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