Why Beauty is Truth Page 9

  As Gauss developed his ideas in number theory and analysis, he became more and more dissatisfied that no one had proved this assumption. Characteristically, he came up with a proof. It was complicated and curiously indirect: any competent mathematician could be convinced that it was correct, but no one could guess how Gauss had come up with it in the first place. The fox of mathematics was wielding his tail with a vengeance.

  The Latin title of Gauss’s dissertation translates as “A new proof that every rational integral function of one variable can be resolved into real factors of the first or second degree.” Unwrapping the jargon of the period, the title asserts that every polynomial (with real coefficients) is a product of terms that are either linear or quadratic polynomials.

  Gauss used the word “real” to make it clear that he was working within the traditional number system, in which negative quantities lack square roots. Nowadays we would state Gauss’s theorem in a logically equivalent but simpler form: every real polynomial of degree n has n real or complex roots. But Gauss chose his terminology carefully, so that his work did not rely on the still puzzling system of complex numbers. Complex roots of a real polynomial can always be combined in pairs to yield real quadratic factors, whereas linear factors correspond to real roots. By phrasing the title in terms of these two types of factor (“factors of the first or second degree”), Gauss circumvented the contentious issue of complex numbers.

  One word in the title was unjustified: “new,” which implies that there are “old” proofs. Gauss gave the first rigorous proof of this basic theorem in algebra. But to avoid offending illustrious predecessors who had already claimed proofs—all of them faulty—Gauss presented his breakthrough as merely the latest proof, using new (that is, correct) methods.

  This theorem came to be known as the Fundamental Theorem of Algebra. Gauss considered it so important that he gave four proofs in all, the last at the age of 70. He personally had no qualms about complex numbers: they played a big role in his thinking, and he subsequently developed his own explanation of their meaning. But he disliked controversy. Over the years he suppressed many of his most original ideas—non-Euclidean geometry, complex analysis, and a rigorous approach to complex numbers themselves—because he did not want to attract what he referred to as “the cries of the Boeotians.”

  Gauss did not confine himself to pure mathematics. Early in 1801, the Italian priest and astronomer Giuseppe Piazzi had discovered a new planet, or so he thought—a faint patch of light in his telescope that moved against the background of the stars from one night to the next, a sure sign that it was a body in the solar system. It was duly given the name Ceres, but it was actually an asteroid, the first to be found. Having found the new world, Piazzi promptly lost it in the glare of the Sun. He had made so few observations that astronomers hadn’t been able to work out the new body’s orbit and worried that they wouldn’t be able to locate it again when it emerged from behind the Sun.

  This was a problem worthy of Gauss, and he set to with a will. He invented better ways to determine orbits from small numbers of observations, and predicted where Ceres would reappear. When it duly did so, Gauss’s fame spread far and wide. The explorer Alexander von Humboldt asked Pierre-Simon de Laplace, an expert in celestial mechanics, to name the greatest mathematician in Germany, and got the reply “Pfaff.” When a startled Humboldt asked, “What about Gauss?” Laplace replied, “Gauss is the greatest mathematician in the world.”

  Unfortunately, this newfound celebrity diverted him from pure mathematics into lengthy calculations in celestial mechanics—generally felt to be a waste of his considerable talents. It’s not that celestial mechanics was unimportant, but other, less able mathematicians could have done the same work. On the other hand, it also set him up for life. Gauss had been looking for a prominent position that offered the opportunity for public service, to reward his sponsor, the duke. His work on Ceres landed him the directorship of the Göttingen observatory, a post that he held for the rest of his academic life.

  He married Johanna Osthoff in 1805. Writing to Bolyai, he described his new wife: “The beautiful face of a Madonna, a mirror of peace of mind and health, tender, somewhat fanciful eyes, a blameless figure—this is one thing; a bright mind and an educated language—this is another; but the quiet, serene, modest and chaste soul of an angel who can do no harm to any creature—that is the best.” Johanna bore him two children, but in 1809 she died in childbirth, and a devastated Gauss “closed her angelic eyes in which I have found heaven for the last five years.” He became lonely and depressed, and life for him was never quite the same. He did find a new wife, Johanna’s best friend Minna Waldeck, but the marriage was not terribly happy despite the birth of three more children. Gauss was always arguing with his sons and telling his daughters what to do, and the boys got so fed up that they left Europe for the United States, where they prospered.

  Soon after taking up the directorship at Göttingen, Gauss returned to an old idea, the possibility of a new type of geometry that satisfies all of Euclid’s axioms except the parallel axiom. He eventually became convinced that logically consistent non-Euclidean geometries are possible, but never published his results for fear that they would be considered too radical. János Bolyai, the son of his old friend Wolfgang, later made similar discoveries, but Gauss felt unable to praise the work because he had anticipated most of it. Still later, when Nikolai Ivanovich Lobachevsky independently rediscovered non-Euclidean geometry, Gauss had him made a corresponding member of the Göttingen Academy, but again offered no public praise.

  Years later, as mathematicians studied these new geometries in more detail, they came to be interpreted as geometries of “geodesics”—shortest paths—on curved surfaces. If the surface had constant positive curvature, like a sphere, the geometry was called elliptic. If the curvature was constant and negative (shaped like a saddle near any point) the geometry was hyperbolic. Euclidean geometry corresponded to zero curvature, flat space. These geometries could be characterized by their metric, the formula for the distance between two points.

  These ideas may have led Gauss to a more general study of curved surfaces. He developed a beautiful formula for the amount of curvature, and proved that it gave the same result in any coordinate system. In this formulation, the curvature did not have to be constant: it could vary from one place to another.

  In a move that is not unusual in mathematics, Gauss in middle age turned to practical applications. He assisted several surveying projects, the biggest being the triangulation of the region of Hanover. He did a lot of fieldwork, followed by data analysis. To aid the work he invented the heliotrope, a device for sending signals by reflected light. But when his heart began to show signs of failure, he stopped surveying and decided to spend his remaining years in Göttingen.

  During this unhappy period a young Norwegian named Abel wrote to him about the impossibility of solving the quintic equation by radicals, but received no reply. Probably Gauss was too depressed even to look at the paper.

  Around 1833, he became interested in magnetism and electricity, collaborating with the physicist Wilhelm Weber on a book, General Theory of Terrestrial Magnetism, published in 1839. They also invented a telegraph linking Gauss’s observatory to the physics laboratory where Weber worked, but the wires kept breaking, and other inventors came up with more practical designs. Then Weber was fired from Göttingen along with six others because they refused to swear allegiance to the new king of Hanover, Ernst August. Gauss was very upset by this, but his political conservatism and reluctance to make waves prevented him from raising any public protest, though he may have made efforts behind the scenes on Weber’s behalf.

  In 1845, Gauss produced a report on the pension fund for widows of Göttingen professors, examining the likely effect of a sudden increase in the number of members. He invested in railway and government bonds and amassed a tidy fortune.

  After 1850, troubled by the onset of heart problems, Gauss cut back on work. The
most important event of that period, for our story, was the habilitation thesis of his student Georg Bernhard Riemann. (In the German academic system, habilitation is the next step up after a PhD.) Riemann generalized Gauss’s work on surfaces to multidimensional spaces, which he called “manifolds.” In particular, he extended the concept of a metric, and found a formula for the curvature of a manifold. In effect, he created a theory of curved multidimensional spaces. Later this idea was to prove crucial in Einstein’s work on gravity.

  Gauss, now being regularly seen by his doctor, attended Riemann’s public lecture on the topic and was impressed. As his health deteriorated further, he spent more and more time in bed but continued to write letters, read, and manage his investments. Early in 1855, Gauss died peacefully in his sleep, the greatest mathematical mind the world has ever known.

  6

  THE FRUSTRATED DOCTOR AND THE SICKLY GENIUS

  The first significant advance over Cardano’s The Great Art came about halfway through the eighteenth century. Although the Renaissance mathematicians could solve cubics and quartics, their methods were basically a series of tricks. Each trick worked, but more, it seemed, by a series of coincidences than for any systematic reason. That reason was finally pinned down around 1770 by two mathematicians: Joseph-Louis Lagrange, a native of Italy who always considered himself French, and Alexandre-Théophile Vandermonde, who definitely was.

  Vandermonde was born in Paris in 1735. His father wanted him to become a musician, and Vandermonde became proficient on the violin and followed a musical career. But in 1770 he became interested in mathematics. His first mathematical publication was about symmetric functions of the roots of a polynomial—algebraic formulas like the sum of all the roots, which do not change if the roots are interchanged. Its most original contribution was to prove that the equation xn – 1 = 0, associated with the regular n-gon, can be solved by radicals if n is 10 or smaller. (Actually, it is solvable by radicals for any n.) The great French analyst Augustin-Louis Cauchy later cited Vandermonde as the first to realize that symmetric functions can be applied to the solution of equations by radicals.

  In Lagrange’s hands, this idea would form the starting point for an attack on all algebraic equations.

  Lagrange was born in the Italian city of Turin and baptized Giuseppe Lodovico Lagrangia. His family had strong French links—his great-grandfather had been a captain in the French cavalry before moving to Italy to serve the duke of Savoy. When Giuseppe was quite young he started using Lagrange as a surname, but combined it with Lodovico or Luigi as his first name. His father was the treasurer of the Office of Public Works and Fortifications in Turin; his mother, Teresa Grosso, was a doctor’s daughter. Lagrange was their first child out of an eventual total of eleven, but only two survived beyond childhood.

  Although the family was in the upper levels of Italian society, they were strapped for cash, thanks to some bad investments. They decided that Lagrange should study law, and he attended the College of Turin. He enjoyed law and classics but found the mathematics classes, which consisted largely of Euclidean geometry, rather boring. Then he came across a book on algebraic methods in optics by the English astronomer Edmond Halley, and his opinion of mathematics changed dramatically. Lagrange was set on the course that would dominate his early research: the application of mathematics to mechanics, especially celestial mechanics.

  He married a cousin, Vittoria Conti. “My wife, who is one of my cousins and who even lived for a long time with my family, is a very good housewife and has no pretensions at all,” he wrote to his friend Jean le Rond D’Alembert, also a mathematician. He also confided that he did not want any children, an ambition he achieved.

  Lagrange took a position in Berlin, wrote numerous research papers, and won the French Academy’s annual prize on several occasions—sharing the 1772 prize with Euler, winning the 1774 prize for work on the dynamics of the Moon and the 1780 prize for work on the influence of planets on cometary orbits. Another of his loves was number theory, and in 1770 he proved a classic of the genre, the Four Squares Theorem, which asserts that every positive whole number is a sum of four perfect squares. For instance, 7 = 22 + 12 + 12 + 12, 8 = 22 + 22 + 02 + 02, and so on.

  He became a member of the French Academy of Sciences and moved to Paris, where he remained for the rest of his life. He believed that it was wise to obey the laws of the country where you lived even if you disagreed with them, a point of view that probably helped him escape the fate of many other intellectuals during the French Revolution. In 1788 Lagrange published his masterpiece, Analytical Mechanics, which rewrote mechanics as a branch of analysis. He was proud that his massive book contained no diagrams whatsoever; in his view this made the logic more rigorous.

  He married his second wife, Renée-Françoise-Adélaide Le Monnier, the daughter of an astronomer, in 1792. In August 1793, during the Reign of Terror, the Academy was shut down, and the only part that remained active was the commission on weights and measures. Many leading scientists were removed—the chemist Antoine Lavoisier, the physicist Charles Augustin Coulomb, and Pierre Simon Laplace. Lagrange became the new chairman of weights and measures.

  At that point his Italian birth became a problem. The revolutionary government passed a law requiring any foreigner born in an enemy nation to be arrested. Lavoisier, who at that point retained some influence, arranged for Lagrange to be exempted from the new law. Soon afterward, a revolutionary tribunal condemned Lavoisier to death; he was guillotined the next day. Lagrange remarked, “It took only a moment to cause this head to fall, and a hundred years will not suffice to produce its like.”

  Under Napoleon, Lagrange was granted several honors: the Legion of Honor and Count of the Empire in 1808, and the Grand Cross of the Imperial Order of the Reunion in 1813. A week after receiving the Grand Cross, he was dead.

  In 1770, the same year that he discovered his Four Squares Theorem, Lagrange embarked upon a vast treatise on the theory of equations, saying, “I propose in this memoir to examine the various methods found so far for the algebraic solution of equations, to reduce them to general principles, and to explain a priori why these methods succeed for the third and fourth degree, and fail for higher degrees.” As Jean-Pierre Tignol put it in his book Galois’s Theory of Algebraic Equations, Lagrange’s “explicit aim is to determine not only how these methods work, but why.”

  Lagrange reached a much deeper understanding of Renaissance methods than the methods’ inventors had; he even proved that the general scheme that he had found to explain their success could not be extended to the fifth degree or higher. Yet he failed to take the further step of wondering whether any solution was possible in those cases. Instead, he tells us that his results “will be useful to those who will want to deal with the solution of the higher degrees, by providing them with various views to this end and above all by sparing them a large number of useless steps and attempts.”

  Lagrange had noticed that all of the special tricks employed by Cardano, Tartaglia, and others were based on one technique. Instead of trying to find the roots of the given equations directly, the idea was to transform the problem into the solution of some auxiliary equation whose roots were related to the original ones, but different.

  The auxiliary equation for a cubic was simpler—a quadratic. This “resolvent quadratic” could be solved by the Babylonian method; then the solution of the cubic could be reconstructed by taking a cube root. This is exactly the structure of Cardano’s formula. For a quartic, the auxiliary equation was also simpler—a cubic. This “resolvent cubic” could be solved by Cardano’s method; then the solution of the quartic could be reconstructed by taking a fourth root—that is, a repeated square root. This is exactly the structure of Ferrari’s formula.

  We can imagine Lagrange’s growing excitement. If the pattern continued, then the quintic equation would have a “resolvent quartic”: solve that by Ferrari’s method and then take a fifth root. And the process would continue, with the sextic havi
ng a resolvent quintic, solvable by what would be known as Lagrange’s method. He would be able to solve equations of any degree.

  Harsh reality brought him down to earth. The resolvent equation for the quintic was not a quartic but an equation of higher degree, a sextic. The method that had simplified the cubic and quartic equations complicated the quintic.

  Mathematics does not progress by replacing difficult problems by even harder ones. Lagrange’s unified method failed on the quintic. Still, he had not proved the quintic to be unsolvable, because there might exist different methods.

  Why not?

  To Lagrange, this was a rhetorical question. But one of his successors took it seriously, and answered it.

  His name was Paolo Ruffini, and when I say that he “answered” Lagrange’s rhetorical question, I am cheating slightly. He thought he had answered it, and his contemporaries never found anything wrong with his answer—partly because they never took his work seriously enough to really try. Ruffini spent his life believing that he had proved the quintic unsolvable by radicals. Only after his death did it turn out that his proof had a significant gap. It was easily overlooked among his pages and pages of intricate calculations; it was an “obvious” assumption, one that he had never even noticed he was making.

  As every professional mathematician knows from bitter experience, it is very difficult to notice that you are making an unstated assumption, precisely because it is unstated.

  Ruffini was born in 1765, the son of a doctor. In 1783 he enrolled at the University of Modena, studying medicine, philosophy, literature, and mathematics. He learned geometry from Luigi Fantini and calculus from Paolo Cassiani. When Cassiani moved on to a post with the Este family, managing their vast estates, Ruffini, though still a student, took charge of Cassiani’s analysis course. He obtained a degree in philosophy, medicine, and surgery in 1788, adding a mathematics degree in 1789. Soon afterward, he took over a professorship from Fantini, whose eyesight was failing.