Conversation so far:
what is the method of finding square root
The most familiar such method, most suited for programmatic calculation, is Newton's method, which is based on a property of the derivative in the calculus.
what is used to find calculations of square root
The methods may be those suitable for mental calculation, requiring at least paper and pencil, and those which are implemented as programs to be executed on a digital electronic computer.
since when have the methods of calculating square roots been known
Procedures for finding square roots, particularly the square root of 2, have been known since at least the period of ancient Babylon in the 17th century BCE.
What was the first method ever used to find square root
Heron's method from first century Egypt was the first ascertainable algorithm for computing square roots.

Question: When did the modern analytical methods begin
Self-contained rewrite: When did the modern analytical methods to find square root begin