Wentworth's Plane and Solid Geometry · Book II, §§ 266–285 · Book 3: Book II: The Circle
27. Measurement, ratio and the theory of limits
Goal: Your student can explain commensurable and incommensurable quantities, what a variable and its limit are, and the test for a limit.
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)
What's in the book
Pages 92–98: measurement and ratio, commensurable and incommensurable quantities, then the Theory of Limits with its theorems (§§ 276–285). The author gathers here every principle of limits needed in the whole book.
The lesson, step by step
Warm up
Ask: What is a ratio? What is 3 ft to 6 in as a ratio?
Measuring and ratio
Read §§ 266–270. Two lengths are commensurable if some unit fits a whole number of times in each. The diagonal and side of a square are not: their ratio is √2, which no fraction equals exactly.
Variables and limits
Read §§ 271–272 and act out the book's example: walk halfway across the room, then half the remaining distance, then half again. You get as close as you like but never arrive.
Two more examples
Read § 273 (polygons with more and more sides inside a circle) and § 274 (the decimal 0.333 … approaching ⅓). Draw a square in a circle, then an octagon, to see the perimeter creep toward the circumference.
The test
Read § 275:
1. Can be made less than any assigned value, however small.
2. Cannot be made absolutely equal to zero.
The theorems in §§ 276–285 follow. Read § 284 carefully: if two variables are always equal and each has a limit, the limits are equal. It is used in many later proofs.
Wrap up
Your student explains, in their own words, what it means for a variable to approach a limit.
Tip
The author advises teachers to illustrate limits both with figures and with numbers. If the algebra of §§ 276–283 is heavy going, read it lightly; the key results are §§ 275, 284 and 285.