I have made a video response here: https://youtu.be/m-LwNJmicmU
EDIT: in response to comments I have included a typed explanation below, exclusively for stack exchange...
There are 2 key quantities here:
- $R_{eff}^{2}$ = square of the effective radius of the reactor.
- A = induced activity in the indium foil from the neutron irradiation (counts/minute), measured the next day.
Unfortunately it's a bit of conjecture from here on, but my understanding is that Fermi used $R^{2}$ as a surrogate for the cross sectional area of his reactor, and A as a surrogate for the 'neutron intensity' (presumably, neutron intensity = number of neutrons per unit area).
Fermi was likely most interested in how the value of A changes over time.
- Increasing A = chain reaction (i.e. nuclear bomb)
- Constant A = sustainable reaction that continues until you run out of fissile material.
- Decreasing A = reaction that will create a small bit of heat but die out.
Fermi could have just plotted A against the number of layers and found where his trendline shoots off to infinity, and determined that to be the point of criticality. But A isn't the only influence factor, there's an interplay effect between A and the cross sectional area. Since he's using effective radius ($R_{eff}$) it mightn't be obvious how that behaves, and the result will also be very specific to his setup.
Either way we can see that $R^{2}$ will increase with the square of the number of layers, and we know A will increase exponentially. Hence A grows faster than $R^{2}$, so when we take a ratio of $R^{2}/A$ it trends to $0$ for criticality. Importantly since both cross sectional area and neutrons/area are strictly positive and non-zero, the ratio will never truly reach zero.
There are a few aspects of the experiment that my (rather naive) video solution cannot speak to. For example, I assumed a perfectly spherical reactor with radius '$R$' when in reality it wasn't spherical and Fermi used the effective radius '$R_{eff}$' instead. There is also uncertainty surrounding the indium foil detector, I have assumed that the quantity Fermi was actually interested in was something similar to neutron fluence and his measurements were aimed at evaluating it through a proxy-quantity (induced activity in the foil) since he was unable to measure it directly.