# Greatest Mathematicians born before 400 A.D.

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

 Born before 400 (this page) Born betw. 400 & 1559 Born betw. 1560 & 1699 Born betw. 1700 & 1799 Born betw. 1800 & 1859 Born betw. 1860 & 1975 List of Greatest Mathematicians

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 certainly familiar with the Pythagorean Theorem; the Plimpton 322 artifact from about the time of Hammurabi the Great contained a table of Pythagorean triples (possibly used to assist land mensuration); some of these triplets are so large (one is 12709, 13500, 18541 -- generated by p=125, q=54) that they almost certainly knew the Pythagorean generating formula. Ancient Mesopotamians also had solutions to quadratic equations and even cubic equations (though they didn't have a general solution for these); they eventually 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.

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 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. It seems that Babylonians were aware of Thales Theorem earlier, but if they had a proof it has not been preserved.) 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.

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, 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. (As we mentioned above, this formula was probably known in Babylonia over 1000 years before Pythagoras.) 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'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 grammar is a field of applied 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, 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 (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 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. 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 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 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 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.

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 (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 is an elegant theorem with a non-trivial proof asserting 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 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, and 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 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 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."

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.)

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.)

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 (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, the earliest glimpse at the Principle of Least Action), 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, a trick wine glass (the Pythagorean Cup mentioned in Pythagoras' mini-bio), "Hero's Fountain" (another trick using hydrostatic principles), a windmill-driven organ, and a 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, Heron may not belong on our List.

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.

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, and Copernicus' own writing may never go beyond this. But as the Heliocentrism Revolution evolved, the fixed stars were eventually 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 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 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, 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.

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