Two Hundred Greatest Mathematicians of All Time

Isaac Newton |
Archimedes |
Carl Gauss |
Leonhard Euler |
Bernhard Riemann |
Henri Poincaré |

J.-L. Lagrange |
Euclid |
David Hilbert |
G.W. Leibniz |
Alex. Grothendieck |
Pierre de Fermat |

**The Greatest
Mathematicians of the Past**

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**.

(Fifty additional names to change this List from 'Greatest of the Past'
to 'Greatest of All Time appear *after* the first 150 names.)

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In compiling this list, I've considered contributions
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- Apollonius of Perga
- Charles Hermite
- Diophantus of Alexandria
- Pierre-Simon Laplace
- Carl Ludwig Siegel
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- Stefan Banach
- Jacques Hadamard
- Giuseppe Peano
- Panini of Shalatula
- André Weil
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Einstein, Galileo, Maxwell, Aristotle and Cardano are among the greatest applied mathematicians in history, but lack the importance as pure mathematicians to qualify for The Top 70. Nevertheless I'd want to include them in any longer list, so I've tucked these ambiguous cases into the #71-#75 slots. | ||||

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- George D. Birkhoff
- Omar al-Khayyám
- Adrien M. Legendre
- Pappus of Alexandria
- Thales of Miletus
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The Top 100 represent a list of
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- John T. Tate
- Alfred Clebsch
- Abu Rayhan al-Biruni
- Ptolemy of Alexandria
- Alexis C. Clairaut
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(As mentioned above, ranks #146-150 are reserved, along with #101-105, for the ten mathematicians born after 1930 who "probably belong in the Top 150.") ... But I couldn't stop! :-) I've expanded the List to 200 names, mostly with 20th-century mathematicians. I don't try to "rank" these final fifty (though having two tiers is convenient). I've not written mini-bios for most of these, but have linked (with a slightly different color of link) to the excellent biographies at Wikipedia or the MacTutor site. Please note that, although I've placed them here after the "Top 150" some of these recent mathematicians may belong among the Top 100 or the Top 150. | ||||

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Al-Farisi Al-Karaji Aleksandrov Apastambha Babbage Bháskara.I Chang Copernicus DeMoivre L.Gersonides Gibbs Heron Ibn.Sinan Madhava Menelaus Plato J.Robinson Russell Seki Wittgenstein Zeno

Click for a discussion of certain omissions. Please send me e-mail 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 e-mail and tell me!

Following are the top mathematicians in
chronological (birth-year) order.
(By the way, the ranking assigned to a mathematician will appear if you
place the cursor atop the name at the top of his mini-bio.):

Little is known of the earliest mathematics, but the famousIshango Bonefrom Early Stone-Age Africa has tally marks suggesting arithmetic. The markings include six prime numbers (5, 7, 11, 13, 17, 19) in order, though this is probably coincidence.The advanced artifacts of Egypt's Old Kingdom and the Indus-Harrapa civilization imply strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where 4500-year old clay tablets show multiplication and division problems; the first abacus may be about this old. By 3600 years ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, and even logarithms and trig functions, using a primitive place-value system (in base 60, not 10). Babylonians were familiar with the Pythagorean Theorem, solutions to 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. The Greeks borrowed from Babylonian mathematics, which was the most advanced of any before the Greeks; but there is no ancient Babylonian mathematician whose name is known.

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, though little written evidence survives. Babylon was much more advanced than Egypt at arithmetic and algebra; this was probably due, at least in part, to their place-value system. But although their base-60 system survives (e.g. in the division of hours and degrees into minutes and seconds) the Babylonian notation, which used the equivalent of IIIIII XXXXXIIIIIII XXXXIII to denote 417+43/60, was unwieldy compared to the "ten digits of the Hindus." (In 2016 historians were surprised to decode ancient Babylonian texts and find very sophisticated astronomical calculations of Jupiter's orbit.)

The Egyptians used the approximation

π ≈ (4/3)(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^{4}√2, and the ancient Babylonians an ever better√2, neither of these ancient cultures achieved aπapproximation as good as Egypt's, or better thanπ ≈ 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 then-best approximation to

π. Apastambha did work summarized below; other early Vedic mathematicians solved quadratic and simultaneous equations.Other early cultures also developed some mathematics. The ancient Mayans apparently had a place-value system with zero before the Hindus did; Aztec architecture implies practical geometry skills. Ancient China certainly developed mathematics, in fact the first known proof of the Pythagorean Theorem is found in a Chinese book (

Zhoubi Suanjing) whichmighthave been written about 1000 BC.

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Thales of Miletus__
(ca 624 - 546 BC) Greek domain

Thales was the Chief of the "Seven Sages" of ancient Greece, and has been called the "Father of Science," the "Founder of Abstract Geometry," and the "First Philosopher." Thales is believed to have studied mathematics under Egyptians, who in turn were aware of much older mathematics from Mesopotamia. Thales may have invented the notion of compass-and-straightedge construction. Several fundamental theorems about triangles are attributed to Thales, including the law of similar triangles (which Thales used famously to calculate the height of the Great Pyramid) and "Thales' Theorem" itself: the fact that any angle inscribed in a semicircle is a right angle. (The other "theorems" were probably more like well-known axioms, but Thales proved Thales' Theorem using two of his other theorems; it is said that Thales then sacrificed an ox to celebrate what might have been the first mathematical proof in Greece.) Thales noted that, given a line segment of lengthx, a segment of lengthx/kcan be constructed by first constructing a segment of lengthkx.Thales was also an astronomer; he invented the 365-day calendar, introduced the use of Ursa Minor for finding North, invented the gnomonic map projection (the first of many methods known today to map (part of) the surface of a sphere to a plane, and is the first person believed to have correctly predicted a solar eclipse. His theories of physics would seem quaint today, but he seems to have been the first to describe magnetism and static electricity. Aristotle said, "To Thales the primary question was not what do we know, but how do we know it." Thales was also a politician, ethicist, and military strategist. It is said he once leased all available olive presses after predicting a good olive season; he did this not for the wealth itself, but as a demonstration of the use of intelligence in business. Thales' writings have not survived and are known only second-hand. Since his famous theorems of geometry were probably already known in ancient Babylon, his importance derives from imparting the notions of mathematical proof and the scientific method to ancient Greeks.

Thales' student and successor was Anaximander, who is often called the "First Scientist" instead of Thales: his theories were more firmly based on experimentation and logic, while Thales still relied on some animistic interpretations. Anaximander is famous for astronomy, cartography and sundials, and also enunciated a theory of evolution, that land species somehow developed from primordial fish! Anaximander's most famous student, in turn, was Pythagoras. (The methods of Thales and Pythagoras led to the schools of Plato and Euclid, an intellectual blossoming 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.)

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Apastambha __
(ca 630-560 BC) India

TheDharmasutracomposed by Apastambha contains mensuration techniques, novel geometric construction techniques, a method of elementary algebra, and what may be an early 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.

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Pythagoras of Samos__
(ca 578-505 BC) Greek domain

Pythagoras, who is sometimes called the "First Philosopher," studied under Anaximander, Egyptians, Babylonians, and the mystic Pherekydes (from whom Pythagoras acquired a belief in reincarnation); he became the most influential of early Greek mathematicians. He is credited with being first to use axioms and deductive proofs, so his influence on Plato and Euclid may be enormous; he is generally credited with much of Books I and II of Euclid'sElements. 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 seems to have been the first man to realize 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 wordsphilosophyandmathematicsare said to have been coined by Pythagoras.Despite Pythagoras' historical importance I may have ranked him too high: many results of the Pythagoreans were due to his students; none of their writings survive; and what is known is reported second-hand, and possibly exaggerated, by Plato and others. Some ideas attributed to him were probably first enunciated by successors like Parmenides of Elea (ca 515-440 BC). Archaeologists now believe that he was not first to invent the diatonic scale: Here is a diatonic-scale song from Ugarit which predates Pythagoras by eight centuries.

Pythagoras' students included Hippasus of Metapontum, the famous anatomist and physician Alcmaeon, Milo of Croton, and Croton's daughter Theano (who may have been Pythagoras's wife). The term

Pythagoreanwas 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.) Given any numbers

aandbthe Pythagoreans were aware of the three distinct means:(a+b)/2(arithmetic mean),√(ab)(geometric mean), and2ab/(a+b)(harmonic mean).The Pythagorean Theorem was known long before Pythagoras, but he was often credited (before discovery of an ancient Chinese text) with the first

proof. He may have discovered the simple parametric form of primitive Pythagorean triplets(xx-yy, 2xy, xx+yy), although the first explicit mention of this may be in Euclid'sElements. Other discoveries of the Pythagorean school include the construction of the regular pentagon, concepts of perfect and amicable numbers, polygonal numbers, golden ratio (attributed to Theano), three of 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 428-348 BC), Theaetetus (see below), and Archytas (see below). These early Greeks ushered in a Golden Age of Mathematics and Philosophy unequaled in Europe until the Renaissance. The emphasis was on pure, rather than practical, mathematics. Plato (who ranks #40 on Michael Hart's famous list of the Most Influential Persons in History) decreed that his scholars should do geometric construction solely with compass and straight-edge rather than with "carpenter's tools" like rulers and protractors.

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Panini (of Shalatula)__
(ca 520-460 BC) Gandhara (India)

Panini's great accomplishment was his study of the Sanskrit language, especially in his textAshtadhyayi. Although this work might be considered the very first study of linguistics or grammar, it used a non-obvious 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 20th-century 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;" his grammar "one of the greatest monuments of human intelligence."

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Zeno of Elea__
(ca 495-435 BC) Greek domain

Zeno, a student of Parmenides, had great fame in ancient Greece. This fame, which continues to the present-day, is largely due to his paradoxes of infinitesimals, e.g. his argument that Achilles can never catch the tortoise (whenever Achilles arrives at the tortoise's last position, the tortoise has moved on). Although some regard these paradoxes as simple fallacies, they have been contemplated for many centuries. It is due to these paradoxes that the use of infinitesimals, which provides the basis for mathematical analysis, has been regarded as a non-rigorous heuristic and is finally viewed as sound only after the work of the great 19th-century rigorists, Dedekind and Weierstrass. Zeno's Arrow Paradox (at any single instant an arrow is at a fixed position, so where does its motion come from?) has lent its name to the Quantum Zeno Effect, a paradox of quantum physics.Eubulides of Miletus was another ancient Greek famous for paradoxes, e.g. "This statement is a lie" -- the sort of inconsistency later used in proofs by Gödel and Turing.

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Hippocrates of Chios__
(ca 470-410 BC) Greek domain

Hippocrates (no known relation to Hippocrates of Cos, the famous physician) wrote his ownElementsmore than a century before Euclid. Only fragments survive but it apparently used axiomatic-based proofs similar to Euclid's and contains many of the same theorems. Hippocrates is said to have invented thereductio ad absurdemproof method. Hippocrates is most famous for his work on the three ancient geometric quandaries: his work on cube-doubling (theDelian Problem) laid the groundwork for successful efforts by Archytas and others; and some claim Hippocrates was first to trisect the general angle. His circle quadrature was of course ultimately unsuccessful but he did prove ingenious theorems about "lunes" (crescent-shaped circle fragments). For example, the area of any right triangle is equal to the sum of the areas of the two lunes formed when semi-circles are drawn on each of the three edges of the triangle. 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 collapsing compass and unmarkable straightedge. There are ingenious solutions available with other tools. Construction of the regular heptagon is another such task, with solutions published by four of the men on this List: Thabit, Alhazen, Vieta, Conway.)

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Archytas of Tarentum__
(ca 420-350 BC) Greek domain

Archytas was an important statesman as well as philosopher. He studied under Philolaus of Croton, was a friend of Plato, and tutored Eudoxus. 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 magnificentMechanical Problemsattributed 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. He appears on Cardano's List of 12 Greatest Geniuses. (Euclid, Aristotle, Archimedes, Apollonius, Ptolemy, and the physician Galen of Pergamum are the other Greeks on that List.) Archytas' most famous mathematical achievement was "doubling the cube" (constructing a line segment larger than another by the factor cube-root of two). Although others solved the problem with other techniques, Archytas' solution for cube doubling was astounding because it wasn't achieved in the plane, but involved the intersection of three-dimensional bodies. This construction (which introduced the

Archytas Curve) has been called "a tour de force of the spatial imagination." He invented the termharmonic meanand worked with geometric means as well (proving that consecutive integers never have rational geometric mean). He was a true polymath: he advanced the theory of music far beyond Pythagoras; studied sound, optics and cosmology; invented the pulley (and a rattle to occupy infants); wrote about the lever; developed the curriculum called quadrivium; and is supposed to have built a steam-powered wooden bird which flew for 200 meters. Archytas is sometimes called the "Father of Mathematical Mechanics."Some scholars think Pythagoras and Thales are partly mythical. If we take that view, Archytas (and Hippocrates) should be promoted in this list.

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Theaetetus__ of Athens
(417-369 BC) Greece

Theaetetus is presumed to be the true author of Books X and XIII of Euclid'sElements, as well as some work attributed to Eudoxus. He was considered one of the brightest of Greek mathematicians, and is the central character in two of Plato's Dialogs. It was Theaetetus who discovered the final two of the five "Platonic solids" and proved that there were no more. He may have been first to note that the square root ofanyinteger, if not itself an integer, must be irrational. (The case √2 is attributed to a student of Pythagoras.)

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Eudoxus of Cnidus__
(408-355 BC) Greek domain

Eudoxus journeyed widely for his education, despite that he was not wealthy, studying mathematics with Archytas in Tarentum, medicine with Philiston in Sicily, philosophy with Plato in Athens, continuing his mathematics study in Egypt, touring the Eastern Mediterranean with his own students and finally returned to Cnidus where he established himself as astronomer, physician, and ethicist. What is known of him is second-hand, through the writings of Euclid and others, but he was one of the most creative mathematicians of the ancient world.Many of the theorems in Euclid's

Elementswere 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; Archimedes credits Eudoxus with inventing a principle eventually called theAxiom of Archimedes: it avoids Zeno's paradoxes by, in effect, forbidding infinities and infinitesimals. Eudoxus' work with irrational numbers, infinitesimals and limits eventually inspired masters like Dedekind. Eudoxus also introduced anAxiom of Continuity; he was a pioneer in solid geometry; and he developed his own solution to the Delian cube-doubling problem. Eudoxus was the first great mathematical astronomer; he developed the complicated ancient theory of planetary orbits; and may have invented the astrolabe. He may have invented the 365.25-day calendar based on leap years, though it remained for Julius Caesar to popularize it. (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.") One of Eudoxus' students was Menaechmus, who was first to describe the conic sections and used them to devise a non-Platonic solution to the cube-doubling problem (and perhaps the circle-squaring problem as well).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

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."π

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Aristotle of Stagira__
(384-322 BC) Macedonia

Aristotle is considered the greatest scientist of the ancient world, and the most influential philosopher and logician ever; he ranks #13 on Michael Hart's list of the Most Influential Persons in History. His science was a standard curriculum for almost 2000 years. Although the physical sciences couldn't advance until the discoveries by great men like Newton and Lavoisier, Aristotle's work in the biological sciences was superb, and served as paradigm until modern times.Aristotle 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.

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Euclid of Alexandria__
(ca 322-275 BC) Greece/Egypt

Euclid of Alexandria (not to be confused with Socrates' student, Euclid of Megara, who lived a century earlier), directed the school of mathematics at the great university of Alexandria. Little else is known for certain about his life, but several very important mathematical achievements are credited to him. He was the first to prove that there are infinitely many prime numbers; he stated and proved the Unique Factorization Theorem; and he devisedEuclid's algorithmfor computing gcd. He introduced the Mersenne primes and observed that(Mis always perfect (in the sense of Pythagoras) if^{2}+M)/2Mis Mersenne. (The converse, that any even perfect number has such a corresponding Mersenne prime, was tackled by Alhazen and proven by Euler.) His books contain many famous theorems, though much of theElementswas due to predecessors like Pythagoras (most of Books I and II), Hippocrates (Book III), Theodorus, Eudoxus (Book V), Archytas (perhaps Book VIII) and Theaetetus. Book I starts with an elegant proof that rigid-compass constructions can be implemented with a collapsing compass. (Given A, B, C, find CF = AB by first constructing equilateral triangle ACD; then use the compass to find E on AD with AE = AB; and finally find F on DC with DF = DE.) 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 areThe 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 textbookThe 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 Elementsintroduced 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."

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Archimedes of Syracuse__
(287-212 BC) Greek domain

Archimedes is universally acknowledged to be the greatest of ancient mathematicians. He studied at Euclid's school (probably after Euclid's death), but his work far surpassed, and even leapfrogged, the works of Euclid. (For example, some of Euclid's 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√3indicates that he'd partially anticipated the method of continued fractions. He developed a recursive method of representing large integers, and was first to note the law of exponents,10. He found a method to trisect an arbitrary angle (using a^{a}·10^{b}= 10^{a+b}markablestraightedge — 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 hisPrinciple of the Lever, the other using a geometric series. Some of Archimedes' work survives only because Thabit ibn Qurra translated the otherwise-lostBook of Lemmas; it contains the angle-trisection 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' angle-trisection method.) Other discoveries known only second-hand include theArchimedean semiregular solidsreported by Pappus, and the Broken-Chord Theorem reported by Alberuni.Archimedes and Newton might be the two best geometers ever, but although each produced ingenious geometric proofs, often they used non-rigorous calculus to

discoverresults, 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: he laid a mathematical foundation for the principles of leverage; discovered the first law of hydrostatics; and invented the compound pulley, the hydraulic screwpump (called Archimedes' screw), a miniature planetarium, and war machines (e.g. catapult and ship-burning mirrors). (Some of these inventions may predate Archimedes. On the other hand, some scholars attribute theAntikythera mechanismto Archimedes or his inspiration.) His books includeFloating Bodies,Spirals,The Sand Reckoner,Measurement of the Circle,Sphere and Cylinder,Plane Equilibriums,Conoids and Spheroids,Quadrature of Parabola, various now-lost works cited by Pappus or others, possiblyThe Book of Lemmas, and (discovered only recently, and often called his most important work)The Method. He developed theStomachionpuzzle (and solved a difficult enumeration problem involving it); other famous gems includeThe Cattle-Problem.The Book of Lemmascontains various geometric gems ("the Salinon," "the Shoemaker's Knife", etc.) and is credited to Archimedes by Thabit ibn Qurra but the attribution is disputed.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,

πis often calledArchimedes' constant. His approximation223/71 < π < 22/7was the best of his day. (Apollonius soon surpassed it, but by using Archimedes' method.) Archimedes' Equiarea Map Theorem asserts that a sphere and its enclosing cylinder have equal surface area (as do the figures' truncations). Archimedes also proved that the volume of that sphere is two-thirds the volume of the cylinder. He requested that a representation of such a sphere and cylinder be inscribed on his tomb.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 MethodandFloating Bodies) are preserved only on a palimpsest rediscovered in 1906 and mostly deciphered only after 1998. Ideas unique to that work are an anticipation of Riemann integration, calculating the volume of a cylindrical wedge (previously first attributed to Kepler); along with Oresme and Galileo he was among the few to comment on the "equinumerosity paradox" (the fact that are as many perfect squares as integers). Although Euler and Newton may have been the most important mathematicians, and Gauss, Weierstrass and Riemann the greatest theorem provers, 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 simplytoofar ahead of his time to have great historical significance. (Some think the Scientific Revolution would have begun sooner hadThe Methodbeen discovered four or five centuries earlier. You can read a 1912 translation of parts ofThe Methodon-line.)

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Eratosthenes of Cyrene__
(276-194 BC) Greek domain

Eratosthenes was one of the greatest polymaths; he is called the Father of Geography, was Chief Librarian at Alexandria, was a poet, music theorist, astronomer (e.g. calculating the Earth's diameter, distance to the Sun, etc.), mechanical engineer (anticipating laws of elasticity, etc.), and was an outstanding mathematician. He is famous for his prime number Sieve, but more impressive was his work on the cube-doubling problem which he related to the design of siege weapons (catapults) where a cube-root calculation is needed.Eratosthenes had the nickname

Beta; he was a master of several fields, but was only second-best of his time. His better was also his good friend: Archimedes of Syracuse dedicatedThe Methodto Eratosthenes.

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Apollonius of Perga__
(262-190 BC) Greek domain

Apollonius Pergaeus, called "The Great Geometer," is sometimes considered the second greatest of ancient Greek mathematicians. (Euclid, Eudoxus and Archytas are other candidates for this honor.) His writings on conic sections have been studied until modern times; he developed methods for normals and curvature. (He is often credited with inventing the names for parabola, hyperbola and ellipse; but these shapes were previously described by Menaechmus, and their names may also predate Apollonius.) 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." The following generalization of the Pythagorean Theorem, where M is the midpoint of BC, is called Apollonius' Theorem:AB.^{ 2}+ AC^{ 2}= 2(AM^{ 2}+ BM^{ 2})Many of his works have survived only in a fragmentary form, and the proofs were completely lost. Most famous was the

Problem of Apollonius, which is to find a circle tangent to three objects, with the objects being points, lines, or circles, in any combination. Constructing the eight circles each tangent to three other circles is especially challenging, but just finding the two circles containing two given points and tangent to a given line is a serious challenge. Vieta was renowned for discovering methods for all ten cases of this Problem. Other great mathematicians who have enjoyed reconstructing Apollonius' lost theorems include Fermat, Pascal, Newton, Euler, Poncelet and Gauss.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 Descartes, 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."

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Chang Tshang __
(ca 200-142 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 (as evidenced by the ancientZhoubi Suanjing), and innovations continued for centuries after Han, the textbookNine Chapters on the Mathematical Arthas special importance.Nine Chapters(known in Chinese asJiu Zhang Suan ShuorChiu 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 mini-bio; other famous contributors are Jing Fang and Zhang Heng.)

Nine Chapterswas 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 220-280). Although Liu Hui mentions Chang's skill, it isn't clear Chang had the mathematical genius to qualify for this list, but he would still be a strong candidate due to his 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 likeNine 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.)

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Hipparchus of Nicaea and Rhodes__
(ca 190-127 BC) Greek domain

Ptolemy may be the most famous astronomer before Copernicus, but he borrowed heavily from Hipparchus, who should thus be considered (along with Galileo and Edwin Hubble) to be one of the three greatest astronomers ever. Careful study of theerrorsin 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 6th-century 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 circle-conformal stereographic and orthographic map projections which carry 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 Babylonian astronomers.) Hipparchus' work implies a better approximation to

πthan that of Apollonius, perhaps it wasπ ≈ 377/120as Ptolemy used.The

Antikythera mechanismis an astronomical clock considered amazing for its time. It may have built about the time of Hipparchus' death, but lost after a few decades (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. (Recent studies suggest that the mechanism may have been designed by Archimedes.)

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Menelaus
of Alexandria__
(ca 70-135) Egypt, Rome

Menelaus wrote several books on geometry and trigonometry, mostly lost except for his works on solid geometry. His work was cited by Ptolemy, Pappus, and Thabit; especially the Theorem of Menelaus itself which is a fundamental and difficult theorem very useful in projective geometry. He also contributed much to spherical trigonometry. Disdaining indirect proofs (anticipating later-day constructivists) Menelaus found new, more fruitful proofs for several of Euclid's results.

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Tiberius(?) Claudius Ptolemaeus
of Alexandria__
(ca 90-168) Egypt (in Greco-Roman domain)

Ptolemy, the Librarian of Alexandria, 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·ADif and only ifABCDis a cyclic quadrilateral). This theorem has many useful corollaries; it was frequently applied in Copernicus' work. Ptolemy also wrote on trigonometry, optics, geography, map projections, and astrology; but is most famous for his astronomy, where he perfected the geocentric model of planetary motions. For this work, Cardano included Ptolemy on his List of 12 Greatest Geniuses, but removed him from the list after learning of Copernicus' discovery. Interestingly, Ptolemy wrote that the fixed point in a model of planetary motion was arbitrary, but rejected the Earth spinning on its axis since he thought this would lead to powerful winds. Ptolemy discussed and tabulated the 'equation of time,' documenting the irregular apparent motion of the Sun. (It took fifteen centuries before this irregularity was correctly attributed to Earth's elliptical orbit.)

The mystery of celestial motions directed scientific inquiry for thousands of years. With the notable exception of the Pythagorean Philolaus of Croton, thinkers generally assumed that the Earth was the center of the universe, but this made it very difficult to explain the orbits of the other planets. This 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, introducing 'equants' to further fine-tune the orbital speeds; this model was the standard for 14 centuries. While some Greeks, notably Aristarchus and Seleucus of Seleucia (and perhaps also Heraclides of Pontus or ancient Egyptians), proposed heliocentric models, these were rejected because there was no parallax among stars. (Aristarchus guessed that the stars were at an almost unimaginable distance, explaining the lack of parallax. Aristarchus would be almost unknown except that Archimedes mentions, and assumes, Aristarchus' heliocentrism inThe Sand Reckoner. I suspect that Archimedes accepted heliocentrism, but thought saying so openly would distract from his work. Hipparchus was another ancient Greek who considered heliocentrism but, because he never guessed that orbits were ellipses rather than cascaded circles, was unable to come up with a heliocentric model that fit his data.) Aryabhata, Alhazen, Alberuni, Omar Khayyám, (perhaps some other Islamic mathematicians like al-Tusi), and Regiomontanus are other great pre-Copernican mathematicians who may have accepted the possibility of heliocentrism.The great skill demonstrated by Ptolemy and his predecessors in developing their complex geocentric cosmology may have

set backscience since in fact 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. (Ptolemy's model predicted phases, but timed quite differently from Galileo's observations.)Since the planets move without friction, their motions offer a pure view of the Laws of Motion; this is one reason that the heliocentric breakthroughs of Copernicus, Kepler and Newton triggered the advances in mathematical physics which led to the Scientific Revolution.

Heliocentrism offered an even more key understandingthat lead to massive change in scientific thought. For Ptolemy and other geocentrists, the "fixed" stars were just lights on a sphererotating around the earth, but after the Copernican Revolution the fixed stars were understood to be immensely far away; this made it possible to imagine that they were themselves suns, perhaps with planets of their own. (Nicole Oresme and Nicholas of Cusa were pre-Copernican thinkers who wrote on both the geocentric question and the possibility of other worlds.) The Copernican perspective led Giordano Bruno and Galileo to posit a single common set of physical laws which ruled both on Earth and in the Heavens. (It was this, rather than just the happenstance of planetary orbits, that eventually most outraged the Roman Church.... And we're getting ahead of our story: Copernicus, Bruno, Galileo and Kepler lived 14 centuries after Ptolemy.)

Liu Hui made major improvements to Chang's influential textbookNine 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's-play for any of them, Archimedes, Apollonius, and Hipparchus had all improved precision of

π's estimate. It seems fitting that Liu Hui did join that select company of record setters: He developed a recurrence formula for regular polygons allowing arbitrarily-close approximations forπ. He also devised an interpolation formula to simplify that calculation; this yielded the "good-enough" value 3.1416, which is still taught today in primary schools. (Liu's successors in China included Zu Chongzhi, whodiddetermine sphere's volume, and whose approximation forπheld the accuracy record for nine centuries.)

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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 rational-number solutions to his problems rather than just integers, and was aware of results like the Brahmagupta-Fibonacci Identity; for these reasons he is often called the "Father of Algebra." His work, however, may seem quite limited to a modern eye: his methods were not generalized, he knew nothing of negative numbers, and, though he often dealt with quadratic equations, never seems to have commented on their second solution. His notation, clumsy as it was, was used for many centuries. (The shorthandxfor "x cubed" was not invented until Descartes.)^{3}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,bwitha>b, there exist positive rationalsc,dsuch thata." (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.")^{3}-b^{3}= c^{3}+d^{3}

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Pappus of Alexandria__
(ca 300) Egypt, Greece

Pappus, along with Diophantus, may have been one of the two greatest Western mathematicians during the 13 centuries that separated Hipparchus 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' best and most original result, and the one which gave him most pride, may be the Pappus Centroid theorems (fundamental, difficult and powerful theorems of solid geometry later rediscovered by Paul Guldin). His other ingenious geometric theorems include Desargues' Homology Theorem (which Pappus attributes to Euclid), an early form of Pascal's Hexagram Theorem, called Pappus' Hexagon Theorem and related to a fundamental theorem: Two projective pencils can always be brought into a perspective position. For these theorems, Pappus is sometimes called the "Father of Projective Geometry." Pappus also demonstrated how to perform angle trisection and cube doubling if one can use mechanical curves like a conchoid or hyperbola. He stated (but didn't prove) the

Isoperimetric Theorem, also writing "Bees know this fact which is useful to them, that the hexagon ... will hold more honey for the same material than [a square or triangle]." (That a honeycomb partition minimizes material for an equal-area partitioning was finally proved in 1999 by Thomas Hales, who also proved the related Kepler Conjecture.) Pappus stated, but did not fully solve, theProblem of Pappuswhich, 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 Descartes 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.

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Mathematicians after Classical Greece__

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

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Decimal system -- from India? China?? Persia???__

Laplace called the decimal system "a profound and important idea [given by India] which appears so simple to us now that we ignore its true merit ... in the first rank of useful inventions [but] it escaped the genius of Archimedes and Apollonius." But even after Fibonacci introduced the system to Europe, it was another 400 years before it came into common use.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

sameas 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

didhave place-value notationwithzero symbols, but neither qualify as inventing a base-10 decimal system: Persia used the base-60 Babylonian system; Mayans used base-20. (Another difference is that the Hindus had nine distinct digit symbols to go with their zero, while earlier place-value systems built up from just two symbols: 1 and either 5 or 10.) The Old Kingdom Egyptians did use a base-ten system, but it was similar to that of Greece and Vedic India: 1, 10, 100 were depicted as separate symbols.Conclusion: The decimal place-value 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!

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

Indian mathematicians excelled for thousands of years, and eventually even developed advanced techniques like Taylor series before Europeans did, but they are denied credit because of Western ascendancy. Among the Hindu mathematicians, 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 al-Khowârizmi (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, e.g. the nature of moonlight, and concept of sidereal year; 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 al-Biruni. Aryabhata is said to have introduced the constante. He usedπ ≈ 3.1416; it is unclear whether he discovered this independently or borrowed it from Liu Hui of China. Although it was first discovered by Nicomachus three centuries earlier, Aryabhata is famous for the identity

Σ (k^{3}) = (Σ k)^{2}Some of Aryabhata's achievements, e.g. an excellent approximation to the sine function, are known only from the writings of Bhaskara I, who wrote: "Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world."

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

No one person gets unique credit for the invention of the decimal system but Brahmagupta's textbook

Brahmasphutasiddhantawas 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+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)

Another famous Brahmagupta theorem dealing with such quadrilaterals can be phrased "In a circle, if the chords AB and CD are perpendicular and intersect at E, then the line from E which bisects AC will be perpendicular to BD." 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 Brahmagupta-Fibonacci Identity (the set of sums of two squares is closed under multiplication). He applied mathematics to astronomy, predicting eclipses, etc.

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Bháskara I__
(600?-680?) Saurastra (India)

The astronomer Bháskara I, who takes the suffix "I" to distinguish him from the more famous Bháskara who lived five centuries later, made key advances to the positional decimal number notation, and was the first known to use the zero symbol. He preserved some of the teachings of Aryabhata which would otherwise have been lost; these include a famous formula giving an excellent approximation to thesinfunction, as well as, probably, the zero symbol itself.Among other original contributions to mathematics, Bháskara I was first to state Wilson's Theorem (which should perhaps be called Bháskara's Conjecture):

(n-1)! ≡ -1 (mod n)if and only if n is prime

Bháskara's Conjecture was rediscovered by Alhazen, Fibonacci, Leibniz and Wilson. The "only if" is easy but the difficult "if" part was finally proved by Lagrange in 1771. Since Lagrange has so many other Theorems named after him, Bháskara's Conjecture is always called "Wilson's Theorem."

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Muhammed `Abu Jafar' ibn Musâ al-Khowârizmi__
(ca 780-850) Khorasan (Uzbekistan), Iraq

Al-Khowârizmi (aka Mahomet ibn Moses) was a Persian who worked as a mathematician, astronomer and geographer early in the Golden Age of Islamic science. He introduced the Hindu decimal system to the Islamic world and Europe; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography. He wrote the bookAl-Jabr, which demonstrated simple algebra and geometry, and several other influential books. Unlike Diophantus' work, which dealt in specific examples, Al-Khowârizmi 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 wordalgorithmis borrowed from Al-Khowârizmi's name, andalgebrais taken from the name of his book. He also coined the wordcipher, which became Englishzero(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. Al-Khowârizmi's texts on algebra and decimal arithmetic are considered to be among the most influential writings ever.

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Ya'qub `Abu Yusuf' ibn Ishaq al-Kindi__
(803-873) Iraq

Al-Kindi (called Alkindus or Achindus in the West) wrote on diverse philosophical subjects, physics, optics, astronomy, music, psychology, medicine, chemistry, and more. He invented pharmaceutical methods, perfumes, and distilling of alcohol. In mathematics, he popularized the use of the decimal system, developed spherical geometry, wrote on many other topics and was a pioneer of cryptography (code-breaking). (Al-Kindi, calledThe 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.) He appears on Cardano's List of 12 Greatest Geniuses. (Al-Khowârizmi and Jabir ibn Aflah are the other Islamic scientists on that list.)

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Al-Sabi Thabit ibn Qurra__ al-Harrani
(836-901) Harran, Iraq

Thabit produced important books in philosophy (including perhaps the famous mystic workDe Imaginibus), medicine, mechanics, astronomy, and especially several mathematical fields: analysis, non-Euclidean geometry, trigonometry, arithmetic, number theory. As well as being an original thinker, Thabit was a key translator of ancient Greek writings; he translated Archimedes' otherwise-lostBook of Lemmasand 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. (Thabit ibn Qurra's Theorem was rediscovered by Fermat and Descartes, and later generalized by Euler.) 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.

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Ibrahim ibn Sinan__ ibn Thabit ibn Qurra
(908-946) Iraq

Ibn Sinan, grandson of Thabit ibn Qurra, was one of the greatest Islamic mathematicians and might have surpassed his famous grandfather had he not died at a young age. He was an early pioneer of analytic geometry, advancing the theory of integration, applying algebra to synthetic geometry, and writing on the construction of conic sections. He produced a new proof of Archimedes' famous formula for the area of a parabolic section. He worked on the theory of area-preserving transformations, with applications to map-making. He also advanced astronomical theory, and wrote a treatise on sundials.

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Mohammed ibn al-Hasn (Alhazen) `Abu Ali' ibn al-Haytham__ al-Basra
(965-1039) Iraq, Egypt

Al-Hassan ibn al-Haytham (Alhazen) made contributions to math, optics, and astronomy which eventually influenced Roger Bacon, Regiomontanus, da Vinci, Copernicus, Kepler, Galileo, Huygens, Descartes and Wallis, thus affecting Europe's Scientific Revolution. He's been called the best scientist of the Middle Ages; hisBook of Opticshas 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 studied optical illusions and was first to explain psychologically why the Moon appears to be larger when near the horizon. He also did work in human anatomy and medicine. (In a famous leap of over-confidence he claimed he could control the Nile River; when the Caliph ordered him to do so, he then had to feign madness!) Alhazen has been called the "Father of Modern Optics," 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, and the Chinese Remainder Theorem. He stated Wilson's Theorem (which is sometimes called Al-Haytham's Theorem). Alhazen 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 Descartes' 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 non-Euclidean geometry.

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Abu al-Rayhan Mohammed ibn Ahmad al-Biruni__
(973-1048) Khorasan (Uzbekistan)

Al-Biruni (Alberuni) was an extremely outstanding scholar, far ahead of his time, sometimes shown with Alkindus and Alhazen as one of the greatest Islamic polymaths, and sometimes compared to Leonardo da Vinci. He is less famous in part because he lived in a remote part of the Islamic empire. He was a great linguist; studied the original works of Greeks and Hindus; is famous for debates with his contemporary Avicenna; studied history, biology, mineralogy, philosophy, sociology, medicine and more; is called the Father of Geodesy and the Father of Arabic Pharmacy; and was one of the greatest astronomers. He was an early advocate of the Scientific Method. He was also noted for his poetry. He invented (but didn't build) a geared-astrolabe 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, andShadows, which starts from notions about shadows but develops much astronomy and mathematics. He 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 may have 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; invented the azimuthal equidistant map projection in common use today, as well as a polyconic method now called the Nicolosi Globular Projection; found trigonometric solutions to polynomial equations; did geometric constructions including angle trisection; and wrote on arithmetic, algebra, and combinatorics as well as plane and spherical trigonometry and geometry. (Al-Biruni's contemporary Avicenna was not particularly a mathematician but deserves mention as an advancing scientist, as does Avicenna's disciple Abu'l-Barakat al-Baghdada, who lived about a century later.)Al-Biruni 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, Al-Biruni provided several novel proofs for, and useful corollaries of, this famous geometric gem. While Al-Biruni may lack the influence and mathematical brilliance to qualify for the Top 100, he deserves recognition as one of the greatest applied mathematicians before the modern era.

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Omar al-Khayyám__
(1048-1123) Persia

Omar Khayyám (aka Ghiyas od-Din Abol-Fath Omar ibn Ebrahim Khayyam Neyshaburi) was one of the greatest Islamic mathematicians. He did clever work with geometry, developing an alternate to Euclid's Parallel Postulate and then deriving the parallel result using theorems based on theKhayyam-Saccheri quadrilateral. He derived solutions to cubic equations using the intersection of conic sections with circles. Remarkably, he stated that the cubic solution could not be achieved with straightedge and compass, a fact that wouldn't be proved until the 19th century. Khayyám did even more important work in algebra, writing an influential textbook, and developing new solutions for various higher-degree equations. He may have been first to develop Pascal's Triangle (which is still called Khayyám's Triangle in Persia), along with the essential Binomial Theorem (Al-Khayyám's Formula):(x+y)^{n}= n! Σ x^{k}y^{n-k}/ k!(n-k)!Khayyám was also an important astronomer; he measured the year far more accurately than ever before, improved the Persian calendar, built a famous star map, and believed that the Earth rotates on its axis. He was a polymath: in addition to being a philosopher of far-ranging scope, he also wrote treatises on music, mechanics and natural science. He was noted for deriving his theories from science rather than religion. Despite his great achievements in algebra, geometry, astronomy, and philosophy, today Omar al-Khayyám is most famous for his rich poetry (

The Rubaiyat of Omar Khayyám).

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Bháscara (II) Áchárya__
(1114-1185) India

Bháscara II (also called Bhaskaracharya) may have been the greatest of the Hindu mathematicians. He made achievements in several fields of mathematics including some Europe wouldn't learn until the time of Euler. His textbooks dealt with many matters, including solid geometry, combinations, and advanced arithmetic methods. He was also an astronomer. (It is sometimes claimed that his equations for planetary motions anticipated the Laws of Motion discovered by Kepler and Newton, but this claim is doubtful.) In algebra, he solved various equations including 2nd-order Diophantine, quartic, Brouncker's and Pell's equations. HisChakravala method, an early application of mathematical induction to solve 2nd-order equations, has been called "the finest thing achieved in the theory of numbers before Lagrange" (although a similar statement was made about one of Fibonacci's theorems). (Earlier Hindus, including Brahmagupta, contributed to this method.) In several ways he anticipated calculus: he used Rolle's Theorem; he may have been first to use the fact thatd; and he once wrote that multiplication bysinx =cosx · dx0/0could 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

sina+b =sina ·cosb +sinb ·cosaBháscara'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.

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

Leonardo (known today as Fibonacci) introduced the decimal system and other new methods of arithmetic to Europe, and relayed the mathematics of the Hindus, Persians, and Arabs. Others, especially Gherard of Cremona, had translated Islamic mathematics, e.g. the works of al-Khowârizmi, into Latin, but Leonardo was the influential teacher. (Two centuries earlier, the mathematician-Pope, Gerbert of Aurillac, had tried unsuccessfully to introduce the decimal system to Europe.) Leonardo also re-introduced older Greek ideas like Mersenne numbers and Diophantine equations. His 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 mosttalentedmathematician of the Middle Ages."Leonardo is most famous for his book

Liber Abaci, but hisLiber Quadratorumprovides the best demonstration of his skill. He definedcongruumsand 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 II's theorems). Although often overlooked, this work includes a proof of then = 4case 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. Al-Farisi was another ancient mathematician who noted FLT4, although attempting no proof.) 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 hisElements. He also wrote on, but didn't prove, Wilson's Theorem.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 like5 yards, 2 feet, and 3 inchesis more efficient than today's notation. It seems hard to believe but before the decimal system, mathematicians had no notation for zero. Referring to this system, Gauss was later to exclaim "To what heights would science now be raised if Archimedes had made that discovery!"Some histories describe him as bringing Islamic mathematics to Europe, but in Fibonacci's own preface to

Liber Abaci, he specifically credits the Hindus:... as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods;

... But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, ...Had the Scientific Renaissance begun in the Islamic Empire, someone like al-Khowârizmi would have greater historic significance than Fibonacci, but the Renaissance did happen in Europe.

Liber Abaci's summary of the decimal system has been called "the most important sentence ever written." Even granting this to be an exaggeration, there is no doubt that the Scientific Revolution owes a huge debt to Leonardo `Fibonacci' Pisano.

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Abu Jafar Muhammad Nasir al-Din al-Tusi__
(1201-1274) 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 Tusi-couple, 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 Chiu-Shao) may have had particularly outstanding breadth and genius. Qin's textbook discusses various algebraic procedures, includes word problems requiring quartic or quintic equations, explains a version of Horner's Method for finding solutions to such equations, includes Heron's Formula for a triangle's area, and introduces the zero symbol and decimal fractions. 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. Their teachings did not make their way to Europe, but were read by the Japanese mathematician Seki, and possibly by Islamic mathematicians like Al-Kashi. 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.

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Zhu Shiejie__
(ca 1265-1303+) China

Zhu Shiejie (Chu Shih-Chieh) was more famous and influential than Qin; historian George Sarton called him "one of the greatest mathematicians ... of all time." His bookJade Mirror of the Four Unknownsstudied multivariate polynomials and is considered the best mathematics in ancient China and describes methods not rediscovered for centuries; for example Zhu anticipated the Sylvester matrix method for solving simultaneous polynomial equations.

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Levi ben Gerson `Gersonides'__
(1288-1344?) France

Gersonides (aka Leo de Bagnols, aka RaLBaG) was a Jewish scholar of great renown, preferring science and reason over religious orthodoxy. He wrote important commentaries on Aristotle, Euclid, the Talmud, and the Bible; he is most famous for his bookMilHamot Adonai("The Wars of the Lord") which touches on many theological questions. He was likely the most talented scientist of his time: he invented the "Jacob's Staff" which became an important navigation tool; described the principles of the camera obscura; etc. In mathematics, Gersonides wrote texts on trigonometry, calculation of cube roots, rules of arithmetic, etc.; and gave rigorous derivations of rules of combinatorics. He was first to make explicit use of mathematical induction. At that time, "harmonic numbers" referred to integers with only 2 and 3 as prime factors; Gersonides solved a problem of music theory with an ingenious proof that there were no consecutive harmonic numbers larger than (8,9). Levi ben Gerson published only in Hebrew so, although some of his work was translated into Latin during his lifetime, his influence was limited; much of his work was re-invented three centuries later; and many histories of math overlook him altogether.Gersonides was also an outstanding astronomer. He proved that the fixed stars were at a huge distance, and found other flaws in the Ptolemaic model. But he specifically rejected heliocentrism, noteworthy since it implies that heliocentrism was under consideration at the time.

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Nicole Oresme__
(ca 1322-1382) France

Oresme was of lowly birth but excelled at school (where he was taught by the famous Jean Buridan), became a young professor, and soon personal chaplain to King Charles V. The King commissioned him to translate the works of Aristotle into French (with Oresme thus playing key roles in the development of both French science and French language), and rewarded him by making him a Bishop. He wrote several books; was a renowned philosopher and natural scientist (challenging several of Aristotle's ideas); contributed to economics (e.g. anticipating Gresham's Law) and to optics (he was first to posit curved refraction). Although the Earth's annual orbit around the Sun was left to Copernicus, Oresme was among the pre-Copernican thinkers to claim clearly that the Earth spun daily on its axis.In mathematics, Oresme observed that the integers were equinumerous with the odd integers; was first to use fractional (and even irrational) exponents; introduced the symbol

for addition; was first to write about general curvature; and, most famously, first to prove the divergence of the harmonic series. 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 Descartes' coordinate geometry and Galileo.) Oresme was aware of Gersonides' work on harmonic numbers and was among those who attempted to link music theory to the ratios of celestial orbits, writing "the heavens are like a man who sings a melody and at the same time dances, thus making music ... in song and in action." Oresme's work was influential; with several discoveries ahead of his time, Oresme deserves to be better known.+

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Madhava of Sangamagramma__
(1340-1425) India

Madhava, also known as Irinjaatappilly Madhavan Namboodiri, founded the important Kerala school of mathematics and astronomy. If everything credited to him was his own work, he was a truly great mathematician. His analytic geometry preceded and surpassed Descartes', 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, formulae forsinq = q - q^{3}/3! + q^{5}/5! - ..., including the one attributed to Leibniz, and the then-best known approximationππ ≈ 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.

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Ghiyath al-Din Jamshid Mas'ud Al-Kashi__
(ca 1380-1429) Iran, Transoxania (Uzbekistan)

Al-Kashi was among the greatest calculaters in the ancient world; wrote important texts applying arithmetic and algebra to problems in astronomy, mensuration and accounting; and developed trig tables far more accurate than earlier tables. He worked with binomial coefficients, invented astronomical calculating machines, developed spherical trig, and is credited with various theorems of trigonometry including the Law of Cosines, which is sometimes called Al-Kashi'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 decimal fractions had been used earlier, e.g. by Qin Jiushao; and Al-Kashi's root calculations may also have been derived from Chinese texts by Qin Jiushao or Zhu Shiejie.Using his methods, al-Kashi calculated

πcorrectly 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 thecalculation record was held briefly by John von Neumann and the ENIAC.)π

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Johannes Müller von Königsberg `Regiomontanus'__
(1436-1476) Bavaria, Italy

Regiomontanus was a prodigy who entered University at age eleven, studied under the influential Georg von Peuerbach, and eventually collaborated with him. He was an important astronomer; he found flaws in Ptolemy's system (thus influencing Copernicus), realized lunar observations could be used to determine longitude, and may have believed in heliocentrism. His ephemeris was used by Columbus, when shipwrecked on Jamaica, to predict a lunar eclipse, thus dazzling the natives and perhaps saving his crew. More importantly, Regiomontanus was one of the most influential mathematicians of the Middle Ages; he published trigonometry textbooks and tables, as well as the best textbook on arithmetic and algebra of his time. (Regiomontanus lived shortly after Gutenberg, and founded the first scientific press.) He was a prodigious reader of Greek and Latin translations, and most of his results were copied from Greek works (or indirectly from Arabic writers like Jabir ibn Aflah); however he improved or reconstructed many of the proofs, and often presented solutions in both geometric and algebraic form. His algebra was more symbolic and general than his predecessors'; he solved cubic equations (though not the general case); applied Chinese remainder methods, and worked in number theory. He posed and solved a variety of clever geometric puzzles, including his famous angle maximization problem. Regiomontanus was also an instrument maker, astrologer, and Catholic bishop. He died in Rome where he had been called to advise the Pope on the calendar; his early death may have delayed the needed reform until the time of Pope Gregory.

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Leonardo da Vinci__
(1452-1519) Italy

Leonardo da Vinci is most renowned for his paintings --Mona LisaandThe Last Supperare among the most discussed and admired paintings ever -- but he did much other work and was probably the most talented, versatile and prolific polymath ever to live; his writings exceed 13,000 folios. He developed new techniques, and principles of perspective geometry, for drawing, painting and sculpture; he was also an expert architect and engineer; and surely the most prolific inventor of all time. Although most of his paper designs were never built, Leonardo's inventions include reflecting and refracting telescope, adding machine, parabolic compass, improved anemometer, parachute, helicopter, flying ornithopter, several war machines (multi-barreled gun, steam-driven cannon, tank, giant crossbow, finned mortar shells, portable bridge), pumps, an accurate spring-operated clock, bobbin winder, robots, scuba gear, an elaborate musical instrument he called the 'viola organista,' and more. (Some of his designs, including the viola organista, his parachute, and a large single-span bridge, were finally built five centuries later.) He developed the mechanical theory of the arch; made advances in anatomy, botany, and other fields of science; developed an octant-based map projection; and he was first to conceive of plate tectonics. He was also a poet and musician.He had little formal training in mathematics until he was in his mid-40's, when he and Luca Pacioli (the other great Italian mathematician of that era) began tutoring each other. Despite this slow start, he did make novel achievements in mathematics: he was first to note the simple classification of symmetry groups on the plane, achieved interesting bisections and mensurations, advanced the craft of descriptive geometry, and developed an approximate solution to the circle-squaring problem. He was first to discover the 60-vertex shape now called "buckyball." (Leonardo is also widely credited with the elegant two-hexagon proof of the Pythagorean Theorem, but this authorship appears to be a myth.) Along with Archimedes, Alberuni, Leibniz, and J. W. von Goethe, Leonardo da Vinci was among the greatest geniuses ever; but none of these appears on Hart's List of the Most Influential Persons in History: genius doesn't imply influence. (However, M.I.T.'s Pantheon project, using the statistics of on-line biographies, prepared a list of the Thirty-Five (or Eighty) Most Influential Persons in History; in addition to five (six) names already on our list and Hart's -- Aristotle, Newton, Einstein, Galileo, Euclid (and Descartes) -- their list includes four (seven) other mathematicians missing from Hart's list: Plato, Leonardo, Pythagoras, Archimedes (and Thales, Pascal, Ptolemy).

Leonardo was also a writer and philosopher. Among his notable adages are "Simplicity is the ultimate sophistication," and "The noblest pleasure is the joy of understanding," and "Human ingenuity ... will never discover any inventions more beautiful, more simple or more practical than those of nature."

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Nicolaus Copernicus__
(1472-1543) Poland

The European Renaissance developed in 15th-century 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, and Leonardo da Vinci began serious math study only very late in life, so the best candidates for mathematical greatness in the Italian Renaissance were foreigners. Along with Regiomontanus from Bavaria, there was an even more famous man from Poland.Nicolaus Copernicus (Mikolaj Kopernik) was a polymath: he studied law and medicine; published poetry; contemplated astronomy; worked professionally as a church scholar and diplomat; and was also a painter. He studied Islamic works on astronomy and geometry at the University of Bologna, and eventually wrote a book of great impact. Although his only famous theorem of mathematics (that certain trochoids are straight lines) may have been derived from Oresme's work, or copied from Nasir al-Tusi, it was mathematical thought that led Copernicus to the conclusion that the Earth rotates around the Sun. 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; however I think there are several reasons why Copernicus' importance may be exaggerated: (1) Copernicus' system still used circles and epicycles, so it was left to Kepler to discover the facts of elliptical orbits; (2) he retained the notion of a sphere of fixed stars, thus missing the unifying insight that our sun is one of many; (3) Giordano Bruno (1548-1600), who built on Copernicus' discovery, was a better and more influential scientist, anticipating some of Galileo's concepts; and (4) the Scientific Revolution didn't really get underway until the invention of the telescope, which would have soon led to the discovery of heliocentrism in any event.

Until the Protestant Reformation, which began about the time of Copernicus' discovery, European scientists were reluctant to challenge the Catholic Church and its belief in geocentrism. Copernicus' book was published only posthumously. It remains controversial whether earlier Islamic or Hindu mathematicians (or even Archimedes with his

The Sand Reckoner) believed in heliocentrism, but were also inhibited by religious orthodoxy.

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Girolamo Cardano__
(1501-1576) Italy

Girolamo Cardano (or Jerome Cardan) was a highly respected physician and was first to describe typhoid fever. He was also an accomplished gambler and chess player and wrote an early book on probability. He was also a remarkable inventor: the combination lock, an advanced gimbal, a ciphering tool, and the Cardan shaft with universal joints are all his inventions and are in use to this day. (The U-joint is sometimes called the Cardan joint.) He also helped develop the camera obscura. Cardano made contributions to physics: he noted that projectile trajectories are parabolas, and may have been first to note the impossibility of perpetual motion machines. He did work in philosophy, geology, hydrodynamics, music; he wrote books on medicine and an encyclopedia of natural science.But Cardano is most remembered for his achievements in mathematics. He was first to publish general solutions to cubic and quartic equations, and first to publish the use of complex numbers in calculations. (Cardano's Italian colleagues deserve much credit: Ferrari first solved the quartic, he or Tartaglia the cubic; and Bombelli first treated the complex numbers as numbers in their own right. Cardano may have been the

lastgreat mathematician unwilling to deal with negative numbers: his treatment of cubic equations had to deal withaxand^{3}- bx + c = 0axas 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^{3}- bx = cCardano's Ring Puzzle, still manufactured today and related to theTower of Hanoipuzzle. (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."

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Rafael Bombelli__
(1526-1572) Italy

Bombelli was a talented engineer who wrote an algebra textbook sometimes considered one of the foremost achievements of the 16th century. Although incorporating work by Cardano, Diophantus and possibly Omar al-Khayyám, the textbook was highly original and extremely influential. Leibniz and Huygens were among many who praised his work. Although noted for his new ideas of arithmetic, Bombelli based much of his work on geometric ideas, and even pursued complex-number arithmetic to an angle-trisection method. In his textbook he introduced new symbolic notations, gave a new square-root procedure based on continued fractions, allowed negative and complex numbers, and gave the rules for manipulating these new kinds of numbers. Bombelli is often called the Inventor of Complex Numbers.

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François Viète__
(1540-1603) France

François Viète (or Franciscus Vieta) was a French nobleman and lawyer who was a favorite of King Henry IV and eventually became a royal privy councillor. In one notable accomplishment he broke the Spanish diplomatic code, allowing the French government to read Spain's messages and publish a secret Spanish letter; this apparently led to the end of the Huguenot Wars of Religion.More importantly, Vieta was certainly the best French mathematician prior to Descartes and Fermat. He laid the groundwork for modern mathematics; his works were the primary teaching for both Descartes 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, including all ten cases of the general

Problem of Apollonius. Vieta also used his new algebraic techniques to construct a regular heptagon. He discovered several trigonometric identities including a generalization of Ptolemy's Formula, the latter (then calledprosthaphaeresis) providing a calculation shortcut similar to logarithms in that multiplication is reduced to addition (or exponentiation reduced to multiplication). Vieta also used trigonometry to find real solutions to cubic equations for which the Italian methods had required complex-number arithmetic; he also used trigonometry to solve a particular 45th-degree equation that had been posed as a challenge. Such trigonometric formulae revolutionized calculations and may even have helped stimulate the development and use of logarithms by Napier and Kepler. He developed the first infinite-product formula forπ. 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 Descartes 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.Descartes, who once wrote "I began where Vieta finished," is now extremely famous, while Vieta is much less known. (He isn't

even mentioned oncein Bell's famousMen of Mathematics.) Many would now agree this is due in large measure to Descartes' 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

πis 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 toπthat was best-yet by a European, though not as accurate as al-Kashi's two centuries earlier.

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Simon Stevin__
(1549-1620) Flanders, Holland

Stevin was one of the greatest practical scientists of the Late Middle Ages. He worked with Holland's dykes and windmills; as a military engineer he developed fortifications and systems of flooding; he invented a carriage with sails that 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 (though also invented at about the same time by Chinese mathematician Zhu Zaiyu and anticipated by Galileo's father, Vincenzo Galilei) the equal-temperament music scale. He also did work in descriptive geometry, trigonometry, optics, geography, and astronomy.In mathematics, Stevin is best known for the notion of real numbers (previously integers, rationals and irrationals were treated separately; negative numbers and even zero and one were often not considered numbers). He introduced (a clumsy form of) 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 special cases of the Intermediate Value Theorem later attributed to Bolzano and 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. √

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John Napier 8th of
Merchistoun__
(1550-1617) Scotland

Napier was a Scottish Laird who was a noted theologian and thought by many to be a magician (his nickname was Marvellous Merchiston). Today, however, he is best known for his work withlogarithms, 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 inventedNapier'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, especially several different kinds of war machine.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 in the Top 100. Nevertheless, his revolutionary methods of arithmetic had immense historical importance; his tables were used by Johannes Kepler himself, and led to the Scientific Revolution.

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Galileo Galilei__
(1564-1642) Italy

Galileo discovered the laws of inertia (including rudimentary forms of all three of Newton's laws of motion), falling bodies (including parabolic trajectories), and the pendulum; he also introduced the notion of relativity which later physicists found so fruitful. (Although he admired Galileo greatly, Einstein's famous result was somewhat misnamed: it depended on theabsoluteness, not relativity, of the speed of light.) Galileo discovered important priciples of dynamics and may have been first to note that a larger body has less relative cohesive strength than a smaller body. 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 astronomical discoveries also included sunspots and lunar craters. His discovery that Venus, like the Moon, had phases was the critical fact which forced acceptance of Copernican heliocentrism. Galileo's contributions outside physics and astronomy were also enormous: He invented the compound microscope and made early discoveries with it. He also made very important contributions to the early development of biology; but perhaps Galileo's most important contribution was to postulateuniversallaws of mechanics, in contrast to Aristotelian and religious notions of separate laws for heaven and earth.Galileo is often called the "Father of Modern Science" because of his emphasis on experimentation. His use of a ramp to discover his Law of Falling Bodies was ingenious. (For his experiments he started with a water-clock to measure time, but found the beats reproduced by trained musicians to be more convenient.) He understood that results needed to be repeated and averaged (he minimized mean absolute-error for his curve-fitting criterion, two centuries before Gauss and Legendre introduced the mean squared-error criterion). For his experimental methods and discoveries, his laws of motion, and for (eventually) helping to spread Copernicus' heliocentrism, Galileo may have been the most influential scientist 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, Galileo did do work in pure mathematics; he derived certain centroids and the parabolic shape of trajectories using a rudimentary calculus before Cavalieri did; he named (and may have been first to discover) the cycloid curve. Moreover, Galileo was one of the 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."

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Johannes Kepler__
(1571-1630) Germany

Kepler was interested in astronomy from an early age, studied to become a Lutheran minister, became a professor of mathematics instead, then Tycho Brahe's understudy, and, on Brahe's death, was appointed Imperial Mathematician at the age of twenty-nine. His observations of the planets with Brahe, along with his study of Apollonius' 1800-year old work, led to Kepler's three Laws of Planetary Motion, which in turn led directly to Newton's Laws of Motion. Beyond his discovery of these Laws (one of the most important achievements in all of science), Kepler is also sometimes called the "Founder of Modern Optics." He furthered the theory of the camera obscura, telescopes built fromtwoconvex lenses, and atmospheric refraction. The question of human vision had been considered by many great scientists including Aristotle, Euclid, Ptolemy, Galen, Alkindus, Alhazen, and Leonardo da Vinci, but it was Kepler who was first to explain the operation of the human eye correctly and to note that retinal images will be upside-down. Kepler developed a rudimentary notion of universal gravitation and applied it to explain tides correctly; however he seems not to have noticed that his empirical laws implied inverse-square gravitation. (Galileo dismissed Kepler's tides and 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 Earth-bound observer is himself describing such an orbit and in almost the same plane as the planets; thus discovering the Laws would be a difficult challenge even for someone armed with computers and modern mathematics. (The very famous Kepler Equation relating a planet's eccentric and anomaly is just one tool Kepler needed to develop.) Kepler understood the importance of his remarkable discovery, even if contemporaries like Galileo did not, writing:

"I give myself up to divine ecstasy ... My book is written. It will be read either by my contemporaries or by posterity — I care not which. It may well wait a hundred years for a reader, as God has waited 6,000 years for someone to understand His work."Kepler also once wrote "Mathematics is the archetype of the beautiful."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 theGolden 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 on stationary points. Kepler once had an opportunity to buy wine, which merchants measured using a shortcut; with the famousKepler's Wine Barrel Problem, he used his rudimentary calculus to deduce which barrel shape would be the best bargain.Kepler reasoned that the structure of snowflakes was evidence for the then-novel atomic theory of matter. He noted that the obvious packing of cannonballs gave maximum density (this became known as

Kepler's Conjecture; optimality was proved among regular packings by Gauss, but it wasn't until 1998 that the possibility of denserirregularpackings 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 Descartes) 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.)

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Gérard Desargues__
(1591-1661) France

Desargues invented projective geometry and found the relationship among conic sections which inspired Blaise Pascal. Among several ingenious and rigorously proven theorems are Desargues' Involution Theorem and his Theorem of Homologous Triangles. Desargues was also a noted architect and inventor: he produced an elaborate spiral staircase, invented an ingenious new pump based on the epicycloid, and had the idea to use cycloid-shaped teeth in the design of gears.Desargues' projective geometry may have been too creative for his time; Descartes admired Desargues but was disappointed his friend didn't apply algebra to his geometric results as Descartes did; Desargues' writing was poor; and one of his best pupils (Blaise Pascal himself) turned away from math, so Desargues' work was largely ignored (except by Philippe de La Hire, Desargues' other prize pupil) until Poncelet rediscovered it almost two centuries later. (Copies of Desargues' own works surfaced about the same time.) For this reason, Desargues may not be important enough to belong in the Top 100, despite that he may have been among the greatest natural geometers ever.

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René Descartes__
(1596-1650) France

Descartes' early career was that of soldier-adventurer and he finished as tutor to royalty, but in between he achieved fame as the preeminent intellectual of his day. He 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. His famous mathematical theorems include the Rule of Signs (for determining the signs of polynomial roots), the elegant formula relating the radii ofSoddy kissing circles, his theorem on total angular defect (an early form of the Gauss-Bonnet result so key to much mathematics), and an improved solution to the Delian problem (cube-doubling). While studying lens refraction, he invented theOvals of Descartes. He improved mathematical notation (e.g. the use of superscripts to denote exponents). He also discovered Euler's Polyhedral Theorem,F+V = E+2. Descartes was very influential in physics and biology as well, e.g. developing laws of motion which included a "vortex" theory of gravitation; but most of his scientific work outside mathematics was eventually found to be incorrect.Descartes has an extremely high reputation and would be ranked even higher by many list makers, but whatever his historical importance his mathematical skill was not in the top rank. Some of his work was borrowed from others, e.g. from Thomas Harriot. He had only insulting things to say about Pascal and Fermat, each of whom was much more brilliant at mathematics than Descartes. (Some even suspect that Descartes arranged the destruction of Pascal's lost

Essay on Conics.) And Descartes made numerous errors in his development of physics, perhaps even delaying science, with Huygens writing "in all of [Descartes'] physics, I find almost nothing to which I can subscribe as being correct." Even the historical importance of his mathematics may be somewhat exaggerated since others, e.g. Fermat, Wallis and Cavalieri, were making similar discoveries independently.

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Francesco Bonaventura de Cavalieri__
(1598-1647) 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."

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Pierre de Fermat__
(1601-1665) France

Pierre de Fermat was the most brilliant mathematician of his era and, along with Descartes, 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. Fermat was first to study certain interesting curves, e.g. the"Witch of Agnesi". He was also an excellent geometer (e.g. discovering a triangle'sFermat point), and (in collaboration with Blaise Pascal) discovered probability theory. Fellow geniuses are the best judges of genius, and Blaise Pascal had this to say of Fermat: "For my part, I confess that [Fermat's researches about numbers] are far beyond me, and I am competent only to admire them." E.T. Bell wrote "it can be argued that Fermat was at least Newton's equal as a pure mathematician."Fermat's most famous discoveries in number theory include the ubiquitously-used

Fermat's Little Theorem; then = 4case of his conjecturedFermat's Last Theorem(he may have proved then = 3case as well); andFermat'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 theFermat-Euler 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 k-gonal 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 so-called "Last Theorem" was actually just a private scribble.)Fermat developed a system of analytic geometry which both preceded and surpassed that of Descartes; he developed methods of differential and integral calculus which Newton acknowledged as an inspiration. Although Kepler anticipated it, Fermat is credited with Fermat's Theorem on Stationary Points (

df(x)/dx = 0at function extrema), the key to many problems in applied analysis. 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 Descartes is more famous than Fermat, and Descartes' writings were more influential. Whatever one thinks of Descartes as a

philosopher, however, it seems clear that Fermat was the bettermathematician. Fermat and Descartes 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 Maupertuis (though Maupertuis himself, like Descartes, had an incorrect explanation of refraction). Fermat and Descartes independently discovered analytic geometry, but it was Fermat who extended it to more than two dimensions, and followed up by developing elementary calculus.

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Gilles Personne de Roberval__
(1602-1675) France

Roberval was an eccentric genius, underappreciated because most of his work was published only long after his death. He did early work in integration, following Archimedes rather than Cavalieri; he worked on analytic geometry independently of Descartes. With his analysis he was able to solve several difficult geometric problems involving curved lines and solids, including results about the cycloid which were also credited to Pascal and Torricelli. Some of these methods, published posthumously, led to him being called the Founder of Kinematic Geometry. He excelled at mechanics, worked in cartography, helped Pascal with vacuum experiments, and invented the Roberval balance, still in use in weighing scales to this day. He opposed Huygens in the early debate about gravitation, though neither fully anticipated Newton's solution.

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Evangelista Torricelli__
(1608-1647) Italy

Torricelli was a disciple of Galileo (and succeeded him as grand-ducal mathematician of Tuscany). He was first to understand that a barometer measures atmospheric weight, and used this insight to invent the mercury barometer and to create a sustained vacuum (then thought impossible). He was a skilled craftsman who built the best telescopes and microscopes of his day. As mathematical physicist, he extended Galileo's results, was first to explain winds correctly, and discovered several key principles including Torricelli's Law (water drains through a small hole with rate proportional to the square root of water depth). In mathematics, he applied Cavalieri's methods to solve difficult mensuration problems; he also wrote on possible pitfalls in applying the new calculus. He discoveredGabriel's Hornwith infinite surface area but finite volume; this "paradoxical" result provoked much discussion at the time. He also solved a problem due to Fermat by locating the isogonic center of a triangle. Torricelli was a significant influence on the early scientific revolution; had he lived longer, or published more, he would surely have become one of the greatest mathematicians of his era.

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John Brehaut Wallis__
(1616-1703) England

Wallis began his life as a savant at arithmetic (it is said he once calculated the square root of a 53-digit number to help him sleep and remembered the result in the morning), a medical student (he may have contributed to the concept of blood circulation), and theologian, but went on to become perhaps the most brilliant and influential English mathematician before Newton. 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 Descartes 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 code-breaker, helping the Commonwealth side (though he later petitioned against the beheading of King Charles I). He was the first great mathematician to consider complex numbers legitimate; and first to use the symbol∞. Wallis coined several terms includingcontinued fraction,induction,interpolation,mantissa, andhypergeometric 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}

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Blaise Pascal__
(1623-1662) France

Pascal was an outstanding genius who studied geometry as a child. At the age of sixteen he stated and proved Pascal's Theorem, a fact relating any six points on any conic section. The Theorem is sometimes called the "Cat's Cradle" or the "Mystic Hexagram." Pascal followed up this result by showing that each of Apollonius' famous theorems about conic sections was a corollary of the Mystic Hexagram; along with Gérard Desargues (1591-1661), 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.

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Christiaan Huygens__
(1629-1695) Holland, France

Christiaan Huygens (or Hugens, Huyghens) was second only to Newton as the greatest mechanist of his era. Although an excellent mathematician, he is much more famous for his physical theories and inventions. He developed laws of motion before Newton, including the inverse-square law of gravitation, centripetal force, and treatment of solid bodies rather than point approximations; he (and Wallis) were first to state the law of momentum conservation correctly. He advanced the wave ("undulatory") theory of light, a key concept beingHuygen's Principle, that each point on a wave front acts as a new source of radiation. His optical discoveries include explanations for polarization and phenomena like haloes. (Because of Newton's high reputation and corpuscular theory of light, Huygens' superior wave theory was largely ignored until the 19th-century work of Young, Fresnel, and Maxwell. Later, Planck, Einstein and Bohr, partly anticipated by Hamilton, developed the modern notion of wave-particle duality.)Huygens is famous for his inventions of clocks and lenses. He invented the escapement and other mechanisms, leading to the first reliable pendulum clock; he built the first balance spring watch, which he presented to his patron, King Louis XIV of France; he was first to give the correct "equation of time" relating sundial time to absolute time. He invented superior lens grinding techniques, the achromatic eye-piece, and the best telescope of his day. He was himself a famous astronomer: he discovered Titan and was first to properly describe Saturn's rings and the Orion Nebula. He also designed, but never built, an internal combustion engine. He promoted the use of an equal-tempered 31-tone music scale to avoid the tuning errors in Stevin's 12-tone scale; a 31-tone organ was in use in Holland as late as the 20th century. Huygens was an excellent card player, billiard player, horse rider, and wrote a book speculating about extra-terrestrial life.

As a mathematician, Huygens did brilliant work in analysis; his calculus, along with that of Wallis, is considered the best prior to Newton and Leibniz. He also did brilliant work in geometry, proving theorems about conic sections, the cycloid and the catenary. He was first to show that the cycloid solves the tautochrone problem; he used this fact to design pendulum clocks that would be more accurate than ordinary pendulum clocks. He was first to find the flaw in Saint-Vincent's then-famous circle-squaring method; Huygens himself solved some related quadrature problems. He introduced the concepts of evolute and involute. His friendships with Descartes, Pascal, Mersenne and others helped inspire his mathematics; Huygens in turn was inspirational to the next generation. At Pascal's urging, Huygens published the first real textbook on probability theory; he also became the first practicing actuary.

Huygens had tremendous creativity, historical importance, and depth and breadth of genius, both in physics and mathematics. He also was important for serving as tutor to the otherwise self-taught Gottfried Leibniz (who'd "wasted his youth" without learning any math). Before agreeing to tutor him, Huygens tested the 25-year old Leibniz by asking him to sum the reciprocals of the triangle numbers.

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Takakazu Seki (Kowa)__
(ca 1637-1708) Japan

Seki Takakazu (aka Shinsuke) was a self-taught prodigy who developed a new notation for algebra, and made several discoveries before Western mathematicians did; these include determinants, the Newton-Raphson method, Newton's interpolation formula, Bernoulli numbers, discriminants, methods of calculus, and probably much that has been forgotten (Japanese schools practiced secrecy). He calculated

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

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James Gregory__
(1638-1675) 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

(fortan^{-1}x = x - x^{3}/3 + x^{5}/5 - x^{7}/7 + ...|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 thatandπwere transcendental. He produced a partial proof that the ancient "Squaring the Circle" problem was impossible.eGregory 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 the Top 100.

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Isaac (Sir) Newton__
(1642-1727) England

Newton was an industrious lad who built marvelous toys (e.g. a model windmill powered by a mouse on treadmill). At about age 22, on leave from University, this genius began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. He is famous for his Three Laws of Motion (inertia, force, reciprocal action) but, as Newton himself acknowledged, these Laws weren't fully novel: Hipparchus, Ibn al-Haytham, Descartes, Galileo and Huygens had all developed much basic mechanics already; and Newton credits the First Law to Aristotle. However Newton was 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 inverse-square 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 writings also made important contributions to the general scientific method. His other intellectual interests included theology, and mysticism. He studied ancient Greek writers like Pythagoras, Democritus, Lucretius, Plato; and claimed that the ancients knew much, including the law of gravitation.

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, observed that white light is a mixture of all the rainbow's colors, noted that purple is created by combining red and blue light and, starting from that observation, was first to conceive of a color hue "wheel." He noted that his dynamical laws were symmetric in time; that just as the past determines the future, so the future might, in principle, determine the past. Newton almost anticipated Einstein's mass-energy equivalence, writing "Gross Bodies and Light are convertible into one another... [Nature] seems delighted with Transmutations." 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. Although Descartes is renowned as the inventor of analytic geometry, he and followers like Wallis were reluctant even to use negative coordinates, so one historian declares Newton to be "the first to work boldly with algebraic equations." 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

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 Descartes' 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 same-mass spheres (or hollowed spheres) of any radius have equal gravitational attraction: this fact is key to celestial motions. (He also proved that objectse^{x}=∑x^{k}/ k!insidea hollowed sphere experience zero net attraction.) He discovered Puiseux series almost two centuries before they were re-invented by Puiseux. (Like some of the greatest ancient mathematicians, Newton took the time to compute an approximation toπ; his was better than Vieta's, though still not as accurate as al-Kashi's.)Newton is so famous for his calculus, optics, and laws of gravitation and motion, it is easy to overlook that he was also one of the very greatest geometers. He was first to fully solve the famous Problem of Pappus, and did so with pure geometry. Building on the "neusis" (non-Platonic) constructions of Archimedes and Pappus, he demonstrated cube-doubling and that angles could be

k-sected for anyk, if one is allowed a conchoid or certain other mechanical curves. He also built on Apollonius' famous theorem about tangent circles to develop the technique now called hyperbolic trilateration. Despite the power of Descartes' 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 circum-scribing 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 non-Platonic tools.) He anticipated Poncelet's Principle of Continuity. An anecdote often cited to demonstrate his brilliance is the problem of thebrachistochrone, 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 theheliocentricsystem allowed Kepler to describe the orbits. InPrincipiaNewton 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 inverse-cubeforces, work largely unappreciated until Chandrasekhar's modern-day 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."

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Gottfried Wilhelm von Leibniz__
(1646-1716) Germany

Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe'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 self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Fifteen who was never the greatest living algorist or theorem prover. I 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 (Electress of Hanover), 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∛x, and even the use of(instead ofa·b) for multiplication; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He worked in number theory, conjecturing Wilson's Theorem. He invented more mathematical terms than anyone, includingaX bfunction,analysis situ,variable,abscissa,parameterandcoordinate. He also coined the wordtranscendental, proving that sin() was not an algebraic function. His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was thenotation("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 was one of the first to articulate the law of energy conservation and may have written on the principle of least action. He developed laws of motion that gave different insights from those of Newton; his views on cosmology anticipated theories of Mach and Einstein and are more in accord with modern physics than are Newton's views. 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 - ...

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Jacob Bernoulli__
(1654-1705) 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 wordintegral. He and his brother were the key pioneers in mathematics during the generations between the era of Newton-Leibniz and the rise of Leonhard Euler.Jacob liked to pose and solve physical optimization problems. His "catenary" problem (what shape does a clothesline take?) became more famous than the "tautochrone" solved by Huygens. Perhaps the most famous of such problems was the brachistochrone, wherein Jacob recognized Newton's "lion's paw", and about which Johann Bernoulli wrote: "You will be petrified with astonishment [that] this same cycloid, the tautochrone of Huygens, is the brachistochrone we are seeking." Jacob did significant work outside calculus; in fact his most famous work was the

Art of Conjecture, a textbook on probability and combinatorics which proves the Law of Large Numbers, the Power Series Equation, and introduces the Bernoulli numbers. 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.

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Johann Bernoulli__
(1667-1748) 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.

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Abraham De Moivre__
(1667-1754) France, England

De Moivre was an important pioneer of analytic geometry and, especially, probability theory. (He and Laplace may be regarded as the two most important early developers of probability theory.) In probability theory he developed actuarial science, posed interesting problems (e.g. about derangements), discovered the normal and Poisson distributions, and proposed (but didn't prove) the Central Limit Theorem. He was first to discover a closed-form formula for the Fibonacci numbers; and he developed an early version of Stirling's approximation ton!. He discovered De Moivre's Theorem: (cosx +i sinx)^{n}=cosnx +i sinnxHe was a close friend and muse of Isaac Newton, who allegedly told people who asked about

Principia: "Go to Mr. De Moivre; he knows these things better than I do."

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Brook Taylor__
(1685-1731) England

Brook Taylor invented integration by parts, developed what is now called the calculus of finite differences, developed a new method to compute logarithms, made several other key discoveries of analysis, and did significant work in mathematical physics. His love of music and painting may have motivated some of his mathematics: He studied vibrating strings; and also wrote an important treatise on perspective 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 Maclaurin and Lagrange.

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Colin Maclaurin__
(1698-1746) Scotland

Maclaurin received a University degree in divinity at age 14, with a treatise on gravitation. He became one of the most brilliant mathematicians of his era. He wrote extensively on Newton's method of fluxions, and the theory of equations, advancing these fields; worked in optics, and other areas of mathematical physics; but is most noted for his work in geometry. Lagrange said Maclaurin's geometry was as beautiful and ingenious as anything by Archimedes. Clairaut, seeing Maclaurin's methods, decided that he too would prove theorems with geometry rather than analysis. Maclaurin did important work on ellipsoids; for his work on tides he shared the Paris Prize with Euler and Daniel Bernoulli. As Scotland's top genius, he was called upon for practical work, including politics. Although Maclaurin's work was quite influential, his influence didn't really match his outstanding brilliance: he failed to adopt Leibnizian calculus with which great progress was being made on the Continent, and much of his best work was published posthumously. Many of his famous results duplicated work by others: Maclaurin's Series was just a form of Taylor's series; the Euler-Maclaurin Summation Formula was also discovered by Euler; and he discovered the Newton-Cotes Integration Formula after Cotes did. His brilliant results in geometry included the construction of a conic from five points, but Braikenridge made the same discovery and published before Maclaurin did. He discovered the Maclaurin-Cauchy Test for Integral Convergence before Cauchy did. He was first to discover Cramer's Paradox, as Cramer himself acknowledged. Colin Maclaurin found a simpler and more powerful proof of the fact that the cycloid solves the famous brachistochrone problem.

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Daniel Bernoulli__
(1700-1782) Switzerland

Johann Bernoulli had a nephew, three sons and some grandsons who were all also outstanding mathematicians. Of these, the most important was his 2nd-oldest son Daniel. Johann insisted that Daniel study biology and medicine rather than mathematics, so Daniel specialized initially in mathematical biology. He went on to win the Grand Prize of the Paris Academy no less than ten times, and was a close friend of Euler. Daniel developed partial differential equations, preceded Fourier in the use of Fourier series, did important work in statistics and the theory of equations, discovered and proved a key theorem about trochoids, developed a theory of economic risk (motivated by the St. Petersburg Paradox discovered by his cousin Nicholas), but is most famous for his key discoveries in mathematical physics: e.g. the Bernoulli Principle underlying airflight, and the notion that heat is simply molecules' random kinetic energy. Daniel Bernoulli is sometimes called the "Founder of Mathematical Physics."

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Leonhard Euler__
(1707-1783) Switzerland

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

Σ_{n}p(n) x^{n}= 1 / Π_{k}(1 - x^{k})

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

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

Four of the most important constant symbols in mathematics (

,π,e= √-1, andi= 0.57721566...) were all introduced or popularized by Euler, along with operators likeγ. He did important work withΣRiemann's zeta functionζ(s) = ∑ k(although it was not then known by that name); he anticipated the concept of analytic continuation by showing^{-s}ζ(-1) = 1+2+3+4+... = -1/12. Euler started as a young student of the Bernoulli family, and was Daniel Bernoulli's roommate in Saint Petersburg, where Euler was first employed as a teacher of physiology. But at age twenty-eight, Euler discovered the striking identityζ(2) =This catapulted Euler to instant fame, since the left-side infinite sum (π^{2}/61 + 1/4 + 1/9 + 1/16 + ...)was a famous problem of the time. Euler and others developed alternate proofs and generalizations of this "Basel problem," and of course the ζ (zeta) function is now very famous. Here is an elegant geometric proof for this theorem (although the video makes it a little more complicated than necessary). Among many other famous and important identities, Euler proved the Pentagonal Number Theorem alluded to above (a beautiful result which has inspired a variety of discoveries), and the Euler Product Formulaζ(s) = ∏(1-pwhere the right-side product is taken over all primes^{-s})^{-1}p. His most famous identity (which Richard Feynman called an "almost astounding ... jewel") unifies the trigonometric and exponential functions:

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

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

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Alexis Claude Clairaut__
(1713-1765) France

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

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

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

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

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Johann Heinrich Lambert__
(1727-1777) Switzerland, Prussia

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

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

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

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

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

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

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

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

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

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

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

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

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Adrien Marie Legendre__
(1752-1833) France

Legendre was an outstanding mathematician who did important work in plane and solid geometry, spherical trigonometry, celestial mechanics and other areas of physics, and especially elliptic integrals and number theory. He found key results in the theories of sums of squares and sums of k-gonal numbers. (For example, he showed that all integers except4can be expressed as the sum of three squares.) He also made key contributions in several areas of analysis: he invented the Legendre transform and Legendre polynomials; the notation for partial derivatives is due to him. He invented the Legendre symbol; invented the study of zonal harmonics; proved that^{k}(8m+7)πwas irrational (the irrationality of^{2}πhad already been proved by Lambert); and wrote important textbooks in several fields. Although he never accepted non-Euclidean geometry, and had spent much time trying to prove the Parallel Postulate, his inspiring geometry text remained a standard until the 20th century. As one of France's premier mathematicians, Legendre did other significant work, promoting the careers of Lagrange and Laplace, developing trig tables, geodesic projects, etc.There are several important theorems proposed by Legendre for which he is denied credit, either because his proof was incomplete or was preceded by another's. He proposed the famous theorem about primes in a progression which was proved by Dirichlet; proved and used the Law of Least Squares which Gauss had left unpublished; proved the N=5 case of Fermat's Last Theorem which is credited to Dirichlet; proposed the famous Prime Number Theorem which was finally proved by Hadamard; improved the Fermat-Cauchy result about sums of k-gonal numbers but this topic wasn't fruitful; and developed various techniques commonly credited to Laplace. His two most famous theorems of number theory, the Law of Quadratic Reciprocity and the Three Squares Theorem (a difficult extension of Lagrange's Four Squares Theorem), were each enhanced by Gauss a few years after Legendre's work. Legendre also proved an early version of Bonnet's Theorem. Legendre's work in the theory of equations and elliptic integrals directly inspired the achievements of Galois and Abel (which then obsoleted much of Legendre's own work); Chebyshev's work also built on Legendre's foundations.

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Jean Baptiste Joseph
Fourier__
(1768-1830) France

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

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

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

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Johann Carl Friedrich Gauss__
(1777-1855) Germany

Carl Friedrich Gauss, the "Prince of Mathematics," exhibited his calculative powers when he corrected his father's arithmetic before the age of three. His revolutionary nature was demonstrated at age twelve, when he began questioning the axioms of Euclid. His genius was confirmed at the age of nineteen when he proved that the regular n-gon was constructible if and only if it is the product of distinct prime Fermat numbers. (He didn't complete the proof of the only-if part. Click to see construction of regular 17-gon.) Also at age 19, he proved Fermat's conjecture that every number is the sum of three triangle numbers. (He further determined the number of distinct ways such a sum could be formed.) At age 24 he publishedDisquisitiones Arithmeticae, probably the greatest book of pure mathematics ever.Although he published fewer papers than some other great mathematicians, Gauss may be the greatest theorem prover ever. Several important theorems and lemmas bear his name; he extended Euclid's Fundamental Theorem of Arithmetic (prime factorization is unique) to the Gaussian (complex) integers; and he was first to produce a rigorous proof of the Fundamental Theorem of Algebra (that an n-th degree polynomial has n complex roots); his Theorema Egregium ("Remarkable Theorem") that a surface's essential curvature derived from its 2-D geometry laid the foundation of differential geometry. Gauss himself used "Fundamental Theorem" to refer to Euler's Law of Quadratic Reciprocity; Gauss was first to provide a proof for this, and provided eight distinct proofs for it over the years. Gauss proved the n=3 case of Fermat's Last Theorem for Eisenstein integers (the triangular lattice-points on the complex plane); though more general, Gauss' proof was simpler than the real integer proof; this simplification method revolutionized algebra. He also found a simpler proof for Fermat's Christmas Theorem, by taking advantage of the identity

x. Other work by Gauss led to fundamental theorems in statistics, vector analysis, function theory, and generalizations of the Fundamental Theorem of Calculus.^{2}+y^{2}= (x + iy)(x - iy)Gauss built the theory of complex numbers into its modern form, including the notion of "monogenic" functions which are now ubiquitous in mathematical physics. (Constructing the regular 17-gon as a teenager was actually an exercise in complex-number algebra, not geometry.) Gauss developed the arithmetic of congruences and became the premier number theoretician of all time. Other contributions of Gauss include hypergeometric series, foundations of statistics, and differential geometry. He also did important work in geometry, providing an improved solution to Apollonius' famous problem of tangent circles, stating and proving the

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

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

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Siméon Denis Poisson__
(1781-1840) France

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

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Bernard Placidus Johann Nepomuk Bolzano__
(1781-1848) Bohemia

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

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Jean-Victor Poncelet__
(1788-1867) France

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

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Augustin-Louis Cauchy__
(1789-1857) France

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

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

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

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August Ferdinand Möbius__
(1790-1868) Germany

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

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Nicolai Ivanovitch Lobachevsky__
(1793-1856) Russia

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

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Michel Floréal Chasles__
(1793-1880) France

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

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Jakob Steiner__
(1796-1863) Switzerland

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

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

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Julius Plücker__
(1801-1868) Germany

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

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Niels Henrik Abel__
(1802-1829) Norway

At an early age, Niels Abel studied the works of the greatest mathematicians, found flaws in their proofs, and resolved to reprove some of these theorems rigorously. He was the first to fully prove the general case of Newton's Binomial Theorem, one of the most widely applied theorems in mathematics. 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(replacingy = f(x)withx = f) 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 complex-variable functions, which were important to the development of both abstract and applied mathematics.^{-1}(y)Finding the roots of polynomials is a key mathematical problem: the general solution of the quadratic equation was known by ancients; the discovery of general methods for solving polynomials of degree three and four is usually treated as the major math achievement of the 16th century; so for over two centuries an algebraic solution for the general 5th-degree polynomial (quintic) was a Holy Grail sought by most of the greatest mathematicians. Abel proved that most quintics did

nothave such solutions. This discovery, at the age of only nineteen, would have quickly awed the world, but Abel was impoverished, had few contacts, and spoke no German. When Gauss received Abel's manuscript he discarded it unread, assuming the unfamiliar author was just another crackpot trying to square the circle or some such. His genius was too great for him to be ignored long, but, still impoverished, Abel died of tuberculosis at the age of twenty-six. His fame lives on and even the lower-case word 'abelian' is applied to several concepts. Liouville said Abel was the greatest genius he ever met. Hermite said "Abel has left mathematicians enough to keep them busy for 500 years."

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

Jacobi was a prolific mathematician who did decisive work in the algebra and analysis of complex variables, and did work in number theory (e.g. cubic reciprocity) which excited Carl Gauss. He is sometimes described as the successor to Gauss. As an algorist (manipulator of involved algebraic expressions), he may have been surpassed only by Euler and Ramanujan. 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, extending Lagrange's famous Four-Squares Theorem to show in

how many distinct waysa given integer can be expressed as 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, q-series, 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 ill-fated contemporary was the more brilliant genius.

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Johann Peter Gustav
Lejeune Dirichlet__
(1805-1859) Germany

Dirichlet was preeminent in algebraic and analytic number theory, but did advanced work in several other fields as well: He discovered the modern definition of function, the Voronoi diagram of geometry, and important concepts in differential equations, topology, and statistics. 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 life-long companion. Fermat and Euler had proved the impossibility of

xfor 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^{k}+ y^{k}= z^{k}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 L-series to prove the important theorem (Gauss' conjecture) that any arithmetic series (without a common factor) has an infinity of primes. It was Dirichlet who proved the fundamental Theorem of Fourier series: that periodic analytic functions can always be represented as a simple trigonometric series. Although he never proved it rigorously, he is especially noted for the

Dirichlet's Principlewhich 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 hisClass Number Formula.

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

Hamilton was a childhood prodigy. Home-schooled and self-taught, he started as a student of languages and literature, was influenced by an arithmetic prodigy his own age, read Euclid, Newton and Lagrange, found an error by Laplace, and made new discoveries in optics; all this before the age of seventeen when he first attended school. At college he enjoyed unprecedented success in all fields, but his undergraduate days were cut short abruptly by his appointment as Royal Astronomer of Ireland at the age of 22. He soon began publishing his revolutionary treatises on optics, in which he developed Hamilton's Principle of Stationary Action. This Principle refined and corrected the earlier principles of least action developed by Maupertuis, Fermat, and Euler; it (and related principles) are key to much of modern physics. His early writing also predicted that some crystals would have an hitherto unknown "conical" refraction mode; this was soon confirmed experimentally.Hamilton's Principle of Least Action, and its associated equations and concept of configuration space, led to a revolution in mathematical physics. Since Maupertuis had named this Principle a century earlier, it is possible to underestimate Hamilton's contribution. However Maupertuis, along with others credited with anticipating the idea (Fermat, Leibniz, Euler and Lagrange) failed to state the full Principle correctly. Rather than minimizing action, physical systems sometimes achieve a non-minimal but

stationaryaction in configuration space. (Poisson and d' Alembert had noticed exceptions to Euler-Lagrange least action, but failed to find Hamilton's solution. Jacobi also deserves some credit for the Principle, but his work came after reading Hamilton.) Because of this Principle, as well as his wave-particle duality (which would be further developed by Planck and Einstein), Hamilton can be considered a major early influence on quantum theory.Hamilton also made revolutionary contributions to dynamics, differential equations, the theory of equations, numerical analysis, fluctuating functions, and graph theory (he marketed a puzzle based on his

Hamiltonian paths). He invented the ingenious hodograph. He coined several mathematical terms includingvector,scalar,associative, andtensor. In addition to his brilliance and creativity, Hamilton was renowned for thoroughness and produced voluminous writings on several subjects.Hamilton himself considered his greatest accomplishment to be the development of quaternions, a non-Abelian field to handle 3-D rotations. While there is no 3-D analog to the Gaussian complex-number plane (based on the equation

), quaternions derive from a 4-D analog based oni^{2}= -1. (Despite their being "obsoleted" by more general matrix and tensor methods, quaternions are still in wide engineering use because of certain practical advantages.)i^{2}=j^{2}=k^{2}=ijk= -jik= -1Hamilton once wrote: "On earth there is nothing great but man; in man there is nothing great but mind."

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Hermann Günter Grassmann__
(1809-1877) Germany

Grassmann was an exceptional polymath: the termGrassmann's Lawis applied to two separate facts in the fields of optics and linguistics, both discovered by Hermann Grassmann. He also did advanced work in crystallography, electricity, botany, folklore, and also wrote on political subjects. He had little formal training in mathematics, yet single-handedly developed linear algebra, vector and tensor calculus, multi-dimensional geometry, new results about cubic surfaces, 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, single-handedly, a new subject." Important mathematicians inspired directly by Grassmann include Peano, Klein, Cartan, Hankel, Clifford, and Whitehead.

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Joseph Liouville__
(1809-1882) France

Liouville did expert research in several areas including number theory, differential geometry, complex analysis (especially Sturm-Liouville theory, boundary value problems and dynamical analysis), topology and mathematical physics. Several theorems bear his name, including the key result that any boundedentirefunction 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. (His proof was constructive, unlike that of Cantor which came 30 years later). 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. Liouville was hugely prolific in number theory but this work is largely overlooked, e.g. the following remarkable generalization of Aryabhata's identity:

for all N, Σ (d_{a}^{3}) = (Σ d_{a})^{2}

wheredis the number of divisors of_{a}a, and the sums are taken over all divisorsaofN.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.

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Ernst Eduard Kummer__
(1810-1893) Germany

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

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Évariste Galois__
(1811-1832) France

Galois, who died before the age of twenty-one, not only never became a professor, but was barely allowed to study as an undergraduate. His output of papers, mostly published posthumously, is much smaller than most of the others on this list, yet it is considered among the most awesome works in mathematics. He applied group theory to the theory of equations, revolutionizing both fields. (Galois coined the mathematical termgroup.) 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' ideas were very far-reaching; for example he is credited as first to prove that trisecting a general angle with Plato's rules is impossible. Galois is sometimes cited (instead of Archimedes, Gauss or Ramanujan) as "the greatest mathematical genius ever." His 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.

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James Joseph Sylvester__
(1814-1897) England, U.S.A.

Sylvester made important contributions in matrix theory, invariant theory, number theory, partition theory, reciprocant theory, geometry, and combinatorics. He invented the theory of elementary divisors, and co-invented the law of quadratic forms. It is said he coined more new mathematical terms (e.g.matrix,invariant,discriminant,covariant,syzygy,graph,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 Descartes and Newton; and, by advancing the theory of partitions, the system of equations posed by Euler asThe 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?"

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Karl Wilhelm Theodor Weierstrass__
(1815-1897) Germany

Weierstrass devised new definitions for the primitives of calculus, developed the concept of uniform convergence, and was then able to prove several fundamental but hitherto unproven theorems. Starting strictly from the integers, he also applied his axiomatic methods to a definition of irrational numbers. He developed important new insights in other fields including the calculus of variations, elliptic functions, and trigonometry. Weierstrass shocked his colleagues when he demonstrated a continuous function which is differentiable nowhere. (Both this and the Bolzano-Weierstrass Theorem were rediscoveries of forgotten results by the under-published Bolzano.) He found simpler proofs of many existing theorems, including Gauss' Fundamental Theorem of Algebra and the fundamental Hermite-Lindemann Transcendence Theorem. Steiner's proof of the Isoperimetric Theorem contained a flaw, so Weierstrass became the first to supply a fully rigorous proof of that famous and ancient result. Peter Dirichlet was a champion of rigor, but Weierstrass discovered a flaw in the argument for Dirichlet's Principle of of variational calculus.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, Mittag-Leffler, and several other famous mathematicians. Bell called him "probably the greatest mathematical teacher of all time." In 1873 Hermite called Weierstrass "the Master of all of us." Today he is often called the "Father of Modern Analysis."

Weierstrass once wrote: "A mathematician who is not also something of a poet will never be a complete mathematician."

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George Boole__
(1815-1864) England

George Boole was a precocious child who impressed by teaching himself classical languages, but was too poor to attend college and became an elementary school teacher at age 16. He gradually developed his math skills; as a young man he published a paper on the calculus of variations, and soon became one of the most respected mathematicians in England despite having no formal training. He was noted for work in symbolic logic, algebra and analysis, and also was apparently the first to discover invariant theory. When he followed up Augustus de Morgan's earlier work in symbolic logic, de Morgan insisted that Boole was the true master of that field, and begged his friend to finally study mathematics at university. Boole couldn't afford to, and had to be appointed Professor instead!Although very few recognized its importance at the time, it is Boole's work in Boolean algebra and symbolic logic for which he is now remembered; this work inspired computer scientists like Claude Shannon. Boole's book

An Investigation of the Laws of Thoughtprompted 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."

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Pafnuti Lvovich Chebyshev__
(1821-1894) Russia

Pafnuti Chebyshev (Pafnuty Tschebyscheff) was noted for work in probability, number theory, approximation theory, integrals, the theory of equations, and orthogonal polynomials. His famous theorems cover a diverse range; they include a new version of the Law of Large Numbers, 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 anynand2n, and working with the zeta function before Riemann did. He made much progress with the Prime Number Theorem, provingtwodistinct forms of that theorem, each incomplete but in a different way. Chebyshev was very influential for Russian mathematics, inspiring Andrei Markov and Aleksandr Lyapunov among others.Chebyshev was also a premier applied mathematician and a renowned inventor; his several inventions include the Chebyshev linkage, a mechanical device to convert rotational motion to straight-line motion. He once wrote "To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls."

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Arthur Cayley__
(1821-1895) England

Cayley was one of the most prolific mathematicians in history; a list of the branches of mathematics he pioneered will seem like an exaggeration. In addition to being very inventive, he was an excellent algorist; some considered him to be the greatest mathematician of the late 19th century (an era that includes Weierstrass and Poincaré). Cayley was the essential founder of modern group theory, matrix algebra, the theory of higher singularities, and higher-dimensional geometry (building on Plücker's work and anticipating the ideas of Klein), as well as the theory of invariants. Among his many important theorems are the Cayley-Hamilton Theorem, and Cayley's Theorem itself (that any group is isomorphic to a subgroup of a symmetric group). He extended Hamilton's quaternions and developed the octonions, but was still one of the first to realize that these special algebras should be subsumed by general matrix methods. He also did original research in combinatorics (e.g. enumeration of trees), elliptic and Abelian functions, and projective geometry. One of his 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

leasteccentric of the great mathematicians: In addition to his life-long love of mathematics, he enjoyed hiking, painting, reading fiction, and had a happy married life. He easily won Smith's Prize and Senior Wrangler at Cambridge, but then worked as a lawyer for many years. He later became professor, and finished his career in the limelight as President of the British Association for the Advancement of Science. He and James Joseph Sylvester were a source of inspiration to each other. These two, along with Charles Hermite, are considered the founders of the important theory of invariants. Though applied first to algebra, the notion of invariants is useful in many areas of mathematics.Cayley once wrote: "As for everything else, so for a mathematical theory: beauty can be perceived but not explained."

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Charles Hermite__
(1822-1901) France

Hermite studied the works of Lagrange and Gauss from an early age and soon developed an alternate proof of Abel's famous quintic impossibility result. He attended the same college as Galois and also had trouble passing their examinations, but soon became highly respected by Europe's best mathematicians for his significant 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. He was first to note remarkable facts about Heegner numbers, e.g.

e= 262537412640768743.9999999999992...^{π√163}

(Without computers he was able to calculate this number, including the twelve 9's to the right of the decimal point.) Very many elegant concepts and theorems are named after Hermite. Hermite's most famous result may be his intricate proof that(along with a broad class of related numbers) is transcendental. (Extending the proof toeπwas left to Lindemann, a matter of regret for historians, some of whom who regard Hermite as the greatest mathematician of his era.)

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Ferdinand Gotthold Max Eisenstein__
(1823-1852) Germany

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

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Leopold Kronecker__
(1823-1891) Germany

Kronecker was a businessman who pursued mathematics mainly as a hobby, but was still very prolific, and one of the greatest theorem provers of his era. He explored a wide variety of mathematics -- number theory, algebra, analysis, matrixes -- and especially the interconnections between areas. Many concepts and theorems are named after Kronecker; some of his theorems are frequently used as lemmas in algebraic number theory, ergodic theory, and approximation theory. He provided key ideas about foundations and continuity despite that he had philosophic objections to irrational numbers and infinities. He also introduced the Theory of Divisors to avoid Dedekind's Ideals; the importance of this and other work was only realized long after his death. Kronecker's philosophy eventually led to the Constructivism and Intuitionism of Brouwer and Poincaré.

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Georg Friedrich Bernhard Riemann__
(1826-1866) Germany

Riemann was a phenomenal genius whose work was exceptionally deep, creative and rigorous; he made revolutionary contributions in many areas of pure mathematics, and also inspired the development of physics. He had poor physical health and died at an early age, yet is still considered to be among the most productive mathematicians ever. He made revolutionary advances in 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 introduced the Riemann integral which clarified analysis. He developed the theory ofmanifolds, a term which he invented. Manifolds underpin topology. By imposing metrics on manifolds Riemann invented differential geometry and took non-Euclidean geometry far beyond his predecessors. Riemann's other masterpieces include tensor analysis, the theory of functions, and a key relationship between some differential equation solutions and hypergeometric series. His generalized notions of distance and curvature described new possibilities for the geometry of space itself. Several important theorems and concepts are named after Riemann, e.g. the Riemann-Roch Theorem, a key connection among topology, complex analysis and algebraic geometry. He was also first to prove theorems named after others, e.g. Green's Theorem. He was so prolific and original that some of his work went unnoticed (for example, Weierstrass became famous for showing a nowhere-differentiable continuous function; later it was found that Riemann had casually mentioned one in a lecture years earlier). Like his mathematical peers (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. His theory unifying electricity, magnetism and light was supplanted by Maxwell's theory; however modern physics, beginning with Einstein's relativity, relies on Riemann's curvature tensor and other 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 went far beyond 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

exactformula, 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 theHypothesis of Riemann's zeta function, needed for the missing step in his proof. This Hypothesis is considered the most important and famous unsolved problem in mathematics. (Asked what he would first do, if he were magically awakened after centuries, David Hilbert replied "I would ask whether anyone had proved the Riemann Hypothesis.")ζ(.)was defined for convergent cases in Euler's mini-bio, which Riemann extended via analytic continuation for all cases. The Riemann Hypothesis "simply" states that in all solutions ofζ(s = a+b, eitheri) = 0shas real parta=1/2or imaginary partb=0.Despite his great creativity (Gauss praised Riemann's "gloriously fertile originality;" another biographer called him "one of the most profound and imaginative mathematicians of all time [and] a great philosopher"), Riemann once said: "If only I had the theorems! Then I should find the proofs easily enough."

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Henry John Stephen Smith__
(1826-1883) England

Henry Smith (born in Ireland) was one of the greatest number theorists, working especially with elementary divisors; he also advanced the theory of quadratic forms. A famous problem of Eisenstein was, givenandn, in how many different ways cankbe expressed as the sum ofnsquares? Smith made great progress on this problem, subsuming special cases which had earlier been famous theorems. Although most noted for number theory, he had great breadth. He did prize-winning work in geometry, discovered the unique normal form for matrices which now bears his name, anticipated specific fractals including the Cantor set, the Sierpinski gasket and the Koch snowflake, and wrote a paper demonstrating the limitations of Riemann integration.kSmith is sometimes called "the mathematician the world forgot." His paper on integration could have led directly to measure theory and Lebesgue integration, but was ignored for decades. The fractals he discovered are named after people who rediscovered them. The Smith-Minkowski-Siegel mass formula of lattice theory would be called just the Smith formula, but had to be rediscovered. And his solution to the Eisenstein five-squares problem, buried in his voluminous writings on number theory, was ignored: this "unsolved" problem was featured for a prize which Minkowski won two decades later!

Henry Smith was an outstanding intellect with a modest and charming personality. He was knowledgeable in a broad range of fields unrelated to mathematics; his University even insisted he run for Parliament. His love of mathematics didn't depend on utility: he once wrote "Pure mathematics: may it never be of any use to anyone."

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Antonio Luigi Gaudenzio Giuseppe Cremona__
(1830-1903) Italy

Luigi Cremona made many important advances in analytic, synthetic and projective geometry, especially in the transformations of algebraic curves and surfaces. Working in mathematical physics, he developed the new field of graphical statics, and used it to reinterpret some of Maxwell's results. He improved (or found brilliant proofs for) several results of Steiner, especially in the field of cubic surfaces. (Some of this work was done in collaboration with Rudolf Sturm.) He is especially noted for developing the theory of Cremona transformations which have very wide application. He found a generalization of Pascal's Mystic Hexagram. Cremona also played a political role in establishing the modern Italian state and, as an excellent teacher, helped make Italy a top center of mathematics.

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James Clerk Maxwell__
(1831-1879) Scotland

At the age of 14, Maxwell published a remarkable paper on the construction of ovals; these were an independent discovery of theOvals of Descartes, but Maxwell allowed more than two foci, had elaborate configurations (he was drawing the ovals with string and pencil), and identified errors in Descartes' treatment of them. 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, viscosity, knot theory, topology, 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, although his contribution to the kinetic theory of gases (which even led to the first estimate of molecular sizes) would already be enough to make him one of the greatest physicists. But then, in 1864 James Clerk Maxwell stunned the world by publishing the equations of electricity and magnetism and showing that light itself is linked to the electro-magnetic force. Richard Feynman considered this the most significant event of the 19th century (though others might give higher billing to Darwin's theory of evolution). While Einstein, Newton, and Galileo may be the Top Three, Maxwell is a strong candidate for "fourth greatest scientist ever." Recalling Newton's comment about "standing on the shoulders" of earlier greats, Einstein was asked whose shoulders

hestood on; he didn't name Newton: he said "Maxwell." Maxwell has been called the "Father of Modern Physics"; he ranks #24 on Hart's list of the Most Influential Persons in History.

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Julius Wilhelm Richard Dedekind__
(1831-1916) Germany

Dedekind was one of the most innovative mathematicians ever; his clear expositions and rigorous axiomatic methods had great influence. He made seminal contributions to abstract algebra and algebraic number theory as well as mathematical foundations. He was one of the first to pursue Galois Theory, making major advances there and pioneering in the application of group theory to other branches of mathematics. Dedekind also invented a system of fundamental axioms for arithmetic, worked in probability theory and complex analysis, and invented prime partitions and modular lattices. Dedekind may be most famous for his theory of ideals and rings; Kronecker and Kummer had begun this, but Dedekind gave it a more abstract and productive basis, which was developed further by Hilbert, Noether and Weil. Though the termringitself was coined by Hilbert, Dedekind introduced the termsmodule,field, andideal. 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 Cutthen led to a rigorous and useful definition of the real numbers. Dedekind was a key mentor for Georg Cantor: 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 Cantor-Bernstein-Schröder Theorem, though he didn't publish his proof. (Because he spent his career at a minor university, and neglected to publish some of his work, Dedekind's contributions may be underestimated.)

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Rudolf Friedrich Alfred Clebsch__
(1833-1872) 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. Clebsch-Aronhold symbolic notation and 'connex'; and proved key theorems about cubic surfaces (for example, the Sylvester pentahedron conjecture) and other high-degree curves, and representations (bijections) between surfaces. Some of his work, e.g. Clebsch-Gordan 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 co-author on a text published 60 years after his death.

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Eugenio Beltrami__
(1835-1899) Italy

Beltrami was an outstanding mathematician noted for early insights connecting geometry and topology (differential geometry, pseudospherical surfaces, etc.), transformation theory, differential calculus, and especially for proving the equiconsistency of hyperbolic and Euclidean geometry for every dimensionality; he achieved this by building on models of Cayley, Klein, Riemann and Liouville. He was first to invent singular value decompositions. (Camille Jordan and J.J. Sylvester each invented it independently a few years later.) Using insights from non-Euclidean geometry, he did important mathematical work in a very wide range of physics; for example he improved Green's theorem, generalized the Laplace operator, studied gravitation in non-Euclidean space, and gave a new derivation of Maxwell's equations.

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Marie Ennemond Camille Jordan__
(1838-1921) France

Jordan was a great "universal mathematician", making revolutionary advances in group theory, topology, and operator theory; and also doing important work in differential equations, number theory, measure 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. In measure theory he developed Peano-Jordan "content" and proved the Jordan Decomposition Theorem. He also proved the Jordan-Holder Theorem of group theory, invented the notion of homotopy, 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 19th-century teachers; his work inspired such mathematicians as Klein, Lie and Borel.

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Marius Sophus Lie__
(1842-1899) Norway

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

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Jean Gaston Darboux__
(1842-1917) France

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

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William Kingdon Clifford__
(1845-1879) 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 non-Euclidean geometry. Combining Hamilton's quaternions, Grassmann's exterior algebra, and his own geometric intuition and understanding of physics, he developed biquaternions, and generalized this togeometric 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 n-Circle Theorem, and did so with a purely geometric argument. Clifford is especially famous for anticipating, before Einstein, that gravitation could be modeled with a non-Euclidean space. He was a polymath; a talented teacher, noted philosopher, writer of children's fairy tales, 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 thirty-three.

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Georg Cantor__
(1845-1918) Russia, Germany

Cantor did brilliant and important work early in his career, for example he greatly advanced the Fourier-series uniqueness question which had intrigued Riemann. In his explorations of that problem he was led to questions of set enumeration, and his greatest invention: set theory. Cantor created modern Set Theory almost single-handedly, defining cardinal numbers, well-ordering, ordinal numbers, and discovering the Theory of Transfinite Numbers. He defined equality between cardinal numbers based on the existence of a bijection, and was the first to demonstrate that the real numbers have a higher cardinal number than the integers. (He also showed that the rationals have the same cardinality as the integers; and that the reals have the same cardinality as the points of N-space and as the power-set of the integers.) Although there are infinitely many distinct transfinite numbers, Cantor conjectured thatC, the cardinality of the reals, was the second smallest transfinite number. ThisContinuum Hypothesiswas 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" and addressed the critics with "no one shall expel us from the paradise that Cantor has created." 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 (he even involved himself in the Shakespeare authorship controversy!). In addition to his set theory and key discoveries in the theory of trigonometric series, he made advances in number theory, and gave the modern definition of irrational numbers. His

Cantor setwas the early inspiration for fractals. Cantor was also an excellent violinist. He once wrote "In mathematics the art of proposing a question must be held of higher value than solving it."

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Friedrich Ludwig Gottlob Frege__
(1848-1925) 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 ofquantifiers; he distinguished terms from predicates, and simple predicates from 2nd-level predicates. From his second-order 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). (Russell report that when he informed Frege of this flaw, he took it with incomparable integrity, grace, and even intellectual pleasure.) 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."

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Ferdinand Georg Frobenius__
(1849-1917) 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, elliptic functions, biquadratic forms, matrixes, and algebra. He was first to actually prove the important Cayley-Hamilton 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 "Cayley-Hamilton Theorem," many lemmas and concepts are named after him, including Frobenius conjugacy class, Frobenius reciprocity, Frobenius manifolds, the Frobenius-Schur Indicator, etc. He is most noted for hischaracter theory, a revolutionary advance which led to the representation theory of groups, and has applications in modern physics. The middle-aged Frobenius invented this after the aging Dedekind asked him for help in solving a key algebraic factoring problem.

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Christian Felix Klein__
(1849-1925) Germany

Klein's key contribution was an application of invariant theory to unify geometry with group theory. This radical new view of geometry inspired Sophus Lie's Lie groups, and also led to the remarkable unification of Euclidean and non-Euclidean geometries which is probably Klein's most famous result. Klein did other work in function theory, providing links between several areas of mathematics including number theory, group theory, hyperbolic geometry, and abstract algebra. His Klein's Quartic curve and popularly-famousKlein's bottlewere among several useful results from his new approaches to groups and higher-dimensional 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 world-wide influence. Klein once wrote "... mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs."

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Oliver Heaviside__
(1850-1925) England

Heaviside dropped out of high school to teach himself telegraphy and electromagnetism, becoming first a telegraph operator but eventually perhaps the greatest electrical engineer ever. He developed transmission line theory, invented the coaxial cable, predicted Cherenkov radiation, described the use of the ionosphere in radio transmission, and much more. Some of his insights anticipated parts of special relativity, and he was first to speculate about gravitational waves. 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 electro-magnetic equations. Four of the famous Maxwell's Equations are in fact due to Oliver Heaviside, Maxwell having presented a more cumbersome version. Although one of the greatest applied mathematicians, Heaviside is omitted from the Top 100 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?"

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Sofia Vasilyevna Kovalevskaya__
(1850-1891) Russia

Sofia Kovalevskaya (aka Sonya Kowalevski; née Korvin-Krukovskaya) was initially self-taught, sought out Weierstrass as her teacher, and was later considered the greatest female mathematician ever (before Emmy Noether). She was influential in the development of Russian mathematics. Kovalevskaya studied Abelian integrals and partial differential equations, producing the important Cauchy-Kovalevsky Theorem; her application of complex analysis to physics inspired Poincaré and others. Her most famous work was the solution to theKovalevskaya top, which has been called a "genuine highlight of 19th-century mathematics." Other than the simplest cases solved by Euler and Lagrange, exact ("integrable") solutions to the equations of motion were unknown, so Kovalevskaya received fame and a rich prize when she solved theKovalevskaya top. Her ingenious solution might be considered a mere curiosity, but since it is still the only post-Lagrange physical motion problem for which an "integrable" solution has been demonstrated, it remains an important textbook example. Kovalevskaya once wrote "It is impossible to be a mathematician without being a poet in soul." She was also a noted playwright.

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Jules Henri Poincaré__
(1854-1912) 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. He published hundreds of papers on a variety of topics and might have become the most prolific mathematician ever, but he died at the height of his powers. Poincaré was clumsy and absent-minded; 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. He is credited with partial solution of Hilbert's 22nd Problem. Several important results carry his name, for example the famous Poincaré Recurrence Theorem, which almost 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. The Conjecture is that all "simply-connected" closed manifolds are topologically equivalent to "spheres"; it is directly relevant to the possible topology of our universe. 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). 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.^{2}

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Andrei Andreyevich Markov__
(1856-1922) Russia

Markov did excellent work in a broad range of mathematics including analysis, number theory, algebra, continued fractions, approximation theory, and especially probability theory: it has been said that his accuracy and clarity transformed probability theory into one of the most perfected areas of mathematics. Markov is best known as the founder of the theory of stochastic processes. In addition to his Ergodic Theorem about such processes, theorems named after him include the Gauss-Markov Theorem of statistics, the Riesz-Markov Theorem of functional analysis, and the Markov Brothers' Inequality in the theory of equations. Markov was also noted for his politics, mocking Czarist rule, and insisting that he be excommunicated from the Russian Orthodox Church when Tolstoy was.Markov had a son, also named Andrei Andreyevich, who was also an outstanding mathematician of great breadth. Among the son's achievements was Markov's Theorem, which helps relate the theories of braids and knots to each other.

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Giuseppe Peano__
(1858-1932) Italy

Giuseppe Peano is one of the most under-appreciated 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 space-filling 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 (non-paradoxical) 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 counter-examples, 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 single-member set is not the same as its element; this fact is now taught in elementary school.

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Samuel Giuseppe Vito Volterra__
(1860-1946) Italy

Vito Volterra founded the field of functional analysis ('functions of lines'), and used it to extend the work of Hamilton and Jacobi to more areas of mathematical physics. He developed cylindrical waves and the theory of integral equations. He worked in mechanics, developed the theory of crystal dislocations, and was first to propose the use of helium in balloons. Eventually he turned to mathematical biology and made notable contributions to that field, e.g. predator-prey equations.

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David Hilbert__
(1862-1943) 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 ofHilbert 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 year-old 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 non-constructive. 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 Hilbert-Waring Theorem.Any one man can only do so much, so the greatest mathematicians should help nurture their colleagues. Hilbert provided a famous List of 23 Unsolved Problems, which inspired and directed the development of 20th-century mathematics. Hilbert was warmly regarded by his colleagues and students, and contributed to the careers of several great mathematicians and physicists including Georg Cantor, Hermann Minkowski, Hermann Weyl, John von Neumann, Emmy Noether, Alonzo Church, and Albert Einstein.

Eventually Hilbert turned to physics and made key contributions to classical and quantum physics and to general relativity. He published the

Einstein Field Equationsindependently of Einstein (though his writings make clear he treats this as strictly Einstein's invention).

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Hermann Minkowski__
(1864-1909) Lithuania, Germany

Minkowski won a prestigious prize at age 18 for reconstructing Eisenstein's enumeration of the ways to represent integers as the sum of five squares. (The Paris Academy overlooked that Smith had already published a solution for this!) His proof built on quadratic forms and continued fractions and eventually led him to the new field of Geometric Number Theory, for which Minkowski's Convex Body Theorem (a sort of pigeonhole principle) is often called the Fundamental Theorem. Minkowski was also a major figure in the development of functional analysis. With his "question mark function" and "sausage," he was also a pioneer in the study of fractals. Several other important results are named after him, e.g. the Hasse-Minkowski Theorem. He was first to extend the Separating Axis Theorem to multiple dimensions. Minkowski was one of Einstein's teachers, and also a close friend of David Hilbert. He is particularly famous for building on Poincaré's work to inventMinkowski spaceto 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."

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Jacques Salomon Hadamard__
(1865-1963) France

Hadamard made revolutionary advances in several different areas of mathematics, especially complex analysis, analytic number theory, differential geometry, partial differential equations, symbolic dynamics, chaos theory, matrix theory, and Markov chains; for this reason he is sometimes 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 approximatelyn/log nprimes less thann. 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 Vallee-Poussin. (Hadamard's proof is considered more elegant and useful than Vallee-Poussin's.) Several other important theorems are named after Hadamard (e.g. his Inequality of Determinants), and some of his theorems are named after others (Hadamard was first to prove Brouwer's Fixed-Point Theorem for arbitrarily many dimensions). 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."

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Felix Hausdorff__
(1868-1942) 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 point-set 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 Banach-Tarski 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 internment.

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Élie Joseph Cartan__
(1869-1951) 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 hisMethod of Prolongationhe 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, Stokes' Theorem, and fundamental theorems about symmetric Riemann spaces. He made a key contribution to Einstein's general relativity, based on what is now called Riemann-Cartan geometry. Cartan's methods were so original as to be fully appreciated only recently; many now consider him to be one of the greatest mathematicians of his era. In 1938 Weyl called him "the greatest living master in differential geometry."

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Félix Édouard Justin Émile Borel__
(1871-1956) France

Borel exhibited great talent while still in his teens, soon practically founded modern measure theory, and received several honors and prizes. Among his famous theorems is the Heine-Borel Covering Theorem. He also did important work in several other fields of mathematics, including divergent series, quasi-analytic functions, differential equations, number theory, complex analysis, theory of functions, geometry, probability theory, and game theory. Relating measure theory to probabilities, he introduced concepts likenormal numbersand the Borel-Kolmogorov 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.

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Tullio Levi-Civita__
(1873-1941) Italy

Levi-Civita was noted for strong geometrical intuition, and excelled at both pure mathematics and mathematical physics. He worked in analytic number theory, differential equations, tensor calculus, hydrodynamics, celestial mechanics, and the theory of stability. Several inventions are named after him, e.g. the non-archimedean Levi-Civita field, the Levi-Civita parallelogramoid, and the Levi-Civita symbol. His work inspired all three of the greatest 20th-century mathematical physicists, laying key mathematical groundwork for Weyl's unified field theory, Einstein's relativity, and Dirac's quantum theory.

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Henri Léon Lebesgue__
(1875-1941) France

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

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Edmund Georg Hermann Landau__
(1877-1938) Germany

Landau was one of the most prolific and influential number theorists ever and wrote the first comprehensive treatment of analytic number theory. He was also adept at complex function theory. He was especially keen at finding very simple proofs: one of his most famous results was a simpler proof of Hadamard's prime number theorem; being simpler it was also more fruitful and led to Landau's Prime Ideal Theorem. In addition to simpler proofs of existing theorems, new theorems by Landau include important facts about Riemann's Hypothesis; facts about Dirichlet series; key lemmas of analysis; a result in Waring's Problem; a generalization of the Little Picard Theorem; a partial proof of Gauss' conjecture about the density of classes of composite numbers; and key results in the theory of pecking orders, e.g. every flock has at least one "king," never exactly two kings, andexactlyone king only if that king is an "emperor." (If every chicken except X is pecked either by X or someone pecked by X, then X is defined as a "King." Landau's pecking orders are now described as round-robin tournaments.) Landau was also the inventor of big-O notation. Hardy wrote that no one was ever more passionately devoted to mathematics than Landau.

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Godfrey Harold Hardy__
(1877-1947) England

Hardy was an extremely prolific research mathematician who did important work in analysis (especially the theory of integration), number theory, global analysis, and analytic number theory. He proved 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. Hardy provided rigorous proofs for several of Ramanujan's conjectures, including Ramanujan's "Master Theorem" of analysis. Among other results of this collaboration was the Hardy-Ramanujan Formula for partition enumeration, which Hardy later used as a model to develop the Hardy-Littlewood Circle Method; Hardy then used this method to prove stronger versions of the Hilbert-Waring Theorem, and in prime number theory; the method has continued to be a very productive tool in analytic number theory. Hardy was also a mentor to Norbert Wiener, another famous prodigy.

Hardy once wrote "A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." He also wrote "Beauty is the first test; there is no permanent place in the world for ugly mathematics."

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René Maurice Fréchet__
(1878-1973) France

Maurice Fréchet introduced the concept of metric spaces (though not using that term); and also made major contributions to point-set topology. Building on work of Hadamard and Volterra, he generalized Banach spaces to use new (non-normed) metrics and proved that many important theorems still applied in these more general spaces. For this work, and his invention of the notion of compactness, Fréchet is called the Founder of the Theory of Abstract Spaces. He also did important work in probability theory and in analysis; for example he proved the Riesz Representation Theorem the same year Riesz did. Many theorems and inventions are named after him, for example Fréchet Distance, which has many applications in applied math, e.g. protein structure analysis.

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Albert Einstein__
(1879-1955) 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 1905 paper introduced the famous equationE = mc; 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^{2}Miracle Yearwithout bothering to qualify it asEinstein's Miracle Year! (Before his Miracle Year, Einstein had been a mediocre undergraduate, and held minor jobs including patent examiner.) 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, sometimes in collaboration with Leo Szilard, he was co-inventor of several devices, including a gyroscopic compass, hearing aid, automatic camera and, most famously, the Einstein-Szilard refrigerator. He became a very famous and influential public figure. (For example, it was his letter that led Roosevelt to start the Manhattan Project.) Among his many famous quotations is: "The search for truth is more precious than its possession."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. But despite the importance of his other contributions it is Einstein's General Theory which is his most profound contribution. Minkowski had developed a flat 4-dimensional space-time to cope with Einstein's Special Theory; but it was Einstein who had the vision to add curvature to that space to describe acceleration.

Einstein certainly has the breadth, depth, and historical importance to qualify for this list; but his genius and significance were not in the field of pure mathematics. (He acknowledged his limitation, writing "I admire the elegance of your [Levi-Civita's] method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.") Einstein

wasa mathematician, however; he pioneered the application of tensor calculus to physics and invented theEinstein summation notation. That Einstein's equation explained a discrepancy in Mercury's orbit was a discovery made by Einstein personally (a discovery he described as 'joyous excitement' that gave him heart palpitations). He composed a beautiful essay about mathematical proofs using the Theorem of Menelaus as his example. Certainly he belongs on a Top 100 List: 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."

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Oswald Veblen__
(1880-1960) 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 Veblen-Young 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; the first five mathematicians he hired for the Institute were Einstein, von Neumann, Weyl, J.W. Alexander and Marston Morse.

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Luitzen Egbertus Jan Brouwer__
(1881-1966) Holland

Brouwer is often considered the "Father of Topology;" among his important theorems were the Fixed Point Theorem, the "Hairy Ball" Theorem, the Jordan-Brouwer Separation Theorem, and the Invariance of Dimension. He developed the method of simplicial approximations, important to algebraic topology; he also did work in geometry, set theory, measure theory, complex analysis and the foundations of mathematics. He was first to anticipate forms like theLakes of Wada, leading eventually to other measure-theory "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 self-righteous.

Intuitionism has had a significant influence, although few strict adherents. Since only constructive proofs are permitted, strict adherence would slow mathematical work. This didn't worry Brouwer who once wrote: "The construction itself is an art, its application to the world an evil parasite."

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Amalie Emmy Noether__
(1882-1935) 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." Emmy Noether is considered the greatest female mathematician ever.

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Waclaw Sierpinski__
(1882-1969) Poland

Sierpinski won a gold medal as an undergraduate by making a major improvement to a famous theorem by Gauss about lattice points inside a circle. He went on to do important research in set theory, number theory, point set topology, the theory of functions, and fractals. He was extremely prolific, producing 50 books and over 700 papers. He was a Polish patriot: he contributed to the development of Polish mathematics despite that his land was controlled by Russians or Nazis for most of his life. He worked as a code-breaker during the Polish-Soviet War, helping to break Soviet ciphers.Sierpinski was first to prove Tarski's remarkable conjecture that the Generalized Continuum Hypothesis implies the Axiom of Choice. He developed three famous fractals: a space-filling curve; the Sierpinski gasket; and the Sierpinski carpet, which covers the plane but has area zero and has found application in antennae design. Borel had proved that almost all real numbers are "normal" but Sierpinski was the first to actually display a number which is normal in every base. He proved the existence of infinitely many Sierpinski numbers having the property that, e.g.

(78557·2is composite number for every natural number n. It remains an unsolved problem (likely to be defeated soon with high-speed computers) whether 78557 is the smallest such "Sierpinski number."^{n}+1)

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Solomon Lefschetz__
(1884-1972) Russia, U.S.A.

Lefschetz was born in Russia, educated as an engineer in France, moved to U.S.A., was severely handicapped in an accident, and then switched to pure mathematics. He was a key founder of algebraic topology, even coining the wordtopology, and pioneered the application of topology to algebraic geometry. Starting from Poincaré's work, he developed Lefschetz duality and used it to derive conclusions about fixed points in topological mappings. The Lefschetz Fixed-point Theorem left Brouwer's famous result as just a special case. His Picard-Lefschetz theory eventually led to the proof of the Weil conjectures. Lefschetz also did important work in algebraic geometry, non-linear differential equations, and control theory. As a teacher he was noted for a combative style. Preferring intuition over rigor, he once told a student who had improved on one of Lefschetz's proofs: "Don't come to me with your pretty proofs. We don't bother with that baby stuff around here."

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George David Birkhoff__
(1884-1984) U.S.A.

Birkhoff is one of the greatest native-born American mathematicians ever, and did important work in many fields. There are several significant theorems named after him: the Birkhoff-Grothendieck 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 Poincaré-Birkhoff 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 prove the four-color map theorem); 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."

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Hermann Klaus Hugo (Peter) Weyl__
(1885-1955) 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, Diophantine approximations, and the foundations of mathematics, but he is most respected for his revolutionary advances in geometric function theory (e.g., differentiable manifolds), the theory of compact groups (incl. representation theory), and theoretical physics (e.g., Weyl tensor, gauge field theory and invariance). For a while, Weyl was a disciple of Brouwer's Intuitionism and helped advance that doctrine, but he eventually found it too restrictive. Weyl was also a very influential figure in all three major fields of 20th-century physics: relativity, unified field theory and quantum mechanics. Because of his contributions to Schrödinger, many think the latter's famous result should be named Schrödinger-Weyl 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."

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John Edensor Littlewood__
(1885-1977) England

John Littlewood was a very prolific researcher. (This fact is obscured somewhat in that many papers were co-authored with Hardy, and their names were always given in alphabetic order.) The tremendous span of his career is suggested by the fact that he won Smith's Prize (and Senior Wrangler) in 1905 and the Copley Medal in 1958. He specialized in analysis and analytic number theory but also did important work in combinatorics, Fourier theory, Diophantine approximations, differential equations, and other fields. He also did important work in practical engineering, creating a method for accurate artillery fire during the First World War, and developing equations for radio and radar in preparation for the Second War. He worked with the Prime Number Theorem and Riemann's Hypothesis; and proved the unexpected fact that Chebyshev's bias, and, while true for most, and all but very large, numbers, are violated infinitely often. (Building on this result, it is now known that there is a big patch of primes near 10Li(x)>π(x)^{9608}that exceed the Li(x) prediction, though few if any of those primes are actually known.) Some of his work was elementary, e.g. his elegant proof that a cube cannot be dissected into unequal cubes; but most of his results were too specialized to state here, e.g. his widely-applied4/3 Inequalitywhich 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's response was that it was possible to betoostrong of a mathematician, "forcing through, where another might be driven to a different, and possibly more fruitful, approach."

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Srinivasa Ramanujan Iyengar__
(1887-1920) India

Like Abel, Ramanujan was a self-taught prodigy who lived in a country distant from his mathematical peers, and suffered from poverty: childhood dysentery and vitamin deficiencies probably led to his early death. Yet he produced 4000 theorems or conjectures in number theory, algebra, and combinatorics. While some of these were old theorems or just curiosities, many were brilliant new theorems with very difficult proofs. For example, he found a beautiful identity connecting Poisson summation to the Möbius function. He also found a brilliant generalization of Lagrange's Four Square Theorem; and much more. Ramanujan 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, the divisor function, gamma functions, "mock theta" functions, hypergeometric series, and "highly composite" numbers. Ramanujan's "Master Theorem" has wide application in analysis, and has been applied to the evaluation of Feynman diagrams. Much of his best work was done in collaboration with Hardy, for example a proof that almost all numbersnhave aboutlog log nprime 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 that1+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 top(), although Hardy was initially awed by Ramanujan's intuitive certainty about the existence of such a formula, and even the form it would have. (Rademacher and Selberg later discovered an exact expression to replace the Hardy-Ramanujan approximation; 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. Mathematicians are baffled that Ramanujan could make these conjectures, which they confirmed only with difficulty using methods not available in Ramanujan's day.

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, some consider Ramanujan to be the greatest mathematical genius ever; but he ranks as low as #19 because his work lacked great influence.

Because of its fast convergence, an odd-looking formula of Ramanujan is sometimes used to calculate

π:

99^{2}/ π = √8 ∑_{k=0,∞}((4k)! (1103+26390 k) / (k!^{4}396^{4k}))

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Thoralf Albert Skolem__
(1887-1963) Norway

Thoralf Skolem proved fundamental theorems of lattice theory, proved the Skolem-Noether Theorem of algebra, also worked with set theory and Diophantine equations; but is best known for his work in logic, metalogic, and non-standard models. Some of his work preceded similar results by Gödel. He developed a theory of recursive functions which anticipated some computer science. He worked on the famous Löwenheim-Skolem Theorem which has the "paradoxical" consequence that systems with uncountable sets can have countable models. ("Legend has it that Thoralf Skolem, up until the end of his life, was scandalized by the association of his name to a result of this type, which he considered an absurdity, nondenumerable sets being, for him, fictions without real existence.")

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George Pólya__
(1887-1985) 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 teachingHow to Solve It, the craft of problem posing and proof. He is also famous for the Pólya Enumeration Theorem. Several other important theorems he proved include the Pólya-Vinogradov Inequality of number theory, the Pólya-Szego Inequality of functional analysis, and the Pólya Inequality of measure theory. He introduced the Hilbert-Pólya Conjecture that the Riemann Hypothesis might be a consequence of spectral theory (in 2017 this Conjecture was partially proved by a team of physicists, and the Riemann Hypothesismightbe close to solution!). 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. His work on plane symmetry groups directly inspired Escher's drawings. Having huge breadth and influence, Pólya has been called "the most influential mathematician of the 20th century."

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Stefan Banach__
(1892-1945) Poland

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

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Norbert Wiener__
(1894-1964) 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 one of the first to show a definition of ordered pair as a set. (Hausdorff also proposed such a definition; both Wiener's and Hausdorff's definitions have been superceded by Kuratowski's(a, b) = {{a}, {a, b}}despite that it leads to a singleton when a=b.) He then did important work in several topics in applied mathematics, including stochastic processes (beginning with Brownian motion), potential theory, Fourier analysis, the Wiener-Hopf 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.

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Carl Ludwig Siegel__
(1896-1981) 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, transcendental number theory, discontinuous groups, the 3-body problem in celestial mechanics, and symplectic geometry. In complex analysis he developed Siegel modular forms, which have wide application in math and physics. He may share credit with Alexander Gelfond for the solution to Hilbert's 7th Problem. 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.

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Pavel Sergeevich Aleksandrov__
(1896-1962) Russia

Aleksandrov worked in set theory, metric spaces and several fields of topology, where he developed techniques of very broad application. He pioneered the studies of compact and bicompact spaces, and homology theory. He laid the groundwork for a key theorem of metrisation. His most famous theorem may be his discovery about "perfect subsets" when he was just 19 years old. Much of his work was done in collaboration with Pavel Uryson and Heinz Hopf. Aleksandrov was an important teacher; his students included Lev Pontryagin.

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Emil Artin__
(1898-1962) Austria, Germany, U.S.A.

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

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Paul Adrien Maurice Dirac__
(1902-1984) England, U.S.A.

Dirac had a severe father and was bizarrely taciturn (perhaps autistic), but became one of the greatest mathematical physicists ever. He developed Fermi-Dirac statistics, applied quantum theory to field theory, predicted the existence of magnetic monopoles, and was first to note that some quantum equations lead to inexplicable infinities. His most important contribution was to combine relativity and quantum mechanics by developing, with pure thought, the Dirac Equation. From this equation, Dirac deduced the existence of anti-electrons, a prediction considered so bizarre it was ignored -- until anti-electrons were discovered in a cloud chamber four years later. For this work he was awarded the Nobel Prize in Physics at age 31, making him one of the youngest Laureates ever. Dirac's mathematical formulations, including his Equation and the Dirac-von Neumann axioms, underpin all of modern particle physics. After his great discovery, Dirac continued to do important work, some of which underlies modern string theory. He was also adept at more practical physics; although he declined an invitation to work on the Manhattan Project, he did contribute a fundamental result in centrifuge theory to that Project.The Dirac Equation was one of the most important scientific discoveries of the 20th century and Dirac was certainly a superb mathematical genius, but I've left Dirac off of the Top 100 since his work didn't advance "pure" mathematics. Like many of the other greatest mathematical physicists (Kepler, Einstein, Weyl), Dirac thought the true equations of physics must have beauty, writing "... it is more important to have beauty in one's equations than to have them fit experiment ... [any discrepancy may] get cleared up with further development of the theory."

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Alfred Tarski__
(1902-1983) 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 Banach-Tarski 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 Tarski-Grothendieck Set Theory, Tarski's Fixed-Point Theorem of lattice theory (from which the famous Cantor-Bernstein-Schröder 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 was first to enunciate the remarkable fact that the Generalized Continuum Hypothesis implies the Axiom of Choice, although proof had to wait for Sierpinski. Tarski's other notable accomplishments include his cylindrical algebra, ordinal algebra, universal algebra, and an elegant and novel axiomatic basis of geometry.

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

John von Neumann (born Neumann Janos Lajos) was a childhood prodigy who could do very complicated mental arithmetic at an early age. As an adult he was noted for hedonism and reckless driving but also became one of the most prolific geniuses in history, making major contributions in many branches of 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; and led von Neumann to propose the Doctrine of Mutual Assured Destruction which was a basis for Cold War strategy. He developed cellular automata (first invented by Stanislaw Ulam), famously constructing a self-reproducing automaton. He worked in mathematical foundations: he formulated the Axiom of Regularity and invented elegant definitions for the counting numbers (

0 = {},n+1 = n ∪ {n}). He also worked in analysis, matrix theory, measure theory, numerical analysis, ergodic theory, group representations, continuous geometry, statistics and topology. Von Neumann discovered an ingenious area-conservation paradox related to the famous Banach-Tarski volume-conservation paradox. He inspired some of Gödel's famous work (and independently proved Gödel's Second Theorem). He is credited with (partial) solution to Hilbert's 5th Problem using the Haar Theorem; this also relates to quantum physics. George Pólya once said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper."Von Neumann did very important work in fields other than pure mathematics. By treating the universe as a very high-dimensional phase space, he constructed an elegant mathematical basis (now called von Neumann algebras) for the principles of quantum physics. He advanced philosophical questions about time and logic in modern physics. He played key roles in the design of conventional, nuclear and thermonuclear bombs; he also advanced the theory of hydrodynamics. He applied game theory and Brouwer's Fixed-Point Theorem to economics, becoming a major figure in that field. His contributions to computer science are many: in addition to co-inventing the stored-program computer, he was first to use pseudo-random number generation, finite element analysis, the merge-sort algorithm, a "biased coin" algorithm, and (though Ulam first conceived the approach) Monte Carlo simulation. By implementing wide-number software he joined several other great mathematicians (Archimedes, Apollonius, Liu Hui, Hipparchus, Madhava, and (by proxy), Ramanujan) in producing the best approximation to

of his time. At the time of his death, von Neumann was working on a theory of the human brain.π

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Andrey Nikolaevich Kolmogorov__
(1903-1987) Russia

Kolmogorov had a powerful intellect and excelled in many fields. As a youth he dazzled his teachers by constructing toys that appeared to be "Perpetual Motion Machines." At the age of 19, he achieved fame by finding a Fourier series that diverges almost everywhere, and decided to devote himself to mathematics. He is considered the founder of the fields of intuitionistic logic, algorithmic complexity theory, and (by applying measure theory) modern probability theory. He also excelled in topology, set theory, trigonometric series, and random processes. 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" Kolmogorov-Arnold-Moser (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 Chapman-Kolmogorov equations, the Borel-Kolmogorov Paradox, and the intriguing Zero-One Law of "tail events" among random variables.

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Henri Paul Cartan__
(1904-2008) France

Henri Cartan, son of the great Élie Cartan, is particularly noted for his work in algebraic topology, and analytic functions; but also worked with sheaves, and many other areas of mathematics. He was a key member of the Bourbaki circle. (That circle was led by Weil, emphasized rigor, produced important texts, and introduced terms likein-, sur-,andbi-jection, as well as theØsymbol.) Working with Samuel Eilenberg (also a Bourbakian), Cartan advanced the theory of homological algebra. He is most noted for his many contributions to the theory of functions of several complex variables. Henri Cartan was an important influence on Grothendieck and others, and an excellent teacher; his students included Jean-Pierre Serre.

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Kurt Gödel__
(1906-1978) Germany, U.S.A.

Gödel, who had the nicknameHerr Warum("Mr. Why") as a child, was perhaps the foremost logic theorist ever, clarifying the relationships between various modes of logic. He partially resolved both Hilbert's 1st and 2nd Problems, the latter with a proof so remarkable that it was connected to the drawings of Escher and music of Bach in the title of a famous book. He was a close friend of Albert Einstein, and was first to discover "paradoxical" solutions (e.g. time travel) to Einstein's equations. About his friend, Einstein later said that he had remained at Princeton's Institute for Advanced Study merely "to have the privilege of walking home with Gödel." (Like a few of the other greatest 20th-century mathematicians, Gödel was very eccentric.)Two of the major questions confronting mathematics are: (1) are its axioms consistent (its theorems all being true statements)?, and (2) are its axioms complete (its true statements all being theorems)? Gödel turned his attention to these fundamental questions. He proved that first-order logic was indeed

complete, but that the more powerful axiom systems needed for arithmetic (constructible set theory) were necessarilyincomplete. He also proved that the Axioms of Choice (AC) and the Generalized Continuum Hypothesis (GCH) wereconsistentwith set theory, but that set theory's own consistency could not be proven. He may have established that the truths of AC and GCH wereindependentof 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 "" IfG(this statement itself) cannot be proven.Gcould be proven it would be a contradictory true statement, so consistency dictates that it indeedcannotbe proven. But that's whatGsays, soGis true! This sounds like mere word play, but building from ordinary logic and arithmetic Gödel was able to construct statementGrigorously.

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André Weil__
(1906-1998) 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. HisWeil conjectureswere 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."

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Shiing-Shen Chern__
(1911-2004) China, U.S.A.

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

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Alan Mathison Turing__
(1912-1954) 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 Gödel's Incompleteness Theorem), and developed the concept of machine intelligence (including his famousTuring Testproposal). He also introduced the notions ofdefinable numberandoracle(important in modern computer science), and was an early pioneer in the study of neural networks. For this work he is called the Father of Computer Science and Artificial Intelligence. 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 clever code-breaking algorithms were his most spectacular contributions at Bletchley Park, he was also a key designer of the Bletchley "Bombe" computer. After the war he helped design other physical computers, as well as theoretical designs; and helped inspire von Neumann's later work. He (and earlier, von Neumann) wrote about the Quantum Zeno Effect which is sometimes called the Turing Paradox. He also studied the mathematics of biology, especially theTuring Patternsof 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.

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

Erdös was a childhood prodigy who became a famous (and famously eccentric) mathematician. He is best known for work in combinatorics (especially Ramsey Theory) and partition calculus, but made contributions across a very broad range of mathematics, including graph theory, analytic number theory, probabilistic methods, and approximation theory. He is regarded as the second most prolific mathematician in history, behind only Euler. Although he is widely regarded as an important and influential mathematician, Erdös founded no new field of mathematics: He was a "problem solver" rather than a "theory developer." He's left us several still-unproven intriguing conjectures, e.g. that4/n = 1/x + 1/y + 1/zhas positive-integer solutions for anyn.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

nand2n, and (though the major contributor was Atle Selberg) Hadamard's Prime Number Theorem itself. He also proved many new theorems, such as the Erdös-Szekeres Theorem about monotone subsequences with its elegant (if trivial) pigeonhole-principle proof.

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Samuel Eilenberg__
(1913-1998) Poland, U.S.A.

Eilenberg is considered a founder of category theory, but also worked in algebraic topology, automata theory and other areas. He coined several new terms includingfunctor,category, andnatural isomorphism. Several other concepts are named after him, e.g. a proof method called theEilenberg telescopeorEilenberg-Mazur Swindle. He worked on cohomology theory, homological algebra, etc. By using his category theory and axioms of homology, he unified and revolutionized topology. Most of his work was done in collaboration with others, e.g. Henri Cartan; but he also single-authored an important text laying a mathematical foundation for theories of computation and language. Sammy Eilenberg was also a noted art collector.

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Israel Moiseevich Gelfand__
(1913-2009) Russia

Gelfand was a brilliant and important mathematician of outstanding breadth with a huge number of theorems and discoveries. He was a key figure of functional analysis and integral geometry; he pioneered representation theory, important to modern physics; he also worked in many fields of analysis, soliton theory, distribution theory, index theory, Banach algebra, cohomology, etc. He made advances in physics and biology as well as mathematics. He won the Order of Lenin three times and several prizes from Western countries. Considered one of the two greatest Russian mathematicians of the 20th century, the two were compared with "[arriving in a mountainous country] Kolmogorov would immediately try to climb the highest mountain; Gelfand would immediately start to build roads." In old age Israel Gelfand emigrated to the U.S.A. as a professor, and won a MacArthur Fellowship.

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Claude Elwood Shannon__
(1916-2001) U.S.A.

Shannon's initial fame was for a paper called "possibly the most important master's thesis of the century." That paper founded digital circuit design theory by proving that universal computation was achieved with an ensemble of switches and boolean gates. He also worked with analog computers, theoretical genetics, and sampling and communication theories. Early in his career Shannon was fortunate to work with several other great geniuses including Weyl, Turing, Gödel and even Einstein; this may have stimulated him toward a broad range of interests and expertise. He was an important and prolific inventor, discovering signal-flow graphs, the topological gain formula, etc.; but also inventing the first wearable computer (to time roulette wheels in Las Vegas casinos), a chess-playing algorithm, a flame-throwing trumpet, and whimsical robots (e.g. a "mouse" that navigated a maze). His hobbies included juggling, unicycling, blackjack card-counting. His investigations into gambling theory led to new approaches to the stock market.Shannon worked in cryptography during World War II; he was first to note that a one-time pad allowed unbreakable encryption as long as the pad was as large as the message; he is also noted for Shannon's maxim that a code designer should assume the enemy knows the system. His insights into cryptology eventually led to information theory, or the mathematical theory of communication, in which Shannon established the relationships among bits, entropy, power and noise. It is as the Founder of Information Theory that Shannon has become immortal.

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Atle Selberg__
(1917-2007) Norway, U.S.A.

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

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John Torrence Tate__
(1925-) U.S.A.

Tate, a student of Emil Artin, is a master of algebraic number theory, p-adic theory and arithmetic geometry. Using Fourier analysis and Tate cohomology groups, he revolutionized the treatments of class field theory and algebraic K-theory. In addition to Tate cohomology groups, Tate's key inventions include rigid analytic geometry, Hodge-Tate theory, Tate-Barsotti groups, applications of adele ring self-duality, the Tate module, Tate curve, Tate twists, and much more. His long and productive career earned the Abel Prize for his "vast and lasting impact on the theory of numbers [and] his incisive contributions and illuminating insights ... He has truly left a conspicuous imprint on modern mathematics."

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Jean-Pierre 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, p-adic 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.

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Alexandre Grothendieck__
(1928-2014) 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, the theory of motives, generalized the Riemann-Roch Theorem to revolutionize K-theory, developed Grothendieck categories, crystalline cohomology, infinity-stacks and more. The guiding principle behind much of Grothendieck's work has been Topos Theory, which he invented to harness the methods of topology. These methods and results have redirected several diverse branches of modern mathematics including number theory, algebraic topology, and representation theory. Among Grothendieck's famous results was his Fundamental Theorem in the Metric Theory of Tensor Products, which was inspired by Littlewood's proof of the4/3 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.

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John Forbes Nash, Jr.__
(1928-2015) U.S.A

TheRiemann Embedding Problemswere important puzzles of geometry that baffled many of the greatest minds for a century. Hilbert showed that Lobachevsky's hyperbolic plane could not be embedded into Euclidean 3-space, but what about into Euclidean 4-space? Cartan and Chern were among the great mathematicians who solved various special cases, but using "methods entirely without precedent" John Nash demonstrated a general solution. This was a true highlight of 20th-century mathematics.Nash was a lonely, tormented schizophrenic whose life was portrayed in the film

Beautiful Mind. He achieved early fame with his work in game theory; this eventually led to the Nobel Prize in Economics. Earlier studies in game theory focused on the simplest cases (two-person zero-sum, or cooperative), but Nash demonstrated "Nash equilibria" for n-person or non-zero-sum non-cooperative games. Nash also excelled at several other fields of mathematics, especially topology, algebraic geometry, partial differential equations, elliptic functions, and the theory of manifolds (including singularity theory, the concept of real algebraic manifolds and isotropic embeddings). He proved theorems of great importance which had defeated all earlier attempts. His most famous theorems were the Nash Embedding Theorems, e.g. that any Riemannian manifold of dimensionkcan be embedded isometrically into some n-dimensional Euclidean space. Other important work was in partial differential equations where he proved that strong regularity constraints apply to solutions of the equations of heat and fluid flow.

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Lennart Axel Edvard Carleson__
(1928-) Sweden

Carleson is a master of complex analysis, especially harmonic analysis, and dynamical systems; he proved many difficult and important theorems; among these are a theorem about quasiconformal mapping extension, a technique to construct higher dimensional strange attractors, and the famous Kakutani Corona Conjecture, whose proof brought Carleson great fame. For the Corona proof he introduced Carleson measures, one of several useful tools he's created for his masterful proofs. In 1966, four years after proving Kakutani's Conjecture, he proved the 53-year old Luzin's Conjecture, a strong statement about Fourier convergence. This was startling because of a 38-year old conjecture suggested by Kolmogorovthat Luzin's Conjecture was false.

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Michael Francis (Sir)
Atiyah__
(1929-) Britain

Atiyah's career has had extraordinary breadth and depth. He advanced the theory of vector bundles; this developed into topological K-theory and the Atiyah-Singer Index Theorem. This Index Theorem is considered one of the most far-reaching theorems ever, subsuming famous old results (Descartes' total angular defect, Euler's topological characteristic), important 19th-century theorems (Gauss-Bonnet, Riemann-Roch), and incorporating important work by Weil and especially Shiing-Shen Chern. It is a key to the study of high-dimension spaces, differential geometry, and equation solving. Several other key results are named after Atiyah, e.g. the Atiyah-Bott Fixed-Point Theorem, the Atiyah-Segal Completion Theorem, and the Atiyah-Hirzebruch spectral sequence. Atiyah's work developed important connections not only between topology and analysis, but with modern physics; Atiyah himself has been a key figure in the development of string theory. This work, and Atiyah-inspired work in gauge theory, restored a close relationship between leading edge research in mathematics and physics. Atiyah is known as a vivacious genius in person, inspiring many, e.g. Edward Witten. Along with Serre, Atiyah is often considered to be one of the very greatest living mathematicians.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."

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John Willard
Milnor__
(1931-) U.S.A.

Milnor founded the field of differential topology and has made other major advances in 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" 7-dimensional hyperspheres gave the first examples of homeomorphic manifolds that were not also diffeomorphic, and developed the fields of differential topology andsurgery theory. Milnor invented certain high-dimensional polyhedra to disprove theHauptvermutung("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 low-dimensional systems.As Fields, Presidential and (twice) Putnam Medalist, as well as winner of the Abel, Wolf and

twoSteele Prizes; Milnor can be considered the most "decorated" mathematician of the modern era.

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John Griggs
Thompson__
(1932-) U.S.A.

Thompson is the master of finite groups. He achieved early fame by proving a long-standing conjecture about Frobenius groups. He followed up by proving (with Walter Feit) that all nonabelian finite simple groups are of even order. This result, proved in a 250-page paper, stunned the world of mathematics; it led to the classification of all finite groups. Thompson also made major contributions to coding theory, and to the inverse Galois problem. His work with Galois groups has been called "the most important advance since Hilbert's time."

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Robert Phelan
Langlands__
(1936-) Canada, U.S.A.

Langlands started by studying semigroups and partial differential equations but soon switched his attention to representation theory where he found deep connections between group theory and automorphic forms; he then used these connections to make profound discoveries in number theory. Langlands' methods, collectively called the Langlands Program, are now central to all of these fields. The Langlands Dual Group^{L}G revolutionized representation theory and led to a large number of conjectures. One of these conjectures is the Principle of Functoriality, of which a partial proof allowed Langlands to prove a famous conjecture of Artin, and Wiles to prove Fermat's Last Theorem. Langlands and others have applied these methods to prove several other old conjectures, and to formulate new more powerful conjectures. He has also worked with Eisenstein series, L-functions, Lie groups, percolation theory, etc. He mentored several important mathematicians (including Thomas Hales, mentioned briefly in Pappus' mini-bio).Langlands once wrote "Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world." He was appointed Hermann Weyl Professor at the Institute for Advanced Study and now sits in the office once occupied by Albert Einstein. This seems appropriate since, as the man "who reinvented mathematics," his advances have sometimes been compared to Einstein's.

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Vladimir Igorevich
Arnold__
(1937-2010) Russia

Arnold is most famous for solving Hilbert's 13th Problem; for the "magnificent" Kolmogorov-Arnold-Moser (KAM) Theorem; and for "Arnold diffusion," which identifies exceptions to the stability promised by the KAM Theorem. In addition to dynamical systems theory, Arnold made contributions to catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, mathematical physics; and was the essential founder of modern singularity theory.

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John Horton
Conway__
(1937-) Britain, U.S.A.

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 Problem, but achieved fame when he discovered the largest then-known sporadic group (the symmetry group of the Leech lattice); this sporadic group is now known to be second in size only to theMonster Group, with which Conway also worked. Conway's fertile creativity has produced a cornucopia of fascinating inventions: markable straight-edge construction of the regular heptagon (a feat also achieved by Alhazen, Thabit, Vieta and perhaps Archimedes), a nowhere-continuous function that has the Intermediate Value property, the Conway box function, the rational tangle theorem in know theory, the aperiodic pinwheel tiling, a representation of symmetric polyhedra, the silly but elegantFractranprogramming language, his chained-arrow notation for large numbers, and many results and conjectures in recreational mathematics. He found the simplest proof for Morley's Trisector Theorem (sometimes called the best result in simple plane geometry since ancient Greece). He proved an unusual theorem about quantum physics: "If experimenters have free will, then so do elementary particles." His most famous construction is the computationally complete automaton known as theGame of Life. His most important theoretical invention, however, may be hissurreal numbersincorporating 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.

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Mikhael Leonidovich
Gromov__
(1943-) Russia, France

Gromov is considered one of the greatest geometers ever, but he has a unique "soft" approach to geometry which leads to applications in other fields: Gromov has contributed to group theory, partial differential equations, other areas of analysis and algebra, and even mathematical biology. He is especially famous for his pseudoholomorphic curves; they revolutionized the study of symplectic manifolds and are important in string theory. By applying his geometric ideas to all areas of mathematics, Gromov has become one of the most influential living mathematicians. He has proved a very wide variety of theorems: important results about groups of polynomial growth, theorems essential to Perelman's proof of the Poincaré conjecture, the nonsqueezing theorem of Hamiltonian mechanics, theorems of systolic geometry, and various inequalities and compactness theorems. Several concepts are named after him, including Gromov-Hausdorff convergence, Gromov-Witten invariants, Gromov's random groups, Gromov product, etc.

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Pierre René
Deligne__
(1944-) Belgium, France, U.S.A.

Using new ideas about cohomology, in 1974 Pierre Deligne stunned the world of mathematics with a spectacular proof of the Weil conjectures. Proof of these conjectures, which were key to further progress in algebraic geometry, had eluded the great Alexandre Grothendieck. With his "unparalleled blend of penetrating insights, fearless technical mastery and dazzling ingenuity," Deligne made other important contributions to a broad range of mathematics in addition to algebraic geometry, including algebraic and analytic number theory, topology, group theory, the Langlands conjectures, Grothendieck's theory of motives, and Hodge theory. Deligne also found a partial solution of Hilbert's 21st Problem. Several ideas are named after him including Deligne-Lusztig theory, Deligne-Mumford stacks, Fourier-Deligne transform, the Langlands-Deligne local constant, Deligne cohomology, and at least eight distinct conjectures.

Shelah has advanced logic, model theory, set theory, and especially the theory of cardinal numbers. His work has led to new methods in diverse fields like group theory, topology, measure theory, stability theory, algebraic geometry, Banach spaces, and combinatorics. He solved outstanding problems like Morley's Problem; proved the independence of the the Whitehead problem; found a "Jonsson group" and counterexamples to outstanding conjectures; improved on Arrow's Voting Theorem; and, most famously, proved key results about singular cardinals.Shelah is the founder of the theories of proper forcing, classification of models, and possible cofinalities. He has authored over 800 papers and several books, making him one of the most prolific mathematicians in history. He has been described as a "phenomenal mathematician, ... produc[ing] results at a furious pace."

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William Paul
Thurston__
(1946-2012) U.S.A.

Thurston revolutionized the study of 3-D manifolds; it was his work which eventually led Perelman to a proof of Poincaré's Conjecture. He developed the key results of foliation theory and, with Gromov, co-founded geometric group theory. One of his award citations states "Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry."

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Terence Chi-Shen
Tao__
(1975-) Australia, U.S.A.

Tao was a phenomenal child prodigy who has become perhaps the greatest living mathematician. He has made important contributions to partial differential equations, combinatorics, harmonic analysis, number theory, group theory, model theory, nonstandard analysis, random matrices, the geometry of 3-manifolds, functional analysis, ergodic theory, etc. and areas of applied math including quantum mechanics, general relativity, and image processing. He has been called the first since David Hilbert to be expert across the entire spectrum of mathematics. Among his earliest important discoveries were results about the multi-dimensional Kakeya needle problem, which led to advances in Fourier analysis and fractals. In addition to his numerous research papers he has written many highly regarded textbooks. One of his prize citations commends his "sheer technical power, his other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view."Much of Tao's work has been done in collaboration: for example with Van Vu he proved the circular law of random matrices; with Ben Green he proved the Dirac-Motzkin conjecture and solved the "orchard-planting problem." Especially famous is the Green-Tao Theorem that there are arbitrarily long arithmetic series among the prime numbers (or indeed among any sufficiently dense subset of the primes). This confirmed an old conjecture by Lagrange, and was especially remarkable because the proof fused methods from number theory, ergodic theory, harmonic analysis, discrete geometry, and combinatorics. Tao is also involved in recent efforts to attack the famous Twin Prime Conjecture.

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