Square root (半九九法) the traditional way
Profesor Kato's Method_ The traditional way.pdfMay be out of date
You might be interested in this explanation of Kato's method on the Totton Heffelfinger website. (archive.org)
Murakami's square root tutor screenshot
See also, this Step by step square root calculation by using half-square method with Kijo-ho By Masaaki Murakami, that uses the second way explained in the article.