Wentworth's Plane and Solid Geometry · Book VIII, §§ 737–757 (Prop. I) · Book 9: Book VIII: The Sphere
118. The sphere and its circles
Goal: Your student can define a sphere, its radius, great and small circles, axis and poles, and prove that every plane section of a sphere is a circle.
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)
- A ball (an orange or a rubber ball) and a rubber band or string
What's in the book
The opening of Book VIII: definitions (§§ 737–742), Proposition I (§ 743) with its corollaries, great and small circles, axis and poles (§§ 747–756), and the distance between two points on a sphere (§ 757).
The lesson, step by step
Warm up
Ask: What is a circle? Now imagine the same idea in space.
The new idea
A sphere is a solid bounded by a surface all points of which are equally distant from a point within called the centre.
Every section of a sphere made by a plane is a circle.
A great circle of a sphere is a section made by a plane which passes through the centre of the sphere.
§ 757: the distance between two points on the surface of a sphere is the arc of the great circle that joins them.
Draw the figure
Cut an orange (or imagine it): every flat cut gives a circle; the cut through the middle gives the largest.
On a ball, put a rubber band around the middle (a great circle) and another nearer the top (a small circle). Mark the two poles of each.
Walk through the proof
I: drop OC perpendicular to the section, and join O to any two points A and B of its boundary. The right triangles OAC and OBC have OC common and OA = OB, so CA = CB. All points of the boundary are equally distant from C.
§ 745: circles made by planes equally distant from the centre are equal, since AC² = OA² − OC².
Your student proves it back
Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Read §§ 750–756 together and show each with the ball and rubber bands. Ask: Through two points on a globe, how many great circles? (One, unless the points are opposite ends of a diameter.)
Wrap up
Ask: On a globe, which lines of latitude are great circles? (Only the equator.) Which meridians? (All of them.)
Tip
A globe or ball you can draw on (with washable marker) makes Book VIII far easier. Keep it handy for every lesson.