Led Astray by a Right Triangle: Misconception, Epiphany, and Redemption

Introduction

In studying the history of mathematics, I always seek to make connections, discerning the historical patterns of development between events and establishing cause-and-effect relationships. All mathematical concepts and practices evolve over periods of time and are shaped by various influences. For example, the orbits of the planets, when first realized, were described by the Pythagoreans (ca. 500 BCE) as circles, a satisfying shape, symmetrical never-ending paths of harmony, geometric perfection, in the minds of their observers. When that perceived harmony was interrupted by anomalies, disruptive and retrogressional motions violating the sanctity of the circular orbs, new compensating, epicyclical-based, Ptolemaic (ca. 100) theories were introduced. Finally, over time, more perceptive observation, data collection, and mathematical computation by such sixteenth century stargazers as Tycho Brahe and Johannes Kepler fixed the planetary orbits as ellipses. Mathematical theories usually follow each other in a conceptually hierarchical fashion. However, when following this principle in investigating traditional Chinese problem-solving practices, the appeal of right triangles and my modern mathematical conditioning led me astray.