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Mathematics and Astronomy in Ancient Egypt and Greece

Mathematics and Astronomy in Ancient Egypt and Greece. Steven Edwards Southern Polytechnic State University. Writing in Egypt 3200 BC. Rhind Papyrus 1650 BC. 87 problems involving fractions, area, volume. Except for 2/3, Egyptians used 1/n for fractions.

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Mathematics and Astronomy in Ancient Egypt and Greece

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  1. Mathematics and Astronomy in Ancient Egypt and Greece Steven EdwardsSouthern Polytechnic State University

  2. Writing in Egypt 3200 BC

  3. Rhind Papyrus 1650 BC 87 problems involving fractions, area, volume

  4. Except for 2/3, Egyptians used 1/n for fractions. The Rhind papyrus has a table of calculations of 2/5, 2/7, 2/9, …. 2/99. How to divide n loaves between 10 men where n =1, 2, 6, 7, 8, or 9. For 7 loaves, each man receives 2/3 1/30. Find the value of heap if heap and seventh of heap is 19.

  5. Rhind Papyrus Algebra

  6. Area of a triangle with side 10 and base 4 is 20.

  7. Thales of Miletus (640 BC) The Ionic School A circle is bisected by its diameter. Base angles of an isosceles triangle are equal. Opposite angles of intersecting lines are equal. The angle in a semicircle is a right angle. SAS congruence for triangles.

  8. Mathematics in Greece was spurred by efforts to solve three famous problems, using Euclidean geometry or other methods. 1. Square the circle, i.e. construct a square whose area is equal to a given circle (or vice versa). 2. Duplicate the cube, i.e. construct a cube whose volume is double a given cube. 3. Trisect a given angle.

  9. Pythagoras 580 BC Pythagoras traveled to Egypt. Pythagoras discovered the construction of the cosmic figures, according to Proclus.

  10. Ancient Egyptian Patterns

  11. Euclid of Alexandria’s figure for the proof of thePythagorean Theorem (300 BC)

  12. Area of a circle in the Rhind Papyrus

  13. Euclid X, Proposition 1, The “Method of Exhaustion”: Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. (Due to Eudoxus) Euclid XII, Proposition 2: Circles are to one another as the squares on the diameters.

  14. Archimedes 287 BC Compare the area of a circle to the area of a right triangle with one leg equal to the radius, the other leg equal to the circumference of the circle. If they are not equal, then one is larger. Suppose that the circle has larger area.

  15. Each triangle has area less than 1/2 the radius times the base. The inscribed figure has area less than 1/2 the radius times the circumference, which in turn is less than the previously mentioned triangle. But with enough sides, the inscribed figure has area arbitrarily close to the circle, so greater than the triangle.

  16. Conclusion: the area of a circle is 1/2 r C. This is possibly the first proof that pi shows up in both as the ratio of circumference to diameter and in the formula for the area.

  17. The Law of Sines from Ptolemy’s Almagest (100 AD) Hipparchus (135 BC) provided much of the source material for Ptolemy

  18. From 3000 BC, Egyptians had a 365 day year. After 1460 years, this year resynchronized with the seasons In 239 BC, Ptolemy III tried to introduce the leap year. Augustus imposed it 200 years later.

  19. Relative to the sun, any given star rises later and later each day. Sirius from the Hubble

  20. From Luxor,now in Paris Luxor Rome Central Park Obelisks

  21. Temple of Ramses II

  22. The Inner Sanctum

  23. Illuminated by the sun twice a year

  24. DecanChart2100 BC

  25. How is time measured? Before 1967, one second was defined as 1/31,556,925.9747 of the tropical year 1900. Now: 9,192, 631, 770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom

  26. Thales of Miletus 624 BCBrought the 365 day year from Egypt Anaximander of Miletus 610 BCThe earth is the center of the universe Pythagoras of Samos 572 BCThe earth is a sphere.

  27. Eudoxus 408 BC Motions of each heavenly body moved by several spheres. One sphere for the fixed stars, three for the sun, three for the moon, and four for each planet, for a total of 27 spheres. Heraclides of Pontus 388 BCThe Earth spins on his axisVenus and Mercury orbit the sun

  28. Aristarchus of Samos 310-230 BC On the Sizes and Distances of the Sun and Moon

  29. When the moon is half full, the great circle dividing light from dark is in the same plane as our eyes

  30. The distance from Earth to Sun is 19 timesthe distance from Earth to Moon.

  31. During a solar eclipse, the moon blots out the sun. Earth Earth

  32. The ratio of the distance to the moon to the diameter of the moon is the same as the ratio of the distance to the sun to the diameter of the sun. M/m = S/s S/M = s/m If the sun is 19 times as far away from us as the moon,then the sun’s diameter is 19 times the moon’s.

  33. Positions of Sun, Moon, and Earth at total solar eclipse S/s = M/m Diameter Distance to Earth Sun 1,392,000 km 148,000,000 km Moon 3500 km 400,000 km

  34. Earth Moon Using the angle from our eye to the moon as 2 degrees, Aristarchus gets the distance from us to the moon is between 25 and 33 times the moon’s diameter.

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