Circular reasoning: who first proved that $C/d$ is a constant?

TL;DR

This paper explores Archimedes's implicit proof that the ratio of a circle's circumference to its diameter is a constant, linking it to the area constant, and discusses its historical significance in the development of geometric rigor.

Contribution

It demonstrates that Archimedes implicitly proved the constancy of the circumference-to-diameter ratio and connects this to the area constant, advancing understanding of the origins of geometric analysis.

Findings

Archimedes's work implies the constancy of C/d.

The circumference constant equals the area constant (C/d=A/r^2).

Archimedes's proof required new axioms beyond Euclid's Elements.

Abstract

We answer the question: who first proved that $C/d$ is a constant? We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant ($C/d=A/r^{2}$). 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 \emph{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.