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
-- [ #101 ]

Johann Bernoulli had a nephew, three sons and some grandsons who were all also outstanding mathematicians. Of these, the most important was his 2nd-oldest 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. Daniel 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
-- [ #4 ]

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. Some scholars rank Euler's 1748Introductio in analysin infinitorumabove Descartes'sGéométrie, Gauss'Disquisitiones, and even Newton'sPrincipia Mathematica. (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. Euler wrote the first definitive treatise on continued fractions, establishing several key theorems on that important topic. Although the invention of generating functions is attributed to DeMoivre, Euler took splendid advantage of the concept: 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 now called "one of his most profound discoveries", relevant in the theory of elliptic modular functions. Another marvelous theorem in partition theory due to Euler states that the number of partitions of any n into distinct parts equals the number of partitions of n into odd parts. (Euler first proved this with generating functions; there is also an exquisitely simple proof, mentioned more than a century later by Sylvester, based on a very simple bijection. I think it was Euler himself who first discovered that bijection, but despite much Googling I am unsure of this.)^{2}+m)/2Euler was a very major figure in number theory: He proved that the sum of the reciprocals of primes diverges (and is approx. ln (ln (p)) if the prime reciprocals up to 1/p are summed). He 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, proved

e=2.71828...to be irrational, discovered (though without complete proof) a broad class of transcendental numbers, and much more. One of the most famous unsolved problems was the 2000 year-old conjecture that all even perfect numbers must have the Mersenne number form that Euclid had discovered. Euler proved Euclid's Conjecture which is now called the Euclid-Euler Theorem.Euler was also first to prove several interesting theorems of geometry, including facts about the

9-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 is credited with the first proof of Fermat's Christmas Theorem (a prime of the form

4k+1is the sum of two squares in exactly one way). Along with three other theorems mentioned in this mini-bio, this means Euler is credited with no less than four of the "Ten Most Beautiful Theorems" selected by a mathematics magazine. In a separate list ("Hundred Most Important Theorems") prepared for a 1999 math conference, Euler is credited with seven of the theorems, well ahead of anyone but Euclid. (Two of these seven theorems aren't otherwise mentioned in this mini-bio: his famous solution to the Königsberg Bridges Problem, and his solutions to Pell's Equation.)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. Euler started as a young student of the Bernoulli family, and was Daniel Bernoulli's roommate in Saint Petersburg, where Euler was first employed as a teacher of physiology. But at age twenty-eight, 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. Euler and others developed alternate proofs and generalizations of this "Basel problem," and of course the ζ (zeta) function is now very famous. Here is an elegant geometric proof for this theorem. 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. This Product Formula leads directly to Riemann's Prime Number Theorem, with the associated Riemann Hypothesis.Even more famous is the identity (which Richard Feynman called an "almost astounding ... jewel") which unifies the trigonometric and exponential functions:

. And 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}= 6 ζ(2) = -log^{2}(-1) = 6 ∏_{p∈Prime}(1-p^{-2})^{-1/2}

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Alexis Claude Clairaut__
(1713-1765) France
-- [ #142 ]

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. (His experience in this survey made him aware of chromatic aberration. Clairaut worked on solving such aberration with a two-lens system, an invention that Newton had thought to be impossible.) 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. Clairaut at first thought that the inverse-square law was wrong, that an inverse-quartic term was needed as correction; Euler and d'Alembert agreed with this. But then Clairaut discovered that this was wrong, that the inverse-square law worked if it was applied with great rigor. Euler, the master of mathematical physics, had trouble understanding Clairaut's rigorous method. When Euler finally understood Clairaut's solution he called it "the most important and profound discovery that has ever been made in mathematics."That Halley's Comet was periodic was known in the time of Halley and Newton but the period varied due to the influence of Jupiter and Saturn, so great rigor was needed to predict its exact apparition in 1758. When Halley's Comet did reappear as he had predicted, Clairaut was acclaimed as "the new Thales."

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Jean-Baptiste le Rond d' Alembert__
(1717-1783) France
-- [ #60 ]

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
-- [ #76 ]

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 thatandπwere transcendental. He made advances in analysis (including the introduction ofeLambert'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
-- [ #7 ]

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 fundamental Theorems of Group Theory. (He did not complete the proof of Lagrange's Theorem -- that the order of a subgroup always divides the order of the group. Gauss, Cauchy, and Jordan each broadened the scope of this important theorem. Some other famous ancient theorems turn out to be corollaries of this Lagrange's Theorem.) He wrote an essay on the "Lagrange points" -- five equilbrium solutions for the three-body problem (three of which had previously been discovered by Euler). One of these points, L2, is in the news: the James Webb Telescope is placed there.Lagrange 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), and especially thePrinciple of Least Action(which Hamilton compared to poetry). 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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Antoine-Laurent Lavoisier__
(1743-1791) France
-- [ unranked ]

Antoine Lavoisier made chemistry a quantitative science. He was not a top mathematician, but deserves our attention as one of the most influential scientists ever. This polymath studied metereology, geology, botany, astronomy and more; he even earned a law degree though he never practiced law. He was also a successful businessman, and a major philanthropist. He devoted much successful effort to, and made important discoveries in, public issues like education, agriculture, water quality, air quality, prison hygiene; and earned a gold medal from the King for his essay on urban street lighting. He was a major figure in the design and adoption of the metric system.But it is Lavoisier's discoveries in chemistry which were most important; and he may be considered the greatest chemist ever. His careful experiments demonstrated the principle of conservation of mass. Water had been considered one of the basic "elements" since ancient times; it was Lavoisier who discovered that water could be decomposed into the elements oxygen and hydrogen in an 8:1 weight ratio. In addition to discovering and naming the elements oxygen and hydrogen, he also predicted the existence of elemental silicon. He developed new chemical nomenclature; for example he named copper sulfate and copper sulfite and demonstrated the difference between the two. (Before Lavoisier, copper sulfate was called "vitriol of Venus.") He was first to show the correct explanation of fire. Lavoisier and Laplace worked together on various experiments, e.g. establishing that animal heat derives from food fuel's combustion with oxygen, just as fire's heat does.

Although he donated most of his income to philanthropic causes he had become quite wealthy; and as a rich aristocrat he was guillotined during France's Reign of Terror. About this Lagrange (who had avoided deportation due to Lavoisier's intervention) said "It took them only an instant to cut off this head, and one hundred years might not suffice to reproduce its like."

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Gaspard Monge (Comte de Péluse)__
(1746-1818) France
-- [ #92 ]

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
-- [ #35 ]

Laplace was the preeminent mathematical astronomer, and is often called the "French Newton." His masterpiece wasMécanique Célestewhich 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 work 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 Antoine Lavoisier, helping to discover the elemental composition of water, and the natures of combustion, respiration and heat itself. Laplace noted that the laws of mechanics are the same with time's arrow reversed (though Newton had apparently noted this earlier). 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 introduced the controversial ("Bayesian") rule of succession and made progress toward the Law of Least Squares. 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
-- [ #99 ]

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 discovered and proved important corollaries to the pentagonal-number partition relationship discovered by Euler. He found key results in the theories of sums of squares and sums of k-gonal numbers. (For example, he showed that all integers except4can be expressed as the sum of three squares.) 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 that^{k}(8m+7)πwas 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
-- [ #69 ]

Joseph Fourier had a varied career: precocious but mischievous orphan, theology student (writing sermons at age 12), 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 Fourier-Motzkin Elimination and an early version of the simplex method; and also 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, Daniel Bernoulli, Lagrange and Laplace.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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Marie-Sophie Germain__ (1776-1831) France
-- [ #151 (tied) ]

Germain showed great skill at number theory (making much progress on Fermat's Last Theorem) and mathematical physics, but was hindered by misogyny. (One of the Top 200, but I just link to her bio at MacTutor.)

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Johann Carl Friedrich Gauss__
(1777-1855) Germany
-- [ #3 ]

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; his proof of Euclid's Fundamental Theorem of Arithmetic (Unique Prime Factorization) is considered the first rigorous proof; he extended this Theorem 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); his Theorema Egregium ("Remarkable Theorem") that a surface's essential curvature derived from its 2-D geometry laid the foundation of differential geometry. 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. (This theorem is so special that it has more published proofs than any other theorem except the Pythagorean Theorem. Eisenstein, Kummer, Cauchy, Jacobi, Liouville, and Lebesgue all discovered novel proofs of the Law of Quadratic Reciprocity.) 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. He also found a simpler proof for Fermat's Christmas Theorem, by taking advantage of the identity

x. Other work by Gauss led to fundamental theorems in statistics, vector analysis, function theory, and generalizations of the Fundamental Theorem of Calculus.^{2}+y^{2}= (x + iy)(x - iy)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 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 (a surveying instrument), and co-invented the telegraph. (The heliotrope, along with the heliograph it inspired, were still in use during the late 20th century.)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
-- [ #121 ]

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
-- [ unranked ]

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 theapproach for rigorous proofs in analysis; this work inspired the great Weierstrass.(ε, δ)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
-- [ #98 ]

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
-- [ #27 ]

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 Lemma was first discovered by Cauchy (and is properly called the Cauchy-Frobenius Orbit-Counting Theorem; 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
-- [ #129 ]

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. He may have been first to note that a 4th spatial dimension would allow any 3-D object to be rotated onto its mirror image.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
-- [ #103 ]

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
-- [ #151 (tied) ]

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
-- [ #63 ]

Jakob Steiner made many major advances in synthetic geometry, hoping that classical methods could avoid any need for analytic geometry methods, which he hated. 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 the very top geniuses (Archimedes, Newton, 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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