Adrien-Marie Legendre

Adrien-Marie Legendre
Discipline	Mathematics
Known for	Legendre polynomials; law of quadratic reciprocity
Fields	Number theory; Mathematical analysis
Occupation	Mathematician
Notable works	Essai sur la théorie des nombres
Era	18th–19th century
Nationality	French
Wikidata	Q191021

Adrien-Marie Legendre (1752–1833) was a leading French mathematician whose work spanned number theory, geometry, analysis, and mathematical physics. He introduced fundamental tools in number theory (the Legendre symbol for quadratic residues) and in analysis (the Legendre polynomials, solutions of Legendre’s differential equation). Legendre’s name is attached to many concepts, including an enduring conjecture on prime gaps and the method of least squares. In the late 18th century he made one of the first systematic attempts to prove the law of quadratic reciprocity, a central theorem determining when a quadratic equation has integer solutions modulo a prime. Although some of his proofs (notably of quadratic reciprocity) were incomplete by modern standards, Legendre’s books and results laid groundwork that later mathematicians like Gauss, Dirichlet, Abel, and Jacobi built upon. His Éléments de géométrie (1794) became a standard textbook in geometry for generations.

Early Life and Education

Adrien-Marie Legendre was born in Paris on September 18, 1752, to a well-to-do family, which allowed him to devote himself to study rather than work for income Little is known of his childhood, but he received an excellent early education. In 1770, at the age of 18, using a plan of research rather than a traditional doctorate project, he “defended” a dissertation in mathematics and physics at the Collège Mazarin (also called Collège des Quatre-Nations) in Paris This treatment outlined problems and results Legendre aimed to achieve, reflecting his intent to pursue mathematics full-time. Legendre stayed in Paris and immersed himself in research, free from financial need.

After completing his studies, Legendre began teaching. From 1775 to 1780 he was appointed as an examiner and instructor at the École Militaire (a military school) in Paris, where he taught alongside Pierre-Simon Laplace (on the recommendation of Jean le Rond d’Alembert) This position kept him connected to practical mathematics – for example, many examination problems involved ballistics and surveying. It also gave him time to continue his own research.

In 1782 Legendre entered a competition sponsored by the Royal Prussian Academy of Science (Berlin Academy) and won the prize for his work on projectile motion. His essay Recherches sur la trajectoire des projectiles dans les milieux résistants solved the problem of determining the curve of a cannonball through air resistance This success brought him to the attention of major mathematicians. The following year (1783) he presented research on the gravitational attraction of ellipsoids to the Paris Academy of Sciences. Laplace reported on Legendre’s work, and Legendre was elected an adjunct member of the Academy that year Also in the 1780s he contributed to the Paris–Greenwich geodetic survey, earning membership in the Royal Society of London in 1787.

Legendre lived through the turbulence of the French Revolution. The Academy of Sciences was closed in 1793, and Legendre lost much of his personal fortune. He married around this time and faced financial difficulties, though he continued working on mathematics. In 1791 he was appointed to the committee to reform weights and measures (helping define the meter) and later worked on producing logarithmic and trigonometric tables After the Revolution, the Academy was re-established (1795), and Legendre took one of the mathematics seats. He continued as an examinator of artillery students at the École Militaire into the early 19th century (until 1815)

Major Works and Ideas

During his career, Legendre published on many topics. His work ranged from celestial mechanics to pure number theory. Key areas include:

• Number theory. Legendre’s Théorie des Nombres (first edition 1798, much expanded by 1808 and again published in 1830) systematically presented number-theoretic results He introduced the Legendre symbol $(a/p)$, a notation for determining whether an integer $a$ is a quadratic residue mod $p$. (By definition, $(a/p)=1$ if $a$ is a perfect square modulo an odd prime $p$, $-1$ if it is not, and $0$ if $p$ divides $a$.) For example, whether the congruence $x^{2}\equiv a{\pmod {p}}$ has a solution is encoded by $(a/p)$. Legendre’s symbol became a fundamental tool in number theory

Legendre studied the law of quadratic reciprocity, a deep theorem originally conjectured by Euler. Informally, this law relates the solvability of $x^{2}\equiv p{\pmod {q}}$ and $x^{2}\equiv q{\pmod {p}}$ for two odd primes $p$ and $q$. Legendre formulated the law using his symbol: he showed that $(p/q)(q/p)=(-1)^{{\frac {(p-1)}{2}}{\frac {(q-1)}{2}}}$ (a factor of $-1$ appears whenever both primes are congruent to 3 mod 4) However, his initial proof (in 1785 and again in a polished 1798 version) contained gaps and unproved lemmas. Gauss later provided the first fully rigorous proof in 1801. (Legendre included his own flawed proof in his 1830 Théorie des Nombres, at which point he simply acknowledged that Gauss’s earlier work had settled the law Even though his proof was incomplete, Legendre was one of the first to state the law (Euler had noted special cases earlier) and to systematize its expression with the Legendre symbol.

In the same 1785 paper, Legendre conjectured two other important principles about primes. First, he observed that any arithmetic progression $a,a+m,a+2m,\dots$ with $\gcd(a,m)=1$ seems to contain infinitely many primes (a statement later proved by Dirichlet in 1837). Second, he proposed now-called Legendre’s conjecture (still unproven): there is always at least one prime between $n^{2}$ and $(n+1)^{2}$ for every integer $n\geq 1$ This simple-sounding claim about gaps between primes has defied proof but highlights Legendre’s interest in prime distribution. He also estimated the number of primes ≤$x$ asymptotically by $x/(\ln x-1.08366)$, an idea resembling the prime number theorem (formalized later by Gauss and others).

• Legendre polynomials and spherical problems. In studying the gravitational pull of ellipsoids, Legendre introduced the polynomials $P_{n}(x)$ now named after him. In 1783–84 he published papers (Recherches sur la figure des planètes and related works) in which he used power-series expansions for the potential of an ellipsoid This led him to discover what are now called Legendre polynomials. Each $P_{n}(x)$ is a polynomial of degree $n$; for example $P_{0}(x)=1$, $P_{1}(x)=x$, $P_{2}(x)=(3x^{2}-1)/2$, and so on. These polynomials satisfy orthogonality: $\int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx=0$ whenever $m\neq n$. In solving Laplace’s equation in spherical coordinates (for example in potential theory or in spherical harmonics), one finds that solutions can be written using Legendre polynomials. Thus these functions are ubiquitous in physics and engineering – in gravitational and electrostatic fields, quantum mechanics, acoustics, and elsewhere Today one speaks of the Legendre differential equation and its polynomial solutions; Legendre’s early work discovered and named them.

• Least squares and astronomy. In 1806 Legendre published Nouvelles méthodes pour la détermination des orbites des comètes, a work on determining comet orbits. In it he described what is now called the method of least squares – fitting data by minimizing the sum of squared deviations. Legendre’s exposition was the first published account of least squares, though Carl Friedrich Gauss independently developed the same method around the same time. Gauss later claimed priority (publishing his work in 1809), which caused a public dispute: Gauss acknowledged using Legendre’s published account but insisted he had discovered the idea first. Legendre protested fiercely but history now credits both men – Legendre for first publishing it and Gauss for also having discovered it. Least squares became a cornerstone of data analysis and statistics

• Geometry and the parallel postulate. Legendre was also active in geometry. Encouraged by the Marquis de Condorcet, he reorganized Euclid’s Elements into a modern textbook. In 1794 he published Éléments de géométrie, a carefully structured presentation of plane geometry. This work became the standard geometry text in much of Europe (and later in the United States) for about a century Legendre simplified many of Euclid’s proofs and presented geometry in a user-friendly way. Notably, in Éléments he gave straightforward proofs that the number $\pi$ is irrational and that $\pi ^{2}$ is irrational, and he conjectured that $\pi$ cannot satisfy any algebraic equation with rational coefficients (a conjecture of transcendence proved only in 1882).

Legendre famously attempted to prove the parallel postulate (Euclid’s fifth postulate) from the other axioms. For decades he sought a proof of the theorem that “the angles of a triangle sum to 180°” based only on Euclid’s remaining assumptions. In practice each attempt quietly used some equivalent form of the parallel postulate, making the proofs circular. Mathematicians later showed the parallel postulate is independent of the other axioms (as János Bolyai and others formalized non-Euclidean geometry), but Legendre held to Euclid’s view. In a memoir from 1832 he asserted the 180°-angle-sum theorem was a fundamental truth not provable from other axioms In retrospect, Legendre’s efforts exemplified how hard it is to deduce the parallel postulate: all his proofs (and those by others at the time) implicitly assumed some hidden form of it.

• Elliptic integrals and analytic functions. From 1786 onward Legendre studied elliptic integrals (integrals of the form $\int R(x,{\sqrt {P(x)}})dx$ where $P$ is cubic or quartic). He spent many years classifying and transforming them. Between 1811 and 1819 he published Exercices de Calcul Intégral (three volumes) laying out the theory of elliptic integrals and related beta and gamma functions From 1825 to 1830 he issued a new three-volume Traité des fonctions elliptiques. Legendre reduced elliptic integrals to three “normal forms” (now called Legendre forms) and made extensive tables of their values These works were foundational – they introduced many tools for later researchers – but soon after their completion, the independent work of Niels Abel and Carl Jacobi revolutionized the theory of elliptic functions. Abel and Jacobi introduced deeper algebraic insight (elliptic functions as inverses of elliptic integrals) that went beyond Legendre’s classical treatment. Even so, Legendre’s systematic development of elliptic integrals remained influential for many years.

Throughout his work, Legendre favored systematic, analytic methods. He often used power series expansions and constructed lengthy tables of numerical values (for example, extensive tables of elliptic integrals). He introduced precise notation (like the Legendre symbol) to clarify number-theoretic statements. As a geometer, Legendre tended to seek direct proofs, insisting on algebraic or geometric demonstrations. However, some of his methods were later judged “unduly obvious” or incomplete. For example, in his geometry he uncritically assumed facts equivalent to the parallel postulate In number theory, Gauss noted that Legendre sometimes omitted checking certain cases in his proofs. These methodological issues drew criticism from contemporaries but also spurred Legendre to refine his arguments in later editions.

Influence

Legendre’s contributions shaped mathematics well beyond his lifetime. His work influenced both contemporaries and later generations. Notably, Carl Friedrich Gauss built on Legendre’s formulations: Gauss referred to Legendre’s Théorie des Nombres while developing his Disquisitiones Arithmeticae. Although Gauss was famously critical of Legendre’s proof of quadratic reciprocity, he acknowledged the importance of the theorem and eventually gave Legendre credit for introducing the question Gauss went on to produce several proofs of the theorem and extended Legendre’s ideas on prime distribution. The methods introduced by Legendre opened paths that Gauss and Johann Dirichlet would follow to formalize analytic number theory.

Beyond number theory, Legendre’s work paved the way for advances in other fields. In analysis and physics, the Legendre polynomials became standard tools. The process of reducing elliptic integrals to normal forms anticipated the breakthroughs of Abel and Jacobi in complex function theory. Legendre himself never discovered these deeper structures, but he laid the computational and conceptual groundwork. For example, in solving mechanical or astronomical problems (such as the rotation of the Earth or celestial orbits), later scientists frequently used Legendre’s expansions and tables of special functions.

Legendre also helped disseminate mathematical knowledge. His Éléments de géométrie became the leading geometry textbook in Europe for about 100 years French schoolchildren for generations learned from his simplified Euclid. Translations of Éléments proliferated across Europe and even in the United States. Likewise, Théorie des Nombres (in its expanded 1808 and 1830 editions) compiled many known results in number theory into a single reference, influencing students and researchers. In the 19th century, aspirant mathematicians widely read Legendre’s treatises, gaining exposure to his notation and approaches.

Among mathematicians, Legendre’s circle included major figures. He corresponded with and was admired by Condorcet, one of the Enlightenment’s leading intellectuals. His work earned him honors: election to the Académie des Sciences (1783), the Royal Society of London (1787), and eventually high rank in Napoleon’s Institut (1803) Although he was less famous than some peers like Lagrange or Laplace, his influence permeated 19th-century mathematics.

Critiques

Even as Legendre’s ideas spread, critics pointed out flaws. In number theory, Gauss was a harsh critic of Legendre’s rigor. Gauss discovered a gap in Legendre’s 1785 reciprocity proof and again in the 1798 revision. He wrote privately (and later in publications) that Legendre’s proof was incomplete. When Gauss published his own 1801 proof, he omitted any mention of Legendre, effectively claiming priority. Legendre was incensed: he complained that Gauss had acted with “excessive impudence” in seemingly appropriating discoveries that had appeared in Legendre’s published work Although historical perspective credits Gauss with the first rigorous proof, it is fair to say the reciprocity law had been gestated in Legendre’s pages as well.

Legendre’s geometric work also drew criticism. Mathematicians noted that his attempted proofs of the parallel axiom were circular. Euclid’s fifth postulate is independent of the others, but Legendre originally believed he could derive it. Later geometers recognized that all substantiated proofs essentially smuggled in equivalent assumptions. In short, Legendre’s efforts, while earnest, ultimately failed to settle the issue. His Éléments was nonetheless extremely useful, and his vision that the angle-sum theorem was an “incontrovertible” truth turned out to reflect the eventual understanding of non-Euclidean geometry.

In other cases Legendre was simply overtaken by others. His arithmetic-progression conjecture was proven by Dirichlet; his elliptic integral work was superseded by Abel and Jacobi; his least-squares method was shared with Gauss. Legendre could be proud of inspiring these advances, but he occasionally bristled at not receiving sole credit. Contemporaries recalled Legendre as meticulous but perhaps too ready to assume his own insight was complete. Nevertheless, these scholarly disputes—by 19th-century standards—reflect the normal process by which mathematics progresses: ideas are polished and completed by successive researchers.

Legacy

Today Legendre is remembered as one of the great 18th–19th century mathematicians. Many concepts bear his name. In number theory, the Legendre symbol $(a/p)$ is standard notation for quadratic residues, and the still-unkown Legendre’s conjecture remains a famous open problem. In analysis and mathematical physics, Legendre polynomials (and associated Legendre functions) appear constantly in solutions to Laplace’s equation, in spherical harmonics, in expanding gravitational and electrostatic potentials, and in quantum mechanics. The Legendre differential equation and Legendre transform (a change of variables in calculus) also honor his name. In mathematical statistics, the least-squares method is often taught with mention of both Gauss’s and Legendre’s roles in its discovery.

In geometry, Legendre’s theorem on spherical triangles (relating the excess of angles to area on a sphere) bears his name. His two analytical proofs that $\pi$ and $\pi ^{2}$ are irrational, though superseded by later work, were important early steps in understanding $\pi$. Legendre’s name even appears in modern programming libraries and scientific computing (for example, functions called `legendre(n,x)` or `legendre_symbol(a,p)` implement his ideas).

Legendre’s textbooks had the longest-lasting practical impact. Éléments de géométrie was used in education long after its author’s death; many 19th-century mathematicians learned geometry from it (it went through dozens of editions and translations) His compilations of number theory and elliptic integrals educated generations of researchers. Legendre’s approach—seeking to simplify, reorganize, and make explicit the mathematics of his time—set a standard for mathematical exposition. In summary, although some of his results were replaced or completed by others, Legendre’s work provided a foundation on which later giants of mathematics could build.

Selected Works

• Éléments de géométrie (1794) – An influential reorganization of Euclid’s Elements, including Legendre’s proofs of $\pi$ and $\pi ^{2}$ irrational and examinations of the parallel postulate
• Essai sur la théorie des nombres (1798; expanded 1808, 1830) – The first book titled “number theory,” gathering many results about primes and quadratic residues (introducing the Legendre symbol)
• Nouvelles méthodes pour la détermination des orbites des comètes (1806) – A treatise on computing comet orbits from observations, which first published the method of least squares
• Exercices de calcul intégral, 3 vols. (1811–1819) – Multi-volume work developing elliptic integrals, beta/gamma functions, and applications to mechanics and astronomy
• Traité des fonctions elliptiques, 3 vols. (1825–1830) – A reorganized and more complete theory of elliptic functions and integrals, containing tables and applications

These and countless papers in the Mémoires de l’Académie des Sciences and elsewhere spread Legendre’s ideas. He left an enduring mark: in modern mathematics courses and literature, phrases like Legendre’s Theorem, Legendre polynomial, or Legendre symbol are common, a testament to his influence.

References: Legendre’s life and work are documented in standard historical sources. For example, the MacTutor History of Mathematics site provides a detailed biography The Encyclopædia Britannica gives an overview of his contributions Modern summaries (such as the History and Mathematics blog) discuss his main ideas, including the Legendre symbol and polynomials These and other accounts confirm the dates and mathematical content summarized above.