Spiral of Theodorus - Discussion

Question

The fact that $\sqrt2$ is not rational goes back to Theodorus of Cyrene from the school of Pythagoras, and is discussed in Plato's dialog "Theaetetus".

Of course, $\sqrt n$ is not rational for any square-free positive integer $n$. However, the theorem was stated and proved by Theodorus of Cyrene only for $n<17$ unless $n=1,4,$ and $9$.

What are some plausible explanations of the obstacle which did not allow Theodorus of Cyrene to obtain this important theorem in full generality?

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I thought that it's possibly because since he used the traditional Pythagorean method of odds and evens, $17$ is the first number for which this method breaks down.

Or maybe that the $\sqrt{17}$ adjacent leg belongs to the last triangle that does not overlap the figure, from the Spiral of Theodorus.

What are some other possible reasons why he stopped at $17$?

Answer 1

From the Wikipedia entry for Wilbur Knorr, on one of his books:

The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry (Dordrecht: D. Reidel Publishing Co., 1975).

This work incorporates Knorr's Ph.D. thesis. It traces the early history of irrational numbers from their first discovery (in Thebes between 430 and 410 BC, Knorr speculates), through the work of Theodorus of Cyrene, who showed the irrationality of the square roots of the integers up to 17, and Theodorus' student Theaetetus, who showed that all non-square integers have irrational square roots. Knorr reconstructs an argument based on Pythagorean triples and parity that matches the story in Plato's Theaetetus of Theodorus' difficulties with the number 17, and shows that switching from parity to a different dichotomy in terms of whether a number is square or not was the key to Theaetetus' success.

(Note: This is copied from an answer I gave at MathOverflow a year ago.)