http://www.math.tamu.edu/~don.allen/m629_00c/index.htm

Follow the thread of this course outline, it's excellent. Read Chapter 6 on medieval mathematics. Look at the problems being solved. Algebraic, yes. But also problems of mechanics were of interest. Here mathematicians exceeded the great one, Archimedes. Note how mathematicians "dabble" with infinity. There was still a philosophical chasm to cross, and this would take several more centuries. It seems that mankind need time to become habituated to an idea before it can be embraced and then absorbed. Read Chapter 7 on the mathematics of the Renaissance. Note the fundamental and practical need for mathematics had a considerable effect on developments. This meant practical algebra, symbolism, and more. Algebra in this time exceed by far that of the ancients. The general solution of the cubic was determined in the mid-16th century. Surprisingly, this led to the notion of complex number. Complex numbers, then, arose not a some cerebric invention of theoretical mathematics but as objects needed to "clean up" the cubic formula. Interesting?? As well, the Ptolemaic world was collapsing. This meant better trigonometry and models of the heavens. The three great astronomers of the renaissance, Copernicus, Brahe, and Kepler, lived during this era. At this time the young genius, Galileo was following their work. Read Chapter 8 about the Transition period to calculus. By the end of the 16th century the European algebraists had achieved about as much as possible following the Islamic Tradition. Translating ancient texts took on a priority The desire here, beside the link with the past, was to determine the Greek methods. Among the great mathematicians of this period, we find Vieta, Stevin, Napier, Descartes, Galileo, and Fermat. Vieta produced symbolic algebra; Stevin used limiting processes; Napier invented logarithms; one of the greatest mathematical technologies, Galileo developed an analytical dynamics; Decartes created the philosophy which was to fuel the next century's mathematics; Fermat produced analytic geometry, calculus, and number theory which was to inspire the work of many. In this transition period we see (a) the beginnings of taking limiting process, (b) The emergence of symbolism and its consequences, (g) sophisticated mathematical arguments, and (d) the flourishing of number theory. Hear about this period. (To listen to this you will need the Windows Media Player.) Read Chapter 9, Part 1. This begins a long chapter on the emergence of calculus. In this particular chapter we see the emergence of calculus and probability and the roles played by Blaise Pascal. Note how inventive he was, not just in mathematics, but in physics and technology of his day. You will also see the role played by the Dutch school, particularly Christian Huygens. Following we study the contributions of Pierre Fermat. The calculus he invents is very nearly what we learn and teach. He clearly deals with an infinitesimal, allowing it to cancel out at the correct moment. What is also significant here is the broad band of contributions of others toward calculus. These contributions were not enough to give them standing prominance in the history of the subject, but they impacted the likes of Newton and Leibnitz, who you must understand did not work in a vacuum. Hear an audio introduction to this period. (To listen to this you will need the FREE Windows Media Player. Note this recording uses a different codec than the previous one. So, if you don't hear it let me know and I will rectify the problem.)

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