(This is the long page. Click here for just the List, with links to the biographies.)
Isaac Newton 
Archimedes 
Carl Gauss 
Leonhard Euler 
Bernhard Riemann 
Henri Poincaré 
J.L. Lagrange 
David Hilbert 
Euclid 
G.W. Leibniz 
Alex. Grothendieck 
Pierre de Fermat 
The Greatest Mathematicians
of All Time
ranked in approximate order of "greatness."
To qualify, the mathematician
must be born before 1930
and his work must have
breadth,
depth, and
historical importance.
 
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.
 
 
 
 
 
 
 
 
 
I've appended five additional names to the List of Seventyfive
Greatest Mathematicians.
Maxwell, Einstein, etc. are among the greatest
applied mathematicians in history, but lack the importance as pure mathematicians to
qualify for The Top 75.
Nevertheless I'd want to add them to any longer list.
Because of their ambiguous status, I've left these five without rank numbers.
I think One Hundred (or 105) is a good list size, but will bring it up to 125 (not a very "round" number, but a cubical one :), just to show some "spares." (For this extended list, I relax the birthdate rule slightly to include two greats born in the 1930's.) 

This is primarily a list of Greatest Mathematicians of the Past, but I use 1930 birth as an arbitrary cutoff, and three of the "Top 100" are still alive as I write.
Click for a discussion of certain omissions. Please send me email if you believe there's a major flaw in my rankings (or an error in any of the biographies). Obviously the relative ranks of, say Fibonacci and Ramanujan, will never satisfy everyone since the reasons for their "greatness" are different. I'm sure I've overlooked great mathematicians who obviously belong on this list. Please email and tell me! (Sorry if mathematician "100." displays as "00." Either my html is flawed, or Microsoft IE doesn't like lists longer than 99.)
(By the way, the ranking assigned to a mathematician will appear if you place the cursor atop the name at the top of his minibio.)
Following are the top mathematicians in
chronological (birthyear) order
(with a larger font used for the names of the Top Thirty):
Little is known of the earliest mathematics, but the famous Ishango Bone from Early StoneAge 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 IndusHarrapa civilization imply strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where 4500year 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 placevalue 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.)
The Pyramids demonstrate that Egyptians were adept at geometry. Yet, although they knew the formula for a pyramid's volume, there's no evidence they even knew the Pythagorean Theorem! Babylon was much more advanced than Egypt at arithmetic and algebra; this was probably due, at least in part, to their placevalue system. But although their base60 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.
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 thenbest 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 placevalue 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 compassandstraightedge 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 wellknown 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 noted that, given a line segment of length x, a segment of length x/k can be constructed by first constructing a segment of length kx.Thales was also an astronomer; he invented the 365day 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 secondhand. 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 unequaled 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 630560 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 578505 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 and his followers began to study the question of planetary motions, which would not be resolved for more than two millenia. 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 secondhand, and possibly exaggerated, by Plato and others. Some ideas attributed to him were probably first enunciated by successors like Parmenides of Elea (ca 515440 BC). Pythagoras' 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; some conjecture that Pythagoras journeyed to India and learned of the proof there.) He may have discovered the simple parametric form of Pythagorean triplets (xxyy, 2xy, xx+yy), although the first explicit mention of this may be in Euclid's Elements. 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.)
In addition to Parmenides, the famous successors of Thales and Pythagoras include Zeno of Elea (see below), Hippocrates of Chios (see below), Plato of Athens (ca 428348 BC), Theaetetus (ca 414369 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 straightedge rather than with "carpenter's tools" like rulers and protractors.
Panini (of Shalatula) (ca 520460 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 nonobvious elegance that would not be equaled 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 20thcentury linguists and computer scientists like Chomsky, Backus, Post and Church are seen to resemble Panini's work 25 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 495435 BC) Greek domain
Zeno, a student of Parmenides, had great fame in ancient Greece. This fame, which continues to the presentday, 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 nonrigorous heuristic and is finally viewed as sound only after the work of the great 19thcentury rigorists, Dedekind and Weierstrass.
Hippocrates of Chios (ca 470410 BC) Greek domain
Hippocrates (no known relation to Hippocrates of Cos, the famous physician) wrote his own Elements more than a century before Euclid. Only fragments survive but it apparently used axiomaticbased 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 cubedoubling (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. Hippocrates also did work in algebra and rudimentary analysis.(Doubling the cube and angle trisection are often called "impossible," but they are impossible only when restricted to compass and unmarkable straightedge. There are ingenious solutions available with other tools. Construction of the regular heptagon is another such task.)
Archytas of Tarentum (ca 420350 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 cuberoot 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 threedimensional 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 steampowered wooden bird which flew for 200 meters. Archytas is sometimes called the "Father of Mathematical Mechanics."
Some scholars think Pythagoras and Thales are partly mythical. If we take that view, Archytas (and Hippocrates) should be promoted in this list.
Eudoxus of Cnidus (408355 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 secondhand, 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 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 cubedoubling 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 (384322 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.) He was personal tutor to the young Alexander the Great. Aristotle's 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 322275 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 (M^{2}+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 rigidcompass constructions can be implemented with collapsingcompass constructions. Although notions of trigonometry were not in use, Euclid's theorems include some closely related to the Laws of Sines and Cosines. 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 (287212 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, and even leapfrogged, the works of Euclid. (For example, some of Euclid'e more difficult theorems are easy analytic consequences of Archimedes' Lemma of Centroids.) 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 unequaled 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). 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. Some of Archimedes' work survives only because Thabit ibn Qurra translated the otherwiselost Book of Lemmas; it contains the angletrisection method and several ingenious theorems about inscribed circles. (Thabit shows how to construct a regular heptagon; it may not be clear whether this came from Archimedes, or was fashioned by Thabit by studying Archimedes' angletrisection method.) Other discoveries known only secondhand include the Archimedean solids reported by Pappus, and the BrokenChord Theorem reported by Alberuni.Archimedes and Newton might be the two best geometers ever, but although each produced ingenious geometric proofs, often they used nonrigorous calculus to discover results, and then devised rigorous geometric proofs for publication. He used integral calculus to determine the centers of mass of hemisphere and cylindrical wedge, and the volume of two cylinders' intersection. Although Archimedes didn't develop differentiation (integration's inverse), Michel Chasles credits him (along with Kepler, Cavalieri, and Fermat, who all lived more than 18 centuries later) as one of the four who developed calculus before Newton and Leibniz. He was one of the greatest mechanists ever, discovering the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, a miniature planetarium, and war machines (e.g.. catapult and shipburning mirrors). His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, Sphere and Cylinder, and (discovered only recently, and often called his most important work) The Method. He developed the Stomachion puzzle (and solved a difficult enumeration problem involving it). Archimedes proved that the volume of a sphere is twothirds 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. (Apollonius soon surpassed it, but by using Archimedes' method.) 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."
Some of Archimedes' greatest writings (including The Method) are preserved only on a palimpsest which has been rediscovered and properly studied only since 1998. Ideas unique to that work are an anticipation of Riemann integration, calculating the volume of a cylindrical wedge (previously first attributed to Kepler), and perhaps an implication that Archimedes understood 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. (Some think the Scientific Revolution would have begun sooner had The Method been discovered four or five centuries earlier.)
Apollonius of Perga (262190 BC) Greek domain
Apollonius Pergaeus, called "The Great Geometer," is sometimes considered the second greatest of ancient Greek mathematicians. (Euclid, Eudoxus and perhaps Archytas are 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 200142 BC) China
Chinese mathematicians excelled for thousands of years, and were first to discover various algebraic and geometric principles. There is some evidence that Chinese writings influenced India and the Islamic Empire, and thus, indirectly, Europe. 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 (though zero was represented as just a space, rather than a discrete symbol), 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 minibio; other famous contributors are Jing Fang and Zhang Heng.)
Nine Chapters was probably based on earlier books, lost during the great book burning of 212 BC, and Chang himself may have been a lord who commissioned others to prepare the book. Moreover, important revisions and commentaries were added after Chang, notably by Liu Hui (ca 220280). Although Liu Hui mentions Chang's skill, we cannot be sure that Chang had the mathematical genius to qualify for this list, but he would still be a strong candidate due to the book's 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 Aryabhata), 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 190127 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. Classical Hindu astronomers, including the 6thcentury genius Aryabhata, borrow much from Ptolemy and Hipparchus.) Hipparchus is called the "Father of Trigonometry"; he developed spherical trigonometry, produced trig tables, and more. He produced at least 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 circleconformal 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.The Antikythera mechanism is an astronomical clock considered amazing for its time. It was built a few decades after Hipparchus' death, but soon lost (remaining at the bottom of the sea for 2000 years). The mechanism implemented the complex orbits which Hipparchus had developed to explain irregular planetary motions; it's not unlikely the great genius helped design this intricate analog computer, which may have been built in Rhodes where Hipparchus spent his final decades.
Tiberius(?) Claudius Ptolemaeus of Alexandria (ca 90168) Egypt (in GrecoRoman domain)
Ptolemy was one of the most famous of ancient Greek scientists. Among his mathematical results, most famous may be Ptolemy's Theorem (AC·BD = AB·CD + BC·AD if and only if ABCD is a cyclic quadrilateral). This theorem has many useful corollaries; it was frequently applied in Copernicus' work. Ptolemy also wrote on trigonometry, optics, geography, and astrology; but is most famous for his astronomy, where he perfected the geocentric model of planetary motions.The mystery of celestial motions directed scientific inquiry for thousands of years. The problem had been considered by Eudoxus, Apollonius, and Hipparchus, who developed a very complicated geocentric model involving concentric spheres and epicyles. Ptolemy perfected (or, rather, complicated) this model even further; his model was the standard for 14 centuries. While some Greeks, notably Aristarchus, proposed heliocentric models, these were rejected because there was no parallax among stars. (Only Aristarchus guessed that the stars were at an almost unimaginable distance, explaining the lack of parallax.) The great skill demonstrated by Ptolemy and his predecessors in developing their complex geocentric cosmology may have set back science since, as we now know, the Earth rotates around the Sun. The geocentric models couldn't explain the observed changes in the brightness of Mars or Venus, but it was the phases of Venus, discovered by Galileo after the invention of the telescope, that finally led to general acceptance of heliocentrism. (Since the planets move without friction, their motions offer a pure view of the Laws of Motion; thus the heliocentric breakthroughs of Copernicus, Kepler and Newton triggered the advances in mathematical physics which led to Scientific Revolution.)
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, incorporation of negative numbers into arithmetic, 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'splay 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 arbitrarilyclose approximations for π. He also devised an interpolation formula to simplify that calculation; this yielded the "goodenough" value 3.1416, which is still taught today in primary schools. (Liu's successors in China included Zu Chongzhi, who did determine sphere's volume, and whose approximation for π held the accuracy record for nine centuries.)
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 advanced a rudimentary arithmetic and algebraic notation, allowed rationalnumber solutions to his problems rather than just integers, and was aware of results like the BrahmaguptaFibonacci 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 x^{3} for "x cubed" was not invented until Déscartes.)Very little is known about Diophantus (he might even have come from Babylonia, whose algebraic ideas he borrowed). 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 a^{3}b^{3} = c^{3}+d^{3}." (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 equalarea partitioning was finally proved in 1999 by Thomas Hales, who also proved the related Kepler Conjecture.) Pappus stated, but did not fully solve, the Problem of Pappus which, given an arbitrary collection of lines in the plane, asks for the locus of points whose distances to the lines have a certain relationship. This problem was a major inspiration for Déscartes and was finally fully solved by Newton.
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 placevalue system based on zero and nine other symbols.
Decimal system  from India? China?? Persia???
It's still hard to believe that the "obvious" and soconvenient 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 Aryabhata, 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.The Chinese used a form of decimal abacus as early as 3000 BC; if it doesn't qualify, by itself, as a "decimal system" then pictorial depictions of its numbers would. Yet for thousands of years after its abacus, China had no zero symbol other than plain space; and apparently didn't have one until after the Hindus. Ancient Persians and Mayans did have placevalue notation with zero symbols, but neither qualify as inventing a base10 decimal system: Persia used the base60 Babylonian system; Mayans used base20. (Another difference is that the Hindus had nine distinct digit symbols to go with their zero, while earlier placevalue systems built up from just two symbols: 1 and either 5 or 10.)
Conclusion: The decimal placevalue system with zero symbol seems to be an obvious invention that in fact was very hard to invent. If you insist on a single winner then India might be it. But China, Babylonia, Persia and even the Mayans deserve Honorable Mention!
Aryabhata (476550) 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, Aryabhata (called Arjehir by Arabs) may be most famous.While Europe was in its early "Dark Age," Aryabhata advanced arithmetic, algebra, elementary analysis, and especially plane and spherical trigonometry, using the decimal system. Aryabhata is sometimes called the "Father of Algebra" instead of alKhowârizmi (who himself cites the work of Aryabhata). His most famous accomplishment in mathematics was the Aryabhata Algorithm (connected to continued fractions) for solving Diophantine equations. Aryabhata 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 very few 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 alBiruni. Aryabhata 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 Aryabhata is the famous identity
Σ (k^{3}) = (Σ k)^{2}
Brahmagupta `Bhillamalacarya' (589668) 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 A^{2} = (a+b+cd)(a+bc+d)(ab+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." He also began the study of rational quadrilaterals which Kummer would eventually complete. 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 number theory problems. He was first to find a general solution to the simplest Diophantine form. His work on Pell's equations has been called "brilliant" and "marvelous." He proved the BrahmaguptaFibonacci 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 alKhowârizmi (ca 780850) Persia, Iraq
AlKhowârizmi (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 AlJabr, which demonstrated simple algebra and geometry, and several other influential books. Unlike Diophantus' work, which dealt in specific examples, AlKhowârizmi was the first algebra text to present general methods; he is often called the "Father of Algebra." (Diophantus did, however, use superior "syncopated" notation.) The word algorithm is borrowed from AlKhowârizmi's name, and algebra is taken from the name of his book. He also coined the word cipher, which became English zero (although this was just a translation from the Sanskrit word for zero introduced by Aryabhata). He was an essential pioneer for Islamic science, and for the many Arab and Persian mathematicians who followed; and hence also for Europe's eventual Renaissance which was heavily dependent on Islamic teachings. AlKhowârizmi's texts on algebra and decimal arithmetic are considered to be among the most influential writings ever.
Ya'qub `Abu Yusuf' ibn Ishaq alKindi (803873) Iraq
AlKindi (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 (codebreaking). (AlKindi, 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.)
AlSabi Thabit ibn Qurra alHarrani (836901) 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, nonEuclidean geometry, trigonometry, arithmetic, number theory. As well as being an original thinker, Thabit was a key translator of ancient Greek writings; he translated Archimedes' otherwiselost Book of Lemmas and applied one of its methods to construct a regular heptagon. 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, and with cubic equations. He was an earlier practitioner of calculus and seems to have been first to take the integral of √x. Like Archimedes, he was able to calculate the area of an ellipse, and to calculate the volume of a paraboloid. He produced an elegant generalization of the Pythagorean Theorem:
AC^{ 2} + BC^{ 2} = AB (AR + BS)
(Here the triangle ABC is not a right triangle, but R and S are located on AB to give the equal angles ACB = ARC = BSC.) Thabit 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 impressive. Among the several great and famous Baghdad geometers, Thabit may have had the greatest genius.
Mohammed ibn alHasn (Alhazen) `Abu Ali' ibn alHaytham alBasra (9651039) Iraq, Egypt
AlHassan ibn alHaytham (Alhazen) 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 overconfidence 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," the "Founder of Experimental Psychology" (mainly for his work with optical illusions), and, because he emphasized hypotheses and experiments, "The First Scientist."In number theory, Alhazen worked with perfect numbers, Mersenne primes, the Chinese Remainder Theorem; and stated Wilson's Conjecture (sometimes called AlHaytham's Theorem though it was first proven by Lagrange). He introduced the Power Series Theorem (later attributed to Jacob Bernoulli). His best mathematical work was with plane and solid geometry, especially conic sections; he calculated the areas of lunes, volumes of paraboloids, and constructed a heptagon using intersecting parabolas. 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. To solve it, Alhazen needed to anticipate Déscartes' analytic geometry, anticipate Bézout's Theorem, tackle quartic equations and develop a rudimentary integral calculus. Alhazen's attempts to prove the Parallel Postulate make him (along with Thabit ibn Qurra) one of the earliest mathematicians to investigate nonEuclidean geometry.
Abu alRayhan Mohammed ibn Ahmad alBiruni (9731048) Khwarizm (Uzbekistan)
AlBiruni (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. (AlBiruni's contemporary Avicenna was not particularly a mathematician but deserves mention as an advancing scientist, as does Avicenna's disciple Abu'lBarakat alBaghdada, who lived about a century later.)AlBiruni has left us what seems to be the oldest surviving mention of the Broken Chord Theorem (if M is the midpoint of circular arc ABMC, and T the midpoint of "broken chord" ABC, then MT is perpendicular to BC). Although he himself attributed the theorem to Archimedes, AlBiruni provided several novel proofs for, and useful corollaries of, this famous geometric gem. While AlBiruni 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 alKhayyám (10481123) Persia
Omar Khayyám (aka Ghiyas odDin AbolFath Omar ibn Ebrahim Khayyam Neyshaburi) is sometimes called the greatest Islamic mathematician. He did clever work with geometry, developing an alternate to Euclid's Parallel Postulate and then deriving the parallel result using theorems based on the KhayyamSaccheri 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. Khayyám did even more important work in algebra, writing an influential textbook, and developing new solutions for various higherdegree equations. He may have been first to develop Pascal's Triangle (which is still called Khayyám's Triangle in Persia), along with the essential Binomial Theorem (AlKhayyám's Formula): (x+y)^{n} = n! Σ x^{k}y^{nk} / k!(nk)!Khayyám 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. His symbol ('shay') for an unknown in an algebraic equation might have been transliterated to become our 'x'. He also wrote treatises on philosophy, music, mechanics and natural science. Despite his great achievements in algebra, geometry, and astronomy, today Omar alKhayyám is most famous for his rich poetry (The Rubaiyat of Omar Khayyám).
Bháscara Áchárya (11141185) 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 2ndorder Diophantine, quartic, Brouncker's and Pell's equations. His Chakravala method, an early application of mathematical induction to solve 2ndorder 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 aBháscara'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. (Another mathematician, Bháscara I who lived five centuries before Bháscara II, was also outstanding. He was famous for advancing the positional decimal number notation, for a formula giving an excellent approximation to the sin function, and for being first to state Wilson's Conjecture.)
Leonardo `Bigollo' Pisano (Fibonacci) (ca 11701245) 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 alKhowârizmi, into Latin, but Leonardo was the influential teacher. He also reintroduced 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 widely ignored or considered incomplete. I'm preparing a page to consider that question.) Another of Leonardo's noteworthy achievements was proving that the roots of a certain cubic equation could not have any of the constructible forms Euclid had outlined in Book 10 of his Elements.
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 alKhowârizmi 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.
Abu Jafar Muhammad Nasir alDin alTusi (12011274) Persia
Tusi was one of the greatest Islamic polymaths, working in theology, ethics, logic, astronomy, and other fields of science. He was a famous scholar and prolific writer, describing evolution of species, stating that the Milky Way was composed of stars, and mentioning conservation of mass in his writings on chemistry. He made a wide range of contributions to astronomy, and (along with Omar Khayyám) was one of the most significant astronomers between Ptolemy and Copernicus. He improved on the Ptolemaic model of planetary orbits, and even wrote about (though rejecting) the possibility of heliocentrism.Tusi is most famous for his mathematics. He advanced algebra, arithmetic, geometry, trigonometry, and even foundations, working with real numbers and lengths of curves. For his texts and theorems, he may be called the "Father of Trigonometry;" he was first to properly state and prove several theorems of planar and spherical trigonometry including the Law of Sines, and the (spherical) Law of Tangents. He wrote important commentaries on works of earlier Greek and Islamic mathematicians; he attempted to prove Euclid's Parallel Postulate. Tusi's writings influenced European mathematicians including Wallis; his revisions of the Ptolemaic model led him to the Tusicouple, a special case of trochoids usually called Copernicus' Theorem, though historians have concluded Copernicus discovered this theorem by reading Tusi.
There were several important Chinese mathematicians in the 13th century, of whom Qin Jiushao (Ch'in ChiuShao) 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 (Pascal's Triangle is still called Yang Hui's Triangle in China), and Zhu Shiejie (the most famous and influential mathematician of this era). Their teachings did not make their way to Europe, but were read by the Japanese mathematician Seki, and possibly by Islamic mathematicians like AlKashi. 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 13221382) 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; and was a renowned philosopher, natural scientist (challenging several of Aristotle's ideas), and important economist (anticipating Gresham's Law). Although the Earth's annual orbit around the Sun was left to Copernicus, Oresme was among the preCopernican thinkers to claim clearly that the Earth spun daily on its axis. 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 ahead of his time, Oresme deserves recognition.
Madhava of Sangamagramma (13401425) 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  q^{3}/3! + q^{5}/5!  ... , formulae for π, including the one attributed to Leibniz, and the thenbest known approximation π ≈ 104348 / 33215.Despite the accomplishments of the Kerala school, Madhava probably does not deserve a place on our List. There were several other great mathematicians who contributed to Kerala's achievements, some of which were made 150 years after Madhava's death. More importantly, the work was not propagated outside Kerala, so had almost no effect on the development of mathematics.
Ghiyath alDin Jamshid Mas'ud AlKashi (ca 13801429) Iran, Transoxania (Uzbekistan)
AlKashi 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, which is sometimes called AlKashi's Theorem. 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, alKashi 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.)
Nicolaus Copernicus (14721543) Poland
The European Renaissance developed in 15thcentury Italy, with the blossoming of great art, and as scholars read books by great Islamic scientists like Alhazen. The earliest of these great Italian polymaths were largely not noted for mathematics. Setting aside Leonardo da Vinci, who began serious math study only very late in life, perhaps the best "Italian" candidate for greatness who lived between Fibonacci and Cardano in time would be a foreigner. This certain Pole studied Islamic works on astronomy and geometry at the University of Bologna, and eventually wrote a book of huge impact.Nicolaus Copernicus (Mikolaj Kopernik) was a polymath; he started by studying law and medicine; later published poetry and contemplated astronomy, while working professionally as a church scholar/diplomat. Although his only famous theorem of mathematics (that certain trochoids are straight lines) seems to be copied from Nasir alTusi, it was mathematical thought that led Copernicus to the conclusion that the Earth rotates around the Sun. (It was left to Kepler to discover the true facts of planetary orbits, as Copernicus' system still used circles and epicycles, rather than ellipses.) Despite opposition from the Roman church, this discovery led, via Galileo, Kepler and Newton, to the Scientific Revolution. For this revolution, Copernicus is ranked #19 on Hart's list of the Most Influential Persons in History. It remains controversial whether earlier Islamic or Hindu mathematicians believed in heliocentrism, but were also inhibited by religious orthodoxy.
Girolamo Cardano (15011576) 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 Ujoint 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 unwilling to deal with negative numbers: his treatment of cubic equations had to deal with ax^{3}  bx + c = 0 and ax^{3}  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 (15401603) 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 the primary teaching for both Déscartes and Fermat; Isaac Newton also studied Vieta. 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 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 complexnumber arithmetic; he also used trigonometry to solve a particular 45thdegree 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 infiniteproduct 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 LOOP: Y := SQRT(Y + 2) PI := PI * 2 / Y IF (more precision needed) GOTO LOOPVieta'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 bestyet by a European, though not as accurate as alKashi's two centuries earlier.
Simon Stevin (15491620) 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 traveled faster than with horses and used it to entertain his patron, the Prince of Orange. He discovered several laws of mechanics including those for energy conservation and hydrostatic pressure. He lived slightly before Galileo who is now much more famous, but Stevin discovered the equal rate of falling bodies before Galileo did, and correctly explained the influence of the moon on tides (which Galileo later got wrong). He was first to write on the concept of unstable equilibrium. He invented improved accounting methods, and the equaltemperament 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; negative numbers and even zero and one were often not considered numbers). He introduced decimal fractions to Europe; suggested a decimal metric system, which was finally adopted 200 years later; invented other basic notation like the symbol √. Stevin proved several theorems about perspective geometry, an important result in mechanics, and the Intermediate Value Theorem attributed to Cauchy. Stevin's books, written in Dutch rather than Latin, were widely read and hugely influential. He was a very key figure in the development of modern European mathematics, and may belong on our List.
John Napier 8th of Merchistoun (15501617) 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 (15641642) Italy
Galileo discovered the laws of inertia, falling bodies (including parabolic trajectories), and the pendulum; he also introduced the notion of relativity which Einstein later found so fruitful. 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, Galileo 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.) His discovery that Venus, like the Moon, had phases was the critical fact which forced acceptance of Copernican heliocentrism. 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 curvefitting criterion, two centuries before Gauss and Legendre introduced the mean squareddifference 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 these comments, it does appear that Galileo ignored experimental results that conflicted with his theories. For example, the Law of the Pendulum, based on Galileo's incorrect belief that the tautochrone was the circle, conflicted with his own observations.) 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 (15711630) 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 twentynine. His observations of the planets with Brahe, along with his study of Apollonius' 1800year 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, the theory of the camera obscura, 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.) Kepler ranks #75 on Michael Hart's famous list of the Most Influential Persons in History. This rank, much lower than that of Copernicus, Galileo or Newton, seems to me to underestimate Kepler's importance, since it was Kepler's Laws, rather than just heliocentrism, which were essential to the early development of mathematical physics.According to Kepler's Laws, the planets move at variable speed along ellipses. (Even Copernicus thought the orbits could be described with only circles.) The Earthbound 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 the theory of logarithms and improved on Napier's tables. 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 thennovel 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 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.)
Gérard Desargues (15911661) 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 based on the epicycloid, and had the idea to use cycloidshaped 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 (15961650) France
Déscartes' early career was that of soldieradventurer 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 both analytic geometry and symbolic algebraic notation and is therefore called the "Father of Modern Mathematics." His use of equations to partially solve the geometric Problem of Pappus revolutionized 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 improved solution to the Delian problem (cubedoubling). He improved mathematical notation (e.g. the use of superscripts to denote exponents). He also discovered Euler's Polyhedral Theorem, F+V = E+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 (15981647) Italy
Cavalieri worked in analysis, geometry and trigonometry (e.g. discovering a formula for the area of a spherical triangle), but is most famous for publishing works on his "principle of indivisibles" (calculus); these were very influential and inspired further development by Huygens, Wallis and Barrow. (His calculus was partly anticipated by Galileo, Kepler and Luca Valerio, and developed independently, though left unpublished, by Fermat.) Among his theorems in this calculus was
lim _{(n→∞)} (1^{m}+2^{m}+ ... +n^{m}) / n^{m+1} = 1 / (m+1)
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 (16011665) 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. Lagrange considered Fermat, rather than Newton or Leibniz, to be the inventor of 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 ubiquitouslyused 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); and Fermat's Christmas Theorem (that any prime (4n+1) can be represented as the sum of two squares in exactly one way) which may be considered the most difficult theorem of arithmetic which had been proved up to that date. Fermat proved the Christmas Theorem with difficulty using "infinite descent," but details are unrecorded, so the theorem is often named the FermatEuler Prime Number Theorem, with the first published proof being by Euler more than a century after Fermat's claim. Another famous conjecture by Fermat is that every natural number is the sum of three triangle numbers, or more generally the sum of k kgonal numbers. As with his "Last Theorem" he claimed to have a proof but didn't write it up. (This theorem was eventually proved by Lagrange for k=4, the very young Gauss for k=3, and Cauchy for general k. Diophantus claimed the k=4 case but any proof has been lost.) I think Fermat's conjectures were impressive even if unproven, and that this great mathematician is often underrated. (Recall that his socalled "Last Theorem" 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 (16161703) England
Wallis began his life as a savant at arithmetic (it is said he once calculated the square root of a 53digit 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. He made major advances in analytic geometry, but also contributions to algebra, geometry and trigonometry. Unlike his contemporary, Huygens who took inspiration from Euclid's rigorous geometry, Wallis embraced the new analytic methods of Déscartes and Fermat. He is especially famous for using negative and fractional exponents (though Oresme had introduced fractional exponents three 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 codebreaker, 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+(4k^{2}1)^{1}
Blaise Pascal (16231662) 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 Gérard Desargues (15911661), he was a key pioneer of projective geometry. He also made important early contributions to calculus; indeed it was his writings that inspired Leibniz. 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, wrote the philosophical treatise Pensées ("We arrive at truth, not by reason only, but also by the heart"), 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 (16291695) 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 inversesquare 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 19thcentury work of Young, Fresnel, and Maxwell. Later, Planck, Einstein and Bohr, partly anticipated by Hamilton, developed the modern notion of waveparticle 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 eyepiece, 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 31tone music: a 31tone 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 extraterrestrial 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 SaintVincent's thenfamous circlesquaring 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 selftaught Gottfried Leibniz (who'd "wasted his youth" without learning any math). Before agreeing to tutor him, Huygens tested the 25year old Leibniz by asking him to sum the reciprocals of the triangle numbers.
Takakazu Seki (Kowa) (ca 16371708) Japan
Seki Takakazu (aka Shinsuke) was a selftaught prodigy who developed a new notation for algebra, and made several discoveries before Western mathematicians did; these include determinants, the NewtonRaphson 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.
James Gregory (16381675) Scotland
James Gregory (Gregorie) was the outstanding Scottish genius of his century. Had he not died at the age of 36, or if he had published more of his work, (or if Newton had never lived), Gregory would surely be appreciated as one of the greatest mathematicians of the early Age of Science. Inspired by Kepler's work, he worked in mechanics and optics; invented a reflecting telescope; and is even credited with using a bird feather as the first diffraction grating. But James Gregory is most famous for his mathematics, making many of the same discoveries as Newton did: the Fundamental Theorem of Calculus, interpolation method, and binomial theorem. He developed the concept of Taylor's series and used it to solve a famous semicircle division problem posed by Kepler and to develop trigonometric identities, including
tan^{1}x = x  x^{3}/3 + x^{5}/5  x^{7}/7 + ... (for x < 1)
Gregory anticipated Cauchy's convergence test, Newton's identities for the powers of roots, and Riemann integration. He may have been first to suspect that quintics generally lacked algebraic solutions, as well as that π and e were transcendental. He produced a partial proof that the ancient "Squaring the Circle" problem was impossible.Gregory declined to publish much of his work, partly in deference to Isaac Newton who was making many of the same discoveries. Because the wide range of his mathematics wasn't appreciated until long after his death, Gregory lacks the historic importance to qualify for our List.
Isaac (Sir) Newton (16421727) 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 famous for his Three Laws of Motion (inertia, force, reciprocal action) but, as Newton himself acknowledged, these Laws weren't fully novel: Hipparchus, Ibn alHaytham, Galileo and Huygens had all developed much basic mechanics already, and Newton credits the First Law itself to Aristotle. However Newton was also apparently the first person to conclude that the ordinary gravity we observe on Earth is the very same force that keeps the planets in orbit. His Law of Universal Gravitation was revolutionary and due to Newton alone. (Christiaan Huygens, the other great mechanist of the era, had independently deduced that Kepler's laws imply inversesquare gravitation, but he considered the action at a distance in Newton's theory to be "absurd.") Newton published the Cooling Law of thermodynamics. He also made contributions to chemistry, and was the important early advocate of the atomic theory. (His other intellectual interests included theology, astrology and alchemy.) Although this list is concerned only with mathematics, Newton's greatness is indicated by the huge range of his physics: even without his Laws of Motion, Gravitation and Cooling, he'd be famous just for his revolutionary 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.) Newton's earliest fame came when he designed the first reflecting telescope: by avoiding chromatic aberration, these were the best telescopes of that era. He also designed the 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 interpolation method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (His equation e^{x} = ∑ x^{k} / 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, using a purely geometric argument of awesome ingenuity, that samemass 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 reinvented 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 alKashi's.)
Newton is so famous for his calculus, optics, and laws of gravitation and motion, it is easy to overlook that he was also one of the very greatest geometers. In addition to new constructions for already solved problems like the Delian cubedoubling problem, he was first to fully solve the famous Problem of Pappus, and did so with pure geometry. Despite the power of Déscartes' analytic geometry, Newton's achievements with synthetic geometry were surpassing. 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." His other marvelous geometric theorems included several about quadrilaterals and their in or circumscribing ellipses. He constructed the parabola defined by four given points, as well as various cubic curve constructions. (As with Archimedes, many of Newton's constructions used nonPlatonic tools.) 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. With the key mystery of celestial motions finally resolved, the Great Scientific Revolution began. (In his work Newton also proved important theorems about inversecube forces, work largely unappreciated until Chandrasekhar's modernday work.) 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 or second 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 (16461716) Germany
Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his coinvention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as secondhighest in all of history, behind only Goethe's. 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 selftaught 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 Fifteen 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. Leibniz also had political influence: he consulted to both the Holy Roman and Russian Emperors; another of his patrons was Sophia Wittelsbach, who was only distantly in line for the British throne, but was made Heir Presumptive. (Sophia died before Queen Anne, but her son was 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 notations ∫f(x)dx, df(x)/dx, and even ∛x; 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 selfsimilarity. 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 the 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 (16541705) Switzerland
Jacob Bernoulli studied the works of Wallis and Barrow; he and Leibniz became friends and tutored each other. 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 NewtonLeibniz 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. He is credited with the invention of polar coordinates (though Newton and Alberuni had also discovered them). Jacob also did outstanding work in geometry, for example constructing perpendicular lines which quadrisect a triangle.
Johann Bernoulli (16671748) Switzerland
Johann Bernoulli learned from his older brother and Leibniz, and went on to become principal 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'Hôpital's Rule before L'Hôpital 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. Johann's son Daniel is also a candidate for greatest Bernoulli.
Brook Taylor (16851731) 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 in drawing which helped develop the fields of both projective and descriptive geometry. His work in projective geometry rediscovered Desargues' Theorem, introduced terms like vanishing point, and influenced Lambert.
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 (17001782) Switzerland
Johann Bernoulli had a nephew, three sons and some grandsons who were all also outstanding mathematicians. Of these, the most important was his son Daniel. Johann insisted that Daniel study biology and medicine rather than mathematics, so Daniel specialized initially in mathematical biology. He went on to win the Grand Prize of the Paris Academy no less than ten times, and was a close friend of Euler. He developed partial differential equations, preceded Fourier in the use of Fourier series, did important work in statistics and the theory of equations, discovered and proved a key theorem about trochoids, developed a theory of economic risk (motivated by the St. Petersburg Paradox discovered by his cousin Nicholas), but is most famous for his key discoveries in mathematical physics, including the Bernoulli Principle underlying airflight. Daniel Bernoulli is sometimes called the "Founder of Mathematical Physics."
Leonhard Euler (17071783) 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 NewtonGiraud formulae, etc. He was also supreme at discrete mathematics, inventing graph theory. He also invented the concept of generating functions; for example, letting p(n) denote the number of partitions of n, Euler found the lovely equation: Σ_{n} p(n) x^{n} = 1 / Π_{k} (1  x^{k})
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 thenknown prime and the largest thenknown 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 9point 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, and the famous Euler's Polyhedral Theorem, F+V = E+2 (although it may have been discovered by Déscartes and first proved rigorously by Jordan). Although noted as the first great "pure mathematician," Euler's pump and turbine equations revolutionized the design of pumps; he also made important contributions to music theory, acoustics, optics, celestial motions, fluid dynamics, and mechanics. He extended Newton's Laws of Motion to rotating rigid bodies; and developed the EulerBernoulli 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 threebody problem which had stumped Newton), and he settled an arithmetic dispute involving 50 terms in a long convergent series. Both these feats were accomplished when he was totally blind. (About this he said "Now I will have less distraction.") François Arago said that "Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind."
Four of the most important constant symbols in mathematics (π, e, 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 by that name); he anticipated the concept of analytic continuation by showing ζ(1) = 1+2+3+4+... = 1/12. As a young student of the Bernoulli family, Euler discovered the striking identity ζ(2) = π^{2}/6 This catapulted Euler to instant fame, since the leftside 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 result which has inspired a variety of discoveries), and the Euler Product Formula ζ(s) = ∏(1p^{s})^{1} where the rightside 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:
e^{i x} = cos x + i sin x. (It is almost wondrous how the particular instance e^{i π}+1 = 0 combines the most important constants and operators together.)Some of Euler's greatest formulae can be combined into curiouslooking formulae for π: π^{2} =  log^{2}(1) = 6 ∏_{p∈Prime}(1p^{2})^{1/2}
Alexis Claude Clairaut (17131765) 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 significant contributions in differential equations and mathematical physics. Clairaut supported Newton against the Continental schools, and helped translate Newton's work into French. The theories of Newton and 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 threebody 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 (17171783) 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 CauchyRiemann 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'AlembertGauss 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 EulerLagrange.) He also did creative work in geometry (e.g. anticipating Monge's Three Circle Theorem), and was principal creator of the major encyclopedia of his day. D'Alembert wrote "The imagination in a mathematician who creates makes no less difference than in a poet who invents."
Johann Heinrich Lambert (17271777) 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 e^{x} are both irrational for any nonzero 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." Lambert, whom Kant called "the greatest genius of Germany," was an outstanding polymath: In addition to several areas of mathematics, he made contributions in philosophy, psychology, cosmology (conceiving of star clusters, galaxies and supergalaxies), mapmaking (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 straightedge constructions without compass. He also developed nonEuclidean geometry, long before Bolyai and Lobachevsky did.
JosephLouis (Comte de) Lagrange (17361813) Italy, France
JosephLouis Lagrange (born Giuseppe Lodovico Lagrangia) was a brilliant man who advanced to become a teenage Professor shortly after first studying mathematics. He excelled in all fields of analysis and number theory; he made key contributions to the theories of determinants, continued fractions, and many other fields. He developed partial differential equations far beyond those of D. Bernoulli and d'Alembert, developed the calculus of variations far beyond that of the Bernoullis, discovered the Laplace transform before Laplace did, and developed terminology and notation (e.g. the use 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 Conjecture (p divides (p1)! + 1 when p is prime); Lagrange's FourSquare Theorem (every positive integer is the sum of four squares); and that n·x^{2} + 1 = y^{2} has solutions for every positive nonsquare 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 19thcentury 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) (17461818) France
Gaspard Monge, son of a humble peddler, was an industrious and creative inventor who astounded early with his genius, becoming a professor of physics at age 16. As a military engineer he developed the new field of descriptive geometry, so useful to engineering that it was kept a military secret for 15 years. Monge made early discoveries in chemistry and helped promote Lavoisier's work; he also wrote papers on optics and metallurgy; Monge's talents were so diverse that he became Minister of the Navy in the revolutionary government, and eventually became a close friend and companion of Napoleon Bonaparte. Traveling with Napoleon he demonstrated great courage on several occasions.In mathematics, Monge is called the "Father of Differential Geometry," and it is that foundational work for which he is most praised. He also did work in discrete math, partial differential equations, and calculus of variations. He anticipated Poncelet's Principle of Continuity. Monge's most famous theorems of geometry are the Three Circles Theorem and Four Spheres Theorem. His early work in descriptive geometry has little interest to pure mathematics, but his application of calculus to the curvature of surfaces inspired Gauss and eventually Riemann, and led the great Lagrange to say "With [Monge's] application of analysis to geometry this devil of a man will make himself immortal."
Monge was an inspirational teacher whose students included Fourier, Chasles, Brianchon, Ampere, Carnot, Poncelet, several other famous mathematicians, and perhaps indirectly, Sophie Germain. Chasles reports that Monge never drew figures in his lectures, but could make "the most complicated forms appear in space ... with no other aid than his hands, whose movements admirably supplemented his words." The contributions of Poncelet to synthetic geometry may be more important than those of Monge, but Monge demonstrated great genius as an untutored child, while Poncelet's skills probably developed due to his great teacher.
PierreSimon (Marquis de) Laplace (17491827) 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 twobody 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 socalled 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 (although the Laplace Transform itself was already known to Lagrange). He was the premier expert at differential and difference equations, and definite integrals. He developed spherical harmonics, potential theory, and the theory of determinants; anticipated Fourier's series; and advanced Euler's technique of generating functions. In the fields of probability and statistics he made key advances: he proved the Law of Least Squares, and introduced the controversial ("Bayesian") rule of succession. In the theory of equations, he was first to prove that any polynomial of even degree must have a real quadratic factor.
Others might place Laplace higher on the List, but he proved no fundamental theorems of 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 (17521833) France
Legendre was an outstanding mathematician who did important work in plane and solid geometry, spherical trigonometry, celestial mechanics and other areas of physics, and especially elliptic integrals and number theory. He found key results in the theories of sums of squares and sums of kgonal numbers. He also made key contributions in several areas of analysis: he invented the Legendre transform and Legendre polynomials; the notation for partial derivatives is due to him. He invented the Legendre symbol; invented the study of zonal harmonics; proved that π^{2} was irrational (the irrationality of π had already been proved by Lambert); and wrote important textbooks in several fields. Although he never accepted nonEuclidean geometry, and had spent much time trying to prove the Parallel Postulate, his inspiring geometry text remained a standard until the 20th century. As one of France's premier mathematicians, Legendre did other significant work, promoting the careers of Lagrange and Laplace, developing trig tables, geodesic projects, etc.There are several important theorems proposed by Legendre for which he is denied credit, either because his proof was incomplete or was preceded by another's. He proposed the famous theorem about primes in a progression which was proved by Dirichlet; proved and used the Law of Least Squares which Gauss had left unpublished; proved the N=5 case of Fermat's Last Theorem which is credited to Dirichlet; proposed the famous Prime Number Theorem which was finally proved by Hadamard; improved the FermatCauchy result about sums of kgonal numbers but this topic wasn't fruitful; and developed various techniques commonly credited to Laplace. His two most famous theorems of number theory, the Law of Quadratic Reciprocity and the Three Squares Theorem (a difficult extension of Lagrange's Four Squares Theorem), 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 (17681830) 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 FourierMotzkin Elimination; and did significant work in operator theory. He is also noted for the notion of dimensional analysis, was first to describe the Greenhouse Effect, and continued his earlier brilliant work with equations.Fourier's greatest fame rests on his use of trigonometric series (now called Fourier series) in the solution of differential equations. Since "Fourier" analysis is in extremely common use among applied mathematicians, he joins the select company of the eponyms of "Cartesian" coordinates, "Gaussian" curve, and "Boolean" algebra. Because of the importance of Fourier analysis, many listmakers would rank Fourier much higher than I have done; however the work was not exceptional 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 (17771855) 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 ngon was constructible if and only if is the product of distinct prime Fermat numbers. (He didn't complete the proof of the onlyif part. Click to see construction of regular 17gon.) Also at age 19, he proved Fermat's conjecture that every number is the sum of three triangle numbers. (He further determined the number of distinct ways such a sum could be formed.) At age 24 he 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 nth 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. (Constructing the regular 17gon as a teenager was actually an exercise in complexnumber algebra, not geometry.) Gauss developed the arithmetic of congruences and became the premier number theoretician of all time. Other contributions of Gauss include hypergeometric series, foundations of statistics, and differential geometry. He also did important work in geometry, providing an improved solution to Apollonius' famous problem of tangent circles, stating and proving the Fundamental Theorem of Normal Axonometry, and solving astronomical problems related to comet orbits and navigation by the stars. (The first asteroid was discovered when Gauss was a young man; he famously constructed an 8thdegree 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 nonEuclidean 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 GaussBonnet 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 lineintegral over a closed curve of a monogenic function is zero): he proved this first but let Cauchy take the credit. Gauss was extremely prolific, and is widely considered to be the most brilliant mathematician who ever lived, so 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 (17811840) France
Siméon Poisson was a protégé of Laplace and, like his mentor, is among the greatest applied mathematicians ever. Poisson was an extremely prolific researcher and also an excellent teacher. In addition to important advances in several areas of physics, Poisson made key contributions to Fourier analysis, definite integrals, path integrals, statistics, partial differential equations, calculus of variations and other fields of mathematics. Poisson made improvements to Lagrange's equations of celestial motions, which Lagrange himself found inspirational. Another of Poisson's contributions to mathematical physics was his conclusion that the wave theory of light implies a 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."
JeanVictor Poncelet (17881867) 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 grouptheoretic developments by Lie and Klein. His theorems of geometry include his Closure Theorem about Poncelet Traverses, the PonceletBrianchon Hyperbola Theorem, and Poncelet's Porism (if two conic sections are respectively inscribed and circumscribed by an ngon, then there are infinitely many such ngons). Perhaps his most famous theorem, although it was left to Steiner to complete a proof, is the beautiful PonceletSteiner Theorem about straightedge constructions.
AugustinLouis Cauchy (17891857) France
Cauchy was extraordinarily prodigious, prolific and inventive. Homeschooled, he awed famous mathematicians at an early age. In contrast to Gauss and Newton, he was almost overeager to publish; in his day his fame surpassed that of Gauss and has continued to grow. Cauchy did significant work in analysis, algebra, number theory and discrete topology. His most important contributions included convergence criteria for infinite series, the "theory of substitutions" (permutation group theory), and especially his insistence on rigorous proofs.Cauchy's research also included differential equations, determinants, and probability. He invented the calculus of residues. 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 CauchySchwarz 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 kgonal numbers for any k. (Gauss had proved the case k = 3.)
One of the duties of a great mathematician is to nurture his successors, but Cauchy selfishly dropped the ball on both of the two greatest young mathematicians of his day, mislaying key manuscripts of 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 "mislaid." (For this historical miscontribution perhaps Cauchy should be demoted.)
Nicolai Ivanovitch Lobachevsky (17931856) Russia
Lobachevsky is famous for discovering nonEuclidean 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 DandelinGräffe method of polynomial roots independently of Dandelin and Gräffe. (In his lifetime, Lobachevsky was underappreciated and overworked; his duties led him to learn architecture and even some medicine.)Although Gauss and Bolyai discovered nonEuclidean 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 2100year old axiom; thus William K. Clifford called Lobachevsky "the Copernicus of Geometry."
Jakob Steiner (17961863) Switzerland
Jakob Steiner made many major advances in synthetic geometry, hoping that classical methods could avoid any need for analysis; and indeed, like Isaac Newton, he was often able to equal or surpass methods of 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, and for its "Malfatti circles." Among many famous and important theorems of classic and projective geometry, he proved that the Wallace lines of a triangle lie in a 3pointed 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 PonceletSteiner 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 DoubleElement Theorem about selfhomologous 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 (18011868) 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 CramerEuler 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 underappreciated in his native Germany, but achieved fame in France and England. In addition to his mathematical work in algebraic and analytic geometry, Plücker did significant 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 (18021829) 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. Several important theorems of analysis are named after Abel, including the (deceptively simple) Abel's Theorem of Convergence (published posthumously). Along with Galois, Abel is considered one of the two founders of group theory. 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, and was first to observe (but in a manuscript mislaid by Cauchy) that they were doubly periodic. Elliptic functions quickly became a productive field of mathematics, and led to more general complexvariable 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 5thdegree 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 twentysix. His fame lives on and even the lowercase word 'abelian' is applied to several concepts. Liouville said Abel was the greatest genius he ever met. Hermite said "Abel has left mathematicians enough to keep them busy for 500 years."
Carl G. J. Jacobi (18041851) Germany
Jacobi was a prolific mathematician who did decisive work in the algebra and analysis of complex variables, and did work in number theory (e.g. cubic reciprocity) which excited Carl Gauss. He is sometimes described as the successor to Gauss. As an algorist (manipulator of involved algebraic expressions), he may have been surpassed only by Euler and Ramanujan. He was also a very highly regarded teacher. In mathematical physics, Jacobi perfected Hamilton's principle of stationary action, and made other important advances.Jacobi's most significant early achievement was the theory of elliptic functions, e.g. his fundamental result about functions with multiple periods. 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. He also made important discoveries in many other areas including theta functions (e.g. his Jacobi Triple Product Identity), higher fields, number theory, algebraic geometry, differential equations, qseries, hypergeometric series, determinants, Abelian functions, and dynamics. He devised the algorithms still used to calculate eigenvectors and for other important matrix manipulations. The range of his work is suggested by the fact that the "Hungarian method," an efficient solution to an optimization problem published more than a century after Jacobi's death, has since been found among Jacobi's papers.
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 illfated contemporary was the more brilliant genius.
Johann Peter Gustav Lejeune Dirichlet (18051859) 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. 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. In addition to his own discoveries, Dirichlet played a key role in interpreting the work of Gauss, and was an influential teacher, mentoring famous mathematicians like Bernhard Riemann (who considered Dirichlet second only to Gauss among living mathematicians), Leopold Kronecker and Gotthold Eisenstein.As an impoverished lad Dirichlet spent his money on math textbooks; Gauss' masterwork became his lifelong companion. Fermat and Euler had proved the impossibility of x^{k} + y^{k} = z^{k} 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 did seminal work in analysis and is considered the founder of analytic number theory. He invented a method of Lseries 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 (18051865) Ireland
Hamilton was a childhood prodigy. Homeschooled and selftaught, 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 nonminimal but stationary action in configuration space. (Poisson and d' Alembert had noticed exceptions to EulerLagrange 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 waveparticle 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 nonAbelian field to handle 3D rotations. While there is no 3D analog to the Gaussian complexnumber plane (based on the equation i^{2} = 1 ), quaternions derive from a 4D analog based on i^{2} = j^{2} = k^{2} = 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 (18091877) 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 singlehandedly developed linear algebra, vector and tensor calculus, multidimensional 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.) Grassmann's exterior algebra, and the associated concept of Grassmannian manifold, provide a simplifying framework for many algebraic calculations. Recently their use led to an important simplification in quantum physics calculations.Of his linear algebra, one historian wrote "few have come closer than Hermann Grassmann to creating, singlehandedly, a new subject." Important mathematicians inspired directly by Grassmann include Peano, Klein, Cartan, Hankel, Clifford, and Whitehead.
Joseph Liouville (18091882) France
Liouville did expert research in several areas including number theory, differential geometry, complex analysis (especially SturmLiouville theory, boundary value problems and dynamical analysis), topology and mathematical physics. Several theorems bear his name, including the key result that any bounded entire function must be constant (the Fundamental Theorem of Algebra is an easy corollary of this!), important results in differential equations, differential algebra, differential geometry, a key result about conformal mappings, and an invariance law about trajectories in phase space which leads to the Second Law of Thermodynamics and is key to Hamilton's work in physics. He was first to prove the existence of transcendental numbers; he invented Liouville integrability and fractional calculus; he found a new proof of the Law of Quadratic Reciprocity. In addition to multiple Liouville Theorems, there are two "Liouville Principles": a fundamental result in differential algebra, and a fruitful theorem in number theory. It is said that Liouville made more discoveries in number theory than anyone else, even Ramanujan, but this work is largely unknown, e.g. the following remarkable generalization of Aryabhata's identity:
for all N, Σ (d_{a}^{3}) = (Σ d_{a})^{2}
where d_{a} is the number of divisors of a, and the sums are taken over all divisors a of N.Liouville established an important journal; influenced Catalan, Jordan, Chebyshev, Hermite; and helped promote other mathematicians' work, especially that of Évariste Galois, whose important results were almost unknown until Liouville clarified them. In 1851 Augustin Cauchy was bypassed to give a prestigious professorship to Liouville instead.
Ernst Eduard Kummer (18101893) 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 4degree Kummer Surface, an important algebraic form which inspired Klein's early work. He solved the ancient problem of finding all rational quadrilaterals. His most important discovery was ideal numbers; this led to the theory of ideals and padic numbers; this discovery's revolutionary nature has been compared to that of nonEuclidean 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 (18111832) France
Galois, who died before the age of twentyone, 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 principal 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 (18141897) 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 coinvented 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 solved (or partially solved) a huge variety of rich puzzles including various geometric gems; the enumeration of polynomial roots first tackled by Déscartes and Newton; and, by advancing the theory of partitions, the system of equations posed by Euler as The Problem of the Virgins. Sylvester was also a linguist, a poet, and did work in mechanics (inventing the skew pantograph) and optics. He once wrote, "May not music be described as the mathematics of the sense, mathematics as music of the reason?"
Karl Wilhelm Theodor Weierstrass (18151897) 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 HermiteLindemann 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 equaled or exceeded even that of Cauchy, Abel and Dirichlet. His students included Kovalevskaya, Frobenius, MittagLeffler, 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 (18151864) 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 (18211894) 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, first rigorous proof of the Central Limit Theorem, 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 straightline 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 (18211895) 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 higherdimensional 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 CayleyHamilton 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 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 lifelong 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 (18221901) 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 famous 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 intricate 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 (18231852) 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 has inspired many to find an algebra tutor with the hopes of learning his theories. 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 epochmaking mathematicians of history.
Georg Friedrich Bernhard Riemann (18261866) 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 nonEuclidean 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 RiemannRoch 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 nowheredifferentiable 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 partially proved an exact formula, albeit weirdly complicated. 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 minibio, 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 (18311879) 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 important kinetic theory of gases, explored elasticity, knot theory, soap bubbles, and more. He introduced the "Maxwell's Demon" as a thought experiment for thermodynamics; his paper "On Governors" effectively founded the field of cybernetics; he advanced the theory of color, and produced the first color photograph. One Professor said of him, "there is scarcely a single topic that he touched upon, which he did not change almost beyond recognition." Maxwell was also a poet.Maxwell did little of importance 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 electromagnetic force. Richard Feynman considered this the most significant event of the 19th century (though others might give equal billing to Darwin's theory of evolution). Maxwell, along with Einstein and Newton, was surely one of the three greatest physicists ever. He has been called the "Father of Modern Physics"; he ranks #24 on Hart's list of the Most Influential Persons in History.
Julius Wilhelm Richard Dedekind (18311916) 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 far ahead of his time, so Noether became famous as the creator of modern algebra; but she acknowledged her great predecessor, frequently saying "It is all already in Dedekind."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 was first to prove the CantorBernstein Theorem; he should thus be considered a coinventor of Cantor's set theory.
Rudolf Friedrich Alfred Clebsch (18331872) Germany
Alfred Clebsch began in mathematical physics, working in hydrodynamics and elasticity, but went on to become a pure mathematician of great brilliance and versatility. He started with novel results in analysis, but went on to make important advances to the invariant theory of Cayley and Sylvester (and Salmon and Aronhold), to the algebraic geometry and elliptic functions of Abel and Jacobi, and to the enumerative and projective geometries of Plücker. He was also one of the first to build on Riemann's innovations. Clebsch developed new notions, e.g. ClebschAronhold symbolic notation and 'connex'; and proved key theorems about cubic surfaces and other highdegree curves, and representations (bijections) between surfaces. Some of his work, e.g. ClebschGordan coefficients which are important in physics, was done in collaboration with Paul Gordan. For a while Clebsch was one of the top mathematicians in Germany, and founded an important journal, but he died young. He was a key teacher of Max Noether, Ferdinand Lindemann, Alexander Brill and Gottlob Frege. Clebsch's great influence is suggested by the fact that his name appeared as coauthor on a text published 60 years after his death.
Marie Ennemond Camille Jordan (18381921) 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. Jordan is especially famous for the Jordan Closed Curve Theorem of topology, a simple statement "obviously true" yet remarkably difficult to prove. Jordan also invented the notion of homotopy, proved the Jordan Decomposition Theorem of measure theory and the JordanHolder Theorem of group theory, invented the Jordan Canonical Forms of matrix theory, and supplied the first complete proof of Euler's Polyhedral Theorem, F+V = E+2. Some consider Jordan second only to Weierstrass among great 19thcentury teachers; his work inspired such mathematicians as Klein, Lie and Borel.
Marius Sophus Lie (18421899) Norway
Lie was twentyfive 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 (18421917) 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.
William Kingdon Clifford (18451879) England
Clifford was a versatile and talented mathematician who was among the first to appreciate the work of both Riemann and Grassmann. He found new connections between algebra, topology and nonEuclidean geometry. Combining Hamilton's quaternions, Grassmann's exterior algebra, and his own geometric intuition and understanding of physics, he developed biquaternions, and generalized this to geometric algebra, which paralleled work by Klein. In addition to developing theories, he also produced ingenious proofs; for example he was first to prove Miquel's nCircle Theorem, and did so with a purely geometric argument. Clifford is especially famous for anticipating, before Einstein, that gravitation could be modeled with a nonEuclidean space. He was a polymath; a talented teacher, noted philosopher, and outstanding athlete. With his singular genius, Clifford would probably have become one of the greatest mathematicians of his era had he not died at age thirtythree.
Georg Cantor (18451918) Russia, Germany
Cantor created Set Theory almost singlehandedly, defining cardinal numbers, wellordering, 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 Nspace.) 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 set theory laid the theoretical basis for the measure theory developed by Borel and Lebesgue. Cantor's invention of modern set theory is now considered one of the most important and creative achievements in modern mathematics.
Cantor demonstrated much breadth before turning his attention to set theory. He made advances in number theory; he proved an important theorem of Fourier theory; he gave the modern definition of irrational numbers; his Cantor set was the early inspiration for fractals. He was also an excellent violinist. Cantor once wrote "In mathematics the art of proposing a question must be held of higher value than solving it."
Friedrich Ludwig Gottlob Frege (18481925) Germany
Gottlob Frege developed the first complete and fully rigorous system of pure logic; his work has been called the greatest advance in logic since Aristotle. He introduced the essential notion of quantifiers; he distinguished terms from predicates, and simple predicates from 2ndlevel predicates. From his secondorder logic he defined numbers, and derived the axioms of arithmetic with what is now called Frege's Theorem. His work was largely underappreciated at the time, partly because of his clumsy notation, partly because his system was published with a flaw (Russell's antinomy). He and Cantor were the era's outstanding foundational theorists; unfortunately their relationship with each other became bitter. Despite all this, Frege's work influenced Peano, Russell, Wittgenstein and others; and he is now often called the greatest mathematical logician ever.Frege also did work in geometry and differential equations; and, in order to construct the real numbers with his set theory, proved an important new theorem of group theory. He was also an important philosopher, and wrote "Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician."
Ferdinand Georg Frobenius (18491917) Germany
Frobenius did significant work in a very broad range of mathematics, was an outstanding algorist, and had several successful students including Edmund Landau, Issai Schur, and Carl Siegel. 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 CayleyHamilton Theorem, and first to extend the Sylow Theorems to abstract groups. He anticipated the important and imaginative Prime Density Theorem, though he didn't prove its general case. Although he modestly left his name off the "CayleyHamilton Theorem," many lemmas and concepts are named after him, including Frobenius conjugacy class, Frobenius reciprocity, the FrobeniusSchur 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 (18491925) 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 nonEuclidean 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 popularlyfamous Klein's bottle were among several useful results from his new approaches to groups and higherdimensional geometries and equations. Klein did significant work in mathematical physics, e.g. writing about gyroscopes. He facilitated David Hilbert's early career, publishing his controversial Finite Basis 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 worldwide influence. Klein once wrote "... mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs."
Oliver Heaviside (18501925) 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 Grassmann; and demonstrated the usage of complex numbers for electromagnetic 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 (18501891) Russia
Sofia Kovalevskaya (aka Sonya Kowalevski; née KorvinKrukovskaya) was initially selftaught, 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 CauchyKovalevsky 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 19thcentury 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 postLagrange physical motion problem for which an "integrable" solution has been demonstrated, it remains an important textbook example. Kovalevskaya once wrote "It is impossible to be a mathematician without being a poet in soul." She was also a noted playwright.
Jules Henri Poincaré (18541912) France
Poincaré founded the theory of algebraic (combinatorial) topology, and is sometimes called the "Father of Topology" (a title also used for Euler and Brouwer). He also did brilliant work in several other areas of mathematics; he was one of the most creative mathematicians ever, and the greatest mathematician of the Constructivist ("intuitionist") style. Poincaré was clumsy and absentminded; like Galois, he was almost denied admission to French University, passing only because at age 17 he was already far too famous to flunk.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 3D surfaces in 4space). 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 = mc^{2}). 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 (18561922) 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. In addition to his Ergodic Theorem about such processes, theorems named after him include the GaussMarkov Theorem of statistics, the RieszMarkov Theorem of functional analysis, the Markov Brothers' Inequality in the theory of equations, and Markov's Theorem itself, which helps relate the theories of braids and knots to each other. Markov 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 (18581932) Italy
Giuseppe Peano is one of the most underappreciated of all great mathematicians. He started his career by proving a fundamental theorem in differential equations, developed practical solution methods for such equations, discovered a continuous spacefilling curve (then thought impossible), and laid the foundations of abstract operator theory. He also produced the best calculus textbook of his time, was first to produce a correct (nonparadoxical) definition of surface area, proved an important theorem about Dirichlet functions, did important work in topology, and much more. Much of his work was unappreciated and left for others to rediscover: he anticipated many of Borel's and Lebesgue's results in measure theory, and several concepts and theorems of analysis. He was the champion of counterexamples, and found flaws in published proofs of several important theorems.Most of the preceding work was done when Peano was quite young. Later he focused on mathematical foundations, and this is the work for which he is most famous. He developed rigorous definitions and axioms for set theory, as well as most of the notation of modern set theory. He was first to define arithmetic (and then the rest of mathematics) in terms of set theory. Peano was first to note that some proofs required an explicit Axiom of Choice (although it was Ernst Zermelo who explicitly formulated that Axiom a few years later).
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 singlemember set is not the same as its element; this fact is now taught in elementary school.
David Hilbert (18621943) Prussia, Germany
Hilbert, often considered the greatest mathematician of the 20th century, was unequaled in many fields of mathematics, including axiomatic theory, invariant theory, algebraic number theory, class field theory and functional analysis. He proved many new theorems, including the fundamental theorems of algebraic manifolds, and also discovered simpler proofs for older theorems. 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. His Nullstellensatz Theorem laid the foundation of algebraic geometry. 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 yearold system of Euclid. As a young Professor he proved his Finite Basis Theorem, now regarded as one of the most important results of general algebra. His mentor, Paul Gordan, had sought the proof for many years, and rejected Hilbert's proof as nonconstructive. Later, Hilbert produced the first constructive proof of the Finite Basis Theorem, as well. In number theory, he proved Waring's famous conjecture which is now known as the HilbertWaring 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 20thcentury 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 published the Einstein Field Equations independently of Einstein (though his writings make clear he treats this as strictly Einstein's invention).
Hermann Minkowski (18641909) 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 HasseMinkowski 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 (18651963) 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 called 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, by building on Riemann's work, proved by Hadamard and ValleePoussin. (Hadamard's proof is considered more elegant and useful than ValleePoussin'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 FixedPoint Theorem for arbitrarily many dimensions). Hadamard was also influential in promoting others' work: He is noted for his survey of Poincaré's work; his staunch defense of the Axiom of Choice led to the acceptance of Zermelo's work. Hadamard was a successful teacher, with André Weil, Maurice Fréchet, and others acknowledging him as key inspiration. Like many great mathematicians he emphasized the importance of intuition, writing "The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it."
Felix Hausdorff (18681942) Germany
Hausdorff had diverse interests: he composed music and wrote poetry, studied astronomy, wrote on philosophy, but eventually focused on mathematics, where he did important work in several fields including set theory, measure theory, functional analysis, and both algebraic and pointset topology. His studies in set theory led him to the Hausdorff Maximal Principle, and the Generalized Continuum Hypothesis; his concepts now called Hausdorff measure and Hausdorff dimension led to geometric measure theory and fractal geometry; his Hausdorff paradox led directly to the famous BanachTarski paradox; he introduced other seminal concepts, e.g. Hausdorff Distance. He also worked in analysis, solving the Hausdorff moment problem.As Jews in Hitler's Germany, Hausdorff and his wife committed suicide rather than submit to interment.
Élie Joseph Cartan (18691951) France
Cartan worked 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. This work was so significant that Cartan, rather than Lie, is considered 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. Cartan 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 embedding Riemann metrics, and fundamental theorems about symmetric Riemann spaces. He made a key contribution to Einstein's general relativity, based on what is now called RiemannCartan 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 (18711956) 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 HeineBorel Covering Theorem. He also did important work in several other fields of mathematics, including divergent series, quasianalytic functions, differential equations, number theory, complex analysis, theory of functions, geometry, probability theory, and game theory. Relating measure theory to probabilities, he introduced concepts like normal numbers and the BorelKolmogorov paradox. He also did work in relativity and the philosophy of science. He anticipated the concept of chaos, inspiring Poincaré. Borel combined great creativity with strong analytic power; however he was especially interested in applications, philosophy, and education, so didn't pursue the tedium of rigorous development and proof; for this reason his great importance as a theorist is often underestimated. 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 (18751941) 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 significant contributions in several other fields including complex analysis, topology, set theory, potential theory, and calculus of variations.
Godfrey Harold Hardy (18771947) 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 several 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 the results of this collaboration was the HardyRamanujan Formula for partition enumeration, which Hardy later used as a model to develop the HardyLittlewood Circle Method; Hardy first used this method to prove stronger versions of the HilbertWaring 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 (18791955) Germany, Switzerland, U.S.A.
Albert Einstein was unquestionably 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 = mc^{2}; 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. For example, in a 1917 paper he anticipated the principle of the laser. Also, he was coinventor of several devices, including a gyroscopic compass, hearing device, automatic camera and, most famously, the EinsteinSzilard refrigerator.Einstein is most famous for his Special and General Theories of Relativity, but he should be considered the key pioneer of Quantum Theory as well, drawing inferences from Planck's work that no one else dared to draw. Indeed it was his articulation of the quantum principle in a 1905 paper which has been called "the most revolutionary sentence written by a physicist of the twentieth century." Einstein's discovery of the photon in that paper led to his only Nobel Prize; years later, he was first to call attention to the "spooky" nature of quantum entanglement. Einstein was also first to call attention to a flaw in Weyl's earliest unified field theory.
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 [LeviCivita'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."
Oswald Veblen (18801960) U.S.A.
Oswald Veblen's first mathematical achievement was a novel system of axioms for geometry. He also worked in topology; projective geometry; differential geometry (where he was first to introduce the concept of differentiable manifold); ordinal theory (where he introduced the Veblen hierarchy); and mathematical physics where he worked with spinors and relativity. He developed a new theory of ballistics during World War I and helped plan the first American computer during World War II. His famous theorems include the VeblenYoung Theorem (an important algebraic fact about projective spaces); a proof of the Jordan Curve Theorem more rigorous than Jordan's; and Veblen's Theorem itself (a generalization of Euler's result about cycles in graphs). Veblen, a nephew of the famous economist Thorstein Veblen, was an important teacher; his famous students included Alonzo Church, John W. Alexander, Robert L. Moore, and J.H.C. Whitehead. He was also a key figure in establishing Princeton's Institute of Advanced Study.
Luitzen Egbertus Jan Brouwer (18811966) Holland
Brouwer is often considered the "Father of Topology;" among his important theorems were the Fixed Point Theorem, the "Hairy Ball" Theorem, the JordanBrouwer 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 measuretheory "paradoxes." Several great mathematicians, including Weyl, were inspired by Brouwer's work in topology.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 selfrighteous.
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 (18821935) Germany
Noether was an innovative researcher who was considered the greatest master of abstract algebra ever; her advances included 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 Betti homology groups led to algebraic topology, and thus 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. Several other important theorems are named after her, e.g. Noether's Normalization Lemma, which provided an important new proof of Hilbert's Nullstellensatz. Noether was an unusual and inspiring teacher; her successful students included Emil Artin, Max Deuring, Jacob Levitzki, etc. 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.
George David Birkhoff (18841984) U.S.A.
Birkhoff is one of the greatest nativeborn American mathematicians ever, and did important work in many fields. There are several significant theorems named after him: the BirkhoffGrothendieck Theorem is an important result about vector bundles; Birkhoff's Theorem is an important result in algebra; and Birkhoff's Ergodic Theorem is a key result in statistical mechanics which has since been applied to many other fields. His PoincaréBirkhoff Fixed Point Theorem is especially important in celestial mechanics, and led to instant worldwide fame: the great Poincaré had described it as most important, but had been unable to complete the proof. In algebraic graph theory, he invented Birkhoff's chromatic polynomial (while trying to solve the fourcolor problem); he proved a significant result in general relativity which implied the existence of black holes; he also worked in differential equations and number theory; he authored an important text on dynamical systems. Like several of the great mathematicians of that era, Birkhoff developed his own set of axioms for geometry; it is his axioms that are often found in today's high school texts. Birkhoff's intellectual interests went beyond mathematics; he once wrote "The transcendent importance of love and goodwill in all human relations is shown by their mighty beneficent effect upon the individual and society."
Hermann Klaus Hugo (Peter) Weyl (18851955) Germany, U.S.A.
Weyl studied under Hilbert and became one of the premier mathematicians of the 20th century. His discovery of gauge invariance and notion of Riemann surfaces form the basis of modern physics. He excelled at many fields including integral equations, harmonic analysis, analytic number theory, and the foundations of mathematics, but he is most respected for his revolutionary advances in geometric function theory (e.g., differentiable manifolds), the theory of compact groups (incl. representation theory), and theoretical physics (e.g., Weyl tensor, gauge field theory and invariance). 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 20thcentury physics: relativity, unified field theory and quantum mechanics. Because of his contributions to Schrödinger, many think the latter's famous result should be named SchrödingerWeyl Wave Equation.
Vladimir Vizgin wrote "To this day, Weyl's [unified field] theory astounds all in the depth of its ideas, its mathematical simplicity, and the elegance of its realization." 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 (18851977) England
John Littlewood was a very prolific researcher. (This fact is obscured somewhat in that many papers were coauthored 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 the smallest such example is probably much larger than a googol). Most of his results were too specialized to state here, e.g. his widelyapplied 4/3 Inequality which guarantees that certain bimeasures are finite, and which inspired one of Grothendieck's most famous results. 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 (18871920) India
Like Abel, Ramanujan was a selftaught 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 developed into 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 HardyRamanujan formula; when Ramanujan's notebooks were studied it was found he had anticipated their technique, but had deferred to his friend and mentor.)
In a letter from his deathbed, Ramanujan introduced his mysterious "mock theta functions", gave examples, and developed their properties. Much later these forms began to appear in disparate areas: combinatorics, the proof of Fermat's Last Theorem, and even knot theory and the theory of black holes. It was only recently, more than 80 years after Ramanujan's letter, that his conjectures about these functions were proven; solutions mathematicians had sought unsuccessfully were found among his examples.
Many of Ramanujan's results 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 oddlooking formula of Ramanujan is sometimes used to calculate π:
99^{2} / π = √8 ∑_{k=0,∞} ((4k)! (1103+26390 k) / (k!^{4} 396^{4k}))
George Pólya (18871985) Hungary
George Pólya (Pólya György) did significant work in several fields: complex analysis, probability, geometry, algebraic number theory, and combinatorics, but is most noted for his teaching How to Solve It, the craft of problem posing and proof. He is also especially famous for the Pólya Enumeration Theorem. Several other important theorems he proved include the PólyaVinogradov Inequality of number theory, the PólyaSzego Inequality of functional analysis, and the Pólya Inequality of measure theory. He introduced the HilbertPólya Conjecture that the Riemann Hypothesis might be a consequence of spectral theory; he introduced the famous "All horses are the same color" example of inductive fallacy; he named the Central Limit Theorem of statistics. Pólya was the "teacher par excellence": he wrote top books on multiple subjects; his successful students included John von Neumann; he directly inspired some of Escher's drawings. Having huge breadth and influence, Pólya has been called "the most influential mathematician of the 20th century."
Stefan Banach (18921945) Poland
Stefan Banach was a selftaught 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 (BanachSteinhaus) Theorem, the Open Mapping (BanachSchauder) Theorem, the Contraction Mapping (Banach fixedpoint) Theorem, and the HahnBanach Theorem. Many of these theorems are of practical value to modern physics; however he also proved the paradoxical BanachTarski 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 BanachMazur 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."
Norbert Wiener (18941964) U.S.A.
Norbert Wiener entered college at age 11, studying various sciences; he wrote a PhD dissertation at age 17 in philosophy of mathematics where he was first to show a definition of ordered pair as a set. He then did important work in several topics in applied mathematics, including stochastic processes (beginning with Brownian motion), potential theory, Fourier analysis, the WienerHopf decomposition useful for solving differential and integral equations, communication theory, cognitive science, and quantum theory. Many theorems and concepts are named after him, e.g the Wiener Filter used to reduce the error in noisy signals. His most important contribution to pure mathematics was his generalization of Fourier theory into generalized harmonic analysis, but he is most famous for his writings on feedback in control systems, for which he coined the new word, cybernetics. Wiener was first to relate information to thermodynamic entropy, and anticipated the theory of information attributed to Claude Shannon. He also designed an early analog computer. Although they differed dramatically in both personal and mathematical outlooks, he and John von Neumann were the two key pioneers (after Turing) in computer science. Wiener applied his cybernetics to draw conclusions about human society which, unfortunately, remain largely unheeded.
Carl Ludwig Siegel (18961981) Germany
Carl Siegel became famous when his doctoral dissertation established a key result in Diophantine approximations. He continued with contributions to several branches of analytic and algebraic number theory, including arithmetic geometry and quadratic forms. He also did seminal work with Riemann's zeta function, Dedekind's zeta functions, transcendence theory, discontinuous groups, the 3body problem in celestial mechanics, and symplectic geometry. In complex analysis he developed Siegel modular forms, which have wide application in math and physics. 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. Atle Selberg called him a "devastatingly impressive" mathematician who did things that "seemed impossible." André Weil declared that Siegel was the greatest mathematician of the first half of the 20th century.
Emil Artin (18981962) 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 Lseries, 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 prizewinning students include John Tate and Serge Lang. Artin also did work in physical sciences, and was an accomplished musician.
Alfred Tarski (19021983) Poland, U.S.A.
Alfred Tarski (born Alfred Tajtelbaum) was one of the greatest and most prolific logicians ever, but also made advances in set theory, measure theory, topology, algebra, group theory, computability theory, metamathematics, and geometry. He was also acclaimed as a teacher. Although he achieved fame at an early age with the BanachTarski Paradox, his greatest achievements were in formal logic. He wrote on the definition of truth, developed model theory, and investigated the completeness questions which also intrigued Gödel. He proved several important systems to be incomplete, but also established completeness results for real arithmetic and geometry. His most famous result may be Tarski's Undefinability Theorem, which is related to Gödel's Incompleteness Theorem but more powerful. Several other theorems, theories and paradoxes are named after Tarski including TarskiGrothendieck Set Theory, Tarski's FixedPoint Theorem of lattice theory (from which the famous CantorBernstein Theorem is a simple corollary), and a new derivation of the Axiom of Choice (which Lebesgue refused to publish because "an implication between two false propositions is of no interest"). Tarski's other notable accomplishments include his cylidrical algebra, ordinal algebra, universal algebra, and an elegant and novel axiomatic basis of geometry.
John von Neumann (19031957) 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 both pure 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 selfreproducing 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 areaconservation paradox related to the famous BanachTarski volumeconservation 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 highdimensional 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 FixedPoint Theorem to economics, becoming a major figure in that field. His contributions to computer science are many: in addition to coinventing the storedprogram computer, he was first to use pseudorandom number generation, finite element analysis, the mergesort algorithm, and a "biased coin" algorithm. By implementing widenumber software he joined several other great mathematicians (Archimedes, Apollonius, Liu Hui, Hipparchus, Madhava, and (by proxy) 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 (19031987) 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. He and his student Vladimir Arnold proved the surprising Superposition Theorem, which not only solved Hilbert's 13th Problem, but went far beyond it. He and Arnold also developed the "magnificent" KAM Theorem, which quantifies how strong a perturbation must be to upset a quasiperiodic dynamical system. Kolmogorov's axioms of probability are considered a partial solution of Hilbert's 6th Problem. 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 BorelKolmogorov Paradox, and the intriguing ZeroOne Law of "tail events" among random variables.
Kurt Gödel (19061978) 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 20thcentury 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 firstorder 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 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 (19061998) France, U.S.A.
Weil made profound contributions to several 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 some of Grothendieck's work. 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."
ShiingShen Chern (19112004) China, U.S.A.
ShiingShen 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 ChernWeil theory, the ChernSimons invariants, and especially for his brilliant generalization of the GaussBonnet 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 (19121954) Britain
Turing developed a new foundation for mathematics based on computation; he invented the abstract Turing machine, designed a "universal" version of such a 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), and was an early pioneer in the study of neural networks. 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 codebreaking 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 the Turing Patterns of morphogenesis which anticipated the discovery of BZ reactions. Turing's life ended tragically: charged with immorality and forced to undergo chemical castration, he apparently took his own life. With his outstanding depth and breadth, Alan Turing would qualify for our list in any event, but his decisive contribution to the war against Hitler gives him unusually strong historic importance.
Paul Erdös (19131996) 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 stillunproven intriguing conjectures, e.g. that 4/n = 1/x + 1/y + 1/z has positiveinteger 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 contributor was Atle Selberg) Hadamard's Prime Number Theorem itself. He also proved many new theorems, such as the ErdösSzekeres Theorem about monotone subsequences with its elegant (if trivial) pigeonholeprinciple proof.
Atle Selberg (19172007) 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 RankinSelberg method. These have inspired other mathematicians, e.g. contributing to Deligne's proof of Weil conjectures. Selberg is also famous for groundbreaking work on Riemann's Hypothesis, and the first "elementary" proof of the Prime Number Theorem.
JeanPierre Serre (1926) France
Serre did important work with spectral sequences and algebraic methods, revolutionizing the study of algebraic topology and algebraic geometry, especially 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, and contributed indirectly to the recent proof of Fermat's Last Theorem. His wide range of research areas also includes number theory, bundles, fibrations, padic modular forms, Galois representation theory, and more. 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 especially noted for his invention of the Theory of Schemes, and other methods to unify different branches of mathematics. He applied algebraic geometry to number theory; applied methods of topology to set theory; etc. 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 outstanding student Pierre Deligne were able to prove the Weil Conjectures. Grothendieck also developed the theory of sheafs, generalized the RiemannRoch Theorem to revolutionize Ktheory, developed Grothendieck categories, crystalline cohomology, infinitystacks 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. Among Grothendieck's famous results was his Fundamental Theorem in the Metric Theory of Tensor Products, which was inspired by Littlewood's proof of the 4/3 Inequality.Grothendieck's radical religious and political philosophies led him to retire from public life while still in his prime, but he is widely regarded as the greatest mathematician of the 20th century, and indeed one of the greatest 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 Ktheory and the AtiyahSinger Index Theorem. This Index Theorem is considered one of the most farreaching theorems ever, subsuming famous old results (Déscartes' total angular defect, Euler's topological characteristic), important 19thcentury theorems (GaussBonnet, RiemannRoch), and incorporating important work by Weil and especially ShiingShen Chern. It is a key to the study of highdimension spaces, differential geometry, and equation solving. Several other key results are named after Atiyah, e.g. the AtiyahBott FixedPoint Theorem, the AtiyahSegal Completion Theorem, and the AtiyahHirzebruch spectral sequence. Atiyah's work developed important connections not only between topology and analysis, but with modern physics; Atiyah himself has been a key figure in the development of string theory. This work, and Atiyahinspired 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 Willard Milnor (1931) U.S.A.
Milnor has made major advances in topology (especially differential topology), algebraic geometry, and dynamical systems. He discovered Milnor maps (related to fiber bundles); important theorems in knot theory; the Duality Theorem for Reidemeister Torsion; the Milnor Attractors of dynamical systems; a new elegant proof of Brouwer's "Hairy Ball" Theorem; and much more. He is especially famous for two counterexamples which each revolutionized topology. His "exotic" 7dimensional hyperspheres gave the first examples of homeomorphic manifolds that were not also diffeomorphic, and developed the fields of differential topology and surgery theory. Milnor invented certain highdimensional polyhedra to disprove the Hauptvermutung ("main conjecture") of geometric topology. While most famous for his exotic counterexamples, his revolutionary insights into dynamical systems have important value to practical applied mathematics. Although Milnor has been called the "Wizard of Higher Dimensions," his work in dynamics began with novel insights into very lowdimensional systems.As Fields, Presidential and (twice) Putnam Medalist, as well as winner of the Abel, Wolf and two Steele Prizes; Milnor can be considered the most "decorated" mathematician of the modern era.
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 thenknown 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 straightedge construction of the regular heptagon (a feat also achieved by Alhazen and Archimedes), a nowherecontinuous function that has the Intermediate Value property, the Conway box function, the aperiodic pinwheel tiling, a representation of symmetric polyhedra, the silly but elegant Fractran programming language, his chainedarrow 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.Conway's great creativity and breadth certainly make him one of the greatest living mathematicians. Conway has won the Nemmers Prize in Mathematics, and was first winner of the Pólya Prize.