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  Events interfered with his academic work. Napoleon Bonaparte defeated the armies of Austria and Sardinia in 1796, turned his sights towards Turin, and captured Milan. Soon he had occupied Modena, and Ruffini was forced to become involved in politics. He had planned to go back to the university in 1798, but refused, on religious grounds, to swear allegiance to the republic. The resulting lack of employment left him more time to carry out his researches, and he focused on the vexed question of the quintic.

  Ruffini convinced himself that there was a good reason why no one had managed to find a solution: there wasn’t one. Specifically, there was no formula involving nothing more esoteric than radicals that would solve the general quintic. In his two-volume tome General Theory of Equations, published in 1799, he claimed to be able to prove this, asserting, “The algebraic solution of general equations of degree greater than four is always impossible. Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume. The immortal Lagrange, with his sublime reflections, has provided the basis of my proof.”

  The proof occupied more than 500 pages of largely unfamiliar mathematics. Other mathematicians found it somewhat daunting. Even today, no one is keen to wade through a very long and technical proof unless there is very good reason to do so. If Ruffini had announced a solution to the quintic, his peers would surely have made the effort. But you can understand their reluctance to devote hundreds of hours to the claim of a negative result.

  Especially when it might be wrong. Few things are more annoying than finding an error on page 499 of a 500-page mathematics book.

  Ruffini sent Lagrange a copy in 1801, and after a few months’ silence he sent another copy, with a note: “If I have erred in any proof, or if I have said something which I believed new, and which is in reality not new, finally if I have written a useless book, I pray you point it out to me sincerely.” Still no reply. He tried again in 1802. Nothing.

  Several years passed without the recognition Ruffini felt was his due. Instead, vague rumors circulated, hinting that there were mistakes in his “proof,” but since no one said what those mistakes might be, Ruffini was unable to defend himself. Eventually, he decided, no doubt correctly, that his proof was too complicated, and set about finding something simpler. He achieved this in 1803, writing, “In the present memoir, I shall try to prove the same proposition with, I hope, less abstruse reasoning and with complete rigor.” The new proof fared no better. The world wasn’t ready for Ruffini’s insights or for the further proofs he published in 1808 and 1813. He never stopped trying to get his work recognized by the mathematical community. When Jean Delambre, who predicted the position of the planet Uranus, wrote a report on the state of mathematics since 1789, he included the sentence, “Ruffini proposes to prove that solving the quintic is impossible.” Ruffini promptly replied, “I not only proposed to prove, but in reality did prove.”

  To be fair, a few mathematicians were happy with Ruffini’s proof. Among them was Cauchy, who had a pretty poor track record when it came to giving credit where it was due, unless it was due to himself. In 1821, he wrote to Ruffini, “Your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgment, proves completely the impossibility of solving algebraically equations of higher than the fourth degree.” But by then the praise was far too late.

  Around 1800 Ruffini started teaching applied mathematics in the city’s military school. He continued to practice medicine, looking after patients from the poorest to the richest in society. In 1814, after the fall of Napoleon, he became rector of the University of Modena. The political situation was still extremely complex, and despite his personal skills, the great respect in which he was held, and his reputation for honesty, his time as rector must have been very difficult.

  Simultaneously, Ruffini held the chairs of applied mathematics, practical medicine, and clinical medicine in the University of Modena. In 1817, there was a typhus epidemic and Ruffini continued to treat his patients until he caught the disease himself. He survived but never fully regained his health, and in 1819 he gave up his chair of clinical medicine. But he never gave up his scientific work, and in 1820 he published a scientific article on typhus based on his own experience as both physician and patient. He died in 1822, barely a year after Cauchy had written to praise his work on the quintic.

  One reason Ruffini’s work was not well received may have been its novelty. Like Lagrange, he based his investigations on the concept of a “permutation.” A permutation is a way to rearrange some ordered list. The most familiar example is shuffling a pack of cards. The usual aim here is to achieve some random—that is, unpredictable—order. The number of different permutations of a pack of cards is huge, so the chance of predicting the outcome of random shuffling is negligible.

  Permutations arise in the theory of equations because the roots of a given polynomial can be considered as a list. Some very basic features of equations are directly related to the effect of shuffling that list. The intuition is that the equation “does not know” the order in which you listed its roots, so permuting the roots should not make any important difference. In particular, the coefficients of the equation should be fully symmetric expressions in the roots—expressions that do not change when the roots are permuted.

  But as Lagrange had appreciated, some expressions in the roots may be symmetric with respect to some permutations, but not others. These “partially symmetric” expressions are intimately associated with any formula for solving the equation. This feature of permutations was familiar to Ruffini’s peers. Much less familiar was Ruffini’s systematic use of another of Lagrange’s ideas: that you can “multiply” two permutations to get another one by performing them in turn.

  Consider the three symbols a, b, c. There are six permutations: abc, acb, bac, bca, cab, and cba. Take one of them, say cba. At first sight, this is just an ordered list formed from the three symbols. But we can also think of it as a rule for rearranging the original list abc. In this case, the rule is “reverse the order.” And we can apply this rule not just to that list but to any list. Apply it to bca, say, and you get acb. So there is a sense in which cba × bca = acb.

  This idea, which is central to our story, probably makes more sense if we draw some diagrams. Here are two diagrams for the permutations that rearrange abc into cba and bca:

  Two permutations of the symbols a, b, c.

  We can combine the two rearrangements into one, by stacking these pictures on top of each other. There are two ways to do this:

  Multiplying permutations. The result depends on which comes first.

  Now we can read off the result of “multiplying” the two permutations by writing down the bottom row, which here (left-hand picture) is acb. With this definition of “multiplication” (which is not the usual concept for multiplying numbers) we can make sense of the statement cba × bca = acb. The convention is that the first permutation in the product goes on the bottom of the stack. It matters, because we get a different answer if we swap the two layers of the stack. The right-hand picture shows that when the permutations are multiplied in the opposite order, the result is bca × cba = bac.

  The essence of Ruffini’s impossibility proof was to develop conditions that must be satisfied by any quintic whose roots can be expressed by radicals. If the general quintic does not satisfy these conditions, then it does not have that kind of root—and thus can’t be solved by any natural extension of the methods that worked for the cubic and quartic.

  Taking a leaf out of Lagrange’s book, Ruffini homed in on symmetric functions of the roots and their relation to permutations. The quintic has five roots, and there are 120 permutations of five symbols. Ruffini realized that this system of permutations would have to possess certain structural features, inherited from any hypothetical formula for solutions of the quintic. If those features were absent, ther
e could be no such formula. It was a bit like hunting a tiger in a muddy jungle. If there really was a tiger present, it would leave clear paw-prints in the mud. No paw-prints, no tiger.

  By exploiting mathematical regularities of this new form of multiplication, Ruffini was able to prove—to his own satisfaction, at least—that the multiplicative structure of the 120 permutations is inconsistent with the symmetric functions that have to exist if the equation can be solved by radicals. And he did achieve something significant. Before Ruffini started working on the quintic, virtually every mathematician in the world was convinced that this equation could be solved; the only question was how. One exception was Gauss, who dropped hints that he thought no solution existed—but he also remarked that it wasn’t a very interesting question, one of the few times his instincts let him down.

  After Ruffini there seems to have been a general feeling that the quintic is not solvable by radicals. Very few thought Ruffini had proved this—but his work certainly made a lot of people feel rather doubtful that radicals were up to the job. This change of perception had an unfortunate side effect: mathematicians became much less interested in the whole issue.

  Ironically, it later emerged that Ruffini’s work had a major gap, but no one spotted it at the time. The skepticism of his contemporaries turned out to be justified, in a way. But the real breakthrough was the method: Ruffini found the correct strategy; he just didn’t use quite the right tactics. The subject needed a strategist who could also pay scrupulous attention to the minutiae of tactics. Now it got one.

  After years of carrying out the good Lord’s work, without complaint, as a pastor in some of the poorest and most remote regions of the Norwegian mountains, in 1784 Hans Mathias Abel received his just reward. He got himself appointed to the parish of Gjerstad, near the southern coast of Norway, not far from Oslo Fjord. Gjerstad wasn’t exactly wealthy, but it was much richer than the places where he had previously ministered. His family finances would improve dramatically.

  Spiritually, Pastor Abel’s task was the same as ever: to look after his flock and do his best to keep them happy and virtuous. He came from a well-to-do family. His Danish great-grandfather had been a merchant who did a lucrative trade supplying the Norwegian army. His father, also a merchant, had been an alderman in the town of Bergen. Hans was proud but modest, not particularly intelligent but far from stupid, and prepared to speak his mind whatever the cost.

  To help feed the poor of the parish, he grew new types of plant on his farm: flax, for making linen, and above all a new type of root vegetable, the ground apple, otherwise known as the potato. He wrote poetry, pottered about collecting information for a history of the area, and lived in harmony with his wife Elisabeth. His house was famous for the quality of its food; alcohol was never served. Drinking was a major social problem in Norway, and the pastor was determined to set his flock an example—though on one occasion he arrived in church as drunk as a newt, to show his parishioners how demeaning drunkenness was. He had two children, an unusually small family for the times: a daughter Margaretha, and a son Søren.

  Margaretha was unexceptional, never married, and lived most of her life with her parents. Søren was altogether different: quick, intelligent, and original, with a taste for high society. He lacked his father’s composure and sense of duty, and suffered for it. Still, he followed his father’s profession, becoming first curate, then pastor; married Anne Marie Simonsen, the daughter of a family friend; and accepted a post in Finnøy, on the southwest coast. “The people round here are superstitious, but are filled with knowledge of the Bible,” he wrote. “They support every erroneous opinion by misunderstood divine authority.” Nevertheless, he enjoyed the job.

  In 1801, Søren wrote to a friend, “My domestic joy has recently been increased, for on the third day of Christmas my wife presented me with a healthy son.” This was Hans Mathias. A brother, Niels Henrik, arrived in the summer of 1802. From day one Niels suffered from ill health, and his mother had to spend a lot of time looking after him.

  Military tensions were running high in Europe, and the combined state of Norway-Denmark was sandwiched between the major powers of England and France. Napoleon wanted to ally it with his cause, so when Britain came to an agreement with Sweden, Norway-Denmark instantly became an enemy of the British, who invaded. After three days, Norway-Denmark surrendered to save Copenhagen from destruction. Later, when Napoleon’s grip on power was fading, his aide Jean Baptiste Bernadotte became king of Sweden. When Norway was ceded to Sweden, the Norwegian parliament, the Storting, was forced to accept Bernadotte as monarch.

  The two boys were sent to the Cathedral School in Oslo in 1815. The mathematics teacher, Peter Bader, was the sort who motivated his students with serious physical violence. Nevertheless, both boys did well. Then in 1818, Bader gave one of the pupils—the son of a representative to the Storting—such a beating that the boy died. Amazingly, Bader was not tried, but he was replaced as mathematics teacher by Bernt Holmboe, who had been assistant to Christoffer Hansteen, the applied mathematics professor. This marked a turning point in Niels’s mathematical career, because Holmboe allowed his pupils to tackle interesting problems outside the usual syllabus. Niels was permitted to borrow classic textbooks, among them some by Euler. “From now on,” Holmboe later wrote, “[Niels] Abel devoted himself to mathematics with the most fervent eagerness and progressed in his science with the speed characteristic of a genius.”

  Shortly before finishing at school, Niels convinced himself that he had solved the quintic equation. Neither Holmboe nor Hansteen could find a mistake, so they transmitted the calculations to Ferdinand Degen, a prominent Danish mathematician, for possible publication by the Danish Academy of Science. Degen, too, found no errors in the work, but being an experienced hand who knew a trick or two, he asked Niels to try out the calculations on some specific examples. Niels quickly realized that something was amiss; he was disappointed, but relieved that he had not been allowed to make a fool of himself by publishing an erroneous result.

  Søren’s ambition and lack of tact now combined with embarrassing results. He read out a statement accusing two Storting representatives of unjustly imprisoning the manager of an ironworks, owned by one of them. This attack on their integrity created an uproar. It then transpired that the man concerned was unreliable, but Søren refused to apologize. Depressed and unhappy, he drank himself to death. At the funeral, Søren’s widow, Anne Marie, became extremely drunk and took her favorite servant to bed. Next morning, she received several visiting officials—still in bed, with her lover beside her. An aunt wrote, “The poor boys, I feel sorry for them.”

  Niels graduated from the Cathedral School in 1821 and took the entrance examination to the University of Christiania (now Oslo). He received the highest possible grade in arithmetic and geometry and good grades in the rest of mathematics, and did terribly in everything else. Now desperately poor, he applied for a grant that would give him free accommodation and wood for the fire. He also sought a grant for living expenses, and some of the professors, recognizing his unusual talent, gave money to create a fellowship for him. Thus provided for, Niels devoted himself to mathematics and to solving the quintic, determined to make good his previous abortive attempt.

  In 1823 Niels worked on elliptic integrals, an area of analysis that would be his most lasting monument, outclassing even his work on the quintic. He tried to prove Fermat’s Last Theorem but found neither proof nor disproof, though he did show that any example that disproves the theorem must involve gigantic numbers.

  In the summer of that year, he went to a ball, met a young woman, and asked her to dance. After several failed attempts, they both burst out laughing—neither of them had the foggiest idea how to dance. The lady was Christine Kemp, universally known as “Crelly,” the daughter of a war commissar. Like Niels she had no money, and earned a living as a private tutor in everything from needlework to science. “She is not beautiful, has red hair and freckles, but she is a wond
erful girl,” he wrote. They fell in love.

  These events gave Niels’s mathematics a boost. Toward the end of 1823, he proved the quintic’s impossibility—and unlike Ruffini’s near miss, his did not have any gaps. Its strategy was similar to Ruffini’s but with better tactics. Initially, Niels didn’t know about Ruffini’s work. Later, he certainly must have known of it because he alludes to its incompleteness. But even Niels did not put his finger on the precise gap in Ruffini’s proof—even though his method turned out to be just what was needed to bridge the gap.

  Niels and Crelly became engaged. To marry his sweetheart, Niels had to get a job—which meant his talents had to be recognized by Europe’s leading mathematicians. Publishing his theory would not be enough: he had to beard the lions in their dens. And to do that, he needed enough money to travel.

  After much effort, the University of Christiania was persuaded to grant Niels enough money for a research visit to Paris, where he would meet some of the world’s leading mathematicians. In preparation for the trip, he decided that he needed printed copies of his best work. He believed that his impossibility proof for the quintic would impress his French peers; unfortunately, all of his work had been printed in Norwegian, in an obscure journal. He therefore decided that he should get his work on the theory of equations printed privately in French. Its title was “Memoir on algebraic equations, wherein one proves the impossibility of solving the general equation of the fifth degree.”