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6 – Greek Math After Euclid

6 – Greek Math After Euclid. The student will learn about. Greek mathematics after the time of Euclid. §6-1 Historical Setting. Student Discussion. §6-2 Archimedes. Student Discussion. §6-2 Archimedes 1. 1. Classical method of determining . 2. Quadrature of a parabola – more follows.

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6 – Greek Math After Euclid

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  1. 6 – Greek Math After Euclid The student will learn about Greek mathematics after the time of Euclid.

  2. §6-1 Historical Setting Student Discussion.

  3. §6-2 Archimedes Student Discussion.

  4. §6-2 Archimedes 1 1. Classical method of determining . 2. Quadrature of a parabola – more follows. 3. Spiral of Archimedes, r = k . 4. Spheres and cylinders – more follows. 5. Conchoids and Spheroids 6. Sand reckoning. 7. Plane Equilibrium – centroids. 8. Floating bodies – hydrostatics.

  5. §6-2 Archimedes 2 Quadrature of a parabola – area of a parabolic segment is four-thirds that of the inscribed triangle having the same base. 1. Use calculus to calculate the area bounded by the parabola y = x2 and y = 4 for –2  x  2. 2. Use Archimedes’ method to calculate the area bounded by the parabola y = x2 and y = 4 for –2  x  2.

  6. §6-2 Archimedes 3 Spheres and cylinders 1. Confirm that the surface area of a sphere is equal to 2/3 the surface area of a circumscribed cylinder. 4  r 2 = 2/3 (6  r 2) 2. Confirm that the volume of a sphere is equal to 2/3 the volume of a circumscribed cylinder. 4/3  r 3 = 2/3 (2  r 2)

  7. §6-3 Eratosthenes Student Discussion.

  8. §6-4 Apollonius Student Discussion.

  9. §6-4 Apollonius 1 Problem of Apollonius – Given three circles (degenerate cases permitted) construct a circle tangent to the given circles. Given three points - easy Given three lines - easy Other cases vary in difficulty. If time do two points and a line.

  10. §6-4 Apollonius 2 From “Plane Loci”. If A and B are fixed points and k is a given constant, then the locus of a point P, such that AP/BP = k is either a circle (if k  1) or a straight line (if k = 1.). 1. Case where k = 1. The locus is the perpendicular bisector of AB. 2. Case where k = 2. Homework: Describe the locus circle completely.

  11. §6-5 Hipparchus, Menelaus, Ptolemy, and Trigonometry. Student Discussion.

  12. §6-5 Hipparchus 60 x 18  18  The chord of 36 = 37; 04, 55 Chord 36 = 2 · 60 · sin 18 sin 18 = chord 36 / 120 sin 18 = 37; 04, 55 / 120 sin 18 = 18; 32, 27, 30 / 60 sin 18 = 00 ; 18, 32, 27, 30 sin 18 = 0.309016204ten Too small by 0.0000008 or the thickness of a human hair over the length of a soccer field.

  13. §6-5 Menelaus. C M N B L A Menelaus’ Theorem. If transversal LMN intersects the three sides of a triangle then:

  14. §6-5 Ptolemy. Three Point Problem. Given points A, B, and C, and angles AVB, AVC, and BVC, find point V. . A . B . V ? . C

  15. §6-6 Heron Student Discussion.

  16. §6-6 Heron 1. Area of a triangle of sides a, b, and c, is: 2. Square root approximation – If a1 is an approximation of the square root of n then is a better approximation. Try 45. Note: the Babylonians used this for 2.

  17. §6-7 Ancient Greek Algebra Student Discussion.

  18. §6–8 Diophantus Student Discussion.

  19. §6–9 Pappus Student Discussion.

  20. §6–10 The Commentators Student Discussion.

  21. Time Line 2400-1600-525 B.C. Babylonians 1900-1000-000 B.C. Egyptions 600 B.C. Thales 540 B.C. Pythagoras 450 B.C. Zeno 440 B.C. 2 Irrational 390 B.C. Socrates / Plato 336-323 B.C. Alexander the Great’s Reign

  22. Time Line 300 B.C. Euclid 287-212 B.C. Archimedes 230 B.C. Eratosthenes 225 B.C. Apollonius 44 B.C. Death of Julius Caesar

  23. Time Line 150 Ptolemy 250 Diophantus 300 Pappus 390 Theon of Alexandria 410 Hypatia 529 School of Athens closed

  24. Assignment Read Chapter 7.

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