Theodorus of Cyrene in Wikipedia
Theodorus of Cyrene (Greek: Θεόδωρος ὁ Κυρήνη) was a Greek mathematician of the 5th century BC. The only first-hand accounts of him that we have are in two of Plato's dialogues: the Theaetetus and the Sophist. In the former, his student Theaetetus attributes to him the theorem that the square roots of the non-square numbers up to 17 are irrational:
Theodorus here was drawing some figures for us in illustration of roots, showing that squares containing three square feet and five square feet are not commensurable in length with the unit of the foot, and so, selecting each one in its turn up to the square containing seventeen square feet and at that he stopped.
(The square containing two square units is not mentioned, perhaps because the incommensurability of its side with the unit was already known.) Theodorus's method of proof is not known. It is not even known whether, in the quoted passage, "up to" (μέχρι) means that seventeen is included. If seventeen is excluded, then Theodorus's proof may have relied merely on considering whether numbers are even or odd. Indeed, Hardy and Wright and Knorr suggest proofs that rely ultimately on the following theorem: If x2 = ny2 is soluble in integers, and n is odd, then n must be congruent to 1 modulo 8 (since x and y can be assumed odd, so their squares are congruent to 1 modulo 8).
A possibility suggested earlier by Zeuthen is that Theodorus applied the so-called Euclidean algorithm, formulated in Proposition X.2 of the Elements as a test for incommensurability. In modern terms, the theorem is that a real number with an infinite continued fraction expansion is irrational. Irrational square roots have periodic expansions. The period of the square root of 19 has length 6, which is greater than the period of the square root of any smaller number. The period of √17 has length one (so does √18; but the irrationality of √18 follows from that of √2).
The so-called Spiral of Theodorus is composed of contiguous right triangles with hypotenuse lengths equal √2, √3, √4, …, √17; additional triangles cause the diagram to overlap. Philip J. Davis interpolated the vertices of the spiral to get a continuous curve. He discusses the history of attempts to determine Theodorus' method in his book Spirals: From Theodorus to Chaos, and makes brief references to the matter in his fictional Thomas Gray series.
That Theaetetus established a more general theory of irrationals, whereby square roots of non-square numbers are irrational, is suggested in the eponymous Platonic dialogue as well as commentary on, and scholia to, the Elements.