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

Question

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.

Answer 1

There are errors in the assumptions of your question:

• That history about Hippasus of Metapontum is highly doubtful. There is the version that you mentioned. There is also the version according to which what he discovered (or revealed) was how to construct a dodecahedron.
• You claim that the Pythagoreans were able to find out whether any two ratios of magnitude are equal: they just had to “find out what ratio of natural numbers each one is equal to, and then compare those ratios to each other.” Concerning this idea, David Fowler wrote: “I know of no explicit evidence for this, early or late.” (The Mathematics of Plato's Academy: A new reconstruction, 2^nd edition, Clarendon Press, 1999, § 10.1)
• Finally, I doubt that the Pythagoreans proved any geometrical statement using the idea that all magnitudes are commensurable. And, if they did, no such proof survived.