Posted by: Dave Richeson | March 5, 2013

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

I just uploaded an article “Circular reasoning: who first proved that $C/d$ is a constant?” to the arXiv. The abstract is below. It is on a topic that I’ve been thinking about and reading about off-and-on for the last year and a half. I’d be happy to hear people’s thoughts, reactions, and impressions.

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.

## Responses

1. I would have been in the first (“It’s obvious.”)camp. I’ve begun reading and absorbing your article.

In the past few years I’ve been telling my pre-calc and calc students that pi=C/D is a definition (with the unstated assumption that C/D is a constant) and A=pi*r^2 is a theorem that we can prove. It seemed like a good example for helping them see the difference between definitions and theorems.

Of course, nothing is as simple as that, is it? I will adjust the way I talk about it now.

• REF:
C:\Users\Panagiotis\Documents\Area of circle to its circumference is a constant -approach to Pi.mht

2. It’s easier to find with a link of course: http://arxiv.org/pdf/1303.0904.pdf

• Ha ha—oops! Thanks for catching that Sue! I updated my post.

3. FOR: C:\Users\Panagiotis\Documents\Area of circle to its circumference is a constant -approach to Pi.mht

REF:

http://mathforum.org/kb/message.jspa?messageID=3401996

4. So the assumption is that the Chinese, who knew of this much earlier than any white man, just took it on faith? Or is it more likely that they simply did not write down mathematical, i.e. technological, secrets for the purpose of protecting special knowledge?