# Theodorus Ok Cyrene

# THEODORUS OK CYRENE

Since the main article on Theodorus (*DSB*, XIII. 314–319) was written, Wilbur Richard Knorr has produced a novel and attractive theory (see his work cited in the bibliography) to explain how Theodorus proved and other numbers to be surds and why he stopped with . It is based on “Pythagorean triples” of numbers that may be set out as the sides of a right triangle. Given an odd number *n*, the side of the square of *n* units in area is constructed as the leg of a right triangle the hypotenuse of which is (*n* + 1)/2 units in length; the other leg is (*n* - 1)/2 units. Given an even number *n*, the side of the square of *n* units in area is constructed as half the leg of a right triangle the hypotenuse of which is *n* + 1; the other leg is *n* - 1. If the constructed root is commensurable with the unit length, there will be a ratio of integers *a*:*b* such that the root and the unit are in that same ratio. Knorr shows in the case of , , and so on, that the assumption of commensurability leads to the contradiction that at least one of the pair of numbers is both odd and even.

This is an attractive theory in that it is no simple extension of the method used to prove the irrationality of (see *DSB*, XIII. 315), but employs the same principle, as anyone investigating the higher irrationals might have been expected to do. It also requires a separate proof for each surd, as the Greek text suggests. Finally, the method encounters difficulties with , Whereas previous commentators have taken the Greek to mean that Theodorus demonstrated the irrationality of and ran into trouble afterward, Knorr’s theory requires that he became entangled at ; but this is a possible interpretation of the text.

Contrary to the view taken in the article “Theaetetus,” note 9 (*DSB*, XIII 305), Knorr holds that *δύvαμis* means “power,” as in later Greek usage.

For clarity, the stages in proving that the process of finding the greatest common measure of 1 and 17 is endless (“Theodorus.” 316) should be presented thus:

(a)

(b)

## BIBLIOGRAPHY

See Wilbur Richard Knorr, *The Evolution of the Euclidean Elements* (Dordrecht. 1975). 181– 193.

Ivor Bulmrr-Thomas

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**Theodorus Ok Cyrene**