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

• Published in 2013
• Added on 2013-03-06

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.

Links

• http://arxiv.org/abs/1303.0904
• http://arxiv.org/pdf/1303.0904v2

Other information

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}
}