Biographies of the greatest mathematicians are in separate files by birth year:

Born before 400 | Born betw. 400 & 1559 | Born betw. 1560 & 1699 |

Born betw. 1700 & 1799 (this page)
| Born betw. 1800 & 1859 | Born betw. 1860 & 1975 |

List of Greatest Mathematicians |

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Daniel Bernoulli__
(1700-1782) Switzerland

Johann Bernoulli had a nephew, three sons and some grandsons who were all also outstanding mathematicians. Of these, the most important was his son Daniel. Johann insisted that Daniel study biology and medicine rather than mathematics, so Daniel specialized initially in mathematical biology. He went on to win the Grand Prize of the Paris Academy no less than ten times, and was a close friend of Euler. He developed partial differential equations, preceded Fourier in the use of Fourier series, did important work in statistics and the theory of equations, discovered and proved a key theorem about trochoids, developed a theory of economic risk (motivated by the St. Petersburg Paradox discovered by his cousin Nicholas), but is most famous for his key discoveries in mathematical physics: e.g. the Bernoulli Principle underlying airflight, and the notion that heat is simply molecules' random kinetic energy. Daniel Bernoulli is sometimes called the "Founder of Mathematical Physics."

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Leonhard Euler__
(1707-1783) Switzerland

Euler may be the most influential mathematician who ever lived (though some would make him second to Euclid); he ranks #77 on Michael Hart's famous list of the Most Influential Persons in History. His colleagues called him "Analysis Incarnate." Laplace, famous for denying credit to fellow mathematicians, once said "Read Euler: he is our master in everything." His notations and methods in many areas are in use to this day. Euler was the most prolific mathematician in history and is often judged to be the best algorist of all time. (This brief summary can only touch on a few highlights of Euler's work. The ranking #4 may seem too low for this supreme mathematician, but Gauss succeeded at proving several theorems which had stumped Euler.)Just as Archimedes extended Euclid's geometry to marvelous heights, so Euler took marvelous advantage of the analysis of Newton and Leibniz. He also gave the world modern trigonometry; pioneered (along with Lagrange) the calculus of variations; generalized and proved the Newton-Giraud formulae; and made important contributions to algebra, e.g. his study of hypergeometric series. He was also supreme at discrete mathematics, inventing graph theory. He also invented the concept of generating functions; for example, letting p(n) denote the number of partitions of n, Euler found the lovely equation:

Σ_{n}p(n) x^{n}= 1 / Π_{k}(1 - x^{k})

The denominator of the right side here expands to a series whose exponents all have the(3m"pentagonal number" form; Euler found an ingenious proof of this.^{2}+m)/2Euler was a very major figure in number theory: He proved that the sum of the reciprocals of primes less than x is approx. (

ln lnx), invented the totient function and used it to generalize Fermat's Little Theorem, found both the largest then-known prime and the largest then-known perfect number, provedeto be irrational, discovered (though without complete proof) a broad class of transcendental numbers, proved that all even perfect numbers must have the Mersenne number form that Euclid had discovered 2000 years earlier, and much more. Euler was also first to prove several interesting theorems of geometry, including facts about the9-point Feuerbach circle; relationships among a triangle's altitudes, medians, and circumscribing and inscribing circles; the famous Intersecting Chords Theorem; and an expression for a tetrahedron's volume in terms of its edge lengths. Euler was first to explore topology, proving theorems about theEuler characteristic, and the famous Euler's Polyhedral Theorem,F+V = E+2(although it may have been discovered by Descartes and first proved rigorously by Jordan). Although noted as the first great "pure mathematician," Euler's pump and turbine equations revolutionized the design of pumps; he also made important contributions to music theory, acoustics, optics, celestial motions, fluid dynamics, and mechanics. He extended Newton's Laws of Motion to rotating rigid bodies; and developed the Euler-Bernoulli beam equation. On a lighter note, Euler constructed a particularly "magical" magic square.Euler combined his brilliance with phenomenal concentration. He developed the first method to estimate the Moon's orbit (the three-body problem which had stumped Newton), and he settled an arithmetic dispute involving 50 terms in a long convergent series. Both these feats were accomplished when he was totally blind. (About this he said "Now I will have less distraction.") François Arago said that "Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind."

Four of the most important constant symbols in mathematics (

,π,e= √-1, andi= 0.57721566...) were all introduced or popularized by Euler, along with operators likeγ. He did important work withΣRiemann's zeta functionζ(s) = ∑ k(although it was not then known by that name); he anticipated the concept of analytic continuation by showing^{-s}ζ(-1) = 1+2+3+4+... = -1/12. As a young student of the Bernoulli family, Euler discovered the striking identityζ(2) =This catapulted Euler to instant fame, since the left-side infinite sum (π^{2}/61 + 1/4 + 1/9 + 1/16 + ...)was a famous problem of the time. Among many other famous and important identities, Euler proved the Pentagonal Number Theorem alluded to above (a beautiful result which has inspired a variety of discoveries), and the Euler Product Formulaζ(s) = ∏(1-pwhere the right-side product is taken over all primes^{-s})^{-1}p. His most famous identity (which Richard Feynman called an "almost astounding ... jewel") unifies the trigonometric and exponential functions:

. (It is almost wondrous how the particular instancee^{i x}=cosx + isinxcombines the most important constants and operators together.)e^{i π}+1 = 0Some of Euler's greatest formulae can be combined into curious-looking formulae for

π:π^{2}= -log^{2}(-1) = 6 ∏_{p∈Prime}(1-p^{-2})^{-1/2}

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Alexis Claude Clairaut__
(1713-1765) France

The reputations of Euler and the Bernoullis are so high that it is easy to overlook that others in that epoch made essential contributions to mathematical physics. (Euler made errors in his development of physics, in some cases because of a Europeanist rejection of Newton's theories in favor of the contradictory theories of Descartes and Leibniz.) The Frenchmen Clairaut and d'Alembert were two other great and influential mathematicians of the mid-18th century.Alexis Clairaut was extremely precocious, delivering a math paper at age 13, and becoming the youngest person ever elected to the

Paris Academy of Sciences. He developed the concept of skew curves (the earliest precursor of spatial curvature); he made very significant contributions in differential equations and mathematical physics. Clairaut supported Newton against the Continental schools, and helped translate Newton's work into French. The theories of Newton and Descartes gave different predictions for the shape of the Earth (whether the poles were flattened or pointy); Clairaut participated in Maupertuis' expedition to Lappland to measure the polar regions. Measurements at high latitudes showed the poles to be flattened: Newton was right. Clairaut worked on the theories of ellipsoids and the three-body problem, e.g. Moon's orbit. That orbit was the major mathematical challenge of the day, and there was great difficulty reconciling theory and observation. It was Clairaut who finally resolved this, by approaching the problem with more rigor than others. When Euler finally understood Clairaut's solution he called it "the most important and profound discovery that has ever been made in mathematics." Later, when Halley's Comet reappeared as he had predicted, Clairaut was acclaimed as "the new Thales."

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Jean-Baptiste le Rond d' Alembert__
(1717-1783) France

During the century after Newton, the Laws of Motion needed to be clarified and augmented with mathematical techniques. Jean le Rond, named after the Parisian church where he was abandoned as a baby, played a very key role in that development. His D'Alembert's Principle clarified Newton's Third Law and allowed problems in dynamics to be expressed with simple partial differential equations; his Method of Characteristics then reduced those equations toordinarydifferential equations; to solve the resultant linear systems, he effectively invented the method of eigenvalues; he also anticipated the Cauchy-Riemann Equations. These are the same techniques in use for many problems in physics to this day. D'Alembert was also a forerunner in functions of a complex variable, and the notions of infinitesimals and limits. With his treatises on dynamics, elastic collisions, hydrodynamics, cause of winds, vibrating strings, celestial motions, refraction, etc., the young Jean le Rond easily surpassed the efforts of his older rival, Daniel Bernoulli. He may have been first to speak of time as a "fourth dimension." (Rivalry with the Swiss mathematicians led to d'Alembert's sometimes being unfairly ridiculed, although it does seem true that d'Alembert had very incorrect notions of probability.)D'Alembert was first to prove that every polynomial has a complex root; this is now called the Fundamental Theorem of Algebra. (In France this Theorem is called the D'Alembert-Gauss Theorem. Although Gauss was first to provide a fully rigorous proof, d'Alembert's proof preceded, and was more nearly complete than, the attempted proof by Euler-Lagrange.) He also did creative work in geometry (e.g. anticipating Monge's Three Circle Theorem), and was principal creator of the major encyclopedia of his day. D'Alembert wrote "The imagination in a mathematician who creates makes no less difference than in a poet who invents."

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Johann Heinrich Lambert__
(1727-1777) Switzerland, Prussia

Lambert had to drop out of school at age 12 to help support his family, but went on to become a mathematician of great fame and breadth. He made key discoveries involving continued fractions that led him to prove thatπis irrational. (He proved more strongly thattan xandeare both irrational for any non-zero rational^{x}x. His proof for this was so remarkable for its time, that its completeness wasn't recognized for over a century.) He also conjectured thatπandewere transcendental. He made advances in analysis (including the introduction ofLambert's W function) and in trigonometry (introducing the hyperbolic functionssinhandcosh); proved a key theorem of spherical trigonometry, and solved the "trinomial equation." Lambert, whom Kant called "the greatest genius of Germany," was an outstanding polymath: In addition to several areas of mathematics, he made contributions in philosophy, psychology, cosmology (conceiving of star clusters, galaxies and supergalaxies), map-making (inventing several distinct map projections), inventions (he built the first practical hygrometer and photometer), dynamics, and especially optics (several laws of optics carry his name).Lambert is famous for his work in geometry, proving Lambert's Theorem (the path of rotation of a parabola tangent triangle passes through the parabola's focus). Lagrange declared this famous identity, used to calculate cometary orbits, to be the most beautiful and significant result in celestial motions. Lambert was first to explore straight-edge constructions without compass. He also developed non-Euclidean geometry, long before Bolyai and Lobachevsky did.

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Joseph-Louis (Comte de) Lagrange__
(1736-1813) Italy, France

Joseph-Louis Lagrange (born Giuseppe Lodovico Lagrangia) was a brilliant man who advanced to become a teen-age Professor shortly after first studying mathematics. He excelled in all fields of analysis and number theory; he made key contributions to the theories of determinants, continued fractions, and many other fields. He developed partial differential equations far beyond those of D. Bernoulli and d'Alembert, developed the calculus of variations far beyond that of the Bernoullis, discovered the Laplace transform before Laplace did, and developed terminology and notation (e.g. the use off'(x)andf''(x)for a function's 1st and 2nd derivatives). He proved a fundamental Theorem of Group Theory. He laid the foundations for the theory of polynomial equations which Cauchy, Abel, Galois and Poincaré would later complete. Number theory was almost just a diversion for Lagrange, whose focus was analysis; nevertheless he was the master of that field as well, proving difficult and historic theorems including Wilson's Theorem (pdivides(p-1)! + 1when p is prime); Lagrange's Four-Square Theorem (every positive integer is the sum of four squares); and thatn·xhas solutions for every positive non-square integer^{2}+ 1 = y^{2}n.Lagrange's many contributions to physics include understanding of vibrations (he found an error in Newton's work and published the definitive treatise on sound), celestial mechanics (including an explanation of why the Moon keeps the same face pointed towards the Earth), the

Principle of Least Action(which Hamilton compared to poetry), and the discovery of the Lagrangian points (e.g., in Jupiter's orbit). Lagrange's textbooks were noted for clarity and inspired most of the 19th-century mathematicians on this list. Unlike Newton, who used calculus to derive his results but then worked backwards to create geometric proofs for publication, Lagrange relied only on analysis. "No diagrams will be found in this work" he wrote in the preface to his masterpieceMécanique analytique.Lagrange once wrote "As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection." Both W.W.R. Ball and E.T. Bell, renowned mathematical historians, bypass Euler to name Lagrange as "the Greatest Mathematician of the 18th Century." Jacobi bypassed Newton and Gauss to call Lagrange "perhaps the greatest mathematical genius since Archimedes."

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Gaspard Monge (Comte de Péluse)__
(1746-1818) France

Gaspard Monge, son of a humble peddler, was an industrious and creative inventor who astounded early with his genius, becoming a professor of physics at age 16. As a military engineer he developed the new field of descriptive geometry, so useful to engineering that it was kept a military secret for 15 years. Monge made early discoveries in chemistry and helped promote Lavoisier's work; he also wrote papers on optics and metallurgy; Monge's talents were so diverse that he became Minister of the Navy in the revolutionary government, and eventually became a close friend and companion of Napoleon Bonaparte. Traveling with Napoleon he demonstrated great courage on several occasions.In mathematics, Monge is called the "Father of Differential Geometry," and it is that foundational work for which he is most praised. He also did work in discrete math, partial differential equations, and calculus of variations. He anticipated Poncelet's Principle of Continuity. Monge's most famous theorems of geometry are the Three Circles Theorem and Four Spheres Theorem. His early work in descriptive geometry has little interest to pure mathematics, but his application of calculus to the curvature of surfaces inspired Gauss and eventually Riemann, and led the great Lagrange to say "With [Monge's] application of analysis to geometry this devil of a man will make himself immortal."

Monge was an inspirational teacher whose students included Fourier, Chasles, Brianchon, Ampere, Carnot, Poncelet, several other famous mathematicians, and perhaps indirectly, Sophie Germain. Chasles reports that Monge never drew figures in his lectures, but could make "the most complicated forms appear in space ... with no other aid than his hands, whose movements admirably supplemented his words." The contributions of Poncelet to synthetic geometry may be more important than those of Monge, but Monge demonstrated great genius as an untutored child, while Poncelet's skills probably developed due to his great teacher.

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Pierre-Simon (Marquis de) Laplace__
(1749-1827) France

Laplace was the preeminent mathematical astronomer, and is often called the "French Newton." His masterpiece wasMecanique Celestewhich redeveloped and improved Newton's work on planetary motions using calculus. While Newton had shown that the two-body gravitation problem led to orbits which were ellipses (or other conic sections), Laplace was more interested in the much more difficult problems involving three or more bodies. (Would Jupiter's pull on Saturn eventually propel Saturn into a closer orbit, or was Saturn's orbit stable for eternity?) Laplace's equations had the optimistic outcome that the solar system was stable.Laplace advanced the nebular hypothesis of solar system origin, and was first to conceive of black holes. (He also conceived of multiple galaxies, but this was Lambert's idea first.) He explained the so-called secular acceleration of the Moon. (Today we know Laplace's theories do not fully explain the Moon's path, nor guarantee orbit stability.) His other accomplishments in physics include theories about the speed of sound and surface tension. He worked closely with Lavoisier, helping to discover the elemental composition of water, and the natures of combustion, respiration and heat itself. Laplace may have been first to note that the laws of mechanics are the same with time's arrow reversed. He was noted for his strong belief in determinism, famously replying to Napoleon's question about God with: "I have no need of that hypothesis."

Laplace viewed mathematics as just a tool for developing his physical theories. Nevertheless, he made many important mathematical discoveries and inventions (although the Laplace Transform itself was already known to Lagrange). He was the premier expert at differential and difference equations, and definite integrals. He developed spherical harmonics, potential theory, and the theory of determinants; anticipated Fourier's series; and advanced Euler's technique of generating functions. In the fields of probability and statistics he made key advances: he proved the Law of Least Squares, and introduced the controversial ("Bayesian") rule of succession. In the theory of equations, he was first to prove that any polynomial of even degree must have a real quadratic factor.

Others might place Laplace higher on the List, but he proved no fundamental theorems of

puremathematics (though his partial differential equation for fluid dynamics is one of the most famous in physics), founded no major branch of pure mathematics, and wasn't particularly concerned with rigorous proof. (He is famous for skipping difficult proof steps with the phrase "It is easy to see".) Nevertheless he was surely one of the greatestappliedmathematicians ever.

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Adrien Marie Legendre__
(1752-1833) France

Legendre was an outstanding mathematician who did important work in plane and solid geometry, spherical trigonometry, celestial mechanics and other areas of physics, and especially elliptic integrals and number theory. He found key results in the theories of sums of squares and sums of k-gonal numbers. He also made key contributions in several areas of analysis: he invented the Legendre transform and Legendre polynomials; the notation for partial derivatives is due to him. He invented the Legendre symbol; invented the study of zonal harmonics; proved thatπwas irrational (the irrationality of^{2}πhad already been proved by Lambert); and wrote important textbooks in several fields. Although he never accepted non-Euclidean geometry, and had spent much time trying to prove the Parallel Postulate, his inspiring geometry text remained a standard until the 20th century. As one of France's premier mathematicians, Legendre did other significant work, promoting the careers of Lagrange and Laplace, developing trig tables, geodesic projects, etc.There are several important theorems proposed by Legendre for which he is denied credit, either because his proof was incomplete or was preceded by another's. He proposed the famous theorem about primes in a progression which was proved by Dirichlet; proved and used the Law of Least Squares which Gauss had left unpublished; proved the N=5 case of Fermat's Last Theorem which is credited to Dirichlet; proposed the famous Prime Number Theorem which was finally proved by Hadamard; improved the Fermat-Cauchy result about sums of k-gonal numbers but this topic wasn't fruitful; and developed various techniques commonly credited to Laplace. His two most famous theorems of number theory, the Law of Quadratic Reciprocity and the Three Squares Theorem (a difficult extension of Lagrange's Four Squares Theorem), were each enhanced by Gauss a few years after Legendre's work. Legendre also proved an early version of Bonnet's Theorem. Legendre's work in the theory of equations and elliptic integrals directly inspired the achievements of Galois and Abel (which then obsoleted much of Legendre's own work); Chebyshev's work also built on Legendre's foundations.

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Jean Baptiste Joseph
Fourier__
(1768-1830) France

Joseph Fourier had a varied career: precocious but mischievous orphan, theology student, young professor of mathematics (advancing the theory of equations), then revolutionary activist. Under Napoleon he was a brilliant and important teacher and historian; accompanied the French Emperor to Egypt; and did excellent service as district governor of Grenoble. In his spare time at Grenoble he continued the work in mathematics and physics that led to his immortality. After the fall of Napoleon, Fourier exiled himself to England, but returned to France when offered an important academic position and published his revolutionary treatise on the Theory of Heat. Fourier anticipated linear programming, developing the simplex method and Fourier-Motzkin Elimination; and did significant work in operator theory. He is also noted for the notion of dimensional analysis, was first to describe the Greenhouse Effect, and continued his earlier brilliant work with equations.Fourier's greatest fame rests on his use of trigonometric series (now called

Fourier series) in the solution of differential equations. Since "Fourier" analysis is in extremely common use among applied mathematicians, he joins the select company of the eponyms of "Cartesian" coordinates, "Gaussian" curve, and "Boolean" algebra. Because of the importance of Fourier analysis, many listmakers would rank Fourier much higher than I have done; however the work was not exceptional aspuremathematics. Fourier's Heat Equation built on Newton's Law of Cooling; and the Fourier series solution itself had already been introduced by Euler, Lagrange and Daniel Bernoulli.Fourier's solution to the heat equation was counterintuitive (heat transfer doesn't seem to involve the oscillations fundamental to trigonometric functions): The brilliance of Fourier's imagination is indicated in that the solution had been

rejectedby Lagrange himself. Although rigorous Fourier Theorems were finally proved only by Dirichlet, Riemann and Lebesgue, it has been said that it was Fourier's "very disregard for rigor" that led to his great achievement, which Lord Kelvin compared to poetry.

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Johann Carl Friedrich Gauss__
(1777-1855) Germany

Carl Friedrich Gauss, the "Prince of Mathematics," exhibited his calculative powers when he corrected his father's arithmetic before the age of three. His revolutionary nature was demonstrated at age twelve, when he began questioning the axioms of Euclid. His genius was confirmed at the age of nineteen when he proved that the regular n-gon was constructible if and only if it is the product of distinct prime Fermat numbers. (He didn't complete the proof of the only-if part. Click to see construction of regular 17-gon.) Also at age 19, he proved Fermat's conjecture that every number is the sum of three triangle numbers. (He further determined the number of distinct ways such a sum could be formed.) At age 24 he publishedDisquisitiones Arithmeticae, probably the greatest book of pure mathematics ever.Although he published fewer papers than some other great mathematicians, Gauss may be the greatest theorem prover ever. Several important theorems and lemmas bear his name; he extended Euclid's Fundamental Theorem of Arithmetic (prime factorization is unique) to the Gaussian (complex) integers; and he was first to produce a rigorous proof of the Fundamental Theorem of Algebra (that an n-th degree polynomial has n complex roots). Gauss himself used "Fundamental Theorem" to refer to Euler's Law of Quadratic Reciprocity; Gauss was first to provide a proof for this, and provided eight distinct proofs for it over the years. Gauss proved the n=3 case of Fermat's Last Theorem for Eisenstein integers (the triangular lattice-points on the complex plane); though more general, Gauss' proof was simpler than the real integer proof; this simplification method revolutionized algebra. Other work by Gauss led to fundamental theorems in statistics, vector analysis, function theory, and generalizations of the Fundamental Theorem of Calculus.

Gauss built the theory of complex numbers into its modern form, including the notion of "monogenic" functions which are now ubiquitous in mathematical physics. (Constructing the regular 17-gon as a teenager was actually an exercise in complex-number algebra, not geometry.) Gauss developed the arithmetic of congruences and became the premier number theoretician of all time. Other contributions of Gauss include hypergeometric series, foundations of statistics, and differential geometry. He proved a surprising fundamental theorem about the curvature of manifolds. He also did important work in geometry, providing an improved solution to Apollonius' famous problem of tangent circles, stating and proving the

Fundamental Theorem of Normal Axonometry, and solving astronomical problems related to comet orbits and navigation by the stars. Ceres, the first asteroid, was discovered when Gauss was a young man; but only a few observations were made before it disappeared into the Sun's brightness. Could its orbit be predicted well enough to rediscover it on re-emergence? Laplace, one of the most respected mathematicians of the time, declared it impossible. Gauss became famous when he used an 8th-degree polynomial equation to successfully predict Ceres' orbit. Gauss also did important work in several areas of physics, developed an important modification to Mercator's map projection, invented the heliotrope, and co-invented the telegraph.Much of Gauss's work wasn't published: unbeknownst to his colleagues it was Gauss who first discovered non-Euclidean geometry (even anticipating Einstein by suggesting physical space might not be Euclidean), doubly periodic elliptic functions, a prime distribution formula, quaternions, foundations of topology, the Law of Least Squares, Dirichlet's class number formula, the key Bonnet's Theorem of differential geometry (now usually called Gauss-Bonnet Theorem), the butterfly procedure for rapid calculation of Fourier series, and even the rudiments of knot theory. Gauss was first to prove the Fundamental Theorem of Functions of a Complex Variable (that the line-integral over a closed curve of a monogenic function is zero), but he let Cauchy take the credit. Gauss was very prolific, and may be the most brilliant theorem prover who ever lived, so many would rank him #1. But several others on the list had more

historicalimportance. Abel hints at a reason for this: "[Gauss] is like the fox, who effaces his tracks in the sand."Gauss once wrote "It is not knowledge, but the act of learning, ... which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again ..."

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Siméon Denis Poisson__
(1781-1840) France

Siméon Poisson was a protégé of Laplace and, like his mentor, is among the greatest applied mathematicians ever. Poisson was an extremely prolific researcher and also an excellent teacher. In addition to important advances in several areas of physics, Poisson made key contributions to Fourier analysis, definite integrals, path integrals, statistics, partial differential equations, calculus of variations and other fields of mathematics. Dozens of discoveries are named after Poisson; for example the Poisson summation formula which has applications in analysis, number theory, lattice theory, etc. He was first to note the paradoxical properties of the Cauchy distribution. He made improvements to Lagrange's equations of celestial motions, which Lagrange himself found inspirational. Another of Poisson's contributions to mathematical physics was his conclusion that the wave theory of light implies a brightArago spotat the center of certain shadows. (Poisson used this paradoxical result to argue that the wave theory was false, but instead the Arago spot, hitherto hardly noticed, was observed experimentally.) Poisson once said "Life is good for only two things, discovering mathematics and teaching mathematics."

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Bernard Placidus Johann Nepomuk Bolzano__
(1781-1848) Bohemia

Bolzano was an ordained Catholic priest, a religious philosopher, and focused his mathematical attention on fields like metalogic, writing "I prized only ... mathematics which was ... philosophy." Still he made several important mathematical discoveries ahead of his time. His liberal religious philosophy upset the Imperial rulers; he was charged with heresy, placed under house arrest, and his writings censored. This censorship meant that many of his great discoveries turned up only posthumously, and were first rediscovered by others. He was noted for advocating great rigor, and is appreciated for developing the(ε, δ)approach for rigorous proofs in analysis; this work inspired the great Weierstrass.Bolzano gave the first analytic proof of the Fundamental Theorem of Algebra; the first rigorous proof that continuous functions achieve any intermediate value (Bolzano's Theorem, rediscovered by Cauchy); the first proof that a bounded sequence of reals has a convergent subsequence (Bolzano-Weierstrass theorem); was first to describe a nowhere-differentiable continuous function; and anticipated Cantor's discovery of the distinction between denumerable and non-denumerable infinities. If he had focused on mathematics and published more, he might be considered one of the most important mathematicians of his era.

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Jean-Victor Poncelet__
(1788-1867) France

After studying under Monge, Poncelet became an officer in Napoleon's army, then a prisoner of the Russians. To keep up his spirits as a prisoner he devised and solved mathematical problems using charcoal and the walls of his prison cell instead of pencil and paper. During this time he reinvented projective geometry. Regaining his freedom, he wrote many papers, made numerous contributions to geometry; he also made contributions to practical mechanics. Poncelet is considered one of the most influential geometers ever; he is especially noted for his Principle of Continuity, an intuition with broad application. His notion of imaginary solutions in geometry was inspirational. Although projective geometry had been studied earlier by mathematicians like Desargues, Poncelet's work excelled and served as an inspiration for other branches of mathematics including algebra, topology, Cayley's invariant theory and group-theoretic developments by Lie and Klein. His theorems of geometry include his Closure Theorem about Poncelet Traverses, the Poncelet-Brianchon Hyperbola Theorem, and Poncelet's Porism (if two conic sections are respectively inscribed and circumscribed by an n-gon, then there are infinitely many such n-gons). Perhaps his most famous theorem, although it was left to Steiner to complete a proof, is the beautiful Poncelet-Steiner Theorem about straight-edge constructions.

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Augustin-Louis Cauchy__
(1789-1857) France

Cauchy was extraordinarily prodigious, prolific and inventive. Home-schooled, he awed famous mathematicians at an early age. In contrast to Gauss and Newton, he was almost over-eager to publish; in his day his fame surpassed that of Gauss and has continued to grow. Cauchy did significant work in analysis, algebra, number theory and discrete topology. His most important contributions included convergence criteria for infinite series, the "theory of substitutions" (permutation group theory), and especially his insistence on rigorous proofs.Cauchy's research also included differential equations, determinants, and probability. He invented the calculus of residues, rediscovered Bolzano's Theorem, and much more. Although he was one of the first great mathematicians to focus on abstract mathematics (another was Euler), he also made important contributions to mathematical physics, e.g. the theory of elasticity. Cauchy's theorem of solid geometry is important in rigidity theory; the Cauchy-Schwarz Inequality has very wide application (e.g. as the basis for Heisenberg's Uncertainty Principle); several important lemmas of analysis are due to Cauchy; the famous Burnside's Counting Theorem was first discovered by Cauchy; etc. He was first to prove Taylor's Theorem rigorously, and first to prove Fermat's conjecture that every positive integer can be expressed as the sum of k k-gonal numbers for any k. (Gauss had proved the case k = 3.)

One of the duties of a great mathematician is to nurture his successors, but Cauchy selfishly dropped the ball on both of the two greatest young mathematicians of his day, mislaying key manuscripts of

bothAbel and Galois. Cauchy is credited with group theory, yet it was Galois who invented this first, abstracting it far more than Cauchy did, some of this in a work which Cauchy "mislaid." (For this historicalmiscontributionperhaps Cauchy should be demoted.)

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August Ferdinand Möbius__
(1790-1868) Germany

Möbius worked as a Professor of physics and astronomy, but his astronomy teachers included Carl Gauss and other brilliant mathematicians, and Möbius is most noted for his work in mathematics. He had outstanding intuition and originality, and prepared his books and papers with great care. He made important advances in number theory, topology, and especially projective geometry. Several inventions are named after him, such as the Möbius transformation and Möbius net of geometry, and the Möbius function and Möbius inversion formula of algebraic number theory. He is most famous for the Möbius strip; this one-sided strip was first discovered by Lister, but Möbius went much further and developed important new insights in topology.Möbius' greatest contributions were to projective geometry, where he introduced the use of homogeneous barycentric coordinates as well as signed angles and lengths. These revolutionary discoveries inspired Plücker, and were declared by Gauss to be "among the most revolutionary intuitions in the history of mathematics."

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Nicolai Ivanovitch Lobachevsky__
(1793-1856) Russia

Lobachevsky is famous for discovering non-Euclidean geometry. He did not regard this new geometry as simply a theoretical curiosity, writing "There is no branch of mathematics ... which may not someday be applied to the phenomena of the real world." He also worked in several branches of analysis and physics, anticipated the modern definition of function, and may have been first to explicitly note the distinction between continuous and differentiable curves. He also discovered the important Dandelin-Gräffe method of polynomial roots independently of Dandelin and Gräffe. (In his lifetime, Lobachevsky was under-appreciated and over-worked; his duties led him to learn architecture and even some medicine.)Although Gauss and Bolyai discovered non-Euclidean geometry independently about the same time as Lobachevsky, it is worth noting that both of them had strong praise for Lobachevsky's genius. His particular significance was in daring to reject a 2100-year old axiom; thus William K. Clifford called Lobachevsky "the Copernicus of Geometry."

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Michel Floréal Chasles__
(1793-1880) France

Chasles was a very original thinker who developed new techniques for synthetic geometry. He introduced new notions likepencilandcross-ratio; made great progress with thePrinciple of Duality; and showed how to combine the power of analysis with the intuitions of geometry. He invented atheory of characteristicsand used it to become the Founder of Enumerative Geometry. He proved a key theorem about solid body kinematics. His influence was very large; for example Poincaré (student of Darboux, who in turn was Chasles' student) often applied Chasles' methods. Chasles was also a historian of mathematics; for example he noted that Euclid had anticipated the method of cross-ratios.

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Jakob Steiner__
(1796-1863) Switzerland

Jakob Steiner made many major advances in synthetic geometry, hoping that classical methods could avoid any need for analysis; and indeed, like Isaac Newton, he was often able to equal or surpass methods of analysis or the calculus of variations using just pure geometry; for example he had pure synthetic proofs for a notable extension to Pascal's Mystic Hexagram, and a reproof of Salmon's Theorem that cubic surfaces have exactly 27 lines. (He wrote "Calculating replaces thinking while geometry stimulates it.") One mathematical historian (Boyer) wrote "Steiner reminds one of Gauss in that ideas and discoveries thronged through his mind so rapidly that he could scarcely reduce them to order on paper." Although thePrinciple of Dualityunderlying projective geometry was already known, he gave it a radically new and more productive basis, and created a new theory of conics. His work combined generality, creativity and rigor.Steiner developed several famous construction methods, e.g. for a triangle's smallest circumscribing and largest inscribing ellipses, and for its "Malfatti circles." Among many famous and important theorems of classic and projective geometry, he proved that the Wallace lines of a triangle lie in a 3-pointed hypocycloid, developed a formula for the partitioning of space by planes, a fact about the surface areas of tetrahedra, and proved several facts about his famous Steiner's Chain of tangential circles and his famous "Roman surface." Perhaps his three most famous theorems are the Poncelet-Steiner Theorem (lengths constructible with straightedge and compass can be constructed with straightedge alone as long as the picture plane contains the center and circumference of some circle), the Double-Element Theorem about self-homologous elements in projective geometry, and the Isoperimetric Theorem that among solids of equal volume the sphere will have minimum area, etc. (Dirichlet found a flaw in the proof of the Isoperimetric Theorem which was later corrected by Weierstrass.) Steiner is often called, along with Apollonius of Perga (who lived 2000 years earlier), one of the two greatest pure geometers ever. (The qualifier "pure" is added to exclude such geniuses as Archimedes, Newton and Pascal from this comparison. I've included Steiner for his extreme brilliance and productivity: several geometers had much more historic influence, and as

solelya geometer he arguably lacked "depth.")Steiner once wrote: "For all their wealth of content, ... music, mathematics, and chess are resplendently useless (applied mathematics is a higher plumbing, a kind of music for the police band). They are metaphysically trivial, irresponsible. They refuse to relate outward, to take reality for arbiter. This is the source of their witchery."

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