Let Vo be a discrete real valuation of a field K and x an indeterminate. In 1936, MacLane  gave a method of constructing all real valuations of K(x) which are extensions of Vo. In this paper, we determine explicitly all rank 2 valuations of K(x) which extend Vo. One can thereby describe all rank 2 valuations of K(x, y) which are trivial on an arbitrary K; x, y being algebraically independent over the field K. The latter valuations have been considered by Zariski  in the case when K is an algebraically closed field of characteristic zero.
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