Most words have several senses and connotations which evolve in time due to semantic shift, so that closely related words may gain different or even opposite meanings over the years. This evolution is very relevant to the study of language and of cultural changes, but the tools currently available for diachronic semantic analysis have significant, inherent limitations and are not suitable for real-time analysis. In this article, we demonstrate how the linearity of random vectors techniques enables building time series of congruent word embeddings (or semantic spaces) which can then be compared and combined linearly without loss of precision over any time period to detect diachronic semantic shifts. We show how this approach yields time trajectories of polysemous words such as amazon or apple, enables following semantic drifts and gender bias across time, reveals the shifting instantiations of stable concepts such as hurricane or president. This very fast, linear approach can easily be distributed over many processors to follow in real time streams of social media such as Twitter or Facebook; the resulting, time-dependent semantic spaces can then be combined at will by simple additions or subtractions.