Zeno of Sidon vindicatus: a mereological analysis of the bisection of the circle

I provide a mereological analysis of Zeno of Sidon's objection that in Euclid's Elements we need to supplement the principle that there are no common segments of straight lines and circumferences. The objection is based on the claim that such a principle is presupposed in the proof that the diameter cuts the circle in half. Against Zeno, Posidonius attempts to prove the bisection of the circle without resorting to Zeno's principle. I show that Posidonius' proof is flawed as it fails to account for the case in which one of the two circumferences cut by the diameter is a proper part of the other. When such a case is considered, then either the bisection of the circle is false or it presupposes Zeno's principle, as claimed by Zeno.

Automated summary

This article provides a mereological analysis of Zeno of Sidon's objection to the principles in Euclid's Elements, specifically the claim that there are no common segments of straight lines and circumferences. The objection centers around the proof that the diameter cuts the circle in half, and Posidonius attempts to prove the bisection without relying on Zeno's principle. The author demonstrates that Posidonius' proof is flawed as it fails to consider cases where one circumference is a proper part of the other, leading to the conclusion that the bisection either false or presupposes Zeno's principle as Zeno claimed.