I'm reading Merzbach and Boyer's A History of Mathematics and finished the section on Hellenic traditions wherein the method of exhaustion is introduced. The section from the text reads,
If from any magnitude there be subtracted a part not less than its half (emphasis mine), and if from the remainder one again subtracts not less than its half, and if this process of subtraction is continued, ultimately there will remain a magnitude less than any preassigned magnitude of the same kind.
In modern notation: if $M$ is a given magnitude, $\varepsilon$ a preassigned magnitude of the same kind, and $r$ is a ratio such that $\frac{1}{2}\le r < 1$, then there is a positive integer $N$ for which $M(1-r)^N < \varepsilon$.
An application of this is immediately given by showing the ratio of areas of circles is equal to the squares of the diameters (the proposition being used to dispense with the $<$ and $>$ cases).
My question is: why the need for the condition $\frac{1}{2}\le r <1$? Even without modern notation, clearly for $\frac{1}{2}\le r <1$, $(1-r)\le \frac{1}{2}$; then by bisections it is easy to show there is some $N_0$ for which $M (1/2)^{N_0}<\varepsilon$. In fact, for $0\le c<1$, it seems easy to give a proof that there is some ${N_1}$ such that $c^{N_1}<\frac{1}{2}$, and then taking $N_2=N\cdot N_1$, the result is the same as before. But again, I don't understand the necessity of restricting to half-magnitudes.
This isn't meant to denigrate the ancients; rather, I'm wondering if it comes from a geometric or cultural stipulation that isn't explicitly mentioned. For instance, in the next chapter on Euclid, the equation $ax-x^2=b^2$ is considered, with the proviso that $a>2b$; this is to ensure the quadratic has two real solutions, as the Greeks would have expected. I'm wondering if there is a factor, besides maybe convenience, that led to this proviso.