Irrational Lengths and The Root Spiral

When calculators were not yet invented, it was hard for mathematicians to approximate irrational numbers. The irrational numbers was the offshoot of the discovery that the side length of the diagonal of a square with side length 1 is irrational. But how do mathematicians of the ancient time approximate a segment with length, say, \sqrt{5}? Can they draw a segment whose length is exactly \sqrt{5}?

root spiral

With the knowledge of the Pythagorean theorem, it is possible to create a right triangle with side 1 unit making the diagonal \sqrt{2}. The diagonal can then be used as the side of another right triangle whose shorter side length is 1. This process can go on producing the figure above. » Read more