Noble's Square Root Calculator
Given the chart, a person would use something with a right angle (such as a rectangular piece of paper or a carpenter's square), place its edges on 10 and the target number on the right horizontal axis, align the right angle on the vertical bar, and read the square root from the vertical bar.
Let and be the lengths of the line segments on the left and right horizontal axis, be the height of the big triangle on the vertical axis, and and be the lengths of the two sloped line segments on the left and right, respectively. There are three right-angled triangles, so by Pythagoras,
Eliminate and in the last equation:
Multiply out the right-hand side:
Cancel and :
By taking , the units work out because the right-hand horizontal axis is scaled to be 10 times as fine as the vertical and left horizontal scales.
 L. A. Graham, Ingenious Mathematical Problems and Methods, New York: Dover Publications, 1959 pp. 72–73.