Wentworth's Plane and Solid Geometry · Book IX, §§ 896–910 (Prop. XI) · Book 10: Book IX: Conic Sections
138. The ellipse
Goal: Your student can define an ellipse, draw one with a string and by points, and name its foci, centre, axes, eccentricity and latus rectum.
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
- Two pins, a piece of cardboard and a loop of string
What's in the book
The definition of the ellipse (§§ 896–901), Proposition XI (§ 902), and the corollaries and definitions (§§ 903–910).
The lesson, step by step
Warm up
Ask: Define a parabola.
The new idea
An ellipse is a curve which is the locus of a point that moves in a plane so that the sum of its distances from two fixed points in the plane is constant.
§ 907: the values of a, b, c are so related that a² = b² + c².
Draw the figure
Push two pins into cardboard about 8 cm apart, loop a string of 12 cm around them, and draw with a pencil keeping the string tight (§ 901). Then measure: the long axis should be 12 cm.
Walk through the proof
XI: A and A' are points since AA' = 2a. For any X between the foci, arcs of radii AX and A'X from F and F' meet at points whose focal radii add to AX + A'X = 2a.
§ 907: the end B of the minor axis is a from each focus, so in the right triangle BOF, a² = b² + c².
Your student proves it back
Close the book. Your student draws the curve and the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason. Point to each line as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Change the distance between the pins and draw again. What happens as the pins get closer? (The ellipse becomes rounder; eccentricity c/a gets smaller.)
Wrap up
Ask: What shape do the planets travel in around the sun? (Ellipses, with the sun at one focus.)