April 11

Plan:

  • Questions
  • Archimedes
  • What's a parabola? A conic section?
  • Quadrature
  • Archimedes' Method; Archimedes' proof by exhaustion

Student Questions:

Discussion:
The discussion page can be found here, for observations and remarks.


** Archimedes Quadrature of the Parabola**

Archimedes (287 - 212 BCE)

  • Greek mathematician, lived in Syracuse (Italy)
  • Some of what he did (displacement theory) is real while others may be parables.

Parabola - Conic Section

  • Conic section - Section of a cone cut by a plane -> can form parabolas, hyperbolas, point, etc.
  • Greeks new them all -> studied them -> tricky to derive properties as plane figures without any algebra and a well-defined three-dimensional space.

Ellipse - special connection between two points in the ellipse

  • Define and prove ellipse -> "Ice Cream Cone Proof"
    • Start with a cone and cut it using a plane to form the ellipse
    • Now, drop a sphere in the top of the cone such that it fits into the cone like ice cream in a cone
    • Gradually shrink the sphere until it touches a point in the ellipse
    • Now, start with a sphere at the bottom of the cone, and increase it in size until it touches the second point in the ellipse
    • These will be the foci in the ellipse
      • We can prove the important properties by exploiting the fact that two lines tangent to a sphere are equal. This will lead to the conclusion that the sum of the distances between two points is constant.

Euclid -> Involved in conic sections too

  • Wrote a number of books on conic sections

Question for the class

  • Analog of the "Ice Cream Cone Proof" for parabola
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