Theodorus of Cyrene

Theodorus of Cyrene was a 5th century B.C. mathematician and was born around 100 years after Pythagoras. (Cyrene is now called Shahhat, in Libya.)

He apparently proved that the square roots of 2, 3, 5, 6 and so on up to 17 were all irrational, except the perfect squares 4, 9, 16. (Unfortunately we no longer have the proofs.) He also went on to construct these supposedly non-existent distances.

He proceeded as follows.

Start with a right triangle with equal sides 1, giving a hypotenuse of √2 (which of course was a problem, because this distance didn’t officially exist):

Then, extend a line with length 1 unit (using your 1-unit measuring stick) at right angles to the first hypotenuse as follows. This gives us the length √3 after we apply Pythagoras’ Theorem to the new triangle.

Do it again, and you now get the length √4 = 2. Theodorus had discovered one hypotenuse with a rational number length.

He kept going and found that the next one to have a "rational" length was √9 = 3.

He continued on to √16 = 4, constructed one more, √17, then stopped.