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We generalise the approximation theory described in Mahler's paper "Perfect Systems" to linked simultaneous approximations and prove the existence of nonsingular approximation and of transfer matrices by generalising Coates' normality zig-zag theorem. The theory sketched here may have application to constructions important in the theory of diophantine approximation. In 1934, at the University of Groningen, Kurt Mahler gave a course in which he generalised the approximation theory for thedoi:10.1017/s1446788700033292 fatcat:sg5ohfo7h5f2pllwz2er5zxp6q