Before Eudoxus's theory of proportion there was a theory of irrationals based on continued fraction expansions, which Fowler calls anthyphairesis. Theaetetus is said to develop a classification of quadratic irrationals in this setting. Given positive $x$ and $y$, with $x/y$ being a square root of a rational number, $x+y$ is called a binomial, $x-y$ an apotome (if $x>y$), and $\sqrt{xy}$ a medial (if it is irrational). There are some interesting results described in book X of Euclid's Elements: the classes are mutually exclusive, for any binomial or apotome the split into $x$ and $y$ is unique, etc. Further classes are considered, apotome of a medial, binomial of a medial, and so on.

Theaetetus seems to deal with positive elements of $\mathbb{Q}(\sqrt{2},\sqrt{3},\dots)$, but I can not think of any algebraic or number theoretic classification scheme that would produce his classes. Are they related to something in modern algebra?

  • I'm making this a comment because of the "modern algebra" part, but they sure are related to the classification of quadrics (but I guess this classifies as "geometry" rather than algebra, although the frontier is dim between the two fields). – VicAche Nov 5 '14 at 20:51
  • @VicAche This is interesting, I thought quadrics were classified by types of quadratic forms rather than number theoretic properties. Could you elaborate. – Conifold Nov 7 '14 at 15:55
  • see "Quadric forms" entry in The Princeton Companion to Mathematics. (just search for it on google if you don't want to buy the book). I don't have enough time to sum it up here now. – VicAche Nov 8 '14 at 17:43
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    I read the interesting paper of Fowler you refer to. He mentions several prominent mathematicians he consulted, and it looks like if he knew the answer to your question he would mention it. I recommend that you post your question on Math Overflow. – Alexandre Eremenko Nov 28 '14 at 13:40
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    @Franz Lemmermeyer There is Plato's dialogue entirely devoted to Theaetetus and named after him, he is also mentioned by many other sources. So we know that he is as historical as Pericles. Euclid does not credit anyone in the Elements, not even himself, but books X and XII are ascribed to Theaetetus by others, just like book V is to Eudoxus. And I only wrote that he seems to deal with (what we now call) elements of a field, not that he knowingly "worked inside" a field. The details are described under the book X link. – Conifold May 7 '15 at 23:04

Here are some modern commentaries on Euclid's book X:

C. M. Taisbak, Coloured quadrangles: a guide to tenth book of Euclid's Elements, Museum Tusculanum Press, Copenhagen, 1962.

D. H. Fowler, An invitation to read book X of Euclid's Elements, Hist. Math. 19 (1992) 233-364.

W. R. Knorr, Euclid’s Tenth Book: An Analytic Survey, Historia Scientiarum 29 (1985), 17–35.

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