1. Twenty-one

On the morning of May 31, 1832, a young man died in the Hôpital Cochin in Paris from a gunshot wound received in a duel the previous day. Hospital records, a death registration, and evidence concerning the postmortem examination survive. The official paperwork gave his age as twenty-one. He had not yet reached his twenty-first birthday.

His name was Évariste Galois. Modern algebra textbooks carry his name everywhere, although the central memoir that made his reputation was not published during his lifetime. A document made by an official almost immediately after his death got an elementary fact wrong. Nothing momentous follows from this particular error. It does, however, establish the rule we shall need throughout this chapter: proximity to an event matters, but proximity does not make every statement infallible.

Our evidence is quite different from that available for the early figures in this series. The first accounts of Hippasus are separated from his supposed lifetime by centuries. Even the familiar dates assigned to Archimedes depend partly on much later calculations. With Galois we have civil registers, school reports, teachers' assessments, Academy minutes, newspaper coverage of trials, prison documentation, hospital papers, letters, and mathematical manuscripts. He passed through an extraordinary number of the institutions that made nineteenth-century France a society of paperwork.

He attended school, failed entrance examinations, was expelled, was arrested twice, went to court, spent time in prison, and was transferred to a medical establishment during a cholera epidemic. Eventually he died in a hospital. Each institution recorded something, often allowing one account to be checked against another. He also wrote for himself: published articles, submissions to the Academy, public letters, correspondence from prison, a final scientific letter, and manuscripts covered with alterations.

The survival of those manuscripts was not inevitable. After his death Auguste Chevalier took charge of papers and copied some of them. In 1843 he passed material to Joseph Liouville. Liouville's library and papers subsequently went to his son-in-law; that man's widow, Liouville's daughter, gave the collection to the Academy around 1905–1906. A private chain of custody preceded public preservation. Carelessness at any link could have changed what we are able to say today.

The collection is roughly 285 leaves, a substantial proportion in Galois's hand. Its original twenty-five dossiers were supplemented by two further dossiers; the current catalogue describes twenty-seven. Digital access has made much of this fragile material far easier to inspect than it was when researchers depended on microfilm. Yet the legend remains thicker than almost any other in mathematics. A board eraser hurled at an examiner. Papers lost by famous mathematicians. A referee dismissing a memoir with a single word. An entire theory invented on the night before a duel, accompanied by repeated cries of having no time. Fourteen years required for someone to understand it.

None of these familiar formulations should be accepted as it stands. That does not mean every incident was invented from nothing. It means that documented facts, later recollections, plausible reconstructions, and theatrical additions have been welded into sentences that conceal their different evidential status. Respectable books helped perform the welding. A doubtful story can survive for generations when each writer inherits a predecessor's authority instead of rechecking the underlying claim.

The deeper question is therefore not merely which anecdotes are false. Why did they flourish around someone whose short life is so extensively documented? Archives can often tell us when a paper arrived or who signed a report. They are much less able to record the instant when someone understood a mathematical structure. That gap will matter at the end. First we must return almost three centuries.

2. The hunt

After the publication of Cardano's great algebraic work in 1545, the problem had a new shape. Cubic and quartic equations could be solved by formulas involving their coefficients, arithmetic operations, and radicals. Could a comparable procedure solve the general equation of degree five? The issue was not whether a particular quintic possessed a root. It was whether one finite algebraic recipe would produce the roots for arbitrary coefficients.

It is misleading to picture three hundred years of mathematicians lining up to attack a single advertised prize problem. There was a long research tradition concerning the algebraic solution of equations, and the quintic remained an obstinate unfinished item within it. Many contributors were developing techniques with wider purposes. Their work entered the eventual toolbox without having been conceived as one more unsuccessful attempt at the modern quintic problem.

During the sixteenth and seventeenth centuries, much of the essential work was the making of that toolbox. Bombelli investigated the unfamiliar quantities encountered in cubic calculations. Viète organized relations between roots and coefficients and methods of transforming equations. Descartes strengthened symbolic manipulation and factorization. By the later seventeenth century, transformations designed to remove intermediate terms offered ways of reducing equations to simpler standard forms. These developments made later questions possible, but we should not retrospectively assign every author the same objective.

Lagrange's investigations of 1770–1771 marked a decisive change in emphasis. Instead of producing another proposed formula, he examined why the existing cubic and quartic methods worked. Form expressions from the roots and then permute those roots. Some expressions remain unchanged; others assume only a limited number of different values. Those values can themselves serve as the roots of an auxiliary equation. The effectiveness of apparently unrelated tricks could thus be understood through permutations and the number of values an expression takes.

For degrees three and four, appropriate auxiliary expressions lead to equations of manageable degree. At degree five, the analogous procedure does not conveniently collapse to a lower degree already under control. Lagrange explained the old successes and exposed a difficulty in extending them. He did not prove that every imaginable radical method must fail. That distinction is the hinge of this history: the failure of a family of methods is not yet a theorem excluding all methods of the prescribed kind.

Nevertheless, disassembling a successful procedure makes a new question available. Perhaps the obstacle is not simply that nobody has found the right formula. Perhaps the proposed kind of formula cannot exist. The long history of attempts to prove Euclid's parallel postulate will present a related shift: failure can become an object of mathematical explanation rather than an embarrassment to be overcome. The analogy concerns the change of question, not an identity between algebraic impossibility and geometrical independence.

In 1799 Paolo Ruffini published an extensive work whose title explicitly announced the impossibility of algebraically solving general equations above the fourth degree. This was not a report of personal defeat. It claimed a theorem. The work ran to several hundred pages, developed unfamiliar arguments, and demanded sustained attention from readers who had to be persuaded that an impossibility result was a legitimate answer to the traditional problem.

Ruffini took permutation methods well beyond Lagrange. Cycles, orders, conjugate arrangements, and distinctions later expressed through transitivity and primitivity became technical objects in their own right. Calling this modern abstract group theory would obscure the historical difference: he did not possess our general axiomatic definition of a group. Yet much of the structural thinking grew inside his study of equations.

The gap in his argument was more specific than a vague shortage of rigor. To control every possible radical expression, he needed to connect the auxiliary radicals introduced at successive stages with rational expressions in the original roots. His treatment assumed a reduction of this kind without adequately proving it. One cannot simply suppose that every intermediate quantity stays inside the field generated by the roots. The older cubic problem had already made the dangers of assumptions about intermediate quantities vivid, although that example by itself does not identify the exact missing algebraic lemma.

Reception was not unanimous neglect. Ruffini encountered substantial objections, revised his proofs repeatedly between 1802 and 1813, failed to obtain the response he wanted from Lagrange, and later received appreciative attention from Cauchy. The length of the work, its novel organization, and the unproved premise all mattered. So did the practical difficulty of transmitting a large book from Modena into the major mathematical networks. A further conceptual obstacle was that a negative theorem asked readers to reconsider what counted as solving a problem. We should treat that as a historical interpretation, not a measurable explanation of every reader's reaction.

In 1824 Niels Henrik Abel privately financed a very short French pamphlet establishing the impossibility of a general radical solution of the quintic. He was twenty-one or twenty-two, depending on the date within that year. The six-page presentation was exceptionally compressed, partly to reduce printing expense. It is reasonable to connect its brevity with economy. It is another matter to imagine a bill proving that he literally could not afford a seventh page. The latter turns a publishing circumstance into a camera-ready scene.

A fuller paper followed in 1826 in Crelle's newly founded journal, addressing general equations beyond the fourth degree. Abel knew of Ruffini and acknowledged him as a predecessor while finding the earlier argument difficult to assess. Abel died of tuberculosis on April 6, 1829, aged twenty-six. Crelle wrote on April 8 to report that his efforts to obtain a Berlin appointment had succeeded, not yet knowing that Abel was dead. Those two days separate the death from the writing of the letter; they do not establish the day on which a postal delivery arrived. A formal appointment document reaching the house just after a funeral is a different claim, produced by blending correspondence, employment negotiations, and imagined delivery details.

3. What does not exist

We must be exact about what Abel's result denies. The quadratic formula is a familiar model: take the coefficients, perform specified additions, subtractions, multiplications, and divisions, and extract a square root. After finitely many operations the roots are available. Cubic and quartic formulas are more elaborate, but their general character is the same. For centuries the desired object was the corresponding formula for degree five.

That object does not exist. This is not the assertion that quintic equations have no answers. Over the complex numbers, every nonconstant polynomial has roots, and a degree-five polynomial has five when multiplicities are counted. With suitable computable input, numerical methods can approximate roots to prescribed accuracy. Other forms of exact representation are also possible. The impossibility concerns one specific class of finite expressions built from coefficients by arithmetic and extraction of radicals.

Nor does the theorem say that every individual quintic resists radicals. Many do not. The equation x⁵ = 2 is enough to expose the confusion: its roots are expressible using a fifth root and roots of unity. What fails is a universal radical formula for the general quintic. The qualifying words carry the content of the theorem. Erase them, and an important statement about the limits of a method becomes a plainly false statement about equations.

There is an enormous difference between not having found a formula and proving that no formula of the stipulated sort can exist. The first invites another search of the same kind. The second redirects the search. It does not end mathematics around the equation; it makes the boundary of one method part of the mathematics. This is the precise sense in which the hunt ended.

Galois pushed toward a more discriminating question: given an equation, under what conditions is it solvable by radicals? Abel's general impossibility result and Galois's structural criterion should not be confused. One closes a universal expectation; the other tells us what governs individual cases. Abel himself also investigated particular classes of solvable equations, so the contrast is a guide to their best-known achievements, not a complete division of their interests.

The new viewpoint studies the symmetries of the roots. Which permutations preserve the algebraic relations over the field of coefficients? The permitted permutations form a structure, and the way that structure can be broken down controls radical solvability. In modern language, in characteristic zero an equation is solvable by radicals precisely when its Galois group is solvable. That is our later formulation of the insight, not a claim that Galois wrote a present-day textbook.

This change of viewpoint also helps explain the difficulty encountered by readers. Traditional equation theory expected a usable condition starting from known coefficients. A criterion stated through relations among unknown roots could seem to relocate the problem rather than solve it. The referee's concern was not simply an inability to recognize genius. A new way of specifying the problem required a new understanding of what an answer might look like. Neither novelty nor skepticism automatically settles whether a particular exposition has supplied enough proof.

The shift from formulas to symmetries is therefore both mathematical and explanatory. It asks us to stop staring only at the expression whose roots we want and to examine the transformations that leave its relevant relationships intact. That approach eventually created a language far larger than the original hunt. It also demanded more explanation than Galois's compressed manuscripts always supplied.

4. Galois in the records

Galois was born on October 25, 1811, at Bourg-la-Reine, south of Paris. Civil documentation supports the date. His father was a local liberal politician and mayor; his educated mother taught him at home in his early years. His father died by suicide in Paris on July 2, 1829. The death is securely documented. The political humiliation commonly presented as its immediate cause relies more heavily on family testimony than on an equally complete contemporary case file.

Several levels of assertion must be kept apart. That his father held office, that local political conflict was bitter, and that he killed himself are historical claims with contemporary support. That malicious verses falsely attributed to him directly caused the death comes through a narrower evidential channel. That this death made the son hate every authority and therefore lose control at an examination is a psychological reconstruction without a diary or letter establishing the connection. Events close together in time are not automatically joined by a demonstrated cause.

The temptation is particularly strong here. Within less than three years come bereavement, examination failure, revolution, expulsion, arrests, imprisonment, and a duel. Narrative easily changes next into therefore. Surviving school reports describe exceptional mathematical preoccupation and an original, difficult character. They do not supply every later story about a boy devouring a geometry book at one sitting as if it were a novel. The distinction is not whether a detail sounds plausible but what kind of evidence carries it.

Galois twice failed the entrance examination for the École Polytechnique. Accounts differ over some dates and circumstances. The famous story of his throwing a board-cleaning implement at the examiner is reliably traceable to Paul Dupuy's 1896 biography, where it is explicitly offered as a tradition. Dupuy could not even securely identify the examiner in that anecdote. Earlier mathematical commentary had already made Galois an exceptionally gifted rejected candidate without documenting the thrown object.

Notice the characteristic distribution of clarity. The identity of the person supposedly attacked remains uncertain, while the dramatic gesture becomes vivid. The object itself changes in transmission: a cloth used to wipe a board hardens in English into an eraser, then returns in other languages as a familiar classroom prop. Translation can make an uncertain story materially more definite even when no writer deliberately fabricates anything.

Dupuy nevertheless remains indispensable. He consulted educational records, Academy and national archives, municipal papers from Galois's birthplace, police materials, hospital documentation, and records from two prisons. He also acknowledged that some police and judicial originals had been destroyed in 1871, requiring contemporary newspapers to fill gaps. His biography is not a fraudulent source to be discarded. It is a major archival undertaking that also contains literary reconstruction. Dupuy's biography is preserved by Numdam.

The same person can be an excellent archival researcher and an effective maker of myth. Those activities can occur on different pages of one book. Later readers often fail to preserve the distinction: a birth date transcribed from a register and a feverish final-night scene acquire the same appearance of authority because both occur inside an authoritative volume.

This is a recurring condition of historical knowledge. Our information about ancient mathematicians often passes through a much later compiler who combines careful preservation with storytelling. Here the intermediary has access to modern municipal archives, but the interpretive problem has not disappeared. The appropriate response is not universal distrust. It is to ask, claim by claim, whether the writer is reproducing a document, reporting a recollection, proposing an inference, or composing a scene.

5. Three rounds of submission

Galois's central research reached the Academy in three rounds. Calling them rounds avoids the misleading picture of one identical bundle being handed back and forth three times. The first belongs to 1829. Academy records show Cauchy presenting related work on May 25 and June 1. These are generally understood as two connected memoirs in equation theory, not simply copies of the surviving 1831 manuscript. No final formal report survives from that round.

The popular explanation says Cauchy lost the papers. The documentary situation is more complicated. A surviving letter dated January 18, 1830, says that Cauchy had intended to present two items, one concerning the young Galois's work. Illness prevented him from attending, and he asked that they be put on the next meeting's agenda. On January 25 he attended and presented the other item, but not the Galois report. The minutes subsequently fall silent.

That silence does not tell us why the report disappeared from the agenda. It does not license a choice among forgetting, contempt, pressure of other business, or destruction. René Taton's 1971 reconstruction proposed that Cauchy and Galois probably agreed to replace reports on the earlier papers with a consolidated and expanded submission for the Academy's prize. The timing fits: Galois submitted a new memoir in February 1830, before the March deadline. But the reconstruction remains an inference. No surviving instruction from Cauchy says precisely that.

The distinction is worth maintaining sentence by sentence. Cauchy received the work and introduced it to the Academy. He intended to report. He wrote that illness prevented attendance. The expected report was not subsequently delivered. Those are documented events. Advice to rewrite for the prize is a plausible account connecting them. Cauchy simply losing the papers is not established by an equivalent contemporary record. A correction can be persuasive without pretending to possess the missing letter.

The second round was the February 1830 prize submission. Fourier, one of the Academy's permanent secretaries, died on May 16. On June 28 the prize went to Abel, already dead, and Jacobi for work on elliptic functions. Galois's memoir did not enter the final assessment and could not later be recovered from the expected papers. An early explanation associated its disappearance with Fourier's death, but the transmitted wording itself was qualified.

Here again, disappearance after a death, a plausible administrative explanation for the loss, and a scene in which Fourier takes the manuscript home and forgets it are three different claims. An accusation that the Academy deliberately destroyed the work is further removed still. The missing document is a real historical loss. It is not an empty surface on which any desired motive can safely be written.

The third round is better documented. The surviving memoir is dated January 16, 1831, and was submitted the next day. Poisson and Lacroix were assigned to examine it; their report was delivered on July 4. They did not merely return it with a contemptuous word. They explained that, despite efforts to follow the proofs, the exposition was insufficiently clear and developed for them to judge its rigor.

They also raised the substantive difficulty already encountered above: conditions expressed through relationships among the roots were not immediately the coefficient-based criteria customary in equation theory. At the end, they noted Galois's claim that the memoir formed part of a larger theory and suggested that publication of the whole might allow its components to illuminate one another. This was an unfavorable report, but it was also an argument about exposition and a recommendation concerning presentation.

Nor should one forget institutional context. Many submissions never received a substantial written assessment at all. The existence of a considered report does not prove that the Academy treated Galois adequately; it does contradict the claim that nobody would read a word. A historian need not turn either the referees or the young author into villains to recognize that an important theory failed to secure publication and understanding at the time its author needed them.

6. That night

The best-known story says that Galois created his whole theory on the night before the duel. The chronology of the manuscripts rules out that formulation. The first major memoir had already been submitted in January 1831. Related research goes back to 1829 and the missing prize paper of 1830. A second important manuscript is dated by modern scholarship to around May or June 1830. The mathematical labor preceded the final night by years.

On May 29, 1832, he wrote a scientific letter to Auguste Chevalier and made alterations and marginal additions to existing manuscripts. The letter is several pages, not a newly composed book of algebra. It surveys results, unfinished investigations, and directions for further work. He also wrote farewells to republican friends, explaining his understanding of the private quarrel and asking them not to blame him for the duel.

The documented version is no less disturbing. A young mathematician, expecting that he might be killed, tried to improve papers he had already struggled to communicate and to leave someone else a map of work not yet fully set out. The legend replaces the hurried preservation of a research inheritance with its miraculous creation. In one version there must have been years of work; in the other, genius needs only a single night.

The celebrated statement that he had no time also has locations, not merely an atmosphere. One occurrence belongs beside the proof of Proposition II in the first memoir, where he indicates that something still needs completing. Another occurs toward the end of the scientific letter, in connection with ideas for transcendental analysis that were not fully developed. They are not an indefinitely repeated cry written down every few minutes throughout the night.

A marginal admission of an unfinished proof is powerful precisely because it refers to a particular mathematical task. Turn it into a refrain and the reader loses that specificity. The alteration is small at the level of words and large at the level of story: an author revising incomplete arguments becomes a visionary pouring out an entire completed theory while death approaches. The duel occurred the following morning; death came the morning after that. Even this distinction is frequently collapsed for dramatic speed.

The letter itself deserves to be read as a research map. Its early portion concerns equations, including primitive equations and radical solvability. It then moves into elliptic functions, periods, and wider transcendental analysis. Near the end Galois describes an ambition to extend ideas about indeterminacy or ambiguity to relations among transcendental quantities: study the interchanges that preserve a given relation. His own horizon was considerably broader than the quintic to which later biographies often tether him.

His requests were practical. Chevalier should publish the letter. Jacobi and Gauss should be asked for public judgments about the importance of the results, not merely their truth. Others might then organize, continue, and complete the work. The letter appeared in September 1832; Chevalier published a memorial account in November. The requested endorsements did not follow. Assertions that both celebrated mathematicians received the entire manuscript collection and contemptuously rejected it require evidence of transmission that we do not possess.

The main mathematical papers eventually appeared under Liouville's editorship in 1846. The original publication remains available, allowing the letter and memoirs to be distinguished rather than fused into a single deathbed document. Later editorial scholarship also warns that even an apparently definitive edition contains judgments about ordering, transcription, and interpretation. Preserving manuscripts is one task; deciding exactly how readers should encounter them is another.

7. The duel

This part of the story must begin with what is not known. No proposed explanation of the duel has been established by a complete and decisive body of contemporary evidence. Galois's letters of May 29 are the earliest starting point. He blamed a woman and a wretched affair of gossip; elsewhere he described two patriots who had drawn him into the encounter, said he had sought reconciliation, and asked friends to forgive those responsible because he regarded them as acting sincerely.

These statements matter, but they do not supply a neutral account of everything that happened. They show how he presented the crisis at the time. They suggest a private dispute involving a woman, honor, and members of the republican milieu. They do not identify every participant, reconstruct the precipitating conversation, or establish the familiar plot of two rivals fighting over a girlfriend. An actor's account is evidence about an event and about the actor's understanding of it; those uses are related but not identical.

His conciliatory remarks also put pressure on a straightforward political-assassination interpretation, though psychology alone cannot rule one out. It would be too strong to say that a victim of a political plot could never sincerely misread an opponent's motives. What is absent is positive contemporary documentation of a government plan: orders, payments, reports from agents, or an arrangement connecting officials to the duel. Suspicion is not a substitute for such evidence.

A family tradition recorded more than sixty years later described the intervention of men called an uncle and a fiancé of the woman. Alexandre Dumas's memoirs supplied an influential name for the opponent. Dumas knew the republican circle and had attended a notorious banquet involving Galois, but he was not an eyewitness to the duel. A later researcher, in 1968, connected the woman's possible identity with civil records, the medical establishment where Galois had stayed, and a name appearing among his papers. This is more than pure invention, but identification of a person connected with the setting is not the same as proving her role in the final quarrel.

The theory that Galois deliberately arranged a duel as a form of suicide is similarly unestablished. No farewell letter announces such an intention. The medical and administrative evidence supports a narrower reconstruction. On the morning of May 30, 1832, he fought somewhere south of Paris, received a gunshot wound to the abdomen, and was eventually taken to the Hôpital Cochin. The reconstructed admission time is about half past nine. He died at roughly ten the next morning.

These records do not identify the opponent or provide a reliable signed account of the seconds and witnesses. They do not establish the rules, the distance between the men, who fired first, whether shots alternated, or whether Galois intentionally withheld his own shot. Early accounts differ about who found and transported him: family recollections mention one kind of passerby, other biographies another. Choosing the most vivid version would conceal the conflict rather than resolve it.

The secure outline is therefore short: a morning duel, an abdominal wound, removal to hospital, death the next day. Assertions that he fired into the air, lay abandoned until the following day, or stood at a precisely reported number of paces exceed what the surviving documentation can confidently bear. The sparseness of this reconstruction may disappoint a reader, but disappointment is not evidence that a missing scene can be supplied.

We know enough to recognize a terrible loss. We do not need to know the exact dialogue at the dueling ground to understand that. What we lose by refusing the embellished version is cinematic closure, not the event's human significance. An unresolved cause should remain unresolved even when the rest of a life seems to demand a final explanation.

8. Fourteen years is not fourteen years of reading

Galois was not wholly unrecognized while alive. He published five mathematical articles, beginning with work on continued fractions in 1829 and including papers in 1830 related to his developing interests in equations and number theory. Teachers recognized his unusual ability. Even an institutional statement criticizing his conduct acknowledged his mathematical talent. Cauchy presented his work; Poisson encouraged a further submission and, with Lacroix, assessed it.

None of that cancels the central failure: his most important theory of radical solvability did not gain formal publication and sustained communal recognition during his lifetime. Partial recognition is a less convenient story than complete neglect. It requires bringing together school records, articles, Academy presentations, and an unfavorable but substantive report. One rejection can illustrate exclusion; a mixed record takes longer to explain. Yet the mixed record is what we have.

The famous fourteen-year delay also needs to be divided. Galois died in 1832; Liouville published the main papers in 1846. But Liouville did not spend that entire period trying to understand manuscripts already in his possession. The material remained principally with Chevalier before reaching Liouville around the summer of 1843. By July of that year Liouville could announce important results to the Academy. The ensuing interval involved checking, arranging, editing, and preparing publication.

Why had the papers not circulated more effectively during the preceding decade? The evidence does not yield a definite answer. Why did Chevalier transfer them when he did? Again, no securely established single explanation. Government suppression of a republican martyr is an attractive political story, but it cannot be inferred from a gap in the transmission record. Fourteen years between death and publication is a chronological fact; fourteen years required to decipher the mathematics is a different proposition.

The myth was built in layers. Galois himself contributed an early and bitter institutional narrative. In a preface written in prison in December 1831, he complained that his prize memoir had vanished and reduced Poisson's response to a failure to understand. The fuller report, with its stated difficulties and recommendation to publish the whole theory, disappeared inside the author's account of rejection. A first-person source can preserve genuine injury while also compressing the conduct of others.

His anger invoked Abel. Abel had submitted a major memoir to the Academy on October 30, 1826, with Cauchy and Legendre assigned as referees. More than two years passed without a result. Jacobi pressed Legendre about it in March 1829; publication finally came in 1841, twelve years after Abel's death. This history gave Galois's accusations an intelligible background. It did not automatically establish his explanation of every episode involving his own papers.

Chevalier's 1832 memorial connected the death with insufficient official recognition. A portrait drawn from memory by Galois's brother reached a broader public in 1848. Mathematical biographical writing followed in 1849. Dumas added the resources of the literary memoir. Dupuy's archival biography then combined serious documentary recovery with a feverish account of the final night. The collection of facts and the composition of legend did not proceed in separate institutions; they could share a page.

E. T. Bell's 1937 popular biography did not invent every anecdote. Its enormous influence came partly from removing qualifications. A transmitted tradition became an event. A complicated submission history became a succession of obtuse authorities destroying genius. Revision of old manuscripts became the overnight production of a mathematical legacy. The chapter's opposition between genius and stupidity supplied a plot into which almost every incident could be fitted.

Tony Rothman's 1982 study, pointedly titled Genius and Biographers, examined this accumulated tradition by comparing later accounts with earlier evidence. More broadly, scholarship on Galois now studies not only his mathematics but the production of his public identity. Caroline Ehrhardt's research makes the temporal and social processes of mathematical recognition an object of inquiry in their own right. Her study of Galois and mathematical time is a useful entry into that discussion.

Return, finally, to the wrong age in the death record. Abundant records do not eliminate error, but they make particular corrections possible. They still cannot tell us the instant when the symmetry of roots became clear to Galois. Our earlier story of Hippasus had a firm mathematical remainder and an uncertain human victim. Here both the mathematics and much of the life are unusually well documented, yet legend grows around the missing moment of understanding.

The final night offers a ready place to put it. It has a date, surviving papers, an identifiable addressee, and a death close enough to cast its shadow backward. A narrative can deposit the breakthrough there and acquire a beginning, climax, and conclusion. That is an interpretation of why the legend is attractive, not a new archival discovery. The same appetite helps sustain falling apples, overflowing baths, and sudden flashes of illumination elsewhere in the history of thought.

The next time a biography tells us that someone understood everything on a particular night, a modest question is worth asking: who supplied the date? Mathematics continues to calculate. What we count as a satisfactory story about mathematics remains much harder to settle.