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

- Published in 2013
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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 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.

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## Other information

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

### BibTeX entry

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