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

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.
 Comments: 17 pages, 8 figures Subjects: History and Overview (math.HO) MSC classes: Primary: 01A20, 01A45. Secondary: 26B15 Cite as: arXiv:1303.0904 [math.HO] (or arXiv:1303.0904v2 [math.HO] for this version)

## Submission history

From: David Richeson [view email]
[v1] Tue, 5 Mar 2013 01:42:45 UTC (1,100 KB)
[v2] Thu, 14 Mar 2013 03:03:59 UTC (1,105 KB)
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