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Algorithm: This method can be derived from (but predates) Newton-Raphson method. 1 Start with an arbitrary positive start value x (the closer to the. root, the better). 2 Initialize y = 1. 3. Do following until desired approximation is achieved. a) Get the next approximation for root using average of x and y. b) Set y = n/x.. Some archaeological considerations: Square Root Approximations in Old Babylonian Mathematics, by David Fowler and Eleanor Robson, in Historia Mathematica, Volume 25, Issue 4, November 1998. A modern description for the algorithm: The Babylonian method for finding square roots by hand, by Rick Wicklin, on The DO Loop

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The Babylonian square-root algorithm. The iterative method is called the Babylonian method for finding square roots, or sometimes Hero's method. It was known to the ancient Babylonians (1500 BC) and Greeks (100 AD) long before Newton invented his general procedure.. This is a method to find each digit of the square root in a sequence. This method is based on the binomial theorem and basically an inverse algorithm solving (+) = + +. It is slower than the Babylonian method, but it has several advantages: It can be easier for manual calculations.