循环推理：谁首先证明了$C/d$是一个常数？
Circular reasoning: who first proved that $C/d$ is a constant?
摘要：
我们回答这个问题：谁首先证明了$C/d$是一个常数？我们认为，阿基米德证明，圆周率的一个圆，其直径是一个常数独立的圆和周长常数等于面积常数（$C/d=A/r^{2}$）。他没有明确地陈述这两个结果，但这两个结果都隐含在他的工作中。他的证明需要增加两个公理以外的欧几里得的\emph{Elements} ;这是第一步，以严格的理论弧长。我们还讨论了阿基米德的工作是如何与2000年的信念共存的-从亚里士多德到笛卡尔的学者所倡导的-不可能找到曲线与直线的比率。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
