A new test for the detection of linear trends of arbitrary length in normally distributed time series is developed. With this test it is possible to detect and estimate gradual changes of the mean value in a candidate series compared with a homogeneous reference series. The test is intended for studies of artificial relative trends in climatological time series, e.g. an increasing urban heat island effect. The basic structure of the new test is similar to that of a widely used test for abrupt changes, the standard normal homogeneity test. The test for abrupt changes is found to remain unaltered after an important generalization.