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Microbios II History Biographies II (Before Calculus) ------------------------------------------------------------------------------------- Mohammed ibn Musa al-Khowarizmi (Arab, ca 825) Described Hindu number system with base ten positional notation and a zero. This spread to Europe through translations of his work. Wrote treatise on algebra called Hisab al-jabr w'al-muqa-balah (Science of taking apart and joining back). Adelard translated his astronomy tables for Europe. Gherardo of Cremona translated his algebra work into Latin (ca 1150). His work included casting out nines, estimating square and cube roots, fractions, and the rule of three (business math). ------------------------------------------------------------------------------------- Bhaskara (India, 1114 - ca 1185 AD) Lived in Ujjain, India (central India, home of Brahmagupta, 5 centuries before). Wrote Siddhanta Siromani (Diadem of an Astronomical System) in 1150. Proved the Pythagorean theorem by a dissection method (central square with side b–a). Also had the "altitude-on-the-hypotenuse" proof of the Pythagorean theorem. Used the method of inversion/false position. Had identities for square roots and quadratic surds, and worked on Pell's equation. Gave 3927/1250 as π; also 22/7 and square root of 10 as approximates to π. Also gave 754/240 which was known to Ptolemy. Lilavati (The Beautiful) deals with arithmetic and Vijaganita (Seed arithmetic) with algebra. ------------------------------------------------------------------------------------- Brahmagupta (India, ca 628 AD) Lived in Ujjain (central India) in 628. Wrote on "the revised system of Brahma", a work on astronomy. Worked on Pell's equation. Found integral solutions to ax + by = c given a, b, c integers. Had Heron's formula for area of a triangle in terms of three sides and also had a generalization. Area of cyclic quadrilateral. Espoused the rule of three (a ratio formula for business math) Works were taken to Baghdad by the Moslems in the 8th century and translated. Much of Bhaskara's work was probably based on Brahmagupta's work. ------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- Girolamo Cardano (Italian, 1501 - 1576) Published solutions to the cubic and the quartic. Doctor, teacher, mathematician, born in Pavia, illegitimate son of a jurist. Lived a turbulent, cruel life. Traveled widely and held chairs at University of Pavia and Bologna. Published a horoscope of Christ's life. Distinguished astrologer for the papal court. Gambler who wrote on probability and the "problem of points". 1545 published: Ars magna on algebra. The book included a solution to the cubic (stolen from Tartaglia) and one on the quartic discovered by his pupil Ferrari. Died by suicide in 1576 at the time of his predicted death . (Also called Jerome Cardan, Hieronymus Cardanus) ------------------------------------------------------------------------------------- Chu Shi-kie (Chinese, ca 1300 AD.) Supposedly greatest mathematician of China history. Lived during the Sung dynasty. Generalized method of extracting square roots to higher degrees and developed "Horner's method" for solving equations. Speaks of a certain triangle (Pascal's) as already ancient (1303 AD). Used "matrix methods" and substitution and elimination to solve systems of linear equations. Used a circle as a symbol for zero. ------------------------------------------------------------------------------------- Christopher Clavius (German, 1537-1612) German born Jesuit who promoted knowledge of mathematics. Wrote popular textbooks on arithmetic and algebra, and published an edition of the Elements. Wrote on trigonometry and astronomy and played a part in the calendar reform. ------------------------------------------------------------------------------------- Nicolas Copernicus (Polish, 1473 - 1543) Prominent astronomer who suggested the sun was the center of the solar system. He was educated at the University of Cracow and studied at Padua and Bologna. His theory of the universe was published after his death and eventually replaced the Aristotelian view. His astronomy work stimulated the study of trigonometry and he wrote a text on trig. ------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- Gerard Desargues (French, 1591 - 1662) Introduced synthetic projective geometry. Born in Lyons. Engineer, architect, army officer who wrote small ignored treatise on conic sections and synthetic projective geometry (Brouillon projet....) Wrote in eccentric style and overshadowed by Descartes analytic geometry a few years later. Introduced the term "involution". Gave popular lectures in Paris while in his thirties and impressed Descartes and Pascal who credited him as an inspiration; Pascal wrote a treatise on conic sections based on Desargues' work. Knew a few surfaces of second degree. Gave the two-triangle theorem (Desargues configuration). A student, La Hire, tried to show that all of Apollonius' work can be derived from Desargues ' method of central projections. Our knowledge of Desargues' work comes from the writings of La Hire. ------------------------------------------------------------------------------------- Rene Descartes (French, 1596-1650) Introduced analytic geometry. Born near Tours. Late sleeper. Soldier for Prince Maurice of Orange. Traveled. Studied Math and Philosophy at Paris and constructed optical instruments. Moved to Holland where he lived for 20 years as a scientist. In 1649 he tutored Queen Christina and died of disease. In Holland wrote Le Monde, an account of the physical universe. Wrote on analytic geometry, Discours, and three appendices to it. Also wrote Meditationes, on philosophy and La Geometrie. Constructed tangents to curves without the use of calculus. Cartesian oval. Developed "Descartes' rule of signs". Worked on solutions to curves of degree > 2. Other studies: the "folium of Descartes", amicable numbers (giving a third pair), tangent to a cycloid, "near discovery" of Euler's v - e + f = 2. Used curves to duplicate the cube. Claimed earth elongated at poles (false.) Introduced solid analytic geometry. Gave a solution to the quartic equation. Studied optics and the weather, including the rainbow. ------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- Pierre de Fermat (French, 1601? - 1665) Lawyer who developed number theory and analytic geometry. Born near Toulouse in 1601? Died at Castres or at Toulouse on January 12, 1665. Son of leather merchant. Developed equations of conics and "spirals of Fermat". Defined curves analytically. Began with the equation and then found the curve, a novel idea for its time. Understood the rudiments of differentiation and had a method for finding maxima and minima of a curve using infinitesimals. Found number theory, stimulated by Diophantus Arithmetica. Example of theorems are on p. 355 of Eves. Some were written in margin of his translation of Arithmetica. Other things he introduced: Fermat's "last theorem", Fermat's conjecture on primes & Fermat primes, the method of infinite descent, a second pair of amicable numbers: 17296 and 18416. He corresponded with Pascal on probability (gambling). ------------------------------------------------------------------------------------- Leonardo Fibonacci (Italian, ca 1175 - 1250) Best mathematician of the middle ages. Leonardo of Pisa. Born in Pisa where his father was a merchant. Brought up in Boughie on the north coast of Africa. Early interest in algebra. Toured the Mediterranean, learning about Eastern and Arabic mathematics. In 1202 he published Liber abaci which promotes the Hindu-Arabic methods of calculation. Strongly advocated the Hindu-Arabic numerals. (Algebra was still rhetorical.) A number of problems became famous: his sequence. Wrote (1220) Practica geometriae, (1225) Liber quadratorum on analysis. Solved a problem on roots of a cubic where he found nine decimal place approximation of the answer and attempted to prove the root was not constructible! (Some of these problems are on pages 263, 284 of Eves.) ------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- Galileo Galilei (Italian, 1564 - 1642) Mathematician, astronomer, physicist. Dethroned Aristotle. Born in Pisa. Discovered the period of pendulum independent of the size of the arc and weight of the bob. Appointed to University of Pisa. Studied falling bodies and disagreed with Aristotle's laws. Resigned University of Pisa (1591) after strife over his repudiation of Aristotle and then appointed to University of Padua where he lectured for 18 years. In 1609 heard about the spyglass (now two years old) and made a number of them. Made telescopes and watched moons of Jupiter. Wrote a book supporting the Copernican view of the solar system. Noted parabolic curve of flying body (in vacuum). Called attention to the cycloid. Built microscope and scaled ("sector") compass. Devout Catholic. Claimed Bible was not intended as a book of scientific truth. Called to Inquisition in 1633 for opposing the Church's view of the Universe. "Recanted". But his ideas continued to spread. -------------------------------------------------------------------------------------
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