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Theodorus of Cyrene was a Greek mathematician of the 5th century BC who was admired by Plato (who mentions him in several of his works, most notably the Theaetetus) . Little is known about him; however, Plato attributes to him the first proof of the irrationality of the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 and 17. The method he used is not stated, but since he stopped at 17, it seems possible that he used the traditional Pythagorean method of odds and evens, since 17 is the first number for which this method breaks down[1].

One conjecture involves a spiral composed of contiguous right triangles with hypotenuse lengths equal √2, √3, √4,..., up to √17 (where he stopped - possibly because additional triangles would cause the diagram to overlap, although one mathematician humorously suggested that "the bell rang"). This is now called the Spiral of Theodorus. There is no historical evidence to indicate why he stopped.

His pupil Theaetetus made the generalization that the side of any square, represented by a surd, was incommensurable with the linear unit.[2]

Philip J. Davis interpolated the vertices of the spiral to get a continuous curve that he named the Spiral of Theodorus. 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.


  1. ^ James R. Choike (1980). "Theodorus' Irrationality Proofs". The Two-Year College Mathematics Journal.  
  2. ^ James Gow (1884). A Short History of Greek Mathematics. University press.  

See also



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