Top Archimedes of Syracuse (287-212 BC) Greek domain Archimedes is universally acknowledged to be the greatest of ancient mathematicians. He studied at Euclids school (probably after Euclids death), but his work far surpassed, and even leapfrogged, the works of Euclid. (For example, some of Euclids more difficult theorems are easy analytic consequences of Archimedes Lemma of Centroids.) His achievements are particularly impressive given the lack of good mathematical notation in his day. His proofs are noted not only for brilliance but for unequaled clarity, with a modern biographer (Heath) describing Archimedes treatises as without exception monuments of mathematical exposition ... so impressive in their perfection as to create a feeling akin to awe in the mind of the reader. Archimedes made advances in number theory, algebra, and analysis, but is most renowned for his many theorems of plane and solid geometry. He was first to prove Herons formula for the area of a triangle. His excellent approximation to √3 indicates that hed partially anticipated the method of continued fractions. He found a method to trisect an arbitrary angle (using a markable straightedge — the construction is impossible using strictly Platonic rules). One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series. Some of Archimedes work survives only because Thabit ibn Qurra translated the otherwise-lost Book of Lemmas; it contains the angle-trisection method and several ingenious theorems about inscribed circles. (Thabit shows how to construct a regular heptagon; it may not be clear whether this came from Archimedes, or was fashioned by Thabit by studying Archimedes angle-trisection method.) Other discoveries known only second-hand include the Archimedean semiregular solids reported by Pappus, and the Broken-Chord Theorem reported by Alberuni. Archimedes and Newton might be the two best geometers ever, but although each produced ingenious geometric proofs, often they used non-rigorous calculus to discover results, and then devised rigorous geometric proofs for publication. He used integral calculus to determine the centers of mass of hemisphere and cylindrical wedge, and the volume of two cylinders intersection. Although Archimedes didnt develop differentiation (integrations inverse), Michel Chasles credits him (along with Kepler, Cavalieri, and Fermat, who all lived more than 18 centuries later) as one of the four who developed calculus before Newton and Leibniz. He was one of the greatest mechanists ever, discovering the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, a miniature planetarium, and war machines (e.g.. catapult and ship-burning mirrors). His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, Sphere and Cylinder, and (discovered only recently, and often called his most important work) The Method. He developed the Stomachion puzzle (and solved a difficult enumeration problem involving it). His Equiarea Map Theorem asserts that a sphere and its enclosing cylinder have equal surface area (as do the figures truncations). Archimedes also proved that the volume of that sphere is two-thirds the volume of the cylinder. He requested that a representation of such a sphere and cylinder be inscribed on his tomb. Archimedes discovered formulae for the volume and surface area of a sphere, and may even have been first to notice and prove the simple relationship between a circles circumference and area. For these reasons, π is often called Archimedes constant. His approximation 223/71 < π < 22/7 was the best of his day. (Apollonius soon surpassed it, but by using Archimedes method.) That Archimedes shared the attitude of later mathematicians like Hardy and Brouwer is suggested by Plutarchs comment that Archimedes regarded applied mathematics as ignoble and sordid ... and did not deign to [write about his mechanical inventions; instead] he placed his whole ambition in those speculations the beauty and subtlety of which are untainted by any admixture of the common needs of life. Some of Archimedes greatest writings (including The Method) are preserved only on a palimpsest which has been rediscovered and properly studied only since 1998. Ideas unique to that work are an anticipation of Riemann integration, calculating the volume of a cylindrical wedge (previously first attributed to Kepler), and perhaps an implication that Archimedes understood the distinction between countable and uncountable infinities (a distinction which wasnt resolved until Georg Cantor, who lived 2300 years after the time of Archimedes). Although Euler (along with Newton or Leibniz) may have been the most important mathematicians, and Gauss the greatest theorem prover, it is widely accepted that Archimedes was the greatest genius who ever lived. Yet, Hart omits him altogether from his list of Most Influential Persons: Archimedes was simply too far ahead of his time to have great historical significance. (Some think the Scientific Revolution would have begun sooner had The Method been discovered four or five centuries earlier.)
Posted on: Thu, 01 Jan 2015 21:48:01 +0000
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