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.):

Earliest mathematicians Little is known of the earliest mathematics, but the famous Ishango Bone from Early Stone-Age Africa has tally marks suggesting arithmetic. The markings include six prime numbers (5, 7, 11, 13, 17, 19) in order, though this is probably coincidence. The advanced artifacts of Egypt's Old Kingdom and the Indus-Harrapa civilization imply strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where 4500-year old clay tablets show multiplication and division problems; the first abacus may be about this old. By 3600 years ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, and even logarithms 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)4 (derived from the idea that a circle of diameter 9 has about the same area as a square of side 8). Although the ancient Hindu mathematician Apastambha had achieved a good approximation for √2, and the ancient Babylonians an ever better √2, neither of these ancient cultures achieved a π approximation as good as Egypt's, or better than π ≈ 25/8, until the Alexandrian era. The sudden blossoming of math in the Iron Ages of India and Greece owes much to the ancient mathematics of Egypt and Babylonia. Early Vedic mathematicians The greatest mathematics before the Golden Age of Greece was in India's early Vedic (Hindu) civilization. The Vedics understood relationships between geometry and arithmetic, developed astronomy, astrology, calendars, and used mathematical forms in some religious rituals. The earliest mathematician to whom definite teachings can be ascribed was Lagadha, who apparently lived about 1300 BC and used geometry and elementary trigonometry for his astronomy. Baudhayana lived about 800 BC and also wrote on algebra and geometry; Yajnavalkya lived about the same time and is credited with the then-best approximation to π. Apastambha did work summarized below; other early Vedic mathematicians solved quadratic and simultaneous equations. Other early cultures also developed some mathematics. The ancient Mayans apparently had a place-value system with zero before the Hindus did; Aztec architecture implies practical geometry skills. Ancient China certainly developed mathematics, in fact the first known proof of the Pythagorean Theorem is found in a Chinese book (Zhoubi Suanjing) which might have been written about 1000 BC.

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 length x, a segment of length x/k can be constructed by first constructing a segment of length kx. Thales was also an astronomer; he invented the 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. While more ancient mathematicians were concerned with practical calculations, modern mathematics began with the Greek emphasis on proofs and philosophy. Pythagoras and Parmenides of Elea also played key roles in that development which began with Thales. These ideas led to the schools of Plato, Aristotle and Euclid, and an intellectual blossoming unequaled until Europe's Renaissance. Thales is ranked #38 on the Pantheon List of Most Popular and Productive Persons. 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.

Apastambha (ca 630-560 BC) India

The Dharmasutra composed by Apastambha contains mensuration techniques, novel geometric construction techniques, a method of elementary algebra, and what may be 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.

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's Elements. 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 millennia. The words philosophy and mathematics are said to have been coined by Pythagoras. He is supposed to have invented the Pythagorean Cup, a clever wine goblet which punishes a drinker who greedily fills his cup to the top by then using siphon pressure to drain the cup. Pythagoras ranks #10 on the Pantheon Popularity/Productivity List, but despite this 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 (who was first to claim that thinking occurred in the brain rather than heart), Milo of Croton, and Milo's daughter Theano (who may have been Pythagoras's wife). The term Pythagorean was also adopted by many disciples who lived later; these disciples include Philolaus of Croton, the natural philosopher Empedocles, and several other famous Greeks. Pythagoras' successor was apparently Theano herself: the Pythagoreans were one of the few ancient schools to practice gender equality. Pythagoras discovered that harmonious intervals in music are based on simple rational numbers. This led to a fascination with integers and mystic numerology; he is sometimes called the "Father of Numbers" and once said "Number rules the universe." (About the mathematical basis of music, Leibniz later wrote, "Music is the pleasure the human soul experiences from counting without being aware that it is counting." Other mathematicians who investigated the arithmetic of music included Huygens, Euler and Simon Stevin.) Given any numbers a and b the Pythagoreans were aware of the three distinct means: (a+b)/2 (arithmetic mean), √(ab) (geometric mean), and 2ab/(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's Elements. 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.

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

Panini's great accomplishment was his study of the Sanskrit language, especially in his text Ashtadhyayi. Although this work might be considered the very first study of linguistics or grammar, it used a non-obvious elegance that would not be 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."

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.

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

Hippocrates (no known relation to Hippocrates of Cos, the famous physician) wrote his own Elements more than a century before Euclid. Only fragments survive but it apparently used axiomatic-based proofs similar to Euclid's and contains many of the same theorems. Hippocrates is said to have invented the reductio ad absurdem proof method. Hippocrates is most famous for his work on the three ancient geometric quandaries: his work on cube-doubling (the Delian Problem) laid the groundwork for successful efforts by Archytas and others; 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.)

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 magnificent Mechanical Problems attributed to (pseudo-)Aristotle were probably actually written by Archytas, making him one of the greatest mathematicians of antiquity. Archytas introduced "motion" to geometry, rotating curves to produce solids. If his writings had survived he'd surely be considered one of the most brilliant and innovative geometers of antiquity. 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 term harmonic mean and worked with geometric means as well (proving that consecutive integers never have rational geometric mean). He was a true polymath: he advanced the theory of music far beyond Pythagoras; studied sound, optics and cosmology; invented the pulley (and a rattle to occupy infants); wrote about the lever; developed the curriculum called quadrivium; is credited with inventing the screw; 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.

Theaetetus of Athens (417-369 BC) Greece

Theaetetus is presumed to be the true author of Books X and XIII of Euclid's Elements, 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 any integer, if not itself an integer, must be irrational. (The case √2 is attributed to a student of Pythagoras.)

Eudoxus of Cnidus (408-355 BC) Greek domain

Eudoxus journeyed widely for his education, despite that he was not wealthy, studying mathematics with Archytas in Tarentum, medicine with Philiston in Sicily, philosophy with Plato in Athens, continuing his mathematics study in Egypt, touring the Eastern Mediterranean with his own students and finally returned to Cnidus where he established himself as astronomer, physician, and ethicist. What is known of him is second-hand, through the writings of Euclid and others, but he was one of the most creative mathematicians of the ancient world. Many of the theorems in Euclid's Elements were first proved by Eudoxus. While Pythagoras had been horrified by the discovery of irrational numbers, Eudoxus is famous for incorporating them into arithmetic. He also developed the earliest techniques of the infinitesimal calculus; Archimedes credits Eudoxus with inventing a principle eventually called the Axiom 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 an Axiom of Continuity; he was a pioneer in solid geometry; and he developed his own solution to the Delian cube-doubling problem. Eudoxus was the first great mathematical astronomer; he developed the complicated ancient theory of planetary orbits; and may have invented the astrolabe. 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 is often called 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 π as the limit of polygonal perimeters. None of these seems difficult today, but it does seem remarkable that they were all first achieved by the same man. Eudoxus has been quoted as saying "Willingly would I burn to death like Phaeton, were this the price for reaching the sun and learning its shape, its size and its substance."

Aristotle of Stagira (384-322 BC) Macedonia

Aristotle was the most prominent scientist of the ancient world, and perhaps 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 disciple and successor Theophrastus was also a great scientist, as was Theophrastus' successor Strato of Lampsacus. Although Aristotle was probably the greatest biologist of the ancient world, his work in physics and mathematics may not seem enough to qualify for this list. But his teachings covered a very wide gamut and dominated the development of ancient science. His writings on definitions, axioms and proofs may have influenced Euclid; and he was one of the first mathematicians 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. A charge sometimes made against Aristotle is that his wrong ideas held back the development of science. But this charge is unfair; Aristotle himself stressed the importance of observation and experimentation, and to be ready to reject old hypotheses and prepare new ones. Of course many of his ideas turned out to be wrong, but many were correct: e.g. the explanation that sound comes from the vibration of air. And even if, as is widely agreed, Aristotle's geometric theorems were not his own work, his status as the most influential logician and philosopher in all of history makes him a strong candidate for the List.

The Nature of Time and Space

Although he was a disciple of Plato, Aristotle had very different ideas from his tutor on many topics. While Plato felt that time and space were absolute entities that would exist even in a void without objects or events, Aristotle believed that time and space were defined by objects and events, that eventless time or perfect vacuums were absurdities. Aristotle's common-sense view was the default until Newton stipulated notions of absolute mathematical time and space to make his equations of motion sensical. This latter view seems like common-sense today, but it is said that this modern view started with the Newtonian revolution. While Newton took Plato's side in the debate about the nature of time and space, his rival Leibniz took Aristotle's side. These opposing views were finally subsumed by wholly new notions of time and space introduced by one of the greatest of all geniuses: Albert Einstein.

Strato of Lampsacus (ca 335-269 BC) Greece

Although not a mathematician, the natural philosopher Strato is worth mention. He made many improvements to Aristotle's natural philosophy, including improved understandings of time, space, acceleration and continuity; and may have been the true author of the Mechanics text originally attributed to Aristotle. He also wrote about moral philosophy. He equated God with natural law, so was perhaps the first atheist. His thinking has been compared to that of Baruch Spinoza.

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 produced an incomplete proof of the Unique Factorization Theorem (Fundamental Theorem of Arithmetic); and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect (in the sense of Pythagoras) if M is Mersenne. (The converse, that any even perfect number has such a corresponding Mersenne prime, was tackled by Alhazen and proven by Euler.) His books contain many famous theorems, though much of the Elements was 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.) Book III Proposition 35 proves that when chords of a circle intersect, the products of their segments are equal. Although notions of trigonometry were not in use, Euclid's theorems include some closely related to the Laws of Sines and Cosines. Among several books attributed to Euclid are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), a book on spherical geometry, a book on logic fallacies, and his comprehensive math textbook The Elements. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry. Euclid ranks #14 on Michael Hart's famous list of the Most Influential Persons in History. The Elements introduced the notions of axiom and theorem; was used as a textbook for 2000 years; and in fact is still the basis for high school geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem. There are many famous quotations about Euclid and his books. Abraham Lincoln abandoned his law studies when he didn't know what "demonstrate" meant and "went home to my father's house [to read Euclid], and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."

Archimedes of Syracuse (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 might have been first to prove Heron's formula for the area of a triangle. His excellent approximation to √3 indicates that he'd partially anticipated the method of continued fractions. He developed a recursive method of representing large integers, and was first to note the law of exponents, 10a·10b = 10a+b. Working with exponents, he developed simple notation and names for numbers larger than 10^(10^16); this will seem more startling when you recall that it was another 18 centuries before Europeans would invent the word "million." Archimedes found a method to trisect an arbitrary angle (using a markable straightedge — the construction is impossible using strictly Platonic rules). One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series. Some of Archimedes' work survives only because Thabit ibn Qurra translated the otherwise-lost Book 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 the Archimedean semiregular solids reported 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 discover results, and then devised rigorous geometric proofs for publication. Archimedes used integral calculus to determine the centers of mass of hemisphere and cylindrical wedge, and the volume of two cylinders' intersection. He also worked with various spirals, paraboloids of revolution, etc. 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. (Although familiar with the utility of infinitesimals, he accepted the "Theorem of Eudoxus" which bans them to avoid Zeno's paradoxes. Modern mathematicians refer to that "Theorem" as the Axiom of Archimedes.) Archimedes was an astronomer (details of his discoveries are lost, but it is likely he knew the Earth rotated around the Sun). He was one of the greatest mechanists ever, discovering Archimedes' Principle of Hydrostatics. (A body partially or completely immersed in a fluid effectively loses weight equal to the weight of the fluid it displaces. Archimedes is famous for testing the purity of his King's gold crown, but he didn't write up his solution; it was finally Galileo who pointed out that a test based on measuring water displacement, as had been assumed to be Archimedes' "Eureka!" method, would be extremely imprecise. Instead Archimedes must have applied the less trivial corollaries of his Principle of Hydrostatics by comparing a balance scale's reading in and out of water.) Archimedes developed the mathematical foundations underlying the advantage of basic machines: lever, screw and compound pulley. Although Archytas perhaps invented the screw, and Stone-Age man (and even other animals) used levers, it is said that the compound pulley was invented by Archimedes himself. For these achievements he is widely considered to be one of the three or four greatest theoretical physicists ever. Archimedes was a prolific inventor: in addition to inventing the compound pulley, he invented the hydraulic screw-pump (called Archimedes' screw); a miniature planetarium; and several war machines -- catapult, parabolic mirrors to burn enemy ships, a steam cannon, and 'the Claw of Archimedes.' (Some scholars attribute the Antikythera mechanism to Archimedes -- Is it the Archimedean planetarium mentioned by Cicero? However this is unlikely: the detailed motion of the Moon produced by the mechanism was probably unknown until Hipparchus.) His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, Sphere and Cylinder, Plane Equilibriums, Conoids and Spheroids, Quadrature of Parabola, The Book of Lemmas (translated and attributed by Thabit ibn Qurra), various now-lost works (on Mirrors, Balances and Levers, Semi-regular Polyhedra, etc.) cited by Pappus or others, and (discovered only recently, and often called his most important work) The Method. He developed the Stomachion puzzle (and solved a difficult enumeration problem involving it); other famous gems include The Cattle-Problem. The Book of Lemmas contains 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, π is often called Archimedes' constant. His approximation 223/71 < π < 22/7 was the best of his day. (Apollonius soon surpassed it, but by using Archimedes' method.) 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 Method and Floating 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, Archimedes was among the few to comment on the "equinumerosity paradox" (the fact that there 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 simply too far ahead of his time to have great historical significance. (He does rank #11 on the Pantheon List. Some think the Scientific Revolution would have begun sooner had The Method been discovered four or five centuries earlier. You can read a 1912 translation of parts of The Method on-line.)

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, mechanical engineer (anticipating laws of elasticity, etc.), astronomer (he is credited as first to measure the circumference of the Earth), and 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 dedicated The Method to Eratosthenes.

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."

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 ancient Zhoubi Suanjing), and innovations continued for centuries after Han, the textbook Nine Chapters on the Mathematical Art has special importance. Nine Chapters (known in Chinese as Jiu Zhang Suan Shu or Chiu Chang Suan Shu) was apparently written during the early Han Dynasty (about 165 BC) by Chang Tshang (also spelled Zhang Cang). Many of the mathematical concepts of the early Greeks were discovered independently in early China. Chang's book gives methods of arithmetic (including cube roots) and algebra, uses the decimal system (though zero was represented as just a space, rather than a discrete symbol), proves the Pythagorean Theorem, and includes a clever geometric proof that the perimeter of a right triangle times the radius of its inscribing circle equals the area of its circumscribing rectangle. (Some of this may have been added after the time of Chang; some additions attributed to Liu Hui are mentioned in his mini-bio; other famous contributors are Jing Fang and Zhang Heng.) Nine Chapters was probably based on earlier books, lost during the great book burning of 212 BC, and Chang himself may have been a lord who commissioned others to prepare the book. Moreover, important revisions and commentaries were added after Chang, notably by Liu Hui (ca 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 like Nine Chapters. No one person can be credited with the invention of the decimal system, but key roles were played by early Chinese (Chang Tshang and Liu Hui), Brahmagupta (and earlier Hindus including Aryabhata), and Leonardo Fibonacci. (After Fibonacci, Europe still did not embrace the decimal system until the works of Vieta, Stevin, and Napier.)

Hipparchus of Nicaea 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 the errors in the catalogs of Ptolemy and Hipparchus reveal both that Ptolemy borrowed his data from Hipparchus, and that Hipparchus used principles of spherical trig to simplify his work. Classical Hindu astronomers, including the 6th-century genius Aryabhata, borrow much from Ptolemy and Hipparchus. Most of Hipparchus' work has been lost, and little is known of his life. Even his dates have been deduced from the lunar eclipses he discusses. (Some of his delicate measurements required lunar eclipses.) 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 π than that of Apollonius, perhaps it was π ≈ 377/120 as Ptolemy used. The Antikythera mechanism is an astronomical clock considered amazing for its time. It may have been 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 might have been designed soon after Archimedes' time, but that genius was probably not the designer: the Mechanism doesn't match descriptions of the planetariums that Archimedes built.) (Let's mention another Greek astronomer contemporaneous to Hipparchus, Seleucus of Seleucia (ca 190-145 BC), who is noted for supporting heliocentrism. explaining tides, and proposing that the universe is infinite.)

Titus Lucretius Carus (ca 99-55 BC) Rome

Lucretius was a highly-praised poet rather than a mathematician, but may deserve mention in the history of the development of science. His poem "On the Nature of Things" showed advanced knowledge of the natural world. He postulated that equal weights fall at equal speeds in a vacuum, long before Galileo made the same observation. Albert Einstein is famous for proving the existence of atoms from Brownian Motion, but Lucretius had used the same argument almost 2000 years earlier. Lucretius also had advanced ideas in philosophy, psychology and cosmology. He had a notion of evolution (though not species origin) based on survival of the fittest, and denied that humans were intrinsically superior to other animals. The "three-age" system (dividing human prehistory and history into Stone, Bronze and Iron Ages) is due to Lucretius. (Like all ancient natural philosophers, Lucretius had misconceptions: it appears he didn't accept that the Earth is spherical.)

Heron of Alexandria (ca 10-75) Egypt

Heron (or Hero) was apparently a teacher at the great university of Alexandria, but there is much uncertainty about his life and work. He wrote on mechanics (analysing the five basic machines of mechanical advantage), astronomy (determining longitudes), hydrostatics, architecture, surveying, optics (he introduced the 'shortest-distance' explanation for mirror reflection), arithmetic (finding square roots and cube roots), and geometry (finding the areas and volumes of various shapes). He was an inventor; he was first to describe a syringe, a windmill, a pump for extinguishing fires, and some very primitive counters and computers. He is especially famous for his invention of the aeolipile which rotated using steam from an attached cauldron, and is considered the first steam engine. (Vitruvius may have described such a machine before Heron.) He is noted for designing various toys (probably developed as teaching aids for his lectures); these included a puppet theater driven by strings and weights, a robot trumpet, trick wine glasses, a windmill-driven organ, and coin-operated vending machine. His most famous discovery in mathematics was Heron's Formula for the area A of a triangle with sides a,b,c:

A2 = s(s-a)(s-b)(s-c) where s = (a+b+c)/2 But there is some controversy about the actual authorship of Heron's books; and much of Heron's best physics and mathematics (possibly including Heron's Formula) appear to repeat discoveries by Archimedes. Thus, despite his fame, we do not include Heron on our List.

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.

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

Ptolemy `the Wise,' Librarian of Alexandria, was one of the most famous of ancient Greek scientists, ranking #80 on the Pantheon List of Most Popular and Productive Persons. His textbooks were among the most important of the ancient world, perhaps because they supplanted most that had come before. He provided new insights into optics and music; he was the best geographer of his day. Among his mathematical results, most famous may be Ptolemy's Theorem (AC·BD = AB·CD + BC·AD if and only if ABCD is a cyclic quadrilateral). This theorem has many useful corollaries; it was frequently applied in Copernicus' work. Ptolemy wrote on trigonometry, optics, geography, and map projections; 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.)

Geocentrism vs. Heliocentrism

The mystery of celestial motions directed scientific inquiry for thousands of years. Except for some Pythagoreans like 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 epicycles. 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, proposed heliocentric models, these were usually rejected due to the lack of parallax among stars, and the lack of the ferocious winds that might be expected if the Earth were in motion. (However some historians believe that many ancient Greek scholars, perhaps including Plato, accepted the possibility of Earth's motion, and that geocentrism became dogma only after Ptolemy's work.) 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. Several thinkers proposed a hybrid system with Mercury and Venus rotating the Sun but the outer planets and the Sun itself rotating Earth; these thinkers may have included ancient Egyptians, the Greek Heraclides of Pontus, some of the Islamic scientists, a member of Madhava's Kerala school, and Tycho Brahe -- the astronomer who linked Copernicus to Kepler. A related question is: Does the Earth spin daily on its axis? All heliocentrists, beginning with Heraclides of Pontus, seem to have accepted that, as well as some who were more doubtful about the Earth's annual orbit. Another related question is: Is the universe finite, or is it infinite? Democritus, Seleucus, Nicholas of Cusa and Giordano Bruno were four who proposed an infinite universe prior to Galileo. 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), Regiomontanus, and Leonardo da Vinci are other great pre-Copernican mathematicians who may have accepted the possibility of heliocentrism. Another reason to doubt that the Earth moves, is that we don't feel that motion, or see its effect on falling bodies. This difficulty, which almost disappears once Newton's First Law of Motion is accepted, was addressed before Newton by Jean Buridan, Nicole Oresme, Giordano Bruno, Pierre Gassendi (1592-1655) and, of course, Galileo Galilei. The great skill demonstrated by Ptolemy and his predecessors in developing their complex geocentric cosmology may have set back science 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 understanding that lead to massive change in scientific thought. For Ptolemy and other geocentrists, the "fixed" stars were just lights on a sphere rotating 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.... But we're getting way ahead of our story: Copernicus, Bruno, Galileo and Kepler lived 14 centuries after Ptolemy.)

Liu Hui (ca 220-280) China

Liu Hui made major improvements to Chang's influential textbook Nine Chapters, making him among the most important of Chinese mathematicians ever. (He seems to have been a much better mathematician than Chang, but just as Newton might have gotten nowhere without Kepler, Vieta, Huygens, Fermat, Wallis, Cavalieri, etc., so Liu Hui might have achieved little had Chang not preserved the ancient Chinese learnings.) Among Liu's achievements are an emphasis on generalizations and proofs, 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 for π. He also devised an interpolation formula to simplify that calculation; this yielded the "good-enough" value 3.1416, which is still taught today in primary schools. (Liu's successors in China included Zu Chongzhi, who did determine sphere's volume, and whose approximation for π held the accuracy record for nine centuries.)

Diophantus of Alexandria (ca 250) Greece, Egypt

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

Pappus of Alexandria (ca 300) Egypt, Greece

Pappus, along with Diophantus, may have been one of the two greatest Western mathematicians during the 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, the Problem of Pappus which, given an arbitrary collection of lines in the plane, asks for the locus of points whose distances to the lines have a certain relationship. This problem was a major inspiration for 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.

Mathematicians after Classical Greece

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

Decimal system -- from 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 same as their alphabet's letters; this might have hindered the development of "syncopated" notation. The most ancient Hindu records did not use the ten digits of Aryabhata, but rather a system similar to that of the ancient Greeks, suggesting that China, and not India, may indeed be the "ultimate" source of the modern decimal system. The Chinese used a form of decimal abacus as early as 3000 BC; if it doesn't qualify, by itself, as a "decimal system" then pictorial depictions of its numbers would. Yet for thousands of years after its abacus, China had no zero symbol other than plain space; and apparently didn't have one until after the Hindus. Ancient Persians and Mayans did have place-value notation with zero 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!

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 constant e. He used π ≈ 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 (k3) = (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."

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

No one person gets unique credit for the invention of the decimal system but Brahmagupta's textbook Brahmasphutasiddhanta was very influential, and is sometimes considered the first textbook "to treat zero as a number in its own right." It also treated negative numbers. (Others claim these were first seen 800 years earlier in Chang Tshang's Chinese text and were implicit in what survives of earlier Hindu works, but Brahmagupta's text discussed them lucidly.) Along with Diophantus, Brahmagupta was also among the first to express equations with symbols rather than words. Brahmagupta Bhillamalacarya (`The Teacher from Bhillamala') made great advances in arithmetic, algebra, numeric analysis, and geometry. Several theorems bear his name, including the formula for the area of a cyclic quadrilateral:

16 A2 = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)

Another famous Brahmagupta theorem dealing with such quadrilaterals can be phrased "In a circle, if the chords AB and CD are perpendicular and intersect at E, then the line from E which bisects AC will be perpendicular to BD." 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), and generalized it further to assist in the Pell's solutions. He applied mathematics to astronomy, predicting eclipses, etc.

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 the sin function, 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."

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 book Al-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 word algorithm is borrowed from Al-Khowârizmi's name, and algebra is taken from the name of his book. He also coined the word cipher, which became English zero (although this was just a translation from the Sanskrit word for zero introduced by Aryabhata). He was an essential pioneer for Islamic science, and for the many Arab and Persian mathematicians who followed; and hence also for Europe's eventual Renaissance which was heavily dependent on Islamic teachings. Al-Khowârizmi's texts on algebra and decimal arithmetic are considered to be among the most influential writings ever.

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). (His work with code-breaking also made him a pioneer in basic concepts of probability.) (Al-Kindi, called The Arab Philosopher, can not be considered among the greatest of mathematicians, but was one of the most influential general scientists between Aristotle and da Vinci.) He appears on Cardano's List of 12 Greatest Geniuses. (Al-Khowârizmi and Jabir ibn Aflah are the other Islamic scientists on that list.)

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

Thabit produced important books in philosophy (including perhaps the famous mystic work De Imaginibus), medicine, mechanics, astronomy, and especially several mathematical fields: analysis, non-Euclidean geometry, trigonometry, arithmetic, number theory. As well as being an original thinker, Thabit was a key translator of ancient Greek writings; he translated Archimedes' otherwise-lost Book of Lemmas and applied one of its methods to construct a regular heptagon. He developed an important new cosmology superior to Ptolemy's (and which, though it was not heliocentric, may have inspired Copernicus). He was perhaps the first great mathematician to take the important step of emphasizing real numbers rather than either rational numbers or geometric sizes. He worked in plane and spherical trigonometry, and with cubic equations. He was an early 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.

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.

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. While Aristotle thought vision arose from rays sent from the eye to the viewed object, and Ptolemy thought light rays originated from objects, Alhazen understand that an object's light is reflected sunlight. He's been called the best scientist of the Middle Ages; his Book of Optics has been called the most important physics text prior to Newton; his writings in physics anticipate the Principle of Least Action, Newton's First Law of Motion, and the notion that white light is composed of the color spectrum. (Like Newton, he favored a particle theory of light over the wave theory of Aristotle.) His other achievements in optics include improved lens design, an analysis of the camera obscura, Snell's Law, an early explanation for the rainbow, a correct deduction from refraction of atmospheric thickness, and experiments on visual perception. He 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 regular 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.

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, and Shadows, 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.

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 the Khayyam-Saccheri quadrilateral. He derived solutions to cubic equations using the intersection of conic sections with circles. Remarkably, he stated that the cubic solution could not be achieved with straightedge and compass, a fact that wouldn't be proved until the 19th century. 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 xkyn-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).

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 and advanced arithmetic methods; and are the first texts to present clearly the formulae to enumerate combinations and permutations. He was also an astronomer. (It is sometimes claimed that his equations for planetary motions anticipated the Laws of Motion discovered by Kepler and Newton, but this claim is doubtful.) In algebra, he solved various equations including 2nd-order Diophantine, quartic, Brouncker's and Pell's equations. His Chakravala method, an early application of mathematical induction to solve 2nd-order equations, has been called "the finest thing achieved in the theory of numbers before Lagrange" (although a similar statement was made about one of Fibonacci's theorems). (Earlier Hindus, including Brahmagupta, contributed to this method.) In several ways he anticipated calculus: he used Rolle's Theorem; he may have been first to use the fact that dsin x = cos x · dx; and he once wrote that multiplication by 0/0 could be "useful in astronomy." In trigonometry, which he valued for its own beauty as well as practical applications, he developed spherical trig and was first to present the identity

sin a+b = sin a · cos b + sin b · cos a Bháscara's achievements came centuries before similar discoveries in Europe. It is an open riddle of history whether any of Bháscara's teachings trickled into Europe in time to influence its Scientific Renaissance.

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

Leonardo (known today as Fibonacci) introduced the decimal system and other new methods of arithmetic to Europe, and relayed the mathematics of the Hindus, Persians, and Arabs. Others, 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 most talented mathematician of the Middle Ages." Leonardo is most famous for his book Liber Abaci, but his Liber Quadratorum provides the best demonstration of his skill. He defined congruums and proved theorems about them, including a theorem establishing the conditions for three square numbers to be in consecutive arithmetic series; this has been called the finest work in number theory prior to Fermat (although a similar statement was made about one of Bhaskara II's theorems). Although often overlooked, this work includes a proof of the n = 4 case of Fermat's Last Theorem. (Leonardo's proof of FLT4 is widely ignored or considered incomplete. I'm preparing a page to consider that question. 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 his Elements. 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 like 5 yards, 2 feet, and 3 inches is more efficient than today's notation. It seems hard to believe but before the decimal system, mathematicians had no notation for zero. Referring to this system, Gauss was later to exclaim "To what heights would science now be raised if Archimedes had made that discovery!" Some histories describe him as bringing Islamic mathematics to Europe, but in Fibonacci's own preface to Liber Abaci, he specifically credits the Hindus: ... as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods;

... But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, ... Had the Scientific Renaissance begun in the Islamic Empire, someone like al-Khowârizmi would have greater historic significance than Fibonacci, but the Renaissance did happen in Europe. Liber Abaci's summary of the decimal system has been called "the most important sentence ever written." Even granting this to be an exaggeration, there is no doubt that the Scientific Revolution owes a huge debt to Leonardo `Fibonacci' Pisano.

Abu Jafar Muhammad Nasir al-Din al-Tusi (1201-1274) Persia

Al-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. (Allegedly one of his students hypothesized elliptical orbits.) 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.

Qin Jiushao (1202-1261) China

There were several important Chinese mathematicians in the 13th century, of whom Qin Jiushao (Ch'in Chiu-Shao) may have had particularly outstanding breadth and genius. Qin's textbook discusses various algebraic procedures, includes word problems requiring quartic or quintic equations, explains a version of Horner's Method for finding solutions to such equations, includes Heron's Formula for a triangle's area, and introduces the zero symbol 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.

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 book Jade Mirror of the Four Unknowns studied 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.

Kamal al-Din Abu'l Hasan Muhammad Al-Farisi (1267-1319) Iran

Al-Farisi was a student of al-Shirazi who in turn was a student of al-Tusi. He and al-Shirazi are especially noted for the first correct explanation of the rainbow. Al-Farisi made several other corrections in his comprehensive commentary on Alhazen's textbook on optics. Al-Farisi made several contributions to number theory. He improved Thabit's theorem about amicable numbers, made important new observations about the binomial coefficients, and noted the N=4 case of Fermat's Last Theorem. In addition to his work with amicable numbers, he is especially noted for his improved proof of Euclid's Fundamental Theorem of Arithmetic.

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 book MilHamot 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). (In fact, (2^3,3^2) is the only pair of consecutive powers; this is Catalan's Conjecture and was first proved in the 21st century.) 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.

Nicole Oresme (ca 1322-1382) France

Oresme was of lowly birth but excelled at school, became a young professor, and soon personal chaplain to King Charles V. (One of his early teachers was Jean Buridan (ca 1300-ca 1358), famous for positing what became Newton's First Law of Motion, contrary to Aristotelian dogma.) 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. His contributions to political science (on human rights and popular monarchy) influenced King Charles V. He wrote several books; was a renowned philosopher and natural scientist (challenging several of Aristotle's ideas); he 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.

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 sin q = q - q3/3! + q5/5! - ... , formulae for π, including the one attributed to Leibniz, and the then-best known approximation π ≈ 104348 / 33215. Despite the accomplishments of the Kerala school, Madhava probably does not deserve a place on our List. There were several other great mathematicians who contributed to Kerala's achievements, some of which were made 150 years after Madhava's death. More importantly, the work was not propagated outside Kerala, so had almost no effect on the development of mathematics.

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

Al-Kashi was among the greatest calculators 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 π correctly to 17 significant digits, breaking Madhava's record. (This record was subsequently broken by relative unknowns: a German ca. 1600, John Machin 1706. In 1949 the π calculation record was held briefly by John von Neumann and the ENIAC.)

Nicholas Kryffs of Cusa (1401-1464) Palatinate, Italy

Nicholas Kryffs (aka Nikolaus Cusanus) was an astronomer and ordained priest. His main work was in philosophy (he wrote "All we know of the truth is that the absolute truth, such as it is, is beyond our reach") and theology; but he also made proposals for scientific experimentation, e.g. the use of water-clocks to investigate the speed of falling bodies, and the use of weighing balances to quantify the transfer of mass from soil to growing plant. Nicholas was also a mathematician: he wrote on geometric problems, calendar reform, and invented the Borda Count method of balloting (although it is named after its 18th-century rediscoverer), but his work doesn't qualify him for our list. He belongs in our story because he deduced that the Earth orbited the Sun before Copernicus did, and made a deduction that Copernicus missed: that the universe is immensely huge or infinite; and the fixed stars may have their own planets with their own life forms. He was far ahead of his time, but his writings eventually influenced Galileo, Leibniz, and another mathematician-priest more famous than himself: Giordano Bruno, who wrote "If Nicholas [of Cusa] had not been hindered by his priest's vestment, he would have even been greater than Pythagoras!"

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, especially 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.

Leonardo da Vinci (1452-1519) Italy

Leonardo da Vinci is most renowned for his paintings -- Mona Lisa and The Last Supper are 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; and worked as intended.) Like another genius (Albert Einstein) da Vinci was intrigued by the science of river meanders; there is a meandering river in the background of Mona Lisa. His scientific writings are much more extensive and thorough than is generally appreciated; he made advances in anatomy, botany, and many other fields of science; he developed much mechanics including the theory of the arch; he developed an octant-based map projection; he was first to concei