Doing square root on the abacus is a lot like doing it on paper. The big difference? It’s actually *easier* on the abacus.
The one difficult step in the paper square root is guessing the approximate digits; as you get beyond the third or fourth digit, the numbers start getting a bit large, and it can be hard to guess the correct estimate. On the abacus, you can very rapidly do repeated subtraction, so you deliberately guess low, and then add on. You’ll see what I mean as we work through an example.
One thing about the square root is you need a bigger abacus. So far, we’ve used very small ones for the examples here. The more you want to do with an abacus, the bigger you want it to be. A small abacus typically has something like 9 columns; a medium abacus has 13 digits. But for more interesting calculations, the kind of thing that we westerners would have a slide rule or scientific calculator, the abacus equivalent is a *27* digit abacus with a couple of sliding markers for helping keep track of things. You want a nice big abacus for doing things like roots, because you’re going to partition it into multiple sections.
To do the square root on the abacus, you need at least three sections: one to hold the number you’re taking the root of; one to hold the answer you’re building; and one do use as a sort of scratchpad for multiplication