Contributions of Hindus Mathematicians

Earliest Mathematicians

Ø  Little is known of the earliest mathematics, but the famous Ishango Bone from Early Stone-Age Africa has tally marks suggesting arithmetic. The markings include six prime numbers (5, 7, 11, 13, 17, 19) in order, though this is probably coincidence (lich, chance, accident).

Ø  The advanced artifacts (object, work of art) of Egypt's Old Kingdom and the Indus-Harrapa civilization imply (involve, mean) strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where 4500-year old clay (soil, mud) tablets show multiplication and division problems; the first abacus may be about this old.

Ø  By 3600 years ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, and even logarithms, using a primitive place-value system (in base 60, not 10).

Ø  Babylonians were familiar with the Pythagorean Theorem, quadratic equations, even cubic equations (though they didn't have a general solution for these), and eventually even developed methods to estimate terms for compound interest.

Ø  Also at least 3600 years ago, the Egyptian showed simple algebra methods and included a table giving optimal expressions using Egyptian fractions.

Ø  Egyptians may have had more advanced geometry; Babylon was much more advanced at arithmetic and algebra.

Ø  This was probably due, at least in part, to their place-value system. But although their base-60 system survives (e.g. in the division of hours and degrees into minutes and seconds) the Babylonian notation, which used the equivalent of IIIIII XXXXXIIIIIII XXXXIII to denote 417+43/60, was unwieldy compared to the "ten digits of the Hindus."

Ø  The Egyptians used the approximation π ≈ (4/3)4 (derived from the idea that a circle of diameter 9 has about the same area as a square of side 8). Although the ancient Hindu mathematician Apastamba had achieved a good approximation for √2, and the ancient Babylonians an ever better √2, neither of these ancient cultures achieved a π approximation as good as Egypt's, or better than π ≈ 25/8, until the Alexandrian era.

Early Vedic mathematicians

The greatest mathematics before the Golden Age of    Greece was in India's early Vedic (Hindu) civilization.

The Vedics understood relationships between geometry and arithmetic, developed astronomy, astrology, calendars,           and used mathematical forms in some religious rituals.

Ø  The earliest mathematician to whom definite teachings can be ascribed was Lagadha, who apparently lived about 1300 BC and used geometry and elementary trigonometry for his astronomy.

Ø  Baudhayana lived about 800 BC and also wrote on algebra and geometry

Ø  Yajnavalkya lived about the same time and is credited with the then-best approximation to π.

Ø  A famous early Vedic mathematician was Apastamba, who lived slightly before Pythagoras, did work in geometry, advanced arithmetic, and may have proved the Pythagorean Theorem. (Apastamba used an excellent approximation for the square root of 2 (577/408, one of the continued fraction approximants); a 20th-century scholar has "reverse-engineered" a plausible geometric construction that led to this approximation.)

Ø  Other early Vedic mathematicians solved quadratic and simultaneous equations.

Thales

Ø  He invented the notion of compass-and-straightedge construction.

Ø  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 the fact that any angle inscribed in a semicircle is a right angle.

Ø  He is called the "Father of Science," the "Founder of Abstract Geometry," and the "First Philosopher."

Ø  He was also an astronomer.

Ø  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.

Ø  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.

Ø  Anaximander is famous for astronomy, cartography and sundials, and also enunciated a theory of evolution, that land species somehow developed from primordial fish

Ø  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.

Pythagoras of Samos (ca 578-505 BC) Greece

Ø  Pythagoras, who is sometimes called the "First Philosopher," studied under Anaximander,

Ø  Pythagoras as a wizard and founding mystic philosopher.

Ø   Pythagoras was very interested in astronomy and recognized that the Earth was a globe similar to the other planets.

Ø  He believed thinking was located in the brain rather than heart. The words "philosophy" and "mathematics" are said to have been coined by Pythagoras.

Ø  He discovered that harmonious intervals in music are based on simple rational numbers.

Ø  Leibniz later wrote, "Music is the pleasure the human soul experiences from counting without being aware that it is counting."

Ø  Other mathematicians who investigated the arithmetic of music included Huygens, Euler and Simon Stevin

Ø  He also discovered the simple parametric form of Pythagorean triplets (xx- yy, 2xy, xx+ yy).

Eudoxus of Cnidus (408-355 BC) Asia Minor, Greece

Ø  Many of the theorems in Euclid's Elements were first proved by Eudoxus.

Ø  He developed the earliest techniques of the infinitesimal calculus.

Ø  He is sometimes credited with first use of the Axiom of Archimedes, which avoids Zeno's paradoxes by, in effect, forbidding infinities and infinitesimals.

 Eudoxus developed the ancient theory of planetary orbits. (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.")

Euclid of Megara & Alexandria (ca 322-275 BC) Greece/Egypt

Ø  Euclid may have been a student of Aristotle.

Ø  He proved the unique factorization theorem ("Fundamental Theorem of Arithmetic").

Archimedes of Syracuse (287-212 BC) Greece/Sicily

Ø  Archimedes made advances in number theory, algebra, and analysis, but his greatest contributions were in geometry.

Ø  One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series.

Ø  Archimedes discovered formulae for the volume and surface area of a sphere, and may even have been first to notice and prove the simple relationship between a circle's circumference and area. For these reasons, &pi is often called Archimedes' constant.

Apollonius of Perga (262-190 BC) Greece

Ø  He is called "The Great Geometer,"

Ø  He is the second greatest of ancient Greek mathematicians

Ø  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."

Diophantus of Alexandria (ca 250) Greece, Egypt

Ø  He wrote several books on arithmetic and algebra,

Ø  He is called the "Father of Algebra."

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