CIRCULAR REASONING: WHO FIRST PROVED THAT C/d IS A CONSTANT?

Abstract. We answer the question: who first proved that C/d is a con-stant? We argue that Archimedes proved that the ratio of the circumfer-ence of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant (C/d = A/r2). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid’s Elements; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes’s work coexisted with the 2000-year belief—championed by scholars from Aristotle to Descartes—that it is impossible to find the ratio of a curved line to a straight line. For a long time I was too embarrassed to ask the question in the title. The expression C = pid, which gives the relationship between the circumference and the diameter of a circle, is one of the few formulas known to almost all children and adults, regardless of how long they have been out of school. But who first proved it? I was sure that merely asking the question would tarnish my reputation as a mathematician and as a student of the history of mathematics. Surely the answer is well-known and easy to find. But I could not locate it—at least none of the standard reference books discussed the matter. When I finally summoned the courage to ask others I received two types of replies: either “It is obvious; all circles are similar ” or “I don’t know. ” The first of these is intriguing and has some merit; it may be why the invariance of C/d was discovered in so many different cultures. But it does not easily lead to a rigorous proof. So I began to dig deeper, and the more I looked, the more confusing were the facts. In short: • The existence of the circumference constant (C/d) was known to many ancient civilizations, and thus one must expect it was known to the Greeks. • Aristotle (384–322 BCE) asserted that it was impossible to compare the lengths of curves and line segments, and this belief was widely held until the seventeenth century. Thus one cannot discuss the ratio of the circumference to the diameter