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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