Descartes, Fermat, and Analytic Geometry PDF
Preview Descartes, Fermat, and Analytic Geometry
Analysis, Synthesis, the Infinite, and Numbers Chapter 15 Getting Our Bearings • Where are we in history? • We’ve finished the middle ages, and have embarked on the Early Modern period (about 1500 – 1800 or so). • In fact, we’ve finished the 1500’s as well. Not Europe Place Dates Names Mathematics China Ancient & Medieval The Nine Chapters, Square and cube roots, Sea Island indeterminate equations, Mathematical systems of linear Manual, others equations and linear congruences, polynomial equations India Ancient & Medieval Aryabhata, Square and cube roots, Bramagupta, indeterminate equations, others linear congruences, combinatorics. Islamic Empire Medieval Al Khwarizmi, Algebra, solutions of Omar Khayyam, al cubics by conics, Euclid’s Tusi, others 5th postulate, Hindu‐ Arabic numeration, Trigonometry Europe Place Dates Names Mathematics Europe Medieval Leonardo of Pisa, bar Islamic Methods introduced to Hiyya, ibn Ezra, ben Europe Gerson Europe Renaissance / Pacioli, Cardano, Solution of the cubic, Early Modern Targaglia, Ferrari, del systematic algebra & theory of Ferro, Bombelli, Viète, solving equations, Stevin, Regiomontanus, trigonometry, decimal Copernicus, Kepler, fractions, logarithms, Brahe, Napier, others heliocentric astronomy. Getting Our Bearings • Where are we in history? • We’ve finished the middle ages, and have embarked on the Early Modern period (about 1500 – 1800 or so). • In fact, we’ve finished the 1500’s as well. • We are now entering the 1600’s. Next up Place Dates Names Mathematics Europe 1600’s Fermat, Descartes, Analytic geometry, theory of equations, Newton, Liebniz, area, normals, tangents, max/min, Barrow, Pascal, systematic calculus Oughtred, Harriot, Wallis Europe 1700’s Bernoulli, Further development of the calculus: Bernoulli, differential equations, brachistochrone Bernoulli, …. problem, tautochrone problem, catenary Maclaurin, problem, calculus of variations, multi‐ L’Hopital, dimensional calculus, transcendental Euler functions (logs, exp, trig), complex numbers, multi‐dimensional calculus, partial differential equations, calculus texts, the foundations of calculus, theory of equations, number theory. Galileo and Cavalieri • For us, the major mathematical interest in his work is his treatment of the infinite, and we’ll discuss that later when we talk about Cantor. • We will also discuss Cavalieri’s work when we talk about the development of calculus prior to Newton and Leibniz. Pierre de Fermat • 1601‐1665 • Trained as a lawyer at the University of Toulouse. • Appointed as a judge in Toulouse in 1638. • Married and had five children. Pierre de Fermat • Considered math his hobby, and never really published any of his works. • Like many others, he was interested in “restoring” lost works of ancient Greek mathematics. • His work is known mainly because of his correspondences with other notable mathematicians, such as Mersenne (the “walking scientific journal of France.”) Pierre de Fermat • Made contributions to – Number theory – Analytic Geometry – Probability – Analysis (calculus) • Often didn’t provide proofs. Didn’t like to “polish” his work. This annoyed some of his contemporaries. He was often correct, but not always.
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