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Archimedes

Archimedes. The Archimedes Portrait XVII century Domenico Fetti. Ancient Greece. At around 600 BC, the nature of Mathematics began to change in Greece…. - Polis, city-states;. - Exposure to other cultures;. The Greek were eager to learn.

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Archimedes

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  1. Archimedes The Archimedes Portrait XVII century Domenico Fetti

  2. Ancient Greece At around 600 BC, the nature of Mathematics began to change in Greece… - Polis, city-states; - Exposure to other cultures; The Greek were eager to learn. - Mathematics valued as the basis for the study of the physical world; - Idea of mathematical proof;

  3. Life of Archimedes (287-212 BC) • son of an Phidias, an astronomer • related to King Hierro II of Syracuse? Dont disturb my circles! • spent time in Alexandria? • Archimedes Screw • Second Punic War; Plutarch’s biography of General Marcellus

  4. a b A B Lever • equal weights at equal distances from the fulcrum of a lever balance • Archimedes created a mathematical model of the lever; he was the first mathematician to derive quantitative results from the creation of mathematical models of physical problems. • lever is rigid, but weightless • fulcrum and weights are points Law of the lever: Magnitudes are in equilibrium at distances reciprocally proportional to their weights: A x a = B x b

  5. Golden Crown Law of buoyancy (Archimedes’ Principle): The buoyant force is equal to the weight of the displaced fluid.

  6. Estimation of π Proposition 3: The ratio of the circumference of any circle to its diameter is less than but greater than . Circle of diameter 1 – circumference π Inscribed polygon of perimeter p Circumscribed polygon of perimeter P p ≤π≤P

  7. Estimation of π(adapted) sn = sin( ) Circle of diameter 1 - circumference π Inscribed 2n-gon - 2n sides each of length sn - perimeter pn=2n sn Circumscribed 2n-gon - 2n sides each of length Tn - perimeter Pn=2n Tn Tn = tan( ) pn≤π≤Pn

  8. Estimation of π(adapted) pn≤π≤Pn

  9. 3 : 2 : 1 Archimedes’ tombstone Known to Democritus:

  10. E C 2r B A r 2r D F

  11. P H E C 2r R y T B A X r 2r D x F G Q

  12. P H E R y C 2r B A R X x T 2r B A X r 2r D x F G Q

  13. P H E C 2r R Sphere - circle C2 of radius y T A B X r Cone - circle C1 of radius x 2r Cylinder - circle C of radius 2r 2r D x F G Q

  14. P H E C R T A B X r 2r D x F G Q A Sphere - circle C2 of radius y x Cone - circle C1 of radius x C1 Cylinder - circle C of radius 2r C C2

  15. Sphere - circle C2 of radius y Cone - circle C1 of radius x Cylinder - circle C of radius 2r

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