Here is an old poem (1849) by George Van Waters that offers instruction on finding a square root. This process was part of my junior high learning at the Keith School in Indiana, PA lots of years ago but I suppose the algorithm is seldom taught in 21st century classrooms. (In case the poem's directions are unclear, additional instruction is offered here.)
Square root by George Van Waters
Divide into periods of two figures each,
The number you know, as the pedagogues teach, --
In the left hand period find the greatest square,
Which from it subtract, and to what remains there
Bring the next period down for a Dividend (fair):
Place the root of the square at the right hand of all,
And two times the root a Divisor we call.
Then try the Divisor, see how many times
The Dividend holds it (by prose or by rhymes).
Of its right hand figure exclusive, you know,
And write in the root the number't will go,
Then to the Divisor the same figure tie,
And by the same figure the whole multiply;
The product then take from the Dividend (penned),
And of that which remains, make a new dividend;
By bringing the period that's next, along side, --
And for a Divisor that's new and untried,
Just double the figures that stand in the root,
And work as before, till the answer is got.
The poem is taken from the 1852 edition of The poetical geography, designed to accompany outline maps or school atlases : to which are added the rules of arithmetic in rhyme by George Van Waters. This link offers the complete text of Van Waters' book. In it Van Waters comments on how difficult it is to remember important information -- indeed, in the past days when poetry was passed from person to person primarily by oral recitation, rhymes were an important memory aid. Now that printed (and online) versions make poems readily accessible, we need not rely on memory -- and rhymes are not necessary. A recent Washington Post OpEd piece by Alexandra Petri calls attention to the fact that many of us now lean heavily on Google as our memory aid.