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A Euclidean number is a number which can be obtained by repeatedly solving the Quadratic Equation. Euclidean numbers, together with the Rational Numbers, can be constructed using classical Geometric Constructions. However, the cases for which the values of the Trigonometric Functions Sine, Cosine, Tangent, etc., can be written in closed form involving square roots of Real Numbers are much more restricted.
See also Algebraic Integer, Algebraic Number, Constructible Number, Radical Integer
References
Conway, J. H. and Guy, R. K. ``Three Greek Problems.'' In The Book of Numbers. New York: Springer-Verlag,
pp. 192-194, 1996.