Early Middle Ages

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Pierre de Fermat 1601-1665: lawyer at the Parliament of Toulouse, France, and a mathematician: Analytic Geometry, Calculus, Number theory, Probability, Optics. Fermat's manuscript in analytic geometry ("Introduction to Plane and Solid Loci") published in 1679 was circulated already in 1636, one year before the publication of Descartes' La géométrie. Developed Calculus: maxima and minima of functions, tangents to curves, area, center of mass, least action, etc. He evaluated the integral of power functions using an ingenious trick, reducing it to the sum of geometric series (used later by Newton, and then Leibniz for the fundamental theorem of calculus).

In number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. While researching perfect numbers he discovered the little theorem. He invented a factorization method—Fermat's factorization method—as well as the proof technique of infinite descent, which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4. Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, etc.

In 1654, Fermat and Pascal laied foundations of probability theory. Fermat made the first ever rigorous probability calculation to answer a question he was asked by a professional gambler: why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in his losing.

Fermat's principle of least time or Fermat's principle (enunciated by Hero of Alexandria in the 1st century CE) leaded to the principle of least action in physics.

Blaise Pascal 1623-1662 mathematician, physicist, inventor, writer, philosopher. Study of fluids: the concepts of pressure and vacuum clarified after the work of Torricelli (many disputes before acceptance). Treatise on projective geometry at the age of 16. Probability theory: joint with Fermat (in letters). Economic and social sciences. At age 19 constructed a mechanical calculator capable of addition and subtraction, called Pascal's calculator or the Pascaline.

1654: works on philosophy and theology, Lettres provinciales and the Pensées, Math: on the arithmetical triangle, and in 1658-1659 on the cycloid and its use in calculating the volume of solids.Pascal had poor health and died right after his 39th birthday.

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