Tuesday, January 17, 2012

The Greatest Mathematicians of All Time world wide

  1. Isaac Newton
  2. Archimedes
  3. Carl F. Gauss
  4. Leonhard Euler
  5. Bernhard Riemann
  1. Euclid
  2. Henri Poincaré
  3. Joseph-Louis Lagrange
  4. David Hilbert
  5. Gottfried W. Leibniz
  1. Alexandre Grothendieck
  2. Pierre de Fermat
  3. Niels Abel
  4. Évariste Galois
  5. John von Neumann

  1. Karl W. T. Weierstrass
  2. René Déscartes
  3. Brahmagupta
  4. Carl G. J. Jacobi
  5. Srinivasa Ramanujan
  1. Augustin Cauchy
  2. Peter G. L. Dirichlet
  3. Hermann K. H. Weyl
  4. Eudoxus of Cnidus
  5. Georg Cantor
  1. Muhammed al-Khowârizmi
  2. Arthur Cayley
  3. Emma Noether
  4. Pythagoras of Samos
  5. Leonardo `Fibonacci'

At some point a longer list will become a List of Great Mathematicians rather than a List of Greatest Mathematicians. I've expanded the List to an even Hundred, but you may prefer to reduce it to a Top Seventy, Top Sixty, Top Fifty, Top Forty or Top Thirty list, or even Top Twenty, Top Fifteen or Top Ten List.
  1. Kurt Gödel
  2. Aryabhatta
  3. Charles Hermite
  4. Apollonius of Perga
  5. Blaise Pascal
  1. Pierre-Simon Laplace
  2. Richard Dedekind
  3. Diophantus of Alexandria
  4. William R. Hamilton
  5. Bháscara Áchárya
  1. Gaspard Monge
  2. George Boole
  3. Stefan Banach
  4. Jean le Rond d'Alembert
  5. Felix Christian Klein

  1. Ferdinand Eisenstein
  2. Jacques Hadamard
  3. François Viète
  4. Johannes Kepler
  5. Élie Cartan
  1. Jean-Victor Poncelet
  2. Archytas of Tarentum
  3. Jacob Bernoulli
  4. Hipparchus of Nicaea
  5. André Weil
  1. Godfrey H. Hardy
  2. Andrey N. Kolmogorov
  3. Joseph Fourier
  4. Siméon-Denis Poisson
  5. Julius Plücker

  1. Alhazen ibn al-Haytham
  2. Liu Hui
  3. F.E.J. Émile Borel
  4. Christiaan Huygens
  5. John Wallis
  1. L.E.J. Brouwer
  2. Michael F. Atiyah
  3. Carl Ludwig Siegel
  4. Henri Léon Lebesgue
  5. Hermann G. Grassmann
  1. Jakob Steiner
  2. Pappus of Alexandria
  3. John E. Littlewood
  4. Jean-Pierre Serre
  5. M. E. Camille Jordan

  1. Atle Selberg
  2. Pafnuti Chebyshev
  3. Shiing-Shen Chern
  4. Giuseppe Peano
  5. Panini (of Shalatula)
  1. Johann Bernoulli
  2. Francesco B. Cavalieri
  3. Adrien M. Legendre
  4. Ernst E. Kummer
  5. James J. Sylvester
  1. Hippocrates of Chios
  2. Emil Artin
  3. Hermann Minkowski
  4. Alan M. Turing
  5. Omar al-Khayyám

  1. Girolamo Cardano
  2. Joseph Liouville
  3. Johann H. Lambert
  4. Thales of Miletus
  5. Paul Erdös
  1. Alexis C. Clairaut
  2. Marius Sophus Lie
  3. Nicolai Lobachevsky
  4. Thabit ibn Qurra
  5. Jean Gaston Darboux

Earliest mathematicians
Little is known of the earliest mathematics, but the famous Ishango Bone from Early Stone-Age Africa has tally marks suggesting arithmetic. The markings include six prime numbers (5, 7, 11, 13, 17, 19) in order, though this is probably coincidence. The advanced artifacts of Egypt's Old Kingdom and the Indus-Harrapa civilization imply strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where 4500-year old clay tablets show multiplication and division problems; the first abacus may be about this old. By 3600 years ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, and even logarithms, using a primitive place-value system (in base 60, not 10). Babylonians were familiar with the Pythagorean theorem, quadratic equations, even cubic equations (though they didn't have a general solution for these), and eventually even developed methods to estimate terms for compound interest.
Also at least 3600 years ago, the Egyptian scribe Ahmes produced a famous manuscript (now called the Rhind Papyrus), itself a copy of a late Middle Kingdom text. It showed simple algebra methods and included a table giving optimal expressions using Egyptian fractions. (Today, Egyptian fractions lead to challenging number theory problems with no practical applications, but they may have had practical value for the Egyptians. To divide 17 grain bushels among 21 workers, the equation 17/21 = 1/2 + 1/6 + 1/7 has practical value, especially when compared with the "greedy" decomposition 17/21 = 1/2 + 1/4 + 1/17 + 1/1428.)
While Egyptians may have had more advanced geometry, Babylon was much more advanced at arithmetic and algebra. This was probably due, at least in part, to their place-value system. But although their base-60 system survives (e.g. in the division of hours and degrees into minutes and seconds) the Babylonian notation, which used the equivalent of IIIIII XXXXXIIIIIII XXXXIII to denote 417+43/60, was unwieldy compared to the "ten digits of the Hindus."
The Egyptians used the approximation π ≈ (4/3)4 (derived from the idea that a circle of diameter 9 has about the same area as a square of side 8). Although the ancient Hindu mathematician Apastambha had achieved a good approximation for √2, and the ancient Babylonians an ever better √2, neither of these ancient cultures achieved a π approximation as good as Egypt's, or better than π ≈ 25/8, until the Alexandrian era.
Early Vedic mathematicians
The greatest mathematics before the Golden Age of Greece was in India's early Vedic (Hindu) civilization. The Vedics understood relationships between geometry and arithmetic, developed astronomy, astrology, calendars, and used mathematical forms in some religious rituals. The earliest mathematician to whom definite teachings can be ascribed was Lagadha, who apparently lived about 1300 BC and used geometry and elementary trigonometry for his astronomy. Baudhayana lived about 800 BC and also wrote on algebra and geometry; Yajnavalkya lived about the same time and is credited with the then-best approximation to π. Apastambha did work summarized below; other early Vedic mathematicians solved quadratic and simultaneous equations.
Other early cultures also developed some mathematics. The ancient Mayans apparently had a place-value system with zero before the Hindus did; Aztec architecture implies practical geometry skills. Ancient China certainly developed mathematics, though little written evidence survives prior to Chang Tshang's famous book.

Thales of Miletus (ca 624 - 546 BC) Greek domain
Thales was the Chief of the Seven Sages of ancient Greece, and has been called the "Father of Science," the "Founder of Abstract Geometry," and the "First Philosopher." Thales is believed to have studied mathematics under Egyptians, who in turn were aware of much older mathematics from Mesopotamia. Thales may have invented the notion of compass-and-straightedge construction. Several fundamental theorems about triangles are attributed to Thales, including the law of similar triangles (which Thales used famously to calculate the height of the Great Pyramid) and "Thales' Theorem" itself: the fact that any angle inscribed in a semicircle is a right angle. (The other "theorems" were probably more like well-known "axioms", but Thales proved Thales' Theorem using two of his other theorems; it is said that Thales then sacrificed an ox to celebrate what might have been the very first mathematical proof!) Thales was also an astronomer; he invented the 365-day calendar, introduced the use of Ursa Minor for finding North, and is the first person believed to have correctly predicted a solar eclipse. His theories of physics would seem quaint today, but he seems to have been the first to describe magnetism and static electricity. Aristotle said, "To Thales the primary question was not what do we know, but how do we know it." Thales was also a politician, ethicist, and military strategist. It is said he once leased all available olive presses after predicting a good olive season; he did this not for the wealth itself, but as a demonstration of the use of intelligence in business. Thales' writings have not survived and are known only second-hand. Since his famous theorems of geometry were probably already known in ancient Babylon, his importance derives from imparting the notions of mathematical proof and the scientific method to ancient Greeks.
Thales' student and successor was Anaximander, who is often called the "First Scientist" instead of Thales: his theories were more firmly based on experimentation and logic, while Thales still relied on some animistic interpretations. Anaximander is famous for astronomy, cartography and sundials, and also enunciated a theory of evolution, that land species somehow developed from primordial fish! Anaximander's most famous student, in turn, was Pythagoras. (The methods of Thales and Pythagoras led to the schools of Plato and Euclid, an intellectual blossoming unequalled until Europe's Renaissance. For this reason Thales may belong on this list for his historical importance despite his relative lack of mathematical achievements.)

Apastambha (ca 630-560 BC) India
The Dharmasutra composed by Apastambha contains mensuration techniques, novel geometric construction techniques, a method of elementary algebra, and what may be the first known proof of the Pythagorean Theorem. Apastambha's work uses the excellent (continued fraction) approximation √2 ≈ 577/408, a result probably derived with a geometric argument. Apastambha built on the work of earlier Vedic scholars, especially Baudhayana, as well as Harappan and (probably) Mesopotamian mathematicians. His notation and proofs were primitive, and there is little certainty about his life. However similar comments apply to Thales of Miletus, so it seems fair to mention Apastambha (who was perhaps the most creative Vedic mathematician before Panini) along with Thales as one of the earliest mathematicians whose name is known.

Pythagoras of Samos (ca 578-505 BC) Greek domain
Pythagoras, who is sometimes called the "First Philosopher," studied under Anaximander, Egyptians, Babylonians, and the mystic Pherekydes (from whom Pythagoras acquired a belief in reincarnation); he became the most influential of early Greek mathematicians. He is credited with being first to use axioms and deductive proofs, so his influence on Plato and Euclid may be enormous. He and his students (the "Pythagoreans") were ascetic mystics for whom mathematics was partly a spiritual tool. (Some occultists treat Pythagoras as a wizard and founding mystic philosopher.) Pythagoras was very interested in astronomy and recognized that the Earth was a globe similar to the other planets. He believed thinking was located in the brain rather than heart. The words "philosophy" and "mathematics" are said to have been coined by Pythagoras. Despite Pythagoras' historical importance I may have ranked him too high: many results of the Pythagoreans were due to his students; none of their writings survive; and what is known is reported second-hand, and possibly exaggerated, by Plato and others. His students included Hippasus of Metapontum, perhaps the famous physician Alcmaeon, Milo of Croton, and Croton's daughter Theano (who may have been Pythagoras's wife). The term "Pythagorean" was also adopted by many disciples who lived later; these disciples include Philolaus of Croton, the natural philosopher Empedocles, and several other famous Greeks. Pythagoras' successor was apparently Theano herself: the Pythagoreans were one of the few ancient schools to practice gender equality.
Pythagoras discovered that harmonious intervals in music are based on simple rational numbers. This led to a fascination with integers and mystic numerology; he is sometimes called the "Father of Numbers" and once said "Number rules the universe." (About the mathematical basis of music, Leibniz later wrote, "Music is the pleasure the human soul experiences from counting without being aware that it is counting." Other mathematicians who investigated the arithmetic of music included Huygens, Euler and Simon Stevin.)
The Pythagorean Theorem was known long before Pythagoras, but he is often credited with the first proof. (Apastambha proved it in India at about the same time, and some theorize that Pythagoras journeyed to India and learned of the proof there.) He also discovered the simple parametric form of Pythagorean triplets (xx-yy, 2xy, xx+yy). Other discoveries of the Pythagorean school include the concepts of perfect and amicable numbers, polygonal numbers, golden ratio (attributed to Theano), the five regular solids (attributed to Pythagoras himself), and irrational numbers (attributed to Hippasus). It is said that the discovery of irrational numbers upset the Pythagoreans so much they tossed Hippasus into the ocean! (Another version has Hippasus banished for revealing the secret for constructing the sphere which circumscribes a dodecahedron.)
The famous successors of Thales and Pythagoras included Parmenides of Elea (ca 515-440 BC), Zeno of Elea (see below), Hippocrates of Chios (see below), Plato of Athens (ca 428-348 BC), Theaetetus (ca 414-369 BC), and Archytas (see below). These early Greeks ushered in a Golden Age of Mathematics and Philosophy unequaled in Europe until the Renaissance. The emphasis was on pure, rather than practical, mathematics. Plato (who ranks #40 on Michael Hart's famous list of the Most Influential Persons in History) decreed that his scholars should do geometric construction solely with compass and straight-edge rather than with "carpenter's tools" like rulers and protractors.

Panini (of Shalatula) (ca 520-460 BC) Gandhara (India)
Panini's great accomplishment was his study of the Sanskrit language, especially in his text Ashtadhyayi. Although this work might be considered the very first study of linguistics or grammar, it used a non-obvious elegance that would not be equalled in the West until the 20th century. Linguistics may seem an unlikely qualification for a "great mathematician," but language theory is a field of mathematics. The works of eminent 20th-century linguists and computer scientists like Chomsky, Backus, Post and Church are seen to resemble Panini's work 24 centuries earlier. Panini's systematic study of Sanskrit may have inspired the development of Indian science and algebra. Panini has been called "the Indian Euclid" since the rigor of his grammar is comparable to Euclid's geometry. Although his great texts have been preserved, little else is known about Panini. Some scholars would place his dates a century later than shown here; he may or may not have been the same person as the famous poet Panini. In any case, he was the very last Vedic Sanskrit scholar by definition: his text formed the transition to the Classic Sanskrit period. Panini has been called "one of the most innovative people in the whole development of knowledge."

Zeno of Elea (ca 495-435 BC) Greek domain
Zeno, a student of Parmenides, had great fame in ancient Greece. This fame, which continues to the present-day, is largely due to his paradoxes of infinitesimals, e.g. his argument that Achilles can never catch the tortoise (whenever Achilles arrives at the tortoise's last position, the tortoise has moved on). Although some regard these paradoxes as simple fallacies, they have been contemplated for many centuries. It is due to these paradoxes that the use of infinitesimals, which provides the basis for mathematical analysis, has been regarded as a non-rigorous heuristic and is finally viewed as sound only after the work of the great 19th-century rigorists, Dedekind and Weierstrass.

Hippocrates of Chios (ca 470-410 BC) Greek domain
Hippocrates (no relation to the famous physician) wrote his own Elements more than a century before Euclid. Only fragments survive but it apparently used axiomatic-based proofs similar to Euclid's and contains many of the same theorems. Hippocrates is said to have invented the reductio ad absurdem proof method. Hippocrates is most famous for his work on the three ancient geometric quandaries: his work on cube-doubling (the Delian Problem) laid the groundwork for successful efforts by Archytas and others; his circle quadrature was of course ultimately unsuccessful but he did prove ingenious theorems about "lunes" (certain circle fragments); and some claim Hippocrates was first to trisect the general angle. (Doubling the cube and angle trisection are often called "impossible," but they are impossible only when restricted to collapsing compass and unmarkable straightedge. There are ingenious solutions available with other tools.) Hippocrates also did work in algebra and rudimentary analysis.

Archytas of Tarentum (ca 420-350 BC) Greek domain
Archytas was an important statesman as well as philosopher. He studied under Philolaus of Croton, was a friend of Plato, and tutored Eudoxus and Menaechmus. In addition to discoveries always attributed to him, he may be the source of several of Euclid's theorems, and some works attributed to Eudoxus and perhaps Pythagoras. Recently it has been shown that the magnificent Mechanical Problems attributed to (pseudo-)Aristotle were probably actually written by Archytas, making him one of the greatest mathematicians of antiquity. Archytas introduced "motion" to geometry, rotating curves to produce solids. If his writings had survived he'd surely be considered one of the most brilliant and innovative geometers of antiquity. Archytas' most famous mathematical achievement was "doubling the cube" (constructing a line segment larger than another by the factor cube-root of two). Although others solved the problem with other techniques, Archytas' solution for cube doubling was astounding because it wasn't achieved in the plane, but involved the intersection of three-dimensional bodies. This construction (which introduced the "Archytas Curve") has been called "a tour de force of the spatial imagination." He invented the term "harmonic mean" and worked with geometric means as well (proving that consecutive integers never have rational geometric mean). He was a true polymath: he advanced the theory of music far beyond Pythagoras; studied sound, optics and cosmology; invented the pulley (and a rattle to occupy infants!); wrote about the lever; developed the curriculum called quadrivium; and is supposed to have built a steam-powered wooden bird which flew for 200 meters. Archytas is sometimes called the Father of Mathematical Mechanics.

Eudoxus of Cnidus (408-355 BC) Greek domain
Eudoxus journeyed widely for his education, despite that he was not wealthy, studying mathematics with Archytas in Tarentum, medicine with Philiston in Sicily, philosophy with Plato in Athens, continuing his mathematics study in Egypt, touring the Eastern Mediterranean with his own students and finally returned to Cnidus where he established himself as astronomer, physician, and ethicist. What is known of him is second-hand, through the writings of Euclid and others, but he was one of the most creative 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; he is sometimes credited with first use of the Axiom of Archimedes, which avoids Zeno's paradoxes by, in effect, forbidding infinities and infinitesimals; yet he also developed a method of taking limits. Eudoxus' work with irrational numbers and infinitesimals may have helped inspire such masters as Archimedes and Dedekind. Eudoxus also introduced an Axiom of Continuity; he was a pioneer in solid geometry; and he developed his own solution to the Delian cube-doubling problem. Eudoxus was the first great mathematical astronomer; he developed the complicated ancient theory of planetary orbits; and may have invented the astrolabe. (It is sometimes said that he knew that the Earth rotates around the Sun, but that appears to be false; it is instead Aristarchus of Samos, as cited by Archimedes, who may be the first "heliocentrist.")
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 π 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. Eudoxus has been quoted as saying "Willingly would I burn to death like Phaeton, were this the price for reaching the sun and learning its shape, its size and its substance."

Aristotle of Stagira (384-322 BC) Macedonia
Aristotle is considered the greatest scientist of the ancient world, and the most influential philosopher and logician ever; he ranks #13 on Michael Hart's list of the Most Influential Persons in History. (His science was a standard curriculum for almost 2000 years, unfortunate since many of his ideas were quite mistaken.) His writings on definitions, axioms and proofs may have influenced Euclid. He was also the first mathematician to write on the subject of infinity. His writings include geometric theorems, some with proofs different from Euclid's or missing from Euclid altogether; one of these (which is seen only in Aristotle's work prior to Apollonius) is that a circle is the locus of points whose distances from two given points are in constant ratio. Even if, as is widely agreed, Aristotle's geometric theorems were not his own work, his status as the most influential logician and philosopher makes him a candidate for the List.

Euclid of Megara & Alexandria (ca 322-275 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; he stated and proved the unique factorization theorem; and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect (in the sense of Pythagoras) if M is Mersenne. (The converse, that any even perfect number has such a corresponding Mersenne prime, was tackled by Alhazen and proven by Euler.) He proved that there are only five "Platonic solids," as well as theorems of geometry far too numerous to summarize; among many with special historical interest is the proof that rigid-compass constructions can be implemented with collapsing-compass constructions. Among several books attributed to Euclid are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), a book on spherical geometry, a book on logic fallacies, and his comprehensive math textbook The Elements. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry. 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; 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. There are many famous quotations about Euclid and his books. Abraham Lincoln abandoned his law studies when he didn't know what "demonstrate" meant and "went home to my father's house [to read Euclid], and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."

Archimedes of Syracuse (287-212 BC) Greek domain
Archimedes is universally acknowledged to be the greatest of ancient mathematicians. He studied at Euclid's school (probably after Euclid's death), but his work far surpassed the works of Euclid. His achievements are particularly impressive given the lack of good mathematical notation in his day. His proofs are noted not only for brilliance but for unequalled clarity, with a modern biographer (Heath) describing Archimedes' treatises as "without exception monuments of mathematical exposition ... so impressive in their perfection as to create a feeling akin to awe in the mind of the reader." Archimedes made advances in number theory, algebra, and analysis, but is most renowned for his many theorems of plane and solid geometry. He was first to prove Heron's formula for the area of a triangle. His excellent approximation to √3 indicates that he'd partially anticipated the method of continued fractions. He found a method to trisect an arbitrary angle (using a markable straightedge — the construction is impossible using strictly Platonic rules). Although it doesn't survive in his writings, Pappus reports that he discovered the Archimedean solids. One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series. Archimedes anticipated integral calculus, most notably by determining the centers of mass of hemisphere and cylindrical wedge, and the volume of two cylinders' intersection. Although Archimedes made little use of differential calculus, Chasles credits him (along with Kepler, Cavalieri, and Fermat) as one of the four who developed calculus before Newton and Leibniz. He was similar to Newton in that he used his (non-rigorous) calculus to discover results, but then devised rigorous geometric proofs for publication. 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 books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder. He developed the Stomachion puzzle (and solved a difficult enumeration problem involving it). 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.
Archimedes discovered formulae for the volume and surface area of a sphere, and may even have been first to notice and prove the simple relationship between a circle's circumference and area. For these reasons, π is often called Archimedes' constant. His approximation 223/71 < π < 22/7 was the best of his day, though Apollonius soon surpassed it. That Archimedes shared the attitude of later mathematicians like Hardy and Brouwer is suggested by Plutarch's comment that Archimedes regarded applied mathematics "as ignoble and sordid ... and did not deign to [write about his mechanical inventions; instead] he placed his whole ambition in those speculations the beauty and subtlety of which are untainted by any admixture of the common needs of life."
In the 20th century, modern technology led to the discovery of new writings by Archimedes, hitherto hidden on a palimpsest, including a note that implies 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). Although Newton may have been the most important mathematician, and Gauss the greatest theorem prover, it is widely accepted that Archimedes was the greatest genius who ever lived. Yet, Hart omits him altogether from his list of Most Influential Persons: Archimedes was simply too far ahead of his time to have great historical significance.

Apollonius of Perga (262-190 BC) Greek domain
Apollonius Pergaeus, called "The Great Geometer," is sometimes considered the second greatest of ancient Greek mathematicians (Euclid and Eudoxus are the other candidates for this honor). His writings on conic sections have been studied until modern times; he invented the names for parabola, hyperbola and ellipse; he developed methods for normals and curvature. Although astronomers eventually concluded it was not physically correct, Apollonius developed the "epicycle and deferent" model of planetary orbits, and proved important theorems in this area. He deliberately emphasized the beauty of pure, rather than applied, mathematics, saying his theorems were "worthy of acceptance for the sake of the demonstrations themselves." Since many of his works have survived only in a fragmentary form, several great Renaissance and Modern mathematicians (including Vieta, Fermat, Pascal and Gauss) have enjoyed reconstructing and reproving his "lost" theorems. (Among these, the most famous is to construct a circle tangent to three other circles.)
In evaluating the genius of the ancient Greeks, it is well to remember that their achievements were made without the convenience of modern notation. It is clear from his writing that Apollonius almost developed the analytic geometry of Déscartes, but failed due to the lack of such elementary concepts as negative numbers. Leibniz wrote "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times."

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, uses the decimal system with zero and negative numbers, 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; some additions attributed to Liu Hui are mentioned in his mini-bio; other famous contributers are Jing Fang and Zhang Heng.)
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), so Chang may not be appropriate for a Top Mathematicians List (though he was probably not a mere bureaucrat or copyist, as Liu Hui mentions Chang's skill). Nevertheless his book had immense historical importance: It was the dominant Chinese mathematical text for centuries, and had great influence throughout the Far East. After Chang, Chinese mathematics continued to flourish, discovering trigonometry, matrix methods, the binomial theorem, etc. 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.
No one person can be credited with the invention of the decimal system, but key roles were played by early Chinese (Chang Tshang and Liu Hui), Brahmagupta (and earlier Hindus including Aryabhatta), and Leonardo Fibonacci. (After Fibonacci, Europe still did not embrace the decimal system until the works of Vieta, Stevin, and Napier.)

Hipparchus of Nicaea (ca 190-127 BC) Greek domain
Ptolemy may be the most famous astronomer before Copernicus, but he borrowed heavily from Hipparchus, who might be considered the greatest astronomer ever. (Careful study of the errors in the catalogs of Ptolemy and Hipparchus reveal both that Ptolemy borrowed his data from Hipparchus, and that Hipparchus used principles of spherical trig to simplify his work. Late Vedic astronomers, including the 6th-century genius Aryabhatta, borrow much from Ptolemy and Hipparchus.) As a mathematician, Hipparchus developed spherical trigonometry, produced trig tables, and fourteen texts of physics and mathematics nearly all of which have been lost, but which seem to have had great teachings, including much of Newton's Laws of Motion. In one obscure surviving work he demonstrates familiarity with the combinatorial enumeration method now called Schröder's Numbers. He invented the circle-conformal stereographic map projection which carries his name. As an astronomer, Hipparchus is credited with the discovery of equinox precession, length of the year, thorough star catalogs, and invention of the armillary sphere and perhaps the astrolabe. He had great historical influence in Europe, India and Persia, at least if credited also with Ptolemy's influence. (Hipparchus himself was influenced by Chaldean astronomers.) Hipparchus' work implies a better approximation to π than that of Apollonius, perhaps it was π ≈ 377/120 as Ptolemy used. It took much skill for Eudoxus, Apollonius and Hipparchus to develop their complex geocentric cosmology. Their skill may have set back science since, as we now know, the Earth rotates around the Sun!

Liu Hui (ca 220-280) China
Liu Hui made major improvements to Chang's influential textbook Nine Chapters, making him among the most important of Chinese mathematicians ever. (He seems to have been a much better mathematician than Chang, but just as Newton might have gotten nowhere without Kepler, Vieta, Huygens, Fermat, Wallis, Cavalieri, etc., so Liu Hui might have achieved little had Chang not preserved the ancient Chinese learnings.) Among Liu's achievements are an emphasis on generalizations and proofs, an early recognition of the notions of infinitesimals and limits, the Gaussian elimination method of solving simultaneous linear equations, calculations of solid volumes (including the use of Cavalieri's Principle), anticipation of Horner's Method, and a new method to calculate square roots. Like Archimedes, Liu discovered the formula for a circle's area; however he failed to calculate a sphere's volume, writing "Let us leave this problem to whoever can tell the truth." Although it was almost child's-play for any of them, Archimedes, Apollonius, and Hipparchus had all improved precision of π's estimate. It seems fitting that Liu Hui did join that select company of record setters: He developed a recurrence formula for regular polygons allowing arbitrarily-close approximations for π. He also devised an interpolation formula to simplify that calculation; this yielded the "good-enough" value 3.1416, which is still taught today in primary schools. (Liu's successors in China applied his method to produce even better approximations, so that country kept the pi-accuracy record for 1100 years.)

Diophantus of Alexandria (ca 250) Greece, Egypt
Diophantus was one of the most influential mathematicians of antiquity; he wrote several books on arithmetic and algebra, and explored number theory further than anyone earlier. (He probably studied Babylonian algebra, which was more advanced than anything in ancient Greece.) He advanced a rudimentary arithmetic and algebraic notation, allowed rational-number solutions to his problems rather than just integers, and was aware of results like the Brahmagupta-Fibonacci Identity; for these reasons he is often called the "Father of Algebra." His work, however, may seem quite limited to a modern eye: his methods were not generalized, he knew nothing of negative numbers, and, though he often dealt with quadratic equations, never seems to have commented on their second solution. His notation, clumsy as it was, was used for many centuries. (The shorthand x3 for "x cubed" was not invented until Déscartes.) Very little is known about Diophantus. Many of his works have been lost, including proofs for lemmas cited in the surviving work, some of which are so difficult it would almost stagger the imagination to believe Diophantus really had proofs! Among these are Fermat's conjecture (Lagrange's theorem) that every integer is the sum of four squares, and the following: "Given any positive rationals a, b with a>b, there exist positive rationals c, d such that a3-b3 = c3+d3." (This latter "lemma" was investigated by Vieta and Fermat and finally solved, with some difficulty, in the 19th century. It seems unlikely that Diophantus actually had proofs for such "lemmas.")

Pappus of Alexandria (ca 300) Egypt, Greece
Pappus, along with Diophantus, may have been one of the two greatest Western mathematicians during the 14 centuries that separated Apollonius and Fibonacci. He wrote about arithmetic methods, plane and solid geometry, the axiomatic method, celestial motions and mechanics. In addition to his own original research, his texts are noteworthy for preserving works of earlier mathematicians that would otherwise have been lost. Pappus presents several ingenious geometric theorems including Desargues' Homology Theorem (which Pappus attributes to Euclid), a special case of Pascal's Hexagram Theorem, and Pappus' Theorem itself (two projective pencils can always be brought into a perspective position). For these theorems, Pappus is sometimes called the "Father of Projective Geometry." Other ingenious theorems include an angle trisection method using a fixed hyperbola. He stated (but didn't prove) the Isoperimetric Theorem, also writing "Bees know this fact which is useful to them, that the hexagon ... will hold more honey for the same material than [a square or triangle]." (That a honeycomb partition minimizes material for an equal-area partioning was finally proved in 1999 by Thomas Hales, who also proved the related Kepler Conjecture.)
For preserving the teachings of Euclid and Apollonius, as well as his own theorems of geometry, Pappus certainly belongs on a list of great ancient mathematicians. But these teachings lay dormant during Europe's Dark Ages, diminishing Pappus' historical significance.

Mathematicians after Classical Greece
Alexander the Great spread Greek culture to Egypt and much of the Orient; thus even Hindu mathematics may owe something to the Greeks. Greece was eventually absorbed into the Roman Empire (with Archimedes himself famously killed by a Roman soldier). Rome did not pursue pure science as Greece had (as we've seen, the important mathematicians of the Roman era were based in the Hellenic East) and eventually Europe fell into a Dark Age. The Greek emphasis on pure mathematics and proofs was key to the future of mathematics, but they were missing an even more important catalyst: a decimal place-value system based on zero and nine other symbols.

Decimal system -- from China?
It's still hard to believe that the "obvious" and so-convenient decimal system didn't catch on in Europe until almost the Renaissance. Ancient Greeks, by the way, did not use the unwieldy Roman numerals, but rather used 27 symbols, denoting 1 to 9, 10 to 90, and 100 to 900. Unlike our system, with ten digits separate from the alphabet, the 27 Greek number symbols were the same as their alphabet's letters; this might have hindered the development of "syncopated" notation. The most ancient Hindu records did not use the ten digits of Aryabhatta, but rather a system similar to that of the ancient Greeks, suggesting that China, and not India, may indeed be the "ultimate" source of the modern decimal system.
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 (called Arjehir by Arabs) may be most famous. While Europe was in its early "Dark Age," Aryabhatta advanced arithmetic, algebra, elementary analysis, and especially trigonometry, using the decimal system. Aryabhatta is sometimes called the "Father of Algebra" instead of al-Khowârizmi (who himself cites the work of Aryabhatta). His most famous accomplishment in mathematics was the Aryabhatta Algorithm (connected to continued fractions) for solving Diophantine equations. Aryabhatta made several important discoveries in astronomy; for example, his estimate of the Earth's circumference was more accurate than any achieved in ancient Greece. He was among the ancient scholars who realized the Earth rotated daily on an axis; claims that he also espoused heliocentric orbits are controversial, but may be confirmed by the writings of al-Biruni. Aryabhatta is said to have introduced the constant e. He used π ≈ 3.1416; it is unclear whether he discovered this independently or borrowed it from Liu Hui of China. Among theorems first discovered by Aryabhatta is the famous identity
     Σ (k3) = (Σ k)2

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 very 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 earlier Hindu works, but Brahmagupta's text discussed them lucidly.) Along with Diophantus, Brahmagupta was also among the first to express equations with symbols rather than words.
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:
        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 proved the Brahmagupta-Fibonacci Identity (the set of sums of two squares is closed under multiplication). He applied mathematics to astronomy, predicting eclipses, etc.

Muhammed `Abu Jafar' ibn Musâ  al-Khowârizmi (ca 780-850) Persia, Iraq
Al-Khowârizmi (aka Mahomet ibn Moses) 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; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography. He wrote the book Al-Jabr, which demonstrated simple algebra and geometry, and several other influential books. Unlike Diophantus' work, which dealt in specific examples, Al-Khowârizmi presented general methods. The word algorithm is borrowed from Al-Khowârizmi's name. There were several Muslim mathematicians who contributed to the development of Islamic science, and indirectly to Europe's later Renaissance, but Al-Khowârizmi was one of the earliest and most influential.

Ya'qub `Abu Yusuf' ibn Ishaq  al-Kindi (803-873) Iraq
Al-Kindi (called Alkindus in the West) wrote on diverse philosophical subjects, physics, optics, astronomy, music, psychology, medicine, chemistry, and more. He invented pharmaceutical methods, perfumes, and distilling of alcohol. In mathematics, he popularized the use of the decimal system, developed spherical geometry, wrote on many other topics and was a pioneer of cryptography (code-breaking). (Al-Kindi, called The Arab Philosopher, can not be considered among the greatest of mathematicians, but was one of the most influential general scientists between Aristotle and da Vinci.)

Al-Sabi Thabit  ibn Qurra al-Harrani (836-901) Harran, Iraq
Thabit produced important books in philosophy (including perhaps the famous mystic work De Imaginibus), medicine, mechanics, astronomy, and especially several mathematical fields: analysis, non-Euclidean geometry, trigonometry, arithmetic, number theory. (As well as being an original thinker, Thabit was one of the most important translaters of ancient Greek writings.) He developed an important new cosmology superior to Ptolemy's (and which, though it was not heliocentric, may have inspired Copernicus). He was perhaps the first great mathematician to take the important step of emphasizing real numbers rather than either rational numbers or geometric sizes. He worked in plane and spherical trigonometry, in cubic equations. He was an earlier practitioner of calculus and seems to have been first to take the integral of √x. He also may have been first to calculate the area of an ellipse, and first to calculate the volume of a paraboloid. He also worked in number theory where he is especially famous for his theorem about amicable numbers. While many of his discoveries in geometry, plane and spherical trigonometry, and analysis (parabola quadrature, trigonometric law, principle of lever) duplicated work by Archimedes and Pappus, Thabit's list of novel achievements is still impressive. Among the several great and famous Baghdad geometers, Thabit may have had the greatest genius.

Alhazen `Abu Ali'   ibn al-Haytham (965-1039) Iraq, Egypt
Alhazen ibn al-Haytham (Al-asan ibn Haisham) made contributions to math, optics, and astronomy which eventually influenced Roger Bacon, Regiomontanus, da Vinci, Copernicus, Kepler and Wallis, among others, thus affecting Europe's Scientific Revolution. He's been called the best scientist of the Middle Ages; his Book of Optics has been called the most important physics text prior to Newton; his writings in physics anticipate the Principle of Least Action, Newton's First Law of Motion, and the notion that white light is composed of the color spectrum. (Like Newton, he favored a particle theory of light over the wave theory of Aristotle.) His other achievements in optics include improved lens design, an analysis of the camera obscura, Snell's Law, an early explanation for the rainbow, a correct deduction from refraction of atmospheric thickness, and experiments on visual perception. He also did work in human anatomy and medicine. (In a famous leap of over-confidence he claimed he could control the Nile River; when the Caliph ordered him to do so, he then had to feign madness!) Alhazen has been called the "Father of Modern Optics" and, because he emphasized hypotheses and experiments, "The First Scientist." In number theory, Alhazen worked with perfect numbers, Mersenne primes; and stated Wilson's Theorem (eventually proven by Lagrange). He essentially proved the Power Series Theorem (later attributed to Bernoulli). He solved Alhazen's Billiard Problem (originally posed as a problem in mirror design), a difficult construction which continued to intrigue several great mathematicians including Huygens. Alhazen's attempts to prove the Parallel Postulate make him (along with Thabit ibn Qurra) one of the earliest mathematicians to investigate non-Euclidean geometry.

Abu al-Rayhan Mohammed ibn Ahmad  al-Biruni (973-1048) Khwarizm (Uzbekistan)
Al-Biruni (Alberuni) was an extremely outstanding scholar, far ahead of his time, sometimes shown with Alkindus and Alhazen as one of the greatest Islamic polymaths, and sometimes compared to Leonardo da Vinci. He is less famous in part because he lived in a remote part of the Islamic empire. He was a great linguist; studied the original works of Greeks and Hindus; is famous for debates with his contemporary Avicenna; studied history, biology, mineralogy, philosophy, sociology, medicine and more; was called the Father of Arabic Pharmacy; and was one of the greatest astronomers. He was also noted for his poetry. He invented (but didn't build) a mechanical clock, and worked with springs and hydrostatics. He wrote prodigiously on all scientific topics (his writings are estimated to total 13,000 folios); he was especially noted for his comprehensive encyclopedia about India, and Shadows, which starts from notions about shadows but develops much astronomy and mathematics. He applied scientific methods; and anticipated future advances including Darwin's natural selection, Newton's Second Law, the immutability of elements, the nature of the Milky Way, and much modern geology. Among several novel achievements in astronomy, he used observations of lunar eclipse to deduce relative longitude, estimated Earth's radius most accurately, believed the Earth rotated on its axis and accepted heliocentrism as a possibility. In mathematics, he was first to apply the Law of Sines to astronomy, geodesy, and cartography; anticipated the notion of polar coordinates; found trigonometric solutions to polynomial equations; did geometric constructions including angle trisection; and wrote on arithmetic, algebra, and combinatorics as well as plane and spherical trigonometry and geometry. Al-Biruni has left us what seems to be the oldest surviving mention of the Broken Chord Theorem (if M is the midpoint of arc ABMC, and T the midpoint of "broken chord" ABC, then MT is perpendicular to BC). Although he himself attributed the theorem to Archimedes, Al-Biruni provided several novel proofs for, and useful corollaries of, this famous geometric gem. While Al-Biruni may lack the influence and mathematical brilliance to qualify for our List, he deserves recognition as one of the greatest applied mathematicians before the modern era.

Omar  al-Khayyám (1048-1123) Persia
Omar Khayyam (aka Ghiyas od-Din Abol-Fath Omar ibn Ebrahim Khayyam Neyshaburi) did clever work with geometry, developing an alternate to Euclid's Parallel Postulate and then deriving the parallel result using theorems based on the Khayyam-Saccheri quadrilateral. He derived solutions to cubic equations using the intersection of conic sections with circles. Remarkably, he stated that the cubic solution could not be achieved with straightedge and compass, a fact that wouldn't be proved until the 19th century. Khayyam did even more important work in algebra, writing an influential textbook, and developing new solutions for various higher-degree equations. He may have been first to develop the Binomial Theorem and Pascal's Triangle. His symbol ('shay') for an unknown in an algebraic equation was transliterated to become our 'x'. Khayyam was also an important astronomer; he measured the year far more accurately than ever before, improved the Persian calendar, and built a famous star map. He emphasized science over religion and proved that the Earth rotates around the Sun. He also wrote treatises on philosophy, music, mechanics and natural science. Despite his great achievements in algebra, geometry, and astronomy, today Omar al-Khayyám is most famous for his rich poetry (The Rubaiyat of Omar Khayyam).

Bháscara Áchárya (1114-1185) India
Bháscara (also called Bhaskara II or Bhaskaracharya) may have been the greatest of the Hindu mathematicians. He made achievements in several fields of mathematics including some Europe wouldn't learn until the time of Euler. His textbooks dealt with many matters, including solid geometry, combinations, and advanced arithmetic methods. He was also an astronomer. (It is sometimes claimed that his equations for planetary motions anticipated the Laws of Motion discovered by Kepler and Newton, but this claim is doubtful.) In algebra, he solved various equations including 2nd-order Diophantine, quartic, Brouncker's and Pell's equations. His "Chakravala method," an early application of mathematical induction to solve 2nd-order equations, has been called "the finest thing achieved in the theory of numbers before Lagrange" (although a similar statement was made about one of Fibonacci's theorems). (Earlier Hindus, including Brahmagupta, contributed to this method.) In several ways he anticipated calculus: he used Rolle's Theorem; he may have been first to use the fact that dsin x = cos x · dx; and he once wrote that multiplication by 0/0 could be "useful in astronomy." In trigonometry, which he valued for its own beauty as well as practical applications, he developed spherical trig and was first to present the identity
    sin a+b = sin a · cos b + sin b · cos a
Bháscara's achievements came centuries before similar discoveries in Europe. It is an open riddle of history whether any of Bháscara's teachings trickled into Europe in time to influence its Scientific Renaissance.

Leonardo `Bigollo' Pisano (Fibonacci) (ca 1170-1245) Italy
Leonardo (known today as Fibonacci) introduced the decimal system and other 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-Khowârizmi, into Latin, but Leonardo was the influential teacher. He also re-introduced older Greek ideas like Mersenne numbers and Diophantine equations. Leonardo's writings cover a very broad range including new theorems of geometry, methods to construct and convert Egyptian fractions (which were still in wide use), irrational numbers, the Chinese Remainder Theorem, theorems about Pythagorean triplets, and the series 1, 1, 2, 3, 5, 8, 13, .... which is now linked with the name Fibonacci. In addition to his great historic importance and fame (he was a favorite of Emperor Frederick II), Leonardo `Fibonacci' is called "the greatest number theorist between Diophantus and Fermat" and "the most talented mathematician of the Middle Ages." Leonardo is most famous for his book Liber Abaci, but his Liber Quadratorum provides the best demonstration of his skill. He defined congruums and proved theorems about them, including a theorem establishing the conditions for three square numbers to be in consecutive arithmetic series; this has been called the finest work in number theory prior to Fermat (although a similar statement was made about one of Bhaskara's theorems). Although often overlooked, this work includes a proof of the n = 4 case of Fermat's Last Theorem. (Leonardo's proof of FLT4 is generally ignored or considered invalid. I'm preparing a page to consider that question.)
Leonardo provided Europe with the decimal system, algebra and the 'lattice' method of multiplication, all far superior to the methods then in use. He introduced notation like 3/5; his clever extension of this 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!"
Some histories describe him as bringing Islamic mathematics to Europe, but in Fibonacci's own preface to Liber Abaci, he specifically credits the Hindus:
... as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods;
... But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, ...
Had the Scientific Renaissance begun in the Islamic Empire, someone like al-Khowârizmi would have greater historic significance than Fibonacci, but the Renaissance did happen in Europe. Liber Abaci's summary of the decimal system has been called "the most important sentence ever written." Even granting this to be an exaggeration, there is no doubt that the Scientific Revolution owes a huge debt to Leonardo `Fibonacci' Pisano.

Qin Jiushao (1202-1261) China
There were several important Chinese mathematicians in the 13th century, of whom Qin Jiushao (Ch'in Chiu-Shao) may have had particularly outstanding breadth and genius. Qin's textbook discusses various algebraic procedures, includes word problems requiring quartic or quintic equations, explains a version of Horner's Method for finding solutions to such equations, includes Heron's Formula for a triangle's area, and introduces the zero symbol. Qin's work on the Chinese Remainder Theorem was very impressive, finding solutions in cases which later stumped Euler. Other great Chinese mathematicians of that era are Li Zhi, Yang Hui, and Zhu Shiejie (who may be the most famous and influential of these). Although Qin was a soldier and governor noted for corruption, with mathematics just a hobby, I've chosen him to represent this group because of the key advances which appear first in his writings.

Nicole  Oresme (ca 1322-1382) France
Oresme was of lowly birth but excelled at school (where he was taught by the famous Jean Buridan), became a young professor, and soon personal chaplain to King Charles V. The King commissioned him to translate the works of Aristotle into French (with Oresme thus playing key roles in the development of both French science and French language), and rewarded him by making him a Bishop. He wrote several books; was a renowned philosopher, natural scientist (challenging several of Aristotle's ideas, and postulating that the Earth moved), and important economist (anticipating Gresham's Law). Oresme used a graphical diagram to demonstrate the Merton College Theorem (a discovery related to Galileo's Law of Falling Bodies made by Thomas Bradwardine, et al); it is said this was the first abstract graph. (Some believe that this effort inspired Déscartes' coordinate geometry.) Oresme was also first to use fractional exponents; first to write of general curvature; and, most famously, first to prove the divergence of the harmonic series. His work was largely ignored, so may have had little historic importance, but with several discoveries way ahead of his time, Oresme deserves recognition.

Madhava of Sangamagramma (1340-1425) India
Madhava, also known as Irinjaatappilly Madhavan Namboodiri, founded the important Kerala school of mathematics and astronomy. If everything credited to him was his own work, he was a truly great mathematician. His analytic geometry preceded and surpassed Déscartes', and included differentiation and integration. Madhava also did work with continued fractions, trigonometry, and geometry. He has been called "the Founder of Mathematical Analysis." Madhava is most famous for his work with Taylor series, discovering identities like   sin q = q - q3/3! + q5/5! - ...   , formulae for π, including the one attributed to Leibniz, and the then-best known approximation π ≈ 104348 / 33215.

Ghiyath al-Din Jamshid Mas'ud  Al-Kashi (ca 1380-1429) Iran, Transoxania (Uzbekistan)
Al-Kashi was among the greatest calculaters in the ancient world; wrote important texts applying arithmetic and algebra to problems in astronomy, mensuration and accounting; and developed trig tables far more accurate than earlier tables. He worked with binomial coefficients, invented astronomical calculating machines, developed spherical trig, and is credited with various theorems of trigonometry including the Law of Cosines. He is sometimes credited with the invention of decimal fractions (though he worked mainly with sexagesimal fractions), and a method like Horner's to calculate roots. However his decimal fractions built on prior work, and some historians think his method for calculating roots derived from reading Chinese texts by Qin Jiushao or Zhu Shiejie. Using his methods, al-Kashi calculated π correctly to 17 significant digits, breaking Madhava's record. (This record was subsequently broken by relative unknowns: a German ca. 1600, John Machin 1706. In 1949 the π calculation record was held briefly by John von Neumann and the ENIAC.)

Girolamo  Cardano (1501-1576) Italy
Girolamo Cardano (or Jerome Cardan) was a highly respected physician and was first to describe typhoid fever. He was also an accomplished gambler and chess player and wrote an early book on probability. He was also a remarkable inventor: the combination lock, an advanced gimbal, a ciphering tool, and the Cardan shaft with universal joints are all his inventions and are in use to this day. (The U-joint is sometimes called the Cardan joint.) He also helped develop the camera obscura. Cardano made contributions to physics: he noted that projectile trajectories are parabolas, and may have been first to note the impossibility of perpetual motion machines. He did work in philosophy, geology, hydrodynamics, music; he wrote books on medicine and an encyclopedia of natural science. But Cardano is most remembered for his achievements in mathematics. He was first to publish general solutions to cubic and quartic equations, and first to publish the use of complex numbers in calculations. (Cardano's Italian colleagues deserve much credit: Ferrari first solved the quartic, he or Tartaglia the cubic; and Bombelli first treated the complex numbers as numbers in their own right. Cardano may have been the last great mathematician unaware of negative numbers: his treatment of cubic equations had to deal with ax3 - bx + c = 0 and ax3 - bx = c as two different cases.) Cardano introduced binomial coefficients and the binomial theorem, and introduced and solved the geometric hypocyloid problem, as well as other geometric theorems (e.g. the theorem underlying the 2:1 spur wheel which converts circular to reciprocal rectilinear motion). Cardano is credited with Cardano's Ring Puzzle, still manufactured today and related to the Tower of Hanoi puzzle. (This puzzle may predate Cardano, and may even have been known in ancient China.) Da Vinci and Galileo may have been more influential than Cardano, but of the three great generalists in the century before Kepler, it seems clear that Cardano was the most accomplished mathematician.
Cardano's life had tragic elements. Throughout his life he was tormented that his father (a friend of Leonardo da Vinci) married his mother only after Cardano was born. (And his mother tried several times to abort him.) Cardano's reputation for gambling and aggression interfered with his career. He practiced astrology and was imprisoned for heresy when he cast a horoscope for Jesus. (This and other problems were due in part to revenge by Tartaglia for Cardano's revealing his secret algebra formulae.) His son apparently murdered his own wife. Leibniz wrote of Cardano: "Cardano was a great man with all his faults; without them, he would have been incomparable."

François  Viète (1540-1603) France
François Viète (or Franciscus Vieta) was a French nobleman and lawyer who was a favorite of King Henry IV and eventually became a royal privy councillor. In one notable accomplishment he broke the Spanish diplomatic code, allowing the French government to read Spain's messages and publish a secret Spanish letter; this apparently led to the end of the Huguenot Wars of Religion. More importantly, Vieta was certainly the best French mathematician prior to Déscartes and Fermat. He laid the groundwork for modern mathematics; his works were studied by Isaac Newton. In his role as a young tutor Vieta used decimal numbers before they were popularized by Simon Stevin and may have guessed that planetary orbits were ellipses before Kepler. Vieta did work in geometry, reconstructing and publishing proofs for Apollonius' lost theorems. He discovered several trigonometric identities including the Law of Cosines and a generalization of Ptolemy's Formula, the latter (then called prosthaphaeresis) providing a calculation shortcut similar to logarithms in that multiplication is reduced to addition (or exponentiation reduced to multiplication). Vieta also used trigonometry to find real solutions to cubic equations for which the Italian methods had required complex-number arithmetic; he also used trigonometry to solve a particular 45th-degree equation that had been posed as a challenge. Such trigonometric formulae revolutionized calculations and may even have helped stimulate the development and use of logarithms by Napier and Kepler. He developed the first infinite-product formula for π. In addition to his geometry and trigonometry, he also found results in number theory, but Vieta is most famous for his systematic use of decimal notation and variable letters, for which he is sometimes called the Father of Modern Algebra. (Vieta used A,E,I,O,U for unknowns and consonants for parameters; it was Déscartes who first used X,Y,Z for unknowns and A,B,C for parameters.) In his works Vieta emphasized the relationships between algebraic expressions and geometric constructions. One key insight he had is that addends must be homogeneous (i.e., "apples shouldn't be added to oranges"), a seemingly trivial idea but which can aid intuition even today.
Déscartes, who once wrote "I began where Vieta finished," is now extremely famous, while Vieta is much less known. (He isn't even mentioned once in Bell's famous Men of Mathematics.) Many would now agree this is due in large measure to Déscartes' deliberate deprecations of competitors in his quest for personal glory. (Vieta wasn't particularly humble either, calling himself the "French Apollonius.")

PI := 2
Y := 0
Y := SQRT(Y + 2)
PI := PI * 2 / Y
IF (more precision needed) GOTO LOOP

Vieta's formula for π is clumsy to express without trigonometry, even with modern notation. Easiest may be to consider it the result of the BASIC program above. Using this formula, Vieta constructed an approximation to π that was best-yet by a European, though not as accurate as al-Kashi's two centuries earlier.

Simon  Stevin (1549-1620) Flanders, Holland
Stevin was one of the greatest practical scientists of the Late Middle Ages. He worked with Holland's dykes and windmills; as a military engineer he developed fortifications and systems of flooding; he invented a carriage with sails that travelled faster than with horses. He discovered several laws of mechanics including those for energy conservation, hydrostatic pressure, the equal rate of falling bodies attributed to Galileo, and the influence of the moon on tides. He invented improved accounting methods, and the equal-temperament music scale. He also did work in descriptive geometry, trigonometry, optics, geography, and astronomy. In mathematics, Stevin is best known for the notion of real numbers (previously integers, rationals and irrationals were treated separately) and for introducing decimal fractions to Europe; his historical importance may warrant a place on the List despite that he proved no major theorems of pure mathematics.

John  Napier  8th of Merchistoun (1550-1617) Scotland
Napier was a Scottish Laird who was a noted theologian and thought by many to be a magician (his nickname was Marvellous Merchiston). Today, however, he is best known for his work with logarithms, a word he invented. (Several others, including Archimedes, had anticipated the use of logarithms.) He published the first large table of logarithms and also helped popularize usage of the decimal point and lattice multiplication. He invented Napier's Bones, a crude hand calculator which could be used for division and root extraction, as well as multiplication. He also had inventions outside mathematics, including war machines. Napier's noted textbooks also contain an exposition of spherical trigonometry. Although he was certainly very clever (and had novel mathematical insights not mentioned in this summary), Napier proved no deep theorem and may not belong on a list of great mathematicians. Nevertheless, his revolutionary methods of arithmetic had immense historical importance; his tables were used by Johannes Kepler himself, and led to the Scientific Revolution.

Galileo  Galilei (1564-1642) Italy
Galileo discovered the laws of inertia, falling bodies (including parabolic trajectories), and the pendulum. He even anticipated Einstein's special relativity. He was a great inventor: in addition to being first to conceive of the pendulum clock, he developed a new type of pump, and the best telescope, thermometer, hydrostatic balance, and cannon sector of his day. As a famous astronomer, he pointed out that Jupiter's Moons, which he discovered, provide a natural clock and allow a universal time to be determined by telescope anywhere on Earth. (This was of little use in ocean navigation since a ship's rocking prevents the required delicate observations.) Galileo is also renowned for early discoveries with microscope. Galileo is often called the "Father of Modern Science" because of his emphasis on experimentation. He understood that results needed to be repeated and averaged (he used mean absolute difference as his curve-fitting criterion, two centuries before Gauss and Legendre introduced the mean squared-difference criterion). For his experimental methods and discoveries, his laws of motion, and for (eventually) helping to spread Copernicus' heliocentrism, Galileo is considered to be one of the most influential scientists ever; he ranks #12 on Hart's list of the Most Influential Persons in History. Despite his extreme importance to mathematical physics, Galileo doesn't usually appear on lists of greatest mathematicians. However Cavalieri and Torricelli were his students; to encourage them, he probably avoided competing with them. Galileo derived certain centroids using a rudimentary calculus before Cavalieri did, named (and may have been first to discover) the cycloid curve, and did other mathematical work. Moreover, Galileo may have been first to write about infinite equinumerosity (the "Hilbert's Hotel Paradox"). Galileo once wrote "Mathematics is the language in which God has written the universe."

Johannes  Kepler (1571-1630) Germany
Kepler was interested in astronomy from an early age, studied to become a Lutheran minister, became a professor of mathematics instead, then Tycho Brahe's understudy, and, on Brahe's death, was appointed Imperial Mathematician at the age of twenty-nine. His observations of the planets with Brahe, along with his study of Apollonius' 1800-year old work, led to Kepler's three Laws of Planetary Motion, which in turn led directly to Newton's Laws of Motion. Beyond his discovery of these Laws (one of the most important achievements in all of science), Kepler is also sometimes called the Founder of Modern Optics. He was first to study the operation of the human eye, telescopes built from two convex lenses, and atmospheric refraction. Kepler was first to explain tides correctly. (Galileo dismissed this as well as Kepler's elliptical orbits, and later published his own incorrect explanation of tides.) As one of the key figures in the Scientific Revolution, he ranks #75 on Michael Hart's famous list of the Most Influential Persons in History. According to Kepler's Laws, the planets move at variable speed along ellipses. The Earth-bound observer is himself describing such an orbit and in almost the same plane as the planets; thus discovering the Laws would be a difficult challenge even for someone armed with computers and modern mathematics. (The very famous Kepler Equation relating a planet's eccentric and anomaly is just one tool Kepler needed to develop.) Kepler understood the importance of his remarkable discovery, even if contemporaries like Galileo did not, writing:
"I give myself up to divine ecstasy ... My book is written. It will be read either by my contemporaries or by posterity — I care not which. It may well wait a hundred years for a reader, as God has waited 6,000 years for someone to understand His work."
Besides the trigonometric results needed to discover his Laws, Kepler made other contributions to mathematics. He generalized Alhazen's Billiard Problem, developing the notion of curvature. He was first to notice that the set of Platonic regular solids was incomplete if concave solids are admitted, and first to prove that there were only 13 "Archimedean solids." He proved theorems of solid geometry later discovered on the famous palimpsest of Archimedes. He rediscovered the Fibonacci series, applied it to botany, and noted that the ratio of Fibonacci numbers converges to the Golden Mean. He was a key early pioneer in calculus, and embraced the concept of continuity (which others avoided due to Zeno's paradoxes); his work was a direct inspiration for Cavalieri and others. He developed mensuration methods and anticipated Fermat's theorem (df(x)/dx = 0 at function extrema). Kepler once had an opportunity to buy wine, which merchants measured using a shortcut; with the famous Kepler's Wine Barrel Problem, he used his rudimentary calculus to deduce which barrel shape would be the best bargain.
Kepler reasoned that the structure of snowflakes was evidence for the then-novel atomic theory of matter. He noted that the obvious packing of cannonballs gave maximum density (this became known as "Kepler's Conjecture"; optimality was proved among regular packings by Gauss, but it wasn't until 1998 that the possibility of denser irregular packings was disproven). In addition to his physics and mathematics, Kepler wrote a science fiction novel, and was an astrologer and mystic. He had ideas similar to Pythagoras about numbers ruling the cosmos (writing that the purpose of studying the world "should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics"). Kepler's mystic beliefs even led to his own mother being imprisoned for witchcraft.
Johannes Kepler (along with Galileo, Fermat, Huygens, Wallis, Vieta and Déscartes) is among the giants on whose shoulders Newton was proud to stand. Some historians place him ahead of Galileo and Copernicus as the single most important early contributor to the early Scientific Revolution. Chasles includes Kepler on a list of the six responsible for conceiving and perfecting infinitesimal calculus (the other five are Archimedes, Cavalieri, Fermat, Leibniz and Newton). (www.keplersdiscovery.com is a wonderful website devoted to Johannes Kepler's discoveries.)

Gerard   Desargues (1591-1661) France
Desargues invented projective geometry and found the relationship among conic sections which inspired Blaise Pascal. Among several ingenious and rigorously proven theorems are Desargues' Involution Theorem and his Theorem of Homologous Triangles. Desargues was also a noted architect and inventor: he produced an elaborate spiral staircase, invented an ingenious new pump, and had the idea to use cycloid-shaped teeth in the design of gears. Desargues' projective geometry may have been too creative for his time, and was largely ignored (except by Pascal himself) until Poncelet rediscovered it almost two centuries later. (Copies of Desargues' own works surfaced about the same time.) For this reason, Desargues may not belong on our list, despite that he may have been among the greatest natural geometers ever.

René  Déscartes (1596-1650) France
Déscartes' 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 is considered the inventor of analytic geometry and therefore 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. Déscartes developed laws of motion (including a "vortex" theory of gravitation) which were very influential, though largely incorrect. His famous mathematical theorems include the Rule of Signs (for determining the signs of polynomial roots), the elegant formula relating the radii of Soddy kissing circles, his theorem on total angular defect, and an improvement on the ancient construction method for cube-doubling. He improved mathematical notation (e.g. the use of superscripts to denote exponents). He also discovered Euler's Polyhedral Theorem (V - E + F = 2). Déscartes has an extremely high reputation and would be ranked higher by many list makers. I've demoted Déscartes partly because he had only insulting things to say about Pascal and Fermat, each of whom was more brilliant at mathematics than Déscartes. (Some even suspect that Déscartes arranged the destruction of Pascal's lost Essay on Conics.) Déscartes' errors may have set back the cause of science, Huygens writing "in all of [Déscartes'] physics, I find almost nothing to which I can subscribe as being correct." Moreover the historical importance of the Frenchmen may be slightly exaggerated since others, e.g. Wallis and Cavalieri, were developing modern mathematics independently.

Francesco Bonaventura de  Cavalieri (1598-1647) Italy
Cavalieri did work in analysis, geometry and trigonometry; he is most famous for developing a rudimentary calculus. (Because of his calculus, Cavalieri was certainly an influential mathematician; however his work was largely anticipated by Kepler, and was soon surpassed by Fermat and Wallis.) Cavalieri also worked in theology, astronomy, mechanics and optics; he was an inventor, and published logarithm tables. He wrote several books, the first one developing the properties of mirrors shaped as conic sections. His name is especially remembered for Cavalieri's Principle of Solid Geometry. Galileo said of Cavalieri, "Few, if any, since Archimedes, have delved as far and as deep into the science of geometry."

Pierre de  Fermat (1601-1665) France
Pierre de Fermat was the most brilliant mathematician of his era and, along with Déscartes, one of the most influential. Although mathematics was just his hobby (Fermat was a government lawyer), Fermat practically founded Number Theory, and also played key roles in the discoveries of Analytic Geometry and Calculus. He was also an excellent geometer (e.g. discovering a triangle's Fermat point), and (in collaboration with Blaise Pascal) discovered probability theory. Fellow geniuses are the best judges of genius, and Blaise Pascal had this to say of Fermat: "For my part, I confess that [Fermat's researches about numbers] are far beyond me, and I am competent only to admire them." E.T. Bell wrote "it can be argued that Fermat was at least Newton's equal as a pure mathematician." Fermat's most famous discoveries in number theory include the ubiquitously-used Fermat's Little Theorem; the n = 4 case of his conjectured Fermat's Last Theorem (he may have proved the n = 3 case as well); the fact that every natural number is the sum of three triangle numbers; and Fermat's Christmas Theorem (that any prime (4n+1) can be represented as the sum of two squares in exactly one way, also called the Fermat-Euler Prime Number Theorem). As suggested by the "Euler" in the name of this latter theorem (which Fermat records proving with difficulty using "infinite descent"), proofs for this and many other Fermat results had to be rediscovered (most of Fermat's work was never published). However it is wrong to suppose that Fermat's work comprised mostly false or unproven conjectures. (This misconception arises from his so-called "Last Theorem" which was actually just a private scribble.)
Fermat developed a system of analytic geometry which both preceded and surpassed that of Déscartes; he developed methods of differential and integral calculus which Newton acknowledged as an inspiration. Solving   df(x)/dx = 0   to find extrema of f(x) is perhaps the most useful idea in applied mathematics; this technique originated with Fermat. Fermat was also the first European to find the integration formula for the general polynomial; he used his calculus to find centers of gravity, etc.
Fermat's contemporaneous rival René Déscartes is more famous than Fermat, and Déscartes' writings were more influential. Whatever one thinks of Déscartes as a philosopher, however, it seems clear that Fermat was the better mathematician. Fermat and Déscartes did work in physics and independently discovered the (trigonometric) law of refraction, but Fermat gave the correct explanation, and used it remarkably to anticipate the Principle of Least Action later enunciated by Maupertius (though Maupertius himself, like Déscartes, had an incorrect explanation of refraction). Fermat and Déscartes independently discovered analytic geometry, but it was Fermat who extended it to more than two dimensions, and followed up by developing elementary calculus.

John Brehaut  Wallis (1616-1703) England
Wallis began his life as a savant at arithmetic (it is said he once calculated the square root of a 53-digit number to help him sleep and remembered the result in the morning), a medical student (he may have contributed to the concept of blood circulation), and theologian, but went on to become perhaps the most brilliant and influential English mathematician before Newton. (James Gregory is another candidate for this honor, but Gregory's career was truncated by death at age 36.) He made major advances in analytic geometry, but also contributions to algebra, geometry and trigonometry. He is especially famous for using negative and fractional exponents (though Oresme had introduced fractional exponents 3 centuries earlier), taking the areas of curves, and treating inelastic collisions (he and Huygens were first to develop the law of momentum conservation). He was the first European to solve Pell's Equation. Like Vieta, Wallis was a code-breaker, helping the Commonwealth side (though he later petitioned against the beheading of King Charles I). He was the first great mathematician to consider complex numbers legitimate; and first to use the symbol . Wallis coined several terms including "continued fraction," "induction," "interpolation," "mantissa," and "hypergeometric series." Also like Vieta, Wallis created an infinite product formula for pi, which might be (but isn't) written today as:
    π = 2 ∏k=1,∞ 1+(4k2-1)-1

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 Apollonius' famous theorems about conic sections was a corollary of the Mystic Hexagram; along with Gerard Desargues (1591-1661), he was a key pioneer of projective geometry. He also made important early contributions to calculus. Returning to geometry late in life, Pascal advanced the theory of the cycloid. In addition to his work in geometry and calculus, he founded probability theory, and made contributions to axiomatic theory. His name is associated with the Pascal's Triangle of combinatorics and Pascal's Wager in theology. Like most of the greatest mathematicians, Pascal was interested in physics and mechanics, studying fluids, explaining vacuum, and inventing the syringe and hydraulic press. At the age of eighteen he designed and built the world's first automatic adding machine. (Although he continued to refine this invention, it was never a commercial success.) He suffered poor health throughout his life, abandoned mathematics for religion at about age 23, and died at an early age. Many think that had he devoted more years to mathematics, Pascal would have been one of the greatest mathematicians ever.

Christiaan  Huygens (1629-1695) Holland, France
Christiaan Huygens (or Hugens, Huyghens) was second only to Newton as the greatest mechanist of his era. Although an excellent mathematician, he is much more famous for his physical theories and inventions. He developed laws of motion before Newton, including the inverse-square law of gravitation, centripetal force, and treatment of solid bodies rather than point approximations; he (and Wallis) were first to state the law of momentum conservation correctly. He advanced the wave ("undulatory") theory of light, a key concept being Huygen's Principle, that each point on a wave front acts as a new source of radiation. His optical discoveries include explanations for polarization and phenomena like haloes. (Because of Newton's high reputation and corpuscular theory of light, Huygens' superior wave theory was largely ignored until the 19th-century work of Young, Fresnel, and Maxwell. Later, Planck, Einstein and Bohr, partly anticipated by Hamilton, developed the modern notion of wave-particle duality.) Huygens is famous for his inventions of clocks and lenses. He invented the escapement and other mechanisms, leading to the first reliable pendulum clock; he built the first balance spring watch, which he presented to his patron, King Louis XIV of France. He invented superior lens grinding techniques, the achromatic eye-piece, and the best telescope of his day. He was himself a famous astronomer: he discovered Titan and was first to properly describe Saturn's rings and the Orion Nebula. He also designed, but never built, an internal combustion engine. He promoted the use of 31-tone music: a 31-tone organ was in use in Holland as late as the 20th century. Huygens was an excellent card player, billiard player, horse rider, and wrote a book speculating about extra-terrestrial life.
As a mathematician, Huygens did brilliant work in analysis; his calculus, along with that of Wallis, is considered the best prior to Newton and Leibniz. He also did brilliant work in geometry, proving theorems about conic sections, the cycloid and the catenary. He was first to show that the cycloid solves the tautochrone problem; he used this fact to design pendulum clocks that would be more accurate than ordinary pendulum clocks. He was first to find the flaw in Saint-Vincent's then-famous circle-squaring method; Huygens himself solved some related quadrature problems. He introduced the concepts of evolute and involute. His friendships with Déscartes, Pascal, Mersenne and others helped inspire his mathematics; Huygens in turn was inspirational to the next generation. At Pascal's urging, Huygens published the first real textbook on probability theory; he also became the first practicing actuary.
Huygens had tremendous creativity, historical importance, and depth and breadth of genius, both in physics and mathematics. He also was important for serving as tutor to the otherwise self-taught Gottfried Leibniz (who'd "wasted his youth" without learning any math). Before agreeing to tutor him, Huygens tested the 25-year old Leibniz by asking him to sum the reciprocals of the triangle numbers.

Takakazu  Seki (Kowa) (ca 1637-1708) Japan
Seki Takakazu (aka Shinsuke) was a self-taught prodigy who developed a new notation for algebra, and 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 calculated π to ten decimal places using Aitkin's method (rediscovered in the 20th century). He also worked with magic squares. He is remembered as a brilliant genius and very influential teacher.
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 (e.g. a model windmill powered by a mouse on treadmill). At about age 22, on leave from University, this genius began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. He is most famous for his Three Laws of Motion (inertia, force, reciprocal action) and Law of Universal Gravitation. As Newton himself acknowledged, the Laws weren't fully novel: Hipparchus, Ibn al-Haytham, Galileo and Huygens had all developed much basic mechanics already, and Newton credits the First Law itself to Aristotle. (However, since Christiaan Huygens, the other great mechanist of the era and who had also deduced that Kepler's laws imply inverse-square gravitation, considered the action at a distance in Newton's universal gravitation to be "absurd," at least this much of Newton's mechanics must be considered revolutionary. Newton's other intellectual interests included chemistry, theology, astrology and alchemy.) Although this list is concerned only with mathematics, Newton's greatness is indicated by the wide range of his physics: even without his revolutionary Laws of Motion and his Cooling Law of thermodynamics, he'd be famous just for his work in optics, where he explained diffraction and observed that white light is a mixture of all the rainbow's colors. (Although his corpuscular theory competed with Huygen's wave theory, Newton understood that his theory was incomplete without waves, and thus anticipated wave-particle duality.) Newton also designed the first reflecting telescope, first reflecting microscope, and the sextant.
Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (which he called "fluxions"); he shares credit with Leibniz for the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus for several purposes: finding areas, tangents, the lengths of curves and the maxima and minima of functions. In addition to several other important advances in analytic geometry, his mathematical works include the Binomial Theorem, his eponymous numeric method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (His equation   ex = xk / k!   has been called the "most important series in mathematics.") He contributed to algebra and the theory of equations; he was first to state Bézout's Theorem; he generalized Déscartes' rule of signs. (The generalized rule of signs was incomplete and finally resolved two centuries later by Sturm and Sylvester.) He developed a series for the arcsin function. He developed facts about cubic equations (just as the "shadows of a cone" yield all quadratic curves, Newton found a curve whose "shadows" yield all cubic curves). He proved that same-mass spheres of any radius have equal gravitational attraction: this fact is key to celestial motions. He discovered Puiseux series almost two centuries before they were re-invented by Puiseux. (Like some of the greatest ancient mathematicians, Newton took the time to compute an approximation to π; his was better than Vieta's, though still not as accurate as al-Kashi's.)
Newton is so famous for his calculus, optics and laws of motion, it is easy to overlook that he was also one of the greatest geometers. He solved the Delian cube-doubling problem. Even before the invention of the calculus of variations, Newton was doing difficult work in that field, e.g. his calculation of the "optimal bullet shape." Among many marvelous theorems, he proved several about quadrilaterals and their in- or circum-scribing ellipses, and constructed the parabola defined by four given points. He anticipated Poncelet's Principle of Continuity. 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 Jacob 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. (In this work Newton also proved important theorems about inverse-cube forces, work largely unappreciated until Chandrasekhar's modern-day work.) The notion that the Earth rotated about the Sun was introduced in ancient Greece, but Newton explained why it did, and the Great Scientific Revolution began. Newton once wrote "Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things." Sir Isaac Newton was buried at Westminster Abbey in a tomb inscribed "Let mortals rejoice that so great an ornament to the human race has existed."
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 first on any list of physicists, or scientists in general, but some listmakers would demote him slightly on a list of pure mathematicians: his emphasis was physics not mathematics, and the contribution of Leibniz (Newton's rival for the title Inventor of Calculus) lessens the historical importance of Newton's calculus. One reason I've ranked him at #1 is a comment by Gottfried Leibniz himself: "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 von  Leibniz (1646-1716) Germany
Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe. Descriptions which have been applied to Leibniz include "one of the two greatest universal geniuses" (da Vinci was the other); "the most important logician between Aristotle and Boole;" and the "Father of Applied Science." Leibniz described himself as "the most teachable of mortals." Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Ten who was never the greatest living algorist or theorem prover. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication. (And his political influence may have been huge: he was a special consultant to both the Holy Roman and Russian Emperors, and was helped arrange for the son of his patron Sophia Wittelsbach, only distantly in line for the British throne, to be crowned King George I of England.)
Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the symbols , df(x)/dx; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He invented more mathematical terms than anyone, including "function," "analysis situ," "variable," "abscissa," "parameter," and "coordinate." His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was notation ("calculus"), because with "symbols [which] express the exact nature of a thing briefly ... the labor of thought is wonderfully diminished."
Leibniz' thoughts on mathematical physics had some influence. He developed laws of motion that gave different insights from those of Newton. His cosmology was opposed to that of Newton but, anticipating theories of Mach and Einstein, is more in accord with modern physics. Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves.
Although others found it independently (including perhaps Madhava three centuries earlier), Leibniz discovered and proved a striking identity for π:
        π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

Jacob   Bernoulli (1654-1705) Switzerland
Jacob Bernoulli studied the works of Wallis and Barrow; became friends with Leibniz; tutored Leibniz as well as Jacob's own brother Johann. Jacob developed important methods for integral and differential equations, coining the word "integral." He and his brother were the key pioneers in mathematics during the generations between the era of Newton-Leibniz and the rise of Leonhard Euler. Jacob liked to pose and solve physical optimization problems. His "catenary" problem (what shape does a clothesline take?) became more famous than the "tautochrone" solved by Huygens. Perhaps the most famous of such problems was the brachistochrone, wherein Jacob recognized Newton's "lion's paw", and about which Johann Bernoulli wrote: "You will be petrified with astonishment [that] this same cycloid, the tautochrone of Huygens, is the brachistochrone we are seeking." Jacob did significant work outside calculus; in fact his most famous work was the Art of Conjecture, a textbook on probability and combinatorics which proves the Law of Large Numbers, the Power Series Equation, and introduces the Bernoulli numbers. Jacob also did outstanding work in geometry, for example constructing perpendicular lines which quadrisect a triangle.

Johann  Bernoulli (1667-1748) Switzerland
Johann Bernoulli learned from his older brother and Leibniz, and went on to become principle teacher to Leonhard Euler. He developed exponential calculus; together with his brother Jacob, he founded the calculus of variations. Johann solved the catenary before Jacob did; this led to a famous rivalry in the Bernoulli family. (No joint papers were written; instead the Bernoullis, especially Johann, began claiming each others' work.) Although his older brother may have demonstrated greater breadth, Johann had no less skill than Jacob, contributed more to calculus, discovered L'Hopital's Rule before L'Hopital did, and made important contributions in physics, e.g. about vibrations, elastic bodies, optics, tides, and ship sails. It may not be clear which Bernoulli was the "greatest." Johann has special importance as tutor to Leonhard Euler, but Jacob has special importance as tutor to his brother Johann!

Brook  Taylor (1685-1731) England
Brook Taylor invented integration by parts, developed what is now called the calculus of finite differences, developed a new method to compute logarithms, made several other key discoveries of analysis, and did significant work in mathematical physics. His love of music and painting may have motivated some of his mathematics: He studied vibrating strings; and also wrote an important treatise on perspective and vanishing points in drawing which helped develop the fields of both projective and descriptive geometry.
Taylor was one of the few mathematicians of the Bernoulli era who was equal to them in genius, but his work was much less influential. Today he is most remembered for Taylor Series and the associated Taylor's Theorem, but he shouldn't get full credit for this. The method had been anticipated by earlier mathematicians including Gregory, Leibniz, Newton, and, even earlier, Madhava; and was not fully appreciated until the work of later mathematicians like Colin Maclaurin and Lagrange.

Daniel  Bernoulli (1700-1782) Switzerland
Johann Bernoulli had a nephew, three sons and some grandsons who were all also outstanding mathematicians. Of these, the most important was his son Daniel. Johann insisted that Daniel study biology and medicine rather than mathematics, so Daniel specialized initially in mathematical biology. He went on to win the Grand Prize of the Paris Academy no less than ten times, and was a close friend of Euler. He developed partial differential equations, anticipated Fourier series, did important work in statistics and the theory of equations, developed a theory of economic risk (motivated by the St. Petersburg Paradox discovered by his cousin Nicholas), but is most famous for his important discoveries in mathematical physics, including the Bernoulli Principle underlying airflight. Daniel Bernoulli is sometimes called "the Founder of Mathematical Physics."

Leonhard  Euler (1707-1783) Switzerland
Euler may be the most influential mathematician who ever lived (though some would make him second to Euclid); he ranks #77 on Michael Hart's famous list of the Most Influential Persons in History. His colleagues called him "Analysis Incarnate." Laplace, famous for denying credit to fellow mathematicians, once said "Read Euler: he is our master in everything." His notations and methods in many areas are in use to this day. Euler was the most prolific mathematician in history and is often judged to be the best algorist of all time. (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 gave the world modern trigonometry, pioneered (along with Lagrange) the calculus of variations, generalized and proved the Newton-Giraud formulae, etc. He was also supreme at discrete mathematics, inventing graph theory and generating functions. Euler was also a major figure in number theory: He proved that the sum of the reciprocals of primes less than x is approx. (ln ln x), 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 to be irrational, proved that all even perfect numbers must have the Mersenne number form that Euclid had discovered 2000 years earlier, and much more. Euler was also first to prove several interesting theorems of geometry, including facts about the 9-point Feuerbach circle; relationships among a triangle's altitudes, medians, and circumscribing and inscribing circles; and an expression for a tetrahedron's area in terms of its sides. Euler was first to explore topology, proving theorems about the Euler characteristic. Although noted as the first great "pure mathematician," Euler engineered a system of pumps, wrote on philosophy, and made important contributions to music theory, acoustics, optics, celestial motions and mechanics. He extended Newton's Laws of Motion to rotating rigid bodies; and developed the Euler-Bernoulli beam equation. On a lighter note, Euler constructed a particularly "magical" magic square.
Euler combined his brilliance with phenomenal concentration. He developed the first method to estimate the Moon's orbit (the three-body problem which had stumped Newton), and he settled an arithmetic dispute involving 50 decimal places of 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, i = √-1, and γ = 0.57721566...) were all introduced or popularized by Euler, along with operators like Σ. He did important work with Riemann's zeta function   ζ(s) = ∑ k-s   (although it was not then known with that name or notation); he anticipated the concept of analytic continuation by "proving" ζ(-1) = 1+2+3+4+... = -1/12. As a young student of the Bernoulli family, Euler discovered the striking identity   π2/6 = ζ(2)   This catapulted Euler to instant fame, since the right-side infinite sum (1 + 1/4 + 1/9 + 1/16 + ...) was a famous problem of the time. Among many other famous and important identities, Euler proved the Pentagonal Number Theorem (a beautiful little result which has inspired a variety of discoveries), and the Euler Product Formula     ζ(s) = ∏(1-p-s)-1   where the right-side product is taken over all primes p. His most famous identity (which Richard Feynman called an "almost astounding ... jewel") unifies the trigonometric and exponential functions:   ei x = cos x + i sin x.
Some of Euler's greatest formulae can be combined into curious-looking formulae for π:   π2   =   - log2(-1)   =   6 ∏p∈Prime(1-p-2)-1/2

Alexis Claude  Clairaut (1713-1765) France
The reputations of Euler and the Bernoullis are so high that it is easy to overlook that others in that epoch made essential contributions to mathematical physics. (Euler made errors in his development of physics, in some cases because of a Europeanist rejection of Newton's theories in favor of the contradictory theories of Déscartes and Leibniz.) The Frenchmen Clairaut and d'Alembert were two other great and influential mathematicians of the early 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 important 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 Déscartes gave different predictions for the shape of the Earth (whether the poles were flattened or pointy); Clairaut participated in Maupertuis' expedition to Lappland to measure the polar regions. Measurements at high latitudes showed the poles to be flattened: Newton was right. Clairaut worked on the theories of ellipsoids and the three-body problem, e.g. Moon's orbit. That orbit was the major mathematical challenge of the day, and there was great difficulty reconciling theory and observation. It was Clairaut who finally resolved this, by approaching the problem with more rigor than others. When Euler finally understood Clairaut's solution he called it "the most important and profound discovery that has ever been made in mathematics." Later, when Halley's Comet reappeared as he had predicted, Clairaut was acclaimed as "the new Thales."

Jean le Rond  d' Alembert (1717-1783) France
During the century after Newton, the Laws of Motion needed to be clarified and augmented with mathematical techniques. Jean le Rond, named after the Parisian church where he was abandoned as a baby, played a very key role in that development. His D'Alembert's Principle clarified Newton's Third Law and allowed problems in dynamics to be expressed with simple partial differential equations; his Method of Characteristics then reduced those equations to ordinary differential 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 correct than, the attempted proof by Euler-Lagrange.) He also did creative work in geometry (e.g. anticipating Monge's Three Circle Theorem), and was principle 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."

Johann Heinrich  Lambert (1727-1777) Switzerland, Prussia
Lambert had to drop out of school at age 12 to help support his family, but went on to become a mathematician of great fame and breadth. He was first to prove that π is irrational. (He proved more strongly that tan x and ex are both irrational for any non-zero rational x. His proof for this was so remarkable for its time, that its completeness wasn't recognized for over a century.) He also conjectured that π and e were transcendental. He made advances in analysis (including the introduction of Lambert's W function) and in trigonometry (introducing the hyperbolic functions sinh and cosh); proved a key theorem of spherical trigonometry, and solved the "trinomial equation." He also made important contributions in philosophy, cosmology (conceiving of star clusters, galaxies and supergalaxies), map-making (inventing several 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), as well as a famous identity used to calculate cometary orbits which Lagrange declared 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.

Joseph-Louis (Comte de)  Lagrange (1736-1813) Italy, France
Joseph-Louis Lagrange (born Giuseppe Lodovico Lagrangia) was a brilliant man who advanced to become a teen-age Professor shortly after first studying mathematics. He excelled in all fields of analysis and number theory; he made key contributions to the theories of determinants, continued fractions, and many other fields. He developed partial differential equations far beyond those of D. Bernoulli and d'Alembert, developed the calculus of variations far beyond that of the Bernoullis, and developed terminology and notation (e.g. the use of f'(x) and f''(x) for a function's 1st and 2nd derivatives). He proved a fundamental Theorem of Group Theory. He laid the foundations for the theory of polynomial equations which Cauchy, Abel, Galois and Poincaré would later complete. Number theory was almost just a diversion for Lagrange, whose focus was analysis; nevertheless he was the master of that field as well, proving difficult and historic theorems including Wilson's theorem (p divides (p-1)! + 1 when p is prime); Lagrange's Four-Square Theorem (every positive integer is the sum of four squares); and that n·x2 + 1 = y2 has solutions for every positive non-square integer n. Lagrange's many contributions to physics include understanding of vibrations (he found an error in Newton's work and published the definitive treatise on sound), celestial mechanics (including an explanation of why the Moon keeps the same face pointed towards the Earth), the Principle of Least Action (which Hamilton compared to poetry), and the discovery of the Lagrangian points (e.g., in Jupiter's orbit). Lagrange's textbooks were noted for clarity and inspired most of the 19th-century mathematicians on this list. Unlike Newton, who used calculus to derive his results but then worked backwards to create geometric proofs for publication, Lagrange relied only on analysis. "No diagrams will be found in this work" he wrote in the preface to his masterpiece Mécanique 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."

Gaspard  Monge (Comte de Péluse) (1746-1818) France
Gaspard Monge, son of a humble peddler, was an industrious and creative inventor who astounded early with his genius, becoming a professor of physics at age 16. As a military engineer he developed the new field of descriptive geometry, so useful to engineering that it was kept a military secret for 15 years. Monge made early discoveries in chemistry and helped promote Lavoisier's work; he also wrote papers on optics and metallurgy; Monge's talents were so diverse that he became Minister of the Navy in the revolutionary government, and eventually became a close friend and companion of Napoleon Bonaparte. Traveling with Napoleon he demonstrated great courage on several occasions. In mathematics, Monge is called the Father of Differential Geometry, and it is that foundational work for which he is most praised. He also did work in discrete math, partial differential equations, and calculus of variations. He anticipated Poncelet's Principle of Continuity. Monge's most famous theorems of geometry are the "Three Circles Theorem" and "Four Spheres Theorem." His early work in descriptive geometry has little interest to pure mathematics, but his application of calculus to the curvature of surfaces inspired Gauss and eventually Riemann, and led the great Lagrange to say "With [Monge's] application of analysis to geometry this devil of a man will make himself immortal."
Monge was an inspirational teacher whose students included Fourier, Sophie Germain, Chasles, Brianchon, Ampere, Carnot, Poncelet and several other famous mathematicians. 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.

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. (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 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, 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.
Others might place Laplace higher on the List, but he proved no fundamental theorems of pure mathematics (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 greatest applied mathematicians ever.

Adrien Marie  Legendre (1752-1833) France
Legendre was an outstanding mathematician who did important work in plane and solid geometry, spherical trigonometry, celestial mechanics and other areas of physics, and especially elliptic integrals and number theory. He also made important 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 π2 was irrational (the irrationality of π had 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 important 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 important 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; 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), each had slightly flawed proofs left to Gauss to correct. 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.

Jean Baptiste Joseph  Fourier (1768-1830) France
Joseph Fourier had a varied career: precocious but mischievous orphan, theology student, young professor of mathematics (advancing the theory of equations), then revolutionary activist. Under Napoleon he was a brilliant and important teacher and historian; accompanied the French Emperor to Egypt; and did excellent service as district governor of Grenoble. In his spare time at Grenoble he continued the work in mathematics and physics that led to his immortality. After the fall of Napoleon, Fourier exiled himself to England, but returned to France when offered an important academic position and published his revolutionary treatise on the Theory of Heat. Fourier anticipated linear programming, developing the simplex method and Fourier-Motzkin Elimination; and did important 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 as pure mathematics. Fourier's Heat Equation built on Newton's Law of Cooling; and the Fourier series solution itself had already been introduced by Euler, Lagrange and Daniel Bernoulli.
Fourier's solution to the heat equation was counterintuitive (heat transfer doesn't seem to involve the oscillations fundamental to trigonometric functions): The brilliance of Fourier's imagination is indicated in that the solution had been rejected by 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.

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, for odd n, if and only if n is the product of distinct prime Fermat numbers. (Click to see construction of regular 17-gon.) At age 24 he published Disquisitiones Arithmeticae, probably the greatest book of pure mathematics ever. Although he published fewer papers than some other great mathematicians, Gauss may be the greatest theorem prover ever. Several important theorems and lemmas bear his name; he was first to produce a complete proof of Euclid's Fundamental Theorem of Arithmetic (that every natural number has a unique expression as product of primes); and first to produce a rigorous proof of the Fundamental Theorem of Algebra (that an n-th degree polynomial has n complex roots). Gauss himself used "Fundamental Theorem" to refer to Euler's Law of Quadratic Reciprocity; Gauss was first to provide a proof for this, and provided eight distinct proofs for it over the years. Gauss proved the n=3 case of Fermat's Last Theorem for a class of complex integers; though more general, the proof was simpler than the real integer proof, a discovery which revolutionized algebra. Other work by Gauss led to fundamental theorems in statistics, vector analysis, function theory, and generalizations of the Fundamental Theorem of Calculus.
Gauss built the theory of complex numbers into its modern form, including the notion of "monogenic" functions which are now ubiquitous in mathematical physics. Gauss was 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. (The first asteroid was discovered when Gauss was a young man; he famously constructed an 8th-degree polynomial equation to predict its orbit.) Gauss also did important work in several areas of physics, and invented the heliotrope.
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. 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. Gauss is widely agreed to be the most brilliant and productive mathematician who ever lived and many would rank him #1; however several of the others on the list had more historical importance. 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 ..."

Siméon Denis  Poisson (1781-1840) France
Siméon Poisson was a protégé of Laplace and, like his mentor, is among the greatest applied mathematicians ever. Poisson was an extremely prolific researcher and also an excellent teacher. In addition to important advances in several areas of physics, Poisson made important contributions to Fourier analysis, definite integrals, path integrals, statistics, partial differential equations, calculus of variations and other fields of mathematics. Poisson made improvements to Lagrange's equations of celestial motions, which Lagrange himself found inspirational. Another of Poisson's many contributions to mathematical physics was his conclusion that the wave theory of light implies a bright Arago spot at 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."

Jean-Victor  Poncelet (1788-1867) France
After studying under Monge, Poncelet became an officer in Napoleon's army, then a prisoner of the Russians. To keep up his spirits as a prisoner he devised and solved mathematical problems using charcoal and the walls of his prison cell instead of pencil and paper. During this time he reinvented projective geometry. Regaining his freedom, he wrote many papers, made numerous contributions to geometry; he also made contributions to practical mechanics. Poncelet is considered one of the most influential geometers ever; he is especially noted for his Principle of Continuity, an intuition with broad application. His notion of imaginary solutions in geometry was inspirational. Although projective geometry had been studied earlier by mathematicians like Desargues, Poncelet's work excelled and served as an inspiration for other branches of mathematics including algebra, topology, Cayley's invariant theory and group-theoretic developments by Lie and Klein. His theorems of geometry include his Closure Theorem about Poncelet Traverses, the Poncelet-Brianchon Hyperbola Theorem, and Poncelet's Porism (if two conic sections are respectively inscribed and circumscribed by an n-gon, then there are infinitely many such n-gons). Perhaps his most famous theorem, although it was left to Steiner to complete a proof, is the beautiful Poncelet-Steiner Theorem about straight-edge constructions.

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, 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. 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); the famous Burnside's Counting Theorem was first discovered by Cauchy; etc. He was first to prove Taylor's Theorem rigorously, and first to prove Fermat's conjecture that every positive integer can be expressed as the sum of k k-gon numbers for any k.
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. 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 "mislayed." (For this historical miscontribution perhaps Cauchy should be demoted.)

Nicolai Ivanovitch  Lobachevsky (1793-1856) Russia
Lobachevsky is famous for discovering non-Euclidean geometry. He did not regard this new geometry as simply a theoretical curiosity, writing "There is no branch of mathematics ... which may not someday be applied to the phenomena of the real world." He also worked in several branches of analysis and physics, anticipated the modern definition of function, and may have been first to explicitly note the distinction between continuous and differentiable curves. He also discovered the important Dandelin-Gräffe method of polynomial roots independently of Dandelin and Gräffe. (In his lifetime, Lobachevsky was under-appreciated and over-worked; his duties led him to learn architecture and even some medicine.) Although Gauss and Bolyai discovered non-Euclidean geometry independently about the same time as Lobachevsky, it is worth noting that both of them had strong praise for Lobachevsky's genius. His particular significance was in daring to reject a 2100-year old axiom; thus William K. Clifford (a great geometer who anticipated Einstein by suggesting a link between non-Euclidean geometry and gravitation) called Lobachevsky "the Copernicus of Geometry."

Jakob  Steiner (1796-1863) Switzerland
Jakob Steiner made many major advances in synthetic geometry, hoping that classical methods could avoid any need for analysis; and indeed he was often able to equal or surpass methods of the calculus of variations using just pure geometry. (He wrote "Calculating replaces thinking while geometry stimulates it.") Although the Principle of Duality underlying 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. Among many famous and important theorems of classic and projective geometry, he proved that the Wallace lines of a triangle lie in a 3-pointed hypocycloid, developed a formula for the partitioning of space by planes, a fact about the surface areas of tetrahedra, and proved several facts about his famous Steiner's Chain of tangential circles and his famous "Roman surface." Perhaps his three most famous theorems are the Poncelet-Steiner Theorem (lengths constructible with straightedge and compass can be constructed with straightedge alone as long as the picture plane contains the center and circumference of some circle), the Double-Element Theorem about self-homologous elements in projective geometry, and the Isoperimetric Theorem that among solids of equal volume the sphere will have minimum area, etc. (Dirichlet found a flaw in the proof of the Isoperimetric Theorem which was later corrected by Weierstrass.) Steiner is often called, along with Apollonius of Perga (who lived 2000 years earlier), one of the two greatest pure geometers ever. (The qualifier "pure" is added to exclude such geniuses as Archimedes, Newton and Pascal from this comparison. I've included Steiner for his extreme brilliance and productivity: several geometers had much more historic influence, and as solely a 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."

Julius  Plücker (1801-1868) Germany
Plücker was one of the most innovative geometers, inventing line geometry (extending the atoms of geometry beyond just points), enumerative geometry (which considered such questions as the number of loops in an algebraic curve), geometries of more than three dimensions, and generalizations of projective geometry. He also gave an improved theoretic basis for the Principle of Duality. His novel methods and notations were important to the development of modern analytic geometry, and inspired Cayley, Klein and Lie. He resolved the famous Cramer-Euler Paradox and the related Poncelet Paradox by studying the singularities of curves; Cayley described this work as "most important ... beyond all comparison in the entire subject of modern geometry." In part due to conflict with his more famous rival, Jakob Steiner, Plücker was under-appreciated in his native Germany, but achieved fame in France and England. In addition to his mathematical work in algebraic and analytic geometry, Plücker did important work in physics, e.g. his work with cathode rays. Although less brilliant as a theorem prover than Steiner, Plücker's work, taking full advantage of analysis and seeking physical applications, was far more influential.

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. Legendre had spent much of his life 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 was known by ancients; 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. His fame lives on and even the lower-case word 'abelian' is applied to several concepts. Hermite said "Abel has left mathematicians enough to keep them busy for 500 years."
Carl G. J.  Jacobi (1804-1851) Germay
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. He was also a very highly regarded teacher. Jacobi has special importance in the development of the mathematics of physics. 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 dynamics. 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.

Johann Peter Gustav Lejeune  Dirichlet (1805-1859) Germany
Dirichlet was preeminent in algebraic and analytic number theory, but did advanced work in several other fields as well: He discovered the modern definition of function, the Voronoi diagram of geometry, and important concepts in differential equations, topology, and statistics. Although he was one of the foremost mathematicians of the early 19th century, he is often overlooked. (I rank him higher than most Lists of Great Mathematicians do.) Dirichlet was an important teacher, interpreting the work of Gauss and mentoring famous mathematicians like Leopold Kronecker and Ferdinand Eisenstein. His proofs were noted both for great ingenuity and unprecedented rigor. As an example of his careful rigor, he found a fundamental flaw in Steiner's Isoperimetric Theorem proof which no one else had noticed. As an impoverished lad Dirichlet spent his money on math textbooks; Gauss' masterwork became his life-long companion. Fermat and Euler had proved the impossibility of xk + yk = zk for k = 4 and k = 3; Dirichlet became famous by proving impossibility for k = 5 at the age of 20. Later he proved the case k = 14 and, later still, may have helped Kummer extend Dirichlet's quadratic fields, leading to proofs of more cases. More important than his work with Fermat's Last Theorem was his Unit Theorem, considered one of the most important theorems of algebraic number theory. The Unit Theorem is unusually difficult to prove; it is said that Dirichlet discovered the proof while listening to music in the Sistine Chapel. A key step in the proof uses "Dirichlet's Pigeonhole Principle", a trivial idea but which Dirichlet applied with great ingenuity.
Dirichlet also did important work in analysis and is considered the founder of analytic number theory. He invented a method of L-series to prove the important theorem (Gauss' conjecture) that any arithmetic series (without a common factor) has an infinity of primes. It was Dirichlet who proved the fundamental Theorem of Fourier series: that periodic analytic functions can always be represented as a simple trigonometric series. Although he never proved it rigorously, he is especially noted for the Dirichlet's Principle which posits the existence of certain solutions in the calculus of variations, and which Riemann found to be particularly fruitful. Other fundamental results Dirichlet contributed to analysis and number theory include a theorem about Diophantine approximations and his Class Number Formula.

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. At college he enjoyed unprecedented success in all fields, but 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 developed the Principle of Least Action. He predicted that some crystals would have an hitherto unknown "conical" refraction mode; this was confirmed experimentally. Hamilton's Principle of Least Action, and its associated equations and concept of configuration space, led to a revolution in mathematical physics. Since Maupertuis had named this Principle a century earlier, it is possible to underestimate Hamilton's contribution. However Maupertuis, along with others credited with anticipating the idea (Fermat, Leibniz, Euler and Lagrange) failed to state the full Principle correctly. Rather than minimizing action, physical systems sometimes achieve a non-minimal but stationary action in configuration space. (Poisson and d' Alembert had noticed exceptions to Euler-Lagrange least action, but failed to find Hamilton's solution. Jacobi also deserves some credit for the Principle, but his work came after reading Hamilton.) Because of this Principle, as well as his wave-particle duality (which would be further developed by Planck and Einstein), Hamilton can be considered a major early influence on quantum theory.
Hamilton also made revolutionary contributions to dynamics, differential equations, the theory of equations, numerical analysis, fluctuating functions, and graph theory (he marketed a puzzle based on his Hamiltonian paths). He invented the ingenious hodograph. He coined several mathematical terms including "vector," "scalar," "associative," and "tensor." In addition to his brilliance and creativity, Hamilton was renowned for thoroughness and produced voluminous writings on several subjects.
Hamilton himself considered his greatest accomplishment to be the development of quaternions, a non-Abelian field to handle 3-D rotations. While there is no 3-D analog to the Gaussian complex-number plane (based on the equation   i2 = -1  ), quaternions derive from a 4-D analog based on   i2 = j2 = k2 = ijk = -1. (Despite their being "obsoleted" by more general matrix and tensor methods, quaternions are still in wide engineering use because of certain practical advantages.)
Hamilton once wrote: "On earth there is nothing great but man; in man there is nothing great but mind."

Hermann Günter  Grassmann (1809-1877) Germany
Grassmann was an exceptional polymath: the term Grassmann's Law is applied to two separate facts in the fields of optics and linguistics, both discovered by Hermann Grassmann. He also did advanced work in crystallography, electricity, botany, folklore, and also wrote on political subjects. He had little formal training in mathematics, yet single-handedly developed linear algebra, vector and tensor calculus, multi-dimensional geometry, the theory of extension, and exterior algebra; most of this work was so innovative it was not properly appreciated in his own lifetime. (Heaviside rediscovered vector analysis many years later.) Of his linear algebra, one historian wrote "few have come closer than Hermann Grassmann to creating, single-handedly, a new subject." Important mathematicians inspired directly by Grassmann include Peano, Klein, Cartan, Hankel, Clifford, and Whitehead.

Joseph  Liouville (1809-1882) France
Liouville did expert research in several areas including number theory, differential geometry, complex analysis (especially boundary value problems and dynamical analysis), topology and mathematical physics. Several theorems bear his name, including the important result that any bounded entire function must be constant. He was first to prove the existence of transcendental numbers; he found a new proof of the Law of Quadratic Reciprocity. Among his novel inventions were Liouville integrability and fractional calculus. Liouville established an important journal and helped promote other mathematicians' work, especially that of Évariste Galois. In 1851 Augustin Cauchy was bypassed to give a prestigious professorship to Liouville instead.

Ernst Eduard  Kummer (1810-1893) Germany
Despite poverty, Kummer became an important mathematician at an early age, doing work with hypergeometric series, functions and equations, and number theory. He worked on the 4-degree Kummer Surface, an important algebraic form which inspired Klein's early work. His most important discovery was "ideal numbers;" this led to the theory of ideals and p-adic numbers; this discovery's important and revolutionary nature has been compared to that of non-Euclidean geometry. Kummer is famous for his attempts to prove, with the aid of his ideal numbers, Fermat's Last Theorem. He established that Theorem for almost all exponents (including all less than 100) but not the general case. Kummer was an inspirational teacher; his famous students include Cantor, Frobenius, Fuchs, Schwarz, Gordan, Joachimsthal, Bachmann, and Kronecker. (Leopold Kronecker was a brilliant genius sometimes ranked ahead of Kummer in lists like this; that Kummer was Kronecker's teacher at high school persuades me to give Kummer priority.)

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 much smaller than most of the others 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. (Galois coined the mathematical term "group.") While Abel was the first to prove that some polynomial equations had no algebraic solutions, Galois established the necessary and sufficient condition for algebraic solutions to exist. His principle treatise was a letter he wrote the night before his fatal duel, of which Hermann Weyl wrote: "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind." Galois' last words (spoken to his brother) were "Ne pleure pas, Alfred! J'ai besoin de tout mon courage pour mourir à vingt ans!" This tormented life, with its pointless early end, is one of the great tragedies of mathematical history. Although Galois' group theory is considered one of the greatest developments of 19th century mathematics, Galois' writings were largely ignored until the revolutionary work of Klein and Lie.

James Joseph  Sylvester (1814-1897) England, U.S.A.
Sylvester made important contributions in matrix theory, invariant theory, number theory, partition theory, reciprocant theory, geometry, and combinatorics. He invented the theory of elementary divisors, and co-invented the law of quadratic forms. It is said he coined more new mathematical terms (e.g. "matrix," "invariant," "discriminant," "covariant," "syzygy," "Jacobian") than anyone except Leibniz. Sylvester was especially noted for the broad range of his mathematics and his ingenious methods. He was also a linguist, a poet, and did work in mechanics and optics. Sylvester once wrote, "May not music be described as the mathematics of the sense, mathematics as music of the reason?"

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. He developed new insights in several fields including the calculus of variations and trigonometry. He discovered the concept of uniform convergence. Weierstrass shocked his colleagues when he demonstrated a continuous function which is differentiable nowhere. He found simpler proofs of many existing theorems, including Gauss' Fundamental Theorem of Algebra and the fundamental Hermite-Lindemann Transcendence Theorem. Steiner's proof of the Isoperimetric Theorem contained a flaw, so Weierstrass became the first to supply a fully rigorous proof of that famous and ancient result. Starting strictly from the integers, he also applied his axiomatic methods to a definition of irrational numbers. Weierstrass demonstrated extreme brilliance as a youth, but during his college years he detoured into drinking and dueling and ended up as a degreeless secondary school teacher. During this time he studied Abel's papers, developed results in elliptic and Abelian functions, proved the Laurent expansion theorem before Laurent did, 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 and most inspirational mathematicians in the world. His insistence on absolutely rigorous proofs equalled or exceeded even that of Cauchy, Abel and Dirichlet. His students included Kovalevskaya, Frobenius, Mittag-Leffler, and several other famous mathematicians. Bell called him "probably the greatest mathematical teacher of all time." In 1873 Hermite called Weierstrass "the Master of all of us." Today he is often called the "Father of Modern Analysis."
Weierstrass once wrote: "A mathematician who is not also something of a poet will never be a complete mathematician."

George  Boole (1815-1864) England
George Boole was a precocious child who impressed by teaching himself classical languages, but was too poor to attend college and became an elementary school teacher at age 16. He gradually developed his math skills; as a young man he published a paper on the calculus of variations, and soon became one of the most respected mathematicians in England despite having no formal training. He was noted for work in symbolic logic, algebra and analysis, and also was apparently the first to discover invariant theory. When he followed up Augustus de Morgan's earlier work in symbolic logic, de Morgan insisted that Boole was the true master of that field, and begged his friend to finally study mathematics at university. Boole couldn't afford to, and had to be appointed Professor instead! Although very few recognized its importance at the time, it is Boole's work in Boolean algebra and symbolic logic for which he is now remembered; this work inspired computer scientists like Claude Shannon. Boole's book An Investigation of the Laws of Thought prompted Bertrand Russell to label him the "discoverer of pure mathematics."
Boole once said "No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful."

Pafnuti Lvovich  Chebyshev (1821-1894) Russia
Pafnuti Chebyshev (Pafnuty Tschebyscheff) was noted for work in probability, number theory, approximation theory, integrals, the theory of equations, and orthogonal polynomials. His famous theorems cover a diverse range; they include a new version of the Law of Large Numbers, and an important result in integration of radicals first conjectured by Abel. He invented the Chebyshev polynomials, which have very wide application; many other theorems or concepts are also named after him. He did very important work with prime numbers, proving that there is always a prime between any n and 2n, and working with the zeta function before Riemann did. He made much progress with the Prime Number Theorem, proving two distinct forms of that Theorem, each incomplete but in a different way. Chebyshev was very influential for Russian mathematics, inspiring Andrei Markov and Aleksandr Lyapunov among others. Chebyshev was also a premier applied mathematician and a renowned inventor; his several inventions include the Chebyshev linkage, a mechanical device to convert rotational motion to straight-line motion. He once wrote "To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls."

Arthur  Cayley (1821-1895) England
Cayley was one of the most prolific mathematicians in history; a list of the branches of mathematics he pioneered will seem like an exaggeration. In addition to being very inventive, he was an excellent algorist; some considered him to be the greatest mathematician of the late 19th century (an era that includes Weierstrass and Poincaré). Cayley was the essential founder of modern group theory, matrix algebra, the theory of higher singularities, and higher-dimensional geometry (building on Plücker's work and anticipating the ideas of Klein), as well as the theory of invariants. Among his many important theorems are the Cayley-Hamilton Theorem, and Cayley's Theorem itself (that any group is isomorphic to a subgroup of a symmetric group). He extended Hamilton's quaternions and developed the octonions, but was still one of the first to realize that these special algebras should be subsumed by general matrix methods. He also did original research in combinatorics (e.g. enumeration of trees), elliptic and Abelian functions, and projective geometry. One of his many famous geometric theorems is a generalization of Pascal's Mystic Hexagram result; another resulted in an elegant proof of the Quadratic Reciprocity law. Cayley may have been the least eccentric of the great mathematicians: In addition to his life-long love of mathematics, he enjoyed hiking, painting, reading fiction, and had a happy married life. He easily won Smith's Prize and Senior Wrangler at Cambridge, but then worked as a lawyer for many years. He later became professor, and finished his career in the limelight as President of the British Association for the Advancement of Science. He and James Joseph Sylvester 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.
Cayley once wrote: "As for everything else, so for a mathematical theory: beauty can be perceived but not explained."

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 great advances in analytic number theory, elliptic functions, and quadratic forms. Along with Cayley and Sylvester, he founded the important theory of invariants. Hermite's theory of transformation allowed him to connect analysis, algebra and number theory in novel ways. He was a kindly modest man and an inspirational teacher. Among his students was Poincaré, who said of Hermite, "He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures.... Methods always seemed to be born in his mind in some mysterious way." Hermite's other important students included Darboux, Borel, and Hadamard who wrote of "how magnificent Hermite's teaching was, overflowing with enthusiasm for science, which seemed to come to life in his voice and whose beauty he never failed to communicate to us, since he felt it so much himself to the very depth of his being." 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. Hermite's most famous result may be his ingenious proof that e (along with a broad class of related numbers) is transcendental. (Extending the proof to π was left to Lindemann, a matter of regret for historians, some of whom who regard Hermite as the greatest mathematician of his era.)

Ferdinand Gotthold Max  Eisenstein (1823-1852) Germany
Eisenstein was born into severe poverty and suffered health problems throughout his short life, but was still one of the more significant mathematicians of his era. Today's mathematicians who study Eisenstein are invariably amazed by his brilliance and originality. He made revolutionary advances in number theory, algebra and analysis, and was also a composer of music. He anticipated ring theory, developed a new basis for elliptic functions, studied ternary quadratic forms, proved several theorems about cubic and quartic reciprocity, discovered the notion of analytic covariant, and much more. Eisenstein was a young prodigy; he once wrote "As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork." Despite his early death, he is considered one of the greatest number theorists ever. Gauss named Eisenstein, along with Newton and Archimedes, as one of the three epoch-making mathematicians of history.

Georg Friedrich Bernhard  Riemann (1826-1866) Germany
Riemann was a phenomenal genius whose work was exceptionally deep, creative and rigorous; he made revolutionary contributions in many areas of pure mathematics, and also inspired the development of physics. He had poor physical health and died at an early age, yet is still considered to be among the most productive mathematicians ever. He was the master of complex analysis, which he connected to both topology and number theory, He applied topology to analysis, and analysis to number theory, making revolutionary contributions to all three fields. He took non-Euclidean geometry far beyond his predecessors. He introduced the Riemann integral which clarified analysis. Riemann's other masterpieces include differential geometry, tensor analysis, 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. Several important theorems and concepts are named after Riemann, e.g. the Riemann-Roch theorem, a key connection among topology, complex analysis and algebraic geometry. He was so prolific and original that some of his work went unnoticed (for example, Weierstrass became famous for showing a nowhere-differentiable continuous function; later it was found that Riemann had casually mentioned one in a lecture years earlier). Like his mathematical peers (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. His theory unifying electricity, magnetism and light was supplanted by Maxwell's theory; however modern physics, beginning with Einstein's relativity, relies on Riemann's notions of the geometry of space. Riemann's teacher was Carl Gauss, who helped steer the young genius towards pure mathematics. Gauss selected "On the hypotheses that Lie at the Foundations of Geometry" as Riemann's first lecture; with this famous lecture Riemann advanced Gauss' initial effort in differential geometry, extended it to multiple dimensions, and introduced the new and important theory of differential manifolds. Five years later, to celebrate his election to the Berlin Academy, Riemann presented a lecture "On the Number of Prime Numbers Less Than a Given Quantity," for which "Number" he presented and proved an exact formula, albeit weirdly complicated and seemingly intractable. Numerous papers have been written on the distribution of primes, but Riemann's contribution is incomparable, despite that his Berlin Academy lecture was his only paper ever on the topic, and number theory was far from his specialty. In the lecture he posed the "Hypothesis of Riemann's zeta function" which is now considered the most important and famous unsolved problem in mathematics. (Asked what he would first do, if he were magically awakened after centuries, David Hilbert replied "I would ask whether anyone had proved the Riemann Hypothesis.") ζ() was defined for convergent cases in Euler's mini-bio, which Riemann extended via analytic continuation for all cases. The Riemann Hypothesis "simply" states that in all solutions of ζ(s = a+bi) = 0, either s has real part a=1/2 or imaginary part b=0.
Despite his great creativity (Gauss praised Riemann's "gloriously fertile originality"), Riemann once said: "If only I had the theorems! Then I should find the proofs easily enough."

James Clerk  Maxwell (1831-1879) Scotland
Maxwell published a remarkable paper on the construction of novel ovals, at the age of 14; his genius was soon renowned throughout Scotland, with the future Lord Kelvin remarking that Maxwell's "lively imagination started so many hares that before he had run one down he was off on another." He did a comprehensive analysis of Saturn's rings, developed the kinetic theory of gases, explored knot theory, and more. He advanced the theory of color, and produced the first color photograph. Maxwell was also a poet. One Professor said of him, "there is scarcely a single topic that he touched upon, which he did not change almost beyond recognition." Maxwell did little work in pure mathematics, so his great creativity in mathematical physics might not seem enough to qualify him for this list. However, in 1864 James Clerk Maxwell stunned the world by publishing the equations of electricity and magnetism and showing that light itself is linked to the electro-magnetic force. This, along with Darwin's theory of evolution, is considered one of the greatest discoveries of the 19th century; and Maxwell himself, along with Newton and Einstein, is frequently named as one of the three greatest physicists ever. He ranks #24 on Hart's list of the Most Influential Persons in History.

Julius Wilhelm Richard  Dedekind (1831-1916) Germany
Dedekind was one of the most innovative mathematicians ever; his clear expositions and rigorous axiomatic methods had great influence. He made seminal contributions to abstract algebra and algebraic number theory as well as mathematical foundations. He was one of the first to pursue Galois Theory, making major advances there and pioneering in the application of group theory to other branches of mathematics. Dedekind also invented a system of fundamental axioms for arithmetic, worked in probability theory and complex analysis, and invented prime partitions and modular lattices. Dedekind may be most famous for his theory of ideals and rings; Kronecker and Kummer had begun this, but Dedekind gave it a more abstract and productive basis, which was developed further by Hilbert, Noether and Weil. Though the term "ring" itself was coined by Hilbert, Dedekind introduced the terms "module," "field," and "ideal." Dedekind was concerned with rigor, writing "nothing capable of proof ought to be accepted without proof." Before him, the real numbers, continuity, and infinity all lacked rigorous definitions. The axioms Dedekind invented allow the integers and rational numbers to be built and his "Dedekind Cut" then led to a rigorous and useful definition of the real numbers. Dedekind anticipated and inspired Cantor's work: he introduced the notion that a bijection implied equinumerosity, used this to define infinitude (a set is infinite if equinumerous with its proper subset), and proved the Cantor-Bernstein Theorem; he should thus be considered a co-inventor of Cantor's set theory.

Marie Ennemond Camille  Jordan (1838-1921) France
Jordan was a great "universal mathematician", making revolutionary advances in group theory, topology, and operator theory, and also doing important work in differential equations, number theory, matrix theory, combinatorics, algebra and especially Galois theory. He worked as both mechanical engineer and professor of analysis. His work inspired such mathematicians as Klein and Lie. Jordan is especially famous for the Jordan curve theorem of topology, for inventing the notion of homotopy, and for the Jordan-Holder theorem.

Marius Sophus   Lie (1842-1899) Norway
Lie was twenty-five years old before his interest in and aptitude for mathematics became clear, but then did revolutionary work with continuous symmetry and continuous transformation groups. These groups and the algebra he developed to manipulate them now bear his name; they have major importance in the study of differential equations. Lie sphere geometry is one result of Lie's fertile approach and even led to a new approach for Apollonius' ancient problem about tangent circles. Lie became a close friend and collaborator of Felix Klein early in their careers; their methods of relating group theory to geometry were quite similar; but they eventually fell out after Klein became (unfairly?) recognized as the superior of the two. Lie's work wasn't properly appreciated in his own lifetime, but one later commentator was "overwhelmed by the richness and beauty of the geometric ideas flowing from Lie's work."

Jean Gaston   Darboux (1842-1917) France
Darboux did outstanding work in geometry, differential geometry, analysis, function theory, mathematical physics, and other fields, his ability "based on a rare combination of geometrical fancy and analytical power." He devised the Darboux integral, equivalent to Riemann's integral but simpler; developed a novel mapping between (hyper-)sphere and (hyper-)plane; proved an important Envelope Theorem in the calculus of variations; developed the field of infinitesimal geometry; and more. Several important theorems are named after him including a generalization of Taylor series, the foundational theorem of symplectic geometry, and the fact that "the image of an interval is also an interval." He wrote the definitive textbook on differential geometry; he was an excellent teacher, inspiring Borel, Cartan and others.

Georg  Cantor (1845-1918) Russia, Germany
Cantor 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. (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 partly resolved many years later: Cantor's Continuum Hypothesis is an "Undecidable Statement" of Set Theory. Cantor's revolutionary set theory attracted vehement opposition from Poincaré ("grave disease"), Kronecker (Cantor was a "charlatan" and "corrupter of youth"), Wittgenstein ("laughable nonsense"), and even theologians. David Hilbert had kinder words for it: "The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity." Cantor's own attitude was expressed with "The essence of mathematics lies in its freedom." Cantor's invention of modern set theory is now considered one of the most important and creative 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. Cantor once wrote "In mathematics the art of proposing a question must be held of higher value than solving it."

Ferdinand Georg  Frobenius (1849-1917) Germany
Frobenius did important work in a very broad range of mathematics, was an outstanding algorist, and had several successful students including Edmund Landau and Issai Schur. In addition to developing the theory of abstract groups, Frobenius did important work in number theory, differential equations, matrixes, and algebra. He was first to actually prove the important Cayley-Hamilton theorem, and first to extend the Sylow theorems to abstract groups. Although he modestly left his name off the "Cayley-Hamilton theorem," many lemmas and concepts are named after him, including Frobenius conjugacy class, Frobenius reciprocity, the Frobenius-Schur Indicator, etc. He is most noted for his character theory, which led to the representation theory of groups, and has applications in modern physics.

Christian Felix  Klein (1849-1925) Germany
Klein's key contribution was an application of invariant theory to unify geometry with group theory. This radical new view of geometry inspired Sophus Lie's Lie groups, and also led to the remarkable unification of Euclidean and non-Euclidean geometries which is probably Klein's most famous result. Klein did other work in function theory, providing links between several areas of mathematics including number theory, group theory, hyperbolic geometry, and abstract algebra. His Klein's Quartic curve and popularly-famous "Klein's bottle" were among several useful results from his new approaches to groups and higher-dimensional geometries and equations. Klein did important work in mathematical physics, e.g. writing about gyroscopes. He facilitated David Hilbert's early career, publishing his controversial Finiteness Theorem and declaring it "without doubt the most important work on general algebra [the leading German journal] ever published." Klein is also famous for his book on the icosahedron, reasoning from its symmetries to develop the elliptic modular and automorphic functions which he used to solve the general quintic equation. He formulated a "grand uniformization theorem" about automorphic functions but suffered a health collapse before completing the proof. His focus then changed to teaching; he devised a mathematics curriculum for secondary schools which had world-wide influence. Klein once wrote "... mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs."

Oliver   Heaviside (1850-1925) England
Heaviside dropped out of high school to teach himself telegraphy and electromagnetism, becoming first a telegraph operator but eventually perhaps the greatest electrical engineer ever. He developed transmission line theory, invented the coaxial cable, predicted Cherenkov radiation, described the use of the ionosphere in radio transmission, and much more. Some of his insights anticipated parts of special relativity. For his revolutionary discoveries in electromagnetism and mathematics, Heaviside became the first winner of the Faraday Medal. As an applied mathematician, Heaviside developed operational calculus (an important shortcut for solving differential equations); developed vector analysis independently of Grassman; and demonstrated the usage of complex numbers for electro-magnetic equations. The Four Maxwell's Equations are in fact due to Oliver Heaviside, Maxwell having presented a more cumbersome version with twenty equations. Although one of the greatest applied mathematicians, Heaviside is omitted from our List because he didn't provide proofs for his methods. Of this Heaviside said, "Should I refuse a good dinner simply because I do not understand the process of digestion?"

Sofia Vasilyevna  Kovalevskaya (1850-1891) Russia
Sofia Kovalevskaya (aka Sonya Kowalevski; née Korvin-Krukovskaya) was initially self-taught, sought out Weierstrass as her teacher, and was later considered the greatest female mathematician ever (before Emma Noether). She was influential in the development of Russian mathematics. Kovalevskaya studied Abelian integrals and partial differential equations, producing the important Cauchy-Kovalevsky theorem; her application of complex analysis to physics inspired Poincaré and others. Her most famous work was the solution to the Kovalevskaya top, which has been called a "genuine highlight of 19th-century mathematics." Other than the simplest cases solved by Euler and Lagrange, exact ("integrable") solutions to the equations of motion were unknown, so Kovalevskaya received fame and a rich prize when she solved the Kovalevskaya top. Her ingenious solution might be considered a mere curiosity, but since it is still the only post-Lagrange physical motion problem for which an "integrable" solution has been demonstrated, it remains an important textbook example. Kovalevskaya was also a noted playwright.

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 Constructivist ("intuitionist") style. Poincaré founded the theory of algebraic (combinatorial) topology, and is sometimes called the Father of Topology (a title also used for Euler and Brouwer), but produced a large amount of brilliant work in many other areas of mathematics. In addition to his topology, Poincaré laid the foundations of homology; he discovered automorphic functions (a unifying foundation for the trigonometric and elliptic functions), and essentially founded the theory of periodic orbits; he made major advances in the theory of differential equations. Several important results carry his name, for example the famous Poincaré Recurrence Theorem, which seems to contradict the Second Law of Thermodynamics. Poincaré is especially noted for effectively discovering chaos theory, and for posing "Poincaré's conjecture;" that conjecture was one of the most famous unsolved problems in mathematics for an entire century, and can be explained without equations to a layman (provided the layman can visualize 3-D surfaces in 4-space). Recently Grigori Perelman proved Poincaré's conjecture, and is eligible for the first Million Dollar math prize in history.
As were most of the greatest mathematicians, Poincaré was intensely interested in physics. He made revolutionary advances in fluid dynamics and celestial motions; he anticipated Minkowski space and much of Einstein's Special Theory of Relativity (including the famous equation E = mc2). Poincaré also found time to become a famous popular writer of philosophy, writing, "Mathematics is the art of giving the same name to different things;" and "A [worthy] mathematician experiences in his work the same impression as an artist; his pleasure is as great and of the same nature;" and "If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living." With his fame, Poincaré helped the world recognize the importance of the new physical theories of Einstein and Planck.

Andrei Andreyevich  Markov (1856-1922) Russia
Markov did excellent work in a broad range of mathematics including analysis, number theory, algebra, continued fractions, approximation theory, and especially probability theory: it has been said that his accuracy and clarity transformed probability theory into one of the most perfected areas of mathematics. Markov is best known as the founder of the theory of stochastic processes. He was also noted for his politics, mocking Czarist rule, and insisting that he be excommunicated from the Russian Orthodox Church when Tolstoy was.

Giuseppe  Peano (1858-1932) Italy
Peano started his career by proving a fundamental theorem in differential equations, developed practical solution methods for such equations, discovered a continuous space-filling curve (then thought impossible), and laid the foundations of abstract operator theory. But his most important work was in mathematical foundations. He developed rigorous definitions and axioms for set theory, and was first to define arithmetic (and then the rest of mathematics) in terms of set theory. Despite his early show of genius, Peano's quest for utter rigor may have detracted from his influence in mainstream mathematics. Moreover, since he modestly referenced work by predecessors like Dedekind, Peano's huge influence in axiomatic theory is often overlooked. Yet Bertrand Russell reports that it was from Peano that he first learned that a single-member set is not the same as its element; this fact is now taught in elementary school.

David  Hilbert (1862-1943) Prussia, Germany
Hilbert was preeminent 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. He was also the founder of the "Formalist" school which opposed the "Intuitionism" of Kronecker and Brouwer. 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" rather than 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 provided a famous 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. He may have published the "Einstein Field Equations" independently of Einstein. (Since he had already learned of the theory's intuition from personal lectures by Einstein, it is wrong, as some do, to claim Hilbert's publication diminishes Einstein's greatness.)

Hermann  Minkowski (1864-1909) Lithuania, Germany
Minkowski won a prestigious prize at age 18 for reconstructing Eisenstein's enumeration of the ways to represent integers as the sum of five squares. His proof built on quadratic forms and continued fractions and eventually led him to the new field of Geometric Number Theory, for which Minkowski's Convex Body Theorem (a sort of pigeonhole principle) is often called the Fundamental Theorem. Minkowski was also a major figure in the development of functional analysis. With his "question mark function" and "sausage," he was also a pioneer in the study of fractals. Several other important results are named after him, e.g. the Hasse-Minkowski Theorem. He was first to extend the Separating Axis Theorem to multiple dimensions. Minkowski was one of Einstein's teachers, and also a close friend of David Hilbert. He is particularly famous for building on Poincaré's work to invent Minkowski space to deal with Einstein's Special Theory of Relativity. This not only provided a better explanation for the Special Theory, but helped inspire Einstein toward his General Theory. Minkowski said that his "views of space and time ... have sprung from the soil of experimental physics, and therein lies their strength.... Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

Jacques Salomon  Hadamard (1865-1963) France
Hadamard made revolutionary advances in several different areas of mathematics, especially complex analysis, analytic number theory, differential geometry, partial differential equations, symbolic dynamics, chaos theory, matrix theory, and Markov chains; for this reason he is sometimes considered the "Last Universal Mathematician." He also made contributions to physics. One of the most famous results in mathematics is the Prime Number Theorem, that there are approximately n/log n primes less than n. This result was conjectured by Legendre and Gauss, attacked cleverly by Riemann and Chebyshev, and finally proved by Hadamard and Vallee-Poussin. (Hadamard's proof is considered more elegant and useful than Vallee-Poussin's.) Several other important theorems are named after Hadamard (e.g. his Inequality of Determinants), and some of his theorems are named after others (Hadamard was first to prove Brouwer's Fixed-Point Theorem for arbitrarily many dimensions). He is also noted for his survey of Poincaré's work. Hadamard was an influential teacher, with André Weil, Maurice Fréchet, and others acknowledging him as key inspiration.

Élie Joseph  Cartan (1869-1951) France
Cartan did very important work in the theory of Lie groups and Lie algebras, applying methods of topology, geometry and invariant theory to Lie theory, and classifying all Lie groups. His work was so important that Cartan, rather than Lie, can rightfully be viewed as the most important developer of the theory of Lie groups. Using Lie theory and ideas like his "method of prolongation" he advanced the theories of differential equations and differential geometry. He introduced several new concepts including algebraic group, exterior differential forms, spinors, moving frames, Cartan connections. He proved several important theorems, e.g. Schläfli's Conjecture about imbedding Riemann metrics, and fundamental theorems about symmetric Riemann spaces. He made an important contribution to Einstein's general relativity, based on what is now called Riemann-Cartan geometry. Cartan's methods were so original as to be fully appreciated only recently; many now consider him to be one of the greatest mathematicians of his era. In 1938 Weyl called him "the greatest living master in differential geometry."

Félix Édouard Justin Émile  Borel (1871-1956) France
Borel exhibited great talent while still in his teens, soon practically founded modern measure theory, and received several honors and prizes. Among his famous theorems is the Heine-Borel Covering Theorem. He also did important work in several other fields of mathematics, including divergent series, quasi-analytic functions, differential equations, number theory, complex analysis, geometry, probability theory, and game theory. He also did work in relativity and the philosophy of science. Borel was decorated for valor in World War I, entered politics between the Wars, and joined the French Resistance during World War II.

Henri Léon  Lebesgue (1875-1941) France
Lebesgue did groundbreaking work in real analysis, advancing Borel's measure theory; his Lebesgue integral superseded the Riemann integral and improved the theoretical basis for Fourier analysis. Several important theorems are named after him, e.g. the Lebesgue differentiation theorem and Lebesgue's number lemma. He did important work on Hilbert's 19th Problem, and in the Jordan curve theorem for higher dimensions. In 1916, the Lebesgue integral was compared "with a modern Krupp gun, so easily does it penetrate barriers which were impregnable." In addition to his seminal contributions to measure theory and Fourier analysis, Lebesgue made important contributions in several other fields including complex analysis, topology, set theory, potential theory, and calculus of variations.

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 proved many important theorems about numbers, for example that Riemann's zeta function has infinitely many zeros with real part 1/2. He was also an excellent teacher and wrote several excellent textbooks, as well as a famous treatise on the mathematical mind. He abhorred applied mathematics, treating mathematics as a creative art; yet his work has found application in population genetics, cryptography, thermodynamics 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. Hardy was also a mentor to Norbert Wiener, another famous prodigy.
Hardy once wrote "A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." He also wrote "Beauty is the first test; there is no permanent place in the world for ugly mathematics."

Albert  Einstein (1879-1955) Germany, Switzerland, U.S.A.
Albert Einstein is generally agreed to be one of the two greatest physicists in all of history. The atomic theory achieved general acceptance only after Einstein's 1905 paper which showed that atoms' discreteness explained Brownian motion. Another famous 1905 paper introduced the famous equation E = mc2; yet Einstein published other papers that same year, two of which were more important and influential than either of the two just mentioned! No wonder that physicists speak of the Miracle Year without bothering to qualify it as Einstein's Miracle Year! (Altogether Einstein published at least 300 books or papers on physics.) Although most famous for his Special and General Theories of Relativity, Einstein's discovery of the photon made him the key pioneer in quantum theory (and led to his only Nobel Prize). Many years later, he was first to call attention to the "spooky" nature of quantum entanglement.
Einstein certainly has the breadth, depth, and historical importance to qualify for this list; but his genius and significance were not in the field of pure mathematics. (He acknowledged his limitation, writing "I admire the elegance of your [Levi-Civita's] method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.") Einstein was a mathematician, however; he pioneered the application of tensor calculus to physics and invented the "Einstein summation notation." I've chosen to include him on this list because his extreme greatness overrides his focus away from math. Einstein ranks #10 on Michael Hart's famous list of the Most Influential Persons in History. His General Theory of Relativity has been called the most creative and original scientific theory ever. Einstein once wrote "... the creative principle resides in mathematics [; thus] I hold it true that pure thought can grasp reality, as the ancients dreamed."

Luitzen Egbertus Jan  Brouwer (1881-1966) Holland
Brouwer is often considered the Father of Topology; among his important theorems were the Fixed Point Theorem, the "Hairy Ball" Theorem, the Jordan-Brouwer Separation Theorem, and the Invariance of Dimension. He developed the method of simplicial approximations, important to algebraic topology; he also did work in geometry, set theory, measure theory, complex analysis and the foundations of mathematics. He was first to anticipate forms like the Lakes of Wada, leading eventually to other measure-theory "paradoxes." Brouwer is most famous as the founder of Intuitionism, a philosophy of mathematics in sharp contrast to Hilbert's Formalism, but Brouwer's philosophy also involved ethics and aesthetics and has been compared with those of Schopenhauer and Nietzsche. Part of his mathematics thesis was rejected as "... interwoven with some kind of pessimism and mystical attitude to life which is not mathematics ..." As a young man, Brouwer spent a few years to develop topology, but once his great talent was demonstrated and he was offered prestigious professorships, he devoted himself to Intuitionism, and acquired a reputation as eccentric and self-righteous.
Intuitionism has had a significant influence, although few strict adherents. Since only constructive proofs are permitted, strict adherence would slow mathematical work. This didn't worry Brouwer who once wrote: "The construction itself is an art, its application to the world an evil parasite."
Amalie Emma  Noether (1882-1935) Germany
Noether was an innovative researcher who made several major advances in abstract algebra, including a new theory of ideals, the inverse Galois problem, and the general theory of commutative rings. She originated novel reasoning methods, especially one based on "chain conditions," which advanced invariant theory and abstract algebra; her insistence on generalization led to a unified theory of modules and Noetherian rings. Her approaches tended to unify disparate areas (algebra, geometry, topology, logic) and led eventually to modern category theory. Her invention of homology groups revolutionized topology. Noether's work has found various applications in physics, and she made direct advances in mathematical physics herself. Noether's Theorem establishing that certain symmetries imply conservation laws has been called the most important Theorem in physics since the Pythagorean Theorem. Noether was an unusual and inspiring teacher; her successful students include Emil Artin. She was generous with students and colleagues, even allowing them to claim her work as their own. Noether was close friends with the other greatest mathematicians of her generation: Hilbert, von Neumann, and Weyl. Weyl once said he was embarrassed to accept the famous Professorship at Göttingen because Noether was his "superior as a mathematician." Many would agree that Emmy Noether was the greatest female mathematician ever.

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. 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). For a while, Weyl was a disciple of Brouwer's Intuitionism and helped advance that doctrine, but he eventually found it too restrictive. Weyl was also a very influential figure in all three major fields of 20th-century physics: relativity, unified field theory and quantum mechanics.
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."

John Edensor  Littlewood (1885-1977) England
John Littlewood was a very prolific researcher. (This fact is obscured somewhat in that many papers were co-authored with Hardy, and their names were always given in alphabetic order.) The tremendous span of his career is suggested by the fact that he won Smith's Prize (and Senior Wrangler) in 1905 and the Copley Medal in 1958. He specialized in analysis and analytic number theory but also did important work in combinatorics, mathematical physics and other fields. He worked with the Prime Number Theorem and Riemann's Hypothesis, proved that Li(x) underestimates the number of primes infinitely often (although there are no such examples for any x less than a googol). Most of his results were too specialized to state here, e.g. his widely-applied "4/3 Inequality" which guarantees that certain bimeasures are finite. Hardy once said that his friend was "the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power." Littlewood replied that it was possible to be too strong of a mathematician, "forcing through, where another might be driven to a different, and possibly more fruitful, approach."

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. He might be almost unknown today, except that his letter caught the eye of Godfrey Hardy, who saw remarkable, almost inexplicable formulae which "must be true, because if they were not true, no one would have had the imagination to invent them." Ramanujan's 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 best work was done in collaboration with Hardy, for example a proof that almost all numbers n have about log log n prime factors (a result which inspired probabilistic number theory). Much of his methodology, including unusual ideas about divergent series, was his own invention. (As a young man he made the absurd claim that 1+2+3+4+... = -1/12. Later it was noticed that this claim translates to a true statement about the Riemann zeta function, with which Ramanujan was unfamiliar.) Ramanujan's innate ability for algebraic manipulations equaled or surpassed that of Euler and Jacobi. Ramanujan's most famous work was with the partition enumeration function p(), Hardy guessing that some of these discoveries would have been delayed at least a century without Ramanujan. Together, Hardy and Ramanujan developed an analytic approximation to p(). (Rademacher and Selberg later discovered an exact expression to replace the Hardy-Ramanujan formula; when Ramanujan's notebooks were studied it was found he had anticipated their technique, but had deferred to his friend and mentor.)
Many of Ramanujan's other results would also 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. (Ramanujan may have had unrecorded proofs, poverty leading him to use chalk and erasable slate rather than paper.) Unlike Abel, much of whose work depended on the complex numbers, most of Ramanujan's work focused on real numbers. Despite these limitations, Ramanujan is considered one of the greatest geniuses ever.
Because of its fast convergence, an odd-looking formula of Ramanujan is often used to calculate π:
      992 / π = √8 ∑k=0,∞ (4k! (1103+26390 k) / (k!4 3964k))

Stefan  Banach (1892-1945) Poland
Stefan Banach was a self-taught mathematician who is most noted as the Founder of Functional Analysis and for his contributions to measure theory. Among several important theorems bearing his name are the Uniform Boundedness (Banach-Steinhaus) Theorem, the Open Mapping (Banach-Schauder) Theorem, the Contraction Mapping (Banach fixed-point) Theorem, and the Hahn-Banach theorem. Many of these theorems are of practical value to modern physics; however he also proved the paradoxical Banach-Tarski Theorem, which demonstrates a sphere being rearranged into two spheres of the same original size! (Banach's proof uses the Axiom of Choice and is sometimes cited as evidence that that Axiom is false!) The wide range of Banach's work is indicated by the Banach-Mazur results in game theory (which also challenge the axiom of choice). Banach also made brilliant contributions to probability theory, set theory, analysis and topology. Banach once said "Mathematics is the most beautiful and most powerful creation of the human spirit."

Carl Ludwig  Siegel (1896-1981) Germany
Carl Siegel is especially noted for his contributions to several branches of analytic and algebraic number theory, especially arithmetic geometry, quadratic forms and Diophantine equations. He also did seminal work with Dedekind's zeta functions, transcendence theory, discontinuous groups, the 3-body problem in celestial mechanics, Riemann's zeta function, and the history of mathematics. In complex analysis he developed Siegel modular forms, which have wide application in math and physics. Important theorems proved by, and named after him include a theorem on the finitude of integral points (in certain curves of non-zero genus), lemmas about auxiliary functions, etc. Siegel admired the "simplicity and honesty" of masters like Gauss, Lagrange and Hardy and lamented the modern "trend for senseless abstraction." He and Israel Gelfand were the first two winners of the Wolf Prize in Mathematics. André Weil declared that Siegel was the greatest mathematician of the first half of the 20th century.

Emil  Artin (1898-1962) Austria, Germany, U.S.A.
Artin was an important and prolific researcher in several fields of algebra, including algebraic number theory, the theory of rings, field theory, algebraic topology, Galois theory, a new method of L-series, and geometric algebra. Among his most famous theorems were Artin's Reciprocity Law, key lemmas in Galois theory, and results in his Theory of Braids. He also produced two very influential conjectures: his conjecture about the zeta function in finite fields developed into the field of arithmetic geometry; Artin's Conjecture on primitive roots inspired much work in number theory, and was later generalized to become Weil's Conjectures. He is credited with solution to Hilbert's 17th Problem and partial solution to the 9th Problem. His prize-winning students include John Tate and Serge Lang. Artin also did work in physical sciences, and was an accomplished musician.

John  von Neumann (1903-1957) Hungary, U.S.A.
John von Neumann (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 in many branches of pure mathematics, and applied mathematics. He was an essential pioneer of both quantum physics and computer science. Von Neumann pioneered the use of models in set theory, thus improving the axiomatic basis of mathematics. He proved a generalized spectral theorem sometimes called the most important result in operator theory. He developed von Neumann Algebras. He was first to state and prove the minimax theorem and thus invented game theory; this work also advanced operations research. He invented cellular automata, famously constructing a self-reproducing automaton. He invented elegant definitions for the counting numbers (0 = {}, n+1 = n ∪ {n}). He also worked in analysis, matrix theory, measure theory, numerical analysis, ergodic theory, group representations, continuous geometry, statistics and topology. Von Neumann discovered an ingenious area-conservation paradox related to the famous Banach-Tarski volume-conservation paradox. He inspired some of Gödel's famous work (and independently proved Gödel's Second Theorem). He is credited with (partial) solution to Hilbert's 5th Problem using the Haar Theorem; this also relates to quantum physics. George Pólya once said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper."
Von Neumann did very important work in fields other than pure 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 also advanced the theory of hydrodynamics. 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. By implementing wide-number software he joined several other great mathematicians (Archimedes, Apollonius, Liu Hui, Hipparchus, Madhava, Ramanujan) in producing the best approximation to π of his time. At the time of his death, von Neumann 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 (by applying measure theory) modern probability theory. He also excelled in topology, set theory, trigonometric series, and random processes. His surprising and far-reaching Superposition Theorem led to solution of Hilbert's 13th Problem, as just a special case. He made important contributions to the constructivist ideas of Kronecker and Brouwer. 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 notions named after Kolmogorov, such as the "Kolmogorov backward equation," the Borel-Kolmogorov paradox, and the intriguing Zero-One Law of "tail events" among random variables.

Kurt  Gödel (1906-1978) Germany, U.S.A.
Gödel, who had the nickname Herr Warum ("Mr. Why") as a child, was perhaps the foremost logic theorist ever, clarifying the relationships between various modes of logic. He partially resolved both Hilbert's 1st and 2nd Problems, the latter with a proof so remarkable that it was connected to the drawings of Escher and music of Bach in the title of a famous book. He was a close friend of Albert Einstein, and was first to discover "paradoxical" solutions (e.g. time travel) to Einstein's equations. About his friend, Einstein later said that he had remained at Princeton's Institute for Advanced Study merely "to have the privilege of walking home with Gödel." (Like a few of the other greatest 20th-century mathematicians, Gödel was very eccentric.) Two of the major questions confronting mathematics are: (1) are its axioms consistent (its theorems all being true statements)?, and (2) are its axioms complete (its true statements all being theorems)? Gödel turned his attention to these fundamental questions. He proved that first-order logic was indeed complete, but that the more powerful axiom systems needed for arithmetic (constructible set theory) were necessarily incomplete. He also proved that the Axioms of Choice (AC) and the Generalized Continuum Hypothesis (GCH) were consistent with set theory, but that set theory's own consistency could not be proven. He may have established that the truths of AC and GCH were independent of the usual set theory axioms, but the proof was left to his disciple Paul Cohen.
In Gödel's famous proof of Incompleteness, he exhibits a true statement (G) which cannot be proven, to wit "G (this statement itself) cannot be proven." If G could be proven it would be a contradictory true statement, so consistency dictates that it indeed cannot be proven. But that's what G says, so G is true! This sounds like mere word play, but building from ordinary logic and arithmetic Gödel was able to construct statement G rigorously.

André  Weil (1906-1998) France, U.S.A.
Weil made profound contributions to many areas of mathematics, especially algebraic geometry, which he showed to have deep connections with number theory. His "Weil conjectures" were very influential; these and other works laid the groundwork for many of Grothendieck's achievements. Weil proved a special case of the Riemann hypothesis; he contributed, at least indirectly, to the recent proof of "Fermat's last Theorem;" he also worked in group theory, general and algebraic topology, differential geometry, sheaf theory, representation theory, and theta functions. He invented several new concepts including vector bundles, and uniform space. His work has found applications in particle physics and string theory. He is considered to be one of the most influential of modern mathematicians. Weil's biography is interesting. He studied Sanskrit as a child, loved to travel, taught at a Muslim university in India for two years (intending to teach French civilization), wrote as a young man under the famous pseudonym Nicolas Bourbaki, spent time in prison during World War II as a Jewish objector, was almost executed as a spy, escaped to America, and eventually joined Princeton's Institute for Advanced Studies. He once wrote: "Every mathematician worthy of the name has experienced [a] lucid exaltation in which one thought succeeds another as if miraculously."

Shiing-Shen  Chern (1911-2004) China, U.S.A
Shiing-Shen Chern (Chen Xingshen) studied under Élie Cartan, and became perhaps the greatest master of differential geometry. He is especially noted for his work in algebraic geometry, topology and fiber bundles, developing his Chern characters (in a paper with "a tremendous number of geometrical jewels"), developing Chern-Weil theory, the Chern-Simons invariants, and especially for his brilliant generalization of the Gauss-Bonnet Theorem to multiple dimensions. His work had a major influence in several fields of modern mathematics as well as gauge theories of physics. Chern was an important influence in China and a highly renowned and successful teacher: one of his students (Yau) won the Fields Medal, another (Yang) the Nobel Prize in physics. Chern himself was the first Asian to win the prestigious Wolf Prize.

Alan Mathison  Turing (1912-1954) Britain
Turing developed a new foundation for mathematics based on computation; he invented the abstract Turing machine, proved the famous Halting Theorem (related to Godel's Incompleteness Theorem), and developed the concept of machine intelligence (including his famous "Turing Test" proposal). He also introduced the notions of "definable number" and "oracle" (important in modern computer science). Turing also worked in group theory, numerical analysis, and complex analysis; he developed an important theorem about Riemann's zeta function; he had novel insights in quantum physics. During World War II he turned his talents to cryptology; his creative algorithms were considered possibly "indispensable" to the decryption of German Naval Enigma coding, which in turn is judged to have certainly shortened the War by at least two years. Although his contributions to Bletchley Park's hardware were much less important than his code-breaking algorithms, he did paper designs of two other computers himself, and helped inspire von Neumann's later work. After the war he studied the mathematics of biology, especially his "Turing patterns" of morphogenesis. Turing's life ended tragically: he was charged with immorality, forced to undergo chemical castration, and may have later killed himself. Due to his work in computer science and his decisive contribution to the war against Hitler Alan Turing deserves a place on our List.

Paul  Erdös (1913-1996) Hungary, U.S.A., Israel, etc.
Erdös was a childhood prodigy who became a famous (and famously eccentric) mathematician. He is best known for work in combinatorics (especially Ramsey Theory) and partition calculus, but made contributions across a very broad range of mathematics, including graph theory, analytic number theory, probabilistic methods, and approximation theory. He is regarded as the second most prolific mathematician in history, behind only Euler. Although he is widely regarded as an important and influential mathematician, Erdös founded no new field of mathematics: He was a "problem solver" rather than a "theory developer." He's left us several still-unproven intriguing conjectures, e.g. that   4/n = 1/x + 1/y + 1/z   has positive-integer solutions for any n. Erdös liked to speak of "God's Book of Proofs" and discovered new, more elegant, proofs of several existing theorems, including the two most famous and important about prime numbers: Chebyshev's Theorem that there is always a prime between any n and 2n, and (though the major contributer was Atle Selberg) Hadamard's Prime Number Theorem itself. He also proved many new theorems, such as the Erdös-Szekeres Theorem about monotone subsequences with its elegant (if trivial) pigeonhole-principle proof.

Atle  Selberg (1917-2007) Norway, U.S.A.
Selberg may be the greatest analytic number theorist ever. He also did important work in Fourier spectral theory, lattice theory (e.g. introducing and partially proving the conjecture that "all lattices are arithmetic"), and the theory of automorphic forms, where he introduced Selberg's Trace Formula. He developed a very important result in analysis called the Selberg Integral. Other Selberg techniques of general utility include mollification, sieve theory, and the Rankin-Selberg method. These have inspired other mathematicians, e.g. contributing to Deligne's proof of Weil conjectures. Selberg is also famous for ground-breaking work on Riemann's Hypothesis, and the first "elementary" proof of the Prime Number Theorem.

Jean-Pierre  Serre (1926-) France
Serre did important work with spectral sequences and algebraic methods, revolutionizing the study of homotopy groups and sheaves. Hermann Weyl praised Serre's work strongly, saying it gave an important new algebraic basis to analysis. He collaborated with Grothendieck and Pierre Deligne, helped resolve the Weil conjectures, worked in number theory, and contributed indirectly to the recent proof of Fermat's Last Theorem. Serre has been much honored: he is the youngest ever to win a Fields Medal; 49 years after his Fields Medal he became the first recipient of the Abel Prize.

Alexandre  Grothendieck (1928-) Germany, France
Grothendieck has done brilliant work in several areas of mathematics including number theory, geometry, topology, and functional analysis, but especially in the fields of algebraic geometry and category theory, both of which he revolutionized. He is most famous for his methods to unify different branches of mathematics, for example using algebraic geometry in number theory. Grothendieck 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. With these new methods, Grothendieck and his famous student Pierre Deligne were able to prove the Weil Conjectures. Grothendieck also developed the theory of sheafs, invented the theory of schemes, generalized the Riemann-Roch Theorem to revolutionize K-theory, developed Grothendieck categories, crystalline cohomology, infinity-stacks and more. The guiding principle behind much of Grothendieck's work has been Topos Theory, which he invented to harness the methods of topology. These methods and results have redirected several diverse branches of modern mathematics including number theory, algebraic topology, and representation theory. Grothendieck's radical religious and political philosophies led him to retire from public life while still in his prime, but he is widely considered the greatest mathematician of the 20th century, and is sometimes called one of the greatest mathematical geniuses ever.

Michael Francis (Sir)  Atiyah (1929-) Britain
Atiyah's career has had extraordinary breadth and depth. He advanced the theory of vector bundles; this developed into topological K-theory and the Atiyah-Singer Index Theorem. This Index Theorem is considered one of the most far-reaching theorems ever, subsuming famous old results (Déscartes' total angular defect, Euler's topological characteristic), important 19th-century theorems (Gauss-Bonnet, Riemann-Roch), and incorporating important work by Weil and especially Shiing-Shen Chern. It is a key to the study of high-dimension spaces, differential geometry, and equation solving. Several other important results are named after Atiyah, e.g. the Atiyah-Bott Fixed-Point Theorem, the Atiyah-Segal Completion Theorem, and the Atiyah-Hirzebruch spectral sequence. Atiyah's work developed important connections not only between topology and analyis, but with modern physics; Atiyah himself has been a key figure in the development of string theory. This work, and Atiyah-inspired work in gauge theory, restored a close relationship between leading edge research in mathematics and physics. Atiyah is known as a vivacious genius in person, inspiring many, e.g. Edward Witten. With Grothendieck retired, Atiyah is often considered to be the greatest living mathematician. Atiyah once said a mathematician must sometimes "freely float in the atmosphere like a poet and imagine the whole universe of possibilities, and hope that eventually you come down to Earth somewhere else."

John Horton  Conway (1937-) Britain
Conway has done pioneering work in a very broad range of mathematics including knot theory, number theory, group theory, lattice theory, combinatorial game theory, geometry, quaternions, tilings, and cellular automaton theory. He started his career by proving a case of Waring's conjecture, but achieved fame when he discovered the largest then-known sporadic group (the symmetry group of the Leech lattice); this sporadic group is now known to be second in size only to the "Monster Group," with which Conway also worked. Conway's fertile creativity has produced a cornucopia of fascinating inventions: markable straight-edge construction of the regular heptagon (a feat also achieved by Archimedes), a nowhere-continuous function that has the Intermediate Value property, the Conway box function, the aperiodic pinwheel tiling, a representation of symmetric polyhedra, his chained-arrow notation for large numbers, and many results and conjectures in recreational mathematics. He found the simplest proof for Morley's Trisector Theorem (sometimes called the best result in simple plane geometry since ancient Greece). He proved an unusual theorem about quantum physics: "If experimenters have free will, then so do elementary particles." His most famous construction is the computationally complete automaton known as the Game of Life. His most important theoretical invention, however, may be his surreal numbers incorporating infinitesimals; he invented them to solve combinatorial games like Go, but they have pure mathematical significance as the largest possible ordered field. Although Conway hasn't won the highest mathematics prizes, nor perhaps proved deep theoretical theorems, his great creativity and breadth qualifies him as one of the greatest living mathematicians.

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