*"Oddly enough, calculating the square root of an eighty-digit number—whose square root still has forty digits—is harder than calculating the 1,789th root of a seven-thousand-digit number" (!) says Jean-Paul Delahaye in his article Un calcul révolutionnaire*, published in the Quebec journal Accromaths (spring 2018). Lightning calculators therefore attracted more attention for spectacular calculations of nth roots than for simple square roots, although John Wallis (1616–1703), for example, is said to have been able to take the roots of fifty-five-digit numbers (history does not tell us how).
One thing is certain: this kind of calculation, which often requires numerous multiplications and additions, is possible only if the person performing it can carry them out very quickly. To calculate quickly in one's head, one can use Newton's method (see In brief: "Inside your calculators"); starting from x0 = 1, it is relatively easy to obtain values that converge quite rapidly to the desired root: 1.75, then 1.73214… and 1.73205081, with seven correct decimal places already reached by the third term.
It seems that the Italian lightning calculator Giacomo Inaudi (1867–1950), who had exceptional mental-arithmetic skills and knew his table of squares by heart, used trial and error. For he is said to have tried 14,000, then 15,000, followed by 14,600, 14,650, 14,660, 14,670… and, calculating their squares very rapidly, would have calculated the difference between each square and the given number to arrive very quickly at the correct value: 14,672.
