Following Tarski, we view n n -dimensional Euclidean geometry as a first-order theory E n {E_n} with an infinite set of axioms about the relations of betweenness (among points on a line) and equidistance (among pairs of points). We show that for k > n k > n , E n {E_n} does not admit a k k -dimensional interpretation in the theory RCF of real closed fields, and we deduce that E n {E_n} cannot be interpreted r r -dimensionally in E s {E_s} , when r ⋅ s > n r \cdot s > n .