# What geometric results were first proven by assuming all real numbers are rational?

Pythagoras and his followers believed that all magnitudes are commensurable; that is, the ratio of two magnitudes of the same kind, like two lengths or two areas, is equal to the ratio of natural numbers. In modern language, this means that all real numbers are rational numbers. This allowed them to find out whether any two ratios of magnitude are equal: you just find out what ratio of natural numbers each one is equal to, and then compare those ratios to each other.

But Hippassus of Metapontum discovered that the ratio of a diagonal of a square to its side is not equal to a ratio of natural numbers. So a new definition was needed for when two ratios of magnitudes are equal. This was done by Eudoxus of Cnidus. In modern language, his definition says that two real numbers are equal if the same rational numbers are less than, equal to, and greater than them (basically the idea behind Dedekind cuts).

My question is, what results of geometry did the Pythagoreans prove using the incorrect assumption that all real numbers were rational? Long ago, I read a book, whose name I don't recall, which listed some results, and described how they had to be reproved with Eudoxus' new definition.