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Euclid

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For any Johnnies as hopelessly, nerdily in love with Euclid as I am - enjoy!

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Euclid

  1. 1. Euclid’sElements
  2. 2. Euclid’sElements
  3. 3. Euclid’sElements
  4. 4. Thousands of years after its author died, here we are still marveling at this text…Take a moment to appreciate the nuances of this poem of a mathematics book.
  5. 5. By hovering your mouse over a prop, you can see both
  6. 6. By hovering your mouse over a prop, you can see both- the props that went into its proof
  7. 7. By hovering your mouse over a prop, you can see both-the props that went into its proof and - the later props that rely on it.
  8. 8. Happy geometring!9/27/2011 12
  9. 9. 9/27/2011 13
  10. 10. On a given finite straight line - - to construct an equilateral triangle.9/27/2011 14
  11. 11. At a given point, with a given straight line - - to place (as an extremity) a straight line equal to the given straight line.9/27/2011 15
  12. 12. Given two unequal straight lines - - to cut off from the greater a straight line equal to the less.9/27/2011 16
  13. 13. If two triangles each have two of their respective sides and the contained angles equal to each other - - the triangle will be equal to the triangle; - the remaining angles will be equal to the remaining angles, respectively.9/27/2011 17
  14. 14. In isosceles triangles - - the angles at the base will be equal to each other; - as will be the angles under the base.9/27/2011 18
  15. 15. If in a triangle two angles be equal to one another - - the sides which subtend the equal angles will also be equal to one another.9/27/2011 19
  16. 16. If two straight lines (constructed at the extremities of a straight line) meet at a point - - there cannot be constructed, on the same side of the line, two lines equal to the other straight lines which meet at a different point.9/27/2011 20
  17. 17. If two triangles have the two sides and the base equal, respectively - - the angles contained by those straight lines will also be equal.9/27/2011 21
  18. 18. Given a rectilineal angle - - to bisect it.9/27/2011 22
  19. 19. Given a finite straight line - - to bisect it.9/27/2011 23
  20. 20. To a given straight line, and from a given point on it - - to draw a straight line at right angles.9/27/2011 24
  21. 21. To a given straight line, from a given point which is not on it - - to draw a perpendicular straight line.9/27/2011 25
  22. 22. If a straight line set up on a straight line makes angles - - it will make either two right angles, or angles equal to two right angles.9/27/2011 26
  23. 23. If two straight lines, meeting another straight line at the same point, make the adjacent angles equal to two right angles - - the two straight lines will be in a straight line with each other.9/27/2011 27
  24. 24. If two straight lines cut one another - -they make the vertical angles equal to one another.9/27/2011 28
  25. 25. If one of the sides of any triangle be produced - - the resulting exterior angle is greater than either of the interior, opposite angles.9/27/2011 29
  26. 26. Two angles of any triangle, when taken together - - are less than two right angles.9/27/2011 30
  27. 27. The greater side in any triangle - - subtends the greater angle.9/27/2011 31
  28. 28. The greater angle in any triangle - - is subtended by the greater side.9/27/2011 32
  29. 29. Two sides of any triangle, when taken together in any manner, - are greater than the remaining one.9/27/2011 33
  30. 30. If, from the extremities of one side of a triangle, two straight lines meeting within the triangle be constructed - - the straight lines will be less than the triangle’s remaining two sides; - but they will contain a greater angle.9/27/2011 34
  31. 31. Given three straight lines (as long as two taken together are greater than the remaining long) - - to construct a triangle using three straight lines equal to those given.9/27/2011 35
  32. 32. Given a rectilineal angle, as well as a given point on a straight line - - to construct at that point another, equal rectilineal angle.9/27/2011 36
  33. 33. If two triangles have two of their sides respectively equal, but one of the contained angles is larger than the other - - the triangle with the larger angle will also have a larger base.9/27/2011 37
  34. 34. If two triangles have two of their sides respectively equal, but one of the bases is larger than the other - - the triangle with the larger base will also have a larger angle.9/27/2011 38
  35. 35. If two triangles have two of their angles respectively equal, as well as any one of their respective sides equal - -the remaining respective sides will be equal; -as will be the remaining angle.9/27/2011 39
  36. 36. If a straight line falling on two straight lines make the alternate angles equal to one another - - the straight lines will be parallel to one another.9/27/2011 40
  37. 37. If a straight line falling on two straight lines make (a)the exterior angle equal to the interior, opposite angle on the same side, or (b)the interior angles on the same side equal to two right angles - - the straight lines will be parallel to one another.9/27/2011 41
  38. 38. If a straight line falls on parallel straight lines - - alternate angles are equal; - the exterior angle is equal to the interior, opposite angle; - and interior angles on the same side are equal to two right angles.9/27/2011 42
  39. 39. Straight lines parallel to the same straight line - - are also parallel to one another.9/27/2011 43
  40. 40. Given a straight line and a point (not on the line) - - to draw through the point a line parallel to the one given.9/27/2011 44
  41. 41. If one of the sides of any triangle be produced - -the resulting exterior angle is equal to the two interior, opposite angles; - and the triangle’s three interior angles are equal to two right angles.9/27/2011 45
  42. 42. Straight lines that join equal and parallel straight lines, in the same respective directions - -are themselves equal and parallel as well.9/27/2011 46
  43. 43. In parallelogrammic areas - -opposite sides are equal to one another; - opposite angles are equal to one another; -and the diameter bisects the areas.9/27/2011 47
  44. 44. Parallelograms which share a base and are in the same parallels - - are equal to each other.9/27/2011 48
  45. 45. Parallelograms which are on equal (but not shared bases) and in the same parallels - - are equal to one another.9/27/2011 49
  46. 46. Triangles which are share a base and are in the same parallels - - are equal to one another.9/27/2011 50
  47. 47. Triangles which are on equal (not shared) bases and in the same parallels - - are equal to one another.9/27/2011 51
  48. 48. If equal triangles share a base and are on the same side - - they are also in the same parallels.9/27/2011 52
  49. 49. If equal triangles be on equal (not shared bases) and on the same side - - they are also in the same parallels.9/27/2011 53
  50. 50. If a parallelogram share a base with a triangle and be in the same parallels - - the parallelogram is double of the triangle9/27/2011 54
  51. 51. From a given triangle, in a given rectilineal angle - - to construct a parallelogram equal to the triangle.9/27/2011 55
  52. 52. In any parallelogram - - the compliments about the diameter are equal to one another.9/27/2011 56
  53. 53. From a given triangle, within a given rectilineal angle, and to a given straight line - - to apply a parallelogram equal to the triangle.9/27/2011 57
  54. 54. Given any rectilineal figure and a rectilineal angle - - to construct in the angle a parallelogram equal to the figure.9/27/2011 58
  55. 55. On a given straight line - - to describe a square.9/27/2011 59
  56. 56. In right-angled triangles - - the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.9/27/2011 60
  57. 57. If the square on one of a triangle’s sides be equal to the squares on its remaining two sides - - the angle contained by the remaining two sides of the triangle is right.9/27/2011 61

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