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

• Published in 2013
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.

## Other information

arxivId
1303.0904
keywords
and phrases,arc,archimedes,aristotle,circle,descartes,history,pi
pages
17

### BibTeX entry

@article{Richeson2013,
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.},
archivePrefix = {arXiv},
arxivId = {1303.0904},
author = {Richeson, David},
eprint = {1303.0904},
keywords = {and phrases,arc,archimedes,aristotle,circle,descartes,history,pi},
month = {mar},
pages = 17,
title = {Circular reasoning: who first proved that {\$}C/d{\$} is a constant?},
url = {http://arxiv.org/abs/1303.0904 http://arxiv.org/pdf/1303.0904v2},
year = 2013,
primaryClass = {math.HO},
urldate = {2013-03-06}
}