After brief bios of "The Top Thirty" (and a few of the "also-rans") are shown reasons for omitting ancient mathematicians and reasons for certain other omissions.

Here are the Greatest Mathematicians in chronological order:
 

Eudoxus of Cnidus (408-355 BC) Asia Minor, Greece

Although not rich, Eudoxus journeyed widely for his education, studying medicine with Philiston in Sicily, philosophy with Plato in Athens, mathematics in Egypt, touring the Eastern Mediterranean with his own students and finally returning to Cnidus where he established himself as astronomer and physician. What is known of him is second-hand, through the writings of Euclid and others, but he seems to have been one of the great mathematicians of the ancient world.

Many of the theorems in Euclid's Elements were first proved by Eudoxus. While Pythagoras had been horrified by the discovery of irrational numbers, Eudoxus is famous for incorporating them into arithmetic. He also developed the earliest techniques of the infinitesimal calculus. Eudoxus was the first person known to have recognized that the Earth rotates around the Sun.

Four of Eudoxus' most famous discoveries were the volume of a cone, extension of arithmetic to the irrationals, summing formula for geometric series, and viewing pi as the limit of polygonal perimeters. None of these seems difficult today, but it does seem remarkable that they were all first achieved by the same man.

Euclid of Megara & Alexandria (ca. 322 - ca. 275 BC) Greece/Egypt

Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers, and established the relationship between perfect numbers and Mersenne primes. Among several books attributed to him are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), and his comprehensive math textbook The Elements. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Euclid ranks #14 on Michael Hart's famous list of the Most Influential Persons in History. The Elements introduced the notions of axiom and theorem. It was used as a textbook for 2000 years and in fact is still the basis for high school geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem.

Archimedes of Syracuse (287-212 BC) Greece

Archimedes studied at Euclid's school (probably after Euclid's death), but his work far surpassed the works of Euclid. Archimedes made advances in number theory and algebra, but his greatest contributions were in geometry. His methods anticipated both the integral and differential calculus. His achievements are particularly impressive given the lack of good mathematical notation in his day.

His original achievements in physics include the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, and war machines. His works include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder. Archimedes proved that the volume of a sphere is two-thirds the volume of a circumscribing cylinder. He requested that a representation of such a sphere and cylinder be inscribed on his tomb.

Recently, modern technology has led to the discovery of new writings by Archimedes, hitherto hidden on a palimpsest. This has caused Archimedes to rise even higher in the esteem of mathematical historians. These new writings imply an understanding of the distinction between countable and uncountable infinities, a distinction which wasn't resolved until Georg Cantor, who lived 2300 years after the time of Archimedes.

Chang Tshang (ca 200-142 BC) China

Chinese mathematicians excelled for thousands of years, and were first to discover various algebraic and geometric principles, but they are denied credit because of Western ascendancy. Although there were great Chinese mathematicians a thousand years before the Han Dynasty, and innovations continued for centuries after Han, the textbook Nine Chapters on the Mathematical Art has special importance. Nine Chapters (known in Chinese as Jiu Zhang Suan Shu or Chiu Chang Suan Shu) was apparently written during the early Han Dynasty (about 165 BC) by Chang Tshang (also spelled Zhang Cang).

Many of the mathematical concepts of the early Greeks were discovered independently in early China. Chang's book gives methods of arithmetic (including cube roots) and algebra (including matrix-based solution of simultaneous equations), mentions infinitesimals and limits, uses the decimal system with zero and negative numbers, uses Cavalieri's Principle of solid geometry, proves the Pythagorean Theorem, and includes a clever geometric proof that the perimeter of a right triangle times the radius of its inscribing circle equals the area of its circumscribing rectangle. (Some of this may have been added after the time of Chang.)

Nine Chapters was probably based on earlier books, lost during the great book burning of 212 BC, so Chang himself may not have been the major creative genius. Moreover, important revisions and commentaries were added after Chang, notably by Liu Hui (ca 220-280). But even if we suppose Chang was a mere copyist, his book had immense historical importance. It was the dominant Chinese mathematical text for centuries, and had great influence throughout the Far East. Some of the teachings made their way to India, and from there to the Islamic world and Europe. There is some evidence that the Hindus borrowed the decimal system itself from books like Nine Chapters.

Aryabhatta (476-550) Ashmaka & Kusumapura (India)

Indian mathematicians excelled for thousands of years, and eventually even developed advanced techniques like Taylor series before Europeans did, but they are denied credit because of Western ascendancy. Among the Hindu mathematicians, Aryabhatta is probably most famous.

While Europe was in its early "Dark Age," Aryabhatta advanced arithmetic and algebra, using the decimal system; he developed continued fractions; he anticipated elementary calculus. Aryabhatta is sometimes considered the "Father of Algebra" instead of al-Khowarizmi (who himself cites the work of Aryabhatta). He is credited with the Aryabhatta Algorithm for solving Diophantine equations, and may have been first to introduce the constant e.

Aryabhatta's name is closely associated with trigonometry: he may have been the first to introduce inverse trig functions and spherical trigonometry. He calculated improved approximations for pi and trig functions.

Aryabhatta independently concluded that the planets rotate around the Sun but, unlike Eudoxus, realized the orbits were ellipses rather than circles.

Brahmagupta `Bhillamalacarya' (589-668) Rajasthan (India)

No one person gets unique credit for the invention of the decimal system but Brahmagupta's textbook Brahmasphutasiddhanta was most influential, and is sometimes considered the first textbook "to treat zero as a number in its own right." It also treated negative numbers. (Others claim these were first seen 800 years earlier in Chang Tshang's Chinese text and were implicit in what survives of Aryabhatta's work, but Brahmagupta's text discussed them lucidly.)

Brahmagupta Bhillamalacarya (`The Teacher from Bhillamala') made great advances in arithmetic, algebra, numeric analysis, and geometry. Several theorems bear his name, including the formula for the area of a cyclic quadrilateral, which can be written:
        16 A² = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)
Another famous Brahmagupta theorem dealing with such quadrilaterals can be phrased "In a circle, if the chords AB and CD are perpendicular and intersect at E, then the line from E which bisects AC will be perpendicular to BD." Proving Brahmagupta's theorems are good challenges even today.

In addition to his famous writings on practical mathematics and his ingenious theorems of geometry, Brahmagupta solved the general quadratic equation, and worked on Diophantine and Pell's equations. He applied mathematics to astronomy, predicting eclipses, etc.

Muhammed ibn Musa al-Khowarizmi (ca 780-850) Persia, Iraq

Al-Khowarizmi was a Persian who worked as a mathematician, astronomer and geographer early in the Golden Age of Islamic science. He introduced the Hindu decimal system to the Islamic world and Europe. He wrote the book Al-Jabr, which demonstrated simple algebra and geometry, and several other influential books including ones on calculating with the decimal system, trigonometry, geography, astronomy, the Hebrew calendar, etc. The word algorithm is borrowed from Al-Khowarizmi's name. Among several Muslim mathematicians who contributed to the development of Islamic science, and indirectly to Europe's later Renaissance, Al-Khowarizmi was the most famous and most influential.

Leonardo `Fibonacci' Pisano (ca 1170-1245) Italy

Leonardo (usually called Fibonacci today) introduced new methods of arithmetic to Europe, and relayed the mathematics of the Hindus, Persians, and Arabs. Others had translated Islamic mathematics, e.g. the works of al-Khowarizmi, into Latin, but Leonardo was the influential teacher. He re-introduced older Greek ideas like Mersenne numbers and Diophantine equations, and made original contributions in geometry and number theory. His writings cover a broad range including irrational numbers, the Chinese Remainder Theorem, theorems about Pythagorean triples, and the series 1, 1, 2, 3, 5, 8, 13, .... which is now linked with the name Fibonacci.

Leonardo provided Europe with the decimal system, algebra and the 'lattice' method of multiplication, all far superior to the methods then in use. His clever notation for quantities like 5 yards, 2 feet, and 3 inches is more efficient than today's notation. It seems hard to believe but before the decimal system, mathematicians had no notation for zero. Referring to this system, Gauss was later to exclaim "To what heights would science now be raised if Archimedes had made that discovery!" Leonardo Fibonacci is generally regarded as the best and most important European mathematician throughout the 1900 years separating Archimedes and Descartes; he borrowed ideas from many sources besides Islamic scientists and brought them to Europe's attention.

Had the Scientific Renaissance begun in the Islamic Empire, someone like al-Khowarizmi might be considered to have great historic significance, with Fibonacci a mere footnote. But the Renaissance did happen in Europe and there is no doubt that the writings of Leonardo `Fibonacci' Pisano played a key role in that development.

René Déscartes (1596-1650) France

Descartes' early career was that of soldier-adventurer and he finished as tutor to royalty, but in between he achieved fame as the preeminent intellectual of his day. He invented analytic geometry and is therefore called the "Father of Modern Mathematics." Because of his famous philosophical writings ("Cogito ergo sum") he is considered, along with Aristotle, to be one of the most influential thinkers in history. (He ranks #49 on Michael Hart's famous list of the Most Influential Persons in History.) Descartes made important contributions to physics (e.g. the law of conservation of momentum), and mathematical notation (e.g. the use of superscripts to denote exponents). His famous mathematical theorems include the Rule of Signs (for determining the signs of polynomial roots), and the elegant formula relating the radii of "Soddy kissing circles."

(Descartes has an extremely high reputation and would be ranked much higher by most list makers. I've demoted him partly because he had only insulting things to say about Pascal and Fermat, each of whom was more brilliant at mathematics than Descartes.)

Pierre de  Fermat (1601-1665) France

Fermat was a lawyer and government official; mathematics was his hobby, yet he made very major advances in both continuous and discrete mathematics, and practically founded modern number theory. Fermat is most remembered for `Fermat's Little Theorem', ubiquitous in number theory, and for his claim to have proved "Fermat's Last Theorem", but he did much other work as well. He proposed a system of analytic geometry before Descartes proposed his, and developed the elementary methods of differential and integral calculus which inspired Newton. Solving f'(x) = 0 to find extrema of f(x) is perhaps the most useful idea in applied mathematics; this technique is attributed to Fermat. In collaboration with Blaise Pascal, Fermat founded the theory of probability. Fermat also discovered basic principles of optics. (Some will think I've ranked Fermat too high, but he had great historic importance for both calculus and number theory.)

Fermat's contemporaneous rival Rene Descartes is more famous than Fermat, and Descartes' writings were more influential. Whatever one thinks of Descartes as a philosopher, however, it seems clear that Fermat was the better mathematician. Fermat and Descartes independently discovered analytic geometry, but Fermat followed this up by developing elementary calculus to determine minima, maxima and tangents. Fermat and Descartes did work in physics and independently discovered the (trigonometric) law of refraction, but only Fermat had the insight to realize that the refraction law implied that light has a finite speed !

Blaise Pascal (1623-1662) France

Pascal was an outstanding genius who studied geometry as a child. At the age of sixteen he stated and proved Pascal's Theorem, a fact relating any six points on any conic section. The Theorem is sometimes called the "Cat's Cradle" or the "Mystic Hexagram." Pascal followed up this result by showing that each of Appolonius' famous theorems about conic sections was a corollary of the Mystic Hexagram. Returning to geometry late in life, he advanced the theory of the cycloid. In addition to classic and projective geometry, Pascal founded probability theory, made contributions to axiomatic theory, and the invention of calculus. His name is associated with the Pascal's Triangle of combinatorics and Pascal's Wager in theology.

Like most of the greatest mathematicians, he was interested in physics and mechanics, studying fluids, explaining vacuum, and inventing the syringe and hydraulic press. At the age of eighteen Pascal designed and built the world's first automatic adding machine. (Although he continued to refine this invention, it was never a commercial success.)

Pascal abandoned mathematics for religion, suffered poor health, and died at an early age.

Takakazu Seki (Kowa) (1637?-1708) Japan

Seki Takakazu made several discoveries before Western mathematicians did; these include determinants, the Newton-Raphson method, Newton's interpolation formula, Bernoulli numbers, discriminants, methods of calculus, and probably much that has been forgotten (Japanese schools practiced secrecy). He is remembered as a brilliant genius and very influential teacher. He worked with magic squares, developed a new notation for algebra, and calculated pi to ten decimal places using Aitkin's method (rediscovered in the 20th century).

Seki's work was not propagated to Europe, so has minimal historic importance; otherwise Seki might rank high on our list.

Isaac  (Sir)  Newton (1642-1727) England

Newton was an industrious lad who built marvelous toys. His genius seems to have blossomed at about age 22 when, on leave from University, he began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. (Newton's other intellectual interests included theology and alchemy.) Although this list is concerned only with mathematics, Newton's greatness is indicated by the wide range of physical principles he was first to enunciate, including gravitation, and the idea that white light is a mixture of all the rainbow's colors.

Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (what he called the "method of fluxions"); his most crucial insight being what is now called the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus to solve a variety of problems: finding areas, tangents, the lengths of curves and the maxima and minima of functions. Other mathematical works include the Binomial Theorem and the numeric Method which still bears his name. An anecdote often cited to demonstrate his brilliance is the problem of the brachistochrone, which had baffled the best mathematicians in Europe, and came to Newton's attention late in life. He solved it in a few hours and published the answer anonymously. But on seeing the solution Johann Bernoulli immediately exclaimed "I recognize the lion by his footprint."

In 1687 Newton published  Philosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The motion of the planets was not understood before Newton, although the heliocentric system allowed Kepler to describe the orbits. In Principia Newton analyzed the consequences of his Laws of Motion and introduced the Law of Universal Gravitation. The notion that the Earth rotated about the Sun was first introduced by Eudoxus of Cnidus, but Newton explained why it did, and the Great Scientific Revolution began.

Newton ranks #2 on Michael Hart's famous list of the Most Influential Persons in History. (Muhammed the Prophet of Allah is #1.) Whatever the criteria, Newton would certainly rank at the top on any list of physicists, or scientists in general, but I've demoted him on this list: his emphasis was physics not mathematics, and Leibniz's contribution lessens the historical importance of Newton's calculus. A comment by Leibniz, however, persuades me to rank Newton near the top: Despite being a rival for the title of Inventor of Calculus, Leibniz once wrote "Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part."

Gottfried Wilhelm  Leibniz (1646-1716) Germany

As with Fermat, mathematics was a sideline for Leibniz, who was an historian, philosopher, and lawyer. He and Newton were the two undisputed great intellects of their era, and Leibniz might be remembered with a great awe if he had devoted himself to pure science.

Despite that his mathematics was a self-taught hobby, Leibniz was the leading pioneer of all three major branches of modern mathematics: the continuous, the discrete, and the symbolic. Leibniz's contribution to calculus was probably more influential than Newton's: Newton kept his results secret until after Leibniz published, and Leibniz's superior notation is used to this day.

While Leibniz didn't produce as much math as others on the list, he did do brilliant work. He invented the concept of matrix determinant; he designed the first calculator that could do multiplication; and he was the first to discover and prove the striking identity:
        pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

Leonhard  Euler (1707-1783) Switzerland

Euler made decisive contributions in all areas of mathematics. He gave the world modern trigonometry. Just as Archimedes extended Euclid's geometry to marvelous heights, so Euler took marvelous advantage of the analysis of Newton and Leibniz; he made important advances in mathematical physics. Two of the most important advances in 18th century were Lagrange's calculus of variations and Fourier's spectral series: in each case the key initial discovery was actually Euler's. He was the most prolific mathematician in history and the best algorist. His colleagues called him "Analysis Incarnate." He was supreme at discrete mathematics, as well as continuous: He invented graph theory and generating functions. Euler ranks #77 on Michael Hart's famous list of the Most Influential Persons in History. An indication of his importance is that four of the most important constant symbols in mathematics (pi, e, i = sqrt(-1), and gamma = 0.57721566...) were all introduced or popularized by Euler.

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 decimal places of a long convergent series. Both these feats were accomplished when he was totally blind.

As a young man, Euler discovered and proved the following:
        pi²/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ...
This striking identity catapulted Euler to instant fame, since the right-side infinite sum was a famous unsolved problem of the day. Another equation for which Euler is famous is e^{i x} = cos x + i sin x.

Joseph-Louis (Comte de)  Lagrange (1736-1813) Italy/France

Lagrange was a brilliant man who became Professor of Mathematics at an early age. 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 invented partial differential equations, and the calculus of variations. He proved a fundamental Theorem of Group Theory, as well as two number theory theorems of great historic interest: Wilson's prime-number theorem, and the fact that every positive integer is the sum of four squares. He laid the foundations for the theory of polynomial equations which Cauchy, Abel, Galois and Poincaré would later complete. His many contributions to physics include understanding of vibrations and celestial mechanics, the principle of least action (which Hamilton compared to poetry), and the discovery of the Lagrangian points (e.g., in Jupiter's orbit).

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 masterpiece Mécanique analytique. (The metric system, using base 10, may owe its existence to Lagrange: There was a strong movement to use base 12, which Lagrange satirized by proposing base 11.)

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

Laplace was the preeminent mathematical astronomer, and is often called the "French Newton." His masterpiece was Mecanique Celeste which 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 and multiple galaxies. 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 was noted for his strong belief in determinism, famously replying to Napolean'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, most notably the Laplace Transform. He was the premier expert at differential and difference equations, and definite integrals. He developed spherical harmonics, potential theory, the theory of determinants, and advanced Euler's technique of generating functions. In the fields of probability and statistics he made important 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.

While others might place Laplace high on the List, he proved no fundamental theorems of pure mathematics (though his partial differential equation for fluid dynamics is one of the most famous in physics); he 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 greatest applied mathematicians ever.

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 n is the product of prime Fermat numbers. At age 24 he published Disquisitiones Arithmeticae, probably the greatest book of pure mathematics ever.

Gauss built the theory of complex numbers into its modern form, including the notion of "monogenic" functions which are now ubiquitous in mathematical physics. The other contributions of Gauss are quite numerous and include the Fundamental Theorem of Algebra (that an n-th degree polynomial has n complex roots), foundations of statistics (including Law of Least Squares) and differential geometry. He was the premier number theoretician of all time, proving Euler's Law of Quadratic Reciprocity. He also did important work in several areas of physics. Much of Gauss's work wasn't published: unbeknownst to his colleagues it was Gauss who first discovered doubly periodic elliptic functions, non-Euclidean geometry, quaternions, foundations of topology, the butterfly procedure for rapid calculation of Fourier series, and even the rudiments of knot theory. Also in this category is the Fundamental Theorem of Functions of a Complex Variable (that the line-integral over a closed curve of a monogenic function is zero): he proved this first but let Cauchy take the credit.

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 important work in analysis, algebra and number theory. One of his important contributions was the "theory of substitutions" (permutation group theory).

Cauchy's research also included convergence of infinite series, differential equations, determinants, and probability. He invented the calculus of residues. 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. He was the first to prove Fermat's conjecture that every positive integer can be expressed as the sum of k k-gon numbers for any k, and also refined Euler's results in discrete topology. Another of Cauchy's contributions was his insistence on rigorous proofs.

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 the key manuscripts of both Abel and Galois. For this historical miscontribution I've demoted Cauchy slightly.

Niels Henrik Abel (1802-1829) Norway

At an early age, Niels Abel studied the works of the greatest mathematicians, found flaws in their proofs, and resolved to reprove some of these theorems rigorously. He was the first to fully prove the general case of Newton's Binomial Theorem, one of the most widely applied theorems in mathematics. Perhaps his most famous achievement was the (deceptively simple) Abel's Theorem of Convergence (published posthumously), one of the most important theorems in analysis; but there are several other Theorems which bear his name. Abel also made contributions in algebraic geometry and the theory of equations.

Inversion (replacing y = f(x) with x = f^-1(y)) is a key idea in mathematics (consider Newton's Fundamental Theorem of Calculus); Abel developed this insight. One of the most respected mathematicians of Abel's day had spent a lifetime studying elliptic integrals, but Abel inverted these to get elliptic functions, which quickly became a productive field of mathematics, and led to more general complex-variable functions, which were important to the development of both abstract and applied mathematics.

Finding the roots of polynomials is a key mathematical problem: the general solution of the quadratic equation may have been known by ancients, and the discovery of general methods for solving polynomials of degree three and four is usually treated as the major math achievement of the 16th century, so for over two centuries an algebraic solution for the general 5th-degree polynomial (quintic) was a Holy Grail sought by most of the greatest mathematicians. Abel proved that most quintics did not have such solutions. This discovery, at the age of only nineteen, would have quickly awed the world, but Abel was impoverished, had few contacts, and spoke no German. When Gauss received Abel's manuscript he discarded it unread, assuming the unfamiliar author was just another crackpot trying to square the circle or some such. His genius was too great for him to be ignored long, but, still impoverished, Abel died of tuberculosis at the age of twenty-six. (Abel achieved a distinction attained by very few humans: his name in lower-case letters, or the form 'abelian', is applied to several concepts.)

Carl G. J. Jacobi (1804-1851) Germany

Jacobi was a prolific mathematician who did decisive work in the algebra and analysis of complex variables, and did work in number theory (e.g. cubic reciprocity) which excited Carl Gauss. He is sometimes described as the successor to Gauss. As an algorist (manipulator of involved algebraic expressions), he may have been surpassed only by Euler and Ramanujan. Jacobi was also an especially inspirational math teacher.

Jacobi's most important early achievement was the theory of elliptic functions. He also made important advances in many other areas, including higher fields, number theory, algebraic geometry, differential equations, theta functions, q-series, determinants, Abelian functions, and physics. He devised the algorithms still used to calculate eigenvectors and for other important matrix manipulations. Jacobi was the first to apply elliptic functions to number theory, producing a new proof of Fermat's famous conjecture (Lagrange's theorem) that every integer is the sum of four squares.

Like Abel, as a young man, Jacobi attempted to factor the general quintic equation. Unlike Abel, he seems never to have considered proving its impossibility. This fact is sometimes cited to show that despite Jacobi's creativity, his ill-fated contemporary was the more brilliant genius.

William Rowan (Sir) Hamilton (1805-1865) Ireland

Hamilton was a childhood prodigy. Home-schooled and self-taught, he started as a student of languages and literature, was influenced by an arithmetic prodigy his own age, read Euclid, Newton and Lagrange, found an error by Laplace, and made new discoveries in optics; all this before the age of seventeen when he first attended school!

In school he continued to excel, supplementing his mathematics with studies of literature, theology, astronomy and physics. His undergraduate days were cut short abruptly by his appointment as Trinity Professor of Astronomy at the age of 22. He soon began publishing his revolutionary treatises on optics, in which he introduced the Principle of Least Action, which became a major influence on quantum mechanics. He predicted that some crystals would have an hitherto unknown `conical' refraction mode; this was confirmed experimentally.

Hamilton also made revolutionary contributions to dynamics, differential equations, the theory of equations, numerical analysis, fluctuating functions, and graph theory. He invented the hodograph. Hamilton himself considered his greatest accomplishment to be the development of quaternions, a non-Abelian field to handle 3-D rotations. This was once considered a very important method, but has since been superseded by the methods of matrices and tensors.

Évariste Galois (1811-1832) France

Galois, who died before the age of twenty-one, not only never became a professor, but was barely allowed to study as an undergraduate. His output of papers, mostly published posthumously, is by far the smallest of anyone on this list, yet it is considered among the most awesome works in mathematics. He applied group theory to the theory of equations, revolutionizing both fields. While Abel was the first to prove that some polynomial equations had no algebraic solutions, Galois established the necessary and sufficient conditions for algebraic solutions to exist.

Galois' tormented life with its pointless early end is one of the great tragedies of mathematical history.

Karl Wilhelm Theodor Weierstrass (1815-1897) Germany

Weierstrass devised new definitions for the primitives of calculus and was then able to prove several fundamental but hitherto unproven theorems. Weierstrass shocked his colleagues when he demonstrated a continuous function which is differentiable nowhere. He is now called the "Father of Modern Analysis." Starting strictly from the integers, he also applied his axiomatic methods to a definition of irrational numbers.

Although he demonstrated great brilliance as a youth, Weierstrass' early career was as a secondary school teacher. During this time he studied Abel's papers, developed results in elliptic and Abelian functions, and independently proved the Fundamental Theorem of Functions of a Complex Variable. He was interested in power series and felt that others had overlooked the importance of Abel's Theorem. Eventually one of his papers was published in a journal; he was immediately given an honorary doctorate and was soon regarded as one of the best mathematicians in the world.

Arthur Cayley (1821-1895) England

Cayley was one of the most prolific mathematicians ever, but also a well-rounded man: In addition to his life-long love of mathematics, he enjoyed hiking, painting, reading fiction, and had a happy married life. He worked as a lawyer for many years, then as professor, and finished his career in the limelight as President of the British Association for the Advancement of Science. He and the great mathematician James Joseph Sylvester (1814-1897) were a source of inspiration to each other. These two, along with Charles Hermite, are considered the founders of the important theory of invariants. Though applied first to algebra, the notion of invariants is useful in many areas of mathematics.

A list of the branches of mathematics Cayley pioneered will seem like an exaggeration: he was the essential founder of modern group theory, matrix algebra, and higher dimensional geometry, as well as the theory of invariants. He stated and proved the Cayley-Hamilton Theorem. He also did original research in combinatorics, elliptic and Abelian functions, and projective geometry (one of his many famous theorems is a generalization of Pascal's Mystic Hexagram result).

Charles Hermite (1822-1901) France

Hermite studied the works of Lagrange and Gauss from an early age and soon developed an alternate proof of Abel's famous quintic impossibility result. He attended the same college as Galois and also had trouble passing their examinations, but soon became highly respected by Europe's greatest mathematicians for his successes in number theory and elliptic functions. Along with Cayley and Sylvester, he founded the important theory of invariants. He was a kindly modest man who inspired his colleagues.

Although he and Abel had proved that the general quintic lacked algebraic solutions, Hermite introduced an elliptic analog to the circular trigonometric functions and used these to provide a general solution for the quintic equation. He developed the concept of complex conjugate which is now ubiquitous in mathematical physics and matrix theory. He was first to prove that the Stirling and Euler generalizations of the factorial function are equivalent. Perhaps Hermite's most famous result was the proof that e is transcendental.

Georg Friedrich Bernhard  Riemann (1826-1866) Germany

Riemann was a fantastic genius whose work was both novel and rigorous. He had poor physical health and died at an early age, but still made revolutionary contributions in many areas of mathematics. He applied topology to analysis, and applied analysis to number theory. His single paper on the Prime Number distribution conjecture is considered the most important ever on that frequently studied topic. He introduced the clarifying notion of the Riemann integral. He posed the "Hypothesis of Riemann's zeta function," which is regarded as the most important and famous unsolved problem in mathematics. His masterpieces were differential geometry, tensor analysis, non-Euclidean geometry, the theory of functions, and, especially, the theory of manifolds. He generalized the notions of distance and curvature and, therefore, described new possibilities for the geometry of space itself.

Like the greatest mathematicians (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. Although his theory unifying electricity, magnetism and light was supplanted by Maxwell's theory, modern physics, beginning with Einstein's relativity, relies on Riemann's notions of the geometry of space.

Georg  Cantor (1845-1918) Russia, Germany

Cantor single-handedly created modern set theory, defining cardinal numbers, well-ordering, ordinal numbers, and discovering the Theory of Transfinite Numbers. He defined equality between cardinal numbers based on the existence of a bijection, and was the first to demonstrate that the real numbers have a higher cardinal number than the integers. (More surprising is that the rationals have the same cardinality as the integers; the reals have the same cardinality as the points of N-space.)

Although there are infinitely many distinct transfinite numbers, Cantor conjectured that C, the cardinality of the reals, was the second smallest transfinite number. This "Continuum Hypothesis" was included in Hilbert's famous List of Problems, and was finally resolved many years later: Cantor's Continuum Hypothesis is an "Undecidable Statement" of Set Theory.

Cantor's revolutionary set theory attracted vehement opposition from Poincare ("grave disease"), Kronecker (Cantor was a "charlatan" and "corrupter of youth"), Wittgenstein ("laughable nonsense"), and even theologians. Despite this, Cantor's invention of modern set theory is now considered one of the most important achievements in modern mathematics.

Cantor also made advances in number theory and trigonometric series. He gave the modern definition of irrational numbers, and anticipated the theory of fractals.

Jules Henri  Poincaré (1854-1912) France

Poincaré was clumsy and frail and supposedly flunked an IQ test, but he was one of the most creative mathematicians ever, and surely the greatest mathematician of the "intuitionist" style. He produced a large amount of brilliant work in all areas of mathematics, but also found time to become a famous popular writer of philosophy. His masterpieces include combinatorial (or algebraic) topology, the theory of differential equations, foundations of homology, the theory of periodic orbits, and the discovery of automorphic functions (a unifying foundation for the trigonometric and elliptic functions). He anticipated modern chaos theory. He posed "Poincare's conjecture," which for an entire century was one of the most famous unsolved problems in mathematics and which can be explained without equations to a layman (provided the layman can visualize 3-D surfaces in 4-space). Recently Poincare's conjecture was settled and the first Million Dollar math prize in history is likely to be awarded.

As with all the greatest mathematicians, Poincaré was interested in physics. He made revolutionary advances in fluid dynamics and celestial motions. With his fame, he helped the world recognize the importance of the new physical theories of Einstein and Planck.

David  Hilbert (1862-1943) Prussia, Germany

Hilbert excelled in many fields of mathematics, including axiomatic theory, invariant theory, algebraic number theory, class field theory and functional analysis. His examination of calculus led him to the invention of "Hilbert space," considered one of the key concepts of functional analysis and modern mathematical physics. He was a founder of fields like metamathematics and modern logic, and is sometimes considered the founder of the "formalist" school. He developed a new system of definitions and axioms for geometry, replacing the 2200 year-old system of Euclid. As a young Professor he proved his "Finiteness Theorem," now regarded as one of the most important results of general algebra. The methods he used were so novel that, at first, the "Finiteness Theorem" was rejected for publication as being "Theology" not mathematics! In number theory, he proved Waring's famous conjecture which is now known as the Hilbert-Waring theorem.

Any one man can only do so much, so the greatest mathematicians should help nurture their colleagues. Hilbert is most famous for his List of 23 Unsolved Problems, which inspired and directed the development of 20th-century mathematics. Hilbert was warmly regarded by his colleagues and students, and contributed to the careers of several great mathematicians and physicists including Georg Cantor, Hermann Minkowski, Hermann Weyl, John von Neumann, Emmy Noether, Alonzo Church, and Albert Einstein.

Eventually Hilbert turned to physics and made key contributions to classical and quantum physics and to general relativity. (Hilbert was a modest man: some historians believe the "Einstein Field Equations" should carry Hilbert's name.)

Godfrey Harold Hardy (1877-1947) England

Hardy was an extremely prolific research mathematician who did important work in analysis (especially the theory of integration), number theory, global analysis, and analytic number theory. He was also an excellent teacher and wrote several excellent textbooks, as well as a famous treatise on the mathematical mind. Although he emphasised pure mathematics (actually abhorring applied mathematics), his work has found application in population genetics, cryptography, thermodyanamics and particle physics.

Hardy is especially famous (and important) for his encouragement of and collaboration with Ramanujan. Among many results of this collaboration was the Hardy-Ramanujan Formula for partition enumeration, which Hardy later used as a model to develop the Hardy-Littlewood Circle Method; Hardy first used this method to prove stronger versions of the Hilbert-Waring theorem, and in prime number theory; the method has continued to be a very productive tool in analytic number theory.

Hermann Klaus Hugo Weyl (1885-1955) Germany, U.S.A.

Weyl studied under Hilbert and became one of the premier mathematicians of the 20th century. Weyl was also a very influential figure in all three major fields of 20th century physics: relativity, unified field theory and quantum physics. He excelled at many fields including integral equations, harmonic analysis, analytic number theory, and the foundations of mathematics, but he is most respected for his revolutionary advances in geometric function theory (e.g., differentiable manifolds), the theory of compact groups (incl. representation theory), and theoretical physics (e.g., Weyl tensor, gauge field theory and invariance).

Weyl once wrote: "My work always tried to unite the Truth with the Beautiful, but when I had to choose one or the other, I usually chose the Beautiful."

Srinivasa Ramanujan Iyengar (1887-1920) India

Like Abel, Ramanujan was a self-taught prodigy who lived in a country distant from his mathematical peers, and suffered from poverty: childhood dysentery and vitamin deficiencies probably led to his early death. Yet he produced 4000 theorems or conjectures in number theory, algebra, and combinatorics. His specialties included infinite series, elliptic functions, continued fractions, partition enumeration, definite integrals, modular equations, gamma functions, mock theta functions, hypergeometric series, and "highly composite" numbers. Much of his methodology, including unusual ideas about divergent series, was his own invention. Ramanujan's innate ability for algebraic manipulations equaled or surpassed that of Euler and Jacobi. Although many formulae have been discovered to calculate pi, a bizarre formula of Ramanujan is often used, because of its fast convergence. Many of Ramanujan's results would probably never have been discovered without him, and are so inspirational that there is a periodical dedicated to them. The theories of strings and crystals have benefited from Ramanujan's work. (Today some professors achieve fame just by finding a new proof for one of Ramanujan's many results.)

Unlike Abel, who insisted on rigorous proofs, Ramanujan often omitted proofs. Unlike Abel, most of whose work specifically depended on the complex numbers, Ramanujan mostly worked only with real numbers. Despite these limitations, Ramanujan is considered one of the greatest geniuses ever.

John  von Neumann (1903-1957) Hungary, U.S.A.

John von Neuman (born Neumann Janos Lajos) was a childhood prodigy who could do very complicated mental arithmetic at an early age. As an adult he was noted for hedonism and reckless driving but also became one of the most prolific geniuses in history, making major contributions to a large variety of branches of mathematics, as well as to quantum physics, economics and computer science.

Von Neumann pioneered the use of models in set theory, thus improving the axiomatic basis of mathematics; he developed von Neumann Algebras; he invented and developed game theory; he invented cellular automata, famously constructing a self-reproducing automata. He also worked in analysis, operator theory, matrix theory, statistics and topology. He inspired some of Godel's famous work. He is credited with (partial) solution to Hilbert's 5th Problem.

Von Neumann did very important work in fields other than mathematics. By treating the universe as a very-high dimensional phase space, he constructed an elegant mathematical basis (now called von Neumann algebras) for the principles of quantum physics. He advanced philosophical questions about time and logic in modern physics. He played a key role in the design of conventional, nuclear and thermonuclear bombs. He applied game theory and Brouwer's fixed-point theorem to economics, becoming a major figure in that field. His contributions to computer science are many: in addition to co-inventing the stored-program computer, he was first to use pseudo-random number generation, finite element analysis, the merge-sort algorithm, and a "biased coin" algorithm. At the time of his death he was working on a theory of the human brain.

Andrey Nikolaevich  Kolmogorov (1903-1987) Russia

Kolmogorov had a powerful intellect and excelled in many fields. As a youth he dazzled his teachers by constructing toys that appeared to be "Perpetual Motion Machines." At the age of 19, he achieved fame by finding a Fourier series that diverges almost everywhere, and decided to devote himself to mathematics. He is considered the founder of the fields of intuitionistic logic, algorithmic complexity theory, and modern probability theory. He also excelled in topology, set theory, trigonometric series, and random processes. He (and his student) resolved Hilbert's 13th Problem. While Kolmogorov's work in probability theory had direct applications to physics, Kolmogorov also did work in physics directly, especially the study of turbulence. There are dozens of theorems or equations named after Kolmogorov, such as the "Kolmogorov backward equation" and the intriguing Zero-One Law of "tail events" among random variables.

Paul Erdos (1913-1996) Hungary, U.S.A., Israel, etc.

Erdos was a childhood prodigy who became a famous (and famously eccentric) mathematician. He is best known for work in Ramsey Theory, but made contributions in many other fields of mathematics, including graph theory, analytic number theory, probabilistic methods, approximation theory, and combinatorics. He is regarded as the second most prolific mathematician in history, behind only Euler. (Euler actually published fewer papers than Erdos, but most of Erdos' papers had co-authors -- he was famous for collaboration.)

Erdos discovered newer, more elegant, proofs of several existing theorems, including the Prime Number Theorem. He also proved many original theorems, perhaps the most famous being the Erdos-Szekeres Theorem about monotone subsequences with its elegant (or trivial) pigeonhole-principle proof.

Despite that he is widely regarded as an important and influential mathematician, Erdos founded no new field of mathematics: He was a "problem solver" rather than a "theory developer." Nevertheless many mathematicians would want to include him on a List such as ours.

Alexander Grothendieck (1928-present) Germany, France

Grothendieck has done brilliant work in several areas of mathematics including number theory, functional (and topological) analysis, but especially in the fields of algebraic geometry and category theory, both of which he revolutionized. He is considered a master of abstraction, rigor and presentation. He has produced many important and deep results in homological algebra, most notably his etale cohomology. He developed the theory of sheafs, invented the theory of schemes, and much more. He is most famous for his methods to unify different branches of mathematics, for example using algebraic geometry in number theory.

Grothendieck's radical political philosophy led him to retire from public life while still in his prime, but he is still considered one of the most brilliant mathematicians ever.

Mathematicians of Antiquity

Omissions from the List