How Did Pappus Do It?

Abstract

One of the chief joys of mathematics is problem-solving and the greatest thrill of problem-solving is the attainment of an elegant result—preferably by elegant means. Pappus was a master of the elegant theorem, as evidenced by the countless “theorems of Pappus” found in the mathematical literature, and it comes as no surprise that he was successful in finding an ingenious way of proving the truth of what he called “an ancient theorem”. This is a proposition described in Book IV of his Mathematical Collection and has to do with a sequence of consecutively tangent circles inscribed in the Arbelos, or the Shoemaker’s Knife, a geometrical figure first treated by Archimedes in his Book of Lemmas. We may safely presume that this Circle Theorem was probably known empirically in times gone by, but had never before been proved.