Wentworth's Plane and Solid Geometry · Book VIII, §§ 778–790 (Props. IX–XII) · Book 9: Book VIII: The Sphere
121. Spherical angles and spherical polygons
Goal: Your student can define a spherical angle and spherical polygon, measure a spherical angle, draw a perpendicular arc, and prove the first theorems on spherical triangles.
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
Spherical angles (§ 778), Propositions IX–XII (§§ 779–790), the definitions of spherical polygons and triangles (§§ 782–788), and Exercise 736.
The lesson, step by step
Warm up
Ask: What is a trihedral angle? State Prop. XXVI of Book VI.
The new idea
A spherical angle is measured by the arc of the great circle described from its vertex as a pole and included between its sides (produced if necessary).
Each side of a spherical triangle is less than the sum of the other two sides.
The sum of the sides of a spherical polygon is less than 360°.
§ 784: from any property of polyhedral angles we may infer an analogous property of spherical polygons; and conversely.
Draw the figure
On the ball, draw a spherical triangle with three great-circle arcs. Draw lines from its corners to the centre (imagine them): they make a trihedral angle.
Walk through the proof
IX: the tangents to the two arcs at A are perpendicular to AO, and so are OB and OC; so the angle between the tangents equals angle BOC, measured by arc BC.
X: draw the arc from E (on the given arc, a quadrant from its pole) through A.
XI and XII: the sides are the face angles of the trihedral or polyhedral angle at the centre, so use Props. XXVI and XXVII of Book VI.
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
Ex. 736. Every point in a great circle which bisects a given arc of a great circle at right angles is equidistant from the extremities of the given arc.
Wrap up
Ask: Why are the sides of a spherical triangle measured in degrees, not inches? (They are arcs; their degrees are the face angles at the centre.)