Polymer melts show “thinning” in shear deformation and “hardening” in simple elongation; the latter phenomenon is especially pronounced for melts of branched polyethylene (LDPE). Elongation hardening and shear thinning are closely connected with the deformation histories in simple elongational and simple shear flows. These flows are used for material characterization, and they are adequate for purely viscous and purely linear viscoelastic liquids. For elastic liquids with slowly fading memory, however, the first principal stretch rateλ̇1(t)turns out to be a better reference than the components of the strain rate tensor, andλ̇1(t)is very different for the two conventional test modes, viz. simple elongation and simple shear. For a more relevant comparison of the rheological behavior in elongation and shear, we therefore deformed a LDPE melt at 150°C such thatλ̇1(t)was the same in both test modes. The LDPE melt investigated showed hardening in elongation as well as in shear whenλ̇1increased exponentially with time, and the melt response was thinning in both elongation and shear whenλ̇1was kept constant, except for small total strains and strain rates at which the melt exhibited linear viscoelasticity.