You have a head full of cotton.
The ancients computed transcendental functions 4000 years ago or earlier.
You have a head full of irrelevancies.
They computed Pi to some degree of precision. They didn’t, until later realize that there was no ratio of integers that would exactly equal pi.
The discovery of irrational number is the discovery that integer ratios are not sufficient.
Calculating pi to some degree of precision, or even recognizing that a given degree of precision is ‘good enough’ is different from the realization that no degree of precision or calculation means can ever be exact.