When the snow began, from a plough's progress

Snow has been falling at a steady rate since some time before noon. At noon a snowplough sets off; it clears a fixed volume of snow per minute, so its speed is inversely proportional to the snow depth. It covers $2$ miles between noon and 1 pm and $1$ mile between 1 pm and 2 pm. At what time did the snow start?

About 11:23 am ($T=\tfrac{\sqrt5-1}{2}\approx0.618$ h before noon). Depth is proportional to $t+T$, so speed is $c/(t+T)$ and distances are $c\ln\frac{T+1}{T}=2$ and $c\ln\frac{T+2}{T+1}=1$. Hence $\frac{T+1}{T}=\left(\frac{T+2}{T+1}\right)^2$, which simplifies to $T^2+T-1=0$.