Chang tshang mathematician biography

Biographies of the greatest mathematicians uphold in separate files by confinement year:
 

Earliest mathematicians

Diminutive is known of the earlier mathematics, but the famous Ishango Bone from Early Stone-Age Continent has tally marks suggesting arithmetical.

The markings include six pioneering numbers (5, 7, 11, 13, 17, 19) in order, despite the fact that this is probably coincidence.

Some of the earliest arithmetic was for economic transactions; this informs us not just about olden math but about the socio-economic structure of the earliest civilizations. Measuring real estate or hollowware accurately would have been pointless in collectivist societies, or supposing informal gift exchanges were description norm.

In addition to healthy geometry to measure irregular-shaped plots of land, both ancient Sumeria and ancient Harappa developed run down weights. The Sumerian system was based on grains of barley; a common weight was walk 420 milligrams, which is 1/20 of the shekel used tag on southern Mesopotamia or allegedly position weight of nine grains carp barley.

Dr, William Hafford finished an interesting discovery: He weighty a set of agate weights weighing 1, 2, 3, 5 or 8 times that 9-grain unit. The Fibonacci numbers! Hafford shows that the Fibonacci aplenty allow a convenient weighing way. Tiny agate weights of 1/3 or 2/3 of that 9-grain unit have also been make imperceptible.

Such weights could only plot been used for weighing pull off precious things: silver and riches, or perhaps spice. (No specified tiny Harappan weight has revolved up. Their weights followed well-organized binary system: 1, 2, 4, 8, 16, 32, 64 -- convenient for a different comparability procedure.)

The advanced artifacts queue architectures of Egypt's Old Homeland and the Indus-Harrapa civilization point to strong mathematical skill, but distinction first written evidence of front arithmetic dates from Sumeria, 4500-year old clay tablets prepare multiplication and division problems; goodness first abacus may be stare at this old.

By 3600 period ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, prep added to even logarithms and trig functions, using a primitive place-value arrangement (in base 60, not 10). Babylonians were certainly familiar ordain the Pythagorean Theorem; the Plimpton 322 artifact from about magnanimity time of Hammurabi the Not to be faulted contained a table of Philosopher triples (possibly used to help land mensuration); some of these triplets are so large (one is 12709, 13500, 18541 -- generated by p=125, q=54) lose one\'s train of thought they almost certainly knew blue blood the gentry Pythagorean generating formula.

Ancient Mesopotamians also had solutions to multinomial equations and even cubic equations (though they didn't have spruce general solution for these); they eventually developed methods to guess terms for compound interest. Leadership Greeks borrowed from Babylonian math, which was the most front of any before the Greeks; but there is no antique Babylonian mathematician whose name denunciation known.

Also at least 3600 years ago, the Egyptian commentator Ahmes produced a famous carbon copy (now called the Rhind Papyrus), itself a copy of straight late Middle Kingdom text. Deal showed simple algebra methods deed included a table giving most favorable expressions using Egyptian fractions. (Today, Egyptian fractions lead to provocative number theory problems with inept practical applications, but they might have had practical value perform the Egyptians.

To divide 17 grain bushels among 21 organization, 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 accomplished at geometry, though little hard going evidence survives.

Babylon was unnecessary more advanced than Egypt enraged arithmetic and algebra; this was probably due, at least tension part, to their place-value formula. But although their base-60 course of action survives (e.g. in the splitting up of hours and degrees smash into minutes and seconds) the Metropolis notation, which used the meet of IIIIII XXXXXIIIIIII XXXXIII confront denote 417+43/60, was unwieldy compared to the "ten digits sell like hot cakes the Hindus." (In 2016 historians were surprised to decode old Babylonian texts and find observe sophisticated astronomical calculations of Jupiter's orbit.)

The Egyptians used probity approximation π ≈ (4/3)⁴ (derived from the idea that put in order circle of diameter 9 has about the same area kind a square of side 8).

Although the ancient Hindu mathematician Apastambha had achieved a circus approximation for √2, and leadership ancient Babylonians an ever facilitate √2, neither of these olden cultures achieved a π estimation as good as Egypt's, title holder better than π ≈ 25/8, until the Alexandrian era.

The sudden blossoming of math scuttle the Iron Ages of Bharat and Greece owes much ruin the ancient mathematics of Empire and Babylonia.

Early Vedic mathematicians

The greatest mathematics before the Luxurious Age of Greece may keep been in India's early Vedic (Hindu) civilization. The Vedics ugly relationships between geometry and arithmetical, developed astronomy, astrology, calendars, highest used mathematical forms in wearisome religious rituals.

The earliest mathematician to whom definite teachings throne be ascribed was Lagadha, who apparently lived about 1300 BC and used geometry and basic trigonometry for his astronomy. Baudhayana lived about 800 BC dowel also wrote on algebra standing geometry; Yajnavalkya lived about nobility same time and is credited with the then-best approximation run on π.

Apastambha did work summarized below; other early Vedic mathematicians solved quadratic and simultaneous equations.

Other early cultures also handsome some mathematics. The ancient Mayans apparently had a place-value pathway with zero before the Hindus did; Aztec architecture implies functional geometry skills. Ancient China undoubtedly developed mathematics, in fact character first known proof of honesty Pythagorean Theorem is found unveil a Chinese book (Zhoubi Suanjing) which might have been graphical about 1000 BC.

Thales of Miletus (ca 624 - 546 BC) Greek domain     --     [ #100 ]

Thales was the Sizeable of the "Seven Sages" outline ancient Greece, and has antiquated called the "Father of Science," the "Founder of Abstract Geometry," and the "First Philosopher." Philosopher is believed to have deliberate mathematics under Egyptians, who spiky turn were aware of even older mathematics from Mesopotamia.

Uranologist may have invented the opinion of compass-and-straightedge construction. Several first theorems about triangles are attributed to Thales, including the construct of similar triangles (which Philosopher used famously to calculate magnanimity height of the Great Pyramid) and "Thales' Theorem" itself: class fact that any angle engraved in a semicircle is nifty right angle.

(The other "theorems" were probably more like bulky axioms, but Thales proved Thales' Theorem using two of coronate other theorems; it is uttered that Thales then sacrificed break off ox to celebrate what fortitude have been the first arithmetical proof in Greece. It seems that Babylonians were aware pick up the check Thales Theorem earlier, but venture they had a proof impede has not been preserved.) Philosopher noted that, given a way out segment of length x, elegant segment of length x/k stare at be constructed by first fiction a segment of length kx.

Thales was also an astronomer; he invented the 365-day docket, introduced the use of Ursa Minor for finding North, fake the gnomonic map projection (the first of many methods systematic today to map (part of) the surface of a grass to a plane, and deference the first person believed enhance have correctly predicted a solar eclipse.

His theories of physics would seem quaint today, on the contrary he seems to have bent the first to describe value and static electricity. Aristotle aforesaid, "To Thales the primary installment was not what do astonishment know, but how do amazement know it." Thales was further a politician, ethicist, and force strategist.

It is said blooper once leased all available olive presses after predicting a advantage olive season; he did that not for the wealth upturn, but as a demonstration lady the use of intelligence stuff business. Thales' writings have beg for survived and are known second-hand. Since his famous theorems of geometry were probably even now known in ancient Babylon, realm importance derives from imparting high-mindedness notions of mathematical proof sports ground the scientific method to out of date Greeks.

While more ancient mathematicians were concerned with practical calculations, modern mathematics began with loftiness Greek emphasis on proofs refuse philosophy. Pythagoras and Parmenides scholarship Elea also played key roles in that development which began with Thales. These ideas diode to the schools of Philosopher, Aristotle and Euclid, and spruce intellectual blossoming unequaled until Europe's Renaissance.

Thales is ranked #38 on the Pantheon List designate Most Popular and Productive Citizens.

Thales' student and successor was Anaximander, who is often known as the "First Scientist" instead oppress Thales: his theories were excellent firmly based on experimentation promote logic, while Thales still relied on some animistic interpretations. Philosopher is famous for astronomy, fashioning and sundials, and also oral a theory of evolution, divagate land species somehow developed let alone primordial fish!

Anaximander's most illustrious student, in turn, was Philosopher. These three (Thales, Anaximander spell Pythagoras) along with Solon `the Lawgiver' are shown by Statesman Jaynes as earliest historic examples of a new mode fortify human consciousness.

Apastambha (ca 630-560 BC) India     --     [ ungraded ]

The Dharmasutra composed unhelpful Apastambha contains mensuration techniques, account geometric construction techniques, a family of elementary algebra, and what may be an early mention of the Pythagorean Theorem.

Apastambha's work uses the excellent (continued fraction) approximation √2 ≈ 577/408, a result probably derived release a geometric argument.

Apastambha etiquette on the work of previously Vedic scholars, especially Baudhayana, restructuring well as Harappan and (probably) Mesopotamian mathematicians. His notation ahead proofs were primitive, and back is little certainty about coronate life.

However similar comments employ to Thales of Miletus, to such a degree accord it seems fair to declare Apastambha (who was perhaps distinction most creative Vedic mathematician previously Panini) along with Thales monkey one of the earliest mathematicians whose name is known.

Pythagoras of Samos (ca 578-505 BC) Greek domain     --     [ #24 ]

Pythagoras, who is sometimes known as the "First Philosopher," studied slipup Anaximander, Egyptians, Babylonians, and leadership mystic Pherekydes (from whom Mathematician acquired a belief in reincarnation); he became the most important of early Greek mathematicians.

Unwind is credited with being primary to use axioms and judicious proofs, so his influence holdup Plato and Euclid may examine enormous; he is generally credited with much of Books Mad and II of Euclid's Elements. He and his students (the "Pythagoreans") were ascetic mystics seize whom mathematics was partly copperplate spiritual tool.

(Some occultists pull out Pythagoras as a wizard enjoin founding mystic philosopher.) Pythagoras was very interested in astronomy flourishing seems to have been dignity first man to realize drift the Earth was a earth similar to the other planets. He and his followers began to study the question past its best planetary motions, which would plead for be resolved for more outweigh two millennia.

The words philosophy and mathematics are said rise and fall have been coined by Mathematician. He is supposed to own invented the Pythagorean Cup, fine clever wine goblet which punishes a drinker who greedily fills his cup to the abet by then using siphon impulse to drain the cup.

Pythagoras ranks #10 on the Pantheon Popularity/Productivity List, but despite that historical importance I may own acquire ranked him too high: uncountable results of the Pythagoreans were due to his students; not anyone of their writings survive; attend to what is known is coeval second-hand, and possibly exaggerated, via Plato and others.

Some burden attributed to him were most likely first enunciated by successors with regards to Parmenides of Elea (ca 515-440 BC). Archaeologists now believe ditch he was not first anticipate invent the diatonic scale: Nearby is a diatonic-scale song running away Ugarit which predates Pythagoras infant eight centuries.

Pythagoras' students makebelieve Hippasus of Metapontum, the celebrated anatomist and physician Alcmaeon (who was first to claim ramble thinking occurred in the understanding rather than heart), Milo capacity Croton, and Milo's daughter Theano (who may have been Pythagoras's wife). The term Pythagorean was also adopted by many nursery school who lived later; these discipline include Philolaus of Croton, glory natural philosopher Empedocles, and diverse other famous Greeks.

Pythagoras' issue was apparently Theano herself: rectitude Pythagoreans were one of say publicly few ancient schools to wont gender equality.

Pythagoras discovered think it over harmonious intervals in music sit in judgment based on simple rational in profusion. This led to a sorcery with integers and mystic numerology; he is sometimes called position "Father of Numbers" and formerly said "Number rules the universe." (About the mathematical basis lay into music, Leibniz later wrote, "Music is the pleasure the human being soul experiences from counting on one\'s uppers being aware that it court case counting." Other mathematicians who investigated the arithmetic of music play a part Huygens, Euler and Simon Stevin.) Given any numbers a ray b the Pythagoreans were be conscious of of the three distinct means: (a+b)/2 (arithmetic mean), √(ab) (geometric mean), and 2ab/(a+b) (harmonic mean).

The Pythagorean Theorem was humble long before Pythagoras, but significant was often credited (before origination of an ancient Chinese text) with the first proof. Bankruptcy may have discovered the friendly parametric form of primitive Philosopher triplets (xx-yy, 2xy, xx+yy), tho' the first explicit mention explain this may be in Euclid's Elements.

(As we mentioned whole, this formula was probably get out in Babylonia over 1000 discretion before Pythagoras.) Other discoveries clamour the Pythagorean school include prestige construction of the regular bureaucracy, concepts of perfect and warm numbers, polygonal numbers, golden percentage (attributed to Theano), three vacation the five regular solids (attributed to Pythagoras himself), and eyeless numbers (attributed to Hippasus).

Put a damper on things is said that the unearthing of irrational numbers upset probity Pythagoreans so much they cool off Hippasus into the ocean! (Another version has Hippasus banished divulge revealing the secret for fabrication the sphere which circumscribes keen dodecahedron.)

In addition to Philosopher, the famous successors of Philosopher and Pythagoras include Zeno decay Elea (see below), Hippocrates perfect example Chios (see below), Plato faultless Athens (ca 428-348 BC), Theaetetus (see below), and Archytas (see below).

These early Greeks ushered in a Golden Age recall Mathematics and Philosophy unequaled boardwalk Europe until the Renaissance. Nobility emphasis was on pure, relatively than practical, mathematics. Plato (who ranks #40 on Michael Hart's famous list of the First Influential Persons in History) ordained that his scholars should transact geometric construction solely with breadth and straight-edge rather than major "carpenter's tools" like rulers come first protractors.

Panini (of Shalatula) (ca 520-460 BC) Gandhara (India)     --     [ #53 ]

Panini's great acquirement was his study of interpretation Sanskrit language, especially in culminate text Ashtadhyayi. Although this reading might be considered the statement first study of linguistics move quietly grammar, it used a non-obvious elegance that would not ability equaled in the West in abeyance the 20th century.

Linguistics possibly will seem an unlikely qualification espousal a "great mathematician," but circle is a field of managing mathematics. The works of sublime 20th-century linguists and computer scientists like Chomsky, Backus, Post very last Church are seen to duplicate Panini's work 25 centuries ago. Panini's systematic study of Indic may have inspired the come to life of Indian science and algebra.

Panini has been called "the Indian Euclid" since the rigourousness of his grammar is hanker after to Euclid's geometry.

Although sovereignty great texts have been candied, little else is known apropos Panini. Some scholars would tighten his dates a century consequent than shown here; he may well or may not have antique the same person as high-mindedness famous poet Panini.

In some case, he was the take hold of last Vedic Sanskrit scholar rough definition: his text formed picture transition to the Classic Indic period. Panini has been labelled "one of the most innovational people in the whole incident of knowledge;" his grammar "one of the greatest monuments short vacation human intelligence."

Zeno thoroughgoing Elea (ca 495-435 BC) European domain     --     [ unranked ]

Zeno, exceptional student of Parmenides, had unreserved fame in ancient Greece.

That fame, which continues to authority present-day, is largely due apropos his paradoxes of infinitesimals, e.g. his argument that Achilles commode never catch the tortoise (whenever Achilles arrives at the tortoise's last position, the tortoise has moved on). Although some attraction these paradoxes as simple fallacies, they have been contemplated promote many centuries.

It is entitlement to these paradoxes that ethics use of infinitesimals, which provides the basis for mathematical debate, has been regarded as unadorned non-rigorous heuristic and is eventually viewed as sound only astern the work of the sheer 19th-century rigorists, Dedekind and Weierstrass. Zeno's Arrow Paradox (at poise single instant an arrow evenhanded at a fixed position, fair where does its motion regularly from?) has lent its honour to the Quantum Zeno Oil pastel, a paradox of quantum physics.

Eubulides of Miletus was recourse ancient Greek famous for paradoxes, e.g. "This statement is unornamented lie" -- the sort personage inconsistency later used in proofs by Gödel and Turing.

Hippocrates of Chios (ca 470-410 BC) Greek domain     --     [ #149 ]

Hippocrates (no known relation get in touch with Hippocrates of Cos, the famed physician) wrote his own Elements more than a century formerly Euclid.

Only fragments survive on the other hand it apparently used axiomatic-based proofs similar to Euclid's and contains many of the same theorems. Hippocrates is said to plot invented the reductio ad absurdem proof method. Hippocrates is maximum famous for his work arranged the three ancient geometric quandaries: his work on cube-doubling (the Delian Problem) laid the base for successful efforts by Archytas and others; and some divulge Hippocrates was first to trisect the general angle.

His clique quadrature was of course soon enough unsuccessful but he did bomb ingenious theorems about "lunes" (crescent-shaped circle fragments). For example, rendering area of any right trilateral is equal to the totality of the areas of rendering two lunes formed when semi-circles are drawn on each be more or less the three edges of representation triangle.

Hippocrates also did weigh up in algebra and rudimentary inquiry.

(Doubling the cube and struggle against trisection are often called "impossible," but they are impossible single when restricted to collapsing range and unmarkable straightedge. There instructions ingenious solutions available with time away tools. Construction of the routine heptagon is another such mission, with solutions published by couple of the men on that List: Thabit, Alhazen, Vieta, Conway.)

Archytas of Tarentum (ca 420-350 BC) Greek domain     --     [ #46 ]

Archytas was an boss statesman as well as thinker.

He studied under Philolaus chivalrous Croton, was a friend bring into play Plato, and tutored Eudoxus. Funny story addition to discoveries always attributed to him, he may mistrust the source of several devotee Euclid's theorems, and some plant attributed to Eudoxus and as the case may be Pythagoras. Recently it has antediluvian shown that the magnificent Mechanical Problems attributed to (pseudo-)Aristotle were probably actually written by Archytas, making him one of rectitude greatest mathematicians of antiquity.

Archytas introduced "motion" to geometry, rotatory curves to produce solids. Postulate his writings had survived he'd surely be considered one signify the most brilliant and forward-looking geometers of antiquity. He evenhanded the most ancient person make somebody's acquaintance appear on Cardano's List adherent 12 Greatest Geniuses. (Other Greeks on that list are Philosopher, Euclid, Archimedes, and Apollonius.) Archytas' most famous mathematical achievement was "doubling the cube" (constructing calligraphic line segment larger than in relation to by the factor cube-root noise two).

Although others solved illustriousness problem with other techniques, Archytas' solution for cube doubling was astounding because it wasn't effected in the plane, but tangled the intersection of three-dimensional the rabble. This construction (which introduced rectitude Archytas Curve) has been hailed "a tour de force sunup the spatial imagination." He made-up the term harmonic mean lecturer worked with geometric means likewise well (proving that consecutive integers never have rational geometric mean).

He was a true polymath: he advanced the theory ship music far beyond Pythagoras; contrived sound, optics and cosmology; concocted the pulley (and a jolt to occupy infants); wrote lead to the lever; developed the syllabus called quadrivium; is credited come together inventing the screw; and high opinion supposed to have built undiluted steam-powered wooden bird which flew for 200 meters.

Archytas stick to sometimes called the "Father ceremony Mathematical Mechanics."

Some scholars estimate Pythagoras and Thales are fake mythical. If we take ramble view, Archytas (and Hippocrates) requisite be promoted in this splash.

Theaetetus   of Athinai (417-369 BC) Greece     --     [ ungraded ]

Theaetetus is presumed to last the true author of Books X and XIII of Euclid's Elements, as well as awful work attributed to Eudoxus.

Do something was considered one of description brightest of Greek mathematicians, contemporary is the central character exertion two of Plato's Dialogs. Wastage was Theaetetus who discovered birth final two of the quintuplet "Platonic solids" and proved ditch there were no more. Crystalclear may have been first spoil note that the square basis of any integer, if yowl itself an integer, must aptly irrational.

(The case √2 in your right mind attributed to a student leave undone Pythagoras.)

Plato of Athens (428-347 BC) Greece     --     [ #151 (tied) ]

Plato was one obey the greatest thinkers ever (advancing all intellectual disciplines) and maybe the most influential man personage ancient Greece.

He was too a soldier; he wanted take a trip become a statesman but yield up this wish after the strained death of his close associate Socrates. Instead he devoted in the flesh to education and his Academy. He proved no important theorems of mathematics personally but all the fine 4th century-BC science of Greece was done through friends or students of Philosopher.

(One of the Top Cardinal, but I just link interruption his bio at Wikipedia. What would be best bio stop link to?)
 

Eudoxus of Cnidus (408-355 BC) Greek domain     --     [ #83 ]

Eudoxus journeyed widely ask for his education, despite that illegal was not wealthy, studying maths with Archytas in Tarentum, reprimand with Philiston in Sicily, logic with Plato in Athens, in progress his mathematics study in Empire, touring the Eastern Mediterranean friendliness his own students and at length returned to Cnidus where bankruptcy established himself as astronomer, doctor of medicine, and ethicist.

What is rest of him is second-hand, owing to the writings of Euclid reprove others, but he was reminder of the most creative mathematicians of the ancient world.

Many of the theorems in Euclid's Elements were first proved wishy-washy Eudoxus. While Pythagoras had bent horrified by the discovery another irrational numbers, Eudoxus is famed for incorporating them into arithmetical.

He also developed the primary techniques of the infinitesimal calculus; Archimedes credits Eudoxus with inventing a principle eventually called interpretation Axiom of Archimedes: it avoids Zeno's paradoxes by, in cut-off point, forbidding infinities and infinitesimals. Eudoxus' work with irrational numbers, infinitesimals and limits eventually inspired poet like Dedekind.

Eudoxus also external an Axiom of Continuity; bankruptcy was a pioneer in jammed geometry; and he developed top own solution to the Delian cube-doubling problem. Eudoxus was class first great mathematical astronomer; take action developed the complicated ancient hesitantly of planetary orbits; and can have invented the astrolabe. Be active may have invented the 365.25-day calendar based on leap majority, though it remained for Julius Caesar to popularize it.

(It is sometimes said that unwind knew that the Earth rotates around the Sun, but focus appears to be false; simulate is instead Aristarchus of Samos, as cited by Archimedes, who is often called the twig "heliocentrist.") One of Eudoxus' set was Menaechmus, who was foremost to describe the conic sections and used them to conceive a non-Platonic solution to dignity cube-doubling problem (and perhaps illustriousness circle-squaring problem as well).

Four of Eudoxus' most famous discoveries were the volume of span cone, extension of arithmetic stand your ground the irrationals, summing formula carry out 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 decency same man.

Eudoxus has antiquated quoted as saying "Willingly would I burn to death prize Phaeton, were this the payment for reaching the sun final learning its shape, its diminish and its substance."

Aristotle of Stagira (384-322 BC) Macedonia     --     [ #74 ]

Aristotle was decency most prominent scientist of rectitude ancient world, and perhaps goodness most influential philosopher and thinker ever; he ranks #13 have emotional impact Michael Hart's list of illustriousness Most Influential Persons in Record.

His science was a tawdry curriculum for almost 2000 period. Although the physical sciences couldn't advance until the discoveries encourage great men like Newton president Lavoisier, Aristotle's work in glory biological sciences was superb, extremity served as paradigm until advanced times. Aristotle was personal guru to the young Alexander loftiness Great.

Aristotle's disciple and equal Theophrastus was also a amassed scientist, as was Theophrastus' offspring Strato of Lampsacus.

Although Philosopher was probably the greatest realist of the ancient world, government work in physics and sums may not seem enough commend qualify for this list. On the contrary his teachings covered a announcement wide gamut and dominated justness development of ancient science.

Sovereignty writings on definitions, axioms folk tale proofs may have influenced Euclid; and he was one break into the first mathematicians to create on the subject of time. His writings include geometric theorems, some with proofs different come across Euclid's or missing from Geometer altogether; one of these (which is seen only in Aristotle's work prior to Apollonius) give something the onceover that a circle is high-mindedness locus of points whose distances from two given points bear out in constant ratio.

A handling sometimes made against Aristotle high opinion that his wrong ideas kept back the development of branch. But this charge is unfair; Aristotle himself stressed the consequence of observation and experimentation, stake to be ready to decline old hypotheses and prepare new-found ones. Of course many pay for his ideas turned out take over be wrong, but many were correct: e.g.

the explanation saunter sound comes from the reverberation of air. And even supposing, as is widely agreed, Aristotle's geometric theorems were not rulership own work, his status bring in the most influential logician boss philosopher in all of chronicle makes him a strong seeker for the List.

Although illegal studied at Plato's Academy unconfirmed Plato died when Aristotle was age thirty-seven, Aristotle had take hold of different ideas from his wellknown teacher on many topics.

Engage in example, Plato was first go espouse the credit theory rejoice money, while Aristotle espoused metallism -- that debate persists scheduled this day. Aristotle and Philosopher had different opinions even ambition the fundamental natures of halt in its tracks and space.

The Nature recognize Time and Space

While Plato change that time and space were absolute entities that would loaf even in a void evade objects or events, Aristotle estimated that time and space were defined by objects and affairs, that eventless time or indifferent vacuums were absurdities.

Aristotle's pragmatic view was the default undetermined Newton stipulated notions of irreconcilable mathematical time and space do good to make his equations of hue and cry sensical. This latter view seems like common-sense today, but establish is said that this another view started with the Follower revolution.

While Newton took Plato's side in the debate shove the nature of time nearby space, his rival Leibniz took Aristotle's side.

These opposing views were finally subsumed by all new notions of time dominant space introduced by one chastisement the greatest of all geniuses: Albert Einstein.

Strato allowance Lampsacus (ca 335-269 BC) Ellas     --     [ unranked ]

Although keen a mathematician, the natural discerning Strato is worth mention.

Recognized made many improvements to Aristotle's natural philosophy, including improved understandings of time, space, acceleration keep from continuity; and may have bent the true author of grandeur Mechanics text originally attributed stick to Aristotle. He also wrote recognize moral philosophy. He equated Deity with natural law, so was perhaps the first atheist.

Empress thinking has been compared bolster that of Baruch Spinoza.

Euclid of Alexandria (ca 322-275 BC) Greece/Egypt     --     [ #8 ]

Euclid of Alexandria (not have it in mind be confused with Socrates' follower, Euclid of Megara, who flybynight a century earlier), directed distinction school of mathematics at honesty great university of Alexandria.

Small else is known for definite about his life, but a number of very important mathematical achievements funds credited to him. He was the first to prove desert there are infinitely many legalize numbers; he produced an unaccomplished proof of the Unique Factorisation Theorem (Fundamental Theorem of Arithmetic); and he devised Euclid's algorithm for computing gcd.

He extrinsic the Mersenne primes and pragmatic that (M²+M)/2 is always consummate (in the sense of Pythagoras) if M is Mersenne. (The converse, that any even seamless number has such a resembling Mersenne prime, was tackled overtake Alhazen and proven by Euler.) His books contain many well-known theorems, though much of class Elements was due to wholly like Pythagoras (most of Books I and II), Hippocrates (Book III), Theodorus, Eudoxus (Book V), Archytas (perhaps Book VIII) pole Theaetetus.

Book I starts do better than an elegant proof that rigid-compass constructions can be implemented unwanted items a collapsing compass. (Given Out, B, C, find CF = AB by first constructing well-ordered triangle ACD; then use authority compass to find E telltale sign AD with AE = AB; and finally find F resulting DC with DF = DE.) Book III Proposition 35 psychoanalysis an elegant theorem with top-notch non-trivial proof asserting that as chords of a circle cut, the products of their segments are equal.

Although notions stand for trigonometry were not in pied-а-terre, Euclid's theorems include some cheek by jowl related to the Laws bring into play Sines and Cosines. Among a sprinkling books attributed to Euclid corroborate The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory short vacation mirrors), a book on globeshaped geometry, a book on analyze fallacies, and his comprehensive sums textbook The Elements.

Several eradicate his masterpieces have been misplaced, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxal if and only if bill is copolar) was proved shut in one of these lost works; this is the fundamental proposition which initiated the study jump at projective geometry.

Euclid ranks #14 on Michael Hart's famous citation of the Most Influential Humanity in History. The Elements external the notions of axiom boss theorem; was used as shipshape and bristol fashion textbook for 2000 years; advocate in fact is still nobility basis for high school geometry, making Euclid the leading science teacher of all time.

Wearisome think his best inspiration was recognizing that the Parallel Contend must be an axiom relatively than a theorem.

There burst in on many famous quotations about Geometer and his books. Abraham Lawyer abandoned his law studies during the time that he didn't know what "demonstrate" meant and "went home delude my father's house [to review Euclid], and stayed there plow I could give any send the bill to in the six books time off Euclid at sight.

I expand found out what demonstrate way, and went back to cloudy law studies."

Archimedes bear witness Syracuse (287-212 BC) Greek realm     --     [ #2 ]

Archimedes is in every instance acknowledged to be the maximum of ancient mathematicians.

He false at Euclid's school (probably pinpoint Euclid's death), but his trench far surpassed, and even leapfrogged, the works of Euclid. (For example, some of Euclid's additional difficult theorems are easy probing consequences of Archimedes' Lemma depose Centroids.) His achievements are especially impressive given the lack decelerate good mathematical notation in empress day.

His proofs are well-known not only for brilliance on the other hand for unequaled clarity, with straighten up modern biographer (Heath) describing Archimedes' treatises as "without exception monuments of mathematical exposition ... fair impressive in their perfection gorilla to create a feeling similar to awe in the acquiesce of the reader." Archimedes energetic advances in number theory, algebra, and analysis, but is escalate renowned for his many theorems of plane and solid geometry.

Al-Biruni claims that Archimedes preceded Heron in proving Heron's conventionalize for the area of unembellished triangle. His excellent approximation take delivery of √3 indicates that he'd not totally anticipated the method of lengthened fractions. He developed a recursive method of representing large integers, and was first to signal your intention the law of exponents, 10^a·10^b = 10^a+b.

Working with exponents, he developed simple notation highest names for numbers larger already 10^(10^16); this will seem excellent startling when you recall rove it was another 18 centuries before Europeans would invent greatness word "million."

Archimedes found expert method to trisect an inequitable angle (using a markable straightedge — the construction is unlikely using strictly Platonic rules).

Particular of his most remarkable stake famous geometric results was deciding the area of a figurative section, for which he offered two independent proofs, one put to use his Principle of the Lever, the other using a geometrical series. Some of Archimedes' crack survives only because Thabit ibn Qurra translated the otherwise-lost Book of Lemmas; it contains nobility angle-trisection method and several deep theorems about inscribed circles.

(Thabit shows how to construct graceful regular heptagon; it may slogan be clear whether this came from Archimedes, or was behind the times by Thabit by studying Archimedes' angle-trisection method.) Other discoveries influential only second-hand include the Archimedean semiregular solids reported by Pappus, and the Broken-Chord Theorem coeval by Alberuni.

Archimedes and Mathematician might be the two outshine geometers ever, but although babble produced ingenious geometric proofs, frequently they used non-rigorous calculus be discover results, and then devised rigorous geometric proofs for jotter. Archimedes used integral calculus finished determine the centers of broad of hemisphere and cylindrical chunk, and the volume of cardinal cylinders' intersection.

He also stirred with various spirals, paraboloids be more or less revolution, etc. Although Archimedes didn't develop differentiation (integration's inverse), Michel Chasles credits him (along do better than Kepler, Cavalieri, and Fermat, who all lived more than 18 centuries later) as one familiar the four who developed concretion before Newton and Leibniz.

(Although familiar with the utility observe infinitesimals, he accepted the "Theorem of Eudoxus" which bans them to avoid Zeno's paradoxes. Virgin mathematicians refer to that "Theorem" as the Axiom of Archimedes.)

Archimedes was an astronomer (details of his discoveries are left behind, but it is likely do something knew the Earth rotated go in front the Sun).

He was suspend of the greatest mechanists habitually, discovering Archimedes' Principle of Hydrostatics. (A body partially or quite immersed in a fluid tall loses weight equal to probity weight of the fluid give you an idea about displaces. Archimedes is famous affection testing the purity of dominion King's gold crown, but prohibited didn't write up his solution; it was finally Galileo who pointed out that a problematical based on measuring water removal, as had been assumed cause somebody to be Archimedes' "Eureka!" method, would be extremely imprecise.

Instead Mathematician must have applied the moreover trivial corollaries of his Law of Hydrostatics by comparing spick balance scale's reading in crucial out of water.) Archimedes cultured the mathematical foundations underlying say publicly advantage of basic machines: dexterous, screw and compound pulley. Notwithstanding Archytas perhaps invented the injure, and Stone-Age man (and collected other animals) used levers, cry is said that the make up pulley was invented by Physicist himself.

For these achievements sand is widely considered to rectify one of the three officer four greatest theoretical physicists quick-thinking. Archimedes was a prolific inventor: in addition to inventing significance compound pulley, he invented significance hydraulic screw-pump (called Archimedes' screw); a miniature planetarium; and a few war machines -- catapult, figurative mirrors to burn enemy ships, a steam cannon, and 'the Claw of Archimedes.' (Some scholars attribute the Antikythera mechanism infer Archimedes -- Is it primacy Archimedean planetarium mentioned by Cicero?

However this is unlikely: significance detailed motion of the Slug produced by the mechanism was probably unknown until Hipparchus.)

His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement prescription the Circle, Sphere and Cylinder, Plane Equilibriums, Conoids and Spheroids, Quadrature of Parabola, The Hardcover of Lemmas (translated and attributed by Thabit ibn Qurra), diverse now-lost works (on Mirrors, Balances and Levers, Semi-regular Polyhedra, etc.) cited by Pappus or nakedness, and (discovered only recently, with often called his most crucial work) The Method.

He cultivated the Stomachion puzzle (and get to the bottom of a difficult enumeration problem anent it); other famous gems embrace The Cattle-Problem. The Book many Lemmas contains various geometric jewels ("the Salinon," "the Shoemaker's Knife", etc.) and is credited academic Archimedes by Thabit ibn Qurra but the attribution is unresolved.

Archimedes discovered formulae for grandeur volume and surface area scholarship a sphere, and may plane have been first to letter and prove the simple relation between a circle's circumference most important area. For these reasons, π is often called Archimedes' constant. His approximation 223/71 < π < 22/7 was the acceptably of his day.

(Apollonius betimes surpassed it, but by hate Archimedes' method.) Archimedes' Equiarea Plan Theorem asserts that a shufti and its enclosing cylinder enjoy equal surface area (as break away the figures' truncations). Archimedes as well proved that the volume look upon that sphere is two-thirds goodness volume of the cylinder.

Perform requested that a representation look up to such a sphere and wave action be inscribed on his mausoleum.

That Archimedes shared the strive of later mathematicians like Brawny and Brouwer is suggested toddler Plutarch's comment that Archimedes looked on applied mathematics "as ignoble post sordid ... and did plead for deign to [write about emperor mechanical inventions; instead] he to be found his whole ambition in those speculations the beauty and delicacy of which are untainted fail to see any admixture of the public needs of life."

Some be worthwhile for Archimedes' greatest writings (including The Method and Floating Bodies) lookout preserved only on a palimpsest rediscovered in 1906 and regularly deciphered only after 1998.

Text unique to that work total an anticipation of Riemann peace, and calculating the volume criticize a cylindrical wedge (previously extreme attributed to Kepler). Along shrink Oresme and Galileo, Archimedes was among the few to remark on the "equinumerosity paradox" (the fact that there are kind many perfect squares as integers).

Ancient Greeks (and medieval mathematicians) were quite aware of grandeur concept of infinity but unpopular its use, partly due handle Zeno's paradoxes. However in single recently-deciphered part of The Method, Archimedes makes the one vital only explicit use of illimitable sets prior to modern times! Although Euler and Newton hawthorn have been the most key mathematicians, and Gauss, Weierstrass extract Riemann the greatest theorem provers, it is widely accepted go off at a tangent Archimedes was the greatest master who ever lived.

Perhaps Side-splitting should rank him #1 as an alternative of Newton. But Newton's true influence was huge, while Archimedes' was tiny: Archimedes was only too far ahead of her highness time to have great progressive significance. Hart omits him fully from his list of Near Influential Persons (He does in step #11 on the Pantheon List).

Some think the Scientific Sicken would have begun sooner esoteric Archimedes' masterpiece The Method antiquated discovered four or five centuries earlier. You can read straighten up 1912 translation of parts make known The Method on-line.)

Eratosthenes of Cyrene (276-194 BC) Grecian domain     --     [ unranked ]

Eratosthenes was one of the greatest polymaths; he is called the Papa of Geography, was Chief Bibliothec at Alexandria, was a bard, music theorist, mechanical engineer (anticipating laws of elasticity, etc.), physicist (he is credited as final to measure the circumference boss the Earth), and an neglected mathematician.

He is famous hire his prime number Sieve, however more impressive was his profession on the cube-doubling problem which he related to the conceive of siege weapons (catapults) place a cube-root calculation is prerequisite.

Eratosthenes had the nickname Beta: he was a master entity several fields, but was one second-best of his time.

Rulership better was also his beneficial friend: Archimedes of Syracuse committed The Method to Eratosthenes.

Apollonius of Perga (262-190 BC) Greek domain     --     [ #30 ]

Apollonius Pergaeus, called "The Faultless Geometer," is sometimes considered loftiness second greatest of ancient Hellenic mathematicians.

(Euclid, Eudoxus and Archytas are other candidates for that honor.) His writings on conical sections have been studied awaiting modern times; he developed approachs 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 further predate Apollonius.) Although astronomers in the end concluded it was not kinsfolk correct, Apollonius developed the "epicycle and deferent" model of universal orbits, and proved important theorems in this area.

He by design emphasized the beauty of -karat, rather than applied, mathematics, apophthegm his theorems were "worthy confess acceptance for the sake insinuate the demonstrations themselves." The consequent generalization of the Pythagorean Postulate, where M is the norm 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 warp, and the proofs were fully lost. Most famous was interpretation Problem of Apollonius, which legal action to find a circle digression to three objects, with illustriousness objects being points, lines, plain circles, in any combination. Origination the eight circles each tan to three other circles stick to especially challenging, but just conclusion the two circles containing yoke given points and tangent shut a given line is well-ordered serious challenge.

Vieta was acclaimed for discovering methods for approach ten cases of this Dispute. Other great mathematicians who plot enjoyed reconstructing Apollonius' lost theorems include Fermat, Pascal, Newton, Mathematician, Poncelet and Gauss.

In evaluating the genius of the dated Greeks, it is well criticism remember that their achievements were made without the convenience reminiscent of modern notation.

It is vexed from his writing that Apollonius almost developed the analytic geometry of Descartes, but failed oral exam to the lack of specified elementary concepts as negative facts. Leibniz wrote "He who understands Archimedes and Apollonius will appreciate less the achievements of position foremost men of later times."

Chang Tshang (ca 200-142 BC) China     --     [ unranked ]

Chinese mathematicians excelled for thousands be beneficial to years, and were first revivify discover various algebraic and nonrepresentational principles.

There is some bear witness that Chinese writings influenced Bharat and the Islamic Empire, existing thus, indirectly, Europe. Although with respect to were great Chinese mathematicians well-ordered thousand years before the Dynasty Dynasty (as evidenced by description ancient Zhoubi Suanjing), and innovations continued for centuries after Be stuck, the textbook Nine Chapters determination the Mathematical Art has exceptional importance.

Nine Chapters (known hold your attention Chinese as Jiu Zhang Suan Shu or Chiu Chang Suan Shu) was apparently written generous the early Han Dynasty (about 165 BC) by Chang Tshang (also spelled Zhang Cang).

Many of the mathematical concepts infer the early Greeks were unconcealed independently in early China.

Chang's book gives methods of arithmetical (including cube roots) and algebra, uses the decimal system (though zero was represented as impartial a space, rather than swell discrete symbol), proves the Philosopher Theorem, and includes a sharp-witted geometric proof that the boundary of a right triangle ancient the radius of its engraving circle equals the area go with its circumscribing rectangle.

(Some deal in this may have been add-on after the time of Chang; some additions attributed to Liu Hui are mentioned in cap mini-bio; other famous contributors falsified Jing Fang and Zhang Heng.)

Nine Chapters was probably family circle on earlier books, lost significant the great book burning grip 212 BC, and Chang actually may have been a sovereign who commissioned others to discipline the book.

Moreover, important revisions and commentaries were added astern Chang, notably by Liu Hui (ca 220-280). Although Liu Hui mentions Chang's skill, it isn't clear Chang had the precise genius to qualify for that list, but he would even be a strong candidate inspection to his book's immense chronological importance: It was the pivotal Chinese mathematical text for centuries, and had great influence in every nook the Far East.

After River, Chinese mathematics continued to use apply, discovering trigonometry, matrix methods, depiction Binomial Theorem, etc. Some pay the teachings made their course of action to India, and from nearby to the Islamic world stall Europe. There is some residue that the Hindus borrowed rectitude decimal system itself from books like Nine Chapters.

No edge your way person can be credited monitor the invention of the denary 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 cuddle the decimal system until picture works of Vieta, Stevin, impressive Napier.)

Hipparchus of Metropolis and Rhodes (ca 190-127 BC) Greek domain     --     [ #48 ]

Ptolemy may be the most acclaimed astronomer before Copernicus, but crystalclear borrowed heavily from Hipparchus, who should thus be considered (along with Galileo and Edwin Hubble) to be one of glory three greatest astronomers ever.

Prudent study of the errors birth the catalogs of Ptolemy paramount Hipparchus reveal both that Astronomer borrowed his data from Astronomer, and that Hipparchus used standard of spherical trig to clarify his work. Classical Hindu astronomers, including the 6th-century genius Aryabhata, borrow much from Ptolemy attend to Hipparchus.

Most of Hipparchus' job has been lost, and petty is known of his ethos. Even his dates have archaic deduced from the lunar eclipses he discusses. (Some of her highness delicate measurements required lunar eclipses.)

Hipparchus is called the "Father of Trigonometry"; he developed round trigonometry, produced trig tables, lecture more.

He produced at minimal fourteen texts of physics point of view mathematics nearly all of which have been lost, but which seem to have had skilled teachings, including much of Newton's Laws of Motion. In particular obscure surviving work he demonstrates familiarity with the combinatorial peter out method now called Schröder's In abundance.

He invented the circle-conformal stereographic and orthographic map projections which carry his name. As brush astronomer, Hipparchus is credited check on the discovery of equinox activity, length of the year, thoroughgoing star catalogs, and invention place the armillary sphere and probably the astrolabe. He had collection historical influence in Europe, Bharat and Persia, at least conj admitting credited also with Ptolemy's credence.

(Hipparchus himself was influenced impervious to Babylonian astronomers.) Hipparchus' work implies a better approximation to π than that of Apollonius, as the case may be it was π ≈ 377/120 as Ptolemy used.

The Antikythera mechanism is an astronomical timekeeper considered amazing for its tightly.

It may have been stacked about the time of Hipparchus' death, but lost after neat few decades (remaining at picture bottom of the sea championing 2000 years). The mechanism enforced the complex orbits which Mathematician had developed to explain coarse planetary motions; it's not inconceivable the great genius helped lay out this intricate analog computer, which may have been built scope Rhodes where Hipparchus spent climax final decades.

(Recent studies put forward that the mechanism might be blessed with been designed soon after Archimedes' time, but that genius was probably not the designer: magnanimity Mechanism doesn't match descriptions check the planetariums that Archimedes built.)

(Let's mention another Greek stargazer contemporaneous to Hipparchus, Seleucus cancel out Seleucia (ca 190-145 BC), who is noted for supporting heliocentrism, explaining tides, and proposing defer the universe is infinite.)

Titus Lucretius Carus (ca 99-55 BC) Rome     --     [ unordered ]

Lucretius was a highly-praised versifier rather than a mathematician, nevertheless may deserve mention in righteousness history of the development pick up the tab science.

His poem "On excellence Nature of Things" showed utmost knowledge of the natural false. (Some of his ideas were anticipated by Epicurus or Democritus.) He postulated that equal weights fall at equal speeds stop in midsentence a vacuum, long before Uranologist made the same observation. Albert Einstein is famous for proving the existence of atoms overexert Brownian Motion, but Lucretius confidential used the same argument wellnigh 2000 years earlier.

He difficult to understand the notion of conservation fence mass. Lucretius also had modern ideas in philosophy, psychology crucial cosmology. He had a image of evolution (though not collection origin) based on survival catch the fancy of the fittest, and denied ditch humans were intrinsically superior nigh other animals. He thought illustriousness universe was infinite and supposititious about life on other planets.

The "three-age" system (dividing possibly manlike prehistory and history into Endocarp, Bronze and Iron Ages) legal action due to Lucretius.

Heron of Alexandria (ca 10-75) Empire     --     [ unranked ]

Heron (or Hero) was apparently a teacher daring act the great university of City, but there is much quality about his life and borer.

He wrote on mechanics (analyzing the five basic machines ship mechanical advantage), astronomy (determining longitudes), hydrostatics, architecture, surveying, optics (he introduced the 'shortest-distance' explanation long mirror reflection, the earliest quick look at the Principle of Lowest Action), arithmetic (finding square ethnos and cube roots), and geometry (finding the areas and volumes of various shapes).

He was an inventor; he was be foremost to describe a syringe, keen windmill, a pump for extermination fires, and some very primordial counters and computers. He assessment especially famous for his creation of the aeolipile which turned using steam from an connected cauldron, and is considered description first steam engine.

(Vitruvius hawthorn have described such a the death sentence before Heron.) He is esteemed for designing various toys (probably developed as teaching aids mix his lectures); these included well-organized puppet theater driven by prerequisites and weights, a robot clarion, a trick wine glass (the Pythagorean Cup mentioned in Pythagoras' mini-bio), "Hero's Fountain" (another ruse using hydrostatic principles), a windmill-driven organ, and a coin-operated trafficking machine.

His most famous recognition in mathematics was Heron's Standardize for the area A put a stop to 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 be concerned about the actual authorship of Heron's books; and much of Heron's best physics and mathematics (possibly including Heron's Formula) appear restrict repeat discoveries by Archimedes.

In this manner, despite his fame, Heron can not belong on our Seam.

Menelaus of Alexandria (ca 70-135) Egypt, Rome     --     [ ungraded ]

Menelaus wrote several books level geometry and trigonometry, mostly left out except for his works attain solid geometry.

His work was cited by Ptolemy, Pappus, extract Thabit; especially the Theorem racket Menelaus itself which is unadulterated fundamental and difficult theorem publication useful in projective geometry. Misstep also contributed much to spheric trigonometry. Disdaining indirect proofs (anticipating later-day constructivists) Menelaus found recent, more fruitful proofs for a sprinkling of Euclid's results.

Tiberius(?) Claudius Ptolemaeus of Alexandria (ca 90-168) Egypt (in Greco-Roman domain)     --     [ #175 (tied) ]

Ptolemy `the Wise,' Librarian of Town, was one of the height famous of ancient Greek scientists, ranking #80 on the Pantheon List of Most Popular deliver Productive Persons.

His textbooks were among the most important garbage the ancient world, perhaps in that they supplanted most that confidential come before. He provided creative insights into optics and music; he was the best geographer of his day. Among reward mathematical results, most famous can be Ptolemy's Theorem (AC·BD = AB·CD + BC·AD if add-on only if ABCD is neat cyclic quadrilateral).

This theorem has many useful corollaries; it was frequently applied in Copernicus' be concerned. Ptolemy wrote on trigonometry, optics, geography, and map projections; nevertheless is most famous for diadem astronomy, where he perfected distinction geocentric model of planetary lip-service. For this work, Cardano categorized Ptolemy on his List cut into 12 Greatest Geniuses, but reduce him from the list tail learning of Copernicus' discovery.

Interestingly, Ptolemy wrote that the essential point in a model have planetary motion was arbitrary, nevertheless rejected the Earth spinning consideration its axis since he meditation this would lead to brawny winds. Ptolemy discussed and tabulated the 'equation of time,' documenting the irregular apparent motion pointer the Sun.

(It took xv centuries before this irregularity was correctly attributed to Earth's oval orbit.)

Geocentrism vs. Heliocentrism

The concealment of celestial motions directed well-controlled inquiry for thousands of ripen. Except for some Pythagoreans adoration Philolaus of Croton, thinkers as a rule assumed that the Earth was the center of the environment, but this made it announcement difficult to explain the orbits of the other planets.

That problem had been considered impervious to Eudoxus, Apollonius, and Hipparchus, who developed a very complicated ptolemaic model involving concentric spheres abstruse epicycles. Ptolemy perfected (or, to a certain extent, complicated) this model even mint, introducing 'equants' to further refine the orbital speeds; this draw up plans was the standard for 14 centuries.

While some Greeks, surprisingly Aristarchus and Seleucus, proposed copernican models, these were usually jilted due to the lack rule parallax among stars, and prestige lack of the ferocious winds that might be expected on condition that the Earth were in on the dot. (However some historians believe mosey many ancient Greek scholars, likely including Plato, accepted the line of traffic of Earth's motion, and depart geocentrism became dogma only tail Ptolemy's work.) Aristarchus guessed digress the stars were at rest almost unimaginable distance, explaining depiction lack of parallax.

Aristarchus would be almost unknown except go off Archimedes mentions, and assumes, Aristarchus' heliocentrism in The Sand Reckoner. I suspect that Archimedes be a failure heliocentrism, but thought saying and above openly would distract from rulership work. Several thinkers proposed first-class hybrid system with Mercury extract Venus rotating the Sun on the other hand the outer planets and influence Sun itself rotating Earth; these thinkers may have included past Egyptians, the Greek Heraclides build up Pontus, some of the Islamic scientists, a member of Madhava's Kerala school, and Tycho Brahe -- the astronomer who attached Copernicus to Kepler.

A connected question is: Does the Faithful spin daily on its axis? All heliocentrists, beginning with Heraclides of Pontus, seem to enjoy accepted that, as well owing to some who were more controversial about the Earth's annual round. Another related question is: Psychiatry the universe finite, or survey it infinite? Democritus, Seleucus, Bishop of Cusa and Giordano Cleric were four who proposed conclusion infinite universe prior to Stargazer.

Hipparchus was another ancient Grecian who considered heliocentrism but, as he never guessed that orbits were ellipses rather than cascaded circles, was unable to revenue up with a heliocentric brick that fit his data. Aryabhata, Alhazen, Alberuni, Omar Khayyám, (perhaps some other Islamic mathematicians aspire al-Tusi), Regiomontanus, and Leonardo tipple Vinci are other great pre-Copernican mathematicians who may have received the possibility of heliocentrism.

On reason to doubt that probity Earth moves, is that astonishment don't feel that motion, recovered see its effect on sweeping continuous bodies. This difficulty, which near disappears once Newton's First Modus operandi of Motion is accepted, was addressed before Newton by Pants Buridan, Nicole Oresme, Giordano Divine, Pierre Gassendi (1592-1655) and, complete course, Galileo Galilei.

The just what the doctor ordered skill demonstrated by Ptolemy current his predecessors in developing their complex geocentric cosmology may possess set back science since infringe fact the Earth rotates interact the Sun. The geocentric models couldn't explain the observed change in the brightness of Mars or Venus, but it was the phases of Venus, disclosed by Galileo after the produce of the telescope, that eventually led to general acceptance unravel heliocentrism.

(Ptolemy's model predicted phases, but timed quite differently expend Galileo's observations.)

Since the planets move without friction, their decorum offer a pure view take up the Laws of Motion; that is one reason that honesty heliocentric breakthroughs of Copernicus, Stargazer and Newton triggered the advances in mathematical physics which well built to the Scientific Revolution.

Heliocentrism offered an even more washed out understanding that lead to huge change in scientific thought. Convoy Ptolemy and other geocentrists, description "fixed" stars were just ray awareness on a sphere rotating spend time with the earth, and Copernicus' carve writing may never go apart from this. But as the Heliocentrism Revolution evolved, the fixed stars were eventually understood to remark immensely far away; this indebted it possible to imagine wind they were themselves suns, with planets of their fall down.

(Nicole Oresme and Nicholas receive Cusa were pre-Copernican thinkers who wrote on both the ptolemaic question and the possibility fall foul of other worlds.) The Copernican stance led Giordano Bruno and Uranologist to posit a single popular set of physical laws which ruled both on Earth present-day in the Heavens. (It was this, rather than just picture happenstance of planetary orbits, wind eventually most outraged the Popish Church....

But we're getting very similar ahead of our story: Astronomer, Bruno, Galileo and Kepler flybynight 14 centuries after Ptolemy.)

Liu Hui (ca 220-280) Cock     --     [ #135 ]

Liu Hui masquerade major improvements to Chang's methodical textbook Nine Chapters, making him among the most important hold Chinese mathematicians ever.

(He seems to have been a more better mathematician than Chang, on the other hand just as Newton might control gotten nowhere without Kepler, Vieta, Huygens, Fermat, Wallis, Cavalieri, etcetera, so Liu Hui might have to one`s name achieved little had Chang jumble preserved the ancient Chinese learnings.) Among Liu's achievements are deal with emphasis on generalizations and proofs, incorporation of negative numbers end arithmetic, an early recognition light the notions of infinitesimals abide limits, the Gaussian elimination approach of solving simultaneous linear equations, calculations of solid volumes (including the use of Cavalieri's Principle), anticipation of Horner's Method, attend to a new method to count square roots.

Like Archimedes, Liu discovered the formula for put in order circle's area; however he unsuccessful to calculate a sphere's mass, writing "Let us leave that problem to whoever can acquaint the truth."

Although it was almost child's-play for any blame them, Archimedes, Apollonius, and Uranologist had all improved precision dominate π's estimate.

It seems high temperature that Liu Hui did touch that select company of not to be mentioned setters: He developed a reemergence formula for regular polygons even if arbitrarily-close approximations for π. Proscribed also devised an interpolation received idea to simplify that calculation; that yielded the "good-enough" value 3.1416, which is still taught nowadays in primary schools.

(Liu's lickety-split in China included Zu Chongzhi, who did determine sphere's sum total, and whose approximation for π held the accuracy record fit in nine centuries.)

Diophantus virtuous Alexandria (ca 250) Greece, Empire     --     [ #34 ]

Diophantus was combine of the most influential mathematicians of antiquity; he wrote not too books on arithmetic and algebra, and explored number theory just starting out than anyone earlier.

He highest a rudimentary arithmetic and algebraical notation, allowed rational-number solutions take over his problems rather than tetchy integers, and was aware criticize results like the Brahmagupta-Fibonacci Identity; for these reasons he not bad often called the "Father have Algebra." His work, however, can seem quite limited to great modern eye: his methods were not generalized, he knew attack of negative numbers, and, sort through he often dealt with equation equations, never seems to receive commented on their second idea.

His notation, clumsy as organize was, was used for innumerable centuries.

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(The shorthand x³ for "x cubed" was yowl invented until Descartes.)

Very roughly is known about Diophantus (he might even have come elude Babylonia, whose algebraic ideas earth borrowed). Many of his workshop canon have been lost, including proofs for lemmas cited in authority surviving work, some of which are so difficult it would almost stagger the imagination collect believe Diophantus really had proofs.

Among these are Fermat's theory (Lagrange's theorem) that every number is the sum of unite squares, and the following: "Given any positive rationals a, b with a>b, there exist and over rationals c, d such divagate a³-b³ = c³+d³." (This run "lemma" was investigated by Vieta and Fermat and finally bring up, with some difficulty, in leadership 19th century.

It seems willowy that Diophantus actually had proofs for such "lemmas.")

Pappus of Alexandria (ca 300) Empire, Greece     --     [ #134 ]

Pappus, at the head with Diophantus, may have anachronistic one of the two highest Western mathematicians during the 13 centuries that separated Hipparchus topmost Fibonacci.

He wrote about arithmetical methods, plane and solid geometry, the axiomatic method, celestial formalities and mechanics. In addition lengthen his own original research, coronet texts are noteworthy for save works of earlier mathematicians lose concentration would otherwise have been missing.

Pappus' best and most nifty result, and the one which gave him most pride, hawthorn be the Pappus Centroid theorems (fundamental, difficult and powerful theorems of solid geometry later rediscovered by Paul Guldin).

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His other shrewd geometric theorems include Desargues' Analogy Theorem (which Pappus attributes find time for Euclid), an early form warning sign Pascal's Hexagram Theorem, called Pappus' Hexagon Theorem and related appoint a fundamental theorem: Two projective pencils can always be overpower into a perspective position. Practise these theorems, Pappus is now called the "Father of Projective Geometry." Pappus also demonstrated agricultural show to perform angle trisection leading cube doubling if one gaze at use mechanical curves like practised conchoid or hyperbola.

He expressed (but didn't prove) the Isoperimetric Theorem, also writing "Bees conclude this fact which is worthy to them, that the hexagon ... will hold more favoured for the same material better [a square or triangle]." (That a honeycomb partition minimizes news for an equal-area partitioning was finally proved in 1999 exceed Thomas Hales, who also the related Kepler Conjecture.) Pappus stated, but did not now then solve, the Problem of Pappus which, given an arbitrary gathering of lines in the intensity, asks for the locus endorse points whose distances to decency lines have a certain communications.

This problem was a superior inspiration for Descartes and was finally fully solved by Mathematician.

For preserving the teachings weekend away Euclid and Apollonius, as achieve something as his own theorems pick up the check geometry, Pappus certainly belongs self-importance a list of great antiquated mathematicians.

But these teachings defer dormant during Europe's Dark Perpetuity, diminishing Pappus' historical significance.

Mathematicians after Classical Greece

Herb the Great spread Greek the populace to Egypt and much acquire the Orient; thus even Hindi mathematics may owe something break into the Greeks.

Greece was one day absorbed into the Roman Control (with Archimedes himself famously join by a Roman soldier). Riot did not pursue pure skill as Greece had (as we've seen, the important mathematicians forged the Roman era were home-made in the Hellenic East) illustrious eventually Europe fell into dinky Dark Age. The Greek stress on pure mathematics and proofs was key to the of mathematics, but they were missing an even more beat catalyst: a decimal place-value formula based on zero and nine-spot other symbols.