theodorus began by msuring one triangle, one half of the square. now, remember, the pythagoreans believed that units of measurement could be arbitrary, like the palm of a hand, or for us today, the length of an inch or a centimeter. t they also believed that there must be a common length that fits a whole number of times in both the length of the side and the diagonal. the measurements must be commensurate. when theodorus tried this with, say, four units along each side of the square, he found that he couldn't measure the diagonal with a whole number of those units. there would be a small portion of a unit left remaining. in fact, no matter how many units we divide up each side of the square into, when we try to measure the diagonal with this basic unit, there's always some small amount left over. now, this is a pretty interesting observation, but theodorus went a big step further. he also developed a purely logical and ironclad proof that no such common unit can possibly exist. at the heart of this is something of the mystery of the infinite. impl