History Biographies II (Before ) ------Mohammed ibn Musa al-Khowarizmi (Arab, ca 825) Described Hindu system with base ten and a zero. This spread to Europe through translations of his work. Wrote treatise on called Hisab al-jabr w'al-muqa-balah ( of taking apart and joining back). Adelard translated his 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, , and the rule of three (business math).

------Bhaskara (India, 1114 - ca 1185 AD) Lived in Ujjain, India (central India, home of , 5 centuries before). Wrote Siddhanta Siromani (Diadem of an Astronomical System) in 1150. Proved the 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 . Lilavati (The Beautiful) deals with 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 solutions to ax + by = c given a, b, c . Had Heron's formula for 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, , 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 ('s) as already ancient (1303 AD). Used "matrix methods" and substitution and elimination to solve systems of linear equations. Used a as a symbol for zero.

------Christopher Clavius (German, 1537-1612) German born Jesuit who promoted knowledge of . Wrote popular textbooks on arithmetic and algebra, and published an edition of the Elements. Wrote on 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 . Born in Lyons. , 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 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 to 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 (giving a third pair), to a , "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.

------ (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 " of Fermat". Defined curves analytically. Began with the equation and then found the , a novel idea for its time. Understood the rudiments of differentiation and had a method for finding maxima and minima of a curve using . 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 (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 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.)

------ (Italian, 1564 - 1642) Mathematician, astronomer, . 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.

------Thomas Harriot (English, 1560 - 1621) Contributed to notation and codification of algebra. "Founder of English school of algebraists". Was sent by Sir Walter Raleigh as a surveyor to the new world in 1585. Wrote on theory of equations, including up to fourth degree, and transformation of equations and numerical approximations to roots. Used < and > signs and aa, aaa for a2 and a3. His great work appeared after his death. Astronomer who discover sunspots and observed satellites of Jupiter (independent of Galileo). Died of cancer caused by sniffing tobacco.

------ (Dutch, 1629 - 1695) Mathematician, physicist, worked in probability and clock making. At 22 wrote a paper pointing out mistakes in Saint-Vincent's work on the of the circle. Wrote a number of papers on quadrature of conics and Snell's computation of π. In 1654 he & brother created better lenses and examined Saturn in detail. Invented the pendulum clock and made the first watch. In 1657 wrote a treatise on probability; introduced expectation. In 1665 moved to Paris and studied the of two bodies in a collision. Taught Leibniz in 1672. Corresponded with Fermat. 1673: Horologium oscillatorium written as a major work in five parts on ; included a study of the cycloidal pendulum clock. With suggested the earth is flattened at the poles. Studied the centrifugal force. Promoted a wave theory of light (unlike Newton). ------Johann Kepler (German, 1571 - 1630) Announced laws of planetary motion. Mathematician. Born near Stuttgart and educated at the University of Tubingen to become Lutheran minister. 1599 - assistant to . In 1601 inherited Brahe's position at Brahe's death. Worked with Brahe's massive notes to formulate laws of planets which he published in 1609, 1619 (based on conic sections). Had crude forms of integration. Studied polyhedra. Introduced the term "focus"; promoted . Suggested perimeter of = π(a + b), where a and b are lengths of minor and major semiaxes. Introduced a concept of continuity. Tragic personal life: Poor eyes due to smallpox at age 4. Two bad marriages, favorite child died of smallpox, wife went mad and died, expelled from the University of Gratz when Catholics took over, mother imprisoned for witchcraft. Cast horoscopes.

------ (Arab, 1050 - 1123 AD) Astronomer, mathematician, writer. Found geometric solution of cubic, systematically classified cubics but rejected negative roots and did not find all positive ones. Wrote Rubaiyat, undertook calendar reform, wrote commentary on Euclid (Discussions and Difficulties with Euclid).

------ (Scottish, 1550-1617) Inventor of logarithms. Born near (of 16-year old father). Spent much time supporting Knox, James I and opposing Catholicism. Attempted in 1593 to prove that the pope was the antichrist and that the world would end between 1688 and 1700. Wrote prophetically of the gun, submarine and army tank. Stories about him: black rooster and neighbor's pigeon. Invented logarithms, based on Werner's formulas. First logarithms were essentially 107 times the natural log. (Early Napier logs were based on formulas.) Published Rabdologiae, on multiplying with his "rods" or "bones". Napier's are trigonometric identities on proportions of sine, cosine to tangents. Had mnemonic diagram for working with spherical triangles.

------ (English, 1574 - 1660) Popularized the use of symbols in algebra. A clergyman in the parish of Bletchingdon, he gave free private lessons to , and Seth Ward. Described a circular and made the first straight logarithmic slide rule. Used the Greek symbol for π. May have written famous appendix to Napier's Descriptio. Wrote Clavis mathematicae which spread knowledge of arithmetic and algebra in . Wrote Trigonometrie (abbreviated the trig functions) and of Proportion.

------ (French, 1623 - 1662) Philosopher and Geometer. Born in the French province of Auvergne in 1623. Prodigy. Discovered geometry against his father's will. At 16 wrote essay on conic sections; at 18 created a calculating machine. 1650 abandoned math and devoted himself to religious contemplation. 1653 returned to math. Corresponded with Fermat. Manuscript on based on Desargues; now lost. Introduced the "mystic hexagram theorem". Studied probability (motivated by Chevalier de Mere's question on gambling) and used his famous triangle to solve the "problem of points". Invented the wheelbarrow and omnibus. Attacked the cycloid with infinitesimals and the "pre-calculus" of the day. 1654 dropped math again after accident with runaway horses. 1658 toothache; he then produced geometry of the cycloid curve. Died in Paris in 1662 at the age of 39.

------Gilles Persone de Roberval (French, 1602 - 1675) Mathematician, physicist. Born near Roberval in , died in Paris in 1675. Assumed the Roberval name to which he was not entitled. Corresponded extensively with many , notably Fermat. Tangents/higher plane curves. Found tangents by averaging vectors. Argued about priority with Torricelli. Also claimed method of indivisibles of Cavalieri and to have squared the cycloid. Retained his position as chair at the College Royale in France by winning contest held every three years (he posed the questions; this explains the reluctance to publish his works.) Found , volumes, centroids.

------ (Italian, 1608 - 1647) Introduced tangents as ratio of two directions about same time as Roberval. Argued about priority. Squared the cycloid. Student of Galileo. Died at age of 39. Proved that area of the cycloid is three times the area of the generation circle. Used infinitesimals to do so. Was accused of plagiarism by Roberval. Worked on the isogonic center of the triangle (proposed by Fermat) and solved by pupil Viviani. 1641 - discovered that an infinite area can sweep out a finite volume as a solid of revolution. Rectification of the logarithmic in 1640. Contributed to physics, where he worked on the barometer, the acceleration of gravity, motions of fluids, the path of a projectile.

------ (Flemish, 1548 - 1620) Promoted decimal fractions in Europe. As quartermaster general for the Dutch army he contributed to , , and military engineering. Invented a carriage propelled by sails. Found the force due to fluid pressure against a vertical rectangular dam, using methods we now use in calculus.

------Tabit ibn Qorra (Arab, 826-901) Wrote good translation of Euclid's Elements; also translated Conics of Apollonius and writings of , Ptolemy, Theodosius. Wrote on astronomy, conics, algebra, magic squares, amicable numbers and found rules for generating them (see Eves 243). Gave a dissection proof of the Pythagorean theorem. Worked on constructing the heptagon. ------Tartaglia, the "Stammerer" (Italian, ca 1499 - 1557) Solved the cubic equation. Born Nicolo Fontana of Brescia. "Stammerer"; brutally injured at the age of 13 when French soldiers massacred civilians in the local cathedral. His father was killed there; he was found there later by his mother. Sent to school for 15 days where he stole a book and taught himself to read and write. Taught science and math in various Italian cities. Solved equations regarding artillery fire. Wrote 2-volume treatise on commercial math (annuities, etc.). Published editions of works of Euclid and Archimedes. Worked on the "problem of points".

------Francois Viete (1540 - 1603) 277-280 Greatest French mathematician of the 16th century. Lawyer. Member of Parliament. Cryptographer (king of Spain complained that he employed magic). Used trigonometry to solve equations. Used all 6 trig functions. Found formulas for cos 2x, cos 3x, cos 4x, cos 5x, cos 6x, ... Most famous work: In artem which helped develop symbolic algebra. Used vowels for unknowns and consonants for knowns. First to use the same letter for a variable, its square, its cube, etc. Gave a systematic process for approximating the root of an equation. Did much on theory of equations. Showed two problems of antiquity required solving cubics. Calculated π and an infinite product for 2/π. First to introduce polar spherical triangles.

------Sir Christopher Wren (1632 - 1723) Savilian professor of astronomy, mathematician, architect. Pupil of Oughtred. Taught geometry at from 1661 to 1673. A founder of the Royal Society and its president for a time. Wrote on collision of bodies, optics, resistance of fluids, celestial mechanics. 1669 - ruling of hyperboloid of one sheet. 1658 - rectified (found the length of the arc of) the cycloid: the arch of the cycloid is equal in length to eight times the radius of the generating circle. After the Great Fire of London in 1666, he took part in rebuilding St. Paul's cathedral and became a famous as an architect. Buried in St. Paul's with the epitaph "If you seek a monument, look about you". Proposed as a unit of length the length of a pendulum beating half seconds = about half a meter.

------c. Ken W. Smith, January 2010