Wentworth's Plane and Solid Geometry · Book VIII, Props. XXIII–XXV · Book 9: Book VIII: The Sphere
125. Unequal sides and angles; the shortest path on a sphere
Goal: Your student can prove the converse of the isosceles theorem for spherical triangles, compare unequal sides and angles, and prove that the shortest line on a sphere is a great-circle arc.
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 XXIII–XXV (§§ 814–816) and Exercises 746–758.
The lesson, step by step
Warm up
Ask: State Prop. XXII.
The new idea
If two angles of a spherical triangle are equal, the sides opposite these angles are equal and the triangle is isosceles.
The shortest line that can be drawn on the surface of a sphere between two points is the arc of a great circle, joining the two points not greater than a semicircumference.
Draw the figure
Stretch a string tight between two cities on a globe (say, New York and Paris). It lies along a great circle, which is why long flights curve north on a flat map.
Walk through the proof
XXIII: use the polar triangle and Prop. XXII.
XXIV: draw BD making angle CBD equal to angle C; then DC = DB, and AD + DB > AB.
XXV: around A and B draw small circles through a point C of the arc. They touch only at C. Any path from A to B that misses C can be shortened by swinging its two parts to pass through C. So the shortest path passes through every point of the arc.
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
Do two constructions on the ball:
Ex. 746. To bisect a spherical angle.
Ex. 749. To construct a spherical triangle, having given the three sides.
Wrap up
Ask: Why do airplanes from the United States to Europe fly over Canada and Greenland? (The great-circle route is shortest.)
Answers to the book's problems
Ex. 746 and 749: copy the plane constructions, using arcs of great circles drawn with compasses as in § 763. Ex. 757: the planes of such arcs contain the axis of the great circle, which is perpendicular to its plane. Ex. 758: a section through the point is smallest when farthest from the centre, which is when its plane is perpendicular to the radius through the point (§ 746).
Worked out for this site from the scan. If a number in your copy differs, trust the book.