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