- What's a parabola? A conic section?
- Archimedes' Method; Archimedes' proof by exhaustion
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