In the year 1968–1969, Professor Mary Cartwright was a visiting member of the Division of Applied Mathematics at Brown University. This was a year of turmoil at Brown—particularly curricular turmoil—and in the course of one of our division meetings Miss Cartwright remarked that when she was a student all mathematics majors were required to know a proof of the nine-point circle theorem. Since the nine-point circle now has a distinct flavor of beautiful irrelevance, Miss Cartwright seemed to be telling us that we should not be too dogmatic as to what constitutes a proper mathematics curriculum. Fashion is spinach even in mathematics, and time often works to “nine-point circle-ize” many of our most relevant and sophisticated topics that are now insisted upon.

At the time, I was giving a course in Numerical Analysis entitled “Iteration Theory in Banach Spaces”, using notes of L. B. Rall, and I saw that it would be possible—and not too far-fetched—to present several lectures which would trace an unlikely path from the nine-point circle to iteration.

This essay presents such a path. The connecting link is the use of conjugate coordinates and the Schwarz reflection function. The path has been faired, as draftsmen say, to pass in a wide arc near a number of allied topics in complex variable theory that have interested me.

In Chapter Two conjugate coordinates are introduced. In Chapter Three elementary notions of plane analytic geometry are expressed in terms of conjugate coordinates.