Greek ladders for approximating square roots may be more ancient than the ancient Greeks. Students at any level can appreciate their beauty and simplicity. Those who have studied calculus can compare them with Newton's Method for approximating roots.
A method for extracting square roots used in Italy through the 18th century was introduced in a manuscript by the 12th century mathematician al-Hassar. Appendix: When Was Fermat Born? (Not in 1601 ...)