Biographies of the greatest mathematicians are in separate files by birth year:

Born before 400 (this page)
| Born betw. 400 & 1559 | Born betw. 1560 & 1699 |

Born betw. 1700 & 1799 | Born betw. 1800 & 1869 | Born betw. 1870 & 1939 |

List of Greatest Mathematicians |

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, 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.

Also at least 3600 years ago, the Egyptian scribe Ahmes produced a famous manuscript (now called the

Rhind Papyrus), itself a copy of a late Middle Kingdom text. It showed simple algebra methods and included a table giving optimal expressions using Egyptian fractions. (Today, Egyptian fractions lead to challenging number theory problems with no practical applications, but they may have had practical value for the Egyptians. To divide 17 grain bushels among 21 workers, the equation 17/21 = 1/2 + 1/6 + 1/7 has practical value, especially when compared with the "greedy" decomposition 17/21 = 1/2 + 1/4 + 1/17 + 1/1428.)The Pyramids demonstrate that Egyptians were adept at geometry. Yet, although they knew the formula for a pyramid's volume, there's no evidence they even knew the Pythagorean Theorem! Babylon was much more advanced than Egypt at arithmetic and algebra; this was probably due, at least in part, to their place-value system. But although their base-60 system survives (e.g. in the division of hours and degrees into minutes and seconds) the Babylonian notation, which used the equivalent of IIIIII XXXXXIIIIIII XXXXIII to denote 417+43/60, was unwieldy compared to the "ten digits of the Hindus."

The Egyptians used the approximation

π ≈ (4/3)(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, though little written evidence survives prior to Chang Tshang's famous book.

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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 very first mathematical proof!) 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 the first known proof of the Pythagorean Theorem. Apastambha's work uses the excellent (continued fraction) approximation√2 ≈ 577/408, a result probably derived with a geometric argument.Apastambha built on the work of earlier Vedic scholars, especially Baudhayana, as well as Harappan and (probably) Mesopotamian mathematicians. His notation and proofs were primitive, and there is little certainty about his life. However similar comments apply to Thales of Miletus, so it seems fair to mention Apastambha (who was perhaps the most creative Vedic mathematician before Panini) along with Thales as one of the earliest mathematicians whose name is known.

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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 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). 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.)

The Pythagorean Theorem was known long before Pythagoras, but he is often credited with the first

proof. (Apastambha proved it in India at about the same time; some conjecture that Pythagoras journeyed to India and learned of the proof there.) He may have discovered the simple parametric form of 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."

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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.

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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; his circle quadrature was of course ultimately unsuccessful but he did prove ingenious theorems about "lunes" (certain circle fragments); and some claim Hippocrates was first to trisect the general angle. Hippocrates also did work in algebra and rudimentary analysis.(Doubling the cube and angle trisection are often called "impossible," but they are impossible only when restricted to compass and unmarkable straightedge. There are ingenious solutions available with other tools. Construction of the regular heptagon is another such task, with solutions published by four of the men on this List.)

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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 and Menaechmus. In addition to discoveries always attributed to him, he may be the source of several of Euclid's theorems, and some works attributed to Eudoxus and perhaps Pythagoras. Recently it has been shown that the 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. 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 of an integer, if not itself an integer, must be irrational.

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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; he is credited with first use of theAxiom of Archimedes, which avoids Zeno's paradoxes by, in effect, forbidding infinities and infinitesimals; yet he also developed a method of taking limits. Eudoxus' work with irrational numbers and infinitesimals may have helped inspire such masters as Archimedes and Dedekind. Eudoxus also introduced 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.")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, unfortunate since many of his ideas were quite mistaken.) He was personal tutor to the young Alexander the Great. Aristotle's writings on definitions, axioms and proofs may have influenced Euclid. He was also the first mathematician to write on the subject of infinity. His writings include geometric theorems, some with proofs different from Euclid's or missing from Euclid altogether; one of these (which is seen only in Aristotle's work prior to Apollonius) is that a circle is the locus of points whose distances from two given points are in constant ratio. Even if, as is widely agreed, Aristotle's geometric theorems were not his own work, his status as the most influential logician and philosopher makes him a candidate for the List.

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

Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers; he stated and proved the Unique Factorization Theorem; and he 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 many are attributed to predecessors like Hippocrates, Theodorus, Eudoxus, Archytas and Theaetetus. He may have proved that rigid-compass constructions can be implemented with collapsing-compass constructions. Although notions of trigonometry were not in use, Euclid's theorems include some closely related to the Laws of Sines and Cosines. Among several books attributed to Euclid 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 which has been rediscovered and properly studied only since 1998. Ideas unique to that work are an anticipation of Riemann integration, calculating the volume of a cylindrical wedge (previously first attributed to Kepler); 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.)

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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 perhaps Archytas are other candidates for this honor.) His writings on conic sections have been studied until modern times; he invented the names for parabola, hyperbola and ellipse; he developed methods for normals and curvature. Although astronomers eventually concluded it was not physically correct, Apollonius developed the "epicycle and deferent" model of planetary orbits, and proved important theorems in this area. He deliberately emphasized the beauty of pure, rather than applied, mathematics, saying his theorems were "worthy of acceptance for the sake of the demonstrations themselves."Since many of his works have survived only in a fragmentary form, several great Renaissance and Modern mathematicians (including Vieta, Fermat, Pascal and Gauss) have enjoyed reconstructing and reproving his "lost" theorems. (Among these, the most famous is to construct a circle tangent to three other circles.)

In evaluating the genius of the ancient Greeks, it is well to remember that their achievements were made without the convenience of modern notation. It is clear from his writing that Apollonius almost developed the analytic geometry of 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, 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 might be considered the greatest astronomer ever (ahead even of Galileo and Edwin Hubble). 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 Chaldean 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 was probably 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. (Some attrubute the mechanism to 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. Disdaining indirect proofs (in imitation of 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 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.The mystery of celestial motions directed scientific inquiry for thousands of years. The problem had been considered by Eudoxus, Apollonius, and Hipparchus, who developed a very complicated geocentric model involving concentric spheres and epicyles. Ptolemy perfected (or, rather, complicated) this model even further; his model was the standard for 14 centuries. While some Greeks, notably Aristarchus, proposed heliocentric models, these were rejected because there was no parallax among stars. (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 in

The 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, and Regiomontanus are other great mathematicians who at least 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. (Since the planets move without friction, their motions offer a pure view of the Laws of Motion; thus the heliocentric breakthroughs of Copernicus, Kepler and Newton triggered the advances in mathematical physics which led to Scientific Revolution.)

Liu Hui made major improvements to Chang's influential 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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