great essay by alan turing

of Descartes' analytic geometry, Newton's achievements with synthetic geometry were surpassing. (As a young man he made the absurd claim that 1234. Minkowski had developed a flat 4-dimensional space-time to cope with Einstein's Special Theory; but it was Einstein who had the vision to add curvature to that space to describe acceleration.

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Riemann's teacher was Carl Gauss, who helped steer the young genius towards pure mathematics. Henry Smith was an outstanding intellect with a modest and charming personality. Nature seems delighted with Transmutations." Ocean tides had intrigued several of Newton's predecessors; once gravitation was known, the Moon's gravitational attraction provided the explanation - except that there are two high tides per day, one when the Moon is farthest away. We have been concentrating on two of these, "A chip in the sugar" and "A lady of letters". Chebyshev was also a premier applied mathematician and a renowned inventor; his several inventions include the Chebyshev linkage, a mechanical device to convert rotational motion to straight-line motion. Much of his methodology, including unusual ideas about divergent series, was his own invention. Although he achieved fame at an early age with the Banach-Tarski Paradox, his greatest achievements were in formal logic. He also contributed much to spherical trigonometry. His long and productive career earned the Abel Prize for his "vast and lasting impact on the theory of numbers and his incisive contributions and illuminating insights. (Although he continued to refine this invention, it was never a commercial success.) He suffered poor health throughout his life, abandoned mathematics for religion at about age 23, wrote the philosophical treatise Penses We arrive at truth, not by reason only, but also by the. His notion of imaginary solutions in geometry was inspirational.