Introduction

This article presents two self-contained proofs of the Pythagorean Theorem that are strictly geometric, involving neither measurements nor numbers. The first might have been discovered by Pythagoras in the sixth century bc. The second is due to Liu Hui from about 300 ad. The two proofs show how mathematicians in two ancient civilizations—one in the West (ancient Greece) and the other in the East (ancient China)—deduced a result about right triangles from strictly geometric arguments. We also briefly contrast the geometric approaches with an arithmetic method employed by mathematicians from a third great ancient civilization—the Babylonians. The question posed in the title of this article is borrowed freely from a book by Frank J. Swetz and T. I. Kao [7]. Our purpose here is to show how the radically different civilizations in China and Greece regarded right triangles in a remarkably similar way.

The material in this article is appropriate for students taking geometry for the first time in high school (or perhaps earlier); we provide suggestions for using cut-outs to help visualize the process. The only notion that is assumed is the concept of congruence, yet even here it is used in the intuitive sense of placing one figure precisely on top of another. The greatest benefit for beginning students might be an understanding of the nature of mathematical proof, because the historical approach adopted here illustrates a type of intuitive argument (based on obvious properties of figures) that preceded formal chains of reasoning that characterize deduction.