never lost his faith in God. In fact, his belief in the divine order of mathematics was reflected in his encounter with the French philosopher and famous atheist Denis Diderot (1713–84). Diderot visited the academy and worked hard to convert all the scholars and scientists to atheism. According to Augustus de Morgan (Budget of Paradoxes, 1782), Euler leaped at the chance to capitalize on the philosopher’s ignorance of mathematics and challenge him. “Diderot was informed that a learned mathematician was in possession of an algebraic demonstration of the existence of God, and would give it before all the Court, if he desired to hear it. Diderot gladly consented … Euler advanced toward Diderot, and said gravely, and in a tone of perfect conviction: ‘Sir, (a + bn)/n = x, hence God exists—reply!’” Diderot understood nothing about mathematics, so he remained silent, but the wild laughter of everyone around him humiliated him so much that he quickly packed his bags and returned to France.9

  IT IS HARD TO BELIEVE how a mathematician could have produced as much as Euler had. He was especially interested in calculus, on which he wrote a series of books and research papers. He also wrote elementary books on mathematics for the Russian secondary education system. His research on the calculus of variations created an important new field, and he founded analytical mechanics as well. Euler also initiated the mathematical study of the rotations of rigid bodies and discovered the equations of fluid dynamics used in the field of hydrodynamics. He proposed the letter e for the natural number that is the base of the natural logarithms and proposed the Greek letter π for the ratio of the circumference of the circle to its diameter. We also credit Euler with what is considered the most beautiful equation in mathematics: eiπ + 1 = 0. The equation incorporates basic elements of mathematics, including the essential numbers 0, 1, e, and π, as well as i and the basic elements of arithmetic (the signs for addition, multiplication, and exponentiation, as well as the equal sign).

  Euler was a great universalist, contributing to many areas of mathematics with an exceptional intensity. He was most fond of calculations, however, performing them constantly in his mind right up until the moment of his death. On September 18, 1783, Euler spent the afternoon calculating the rate of ascent through the air of a balloon and then dined with his family and friends, discussing the planet Uranus, which had been discovered two years earlier by the German-born British astronomer Sir William Herschel. As an after-dinner exercise, Euler calculated the approximate orbit of this new planet. His grandson then came to play with him and, while playing with the boy and drinking tea, he suffered a stroke, uttered, “I die,” and collapsed. In his eulogy for Euler, the Marquis de Condorcet said that at that moment, “Euler ceased to live and calculate.”10

  CARL GAUSS

  The German prodigy Carl Friedrich Gauss (1777–1855) was perhaps the most versatile mathematical genius in history. Gauss was born in Brunswick, Germany, to a family of bricklayers and gardeners, and his father was determined that young Carl become a bricklayer as well. A strict disciplinarian, Carl’s father was cruel to his children and didn’t encourage them to do anything except hard, menial labor. He especially discouraged his son’s intellectual tendencies. Luckily, Gauss’s mother had a softer and far more sensitive disposition, and throughout his life Gauss remained attached to her.

  Gauss began to show signs of an abnormally high intelligence at age two, so his mother, Dorothea Gauss, constantly asked “experts” to evaluate his development. Once, before he was three years old, he watched his father, who had by then advanced to a supervisory position, calculate the accounts so that he could pay the bricklayers in his group. Pointing to an amount to be paid, Gauss suddenly said, “This is wrong.” The toddler had caught his father making an arithmetical error. When the father rechecked his work, he realized that the child was right; but the son’s fantastic ability didn’t seem to improve the father’s view of intellectual pursuits. Fortunately, his wife’s younger brother, Friedrich Benz, was an intelligent man who, after becoming aware of the young child’s genius, rescued his nephew from a life of hard labor.

  Gauss taught himself to read before he had any schooling, and—apparently by watching adults write numbers—he could decipher their meaning and the rules of arithmetic by logical deduction. As an old man, he would joke that he knew how to calculate before he could talk.11 Very much like Euler, Gauss was a natural calculating machine who would do complicated mathematical computations in his mind as a matter of course.

  At age seven, Gauss entered school. Two years later, in a class on arithmetic, a stern schoolmaster named Buettner gave his students a problem that was supposed to be very time-consuming. He asked them to add the long series of numbers from 1 to 100. Gauss already knew, at that age, that he didn’t have to add up all the numbers, as his schoolmates were painstakingly doing. Instead, he apparently arranged the numbers 0 through 100 in two rows, one under the other. The top row arranged the numbers going “forward” from left to right; the second row arranged them going “backward” from left to right:

  By arranging all the numbers from 0 to 100 thus, the young genius could see that the sum of all the pairs was, identically, 100 and that there were 101 such pairs. The total of 101 × 100 equaled 10,100, and since that number represented twice the sum of the integers from 1 to 100, he divided the sum by 2, which gave him 5,050. Generalizing this procedure gives us the rule for the sum of integers from 1 to any given number, n: 1 + 2 + 3 +…+ n = n(n+1)/2.

  Carl Friedrich Gauss, depicted in this lithograph at the age of fifty, is known for the rigor he brought to the study of mathematics.

  Buettner, known for terrorizing his students, apparently softened when he witnessed the amazing performance of the youngest student in the class and, as a gift, gave him an expensive new book on mathematics. “He is beyond me,” he said of Gauss. “I can teach him nothing more.”12 Buettner’s assistant, Johann Martin Bartels, who was only seventeen years old at the time, became a close friend of Gauss. Together they studied infinite series, which brought Gauss to derive a proof of the binomial theorem when the exponent is not an integer. This was the first rigorous proof in his young career.

/>   MATHEMATICAL RIGOR

  According to the historian of mathematics Eric T. Bell, it was Gauss alone who imposed on post-Greek mathematics the rigor we see in it today.13 The question of “rigor” has been key to the nature of mathematics throughout its history. The Babylonians and Egyptians had no interest in proofs, only results. And their solutions—perhaps with some exceptions, such as the Pythagorean triples—had been centered on applications. With their invention of pure mathematics (i.e., mathematics for its own sake), the Greeks also invented rigor. Rigor is the requirement that all statements be proved conclusively, in a way that is absolute (given basic assumptions), and that cannot be denied or destroyed with some counterexample.

  Many of the Greek mathematicians followed this rigorous approach. An excellent example is Euclid’s rigorous proof regarding prime numbers, which is 2,300 years old. Theorem: There are infinitely many prime numbers. Proof: Assume the theorem is false. Then there must be a largest prime; call it p. Let the product of all the prime numbers, plus 1, be represented by the number n = 2 × 3 × 5 × 7 × 11 × 13 × … × p + 1. Is this number a prime? If it is, then you’ve just exhibited a prime number greater than p. If n is not a prime, then by definition it is divisible by one of the primes 2, 3, 5, 7, 11, 13, …, p. Call that particular prime number q. Dividing n by q cannot result in an integer, since you will have the added factor 1/q, so n is not divisible by any prime number and must therefore be prime. Thus the “contradiction” (reductio ad absurdum, as the ancients called it) establishes the truth of the theorem.

  Rigor, for the most part, was not a major concern in seventeenth-century mathematics. Gauss changed that. His study of infinite series taught him that if one is not careful with assumptions and derivations, one can obtain absurd results. One such example is the absurd conclusion he obtained when he attempted a careless derivation, getting the statement 0 = 1. In order to avoid obtaining such results, which sometimes occur if one is not careful when dealing with the infinite, Gauss imposed strict standards on his work very early on—a practice that became a staple of modern mathematics.

  Johann Bartels, coming from an affluent and well-connected family in Brunswick, introduced his brilliant young friend to the Duke of Brunswick, Carl Wilhelm Ferdinand, in 1792. This encounter changed Gauss’s life. His father, as always, was completely uninterested in education and would have done nothing to further his son’s, despite pleas from his wife and her brother. In any case, the family was poor and could not afford it. Noblemen supported the arts and culture in their realms, often taking artists, writers, intellectuals, and jesters into their entourages and under their sponsorship, but to offer full support for a schoolboy was unusual. Fortunately for Gauss, the Duke of Brunswick was very taken with the amazing intellectual abilities of the fourteen-year-old and, surprisingly, offered to pay for his entire education.

  Gauss attended the Collegium Carolinum (Caroline College) in Brunswick, entering at age fifteen and graduating at age eighteen. Of course, he pursued mathematics on his own because he was so far beyond the pedestrian mathematics taught at the college. When he graduated, the duke continued to support his education at the University of Göttingen starting in 1795. By the time he entered the university, Gauss had already invented the powerful technique we know today as the method of least squares, one of the most important principles in mathematical statistics. Least squares is a routine that uses calculus to find the minimum possible sum of the squared errors of a statistical tool for analyzing data. For example, in order to find the best straight line to describe the movement of a set of points representing observations on two variables, we use Gauss’s method of least squares. The routine is used everywhere in theoretical statistics, and his analysis of how errors are actually distributed later led Gauss to derive the Gaussian law of error, also known as the Gaussian distribution. More commonly, we refer to it as the normal distribution, or bell curve.

  Given that he had already made significant contributions to mathematics, it is surprising that Gauss had not yet made up his mind on what to study. He was still attracted to language and considered concentrating on philology. Only in March of the following year did he decide to major in mathematics. In 1796 he began to keep a mathematical diary, in which he wrote until 1814. The diary contains only nineteen pages, but those nineteen pages exhibit 146 concisely stated key results that the young mathematician derived in a number of mathematical fields, including number theory, elliptic functions, analysis, and other areas. In 1898, when a grandson of Gauss first lent the diary to the Royal Society of Göttingen for analysis, it revealed that Gauss placed a priority on achieving as many important discoveries in mathematics as he possibly could. Like Newton before him, Gauss seemed to be reluctant, for some reason, to publish many of his great findings, so his discoveries came to light only after his death. One example of a cryptic result is Gauss’s July 10, 1796, entry, completed just as he finished his first year at the university.14 The sentence reads:

  What does this mean? Gauss had discovered that every positive integer is the sum of, at most, three triangular numbers. Recall the triangular numbers of Pythagoras (Tn), each of which is obtained by adding, to the previous number (Tn-1), a number greater by one than the difference between the previous number and the one before it (Tn-1 – Tn-2 + 1). Thus, in the triangular sequence 1, 3, 6, 10, 15, 21, 28, 36, 45, …, the next number is 55. We can test Gauss’s finding with examples—29 = 28 + 1; 40 = 36 + 3 + 1—but it is very possible that Gauss, the great rigorous mathematician, had a proof of this relationship that never came to light.

  Historian Eric T. Bell quotes Gauss as saying in old age that he had never published the findings in his notebook because, as a young man, he was deluged by so many ideas and had barely any time to record them. Besides, he didn’t care whether other people saw them or not; he was pursuing theorems for his own amusement. Apparently, the outcomes resulted from weeks of research, so he likely possessed proofs, but rather than bothering to write them down, he was always advancing to the next great mathematical puzzle. Gauss preferred to revise assiduously any mathematical result he did publish, provide careful proofs, and make sure every detail was in place. His published works were “few, but ripe.”15 While studying at Göttingen, Gauss wrote a book called Disquisitiones Arithmeticae (Arithmetical Investigations) about number theory, dedicating the book to his patron, the Duke of Brunswick. The book was a masterpiece. Before he had reached the age of twenty-one, he had completed what would be considered one of the most important mathematics books ever written.

  At the university, Gauss also began a lifelong friendship with a Hungarian student named Wolfgang Bolyai, whom he described as “the rarest spirit I ever knew.”16 Since the age of twelve, Gauss had been obsessed with Euclid’s geometry—especially the fifth postulate. He tried to prove the fifth postulate based on the prior four, as many had attempted to do before him, but eventually came to the conclusion that this was not possible. His friend Bolyai was trying to do the same thing when Gauss informed him that he knew that attempts at a proof were futile and that, in fact, geometries other than Euclid’s were possible!

  Through his good friend’s son, Janos Bolyai (1802–60), born six years later, Gauss’s idea would eventually lead to a generalization of Euclid’s work to what we now call non-Euclidean geometries.17

  THERE ARE DIFFERENT KINDS of non-Euclidean geometries. In elliptic geometry it is acceptable for “parallel” lines to meet. For example, two longitudinal lines of the earth are “parallel” but meet at the poles. In hyperbolic geometry there could be infinitely many parallels to a single line, as on a saddle. The picture at the top of page illustrates both elliptic and hyperbolic geometries, as contrasted with Euclidean.

  Notice that on the sphere, triangles are “fat”—i.e., the sum of their angles is more than 180 degrees, the sum of angles of a triangle on Euclid’s plane. On a saddle, however, triangles are “thin”—i.e., the sum of their angles is less than 180 degrees. A circle on the sphere has r
atio of circumference to diameter less than π, and on a saddle that ratio is greater than π, as can be seen by imagining a circle, and its diameter, placed on each of the geometrical objects in the picture above. These geometries are real and mathematically consistent. Had Euclid lived in a more varied terrain than the flat plains of Egypt, perhaps the curvature of the earth would have led him to adapt his own geometry to realms of more interesting curvature—and perhaps to discard his fifth postulate.

  In elliptical geometry (illustrated by the sphere at left), it is possible for two “parallel” lines to meet; in hyperbolic geometry (illustrated by the saddlelike shape at right), the sum of the angles of a triangle is less than 180 degrees.

  In 1799, when he was twenty-two, Gauss finished his doctoral dissertation at the University of Helstedt. His dissertation was the first rigorous proof of the Fundamental Theorem of Algebra.18 Gauss proved that all the roots of an algebraic equation are what we now call complex numbers—numbers of the form a + bi, where i is the square root of –1. He was the first mathematician to propose the interpretation of complex numbers as points on the complex plane, thus giving them a precise geometrical meaning.

  In Gauss’s complex plane, the real part of a complex number is represented by a displacement along the x axis, and the imaginary part by a displacement along the y axis.

  The complex field is the smallest field in which all quadratic equations have their solutions. (Solutions of quadratic equations can be real, imaginary, or a mixture of both, so they all “live” on the complex plane.) Over his lifetime, Gauss provided four different proofs of the Fundamental Theorem of Algebra, the last one when he was seventy years old.19 He also launched the field of differential geometry.