Wentworth's Plane and Solid Geometry · Book VIII, Props. II–IV · Book 9: Book VIII: The Sphere
119. Poles and polar distance; measuring a material sphere
Goal: Your student can prove that all points of a circle are equally distant from its poles, find a pole by arcs, and explain how to find the diameter of a real ball.
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
Propositions II–IV (§§ 758–764), with polar distance (§§ 759–761) and the scholium on drawing great circles with compasses (§ 763).
The lesson, step by step
Warm up
Ask: What are the poles of a circle of a sphere?
The new idea
The distances of all points in the circumference of a circle of a sphere from its poles are equal.
A point on the surface of a sphere, which is at the distance of a quadrant from each of two other points, not the extremities of a diameter, is a pole of the great circle passing through these points.
Given a material sphere to find its diameter.
Draw the figure
On the ball, set compasses (or a string held at one point) to a fixed opening and swing it around a point P: you draw a circle whose pole is P.
Walk through the proof
II: the straight lines from the pole to points of the circle are equal (§ 514), and equal chords cut off equal arcs.
III: the angles POA and POB are right angles, so PO is perpendicular to the plane of the circle (§ 507).
IV: draw a circle on the ball around P; measure three chords and draw the triangle flat; its circumscribed circle has the same radius. Then build the right triangle on paper to find the diameter.
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.
For Prop. IV, your student explains each step of the construction while doing it.
Exercises
If you have compasses and a ball, try Prop. IV and then check your diameter by measuring the ball between two books.
Wrap up
Ask: What is the polar distance of a great circle? (A quadrant, 90°.)