Measurements of $\alpha_{s}$ with the DELPHI detector at LEP

The data collected with DELPHI during the LEP1 period is used to perform a simultaneous fit to the strong coupling constant alpha /sub s/ in 2/sup nd/ order perturbation theory and to the renormalisation scale x/sub mu /. The results are compared to alpha /sub s/ fits in 2 /sup nd/ order perturbation theory with fixed scale x/sub mu / and to other theoretically motivated scale setting methods. To determine the energy dependence of alpha /sub s/, event shape distributions and their mean values are measured for square root (s)=48 GeV to 189 GeV. The strong coupling alpha /sub s/ is extracted in O( alpha /sub s //sup 2/). Next to leading log approximation (NLLA) and in a combined scheme using - evaluated with fragmentation model generators, and an analytical power ansatz. (16 refs).