Gears, Ladders & Eclipses

Continued fractions in machines and skies

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Gear trains need whole-number tooth ratios: convergents give the best ones (Huygens’s 206/7).
• A ladder of resistors computes a continued fraction; the infinite ladder gives φ ohms.
• Eclipses repeat when whole numbers of two kinds of month line up: convergents again.
• The saros, 223 synodic = 242 draconic months ≈ 18 years 11 days, is a convergent.

Steps

1. Huygens’s planetarium
2. A ladder of resistors
3. An endless ladder
4. Eclipse seasons
5. Eighteen years
6. Find the saros