Eudoxus Meets Cayley

1. INTRODUCTION. The uses of geometry stretch back to the dawn of human history. The earliest written records (from Egypt and Babylonia) contain geometric observations, problems, and solutions. During the classical Greek civilization in the period 600–300 BCE geometry was organized and systemized into what we view today as formal mathematics, culminating in Euclid's Elements. By this time mathematics had encountered and resolved its first crisis, as identified by Eves [1, pp. 15–16]: the discovery of irrational numbers (in the fifth century BCE) and the consequent inval-idation of all proofs that depended on the assumption that any two lengths were com-mensurable. When Euclid wrote the Elements he was able to incorporate (as Book 5) Eudoxus's brilliant resolution of the crisis through the development of proportional lengths and similar triangles. It would be difficult to overstate the impact that this first textbook of axiomatic mathematics had on Western civilization. Everyone with a formal education learned Euclid. It is clear from its organization and presentation that Thomas Jefferson and the other writers of The Declaration of Independence knew the Elements. Abraham Lincoln, in the autobiographical sketch he wrote for the Chicago Press & Tribune when he ran for President in 1860, stated that " He studied and nearly mastered the six books of Euclid, since he was a member of Congress. " Archimedes, Newton, Gauss, and the other giants of mathematics were masters of geometry. Thus it is with some trepidation that we present this article. We begin with a few elementary results from classical geometry that Eudoxus and Euclid could certainly have understood. These results naturally lead to a question that is also easily understood in the framework of classical geometry. It is in searching for the answer to this question that we are led into the mathematics of Cayley.