The projection constants λ(Xn) of real n-dimensional symmetric spacesXnsatisfy asymptotically λ(Xn) ⪯ √n—4/√n which is smaller than for general spaces. For small dimensions, specific numerical values are obtained. For n = 2, we reprove the result of D.R. Lewis that λ(X2) ⪯ λ(l22) = 4/π for 2-dimensional symmetric spaces. If n > 2, however, λ(Xn) is generally larger than λ(ln2) for symmetric spaces. The method used is based on trace-duality and estimates involving symmetrically invariant spherical functions.