Bombelli Method
Bombelli used a method related to continued fractions to calculate square roots. His method for finding begins with with, from which it can be shown that . Repeated substitution of the expression on the right hand side for into itself yields a continued fraction
for the root but Bombelli is more concerned with better approximations for . The value chosen for is either of the whole numbers whose squares lies between. The method gives the following convergents for while the actual value is 3.605551275... :
The last convergent equals 3.605550883... . Bombelli's method should be compared with formulas and results used by Heros and Archimedes. The result used by Archimedes in his determination of the value of can be found by using 1 and 0 for the initial values of .
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