Thales was the Chief of the Seven Sages of ancient Greece, and has been called the “Father of Science,” the “Founder of Abstract Geometry,” and the “First Philosopher. ” Thales is believed to have studied mathematics under Egyptians, who in turn were aware of much older mathematics from Mesopotamia. Thales may have invented the notion of compass-and-straightedge construction.
Several fundamental theorems about triangles are attributed to Thales, including the law of similar triangles (which Thales used famously to calculate the height of the Great Pyramid) and “Thales’ Theorem” itself: the fact that any angle inscribed in a semicircle is a right angle. (The other “theorems” were probably more like well-known “axioms”, but Thales proved Thales’ Theorem using two of his other theorems; it is said that Thales then sacrificed an ox to celebrate what might have been the very first mathematical proof!)
Thales noted that, given a line segment of 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, and is the first person believed to have correctly predicted a solar eclipse. His theories of physics would seem quaint today, but he seems to have been the first to describe magnetism and static electricity. Aristotle said, “To Thales the primary question was not what do we know, but how do we know it.
” Thales was also a politician, ethicist, and military strategist. It is said he once leased all available olive presses after predicting a good olive season; he did this not for the wealth itself, but as a demonstration of the use of intelligence in business. Thales’ writings have not survived and are known only second-hand. Since his famous theorems of geometry were probably already known in ancient Babylon, his importance derives from imparting the notions of mathematical proof and the scientific method to ancient Greeks.
Thales’ student and successor was Anaximander, who is often called the “First Scientist” instead of Thales: his theories were more firmly based on experimentation and logic, while Thales still relied on some animistic interpretations. Anaximander is famous for astronomy, cartography and sundials, and also enunciated a theory of evolution, that land species somehow developed from primordial fish! Anaximander’s most famous student, in turn, was Pythagoras. (The methods of Thales and Pythagoras led to the schools of Plato and Euclid, an intellectual blossoming unequalled until Europe’s Renaissance.
For this reason Thales may belong on this list for his historical importance despite his relative lack of mathematical achievements. ) Apastambha (ca 630-560 BC) India The Dharmasutra composed by Apastambha contains mensuration techniques, novel geometric construction techniques, a method of elementary algebra, and what may be the first known proof of the Pythagorean Theorem. Apastambha’s work uses the excellent (continued fraction) approximation v2 ?
577/408, a result probably derived with a geometric argument. Apastambha built on the work of earlier Vedic scholars, especially Baudhayana, as well as Harappan and (probably) Mesopotamian mathematicians. His notation and proofs were primitive, and there is little certainty about his life. However similar comments apply to Thales of Miletus, so it seems fair to mention Apastambha (who was perhaps the most creative Vedic mathematician before Panini) along with Thales as one of the earliest mathematicians whose name is known. Pythagoras of Samos (ca 578-505 BC) Greek domain.
Pythagoras, who is sometimes called the “First Philosopher,” studied under Anaximander, Egyptians, Babylonians, and the mystic Pherekydes (from whom Pythagoras acquired a belief in reincarnation); he became the most influential of early Greek mathematicians. He is credited with being first to use axioms and deductive proofs, so his influence on Plato and Euclid may be enormous. He 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 recognized that the Earth was a globe similar to the other planets. He believed thinking was located in the brain rather than heart. The words “philosophy” and “mathematics” are said to have been coined by Pythagoras. Despite Pythagoras’ historical importance I may have ranked him too high: many results of the Pythagoreans were due to his students; none of their writings survive; and what is known is reported second-hand, and possibly exaggerated, by Plato and others.
His students included Hippasus of Metapontum, perhaps the famous physician Alcmaeon, Milo of Croton, and Croton’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. ) The Pythagorean Theorem was known long before Pythagoras, but he is often credited with the first proof.
(Apastambha proved it in India at about the same time, and some theorize that Pythagoras journeyed to India and learned of the proof there. ) He also discovered the simple parametric form of Pythagorean triplets (xx-yy, 2xy, xx+yy). Other discoveries of the Pythagorean school include the concepts of perfect and amicable numbers, polygonal numbers, golden ratio (attributed to Theano), 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. ) The famous successors of Thales and Pythagoras included Parmenides of Elea (ca 515-440 BC), Zeno of Elea (see below), Hippocrates of Chios (see below), Plato of Athens (ca 428-348 BC), Theaetetus (ca 414-369 BC), and Archytas (see below).
These early Greeks ushered in a Golden Age of Mathematics and Philosophy unequaled in Europe until the Renaissance. The emphasis was on pure, rather than practical, mathematics. Plato (who ranks #40 on Michael Hart’s famous list of the Most Influential Persons in History) decreed that his scholars should do geometric construction solely with compass and straight-edge rather than with “carpenter’s tools” like rulers and protractors. Panini (of Shalatula) (ca 520-460 BC) Gandhara (India).
Panini’s great accomplishment was his study of the Sanskrit language, especially in his text Ashtadhyayi. Although this work might be considered the very first study of linguistics or grammar, it used a non-obvious elegance that would not be equalled in the West until the 20th century. Linguistics may seem an unlikely qualification for a “great mathematician,” but language theory is a field of mathematics. The works of eminent 20th-century linguists and computer scientists like Chomsky, Backus, Post and Church are seen to resemble Panini’s work 24 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. ” Zeno of Elea (ca 495-435 BC) Greek domain Zeno, a student of Parmenides, had great fame in ancient Greece. This fame, which continues to the present-day, is largely due to his paradoxes of infinitesimals, e. g. his argument that Achilles can never catch the tortoise (whenever Achilles arrives at the tortoise’s last position, the tortoise has moved on). Although some regard these paradoxes as simple fallacies, they have been contemplated for many centuries.
It is due to these paradoxes that the use of infinitesimals, which provides the basis for mathematical analysis, has been regarded as a non-rigorous heuristic and is finally viewed as sound only after the work of the great 19th-century rigorists, Dedekind and Weierstrass. Hippocrates of Chios (ca 470-410 BC) Greek domain Hippocrates (no relation to 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; his circle quadrature was of course ultimately unsuccessful but he did prove ingenious theorems about “lunes” (certain circle fragments); and some claim Hippocrates was first to trisect the general angle.
(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. ) Hippocrates also did work in algebra and rudimentary analysis. Archytas of Tarentum (ca 420-350 BC) Greek domain Archytas was an important statesman as well as philosopher. He studied under Philolaus of Croton, was a friend of Plato, and tutored Eudoxus and Menaechmus.
In addition to discoveries always attributed to him, he may be the source of several of Euclid’s theorems, and some works attributed to Eudoxus and perhaps Pythagoras. Recently it has been shown that the 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. 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; 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 (as well as Thales) was mostly mythical. If we take that view, Archytas (and his mentor Hippocrates) should be promoted in this list. Eudoxus of Cnidus (408-355 BC) Greek domain Eudoxus journeyed widely for his education, despite that he was not wealthy, studying mathematics with Archytas in Tarentum, medicine with Philiston in Sicily, philosophy with Plato in Athens, continuing his mathematics study in Egypt, touring the Eastern Mediterranean with his own students and finally returned to Cnidus where he established himself as astronomer, physician, and ethicist.
What is known of him is second-hand, through the writings of Euclid and others, but he was one of the most creative mathematicians of the ancient world. Many of the theorems in Euclid’s Elements were first proved by Eudoxus. While Pythagoras had been horrified by the discovery of irrational numbers, Eudoxus is famous for incorporating them into arithmetic. He also developed the earliest techniques of the infinitesimal calculus; he is credited with first use of the Axiom of Archimedes, which avoids Zeno’s paradoxes by, in effect, forbidding infinities and infinitesimals;
Yet he also developed a method of taking limits. Eudoxus’ work with irrational numbers and infinitesimals may have helped inspire such masters as Archimedes and Dedekind. Eudoxus also introduced 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.
(It is sometimes said that he knew that the Earth rotates around the Sun, but that appears to be false; it is instead Aristarchus of Samos, as cited by Archimedes, who may be the first “heliocentrist. “) Four of Eudoxus’ most famous discoveries were the volume of a cone, extension of arithmetic to the irrationals, summing formula for geometric series, and viewing ? as the limit of polygonal perimeters. None of these seems difficult today, but it does seem remarkable that they were all first achieved by the same man.
Eudoxus has been quoted as saying “Willingly would I burn to death like Phaeton, were this the price for reaching the sun and learning its shape, its size and its substance. ” Aristotle of Stagira (384-322 BC) Macedonia Aristotle is considered the greatest scientist of the ancient world, and the most influential philosopher and logician ever; he ranks #13 on Michael Hart’s list of the Most Influential Persons in History. (His science was a standard curriculum for almost 2000 years, unfortunate since many of his ideas were quite mistaken.)
His writings on definitions, axioms and proofs may have influenced Euclid. He was also the first mathematician to write on the subject of infinity. His writings include geometric theorems, some with proofs different from Euclid’s or missing from Euclid altogether; one of these (which is seen only in Aristotle’s work prior to Apollonius) is that a circle is the locus of points whose distances from two given points are in constant ratio.
Even if, as is widely agreed, Aristotle’s geometric theorems were not his own work, his status as the most influential logician and philosopher makes him a candidate for the List. Euclid of Megara & Alexandria (ca 322-275 BC) Greece/Egypt Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers; he stated and proved the unique factorization theorem; and he 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. ) He proved that there are only five “Platonic solids,” as well as theorems of geometry far too numerous to summarize; among many with special historical interest is the proof that rigid-compass constructions can be implemented with collapsing-compass constructions.
Although notions of trigonometry were not in use, Euclid’s theorems include some closely related to the Laws of Sines and Cosines. Among several books attributed to Euclid 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. “