I think it's reasonable to assume he did not introduce imaginary and transcendental numbers.

Not really.

Once he introduces the square root he introduces imaginary numbers, sqrt(-1), and transcendental numbers, for example the Gelfond–Schneider constant 2^sqrt(2).

Now we're really down the rabbit hole of someone else's intent, but personally, I assumed he was still trying to maintain some restriction. So, no imaginary numbers, no trascendental numbers. It's easy to restrict what we take the root of, and what we do with the potential irrational result of such roots, to ensure that. As you pointed out, to not do so defeats the purpose of the exercise, and I think my assumption is both reasonable and charitable.