Pythagoras, who is sometimes called the “First Philosopher,” studied under Anaximander, Egyptians, Babylonians, and the mystic Pherekydes 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. Panini 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. Zeno of Elea 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 Hippocrates 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 absurdemproof method.
Hippocrates is most famous for his work on the three ancient geometric quandaries: his work on cube-doubling 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 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..
Eudoxus of Cnidus 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.
Aristotle of Stagira 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. Euclid of Megara & Alexandria 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. (He proved that there are only five “Platonic solids,” as well as theorems of geometry far too numerous to summarize; among y with special historical interest is manthe proof that rigid-compass constructions can be implemented with collapsing-compass constructions. Archimedes of Syracuse 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 the works of Euclid.
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 unequalled 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 Apollonius of Perga Apollonius Pergaeus, called “The Great Geometer,” is sometimes considered the second greatest of ancient Greek mathematicians (Euclid and Eudoxus are the other candidates for this honor).
His writings on conic sections have been studied until modern times; he invented the names for parabola, hyperbola and ellipse; he developed methods for normals and curvature. Although astronomers eventually concluded it was not physically correct, Apollonius developed the “epicycle and deferent” model of planetary orbits, and proved important theorems in this area. He deliberately emphasized the beauty of pure, rather than applied, mathematics, saying his theorems were “worthy of acceptance for the sake of the demonstrations themselves. ” Chang Tshang
Chinese mathematicians excelled for thousands of years, and were first to discover various algebraic and geometric principles. There is some evidence that Chinese writings influenced India and the Islamic Empire, and thus, indirectly, Europe. Although there were great Chinese mathematicians a thousand years before the Han Dynasty, and innovations continued for centuries after Han, the textbook Nine Chapters on the Mathematical Art has special importance. Nine Chapters ) was apparently written during the early Han Dynasty (about 165 BC) by Chang Tshang (also spelled Zhang Cang). Hipparchus of Nicaea.
Ptolemy may be the most famous astronomer before Copernicus, but he borrowed heavily from Hipparchus, who might be considered the greatest astronomer ever. 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 Schroder’s Numbers.
Tiberius(? ) Claudius Ptolemaeus of Alexandria Ptolemy was one of the most famous of ancient Greek scientists. Among his mathematical results, most famous may be Ptolemy’s Theorem (AC·BD =AB·CD + BC·AD if and only if ABCD is a cyclic quadrilateral). This theorem has many useful corollaries; it was frequently applied in Copernicus’ work. Ptolemy also wrote on trigonometry, optics, geography, and astrology; but is most famous for his astronomy, where he perfected the geocentric model of planetary motions. The mystery of celestial motions directed scientific enquiry for thousands of years.
The problem had been considered by Eudoxus, Apollonius, and Hipparchus, who developed a very complicated geocentric model involving concentric spheres and epicyles. Liu Hui 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, 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. “